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Fundamentals of Semiconductor Physics and Devices

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Page 1: Fundamentals of Semiconductor Physics and Devices
Page 2: Fundamentals of Semiconductor Physics and Devices

FUNDAMENTALS OF SEMICONDUCTOR PHYSICS AND DEVICES

Page 3: Fundamentals of Semiconductor Physics and Devices

Rolf Enderlein

NommJ. M. Horing

ihmboldt-bhiuemity BerEin Universi?y of Sao Pauh

stewwas Institute of Technology Hoboken, NJ

World Scientific Singapore New Jersey London Hong Kong

Page 4: Fundamentals of Semiconductor Physics and Devices

Puhli.dzed hy

World Scientific Publishing Co. Re. Ltd.

P 0 Box 128, Farrer Road, Singapore 912805 USA oflice: Suite 1 H, 1060 Main Streei, River Edge, NJ 07661

UK oflcficer 57 Shelton Street, Covcnt Garden, London WCZH 9IiE

British Library Catalogiiing-in-Publication Data A catalogue recurd fur this book is available from the British Library.

First published 1997 Reprinted 1999

FUNDAMENTALS OF SEMICONDIJC'IOK PHYSICS AND DEVICES

Copyright 0 1997 by World Scicntific Publishing Co Pte. Ltd. All rights reserved. Th.is book, or parts thrreof, may ~ O I be reprudrtced in any jurwi or by ony m e w s , electmnir or rnerhirnicirl, incIudinx photocopying. recording or any information storage and retrieval sys:slew now known or m be ii-zvmted, without written permissionfrom !he Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive. Danvers, MA 01923, U S A . In this casc pcmission to photocopy i s not required from the publisher.

ISBN 981-02-2387-0

Printed in Singapore.

Page 5: Fundamentals of Semiconductor Physics and Devices

Dedication

This book is dedicated to the memory of Adele and Werner Enderlein (par ents of R.E.), and to the memory of Joseph and Esther Morgenstern (hfor ganstein)(grandparents of N.J. M. 11.).

Page 6: Fundamentals of Semiconductor Physics and Devices

Vi i

Preface

People come to technical books with a vast array of daerent needs and requirements, arising from their differing educational backgrounds, profes- sional orientations and career objectives. This is particularly evident in the field of semiconductors, which stands at the juncture of physics, chemistry, electronic engineering, material science and mathematics. No longer just an academic discipline. this field is at the heart of an ongoing revolution in communications, computation and electronic device applications, that inno- vate many fields and change modern life in myriad ways, large and small. Its profound impact and further potential command interest and attention from all corners of the earth. and from a wide variety of students and researchers.

The clear need to address a broadly diversified and variously motivated readership has weighed heavily in the authors’ considerations. It poses a pedagogical problem faced by many teachers of intermediate level courses on semiconductor physics. Generally speaking, every student has previously studied about half of the course materiaL The difficulty lies in the fact that each student’s exposure is likely to have been to a &fleerent half, depend- ing on which lower level courses and teachers they have had, and where the emphasis lay. To accommodate readers with varied backgrounds we start from first principles and provide fully detaiIed explanations and proofs, as- suming only that the reader is familiar with the Schrodinger equation. This intensively tutorial treatment of the electronic properties of semiconduc- tors includes recent fundamental developments and is carried through to the physical principles of device operation, to meet the needs of readers inter- ested in engineering aspects of semiconductors, as well as those interested in basic physics. Clarity of explanation and breadth of exposure relating to the electronic properties of semiconductors, from first principles to modern de- vices, are our principal objectives in this fraddy pedagogical book. We offer full mathematical derivations to strengthen understanding and discuss the physical significance of results. avoiding reliance on ‘hand waving arguments alone.

To support the reader’s introduction to the physics of semiconductors, we provide a thorough grounding in the basic principles of solid state physics, assuming no prior knowledge of the field on the part of the reader. An ele mentary discussion of the crystal structure, chemical nature and macroscopic properties characterizing semiconductors is given in Chapter 1. Moreover, we also include an extensive appendix to guide the reader through group theory and its applications in connection with the symmetry properties of semiconductors, which are of major importance. Beside spatially homoge- neous bulk semiconductors, we undertake a full exposition of inhomogeneous semiconductor junctions and heterostructures because of their crucial role

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Preface iu

The book has emerged from lectures which the authors presented for physics students at the Humboldt-University of Berlin. Germany, and the State Uni- versity of Sao Paulo, Brazil, and for physics and engineering physics students at the Stevens Institute of TeFhnology in Hoboken, New Jersey, C.S.A. Part of the book has similarities with the german book "Grundlagen der Halbleit- erphysik" ("Fundamentals of Semiconductor Physics") which was written by one of us (R. E.) together with A. Schenk. We are thankful to Dr. Schenk (now at ETH Zurich) for allowing us to use part of his work in the present volume. In writing this book we have had excellent suppoIt from many of our colleagues at our own and other Universities. We are particularly thankful to Prof. Dr. J. Auth (Humboldt-Lniversity Berlin), Prof. Dr. F. Bechstedt (Friedrich-Schiller University Jena), Prof. Dr. W. A. Harrison (Stanford University), Prof. Dr. M. Scheffler (Fritz-Haber Institut, Berlin), Prof. Dr. J. R. b i t e , Prof. Dr. A. Fazzio, and Prof. Dr. J. L. Alves (State University Sao Paulo), as well as to Prof. Dr. H. L. Cui, Prof. Dr. G. Rothberg, Mr. G. Lichtner (Stevens Institute of Technology), and Prof. Dr. G . Gumbs (Hunter College, CWNY, New York), who read parts of the manuscript and contributed helpful suggestions and critical remarks. The technical assistance of Mrs. Hannelore Enderlein is gratefully acknowledged.

RoIf Enderlein Norman J.M. Horing

Sao Paulo Hoboken, N J

October 1996

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Xi

Contents

1 Characterization of sernico nd uct ors 1 1.1 Inlrnduclion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Atomic structure of ideal crystals . . . . . . . . . . . . . . . . 5

1.2.1 Cryst. al latlices . . . . . . . . . . . . . . . . . . . . . . 6 1.2.2 Point groups of equivalent directions arid crystal classes 12 1.2.3 Space groups and crystal structures . . . . . . . . . . 14 1.2.4 Cubic semiconductor structures . . . . . . . . . . . . . 16 1.2.5 Hexagoiial semiconductor st. ructures . . . . . . . . . . 22

1.3 Chemical nature of semiconductors . Material classes . . . . . 28 1.3.1 Group IV elemental semiconductors . . . . . . . . . . 29 1.3.2 111-V semiconductors . . . . . . . . . . . . . . . . . . . 30 1.3.3 11-VI semiconductors . . . . . . . . . . . . . . . . . . . 31 1.3.4 Group \'I elemental semiconductors . . . . . . . . . . 31 1.3.5 IV-VI semiconductors . . . . . . . . . . . . . . . . . . 32 1.3.6 Other compound semiconductors . . . . . . . . . . . . 32

1.4 hlacroscopic properties and their microscopic implications . . 33 1.4.1 Electrical conductivity . . . . . . . . . . . . . . . . . . 34 1.4.2 Depenclenre of conductivity on the semiconductor state 35 1.4.3 Optical absorption spectrum and the band modcl of

srmicoiiductors . . . . . . . . . . . . . . . . . . . . . . 38 1.4.4 Electrical conductivity in the band model . . . . . . . 43 1.4.5 The Hall effect and the existence of positively charged

freely mobile carriers . . . . . . . . . . . . . . . . . . . 45 1.4.6 Seinicondiictors far from thermodynamic equilibrium . 49

2 Electronic structure of ideal crystals 51 2.1 Abcimic cores and vdcnce electrons . . . . . . . . . . . . . . . 51 2.2 The ciynaniical problem . . . . . . . . . . . . . . . . . . . . . 54

lence dwtl-on system . . . . . . . . . . . . . . . . . . . 54 2.2.1 Schriidiiiger equation for the interacting core and va-

2.2.2 Adiabatic approximation . Lattice dynamics . . . . . . 57 2.2.3 Oneparticle approximation . Oneparticle Schriidinger

equation . . . . . . . . . . . . . . . . . . . . . . . . . . 66 General properties of stationary one-rlectron states in a crystal 82 2,3.1 Syinrnctry properties of the one-electron Schrtidinger

2.3.2 Rbch theorem . . . . . . . . . . . . . . . . . . . . . . 85 2.3.3 Reciprocal v e c h space and the reciprocal latt. ice . . . 89 2.3.4 Relation between energy eigenvalues and quasi-wave

vector . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

2.3

equation . . . . . . . . . . . . . . . . . . . . . . . . . . 82

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2.4 Schrodinger equation solution in the nearly-freeelectron ap- proximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 2.4.1 Kon-degenerate perturbat. ion t. heory . . . . . . . . . . 100 2.4.2 Degenerate perturbation theory . . . . . . . . . . . . . 103

2.5 Bandstructure . . . . . . . . . . . . . . . . . . . . . . . . . . 105 2.5.1 Brillouin zones . . . . . . . . . . . . . . . . . . . . . . 105 2.5.2 Degeneracy of energy bands . . . . . . . . . . . . . . . 116 2.5.3 Critical points and effective masses . . . . . . . . . . . 119 2.5.4 Density of states . . . . . . . . . . . . . . . . . . . . . 123 2.5.5 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 2.5.6 Calculational methods for band structure determination133

2.6 Tight binding approximation . . . . . . . . . . . . . . . . . . 140 2.6.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . 140 2.6.2 TB theory- of diamond and zincblende type semicon-

ductors . . . . . . . . . . . . . . . . . . . . . . . . . . 148 2.6.3 sp3-hybrids, total energy and chemical bonding . . . . 165

2.7 k . p-met. hod . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 2.7.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . 179 2.7.2 Valence bands of diamond structure semiconductors

without spin-orbit interaction . . . . . . . . . . . . . . 184 2.7.3 ht t inge T-Kohn model . . . . . . . . . . . . . . . . . . 189 2.7.4 Kana model . . . . . . . . . . . . . . . . . . . . . . . . 200 Band structure of important semiconductors . . . . . . . . . . 211 2.8.1 Silicou . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 2.8.2 Germanium . . . . . . . . . . . . . . . . . . . . . . . . 218 2.8.3 111-V semiconductors . . . . . . . . . . . . . . . . . . 219 2.8.4 IGVI semiconductors . . . . . . . . . . . . . . . . . . . 221 2.8.5 IV-\'I semiconductors . . . . . . . . . . . . . . . . . . 224 2.8.6 Tellurium and selenium . . . . . . . . . . . . . . . . . 224

2.8

3 Electronic s t ructure of semiconductor crystals with per tur- bations 225 3.1 Atomic structure of red semiconductor crystals . . . . . . . . 226

3.1.1 Classification of perturbations . . . . . . . . . . . . . . 226 3.1.2 Point perturbations . . . . . . . . . . . . . . . . . . . . 227 3.1.3 Formation of point perturbations and their movenient 235 3.1.4 h e and planar defects . . . . . . . . . . . . . . . . . 240

3.2 One-electron Schrodinger equation for point perturbations . . 241 3.2.1 Electron-core interaction . . . . . . . . . . . . . . . . . 242 3.2.2 Electron-elw?c.lron interaction . . . . . . . . . . . . . . 245

3.3 Effective mass equation . . . . . . . . . . . . . . . . . . . . . 252 3.3.1 Effective mass equation for a single band . . . . . . . 253 3.3.2 Multjband effective mass equation . . . . . . . . . . . 259

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3.4 Shallow levels . Donor and acceptor states . . . . . . . . . . . 265 3.4.1 Hydrogen model . . . . . . . . . . . . . . . . . . . . . 266 3.4.2 Improvements upon the hydrogen model . . . . . . . . 272

3.5 Deeplevds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 3.5.1 General characterization of deep levels . . . . . . . . . 281 3.5.2 Defect molecule model . . . . . . . . . . . . . . . . . . 285 3.5.3 Solution methods for the oneelectron Schriidinger q u a -

tion of a crystal with a point perturbation . . . . . . . 293 3.5.4 Correlation effects . . . . . . . . . . . . . . . . . . . . 301 3.5.5 Resu1t.s for se1ecDed deep centtas . . . . . . . . . . . . 308

3.6 Clean semiconductor surfaces . . . . . . . . . . . . . . . . . . 334 3.6.1 The concept of clean surfaces . . . . . . . . . . . . . . 334 3.6.2 Atomic structure of clean surlaces . . . . . . . . . . . 336 3.6.3 Electronic structure of crystals with a surface . . . . . 354 3.6.4 htomic and electronic structure of particular surfaces 371

3.7 Semiconductor microstructures . . . . . . . . . . . . . . . . . 388 3.7.1 Neterojunctions . . . . . . . . . . . . . . . . . . . . . . 388 3.7.2 Microstructures; Fabrication, classifications, examples 396 3.7.3 h*lethods for electronic structure calculations . . . . . 409 3.7.4 Elcctronic structure of particular microstructures . . . 420

3.8 Macroscopic electric fields . . . . . . . . . . . . . . . . . . . . 433 3.8.1 Effective mass equation and stationary electron states 434 3.8.2 Non-stationary states . Bloch oscillations . . . . . . . . 437 3.8.3 Interband tunneling . . . . . . . . . . . . . . . . . . . 440 3.8.4 Photon assisted interband tunneling . . . . . . . . . . 442

3.9.1 Effective mass equation in a magnetic field . . . . . . 444 3.9.2 Solution of the effective mass equation . . . . . . . . . 452

3.9 Macroscopic magnetic fields . . . . . . . . . . . . . . . . . . . 443

4 Electron system in t herrnodynamic equilibrium 457 4.1 Fundamentals of the statistical description . . . . . . . . . . . 457 4.2 Calculation of average particle numbers . . . . . . . . . . . . 460

4.2.1 Configuration-independent oneparticle states . . . . . 460 4.2.2 Configuration-dependent one-particle states . . . . . . 462

4.3 Density of states . . . . . . . . . . . . . . . . . . . . . . . . . 469 4.3.1 Total electron concentration . . . . . . . . . . . . . . . 469 4.3.2 Density of states of ideal semiconductors . . . . . . . . 470 4.3.3 Density of states of real semiconductors . . . . . . . . 474

4.4 Free carrier concentrations . . . . . . . . . . . . . . . . . . . . 477 4.4.1 Conservation of total electron number . . . . . . . . . 477 4.4.2 Free carrier concentration dependence on Fermi en-

ergy . Law of mass action . . . . . . . . . . . . . . . . . 478 4.4.3 Intrinsic semiconductors . . . . . . . . . . . . . . . . . 482

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4.4.4 Extrinsic semiconductors . . . . . . . . . . . . . . . . 484

4.4.6 More complex cases . . . . . . . . . . . . . . . . . . . 492 4.4.5 Compensation of donors and acceptors . . . . . . . . . 489

5 Non-equilibrium processes in semiconductors 499 5.1 Fundamentals of the statistical description of non-equilibrium

processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500 5.2 Systematics of non-equilibrium processes in semiconductors . 505

5.2.1 Temporal inhomogeneity and spatial homogeneity . . 505 5.2.2 Spatial inhomogeneity and temporal homogeneity . . . 506 5.2.3 Space and time inhomogeneities . . . . . . . . . . . . 508

5.3 Generation and annihilation of free charge carriers . . . . . . 509 5.3.1 Generation processes . . . . . . . . . . . . . . . . . . . 510 5.3.2 Unipolar annihilation of free charge carriers: capture

at deep centers . . . . . . . . . . . . . . . . . . . . . . 511 5.3.3 Bipolar annihilation of carriers at deep centers . . . . 517

5.4 Drift current . . . . . . . . . . . . . . . . . . . . . . . . . . . 523 5.5 Diffusion and annihilation of free carriers . . . . . . . . . . . 527 5.6 Equilibrium of free carriers in inhomogeneously doped semi-

conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530

6 Semiconductor junctions in thermodynamic equilibrium 535 6.1 pn-junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537

6.1.1 Establishment of thermodynamic equilibrium . . . . . 539 6.1.2 Diffusion voltage . . . . . . . . . . . . . . . . . . . . . 541 6.1.3 Spatial variation of the electric and chemical poten-

tials: Schottky approximation . . . . . . . . . . . . . . 542 6.2 Heterojunctions . . . . . . . . . . . . . . . . . . . . . . . . . . 549

6.2.1 Equilibrium condition . . . . . . . . . . . . . . . . . . 550 6.2.2 Electrostatic potentid . GaAs/Gal-,Al, As heterojunc-

tion as an example . . . . . . . . . . . . . . . . . . . . 552 6.3 Metal-semiconductor junctions . . . . . . . . . . . . . . . . . 557

6.3.1 Energy level diagram before establishing equilibrium . 357 6.3.2 Electrostatic potential . . . . . . . . . . . . . . . . . . 559 6.3.3 Schottky barrier . . . . . . . . . . . . . . . . . . . . . 563

6.4 Insulator-semiconductor junctions . . . . . . . . . . . . . . . . 567 6.4.1 Thermodynamic equilibrium . . . . . . . . . . . . . . 567 6.4.2 Influence of interface states . . . . . . . . . . . . . . . 570 6.4.3 Semiconductor surfaces . . . . . . . . . . . . . . . . . 572

7 Semiconductor junctions under non-equilibrium conditions 573

7.1.1 Electrostatic potential profile . . . . . . . . . . . . . . 576 7.1 pn-junction in an external voltage . . . . . . . . . . . . . . . 574

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Contents xv

7.1.2 Mechamism of current transport through a pn-junction 577 7.1.3 Chemical potential profiles for electrons and holm . . 580 7.1.4 Dependence of current density on voltage . . . . . . . 583 7.1.5 Bipolar transistor'. . . . . . . . . . . . . . . . . . . . . 585 7.1.6 Tune1 diode . . . . . . . . . . . . . . . . . . . . . . . 593

7.2 yn-junction in interaction with light . . . . . . . . . . . . . . 595 7.2.1 Photocffect at a pn-junction . Photodiode and photo-

voltaic element . . . . . . . . . . . . . . . . . . . . . . 595 7.2.2 Laser diode . . . . . . . . . . . . . . . . . . . . . . . . 599

7.3 Metal-semiconductor junction in an external voltage . Rectificrs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606

7.4 hwulator-semiconductor junction in an external voltage . . . 612 7.4.1 Field effect . . . . . . . . . . . . . . . . . . . . . . . . 612 7.4.2 Inversion layers . . . . . . . . . . . . . . . . . . . . . . 614 7.4.3 MOSFET . . . . . . . . . . . . . . . . . . . . . . . . . 620

Appendices

A Group theory for applications in semiconductor physics 623 A.1 Definitions and concepts . . . . . . . . . . . . . . . . . . . . . 623

A . l . 1 Group definition . . . . . . . . . . . . . . . . . . . . . 623 A.1.2 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . 624

A.2 Rigid displacements . . . . . . . . . . . . . . . . . . . . . . . 627 A.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 621 A.2.2 Translations . . . . . . . . . . . . . . . . . . . . . . . . 628 A.2.3 Orthogonal transformations . . . . . . . . . . . . . . . 629 A.2.4 Geometrical interpretation . . . . . . . . . . . . . . . . 631 -4.2.5 Screw rotations and glide re3ections . . . . . . . . . . 632

A.3 Translation. point and space groups . . . . . . . . . . . . . . 635 A.3.1 Lattice translation groups . . . . . . . . . . . . . . . . 635 -4.3.2 Point groups . . . . . . . . . . . . . . . . . . . . . . . 636 A.3.3 Space groups . . . . . . . . . . . . . . . . . . . . . . . 654

A.4 Representations of groups . . . . . . . . . . . . . . . . . . . . 655 A.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 655 A.4.2 Irreducible representations . . . . . . . . . . . . . . . . 661 4.4.3 Products of representations . . . . . . . . . . . . . . . 667

A.5 Representations of the full rotation group . . . . . . . . . . . 673 4.5.1 Vector representation of the rotation group and gen-

erators of infinitesimal rotations . . . . . . . . . . . . 674 A.5.2 Representations for dimensions other than three . . . 676

A.6 Spinor representations . . . . . . . . . . . . . . . . . . . . . . 682 A.6.1 Space-dependent spinors . . . . . . . . . . . . . . . . . 682

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A.6.2 Representation VI . . . . . . . . . . . . . . . . . . . . 683 A.6.3 Irreducible spinor representations . . . . . . . . . . . . 684 A.6.4 Double group method . . . . . . . . . . . . . . . . . . 685

A.7 Projective representations . . . . . . . . . . . . . . . . . . . . 687 A.7.1 Factor systems . . . . . . . . . . . . . . . . . . . . . . 687 A.7.2 Definitions and theorems . . . . . . . . . . . . . . . . 689 A.7.3 Construction of projective representations . . . . . . . 692

A.8 Time reversal symmetry . . . . . . . . . . . . . . . . . . . . . 692 A.8.1 Time reversal operator . . . . . . . . . . . . . . . . . . 693 A.8.2 Additional degenerac!?~ of energy eigenvalues . . . . . 694 A.8.3 Additional selection rules for matrix elements . . . . . 697

A.9 Irreducible representations of space groups . . . . . . . . . . . 698 A.9.1 Representations of translation groups . . . . . . . . . 698 A.9.2 Star of wavevectors . . . . . . . . . . . . . . . . . . . . 700 A.9.3 Small point groups and their projective representations 702 A.9.4 Representations of the fufl space group . . . . . . . . . 704 A.9.5 Spinor representations of space groups . . . . . . . . . 706 A.9.6 Implications of time reversal symmetry . . . . . . . . 707 A.9.7 Compatibility . . . . . . . . . . . . . . . . . . . . . . . 712

A.10 Irreducible representations of small point groups . . . . . . . 712 A.10.1 Character tables . . . . . . . . . . . . . . . . . . . . . 712 A.10.2 Multiplication tables . . . . . . . . . . . . . . . . . . 731 A.10.3 Compatibility relations . . . . . . . . . . . . . . . . . 734

2

B Corrections to the adiabatic approximation 737

C Occupation number representation '74 1

Bibliography 747

Index 757

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FUNDAMENTALS OF SEMICONDUCTOR PHYSICS AND DEVICES

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1

Chapter 1

Characterization of semiconductors

1.1 Introduction

Semiconductors are identified as a unique material group on the basis of their common macroscopir properties, as is done for metals, dielectrics and magnetic materials. The name ‘semiconductor’ stems from the fact that such materials have moderately good conductivity, higher than that of in- sulators, and lower than that of metals. However, if this were the only property which these materials had in common, the term ‘semiconductor‘ would have only a very weak foundation. But such is not the case. In fact, many materials having conductivity between that of metals and insulators. display simultaneously a series of further common properties. In particular, their conductivity depends very strongly on material staie, for example, on temperature and chemical purity, much more so than in the case of met- als. For sufficiently pure semiconductors, the conductivity decays by orders of magnitude while cooling down from room temperature to liquid helium temperature. ,4t absolute zero temperature, semiconductor conductivity al- most vanishes, in contrast to the conductivity of metals, which rises modestly with falling temperature. The conductivity of metals reaches its maximum at low temperature, and for superconductors it effectively becomes infinitely large. In regard to the dependence of semiconductor conductivity on the degree of purity, it has been found that a given semiconductor in a very pure state can resemble an insulator. while in a highly polluted state it acts like a metal, among other peculiarities. Furthermore, irradiation with light can cause a transition from insulator-like behavior to metal-like behavior of one and the same semiconductor. There are yet other optical properties shared by semiconductors: The op-

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2 Chapter I. Characterization of semiconductors

tical absorption spectra of semiconductors exhibit a threshold - below the threshold frequency, light can pass through practically without losses, while above it the light is strongly absorbed. Moreover, good luminescence proper- ties in the visible and infrared spectral range are also characceristic of many semiconductors. Thus, the identification of a semiconductor material in- volves several characteristic properties, not just the one of moderately good electrical conduction. One has a semiconductor only if all such properties apply. This criterion excludes ionic conductors, for example, which exhibit conductivity values of the right order of magnitude, but do not display the characteristic temperature dependence.

One may justifiably question the extent to which this definition of a semi- conductor really makes sense. For example, it is not yet obvious why just the above properties are selected as defining features while others are not, and what internal connections may exist among them. A full answer can only be given by means of a microscopic theory of semiconductor properties which will be developed systematically in the chapters to follow. For the moment, we invite the reader to join us in the recognition that all macro- scopic properties involved in the definition of a semiconductor can be traced back to a common microscopic origin, namely the nature of the spectrum of allowed energy levels and the particulars of their population by electrons. To be specific, the permissible energy levels of a semiconductor form bands which are separated by forbidden regions, and the Characteristic electron population of allowed energy levels is such that, at absolute zero tempera- ture, a semiconductor i s characterized by having only completely occupied and completely empty energy bands (no partially filled bands). It is this common microscopic feature which underlies the totality of macroscopic ma- terial properties that uniquely define a semiconductor. It also provides the basis for uncovering yet other common macroscopic features of this class of materials, beyond the ones already discussed. For instance, it may be expected that semiconductors should be predominantly solid crystalline ma- terials, since the formation of energy bands with gaps between them is most likely to occur in the crystalline phase. Nevertheless, amorphous and liq- uid semiconductors cannot be completely rxcluded since a certain regularity of the relative positions of neighboring atoms also exists in the amorphous and liquid phases. Actually, in addition to solid crystalline semiconductors, which are the main reprcsentatives, a series of amorphous semiconductors has also been found silicon and selenium being important examples. Liq- uid semiconductors are also possible, with melted tellurium among them. Other semiconducting materials, e.g. , silicon and germanium, are metals in the liquid phase.

In this book we restrict our considerations to solid crystalline semicon- ductors. The discussion also partially applies to amorphous and liquid sexni- conductors, but in most cases modifications are necessary. Even the basic

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1.1. Introduction 3

concept o f the quantum mechanical energy spectrum uf electrons has to be defined in tl different way, and a proper treatment of amorphous and liquid materials cannot be acconimodated within the framework of this in- troduction to semiconductor physics. Interested readers are referred to the extensive literature on this subject (see, for example? Elliot, 1983; Mott and Davis, 11979; Bunch-Bruevich, Enderlein, Ewer, Keiper, Mironov, and Xvyagin, 1984; Adhr, Frhsche and Ovahinsky, 1985).

The microscopic definition discussed above contains no recognizable con- straints with regard to the chemical nature of semiconductors. One may expect, therefore, that semiconductors should be distinguished by having a large chemical diversity. This is in fact true ~ semiconductors may be com- posed of a large variety of chemical elements and compounds. The most fully expAored candidates and those used for technical applications today arc crystalline semiconductors consisting of relatively small chemical un i t s , i.e. either dements or binary and ternary compounds.

Knowledge of thc mxistence of a distinct material group ‘semiconductors’ developed, historically, only relatively late. MPtals have been used by men since antiquity, but semiconductors attracted attention for the Erst time only a century and a half ago. The first reference to a characteristic semiconductor property dates back to Faraday who in 1833 observed an increase of the electric conductivity of silver sulfide with temperature. The exponential form of this increase was discovcrcxl by Hittorf in 1851. The trerm ‘semiconductor’ was introduced in 1911 by Konigsberger and Weiss after a similar term of about the same context had already been employed by Ebert (1789) and Bromme (1851).

The late and, initially, relatively slow development of semiconductor physics, is primarily due to the circnnistance that the characteristic proper- ties of semiconductors depend strongly on their degree of purity, more pre- cisely, on the presence 0’1 absence of certain chemical elements. This is also the reason that many semiconducting materials in their natural form as min- erals do not display the typical properties of a semiconductor ~ they are too heavily polluted and have too many structural defects. Natural diamondrr, for example, are semiconductors only in rare cases. Accordingly, clean fab- rication of the materials in the laboratory and the controlled incorporation of chemical elements played a crucial role from the very beginning. The lack of such control in preparation presented fundamental difficulties which had to be overcame in the early days of semiconductor rasearch. The necessary accuracy of composition control, which amounts to one atom in one hundred thousand or less, excccdcd the accuracy that prevailed in chemistry at the t h e by orders of magnitude. It was necessary to raise the accuracy of chem- ical composition control to a level wherein one could measure one millionth of a mole haction instead of one thousandth. only in this way could reliable results be achieved with semiconductors. Since such accuracy was achieved

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4 Chapter 1. Characterization of semiconductors

only gradually, semiconductor physics, in its early history, was confronted with apparently mysterious phenomena and contradictory results. For ex- ample, silicon and germanium were first thought to be metals, until the recognition that the impurity concentrations of certain elements were too large to achieve semiconducting properties.

From the very beginning semiconductor physics research was impeded by the need for expensive fabrication of material samples. The Geld could develop only to the extent that good samples could be made available. Nat- urally, the first samples weie either materials which could br fabricated rela- tively cheaply or materials which occurred in nature in a suitable form, albeit not ideal. Among these first materials were metal sulfides, which like lead sulfide in its mineral form as Galena and copper oxide (CuzO) and selenium in their artificially grown form, displayed good semiconducting properties even with relatively strong pollution and structural imperfections. In 1874, Braun discovered that contacts between certain metal sulfides and metal tips fxhibited different electrical resistance upon reversal of the polarity of the applied voltage. Such point contact structures were used in radio receivers as rectifiers at the beginning of our century. One can mark these point con- tact rectifiers as the first semiconductor devices. Similar rectifying action was also found for selenium and copper oxide. Moreover, a large change of electrical conductivity could also be achieved in these materials through irradiation by light. For selenium, this property was discovered as early as 1852 by Hittorf. Since the beginning of the 20-th century, this effect has also been used practically in the selenium photocell. The first technical use of copper oxide as a rectifier was accomplished in 1926 by Grondahl, followed by rectifiers using selenium. The first practical application of copper oxide in photocells was accomplished in 1932 by Lang.

Because of their technological importance, selenium and copper oxide were the first semiconductors to be subjected to more detailed physical in- vestigations. The semiconducting metal sulfides, selenides and tellurides were already known earlier because of their good luminescence properties. In the further exploration of these materials’ semiconducting behavior, lu- minescence physics partially merged with semiconductor physics.

In the mid-nineteen-thirties, the search for a solid-state-based electronic switching element which could replace the vacuum tube was extended to the elemental semiconductors germanium and silicon. The most important results of this research, which turned out to be decisive for the whole further development of semiconductor physics, were the invention of the germanium- based bipolar transistor in 1949, and the realization of the field effect tran- sistor with the help of silicon at the end of the nineteen-fifties. With the introduction of silicon, the stage was set for the development of semiconduc- tor microelectronics. Later, a similar role was played by compounds involv- ing elements of the third and fifth groups of the periodic table, like gallium

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1.2. Atornic structure of ideal mystah 5

arsenide or phosphide, making possible the development of semiconductor optoelectronirs.

The broad technical application of its results distinguishes semiconduc- tor physics now from its early days. It is well established that semicondnc- tors arc exceptionally well-suited for necessary functions in electronics and electrical engineering. This is by no means accidental, but is due to the mi- croscopic nature of semiconductors, which permits the controlled variation of characteristic matprial properties by external means over a wide range of parameters. The great technical importance of semiconductors has made thorough physical investigation of these materials necessary, but it has also justified the high cost involved in their fabrication and study. Owing to both of these aspects semiconductors are thc best explored and understood materials of condensrd matter today. Moreover, a multitude of physical phe- nomena which occur in other solid state materials may also be observed in semiconductors, often in the most distinctive way. For this reason studies of semiconductors can also provide knowledge of other solid state mattri- als. Semiconductors have in fact become model systems for basic research in condensed matter physics.

The presentation of the microscopic principles of semiconductor physics will occupy most of this book. The introductory first chapter lies outside of this framework because it involves discussion of the results on a phe- nomenological basis. The characterization of semiconductors by means of their unique macroscopic features, which we have touched upon above, will be continued in Chapter 1. In this regard, atomic structure will be discuss4 in section 1.2, chemical nature in section 1.3 and macroscopic physical prop- erties in section 1.4. In dealing with macroscopic properties we will not restrict ourselves to mere description, but we will also use them to mo- tivate the microscopic model of semiconductors introduced above. In this connection, we will make the first step towards a microscopic theory of semi- conductors in section 1.4. Naturally, this will have to be done in a heuristic way, and many questions postponed until later. The full presentation of the microscopic principles of semiconductor physics will follow in later chapters.

1.2 Atomic structure of ideal crystals

In all solid state materials, including amorphous ones, the neighbors of an arbitrarily selected atom are ordered in a regular way, just as in a molecule. The term short-range order is uscd for this property. The neighbors of a particular atom form its short-range order complex. Semiconductor materi- als, as they are treated in this book, are crystals. ‘Sheir atomic structure is approximately that of ideal crystals. The latter are distinguished by yet an- other order, apart from short-range, which is termed long-range order. This

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6 Chapter 1. Characterization of semiconductors

means that,, for a given atom, there are remote atoms possessing the same short-range order complexes as the original atom, rand that the positions of the remote atoms are related to the position of the origina1 atom by sirn- ple transformations. 'l'hp crystal is considered t o be inhitely large in this context. Atoms having identical short-range order complexes are termed equivalent. Equivalent atoms are necessarily of the same chemical species, but chemically identical atoms need not necessarily be equivalent.

1.2.1 Crystal lattices

The transformations displacing atoms into equivalent ones are translations of the crystal by vectors T which form linear combiaations

of three non-coplanar vectors Al, Az. A3 with arbitrary integer coefficients t l , t 2 , t3. The parallelepipeds of the crystal spanned by the particular dis- placement vectors Ak, A2, A3 are called unit cells. By putting unit cells together the wholc crystal may be constructed. ' h e size of a unit cell and the number of atoms in it is not fixed by the above definition, and can in fact be taken arbitrarily large, as long 8s it remains finite. The pertinent question is not how large a unit cell can be, but rather how small. The answer to this question leads us to the definition of the primitive unit cell and the crystal lattice.

Definition

The smallest possible unit cell is called a primitive unit cell In the extreme case this cell can contain only 1 atom, but as a rule, i t has several atoms. If there is only one atom per cell, then the short-range order merges into the long-range order.

If the unit cell is taken to be a primitive one, the vectors Al, A2, & are some minimal vectors al , az, w. The parallelepiped spanned by these vectors is a primitive unit cell, Each translation by a vector R of the form

with integer coefficients r l , r2, r 3 transforms the crystal into itself. One refers to this property as the trunslation~l symmetry of the crystal. The point set defined by the vectors R forms a spatial lattice called the crystal lattice. The vectors al, a2, are termed primitive lattice vectors, The volume 00 of a primitive unit cell may be written as the triple scalar product of al, ag, a,

Ro = al . [ag x a].

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1.2. Atomic structure of ideal cestals 7

While the lattice of a crystal and the volume Qo of its primitive unit cell are well defined, this is not the case for its primitive lattice vectors al, a2, a3 and also it is not true for the form of its primitive unit cell. Any set of linear combinations of the primitive vectors al, a2, a3 which yields a triple scalar product equal to the volume Qo is again a set of primitive lattice vectors, and the parallelepiped spanned by them forms a primitive unit cell. The corners of the parallelepipeds do not necessarily have to lie on lattice points. Each parallelepiped, shifted arbitrarily in space, is again a primitive unit cell. The 'parallelepiped form is also not imperative, as there are also other forms possible. An especially compact primitive unit cell is the so- called Wzgner3eit.z cell. The center of this cell lies on a lattice point and its surface is formed by the perpendicular bisector planes which divide in half the line segments joining the center to adjacent lattice points.

Translations which transform a crystal into itself, by definition, do the same for the lattice of the crystal. Here, the translations are through lattice vectors R, called lattice tramtations The set of all lattice translations forms a group. This term describes a mathematical set of elements among which a 'multiplication' is defined that results in products which are also elements of the set. Further properties of a set forming a group are listed in A p pendix A. In particular, there must be an identity element, and the inverse of an element must also be an element of the set. In the case of translations the 'multiplication' is the consecutive application of two of these transfor- mations. Since two consecutive lattice translations constitute yet another lattice translation, and also the requirements of Appendix A are satisfied, the set of all lattice translations of a crystal forms, in fact, a group, called the translation symmetrg, group or simply the tramlation group. Groups of symmetry elements play a central role in the description of the atomic struc- ture and other microscopic properties of crystals. Appendix A provides a thorough discussion of groups as needed in this book.

Point symmetry of lattices

We now ask whether there are other possible spatial transformations, besides translations, which transform lattices into themselves. From the outset it is clear that the only transformations which may be considered are those that do not change the distances between lattice points, i.e. mgzd drsplacernents of the lattice (see Appendix A). One can show that, besides translations, there is a second class of transformations fitting this description, namely rotatoom and reflectzons, as well as all products which are compounded from them, such as rotation-reflections, rotatton-znverstons and znverszon itself. Taken together, they are termed orthogonal traasfomattom. These differ from translations inasmuch as they leave one or several lattice points un- changed, while the remaining points are shifted by vectors depending on

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8 Chapter 1 . Characterkstion of semiconductors

their positions. In a t,ranslation, all points are shifted by the same vector, with no points fixed. Rotations transform right-hand4 Cartesian coordi- nate systems into right-handed systems, but the application of reflections and inversion to right-handed system results in left-handed ones.

It turns out that there are orthogonal transformations which transform lattices into themselves. They are called point symmetry operations of lat- tices. The set of all point symmetry operations of a lattice forms a group, as does the set of all lattice translations. The multiglicslion of two of these operations is again understood to represent their consecutive application. Groups of point symmetry operations are termed point gvoups. In Appendix A we describe thwn in detail. Not all of the various point groups listed in Appendix A are allowed as symmetry groups of crystal lailices, but only parlicular ones which are called holohedral point groups. We will derive them now by demonstrating that lhey must have three special properties.

First, all uf these point groups must contain the inversion transformation with reqet t to the lattice point R = 0. This may be seen as fullows: Inversion with respect to 0 transforms a lattice point R into -EL Considered joiutly with R, the point -R is a lattice point having -q, -r2, -7-3 as integer coefficients. Therefore, inversion with respect to 0 transforms an arbitrary lattice into itself. It follows immediately that inversion with respect lo any other lattice point will do h e same.

Second, it turns out that rotation symmetry axes Ohrough lattice points can only be 2-, 3-, 4- and 6-fold while ri, 7- and more-fold axes are not compatible with the translation symmetry of the lattice. One may prove this as follows: Let C, be a rotation about such an axis t,hrough an angle 2?r/n. We consider a lattice plane perpendicular to this axis and denote a primitive lattice vector of the corresponding planar lattice by f (see Figure 1.1). Rotating it through 2a/a, it becomes C,f, and a rotation by - 2 ? r / R

transforms it into Cqlf. If, as we suppose, Cm belongs to the point group of the lattice, them s n d o a C;'. Thus both C,f and C i l f are vectors of the plane lattice. The same holds for the sum Cnf i Cg'f ol t,he two vectors. Moreover, C,,f + CGLf represents a vector parallel to f . This means that C,f + CF'f must be an integer multiple of f. Since the largest possible length of Cnf + Cg'f can only he 2 / fJ , one has Cnf f C'L'f = p,F with pn - -2. -LO, 1 or 2. This is indicative t.hat, the relation

p , = 2 C O S (?) must hold for p,. For p n = -2, equation (1.4) yields n 1 2. For pn -= -1 one has n = 3, for pn - 0 it follows that n = 4 and for p , = 1 lhe solulion is n = 6 . For ppI - 2 equation (1.4) has only the trivial solution n = I . This completes the proof concerning rotation symmetv opexations. For so-caIled gvasi-crystals. which do not exhibit an exact translation symmetry, rotations

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1.2. Atomic structure of ideal crystals 9

Figure 1.1: On the mul- tiplicity of the rotation axes of crysta l lattices.

about &fold and other axes are also possible. Third, one finds that point groups of lattices having 3-, 4- or 6-fold axes

of rotation must also necessarily contain mirror planes parallel to each of these axes. The explicit proof of this assertion will not be presented here.

All three required properties described above are satisfied by each of

D 4 h ( & ; $ ) . D ~ h ( $ ; z ) and Oh ($3;). The point group notations used here are those of Schonflies and the international notations are given in brackets. Both systems of notation are explained in Appendix A. In sum- mary. the above results mean that exactly seven different point groups are possible for spatial lattices. They define the seven crystal systems: tri- clinic (Ct), monoclinic, (C2h), orthorhombic ( D z h ) , trigonal (D3d). tetrago- nal (D*h), hexagonal (DGh), and cubic (Oh).

exactly se’i;en point groups, namely c%(i), c Z ~ ( $ ) , ~ 2 h ( g , e ) > 9 2 q ~ 3 & $ ) >

Bravais lattices

Within a given crystal system, several different types of lattices may exist. Their common property is that they all have the same point symmetry, but they may differ otherwise. Figure 1.2 visualizes them by means of their unit cells. These differences give rise to diflerenf lattice types. The simplest lat- tices ef a given point symmetry are represented at the far left of each row in Figure 1.2. They axe called primitive lattices. Even these simple lattices are not unambiguously &ermined by their ‘point symmetry. If one, for in- stance, multiplies all lattice vectors by the same real number, i.e. stretches or compresses the lattices evenly on all sides, the point symmetry remains unchanged. In less symmetrical crystal system one may even change cer- tain length relationships or angles between primitive lattice vectors without disturbing the point symmetry. In the tetragonal system the height of the

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10 Chapter 1. Characterization of semiconductors

rectangular parallelepiped in relation to its basis may be changed arbitrar- ily. Generally speaking, the primitive lattices are determined only up to continuous point transformations preserving their point symmetries.

Moreover, starting from a primitive lattice one can produce other sets of regularly ordered points by adding new points to each primitive unit cell in equivalent positions. As before, equivalent refers to positions which are either identical or that differ by a lattice vector. If one places the new points in symmetrical positions, i.e. those which are transformed into themselves or equivalent points by the point symmetry operations, snch as the centers of the primitive unit cells, one obtains a new point set having the same point symmetry as the original lattice. The new point set, in general, no longer forms a lattice, but may be thought of as the union of several lattices placed within each other. In some special cases, however, the result may still be a primitive lattice. Whether this happens or not is a question which must be explored separately in each case. If the answer is positive, one has to examine whether the lattice is only another realization of the original primitive lattice, i.e. whether or not it can be brought back to the original one by a continuous and symmetry preserving transformation. It turns out that both cases may occur. If the lattices cannot be transformed into each other by such a transformation, then this implies that there are two different types of lattices with the same point symmetry. One calls them different Bmvais types or Bravais latttices. According to this definition two Bravais lattices are of the same type if they may be transformed into each other by a continuous and point symmetry preserving transformation, otherwise they are Bravais lattices of different types.

As an example, we consider the different Bravais lattices in the case of the cubic crystal system. If one adds the body centers of the primitive unit-cell- cubes as additional points to a primitive cubic lattice, the resulting point set has the cubic point symmetry and it forms again a lattice. The same holds if the added points are the centers of the faces of the primitive unit-cell-cubes instead of the body centers. Neither the spacecentered nor the face-centered cubic lattices can be transformed back to a primitive cubic lattice by means of a continuous and symmetry preserving transformation, nor can the two centered lattices be transformed into each other by such a transformation. Therefore, they represent cubic lattices of two new Bravais types. If one adds both face and body centers to the primitive cubic lattice, then the new point set forms again a primitive cubic lattice, however, with a lattice constant equal to half of that of the original lattice. It may be transformed back to the original primitive cubic lattice by means of a continuous and symmetry preserving transformation, thus it does not represent a cubic lattice of a new Bravais type. Altogether one finds three different cubic Bravais lattices, the primitive (p), the body-centered (bc), and the face-centered (fc) ones. Analogous considerations have to be made for the other 6 primitive Iattices.

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1.2. Atomic structure of ideal crystals 11

Cubic P Cuhic 1 C h i c F

Tetragons t I

Monocfinic P

pj$, .# \‘.

I

Monoclinic 1

Triclinic @ Trigonal R Trigonal and hexagonal P

Figure 1.2: Common unit cells of the 14 Bravais lattices.

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12 Chapter I , Charactm‘zation of semiconductors

In this w q , one finds that, in totul, 14 diEerriit Bravais lattices are possible in the 7 crystal systems. The assignment ol Iht.sr lattices to the crystal systems is indicated in Table 1.1. The Bravais lattices themsehps are shown in Figure 1.2.

1.2.2 Point groups of equivalent directions and crystal classes

The lattice of a crystal serves as a conceptional basis for the illustration of its translation symmetry. Lattice points do not necessarily have to be occupied by atoms, which may be bum1 at general points of the primitive unit cells. Generally, a primitive unit cell contains several atoms which may be either chemically identical or different. We denote them here by an atom index 1 which t,akes the values 1 - 1 , 2 , . . . , L, where L is the total number of atoms in a primitive unit cell. For L 2 2 the set of all L atoms is called the basis uf the crystal. In the case I. = 1 one says t.hat the crystal has no basis. ‘I’he position of the 1-th at,om, relative to the corner R of a primitive unit cell, is drscribed by a vrcbor The position Ri of this atom relative to the origin is then given by

R i = R t i . (1.5)

Without loss of generality one may always set onr of the vectors il, e.g. i l .

equal to zero. If the primitive unit cell canlains only 1 atom, it may be placed in o m o f the roriiers of the cell. A crystal without basis may thiis be describd as a laltice whosp points are dl occupied by ahoms. For L 2 2, the crystal may be generated in such a way that one multiples its crystal lattice L-fold, then shifts the resulting sublattices relative to t,he first by, respcctivcly, the vectors 6, . . . ,?I,, and finally occupies the points of the shift,ed lattices, respectively, with atoms of t,he species 2,. . . , L.

With only one atom per primitive unit cell! any point symmetry opera- tion of the lattice will ncccssatily transform the whole crystal into itsdf. For cryst,als with basis, however, this is not true in general. For this reason it is meaningful to consider, besides the point symmetry operations of the crys- tal lattice, also orthogonal transformations which map physically equivalent direchns of Ihe crystal into each other, without necessarily bringing the crystal back onto itself. We explain the meaning of ‘physical equivalence’ by using the example of the relation between the vectors of the electric current density j and the electric field strength E in a crystal. Generally, j is a non-lincar fimrtioii of E and, because of crystal snisotropy, the direct.ions of the two vectors may be different. If E and j are transformed from their original directions reht.ive to the crystal into new ones, without changing their relative orientation, the relation betwccn the new j and E, in general, will differ from the relation before rotat,ion. Analogous statements hold for reflections and rotation-reflections. If tbcre are orthogonal transformations

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1.2. Atomic structure of ideal crystals 13

Crystal

system

tnclinic

monoclinic

rhombic

trigonal

tetragonal

I

hexagonal

cubic

Table 1.1: Symmetry classification of crystals. The following abbreviations are used p - primit.ive, bc - body centered, fc ~ face centered, bfc - basis face centered.

Holohedral

group

Braiais

l a t t k

Crystal

class + ' 2 h

Number of space

groups

p bc fc I)

r c1 1

C, 1 ! I 1

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14 Chapter 1. Characterization of semiconductors

which leave the relation between the two vectors unchanged - which ulti- mately must be verified experimentally - the original and the transformed directions are called ‘physically equivalent’. The group of orthogonal trans- formations which map physically equivalent directions of a crystal into each other, is called the point group of equivalent directions. It defines the crystal class. Each crystal class corresponds to a particular point group of equivalent directions. In crystals without basis the point group of equivalent directions coincides with the holohedral point group. For crystals with a basis this is no longer true in general, and the point groups of equivalent directions are generally subgroups of the holohedral groups. There are as many different point groups of equivalent directions, or crystal classes, as there are different subgroups of the 7 holohedral groups. Using Appendix A one can easily show that their number amounts to 32 (see Table 1.1).

Not all crystal classes can be realized in all crystal systems - the point group which defines the crystal class has to be a subgroup of the holohedral group of the crystal lattice. Each crystal class is, however, found in at least one crystal system, several in more than one. In assigning the crystal classes to the different crystal systems one proceeds as follows: A class which exists in several systems is attributed to the one with the lowest common symmetry. In this way one obtains the assignment between crystal systems and crystal classes shown in Table 1.1.

1.2.3

It remains for us to explore the symmetry of the crystal as a whole. This consists of the set of all rigid displacements which transform the atoms of the crystal into identical or equivalent positions. It is obvious that symmetry op- erations which transform equivalent atoms into each other must necessarily also do the same with physically equivalent directions. The converse, how- ever, is not always true - after carrying out s rotation, reflection, rotation- reflection or rotation-inversion which transforms a particular direction into a physically equivalent one, the atoms of the crystal are not necessarily also trausformed into equivalent atoms. It may be necessary to add to a rotation yet another translation parallel to its axis (screw-rotation), or to add to a reflection a displacement parallel to the mirror plgraetglide-re~ection), or to add both in the case of rotation-reflections and rotation-inversions. That one records in this way all conceivable symmetry operations follows from a theorem proven in Appendix A, which states that every rigid displacement which is not a pure translation or an orthogonal transformation, must be a screw-rotation or a glidereflection.

The parallel displacement P; which follows a rotation about an n-fold axis in a screw-rotation, must be an integer multiple of one n-th of the smallest lattice vector in the direction of the axis. This follows immediately if one

Space groups and crystal structures

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1.2. Atomic structure of ideal crystals 15

considers the n-th power of the screw-rotation, which is also a symmetry element as well. This describes an n-fold repetition of the rotation by 27r/n, i.e. a rotation by 27r, or no rotation at all, followed by the translation n times +. Since the crystal must pass into an equivalent position by such a transformation, the vector n times p’ must be a whole lattice vector. As it points in the direction of the screw axis, it is necessarily a multiple of the shortest lattice vector in this direction. In the international notation system the multiplicity is described by a lower index afbed to the symbol for the rotation axis. This also indicates that the axis is not an ordinary one but a screw axis (with right-handed thread). The symbol 63, for example, means a 6-fold screw axis with a parallel displacement by half the shortest lattice vector in the direction of the axis. For a glidereflection one can similarly show that the translation in the mirror plane must be a superposition of multiples of halves of the smallest linearly independent lattice vectors in this plane.

In summary, we may state that the symmetry operations on a crystal are of the following types: translations by lattice vectors, proper rotations, reflections, rotary reflections, rotation-inversions as well as screw-rotations and glide-reflections. Furthermore, all combinations of these transformations are allowed. The set of all symmetry operations on a crystal forms a group. It is called a space group. If the crystal has no screw-rotations or glide- reflections as symmetry operations, then its space group necessarily contains the point group of equivalent directions as a subgroup. Such space groups are called symmorphic. Space groups with screw-rotations or glide reflections are called non-symmorphic. The latter do not contain the point group of equivalent directions of the crystal. Each element of a symmorphic space group is the product of an element of its translation group and an element of the point group of equivalent directions. The set of all possible space groups of crystals may be obtained in the following way: One considers, first of all, crystals of the triclinic system. In this case only the primitive Bravais lattice is possible. The corresponding space groups may easily be determined - there are only two. Similarly one proceeds with all other combinations of crystal classes, Bravais lattices and crystal systems, moving in the direction of increasing symmetry. At each stage of counting only the newly occurring space groups are added. In this way one obtains the numbers indicated in Table 1.1. The total is 230. Each of these 230 possible space groups corresponds to a particular crystal structure. By specifying its space group, the structure of a crystal is uniquely determined, except, of course, for changes of the distances between atoms and of angles between lines connecting atoms which do not affect the symmetry.

The majority of semiconductors belong to a small selection of the possible crystal structures or space groups. Five are especially important: Their des- ignations in the international system are Fd3m (diamond structure), F W m

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16 Chapter 1. Characterization of semiconductors

(zincblende st ruct ure) , F m3m (rocksalt st ruct ure) , P63m c (wurt zite st ruc- ture), and P3121 or P3221 (selenium structure). Each of these crystal struc- tures is more or less closely tied to a particular material group. We consider this connection in more detail in the next section, which deals with the chemical nature of semiconductors.

The five crystal structures above belong to just two different crystal sys- tems - the diamond, zincblende and rocksalt structures belong to the cubic system and the wurtzite and selenium structures belong to the hexagonal system. We start with the three cubic structures. Table 1.2 summarizes their properties

1.2.4 Cubic semiconductor structures

Crystals having diamond, zincblende and rocksalt structures not only belong to the same crystal system, but also have the same Bravais lattice, namely the face centered cubic, abbreviated as fcc. The fcc lattice is commonly described in a Cartesian coordinate system whose unit vectors e,, ey, e, are taken in the directions of the cubic crystal axes. The lattice constant a is the distance between the lattice points of a primitive cubic reference lattice, which is obtained from the fcc lattices by omitting the face centers. The primitive lattice vectors al,a2,* of the fcc lattice may be chosen in the form

The lattice constants for a series of semiconductors with fcc lattices are listed in Table 1.3. We stress that the cubic lattice constant is neither the distance between two lattice points (it is u / f i for the fcc lattice), nor the distance between two atoms in a crystal with this structure (which is &/4). For the volume Ro of the primitive unit cell one obtains the value u3/4 from equation (1.3). In the case of silicon this yields Ro = 4.00 x cm3. A silicon crystal of volume 1 cm3, therefore, contains 2.5 x primitive unit cells.

The lattice does not determine the positions of the atoms. This is done by the basis of the crystal, in which respect the three cubic semiconductor structures differ.

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1.2. Atomic structure of ideal crystals 17

Table 1.2: Characteristicz of the five most comnion structure types of seniiconduc tors. 0 < C < h, 0 < n < $. More details are given in the text.

Wiirtzite Seleniuni Itocksalt Diamond Zincblende

F#m

fy P3121

P3221

,3; 4 - 2 43m Gmm 32

hexagonal

P

Bravais

lattice

cubic

f c

cubic

f c

cubic

f c

hexagonal

P

Primitive

lattice

vectors

Basis I

I I

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18 Chapter 1. Characterization of semiconductors

Table 1.3: Lattice constants of semiconductors of various types of structures.

ZnS CdS CdSe MgTe AlN GaN InN

3.82 4.14 4.31 4.53 3.11 3.19 3.54

1.64 1.62 1.63 1.62 1.60 1.62 1.61

Selenium structure

Se Te

4.36 4.47

1.14 1.33

Diamond structure

a

cla

a

cl a

C Si Ge a-Sn I 1: 5.43 5.65 , 6.46 1 : AlP RlAs AlSb GaN Gal’ Cans CaSb I d ’

4.37 5.47 5.66 6.14 4.54 5.45 5.65 6.13 5.87

Zincblende structure

InAs InSb ZnS ZnSe ZnTc CdTe HgSe IlgTe

6.05 6.47 5.43 5.66 6.08 6.42 6.08 6.37

Rocksalt structure I I PbS PbSe PbTe SnTc CdO MgS MgSe

5.93 6.12 6.45 6.3 4.70 5.20 5.46

Wurtzite structure I I

Diamond and zincblende structure

For diamond and zincblende structures, the basis consists of two atoms which, respectively, have the same or different chemical natures. The two atoms are shifted with respect to each other in the direction of the body diagonal of the primitive reference cube by a quarter of its side. If the basis atom 1 is put at a lattice point, then the position of atom 2 is given by

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1.2. Atomic structure of ideal crystals 19

Figure 1.3; Spatial model of a crystal having diamond (a) and zincblende structure (b). In the diamond structure all atoms arc of the same chemical clement, in the zincblendc structure the atoms are of two different chemical elements.

Cmll

P 6

0 1 A 0 2 8

3 A 0 4B

(1111

a I A r, 2 0

3 A ( J 4 0 3 * 5 A

6 0

cfm

Figure 1.4: Projection of a crystal having diamond and zincblende structure in a (100) plane (left) and a (111) plane (right). Atoms of the same size and darkness lie in the same plane. The vertical sequence of atomic layers is indicated on the right hand side of the projections. The sequence for the (100) plane is repeated after 4 layers, and that for the (111) plane after 6 layers.

Crystals with diamond and zincblende structures may also be described as a composite of two interpenetrating superposed fcc lattices displaced with respect to each other by the vector ?2 and with their lattice points occupied by, respectively, chemically equivalent or different atoms. The geometric relations are depicted in Figures. 1.3 and 1.4 in, respectively, a 3-dimensional representation and a plane projection. From Figure 1.4 one can readily see that for both crystal structures the cubic axes put through an atom form 4-fold mirror-rotation or inversion rotation axes (which are the same in this drawing). The body diagonals through an atom are %fold axes of

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20 Chapter I . Characterization of semiconductors

Figuw 1.5: Stereograms of the paint groups Oh (left) and 7'6 (right) of, respectively, the diamond and zinchlmde structiires.

rotation, and the planrs through opposite cube dges are mirror planes. Both structures are theiefore tramformd into themselves by the point group T d , the symmetry group of a tetrahcdron. It follows that their point groups of equivalent directions are at least T d . For the zincblende structure there are no further point symmetry operations which transform equivalent directions into each other. The point group of equivalent directions is therefore T d , and the space group turns out to be symmorphic. It is denoted by F13m.

In the raw of diamond structure, thew i s yet another symmptry oper- ation, namely that which transforms the two chemically equivalent atoms of the basis into cach other. It may be described as reflection in a plane perpendicular to a cubic axis, say e,, which cuts the connecting line be- l w r n two atoms at its center, followrd hy a translation in that plane by (e/4) (ex + ey). One, therefore, has a glidereflection as additional symmetry dement. The space group of diamond structure becomw non-symmorphir in this way The point group of directions of this structure may be obtained from the point group T d of the zincblende structure by adding the reflections in planes perpmdicular to the thrw cubic axes. This yields the full cubic group Oh, as one can easily determine by means of the stereograms of the Iwo groups rn Figure 1.5 (for an introduction to the stereograms of point groups, see Appendix A). The group o h can be generated from the tetra- hedral group in yet anothcr way, namely by adding the inversion operation. As an element of the space group, inversion must be relative to the site of an atom and be followed by a tradation through the vector T2 or -?zr depaid- ing on whether m c considers a 1- or 2-atom. For future reference we bear in mind that for the diamond and zincblende structures, inversion always involves an exchange of the two sublattims.

We now extend our discussion to thc neighborhood-relations in the two crys-

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21

Figure 1.6: Spatial model of a crystal with rocksalt structure.

tal structures, the so-called coordmatatm Consider atom 1 of thc Lesis of the unit cell at R = 0. The basis atom 2 of this cell is one of its nearebt neigh- bors. If all rotation-reflections are executed with respect to the axis through atom I, then the basis atom 2 of the primitive unit cell at R = 0 gives rise to three more basis atoms of type 2. They are all at the same distance from atom 1 as thP 2-atom first considered, but naturally they lie in three other primitive unit cells. This means that each 1-atom has, altogether, four nearest neighboring 2 atoms. Since it would also have been possible to start from a 2-atom in this consideration, the mme also holds with re- spect to a %atom, and, of course, independently of whether t h 2-atom is of the same chemical nature as the 1-atom (diamond structure) or is not (zincblende structure). The four nearest neighbor atoms lie at the corners of a tetrahedron, whose center is occupied by the atom itself (see Figure 1.3). Their relative positions are given by the vectors (a/4)(1, 1, l), (e/4)(T, 1, l), (a/4) (1,7, l), (a/4) (1 ,1 , i ) . Thr distanrc betwrrn nearest neighbor atoms is &a/4. For silicon this is 2+35 A . The second nearest npighbors belong to the same sublattice. Therefore, for both structures, they arc atoms of thP same chemical species. They are located at thc nearest-neighbor lattice points of the fcc lattice. relative to the central atom, thus their positions are ( u / 2 } ( * e 9 f ef), (a /2)(fey f e z ) , (a/2)(*e, f ey). This means that thew are 12 second nearest neighbors in each slructurr.

Rocksalt structure

The basis of this crystal structure consists, like that of zincblende structure, of two atoms of diffprent chemical nature which arc displaced relative to each other in the direction of the space diagonal of the reference cube. In contrast to zincblende structure, the displawmcnl i s not, however, a quarter but half of the length o€ the space diagonal of the reference cube, whence

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22 Chapter 1. Characterization of semiconductors

1 A 0 1 B 4 2 A 0 2 8

(1111

1 A 0 2 8

3 A 0 48 - 5 A 0 6B

Figure 1.7: Projection of a crystal having rocksalt structure in the (100) plane (left) and the (111) plane (right). The vertical sequences of atomic layers is shown on the right hand side of the projections. In the (LOO) case the stacking repeata itself after 4 Iayem, and in the (111) case after 6 layers.

Accordingly, a crystal having rocksalt structure can be described as a com- posite of two fcc lattices, displaced with respect to ~ a c h other by the vector ?'z, and with their lattice points occupied by chemically different atoms. This structure is illustrated in Figures 1.6 and 1.7. As one can see horn these drawings, the crystal has the full symmetry of the primitive reference cube. The point group of equivalent directions is therefore the full cubic group o h (see Figure 1.5). The space group is symmorphic and is denoted by FmSm.

In Ihe direction of a cubic axis, the atoms are separated by a distance of a / 2 , with 1- and 2-atoms alternating. Since shorter distances do not occur, these atoms are ncarest neighbors. Each atom, therefore, has 6 nearest neighbors. The second nearest neighbors are the nearest neighboring atoms of the same fcc sublatticp. Their number amounts to 12, as in the zincblende structure. The relative positions are also the same as in this structure.

1.2.5 Hexagonal semiconductor structures

The primitive lattice vectors of the hexagonal lattice may be written in the form

h a h a h ai y- ae,, a2 - --ex + -&e,", ce,. 2 2

Here ek, I$, e," are unit vectors of a cubic coordinate system whose z-axis points in the direction of the c-axis of the hexagonal lattice and whose z-axis is identifird arbitrarily with one of the three symmetric lattice directions in the plane perpendicular to the c-axis. The two primitive lattice vectors a1

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1.2. Atomic structure of ideal crystals 23

and a2 in this plane have equal lengths and form an angle of 2x13 with each other. The volume 5-20 of a primitive unit cell is &a2c/2. The lattice constants a and c of some semiconductors with a hexagonal lattice are listed in Table 1.3.

Wurtzi te s t ruc ture

The basis of crystals with wurtzite structure consists of 4 atoms. We denote them by M I , X I , Mz and X2. Each pair of these atoms, namely M I , M2, on the one hand, and XI, X2) on the other hand, are chemically equivalent, while the two pairs are of different elements M and X. If atom M I is sited at a lattice point, then the position vectors of the three others are

fx1= cc e," , (1.10)

Here, ( is a parameter which may take any value between 0 and 1/2. A wurtzite type crystal can be understood as a composite superposition of four interpenetrating hexagonal lattices, with the lattice points of two of them occupied by atoms of chemical species M, and two of them by atoms of species X. The two M-lattices are displaced relative to each other by F M ~ , likewise as the two X-lattices. The displacement of the XI-lattice with respect to the MI-lattice is Fx1. In Figures 1.8 and 1.9 these relations are illustrated in, respectively, spatial and planar displays. The M-atoms are shown in black and the X-atoms in white. From these figures one can easily determine the symmetry of the wurtzite structure. The c-axis through the center of one of the empty hexagons in Figure 1.9 forms a 6-fold screw axis 63: a 2n/6-rotation about this axis must be connected with a parallel displacement by c/2 in order to transform the crystal into itself. This also mean8 that this axis represents a %fold proper axis of rotation. The axis contains six inequivalent mirror planes, among them three proper reflection planes and three glide-reflection planes joined with translations by c/2 in the direction of the c-axis. The corresponding point group of directions is c 6 v (see Figure 1.10). The space group of the wurtzite structure is non- symmorphic and is denoted by P6amc.

Consider the atom M I in the unit cell at R - 0. Its neighbor in the same unit cell is atom X 1 at the site ?XI. Another neighbor of M I is the atom Xz at the site -ce,h I Fxz of the unit cell at R - -c$. Since the c-axis through atom M i is a %fold symmetry axis (sep Figure t.9), one obtains from Xz two additional X-atoms at the same distance but in other unit cells. Altogether, there are four neighboring X-atoms. In order that all four be at the same distance from M I , and, correspondingly, all four be

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24 Chapter 1 . Characterization of semiconductors

Figure 1.8: Spatial model of a crystal with ideal wurtzite structure.

0 1 A 0 2 8

3 A 0 4 B

Figure 1.9; Projectionof a crystal with Figure 1.10: Stereograms of the ideal wurtzite structure in the plane point group c6~ of the wurtzite normal to the c-axis. The vertical se- structure. quence of atoniic layers is shown on the right herid side.

nearest neighbors of M I , the distance 1 7x1 I between M I and XI must be equal to the distance I -ce," t Fx2 I between Mi and X2 in the unit cell at R = -ce!. This results in the condition

c = , t 9 ( ; ) 1 1 . (1.11)

If it is fiilfilled, the atom Mi has four nearest neighboru of chemical species X . The same statement is true for M I , with the interchange of M and X, dito for XI and Xa. In semiconductor crystals of the wurtzite type, the condition (1.11) is always satisfied almosl perfectly. This means that the four nearest neighbors of an atom are sited at the corners of a slightly deformed tetrahedron which surrounds the central atom symmetrically, and

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1.2. Atomic structure of ideal crystals 25

which is compressed or stretched in the direction of the c-axis. The degree of deformation depends on the ratio cla. For the deformation to vanish, the distance between the M I - and XI-lattice planes perpendicular to the c-axis must be three times the distance between the X I - and M2-lattice planes perpendicular to this axis, just as in the zincblende structure. This yields < = 3(1/2-<) or < = 318 which, taken together with equation (l.ll), results in

( z ) = fi= 1.633. (1.12)

Table 1.2 shows that all wurtzite type semiconductors listed there also satisfy this condition quite well. This indicates that the ordering of the nearest neighbors in the wurtzite structure is almost the same as in the zincblende structure. If a wurtzite crystal exactly obeys conditions (1.11) and (1.12), one says that it has an ideal wurtz i te structure. The distance between two nearest neighbor atoms in the zincblende structure is &all4 with a‘ the cubic lattice constant. The corresponding distance in the wurtzite structure is Cc = (3/8)@ a. For the two distances to be equal, the equation

a’= h a (1.13)

must hold. This result means that in a crystal having wurtzite structure which fuEUs conditions (1.11) and (1.12), the nearest neighbor atoms are positioned at exactly the same sites as in a crystal having zincblende struc- ture with lattice constant 4 a. If one multiplies the lattice constants a, of thc wurtzitc type crystals in Table 1.2 by 4, then one in fact obtains values which fit well witah the cubic lat,t,icc constants of the zincblendc struc- ture crystals of this table. The volume &a2c/2 of a primitive unit cell oi“ the ideal wurtzitc structure is d 3 / 2 , which is twice that of thc zincblende reference structure given by 8 1 4 .

Of course, this does not at all mean that the wurtzite structurc may be traced back to lhe zincblende slruclure, Indeed, the actual positions of the second nearest neighbors differ in the two crystal structures, although they are all at the same distances a = a’/& This may be seen most easily from the projections of the two crystals on the plane perpendicular to the c-axis (see Figure 1.4) or to the space diagonals e, -t e$ -t e, (see Figure 1.9). In the wurtzite structure the vertical stacking repeats itself after two double layers of nearest neighbor atoms, but in the zincblende structure only after three. The wurtzik structure can be set up, therefore, by stacking one upon another two different double layers A and B, the zincblende structiire by stacking three double layers A’, B’, C’ (see Figure 1.11). The uppermost double layer A’ of the zincbbnde structure in Figure 1.l lb coincides with the

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26 Chapter 1. Characterizrrtion of semiconductors

A B - A’ * A - - B - B’

A - C’ A’ . B - B’

-

- - - - : C’

A’

Figure 1.11: Construction of a crystal having wurtzite structure (top) and zincblende structure (bottom) by stacking of double layers of nearest-neighbor atoms. Atoms shown by full circles form the upper part of a double layer, and atoms shown by open circles form the lower part of such a layer. In stacking, the nets shown in each of the double layers must be placed at the same positions. The stacking sequence is depicted on the upper right. M h e r explanations are given in the text.

uppermost double layer A of the wurtzite structure in Figure L l l a , while B’ and C’ differ from B. By examining the layers in Figure 1.11 we also note the different locations of some of the second nearest neighbor atoms of the two crystal structures. Among the 12 second nearest neighbors of an atom in an A = A’-layer, in the ideal wurtzite structure, three are found in the upper B-layer and three in the lower B-layer, while in the zincblende structure three second nearest neighbors are in the C’-layer above and three in the B’-layer below. In Figure 1.11 this fact is illustrated by emphasis on the second nearest neighbor atoms mentioned. Those in double layer B are not in the same positions as those in double layer B’. Thus the second nearest neighbor sites are partially different in the wurtzite and the ideal zincblende structures.

A comparison of the two crystal structures shows that the same ordering of the nearest neighbor atoms may be consistent with different orderings of more remote atoms. From this observation we may conclude that, in a solid,

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1.2. Atomic structure of ideal crystals 27

Figure 1.12: Spatial model of a crystal with selenium structure.

the long-range order could be perturbed without significantly changing the short-range order of the corresponding crystal. This is just what is observed in amorphous silicon and other amorphous semiconductors.

Selenium structure

The basis of trigonal selenium and tellurium crystals consists of three atoms. The vectors of the basis may be written in the form

2c ~3 = -2craez + - 3 e,h, (1.14)

where a is a parameter ranging between 0 and 112. The atoms are ordered on parallel spirals, in which successive atomic positions are rotated with respect to each other by an angle of 120' (see Figure 1.12). The space group can be determined from the projection of the crystal on the plane perpendicular to the c-axis (see Figure 1.13). It contains one %fold screw axis parallel to the c-axis which goes through a center of one of the empty triangles in Figure 1.13. A rotation by 120° or 240' must be followed by a displacement by c/3 or 2c/3 in the direction of the c-axis, to transform the crystal into itself. Beside these screw operations, rotations are allowed with respect to the three inequivalent 2-fold axes perpendicular to the c- axis. These rotation axes pass through the %fold screw axis and through one of the three surrounding atoms. The symmetry operations mentioned correspond to the space group P312. Beside this, the selenium structure may also have symmetry corresponding to the space group P322, where the spiral differing from P312 has not a right-handed thread, but a left-handed one. The point group of directions is, in both cases, D3 (see Figure 1.14).

Concluding this discussion of the characterization of the most common semiconductor crystal structures, it should be emphasized that these struc-

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28 Chapter 1 . Characterization of semiconductors

(00011

o c c 3

Figure 1.13: Projection of a ciystal Figure 1.14: Stereograms of the having selenium structure on the plane point group D3 of the selenium normal to the c-axis. The vertical se- structure. quence of atomic layers is shown on the right hand side.

tures apply to the ideal case of infinite, structurally perfect, and chemically absolutely pure materials. Real crystals deviate from this ideal case in vary- ing degrees. There are structural defects such as stacking faults, step- or screw-dislocations, vacancies, or atoms on interstitial crystal sites. Such de- fects will be considered in detail in Chapter 3, in the context of the electronic structure of yprturbrrl semiconductors. In the next section we will discuss the chemical composition of idea1 semiconductor crystals.

1.3 Chemical nature of semiconductors. Material classes

According to their chemical nature, most semiconductors are inorganic ma- terials. Examples of organic semiconductors indude anthracene, naphthaline and polyacetylcne. In thiv book, .we restrict our considerations to inorganic semiconductors, because these are much bettcr understood and haw much greater twhnolagical iniporttznce than semiconductors of the organic type. Nevertheless, organic semiconductors are also of interest for science and tcch- nology.

The best way to get an overview of the different &sues of inorganic semiconducting materials is to examine thP periodic table of elements. In 'l'able 1.4. a part of the periodic table is shown with which many elemental and compound semiconductors are associated. W e start with the clernenld semiconductors of group IV.

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1,3 . Chemical nature of semiconductors. Material classes

3 c N 0

Mg A1 Si P S

Zn Ga Ge As Se

Cd In Sn Sb 're Hg T1 Pb Bi

29

Table 1.4: Part of the periodic table to which Inany element.al and compound semiconductors are associated.

1.3.1 Group IV elemental semiconductors

Semiconductors of this group are, starting at the top of column IV? carbon (C) in the form as diamond, silicon (Si), germanium (Ge), and tin (Sn) in its gray modification known as a-Sn. All cited semiconductors of this group crystallize into the structure of diamond. They differ from each other in regard to their position Let,weezl metals and insulators. Diamond behaves much like an insulator! and tin, on the contrary, much like a metal. Retween them one has the two typical semicondnctors silicon and germanium. Both mat.&& play an essential role in micro- and optoelectronics. It is weU- known that silicon is the material of preference in microelectronics and, with this, one of the most important materials in dl modern technology.

Beside the pure elemental semiconductors Si aid Ge, alloys of bath ma- terials also have semiconducting properties. The notation for such alloys is (Si,Ge), or Sil-,Ge, to indicate the rpspcctive mole fractions 1 - z and z of the two alloy components. According to their structure, the alloys may be identified as mixed crystals. These are crystals with the same geomcb- ric order of atom sites as in the case of the alloy components ~ here that of diamond in both CBSW. The regular crystal sites are occupied, however. randamly by Si or Ge atoms. T'he probability of finding a particular el* ment on a given site is determined hy the mole frclctiou of this element. in t,he alloy. In the case of Sil-,Ce, the probability of finding Si is 1 - 2. and of finding Ge is x. The lattice constants of inixcd cryst.als, in many

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30 Chapter 1 . Characterization of semiconductors

cases, follow from the lattice constants of the constituents by means of En- ear interpolation. This result is known as Vegard’s rule. The cubic lattice constant a(si,ce) of a Sil-,Ge, alloy, for example, is given by the relation ~ ( s i , ~ ) = (1 - z)asi + Z U G ~ , where asi and a& are the cubic lattice con- stants of Si and Ge, respectively. A mixed crystal strongly deviates from an ideal crystal, but the semiconducting property of the two pure crystals is often retained in the mixed crystal. This is important, and by no means obvious. It was recognized at the beginning of the sixties in (Si, Ge) alloys. Later it was also shown to be true in many other material combinations. The discovery of semiconducting alloys significantly expanded the family of semiconductor materials. It became possible to tailor materials with desired properties, just by varying the alloy elements and mole fractions.

The common feature of the elements of the main group IV of the peri- odic table is that there are four electrons in the outer shell of their electron clouds, the so-called valence shell. The primitive unit cell, which here has two atoms, therefore contains 8 electrons. Later we will prove that semi- conducting properties are related to the crystal structure and the number of valence electrons per primitive unit cell. Using this result, one may conclude that compounds of elements from main groups III and V, and from main groups 11 and VI of the periodic table should also be semiconductors. pro- vided they have zincblende structure, since the crystals of these compounds also have 8 electrons per primitive unit cell. Actually the GI-V compounds (with a few exceptions) do crystallize into the zincblende structure, In the case of ILVI-compounds the wurtzite structure can also occur, besides the zincblende structure. The two structures are, however, very similar, as has been pointed out above, so that semiconducting behavior may be expected also in their case.

1.3.2 111-V semiconductors

The conjecture that the 111-V compounds should be semiconductors is amply confirmed. Following the group III and group V columns of the periodic table from the top down, one obtains the following compound semiconductors: BN, BP, BAS, AlK, Alp, AlAs, AlSb, GaN, Gap, GaAs, GaSb, ZnN, InP, InAs and InSb. Except for the nitrides all these compounds crystallize into the zincblende structure. The nitrides are stable in the wurtzite structure, BY and GaN also have metastable zincblende phases. Just, as in the case of the elemental semiconductors of group IV, the mixed crystals made of binary 111-V compounds also have semiconducting properties. Examples are (Ga, Al)As, Ga(As, P), (In,Ga)As and (In,Ga)(As,P). The principal applications of the 111-V semiconductors and their alloys lie in the field of optoelectronics. (Ga,Al)As and Ga(As,P) are used, €or example, in light emitting diodes (LEDs) and laser diodes for the near infrared to green spectral region. GaN

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1.3. Chemical nature of semiconductors. Material classes 31

is a promising material for blue LEDs and laser diodes. The quaternary alloy system (Ga, In)(As, P) is used in making laserx and pholodiodes for optical fiber communications at 1.55 p m wavelengths, which provides maximum transmission for SiOa-based fibers. GaAs and, lately, also (Gtr,Al)As are employed for transistors with extremely short time delay. BN and GaN can be used for electronic devices to be operated at high temperatures.

1.3.3 11-VI semiconductors

This group includes compounds like ZnS, ZnSe, ZnTe, CdTe, HgSe and HgTe, which crystallize into the zincblende structure, and compounds like CdS, CdSe and MgTe which have the wurtzite structure. For most of these mate- rials the one of the two structures which has not been mentioned corresponds to a metastable phase, and for ZnS both structures are, in fact, stable. The cubic phase of ZnS is the mineral zincblende, and the hexagonal phase is the mineral ururtzite. It i s from these mineral names that the designations of the two crystal structures are derived. CdO, MgS and MgSe form crys- tals with rocksalt structure, HgS (vermilion) has a rocksalt-like structure. As in the case of the 111-V compounds a large number of semiconducting alloys may also be realized from the IT-V1 compounds, e.g. (Hg,Cd)Te, Zn(S,Se), Cd(S,Se) and others. Up to now, technical applications involve mainly (Hg,Cd)Te. It is utilized for detectors of radiation in the medium infrared spectral region, which is of particular interest since the emission maximum of the thermal radiation of bodies lies in this region for a broad range of temperatures. Some semiconductors of this material class, e.g. CdS, have very good photoelectric properties and are used accordingly. Other such compounds, like ZnS, are good luminescent materials. They make it pos- sible to fabricate largearea electroluminescence displays, an alternative to the traditional vacuum-tube. The operation of blue-green LEDs and injec- tion lasers made of 11-VI compound semiconductors was demonstrated very recently.

1.3.4 Group VI elemental semiconductors

Selenium and tellurium are likewise good semiconductors. Both crystallize into the same trigonal chain structure, which is characteristic of these mate- rials (see Table 1.2). Selenium has a long history of practical use. Before the introduction of power rectifiers based on silicon, selenium ones were used. Today, above all, the photoelectric properties of selenium are exploited. This applies, for example, to photocopying where the photosensitive layer of the photostatic cylinder may be made of selenium or a selenium alloy.

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32 Chapter 1. Characterization of semiconductors

1.3.5 IV-VI semiconductors

EIements of groups IV and VI can also form semiconducting compounds. Examples are PbS, PbSe, PbTe and SnTe, which all crystallize into the rocksalt structure. Important semiconducting alloys of such compounds are (Pb,Sn)Te and Pb(S,Se). These semiconductors are now used primarily for the generation and detection of radiation in the medium infrared spectral region. In addition, there is a second group of semiconducting IV-IrI com- pounds which display an orthorhombic structure, including GeS, GeSe, SnS and SnSe. GeTe is an exception as it crystallizes into a rhombohedra1 struc- ture.

1.3.6 Other compound semiconductors

It is to be expected that a combination of two elements of group IV will be a semiconductor. It seems that S ic is the only stable compound of this kind. It is one of the longest known semiconductors of all. Electroluminescence was observed in this material as earlier as 1907. S ic OCCUIS in many crystal structures, among them the zincblende and wurtzite structures. After a long period of only moderate interest in Sic, much activity is now being devoted to this materiai. The reason is the good thermal stability of Sic. which makes it possible to fabricate electronic devices for high temperature and high power applications.

The elements of groups I1 and V may form semiconducting 1 I 3 - v ~ com- pounds. Examples are ZmAsz and CdsAsz, which possess both tetragonal and cubic crystalline phases.

One also finds semiconducting properties in compounds composed of el- ements of groups V and VI with the stoichiometric composition V2-VI3. In particular, the oxides, sulfides. and selenides of the semimetals As, Sb, and Bi are among them. With regard to crystal structure, a relatively large diversity exists among these compounds. Different structure types of the trigonal, orthorhombic and monoclinic crystal systems have been observed. The very good thermoelectric properties of Bi2Se3 and BizTq have long been used for cooling elements.

Finally, one also finds semiconductors among III-VI compounds. Exam- ples are G ~ q S e 3 ~ GazTe3 or In2Se3, which display a 2:3-stoichiometry. They crystallize into so-called defect stmctuws. These are modifications of cer- tain basic crystallographic structures as, for example, of the zincblende or wurtzite type, which distinguish themselves by having a (ordered or disor- dered) network of cation vacancies. These vacancies are necessary to make possible a stoichiometry which deviates h m that of the basic structure. Among the semiconducting 111-VI compounds, however, there are also those with 1:l-stoichiometry. The layer-structures GaS, GaSe, and GaTe are ex-

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1.4. Macroscopic properties and their microscopic implications 33

amples. A semiconductor of this group displaying ferromagnetic properties is EuSe.

So far we have exclusively considered bznary compound semiconductors. Ilowever, thcrc are also semiconductors generated from ternary and qua ternary compounds. Among the ternary I-IIEVI compounds, one has, in particular, the I-111-VTz materials which crystallize, such as CuGaSz, into the chalcopyrite structure belonging to the hexagonal crystal system. Semi- conductors are also counted among the II-IV-V2 compounds, ZnSiPa as an example. The crystal structure is also of chalcopyrite type.

Concluding this survey of the most important semiconducting material classes, we add two more general remarks. Firstly, it has to br remeinherd that we have stated the chemical composition of ideal semiconductor crys- tals, meaning, in particular, chemically pure materials. Absolute chemical purity, however, occiirs in nature just as seldom as absolute crystallographic perfection. Each real semiconductor material contains chemical impurities. In order that crystals composed of such materials really be semiconductors, their impurity concentrations must not to be too large - bear in mind what has been said about this point in the introduction. On the other hand, semiconductors can not to be too pure. In order to have a sufficiently large electric conductivity, they must often be intentionally polluted with certain chemical elements. We will undertake this discussion in greater detail in the next subsection.

Secondly, it is striking that the semiconducting substance classes which were defined pritnarily by their chemical nature also display, as a riile, the same crystallographic structure. The reason for this relation between chem- ical composition and atomic structure is by no means obvious. It is closely connected with the electronic structure of crystals which we will treat in Chapter 2.

1.4 Macroscopic prapert ies and their microscopic irnplicat ions

In this section we describe characteristic macroscopic properties of semi- conductors and we them as a guide to reveal some important microscopic features of these materials as, for example, the distribution of the allowed electron energy levels in the form of bands being separated by forbidden en- ergy regions. Although the microscopic model of a semiconductor obtained in this way cannot be more than a lucky guess, it will stimulate our intu- ition later in this book when we are going to derive the microscopic physics of semiconductors, including their band structure, from first principles.

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34 Chapter 1. Characterization of semiconductors

Table 1.5: Electrical conductivity 0 and charge carrier concentrations for metals, serniconduct.ors, and insulators.

Material class Metal Semiconductor Insulator

l o 3 . . < 10-10

1021 . . . 1o1O n jcm-3

1.4.1 Electrical conductivity

As already mentioned, the term ‘semiconductor’ refers to a characteristic macroscopic property of these materials, namely their electrical conductiv- ity, The latter is defined in terms of the proportionality factor between the current density j and the electric field E in Ohm’s law

j = u E (1.15)

Here, it is assumed that anisotropies either do not exist or are negligible. The following example gives an idea of the order of magnitude of the con- ductivity: Consider a cube with edge length of 1 cm, and apply a voltage of 1 V between two of its opposite planes. Then a current of 1 A will flow if the cube material has a conductivity of (T = 1 ern-'. The order of mag- nitude ranges spanned by the U-values of the three material classes ‘metals, semiconductors, and insulators’ are assembled in Table 1.5. The striking features of this table are the large changes both between the three classes of substances as well as within the ‘semiconductor’ material class. The question arises how these sharp changes may be understood 011 a physical basis.

To construct an explanation, we use the known representation of the electrical conductivity IT as the product of the electron concentration n, the mobility ,u, and the electric charge -e of an electron,

u = enp . (1.16)

The mobility p is defined as the ratio of the magnitude of the average ve- locit,y of an electron to the strength of the electric field driving it (for a derivation see Chapter 5). Formula (1.16) for the Conductivity o shows that its variations may, in principle, be caused either by different values of the mobility p and/or by different electron concentrations n.

The mobility is actually determined by the scattering processes which electrons undergo due to the perturbations of the crystal lattice. One ex-

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1.4. IlIacrascopic properties and their microscopic implications 35

pecks, of course, differences with respect to the strengths of these processes in metals, semiconductors, and insulators, but variations by several orders of magnitude are very unlikely. The orders of magnitude of the mobilities ME,

psc and in normal metals, semiconductors and insulators, respectively, should be more or less the same:

P M E E? PSC =” P I S . (1.17)

This conjecture is confirmed by experiment, and consequently it follows that the large conductivity differences between the three material groups are es- sentially due to their different electron concentrations. If one assumes for metals one free electron per atom, which amounts to approximately 10” electrons per m3, then the conductivity values in Table 1.5 yield the elec- tron concentrations of semiconductors and insulators listed in this table.

1.4.2 Dependence of conductivity on the semiconductor state

The large range of conductivity values of semiconductors, between lo-’ 0-l cm-l and lo3 R-’ m-’ , does not arise, as one might suppose, from the chemical diversity of semiconductors, but is primarily due to the fact that a given semiconductor material covers the entire range of cr-values by itself if its macroscopic state is varied. By ‘state‘ we mean, firstly, the temperature, and secondly the Concentrations of certain impurities.

Considering first the change of the conductivity with temperature, we note that it is particularly pronounced if the semiconductor contains prac- tically no impurities - one c d s this an int~ninsic semiconductor. In Figure 1.15 the dependence of the conductivity on temperature is shown for very pure silicon and germanium samples. The conductivity increases with rising temperature according to the exponential law

(1.18)

Here, rrg is a factor which depends only weakly on temperature, k is the Boltzmann constant and Eg is an energy of about 0.7 eV in the case of ger- manium and 1.1 el’ in the case of silicon. Formula (1.18) also describes the temperature dependence of the conductivity for other pure semiconductor materials fairly well at sufficiently high temperatures, save that the values for cro and E, must be changed. In Table 1.6 the &.-values are listed for some import ant semiconductors.

There is a second way of varying the conductivity of a semiconductor over a wide range, namely to intentionally pollute it with atoms of certain chemical elements. One calls this process doping wzth impup.ity atoms. In this context, the doped semiconductor materials are called extrinsic semi- conductors. As an example we choose silicon, doped with arsenic a t o m

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36 Chapter 1. Characterization of semiconductors

Material

Figure 1.15: Temperature depen- dence of the conductivity a of very pure (intrinsic) Ge and Si. (Ajter Epzianov, 1979.)

C: Si Ge ,O-SiC BN AlN AIP AlAs AlSb

Figure 1.16: Dependence of the conduc tivit.y u of As-doped Si at 1' = 300 X an tlic arsenic concentration ~VD. (After Morin and Makta, 1954.1

Material

E ,

Material

E,

of concentration N D . In Figure 1.16 the change of conductivity is shown with N o varying from 1014 up to crnA3. At temperature 300 K cor- raponding to the data of Figure 1.16, the conductivity rises monotonically with doping ronrrntration. Again, a change of several orders of magnitude i s exhibited.

GaN GaP GaAs ( M h InN IriP I iub IrlSb ZnS

3.30 2.27 1.43 0.71 1.95 1.26 0.36 0.18 3.56

ZnSe CdS CdTe HgTe PbS PbSe PbTe Se Te

2.67 2.50 1.43 0.00 0.37 0.26 0.30 1.8 0.33

Table 1.6: E,-values for several semiconductor materials at 300 K (in eV).(Aftar Lundoldt-Bumstein, 1982.)

E , 1 5.5 1.11 0.66 2.2 6.2 6.28 2.45 3.14 1.63

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1.4. Macroscopic properties and their microscopic implications 37

Figure 1.17: Temperature de- pendence of conductivity u of As-doped silicon for different ar- senic concentrations N D : 1 - 1.75 x 1014, z - 2.1 1015,3 - 1.75 x 101*,4-1.3 x 1017,5-2.2 x 101B.6-2.7x lo1’ cm ’. (After Morin and Maita, 195d.}

b

The trmperalurr dependence of the conductivity is substantially weaker for doped silicon than for pure as may be seen from Figure 1.17. Each curv-e in this figure corresponds to a particular concentration of arsenic atoms. The temperature rise becomes weaker with increasing doping, and for the highest doping it completely vanishes. At higher temperatures, it even turns downward in a decrease that is more pronounced for purer materials. The rise of curve 1 above 400 K indicates thcrt the corresponding [relatively pure) material starts to behave like an intrinsic semiconductor (see Figure 1.15).

The origin of the difference in temperature characteristics of the con- ductivity of pure and d o p d semiconductors remains to be addressed. In this regard, it will be helpful to discuss yet another rnacrosropic property of semiconductors, namely optical absorption.

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38 Chapter 1 , Characterization of semiconductors

f Figure 1.18: Absorption coefficient a of Si BS II fimction of photon energy Bw. (Aj ter Philipp and Taft, 1960.)

100 -'

10-2~ ' I I ' ' I 1 2 4 1 0 2 0 4 0 m m

hw/eV - 1.4.3 Optical absorption spectrum and the band model of

semiconductors

The measured values of the absorption coefficient of silicon are displayed in Figure 1.18 showing a ( h w ) as a function of the photon energy hw. The par- ticular shape of this spectrum, namely the vanishing of a below a threshold energy, the steep rise at this energy up to valurb which are larger by orders of magnitude, and the weak decrease from these high values with further in- crease of photon energy, i s characteristic of all semiconductors. In simplified terms, one may say that semiconductors are transparent below the threshold photon energy, and absorbing above it. The values of the optical threshold energies coincide so well with the Eg-values known from the temperature dependence of the electrical conductivity (see Table L6), that an intimate relation i s strongly suggested. Below we will analyze this relation in greater detail.

Consider a semiconductor in thermodynamic equilibrium at absolute zero temperature, and let E, be the maximum energy which an electron may have m such a semiconductor. If an electron of that energy absorbs a photon of energy hw, lhen its energy increases to the value E, + hw. The absorption can, however, only occur if the energy E, + hw is in the allowed range of electron energies. It is well-known that in quantum mechanics not all energies arc allowed, but only specific energies which are eigenvalues of the Hamiltonian. For atoms these are the energies corresponding to the various Bohr orbitals. Thus, in semiconductors, absorption may be non-vanishing abovc h'g only if the electrons can take energy values E > E, - E* + E,, i.e. if the energy values E' > E, are allowed. In turn, the vaniahing of absorption for E < E, can be explained if the energy values between E, and & are forbidden. Such a distribution of allow4 energy values is shown schematically in Figure 1.19. The representation of these values as a function

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I .4. Macroscopic properties and their microscopic impIications

Valence bond

O H 0 X-- - L

Figure 1.19: Ordering of the allowed e n e r a v d i m of an ideal semicnnduc- tor (band model). T h e spatial ex- tension of the corresponding electron states is also illtlstrnted.

0 1 f [El--)

39

Figure 1.20: Fermi distribution function f (E) as a function of energy at different temperat#ur&. The energy unit is E,, and EF has been set at

E,P.

of electron coordinate is intended to illustrate the degree of localization of the electron. A straight line across the whole crystal from 0 to d means that the electron at this energy is spread out over the whole crystal. The regions with closely clustered allowed energy values and electron states extended over the whole crystal are called energy bands. The lower band in Figure 1.19 is the valcrice band, and the upper is the conduction band. Between these two bands lies a region of forbidden energy values, called the forbidden zone or the energy gap.

The existence of energy bands done is not in itself siifficient, however, for the explanation of the observed absorption spectrum of semiconductors. For this it is also necessary that the valence band be occupied by electrons and the conduction band be empty. If there were no electrons available with energies in the valence band, then there would &o be no electrons to absorb a photon to make the transition to the conduction band possible. If, on the other hand, all states of the conduction band were to be occupied, then no electron could be excited tu this band, and no photon absorb4 in such a transition since, according to the P a d exclusion principle, the conduction band could host no further electrons.

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40 Chapter I , Characterization of semiconductors

0

0 L X-

0 L X 4

Figure 1.21: Distribution of electrons of an ideal semiconductor over the valence and conduction bands. On the left for T = 0 K , on the right for T > 0 K.

The population of energy levels by electrons at equilibrium is determined by the lkrrnz distribution f inrt ion f ( B ) . This function represents the prob- ability of finding an electron in an energy levcl E , and it is given by the expression

(1.19)

Here EF is the so-calld Fwmz mevyy. Like temperature, the Fermi en- ergy is an intensive thermodynamic state variable, namely the free enthalpy per particle or the chemical potential. A more detailed treatment of the Fermi distribution function will he given in Chapter 4. The probability of orcupation of a particular energy value E: depends decisively on the relative position of E with respect to the Fermi energy. If E < EF holds, and in addition J E - EFJ >> kT, then f ( R ) essentially has the value 1. If, on the contrary, E > E F and again IE - EFI >> k T , then f(E) is approximately zero. The shape of f(E) is shown schematically in Figure 1.20. The width of the enera region where the transition horn 1 to 0 takes place is cf the order of magnitude kT. The lower the temperature, the more abrupt this transition becomes. At T = 0 K thc transition is steplike. Qccesionally, one says that f(E) has the form of an ‘iceblock’ at T 0 K , which melts at higher temperatures. 111 Figure 1.20 we have also implied that Pi, lies below E F and E, above E F , in order for the valence bend to be almost completely occupied and the conduction band to be almost empty. This means that

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1.4 . hfacroscopic properties and their microscopic implications 41

Figure 1.22: Absorption of photons by electron transitions from vaIence to con- duction band.

0 L X - - L

the Fermi energy must be located within the energy gap between Ev and E,, sufficiently far away from the two band edges (measured in units of kT). In this. we have described an essential microscopic property of pure semi- conductors. Their Fermi level lies in the energy gap between the valence and conduction bands. The electrons are distributed over the two bands as shown in Figure 1.21. This knowledge provides a complete explanation of the absorption spectrum in Figure 1.18. The explanation is illustrated in Figure 1.22 and needs no further comment.

The differences in the properties of semiconductors and metals may be traced back, in essence, to the different positions of the Fermi levels in these two different types of materials. In metals the Fermi level lies within the conduction band. Insulators do not differ from semiconductors qualitatively with respect to their Fermi level positions, i.e. the Fermi levels are found in the energy gap in both cases, but the gap of insulators is typically larger than that of semiconductors. If Eg > 3.5 e V , as a rule, one has an insulator.

1.4.4 Electrical conductivity in the band model

We will now show that the temperature dependence of the electric con- ductivity of pme semiconductors also follows from the energy band scheme discussed above and the position of the Fermi level in it. Again, we employ the relation u = enp. The mobility p can be assumed to be a relatively weakly varying function of temperature, T. This means that the strong ex- ponential T-dependence of CT must be due to the electron concentration IZ,

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42 Chapter 1. Characterization of semiconductors

and comparison with equation (1.18j yields, approximately

(1.20)

with n o as a factor of dimension ~ r n - ~ depending weakly on temperature. As we know, the electrons of intrinsic semiconductors are statistically dip tributed over the valence and conduction bands in accordance with the Fermi distribution function f(E). At T 7 0 K the valence band is fully occupied, and the conduction band is completely empty. The electrons in the fully oc- cupied valence band cannot contribute to carrier transport. The reason for this is again the Pauli principlc, according to which an electron state may be occupied by only one electron. Since all valence states are occupied, no stste change is possible in the valence band by redistribut,ing the electrons. This means that the T = 0 K state will remain unchanged in 8x1 electric Eeld, and consequently no current will flow. However, a current will arise from the rel- atively few clectrons which, according to thc Fermi distribution function: are populating the conduction band at ftnite temperature. Their concentration can be easily calculated. For a pure semiconductor Ec - EF >> 6T holds. Thus, in the Fermi distribution function j ( E ) of ecpation (l.l!J), out3 may neglect the ‘1’ of the denominator for emrgies E in the conduction band, i.e. energies with E > EC Iu so doing, one obtains, approximately, the Boltzmann distribution function

(1.21)

The concentration of energy levels in the conduction band having energies between E and E + dE is pc(E) d E , where pc(E) is the so-called density of states of the conduction band, which describes the number of states per unit energy and unit volume. In Chapter 2 we will calculate this quantity explicitly, here its mere existence suffices. In terms of p, (E) one may write the electron concentration n of the conduction band in the form

(1.22)

If one substitutes f(E) from (1.21) into the integrand of (1.221, it follows that

where

(1.23)

(1.24)

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1.4. Macroscopic properties and their microscopic implications 43

may be understood as the effectiae density of states of the conduction band. To ensure the consistency of the expressions (1.20) and (1.23) for n, we identify no = N , and (Ec - E F ) = Eg/2, or

1 EF = Ev + -E (1.25)

2 g '

i.e. the Ferini level lies exactly in the middle between the valence and con- diistion bands. Strictly speaking, this is only true for T = 0 K, as we will see later. The corrections at higher temperatures are, however, quite smaI1, and they can bc ignored here. Another inaccuracy of the above considera- tion that. we have ignored is that, at, finite temperatures, the valence band population also changes. so that no longer are all states of this band ocm- pied. Under these circumstances t.he electrons of the valence band also make a contribution to the current. The temperature dependence of this contri- bution is roughly the same as that of the conduction band. The electric charge transport due to the electrons of a not-completely-occupied valence band will be considered more fully later. It is connetted with a remarkable observation of the Hall effect, which we will discuss below.

However, we will first clarify how one may understand the strong change of conductivity with impurity concentration in extrinsic semiconductors. For this purpose we again m e the energy band model of Figure 1.19 for an ides1 semiconductor. If arsenic impurity atoms are present in a silicon crystal, then this model has to be altered. Wc will later prove (see Chapter 3) that each of the arsenic impurity atoms g i v ~ rise to an energy level in the forbidden gap, just below the conduct.ian hand edge, and that the electrons in these levels are localized at the site of the impurity atom. For this reason, we have marked these levels in Figure 1.23 by short line segments.

At temperature T = 0 K each of these levels is occupied by one dec- tron, namely the fifth valence elcctron of an arsenic atom replacing a silicon atom having only four valence electrons (see Figure 1.23). If temperature i s increased, these electrons are excited into the conduction band. In this way, bound electrons which formerly could not participate in carrier trans- port, become freely mobile electrons (see Figure 1.24) which can contribute to transport. If the concentration N D of arsenic atoms is not too large: and the temperature 7' not too law, then practically all arsenic atoms are ionized and the concentration n of free electrons is equal to the that of the arsenic atoms, i.e. one has R = N D . This corresponds to the approximate propor- tionality between cr and N o observed in Figures 1.16 and 1.17. If the carrier concentration has Ihe value NLI> which at sufficiently high temperatures T i s actually the caw, then the conductivity becomes independent of T. The only remaining source of a temperature dependence for u is the T-dependence of the mobility p. It is this relatively weak temperature effect which shows up

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44

t Figure 1.23: Ordering of the allowed en- ergy values of electrons in a silicon crys-

In tal doped with arsonic (schematically). Each arsenic atom correspond# to 1 lo- cslized energy level. G Eg

a, P W

a W

- 2 I a

0

0 L X - - c

in Figure 1.17 at high T .

With this, the experimental observations relating to carrier transport in semiconductors are explained microscopically, at least in principle. We now proceed to the Hall effect.

T - 0

0 X-

T.0

L X-

Figure 1.24; Distribution of the electrons of a silicon crystal doped with arsenic over the allowed energy values. 011 the left for T 7 0 K , OIL the right for T > 0 K .

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45

Phosphorus Boron *) doped Si 1- doped Si u

Current- a ) U

Current 4

Current - Current- bl

Phosphorus doped Si

~ a i i current H ~ I I current + Cl

HOII current + fie'd Hall current d )

Figurc 1.25: nlrlstration of the Hall effect in two silicon smrples, one doped with phosphorus, and the other with boron.

1.4.5 The Hall effect and the existence of positively charged freely mobile carriers

We consider two semiconductor samples of the extrinsic type hoth madf of the Same material, however, with different types of doping. To be specific, we assume two silicon samples, one doped with phosphorus as a group V dement. and the other with bnroii as a group 111 element (see Figiiie 1.25). Applying a voltage to both samples, as shown in Figure 1.25a and b. a current I will Bow. For both samples it has the same direction, namely from '+' to '-', Consider, ROW, the ~ I I I P two sarriples in a niagnPtic Geld B ( ~ e e Figure 1.25~ and d). As is well-known, the Hall effect will be observed in such circumstances, i.e. a current component I H perpendicular to both the electric field E and the magneliu field B will arise. Gnder the wndilforts

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46 Chapter 1. Characterization of semiconductors

of Figure 1.25 no current can flow in this direction, and therefore there will be a voltage UH such that the current caused by it will just compensate the Hall current. Experimentally one finds that the Hall voltage UH has different polarities in the two samples. This means that the corresponding Hall currents flow in opposite directions. If we assume that the magnetic field is directed normal to the plane of the figure and that it points into this plane, then the Hall current flows upwards in the boron-doped sample, and downward in the phosphorus-doped one.

The Hall effect can also be measured in metals. In this case the direction of the Hall current is the same as in the silicon sample doped with phospho-

be understood? The explanation is quite clear if one assumes that, in contrast to metals

and the phosphorus-doped silicon sample, the current in the boron-doped sample is not carried by negative charge carriers but by positive ones. This is illustrated in Figure 1.25d. A charge q, which moves with velocity v in a magnetic field B, experiences the Lorentz force

How can the dlfemnt behauiior of the siicon sample dupd with born

4 F = - [ v x S ] . C

(1.26)

Since negative charges move to the left, and positive ones to the right, the Lorentz force F, which depends directly on the sign of the charge, has, for both charge signs the same direction, namely upwards. For the boron-doped sample this means a Hall current directed upwards, but in the phosphorus- doped sample the Hall current is downwards, since in this case the charge carriers are negative. This is just the observed behavior, which means that the assumption of positive freely mobile charge carriers is successful in ex- plaining the unusual sign of the Hall current in boron-doped silicon. In this way the experimental observations of the Hall effect reveal a remarkable gen- eral property of semicondiictors: For doping with certain atoms, the current is not carried by negative charges as one would expect considering the nega- tive charge of electrons, but by positive ones. In other words, in addition to the electrons as negative freely mobile charge carriers, one also has positive ones in semiconductors under certain conditions.

This surprising observation may be understood as ~OUQWS. One may demonstrate - as we will do later explicitly - that boron atoms whi& sub- stitute silicon atoms in tf silicon crystal, give rise to energy levels in the forbidden zone just above the valence band edge. This is illustrated in Fig- ure 1.26. Electrons in such energy levels are localized at the sites of the boron atoms. At very low temperatures these states are not occupied, and the boron atonis are electrically neutral. At finite temperatures electrons from the valence band are excited into the boron levels (Figure 1.27), leav- ing behind unoccupied states, or occupation hole3 in the valence band. We

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1.4. Macroscopic properties and their microscopic implications

Figure 1.26: Ordering of the allowed en-

t ergy values of electrons in a silicon crys-

tal doped with boron (schematically). Each boron atom corresponds to 1 local- i d energy level.

47

immediately recognize that these occupation holes of the valence band be- have jubt like freely mobile positive charge carriers in an applied electric field. This is illustrated in Figure 1.28. For simplicity only one hole is as- sumed to be in an otherwisr rompletely occupied valence band, placed at the upper band edge. The population of the bend edge is shown in Figure 1.28 at different points of time. At the beginning, the hole is at the outermost left position. The adjacent electrons experience a force which tends to move them to tho left. Rut only the electron neighboring the hole on its right hand side can follow this force, since all other electrons are blocked because the states to their left are already occupied. The electron immediately on the right hand side of the hole, moves into the hole and leaves behind an- other hole to the right of the first one. The hole has thus moved one site further to the right in this way. 111 the next time interval this process is repeated, and the hole again moves one site further to the right etc. Thus, in an electric field, the hole moves like a freely mobile electron, but in the opposite dirwtion, as if carrying a positive charge. In summary, holes in the valence band behave like freely mobile positive charge carriers. This quali- tative introduction of the concept of holes will later be elaborated by more quantitative considerations.

As we have seen above, the sign of the Hall voltage tells one whether the free carriers of an extrinsic semiconductor are negatively or positively charged, Le. whether they are electrons or holes. In the first case one speaks of a n-typr semzcondurtov (the n stands for ‘negative’), in the second of a p-type semarondector ( p for ‘positive’). The antranszc semiconductors intro-

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48 Chapter 1. Characterization d semiconductors

L X-

0 1 X-

Figure 1.27: Distribution of elect.rons of a silicon crystal doped with boron over allowed energy values.

Figure 1.28: Empty states (halefi) in Ihe valence band behaye like freely mobile poriitive charge carri- em in an electric field.

duced earlier haw neither electrons nor holes from impurity atoms. Their mobile electrons arr generated by thermal excitation of bound electrons from the valence band to the rondiiction band. Since each excited electron leaves behind a hole in the valence bmd. one also has mobile holes in intrinsic semi- conductors. Their number is equal to that of the mobile electrons. Holes are also present in n-type semiconductors, but, in very small numbers com- pared to elwtrons. Analogously, a few electrons occur in p type materials with mob1 of the carriers being holes. One calls the many mobile carriers of extrinsic semiconductors majon ly cariwrs and the few mobile carriers mzrrortty earners.

The Hall effect can also be used for purposes other than the determi- nation of whethrr an extrinsic semiconductor is n- or p-type. The absolute

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valup of the Hall voltage determines the concentration of majority carriers. We will veTify this assertion for an n - t y p ~ semicondiictor. In this case the rurrcnt density j can be expressed in the form j =- --env. which permits us lo rewrite equation (1.26) as

X B ] . 1 F = - [ j RC

(1.27)

This is the same force caused by the electric field E ~ I = F/(-e). The voltage corresponding t,o EH is by dehition the Hall voltage Ufi. With b as the width of the sample normal to j and 3, the expression below for U x follows:

where 1

Rff = - P r t C

(1.28)

(1.29)

is the so-called Hall constant. We can write an analogous expression for holes, only the electron concentration has to be replaced by the hole con- centration p . Thin;, hy measuring thc Hall vultagr, onr ran also det.ermine the majority carrier concentration.

1.4.6 Semiconductors far from thermodynamic equilibrium

All properties considered hitherto were for semiconductors in a state of ther- modynamic equilibrium or in close proximity. However. semiconductors may easily be driven into states far from equilibrium. Here, 'far' means that char- acteristic macroscopic propexties of the semiconductor deviate strongly from those in equilibrium.

To be more specific about these qualitative statements, we examine the example of photo-conduction. Consider a sample of an intrinsic semironduc- tor, shielded against unwanted influence of light. If, in this 'dark state' the conductivity is measured. one obtains a relatively small value. in accordance with the relatively low carrier concentration of an intrinsic semiconductor in thermodynamic equilibrium. However, if one irradiates the sample with light which is absorbed by the semiconductor (see Figure 1.22). the conductivity will rise more or less strongly, depending on the intensity of the light and the magnitude of the absorption coefficient. This is shown schematically in Figure 1.29. assuming realistic conditions. When the exposure of the sample to light ceases, its conductivity decreases to the original low dark value. Ev- idently, electrons in the conduction band and holes in the valence band were created by irradiation with light to such an extent that their equilibrium val- ues were exceeded by orders of magnitude. A simple estimate for Figure 1.29

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50 Chapter 1. Characterization of semiconductors

Figiire 1.29: Conductivity of an intrin- sic semiconductor irradiated with light as a function of intensity I of the ab- sorbed radiation (schematically).

1 M - ~ -

1~-7 - I I I

10-4 10-3 lo-7 10-1 ma r]'

I h o t t om* --+

shows that t.he carrier concentration was increased from about 10'' cm4 (equilibrium value) up to c ~ n . - ~ by light of intensity 10 Wcm-'. The electron-hole pairs created by the radiation decay, however, after a short t,ime. 'lhis follows from the fact that continuous irradiation leads to tt sta- lionary conductivity value, and lienre a constant carricr concentration, and, on the other hand, from the observat.ion that the conducbjvity decays down to the dark value after switching off the light source. The 1at.ter observa- t.ion means that, thermodynamic equilibrium i s reestablished by so-called wxnnbinaiion, p7nuesses.

Rwidr irradiation with light, extreme nun-equilibrium states in semicon- ductors can also be created in other ways, for example, by putting an n-type semiconductor in contact with a p t y p e semiconductor or with a metal, or by applying voltage to a. semiconductor which previously had been isolated by a thin insulating layer from one of the electrodes. The ability to c r e at,? extreme non-equilibrium states in semiconductors i s extensively used in electsonic devices. Almost all applicat.ions of semiconductors in such devices rest on this uniquc pQssibility-.

Non-equilibrium processes in semiconductors and the most important scmiconductor devices based on them, such as electric rect.%ers, bipolar and unipolar transistors, tunnel diodes, photodetectmu: solar cells, as well as luminescence and laser diodes, will be dealt with in the second part of this book, i.e. in Chapters 5 , 6 and 7. In the first part of the book, the basic concept.s, discussed above in a heuristic way, will be developed from first principles. This applies to the stationary electron states of an ideal semi- conductor (Chapt,er 2) , their niodifications by impurity atoms and other deviations from the ideal crystal, as well as by external fields (Chapter 3) and t h e statistical distribution of charge carriers over aiergy levels in ther- modynamic equilibrium (Chapter 4).

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51

Chapter 2

Electronic structure of ideal crystals

In sections 1.2 and 1.3 of the preceding chapter we discussed the spatial order and chemical nature of the atoms in ideal semiconductor crystals. The present chapter is focused on the quantum mechanical energy eigenvalues and corresponding eigenstates of electrons in such crystals. The substance of this subject is summed up under the designation electronic structure. We will show how the simple model of electronic structure of a crystal, the energy band model, which we heuristically introduced in section 1.4, can be rigorously deduced from the Schrodinger equation. The periodic ordering of the atoms of a crystal in space is crucial for this proof. It will also be seen that the eigenstates of electrons in a crystal devolve upon the electron states of the free unbound atoms. This is immediately understandable if one imagines that the crystal is grown from the gas phase, i.e. by chemical bonding of previously isolated individual atoms. In this process the electron states of the atoms will change, of course, but the resulting states, i.e. the electron states of the crystal will also depend on the initial electron states of the isolated atoms prior to crystal formation.

2.1 Atomic cores and valence electrons

Qualitatively, the kind of changes contemplated may be characterized as follows: There is no doubt that the electrons of the outer shells, i.e. the valence electrons, will react most strongly in assembling the isolated atoms of a crystal, for they are the primary agents which bind the atoms into the crystal state (Table 2.1). Whether, and to what extent the electrons of the inner shells also change their states, is harder to predict. One may suppose that such inner shell changes will be comparatively slight, at least for those inner shells which lie much lower energetically than the valence shells (Ta-

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52 Chapter 2. Electronic structure of id& crystsls

Core

Nucleus Core electrons

5 + 6+ 18

7f

Table 2.1: Characterization of atomic cores and valence electrons of main group elements from which semiconducting mat,erials may be formed.

Valenw electrons

2 2 2 p

2s22p3 2922p2

Atom

13+

15-

3 1+

3 3+

14+

32-

B

N C

3 2 3 p

3s23p3

424p

4 A p 3

132222p6 3 2 3 p 2

ls22s22p633s23p63d10 3s24p2

A1

P

Ga

AS

St

Ge

ble 2.2). Because the electrons of these deep shells are localized so close to tlirir rrspertive nuclei. they feel potential changes produced by siirrouud- ing atoms as being almost uniform. Strictly speaking, this means that the wavefunctions of these electrons are essentially unaltered. Their eneqy lev- els shift, however, specifically by the change of the constant potential value arross their localization region. The term d i d d a t P shtjts is iised for t h e s ~ shifts of the inner electron levels. By measuring these shifts m e can obtain information about the chemical nature and geometric striicturc of the envi- ronnient of an atom in B solid. In B way, the inner electrotla threby serve as probes.

Here, we arc intermtd in the electronic strizrture of crystals, dnd in this regard the pertinent feature is that the wavefunctions of the inner shcll plec- trons 01 the atoms in the crystal undeigo only weak changes. This statement is even better justifid for the wavefiinctions of the nucleons in the atomic nu- clei, which remain practically unaffected. Furthermore, if the gown crystal is expoard to certain extrrrial perturbations - heat. prrssure, elwtromagnetir fields - the states of the electrons of the inner shells and those of IhenucIeoas frequently do not change. For the e x h n a l perturbations which are of p r im interest in semiconductor physics, this is even generally true. Therefore, in determining the elertronic stricture of sanironductor crystals and the in- fluence of exterruel perturbations on them, the states of the inner electron shells and those of thc nuclei, as a rule, can be assumcd to be the same as those of the free atoms. This allows one to consider the atomic nuclei and

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2.1. Atomic cores and valence eJectrons

-cP 6.4 7.9 9.5 3.4 4.7 5.9 7.2 8.6 3.5 5.8

53

Atom

Table 2.2: Energy levels -E, and -4 of the valence electrons and - E , of the shal- louwt core electrons for some chemical elements which may occur in semiconducting materials. All energies are given in el/. (Afcer Herm.an. and SkiIlman, 1963.)

3 C N 0 A1 Si P S Zn Ga

-tp

-t,

-ec

6.6 9.00 11.5 14.1 4.9 6.5 8.3 10.3 3.1 4.9

12.5 17.5 23.0 29.1 10.1 13.6 17.1 20.8 8.4 11.4

195 291 405 537 87.5 116 147.4 182 20.7 31.7

the inner shcll drrctruns jointly as subsystems of the crystal, whose intrrnal structure i s of no further interest sirire it does nut change. The structure is, so to speak, frozen. In this sense t.hc subsystcms composed of atomic iuiclei and inner electrons are elementary building blocks of the crystal. One rekm t,o them as atom.ic ~ 0 : ~ s .

Since the crystal aIso contains valence electrons as independent particles, we arrivr at a yic:i.ure which is fundamental for the further analysis - the picture of a cryst,al as system composed of at,omic cores and valence electrons. Somctirnes one refers to this concept, as the frozen-core a p p ~ ~ ~ i i m a t i ~ ~ ~ In Table 2.1 the division into core8 and valence electrons is indicated for some elements from which semiconductor crystals arc made.

The frozen nature of the electron states of the cores of a crystal! and their lack of rcsponse to external influences, generally prevails, but as always, there arc exceptions. In heavy metals such like zinc, t.he inner &shells are, energetically, rather close t,o the oubar valence shells (see Tables 2.1 and 2.2). In this case the d-electrons significantly participate in chemical bonding and can no longer be included in t.he core, which is iinchangeahle by definit,iou, Moreowr, the inner shell eledrons of crystals can be excited by means of electromagnetic radiation in thc far UV and X-ray region. This can also occur by means of an electron beam In solid state nuclear reactions, ~ W R

the states of the nuclei of the crystal atoms change.

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54 Chapter 2. Electronic structure of ideal crystals

2.2 The dynamical problem

2.2.1 Schrodinger equation for the interacting core and va- lence electron system

In dealing with the atomic structure of crystals in Chapter 1, we found that their atoms are not located at arbitrary positions but at well d e h d locations, specifically at points which are consistent with the existence of a lattice and a unit cell. In this, the atoms were assumed to be point-like and the space between them was empty. The crystal itself was imagined to extend to infinity. We now consider a more realistic model of a crystal. First, we replace the point-like picture of atoms by introducing spatially extended atomic cores and take their centers of gravity as the atom sites. Second, we recognize that the centers of gravity can also move to other positions than those prescribed for the ideal crystal. In this way, we also account for the fact that the atomic cores in crystals can execute oscillations around their equilibrium positions, and that only these equilibrium positions form an ideal crystal. Thirdly, the space between the massive elements of the crystal, i.e. the cores, is now no longer assumed to be empty, as was done before, but we acknowledge that valence electrons are present there. The assumption of infinite extension of the crystal which, of course, is also not exact, will be addressed at a later stage. This assumption excludes effects due to the existence of bounding surfaces. As far as the electrons are concerned, these effects are treated in section 3.6. The atomic cores will be marked by an integer subindex 3 , and the valence electrons by an integer subindex d. Both should start with 1 and run upwards, reaching arbitrarily large values since we are considering an infinite crystal.

By means of a conceptual device which we are about to introduce - notwithstanding the infinite extent of the crystal - only finite sets of J cores and N valence electrons need be considered. To understand this, we imag- ine the infinite crystal to be divided into parallelepipeds of macroscopic size in such a way, that their edges are parallel to the primitive lattice vec- tors a,,az,W of the crystal. These edges are to be given by the vectors Gal, Gaz, G a with G a large integer. Each of these parallelepipeds should contain an equal number J of cores and N of electrons. One calls these parallelepipeds periodicity regions. Of all possible motions of the particles of the infinite crystal, we now select those particular ones for which the cores and electrons in different periodicity regions have the same positions relative to the origin of their own region, and also have the same speeds. In this way the infinite crystal becomes a periodic continuation of one particular peri- odicity region, and it suffices to describe the motion of the J cores and N electrons of this particular region. If the periodicity regions are made s a -

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2.2. The dynamical problem 55

L

/; + 0

0

0 \

0

Figure 2.1: 13escription of the positiom of the atomic cores (0) and valence elm trons {a) {left part) as well as the interactions between these particles (right part).

ciently large, they will encompass all types of motions of an infinite crystal with desired accuracy. The concept of the periodicity region makes it possi- ble to pass from the original infinite space problem of motion to a finite one without thereby losing the translational symmetry of the infinite crystal.

We use Xj t o denote the center-of-mass coordinates of the j- th atomic core, and xi for the position of the i-th electron, which is further assumed to be point-like {see Figure 2.1)- The j - th core mass will be denoted by Mj. Of course, there are only as many dif€erent values of M j as there are chemically different types of atoms in the crystal, so most of the Mj-values are identical. In the case of electrons we can omit the index i from their masses since they have the common mass m. The momentum of the j-th core is called P3, and that of the i-th electron pi, such that

We are interested in the motion of the interacting atomic cores and valence electrons of the infinite crystal, which can only be adequately treated by means of quantum mechanics. The state of the system is described by a wavefunction @, which depends-on the coordinates xi of all electrons and Xj of all atomic cores, as well as on the time t. Since we assume periodicity of the motion with respect to a periodicity region, it suffices to consider @ as a function of the coordinates xi of the N-electrons and the coordinates Xj of the J cores of only one periodicity region. The state of the particles in the remaining periodicity regions can then be described by means of a periodic continuation of this function, i.e. by means of the relation

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56 Chapter 2. Electronic structure of ideal crystals

@(XI I Ga,, x2 1 C:%, . . . , XN i G%, XI 1 C%, X2 t G&, . . . , X J t G+, t )

with CY ~ 1,2 ,3 . We will now wt up the Hamiltonian 7-l of the system of the N-electrons

and J atomic cores of a periodicity region. We use Tc and ‘re to denote the kinetic energies of the atomic cores and of the electrons, respectively, and we define V ( X ~ , xz, . . . , XN, Xi, X2,. . . , XJ) to be the potential energy of the system. The Harniltonian is the sum of the kinetic and potential energy operators,

‘H =- Tc + Te + V. (2.3)

The kinetic energies Xc and T, ran be expressed in terms of the momenta Pj and pa of the cores and electrons as fOllOW8:

(2.4)

The potential energy is due to three interactions (see Figure 2.1): (1) ‘lhe repulsive Coulomb interaction of the electrons with each other. The corresponding potential encrgy is denoted by Vce. It depends only on the coordinates of the electrons, as given by

(2) The interaction of the electrons with the atomic cores due to their mutu- ally attractive Coulomb forces, and also due to (repulsive) forces of quantum mechanical origin, which become effective if the valence electron wavefunc- tions overlap the inner electron shells of the atomic cores. The electron-core interaction potential energy V,, depends on the locations of both the elec- trons and cores. With respect to the electrons, it is evidently additive, i.e.

vet = vec(X1, x2,. . . , XN, x1, XZ,. . . I X J ) = v c ( xz, . x1, x2,. . . , XJ),

(2.6) where Vc(xi, XI, X2, . . . , XJ) is the potential energy of the i-th electron in the field of all cores. (3) The mutual interaction of cores, which at sufficiently large distances is again of Coulomb type. If the distances become small, repulsive forces of quantum mechanical origin also occur. The core-core interaction potential energy will be denoted by V,. It depends only on the locations of the atomic cores, i.e.

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2.2. The dynamical problem 57

vcc VCr(X1, XL ’ ‘ ’ 9 XJ). (2.7)

Summing the three potential parts (2.5), (2.6) and (2.7), one gets the total potential

v - v, + v,, + v,, which determines the dynamical problem of the crystal uniquely. This prob- lem i s described by the time-dependent Schrodinger equation

(2.8)

i3 at

iFL-e = H6,

whose solution may be determined in terms of the eigeiivalue problem for the Hamilt,onittn 7f, which is given by the time-independenl Schrodinger equation

%IJ! = EQ. (2.10)

The normalization condition for the wavefunction with refereiicc to a pcri- odicity region is

(@I*) d3X1. . . d 3 x J p q X 1 , x2,. . . , XN, XI, xZ, . . . , xj, t ) l 2 = 1.

(2.11) Attempts to solve this eigenvalue problem exactly are hopeless from the very beginning, because it involves a macroscopic system, i.e. a system with about 10” electrons and a similar number of atomic cores, the motions of which are mutually coupled in a rather complex way. One must therefore resort to approximations. Such approximations must first provide the means to reduce the gigantic number of electrons, and secondly, allow for a proper decoupling of the electron and core motions. The second simplification is achieved by the so-called ‘adiabatic approximation’, and the first by the ‘one-particle approximation’. These two approximations will be elaborated below. We begin with the adiabatic approximation, and in the course of the discussion it will also become clear how the somewhat unexpected designation of the latter arises.

2.2.2

The adiabatic approximation (also known as the Born-Oppenheimer approx- imation) is based on the fact that the mass of the atomic cores is many tens of thousands of times larger than that of the electrons - in Si, e.g., 52 thousand times, and in mercury 368 thousand times. In addition, it takes advantage of the fact that in a crystal the kinetic energy of an atomic core is, on average,

Adiabatic approximat ion. Lattice dynamics

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58 Chapter 2. Electronic structure of ideal cry&&

smaller than that of a valence electron. 'l'his can be seen in the following way: If the cores and valence electrons were fwe particles, i.e. if they did not interact, then the average kinetic enpro of a core would be approximately (3/2) h7'. That of a valencp dzctron would be about (3/6) Eli where E F is defined as the Fwmi energy of an electron gas of the same density. The daerence between the average kinetic energies of the two types of particles arises from the fact that the electrons obey Fermi statistics, whereas atomic cores obey Boltzmann statistics. For typical concentrations of valence elec- trons in a crystal of about 10" C W L - ~ the Frrmi e n e r a EF i~ of the order of m~gnit~ude of eV, while kT reaches only about 0.1 eV below the melt mg point of the crystal. The average kinetic energy of a core is therefore generally smdlcr than that of an electron. This remains true when the in- teractions between the electrons and coresi which were omitted above, are taken into account. Writing (M,/2) < X: > and (m/2) < x: > as the avrragP kinetic enmgies of a core and an electron in a crystal, we thus have

M j 1 7 1 . '

2 2 - < X j > < - < x q > .

and it, follows that

(2.12)

(2.13)

Corrmpondingly, one may say that, on statistical average, the cores move much slower than the electrons. This observation plays an important role in the following considerations.

To simplify the notation, we replace the N-component sequence o€ the vectors (XI, x2,. . . ,XN) by x, and the J-component sequence of vectors (X\,Xz,. . ., XJ) by X, ie. we write

x = (XI, xz, . , , XN), x = (XI, xz,. I . , XJ). (2.14)

To take advantage of the slow motion of the cores we write the solution q{x, X) of the Schrodinger equation (2.10) for the total crystal. in the form of a product

qx, X ) = $(x, X) ' @(XI. (2.15)

The necessary normalization (2.11) of the total wavefunction *(x, X) with respect to a periodicity region is assured if each oI the two factors of (2.15) is normalized with respect to this region, i.e. if

(2.16)

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2.2. The dynarnical problem 59

(2.17)

are assumed. An analogous statement holds for the periodicity condition (2.2) of @(x, X). To assure overall periodicity with respect to a periodicity region we assume it for +(x,X) and 4(X) separately. An ‘Ansatz’ of the form (2.15) is always possible since it does not assert separability of the variables x and X, but merely splits off a factor +(X) from the wavefunction @(x,X) which depends only on one variable, X, while retaining the full dependenre on both Coordinates in thr second factor +(x, X). The ‘Ansatz’ (2.15) becomes non-trivial if we proceed as follows: Firstly, we assume that $(xi X) is the solution of a Schrodinger equation for electrons,

1% + v,,,c] $(x, X) = U(X)@(X, X), (2.18)

where, for brevity, we have set

and U(X) is in the nature of an electron energy eigenvaliie. Secondly, we de- mand that the split-off factor, 4(X), of the total wavefunction (2.15) satisfy a Schrodinger equation in which the coordinates of electrons do not appear. It turns out that such an equation cannot be derived rigorously, but only in a special approximation - the adiabatic approximation which was mentioned above. Yet without any approximation we have

[Te + Tc + K , , + VCCI +(x, X)+(X) = E$(x, X)4(X). (2.20)

The set {$4) of the eigenfunctions of the Schriidinger equation (2.20) forms an orthonormalized basis set in the Hilbert space of the crystal. Therefore, relation (2.20) is satisfied if it holds for the Fourier type coefficients relative to all basis functions Q’d’ of this set, i.e., if the identity

($’4’lTe + Tc + v&ec + Vccl$,O) = E&!+t++ (2.21)

is valid. The necessary simplification concerns the matrix element ($’4’(Tc I$4) of the kinetic energy of cores in this equation. Using relation (2.4) between Tc and the squares Pj of core momenta, and applying the product rule for differentiation we get, first of all,

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60 Chapbrr 2. Electronic strncture of ideal crystals

The first two terms on the right-hand side of this quat ion turn out to be small compared to thr kinetic cuergy term of elwtrons in equation c2.20). One has the order of magnitude relations

(2.23)

(2.24)

Here .n/E is a typird value of the core masses M,. The two equations (2.23) and (2.24) are of fundamental importance in crystal dynamics, because they are ultimately responsible for the drroupling of rloctron dynamics from the dynamics of the cores. Therefore we present the proof of thwc equations in Appendix B. Here we proceed an the assumption that these relations are proven.

l W 4 , will be nPglatrd henceforth. With this the operator T, for the ram- bined kinetic energies of all cores satisfies the approximate relation

The terms of rplativeorders of magnitude (rn/M)1/2 2 lov2 and ( m / h f ) w

Tc[.l$(x, X)9(X)l E=r ${x, XjWGq. (2.25)

This means that T, effectively does not act on V(X, X). In view of this rela- tion we reconsider the SrhriidingFr equation (2.20) for the crystal, replaring the terms which still depend on x by means of the electron Schrodinger equa- tion (2.18) in terms involving the electron energy eigenvalue U ( X ) . Finally, forming the scalar product with VI, we obtain the relation

(Tc + VC,(X) + U ( X ) \ d ( X ) = Ern(X). (2.26) This rcpresents the SchrGdinger equation for the atomic cores in which the coordinates x of thc electrons do not appear. The state of the elcctron system, however, enters this equation, namely via its energy U(X) which plays the role of a potential (referred to as adiabatze potential).

In summary, we h a w reached the following description of the total crys- tal, viewed as an interact,ing system of atomic cores and electrqns: The subsystemof electrons is describcd by R separate wavefunction qi(x, X), which obeys a Schrijdinger equahion in which the coordinates of the cores enter only as parameters in the potential, but do not occur as differential operators in t,hr kinetic energy. In this way, the motion 01 the electrons is treat,& as if the cores were at. red. Core motion, which does, of course, occur despite its neglect with respect to electron motion, is described by the wavefunction &(XI and the Schrodinger equation (2.26). The potential of this qua t ion contains, besides the core-core interaction energy, a second contribution IJ(X). This originsles in the interaction of the electrons with

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2.2. The dynamical problem 61

the cores, for without such an interaction the eigenvalue U in the electron Schrodinger equation (2.18) would be a constant independent of X, which could be omitted. The potential contribution U(X) caused by the electron- core interaction does not depend, however, on the electron coordinates. It is an average value over all their positions X. The weight with which the various positions x enter this average over the probability I + ( x , X) 1’ d 3 N ~ of finding the electron system in a volume element d 3 N ~ at the position x, since equation (2.18) implies that

u(X) = (+(X)ITe + vee,ec(X)l+(X)). (2.27)

One can alternatively express this as follows: The electrons move so fast that they are no longer seen by the cores as point-like particles, but as smeared out over all space. Equation (2.26) for 4(X) thus contains the same assumption as equation (2.18) for + ( x , X), namely that the cores move much slower than the electrons. In so far as this feature is seen from the point of view of the electrons, equation (2.18) follows, whereas from the point of view of the cores one obtains equation (2.26). Since the relation between the velocities of the cores and the electrons, according to (2.13), is determined by the inverse ratio of their masses, it is clear why this ratio must be small for the two Schrodinger equations (2.18) and (2.26) to hold jointly in an approximate sense.

It remains yet to clarlfy what effects are neglected because of the above approximation and why this approximation is called ‘adiabatic’. In quan- tum mechanics, one understands adiabatic temporal changes of potentials in the sense that the changes proceed so slowly that no quantum mechanical transitions will occur between the discrete quantum states of the potential, which themselves evolve slowly from the initial onset of time variation. The state of the system thus conforms continuously to the evolving new potential values as a function of time, without any transitions to other states. That exactly this situation is described by equation (2.18) and (2.26), may be seen from (2.24). If one considers the previously neglected term in (2.24) of relative order of magnitude ( m / M ) l I 2 , then the total Hamiltonian 7-t of the crystal has non-vanishing off-diagonal elements

(2.28)

and quantum mechanical transitions between the different eigenstates $14 and $’+’ of the crystal are recognized to occur. These transitions are caused by the kinetic energy of the cores exclusively. If terms of the order of mag- nitude ( m / M ) I 1 2 are omitted, then the quantum transitions due to core motion are also neglected. This is equivalent to the assumption that the core motion be adiabatically slow, in the quantum mechanical understand-

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62 Chapter 2. Electronic structure of i d 4 crystals

ing of this term. The term ‘adiabatic’ thus refers to the essential character of the approximation in neglecting (rn/M)’/’. This approximation is use- ful, of course, only as long as transitions between different eigenstates $$ play no important dynamical role. This is actually the case in regard to many crystal properties and phenomena. There are, however, also effects for which this does not hold, notably electric current transport. The fact that the electric conductivity of an absolutely pure crystal does not become excessively large is due in large part to the scattering of carriers from the oscillations of the atomic cores, i.e. to non-adiabatic quantum transitions between different electron and core states. Also, in the recombination of electron-hole pairs mentioned in Chapter 1, these transitions play a decisive role, with the lattice of atomic cores absorbing the energy which is released during recombination. Formally, one may understand non-adiabatic transi- tions as the result of an interaction between the electrons and the motion of the atomic cores. Since such core motion, as we will see below, represents a superposition of lattice oscillations, also known as ’phonon’ excitations, this interaction is called the electTon-phonon interactaon

We have yet to explore how the two Schrodinger equations (2.18) and (2.26) for the electrons and cores can be actually solved. The problem is that both equations, are, at the outset, not completely determined - the one for the cores contains the adiabatic potential U(X), which can be known only after the equation for the electrons has been solved; and the electron equation can be fully dehed , however, only if the positions X of cores in the potential Vee,..-,-(x, X) are known. The direct way to overcome this difficulty would be the following: One assumes a particular spatial ordering X’ of the cores and uses it to determine for them the potential Ve,ec(x, X’). The lat- ter is then used to solve the electron Schrodinger equation (2.18) (we will not discuss here how this is accomplished, as it will be the subject of the next subsection, 2.2.3). From the solution of the Schrodinger equation (2.18) one obtains the value of the adiabatic potential U at the position X’ of the cores. The same procedure is then applied to all other possible positions X, whereby the adiabatic potential U(X) and the Schrodinger equation (2.26) for the cores are completely determined. This equation can then be used to calculate the core wavefunction t#(X). It follows that the dynamical problem for the crystal as a whole is solved, since one would know its eigenfunctions @(x, X) = y(x, X)q(X). In reality, however, this procedure is unsuccessful. One cannot solve the electron Schrodinger equation for all possible core po- sitions. Therefore, a simplified procedure is necessary. It contains additional approximations, but has the advantage of being feasible in practical terms. In this approach, one ignores the motion of the atomic cores completely and assumes that they are resting in certain equilibrium positions Xq. In re- ality, they execute oscillations around these positions with amplitudes that become smaller as the temperature of the crystal decreases. However, due to

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2.2. The dynamical problem

xo - X" - I .ke , ec (X ,Xn)

? v,v,(xy

no

x n + 1

I

- 0 ?

J

63

Figure 2.2: Iterative calculation of the equilibrium positions of the atomic cores.

the quantum mechanical phenomenon of zero-point oscillations, such motion remains finite even at absolute zero temperature. The equilibrium positions Xq are unknown at the outset. One can determine them by demanding that the total energy

Vo(X) = U(X) + VC(X) (2.29)

of the crystal in equilibrium have a minimum at Xq. Equivalent to this is the requirement that the forces -VxVo(X) on the cores, the so-called Hellman-Feynman forces, vanish at the equilibrium positions:

--vxvo(x)~x,xeq = 0. (2.30)

Bearing this in mind, we may employ the iteration process below for the solution of the two coupled adiabatic equations (see Figure 2.2): In this pro-

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64 Chapter 2. Electronic structure of ideal crystah

cess, one assumes ccrtain trial equilibrium positions Xo, enters them in the electron Schrodinger equation, and determines the eigerivalue U(Xo). This solution is then used to determine the potential Vo(Xo) and the liellmaii- F e y m m forces. Thanks to the Fe'egrman theorem, taken jointly with appro- priate analytical transformations, one can determine these forces without numerically calculating Lhe potentid in the environment of X'. After the first iteration cycle, t.he Hcllman-Feynman forces will, in g~neral. not yet vanish. signifying that thc cmes arc still not at equilibrium posibians. By nieans of the non-vanishing force one det.ermines new trial positions XI. 'The new positions are then substituted again onto the electron Schrudinger equation (z.18)! to calculate a new eigenvalue U(X1), and the latter deter- mines the corresponding Heban-Feynrnan forces. This procedure is to be repeated until the forces become zero. The corresponding core positions are then the equilibrium positions XeQ. In this way one reaches a very impor- tant result, the determination of the atomic structure of the crystal. Such structure calculations are successfully carried out currently fw many solid state systems, including a series of semiconductor cryst.als. With regard to semiconductors, it can he shown, for instance, that under normal conditions, Si haas the diamond structure, and that its Lattice constant a iR 5.49 A.

];at tice oscillations

In so far a6 atomic structure is concernedd: ouly the equilibrium positions of the atomic cores are of interest. These can be understood as average values (Q I X 14) of the core positions X with respect to the core wavcfmction 4 , However, the wavehnction 9 itself contains considerably more information. It determines the probability distribution for the positions of the atomic cores. That the probabilities of the cores being removed from their equi- librium positions are non-zero is tbe quantum mechanical indication of the existence of lattice oacdllatioru. These oscillations may be describd easily using the S&rodinger eqriatiou (2.29) for the cores. In this: it suffices to expand the potential I/o in a Taylor series with respect to hhe displacements X - Xq from the equilibrium positions Xq, neglecting ternis beyond the square term. The linear term of this expansion vanishes since the potential energy has a minimum at Xq. Thus one obtains

VO(X) = vo(xeQ) i- ' ( X - X e ~ ) V ~ v X r ~ ' o ( X ~ ) ( x - x e 4 1. (2.31) 2

With this potential the Schriidinger equation (2.26) for @(XI reads

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2.2. The dynamical problem 65

where. for simplicity, Vo(Xeq) = 0 has been assumed. Equation (2.32) de- scribes a system of coupled harmonic oscillators. The restoring forces are determined by the second derivatives of the potential Vo(X). Using the eigenvectors and eigenvalues of the matrix of restoring forces (actually, of the so-called dynamical matrix which also includes the kinetic energy term), one can easily transform to a system of uncoupled harmonic oscillators. Their motions are called normal mode osciliataom or lattace oscillations, and their excitation quanta are phonom.

Phonons are a good example of the introduction of a concept which is of fundamental importance for the dynamics of many body systems, includ- ing the many-electron system of a crystal which will engage us in the next subsection. The concept we have in mind here is identified by the terms elementary excztatzons or quast-particles (both terms are commonly used). This concept is based on the possibility of decomposing the motion of a system of mutually interacting particles - in our case of the atomic cores of a crystal - into non-interacting components of motion - phonons in ow case. The phonons or, more generally, the elementary excitations are. so to speak, the elements of notzon of the system, while the atomic cores or, more generally, the actual particles. form the ~ ~ T U C ~ U T - Q ~ elements of the sys- tem. ,413 elementary excitation involves coordinated motion of all structural elements of the system. Conversely, the motion of an individual structural element is a superposition of all elementary excitations - the motion of the atomic cores, for example, is a superposition of all normal mode oscillations or phonons.

Besides the oneelectron and one-hole excitations, the phonons are the most important elementary excitations, or quasi-particles, of a crystal. In this book we will deal mainly with electronzc elementary excitations, and will include phonons only if it is otherwise impossible to properly describe electron dynamics. Relevant phonon information will simply be cited with- out detailed justification, since a thorough development of the theory of phonons is beyond the scope of this book. O w choice of subject matter here is conditioned by the fact that electrons and holes are much more important for understanding the properties of semiconductors as they are used in elec- tronic devices, than are phonons. Readers who are particularly interested in phonons are referred to other books (see, e.g., Born and Huang, 1968; Bilz and Kress, 1979; Bonch-Bruevich and Kalashnikov. 1982).

We return now to the Schrodinger equation (2.18) for electrons. In the sense of Figure 2.2, we approximate the core positions X by their equilibrium values Xeq. As far as the latter are concerned, we take the point of view that they are known from experimental structure investigations, e.g., by means of X-ray diffraction. For common semiconductor crystals this is in fact true in all cases. Taking this approach, the potential b’=,- in the electron Schriidinger equation (2.18) is well-defined from the very beginning. To

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66 Chapter 2. Electronic structure of ideal crystals

simplify the notation, we suppress the core coordinates X in the potential = V, + V, henceforth, writing Vec(x, X) = v,(x). Similarly, WP write

the electron wavefunction @(x,X) as $:,(x). Usingequations (2.19), r1.5) and (2.6), the Hamiltonian H = Te + V e , , of the N-electron system in equation (2.18 j takes the explicit form

2.2.3 One-particle appraximation. One-particle Schrodinger

With the Hamiltonian H of (2.331, the N-electron Schrodinger equation (2.18) can be written as

equation

HVj,(Xl, x2,. . ., X N ) = Ull(X1, x2,. . . , XN). (z -34)

The wavefunction ~+4 (XI, xp, . . . , XN) must be periodic and normalized with respect to a periodicity region. The Schrodinger equation (2.34) is impossible to solve directly since it describes an interacting system of electrons having a tremendously large number of particles ~ of order loz2. The goal of this subsection is to provide an approximate description which allows one to reduce the number of particles down to minimum number 1. This will be done by developing a oneparticle Schrijdinger equation whose solutions are rdated to those of the true many electron Schrodinger equation in a well- defined and sdiciently simple way.

Hartree approximation

In keeping with the remarks above, we assme the existence of an infinite set of one-particle wavefunctions q ~ l , ~ 2 : . . .. 'pm, from which the stationary stat- $(xl, x2) . . . ,XN) of the N-electron system may be constructed. The p I I ( x ) , v = 1,2,. . . ,m, are, firstly, taken to be periodic with respect to a periodicity region, as the wavefunction @(XI, x2,. . . , xN) itself, i.e. they satisfy the condition

pv(x) = pu(x + Cajj j = 1,Z, 3. (2.35)

Secondly, they are assumed to form a complete orthonormal set of functions in Hillert space, sylribolically

(CPv~lPv) = 6uhr. (2.36)

Employing pY(x) we form wavefunctions for the N-electron system in the folIowing way. We Brst associate each of the N electrons with a particular

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2.2. The dynamical problem 67

oneparticle state pv[x), i.e. particle 1 with state pull particle 2 with state puL, etc., up to partick N which is associated with the state yo,,. Alterna- tively, WP may say that we occupy state pw with particle 1, state p, with particle 2, etc. Due to the Pauli exclusion principle, each state can host only 1 particle, ignoring spin (which we do at this stage). Thus a given state p1, p2,. . . ,pm may occur among the papdated ones lpy, 'py, . , , pVN not more than once. Most of the states will not even orcur onr~ , i.a, not at all (bear in mind that there is an i nh i t e number of them). These states remain unoccupied. The set of quantum numbers, (q, vz, . . . , VN), termed configzs- ratzon, definea the state of the N-electron system uniquely if we understand that the h s t number in this set refers to the state of particle 1, the second to the state of particle 2 etc.. Henceforth, we abbreviate the configuration

Thirdly. we assign to each configuration (IJ) of the N-electron system a wavefunction $(y)(xl, x2,. I I XN) which is given by the bllowing product of oneparticle wavefunctions:

( V l , V a l . . . , w ) by (.I.

${y}(xl, x21.. .XN) = 'pvl(XL)Lp65(X2). . . IP,(XN) = ~ L p v J x 3 ) - (2.37) j

Disregarding the miitual interaction of the electrons for the moment, the product (2.37) forms an eigenstate of the N-electron system if the one- particle wavefunctions pV, (xJ) are energy eigenstates of the individual o n e electron subsystem Hamiltonians. This suggests the question whether a similar result might be possible for interacting electrons, i.e. whether it will be possible to choose the py,(xj) in such a way that the product statc +ivi obeys the Schriidinger equation

H${V)(Xl* x2,. ' ' I XN) = ~ { V } ~ { Y } ( X l 7 X 2 r . ' . 1 XN) (2.38)

for the fully interacting N-electron system - if not rigorously. then at least in some reasonable approximation.

To address this question one may use the variational principle of quantum mechanics. In this procedureI the oneparticle wavefunctions pV, (xj) are determined such that the expectation value of the N-electron Hamiltonian H becomes a minimum for N-eIecctran states of the product form (2.37). Here we take a slightly different approach, and start from the Schrodinger equation (2.38). This procedure has the advantage that one ran ai. once determine whether there is a suitable approximation in which d ~ { ~ } may be written in the product form (2.37), and also barn the nature of that approximation.

Considering an M-electron system, we label a particular electron i, and this index can take all values between 1 and N. The Hilbert space of the

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68 Chapter 2. Electronic structure of ideal crysWs

system of t.he N - 1 remaining elect,rons 1,2, . . . , i - 1, i + 1,. . . , N is spanned by the set, of product functions

with PI , . . . , pz-l, po+l . . . . , pjv ranging over all possible values 1,2,, . . , co independently of each other. In this remaining Hilbert space we form the Fourier-type coefficients of the Schrodinger equation (2.381, i.e., we multi- ply t,his equation by the complex conjugated product function (2.39) and integrate over all XI, x2.. . . XN with the exception of xz. In this way we obtain

(2.40)

Due to the orthogonality of the qv. the right-hand side of this equation differs from zero only if the pvalues coincide with the v-values, i.e.. if

= q , .., pz-l = ~ ~ - 1 , pcl,+l = v,+l. .... p~ = v w holds. However, the left-hand side of equation (2.40) differs from zero if p J f v3 for one or sev- eral I # t . Thus equation (2.38) cannot hold rigorously, which means that the eigenfunctions ${,,I of H cannot be written exactly as a product {2.37) of oneparticle wavefunctions. This is only possible under the condition that the non-diagonal elements of the Hamiltonian operator on the left-hand side of (2.40) may be neglected. It is this approximation which makes possible the reduction of the N-particle wavefunctions to products of oneparticle wavefunctions. It is called a one-partzcle approozamatton. Strictly speak- ing, it is the simplest variant of a oneparticle approximation, the so-called Hartwe appronmatzoa. .4 more accurate oneparticle approximation, called the Hartree-Fock apprommatzoa, will be discussed below.

Within the framework of the Hartree approximation the equation system (2.40) involves only the diagonal terms with p 3 = vj for each 1 # 2 , and correspondingly takes the form

The diagonal elements in this equation can easily be evaluated as

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2.2. The dynamical problem 69

(2.42)

On the right-hand side of this equation only the first three terms depend on the electron coordinates x1 while the last two are constants in this regard. If one substitutes the expression (2.42) into equation (2.41), then the last two terms can be grouped together with U{'(.} to h m the new eigenvalur

Using the abbreviation

we rewrit,e (2.41) as

(2.44)

(2.45)

The final relation (2.46) has the form of a Schrodinger equation for the 7-th particle where V'{(")(xp) is the potential energy of this particle and EV, is its energy eigenvalue. Beside the potential energy ITc(&) due t o the atomic cores. V'Iv}(x,) also contains the contribution \#'}(x,). It is caused by the mutual interaction of electrons, and is commonly called the Hartree p o t e n t d In explicit form, ITH %{.I (x,) reads

(2.47)

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70 Chapter 2. Electronic structure of ideal crystals

Here the integration runs over a periodicity region. The Hartree putential V~‘”’(X~) describes the potential energy of the i-tb particle in the Coulomb potential produced by the charge distribution --e Ck+% i(oV,(x’)l2 of the re- maining particles. The factor 4 of the electron-electron interaction potential (2.5) does not occur in expressions (2.45) and (2.47) for the Hartree poten- tial since each electron pair contribntes only once. Obviously. the Hartree potential and the corresponding energy cigenvalues depend OB the configu- ration { v } of the N-electron system, and also on the index i of the particle which was removed.

The one-particle Schrdinger equation (2.46) derived above for the i-th elpctron, holds for each other electron as well. only with a somewhat different potential. This difference will now be removed, together with the dependence of the Har t re potential on the configuration {v} of the N-electron system. We argue as follows: If the number A; of electrons is macroscopically large, as in the case of the electron system of a crystal, and if we consider only oneparticle states which are spatially spread out more or less evenly aver the entire crystal, there is no signifkant difference if we extend the sum over k in equation (2.47) for the potential V;,[”’(x,) to include k - i . Then the 2-dependence of the potential no longer exists. The emor thereby incurred is of relative order of magnitude l/N. If one considers, on the othcr hand, only states ( v } of the &--electron system which are similar to each other, one may also neglect the {v)-dependence of the potential and replace V{”}(x) by the value for a representative configuration {vo). The question is, does such a representative codguration exist in the case of a semiconductor, and if so. what is it. The answer to the former question is ‘under normal conditions, yes’. For a ‘representative configuration’ in the abovementioned sense, we have the state of the N-electron system with lowest total energy, the so- called ground state. In this state all one-particle states p” with energies Ev below a special energy value (the Fermi energy) are occupied, and the states with energies above are empty. Under normal conditions the states of the N-electron system which occur in semiconductors, and also in other solids, deviate very little from the ground state, Non-normal conditions are associated with large deviations, e.g., such as semiconductors which are displaced to a highly excited state by intense laser irradiation. Excluding such extreme cases, the Hartree potential Vrj“}(x) for the configuration {v) is almost the same as that for the ground state configuration {v’}, and correspondingly we have

For brevity, we set

(2.48)

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2.2. The dynamlcal problem 71

V(X,) = V&) t VH(Xt).

The extent to which the approximation of a Configuration independent Hartree potential is valid again depends on the kind of one-particle states involved. For the extended, planewavelike oneparticle states of an ideal crystal this approximation works better than for the localized oncpartick states of a real semiconductor. In the latter case the configuration dependence of the potential may become essential (see Chapter 3 for further discussion). Using (2.50), the Schrodinger equation (2.46) becomes

(2.50)

(2.51)

The Hamiltonian of this equation is the same for all particles and no longer depends on the configuration of the N-particle system. Equation (2.51) is therefore the oneparticle Schrodinger equation par excellence, devoid of any reference to a particular particle or configuration of the N-electron system. We may therefore omit the index i in equation (2.51). TJsing the oneparticle Hamiltonian

P' ff = 2, I v(x), (2.52)

this equation becomes

H'Pdx) = Evcpv(x). (2.53)

The Hamiltonian H of equation (2.52) is Hermitian, and it is natural lo

assume that its eigenfunctions form a complete orthonormal set in Hilbert space. 'lhis assumption has in fact been made at the outset, with respect to the oneparticle states cpu(x) forming the product wavefunctionu of the N-electron system.

In summary, the discussion above has shown the following: Within the framework of the oneparticle approximation, i.e. neglecting non-diagonal elements of thc Hamiltonian, the product wavefunctions $ J { ~ } ( X ~ , x2,. . . , XN)

are eigenstates of the N-electron system provided that the oneparticle wave- functions of the product functions satisfy the oneparticle Schrodinger equa- tion (2.53). Solving this equation and forming thc product wavefunction (2.37), one gets approximate solutions of the N-electron Schrodinger equa- tion. In this way we have reached the goal which was formulated at the outset to replace the N-electron problem by a oneparticle problem whose

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72 Chapter 2. Electronic structure of ideal crystals

solution has a well defined and sufficiently simple connection with the solu- tion of the N-electron problem. The idea that the (py(x) are energy eigenstates and the E, are energies of single electrons, underlying the above consideration, needs to be made more precise. Because of the electron-electron interaction, the motion of a particular electron is always tied to the motion of all others, and the energy of an electron is also, in part, energy of interaction. The latter statement manifests itself clearly in relation (2.43) between the one-particle energies E, and the total energy U{.} of the N-electron system, which we will explore in more detail. First of all, it can be further simplified. Using the one-particle Schrodinger equation, one can re-express the terms on the right-hand side of equation (2.43) by one-particle eigenvalues, leading to

The energy of the N-particle system is therefore not just the sum of all one- particle energies. It is necessary to subtract the Coulomb interaction energy of the particles. Therein is reflected the fact that the E, contain a certain portion of interaction energy with other electrons. This is doubly counted in the sum xi E , of one-particle energies, once in summing over the particles themselves, and once in summing over their interaction partners, which is done in E, automatically. To correct this, one must subtract the Coulomb interaction energy.

This shows that the (p,(x) may be interpreted as states of single elec- trons only in a generalized sense. In reality the (py(x) describe stationary states of the motion of the N-electron system in which all electrons are in- volved. These states of motion are not mutually coupled, as in the case of normal oscillations of a system of interacting atomic cores. Using the ter- minology introduced in that context one may consider the states q , ( x ) as states of quasi-particles or elementary excitations of the N-electron system. The E, are the corresponding quasi-particle or excitation energies. There is, however, a qualitative difference between these elementary excitations of the electron system and the normal oscillations of a crystal. This may be made clear as follows. If one adds to the N-electron system (which we will assume to be in the ground state) one more electron, i.e. if one passes over to a ( N + 1)-electron system, then the one-particle Hamiltonian (2.52) does not change within the framework of the approximations made above. The oneparticle wavefunctions pV of the N-electron system therefore also approximately describe the elementary excitations of the system of ( N + 1) electrons. This means that an eigenstate of the (N + 1)-electron system may be realized by keeping the previously available N electrons in their one-

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2.2. The dynamical problem 73

particle states and adding the ( N + 1)-th electron in one of the oneparticle states pu* of the N-electron system which were previously not occupied. Thus, by adding an electron, the energy of the system rises approximately by Ey*. This means that the eigenvalue Ev* of the one-particle Ilamiltonian may be understood as the energy of an electron added to the system. This statement is called Koopman heorem. From it, one can learn more about the kind of elementary excitations of the N-electron system that are described by the p,. These are states in which, as always, all electrons of the system are involved, but not all in the same way. Only one of the electrons is mov- ing in such states, while the others play a passive role; they determine the potential in which this movement occurs. One therefore refers to these states as one-particle excitations of the N-electron system, and to their energies as one-particle excitations energies or, in short, one-particle energies.

In addition to the one-particle excitations considered above there may yet be others. This can be confirmed by taking a (N -1)-electron system instead of the ( N + 1)-electron system. The missing clectron corresponds to a hole in a previously occupied oneparticle state ( p y ~ . The excitation energy of the hole is -EvO, which did not occur among the oneparticle excitations consid- ered above. It therefore represents an additional one-particle excitation. If an electron is removed from state vy and simultaneously an electron is added in state vT, then this corresponds to the excitation of the N-electron system from state ($, Y:, . , . , v k ) into state (v;, v:, . . . , I&). The energy difference with respect to the ground state amounts to EV; - F 0 . It corresponds to the excitation energy of a11 electron-hole-pair with the electron in state pu; and the hole in state p 0 . If one excites a second electron from state

v1 'p o into state p,,;, the energy difference with respect to the ground state is (Eq - Ey:) + (By; - E e ) , etc. The excitation energies of the N-electron system can thus be written as a linear superposition of oneparticle energics. This is valid only within the one-particle approximation. In a strict seiisc one also has many-particle excitations, which will be considered in more de- tail below. As far as the one-particle excitations are concerned, there are no others than the ones considered above, at least as long as one ignores spin and the magnetic interaction between electrons.

5

u!2

It. is now appropriate to clarify how the oneparticle wavefunctions cpv(x) 111w be calculated from the Schrodinger equation (2.53). The potential in Uhis equation, more specifically the Hartree part VH(X), dcpends on the wavcfunctions pV(x) which are involved in the construction of the ground state of the N-electron system. One must know these functions in order to write down the potential and thus define the oneparticle Schrodinger equa- tion. On the other hand, these functions can only be obtained by solving this equation. The situation is similar to that in the preceding section on the coupled Schrodinger equations for the interacting system of electrons and

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74 Chapter 2. Electronic structure of ideal crystals

n d? - PU V"(X) = t'c(x) + VG(X)

Figure 2.3: Self-consistent solution of the oneparticle Schrijdinger equation.

atomic cores. As was done there. we may also solve the present problem it- eratively (see Figure 2.3). We employ one-particle wavefunctions pf(x) close to the true stationary oneelectron states. rs ing cp:(x) we determine a PO- tential t$(x) according to equation (2.491, form the total potential Vo(x) by means of (2.50), and use this to solve the one-particle Schrodinger equation (2.53). The solutions vt(x) are then substituted into formula (2.49), thereby determining new potentials TG(x) and V l ( x ) . With the latter one recalcu- lates the eigenfunctions p:(x) etc. One continues this iterative procedure until the eigenfunctions, and with them also the potential in the follow- ing iteration step. no longer change within a specified limit of accuracy. The eigenfunctions and potential are then said to be determined self-consistently.

Spin a n d spin-orbit interaction

At this point in our treatment of the oneelectron approximation, it is a p propriate to recognize that electrons hwe a spin, i.e. an internal angular momentum with the two possible values rC/2 and --7i/2 in a given direction. This is to say that electrons are capable of a motion in spin space, beside

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2.2. The dynamical problem 75

their motion in coordinate space, which in this context is called orbatal m o - tion. As orbital motion involves dependence of the wavefunction p ( ~ ) on the space coordinate x, spin motion involves a dependence of p(x, s) on the spin variable s which may take the two possible values s = 4 and s = -z (below the latter value will be written as =_ -;). 'I'hus, in consideration of spin, the wavefunction of an elcrtron changes from - an ordinary vector p(x) in Hilbert space to an element {~(x, i), p(x, ;)} in the product space of the ordinary Hilbert spare and tht. two-component spin space, a so-called two-component spinor. fiLnrtion, To determine the spinor state of an electron uniquely, the quantum number X which defines the statr must also specify the spin state. If the latter i s independent of the state in coordinate space, this may be done by specifying another quantum number u for the spin mo- tion along with the quantum number v of the orbital motion, i.e. by setting - v , o where u may take the two values T (spin up) and 1 (spin down).

The spinor {px(x, i), px(x, i)} can then be represented as a product of only one spatially varying function py(x) and a spinor (~~(4)~ ,yo($)} which does not change in coordinate space. The two spirior components cpx(x, s) can then be written as

1 -

- -

(2.55)

In general, however, the orbital and spin motions are coupled. This is mainly due to the fact that, on the one hand, the spin motion is accompanied by a magnetic moment of the electron, and that, on the other hand, the orbital motion gives rise to a magnetic field which couples that magnetic moment. In quantum theory it i s shown that this interaction, which is called spin-ovhit intemction, can be represented by the following additional term H,, in the oneelectron Hamiltouian:

Tz 4m c

H,, - -[VV(X) x p] . (7. (2.56)

Here V(x) denotes, as before, the periodic crystal potential of equation (2.50)) and 3 is the vector whose three components are Pauli's spin matrices. In spin space one usually refers all quantities to the basis X I = (1,0), X I = (0 , l ) . Then the components of are

(2.57) u.=(; i ) , u y - ( i 0 - 0') 3

1 0

Taking account of spin and the spin-orbit interaction, the one-particle Schro- dingw equation (2.55) in Hartree approximation t,akes the form

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76 Chapter 2. Electronic structure of ideal crystals

(2.58)

Spin-orbit interaction i s in fact an important consideration in determination of t,he energy specha of many serniconduct,ors.

Hartree-Fock approximation

An obvious drawback of the Hartree approximation is that the wavefunction of the IV-particle system is not antisymmetric with respect to the exchange of two particles, a requirements of the Pauli exclusion principle. 'l'his de- fect can be easily remedied by replacing h e product wavefiinction (2.37) of the Ilartree state by a linear combination of product wavefunctions with exchaiigd partirle indices and altered signs. In conjunction with this, the spin of the electrons has to be considered, such that the wavefunction of the t-th particle is given by the spinor p~ , (x , , si). The antisymmetric linear cornhatiom of the product waverunctions may he arit,tcn in the form of a so-called Slater determinant

(2.59)

In this determinant, an exchange of t.he variables of two electrons leads to the exchange of the two corresponding rows. The sign of the determinant thereby changes, SO that the Slater determinant actually has the requisite aiitisymmnctry propcrty. If t.wo of the quantum numbers XI, X2,. . . , AN are qual , then two colunms 01 bhP determinant are identical and vanishes. 'l'his means that no states of the N-electron system are allowed with two electrons in the same oneparticle state. "his is just the Pauli principle, automatically enforcd by the use of the det,erminant,al form of the N-part,icle wawfunr tion.

Employing such SlaPer determinants as N-particle wavefunctions, as op- p o s d to simple producls, a one-particle Schriidinger equation far pv(x) may be derivd in the same way as before, but the potential in this equation ia

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2.2. The dyynmiicd problem 77

somewhat different than that in the Hartree equation. It contains an ad- ditional contribution, the so-called exchange potential Vay (x), and the total potential reads

V(X) = K(x) + VH(4 + W X ) . (2.60)

In the case of negligibly small spin-orbit interaction, the orbital state may be characterized by a separate quantum number vt , and the spin state by a separate cpiantum number mi. Thc spinor components cp~,(x,, 8%) are of thr form (2.55) in such circumstances. The sum VH(X) + Vx(x) of the Hartree and cxchange potentials can then be written in a relatively simple form. It can be shown thilt theii action on the coordinate dependent factor of the oncparticle wavefunction cpl,(x) takes the form

(2.61)

The Erst term on the right-hand side of this cquation is t,he Hartree potential. The factor of 2 results from summing over the t,wo spin states associated with each wavefunction cpvk(x). The second term corrcsponds to the exchange po- tential. Formally, it differs from the first term by exchanging the states at the two positions x and x’. The factor of $ reflects the fact that, firstly, the exchange potential acts only between electrons of the same spin, and, secondly, that for the ground state with total spin 0, half of the electrons are in spin-down states, and half are in spin-up states. In this way the magni- tude of the exchange potential is influenced by the existence of electron spin, although its value is the same for spin-up and spin-down states. Equation (2.61) also shows that, unlike the Hartree potential, the exchange potential is non-local. The effect of the exchange potential on the wavefunction p,(x) is represented by an integral operator. In actual calculations one often uses a local approximation for Vx(x). The exchange potential proofs to be at- tractive, which is to be expected: the anti-symmetric form (2.59) of the total wavefunction ${A} means that the probability of finding two electrons with the same spin at the same position is zero so that one has an ‘exchange hole’ around each electron. This lowering of electron density in the vicinity of an electron results in an attractive potential in addition to the repulsive Hartree potential since the total Hartree wavefunction (2.37) does not account for the exchange hole.

The improved oneparticle Schrodinger equation with the potential of (2.60) and (2.61) is called the Hartree-Fock equation, and the oneparticle approximation, which underlies it, is called Hartvee-Fock approximation Thereby, the Hartree and exchange potentials are understood as those for the

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78 Chapter 2. Electronic structrrre of idea? crystah

ground state configuration vi of the N-electron system. The effects of the electron-electron interaction, which are still neglected within the Hartree- Fock approximation with configuration independent Hartree and exchange potentials. are called correlation effects.

Correlation effects

Correlation effects are, first of all, manifested in the fact that the true one- particle excitation energies of an N-electron system differ from those in the Hartree-Fock approximation. In particular, these excitation energies depend on the configuration of the system, one has tt, configuration dependence. Sec- ondly, Slater determinants which in Hartree-Fock approximation are con- sidered to be eigenstates of the total Hamiltonian, in fact do not diagonal- ize tbis Harniltouian ezactlv; there are non-vanishing offdiagonal elements, an effect which is termed configuration interaction. The exact eigenstates of the total Hamiltonian are linear combinations of diflerent Slater deter- minants, and the corresponding energy eigenvaluea are no longer s u m of oneparticle excitation energies, as had been the case for individual Slater determinants. In other terms, the exart, eigenalates of the N-elwtrron system are not oneparticle excitations! but many-particle excitetians. Examples of many-particle excitations include two-particle ezcitation.5 of an electron and a hole which are bound together by their Coulomb Interaction. The exci- tation energy of such a hound electron-hole pair, the so-called ezcitan, is smaller than that of the excitation energy of a free electron and hole pair, differing by the binding energy of the pair. The reason for the designnation ‘correlation effect’ for this phenomenon is obvious: binding may be under- stood as a correlation between the positions of the electron and the hole, since their separation by a distance of about a Bohr radius is more probable than all others. This interpretation presents the correct concept of correla- tion in other cases also ~ the states of the electrons are no longer independent of each other, but, are correlated contrary to the assumptions implicit in the oneparticle approximation. Collective many-particle excitations are excita- tions of states in which all electrons of the system participate in comparable measure. Examples include the plasma osci l - t iam of an electron system. They form a direct electronic analogy to the lattice vibratious of the atomic cores of a crystal. Their excitation quanta are called plasmons.

The consideration of correlation effects stands along the most difficult problems of solid state thmry which, even today, is not completely solved. A comprehensive analysis of this problem i s far beyond the scope of the present book. Readers who are particularly interested in correlation effects will find discussions in a number of textbooks (see, e . g , hbrikosov, Gorkw, md Dzyaloshinski, 1963; Fetter and Walecka, 1971; Ziman, 1974; Callaway, 1976; Madelung, 1978; Harrison, 1981). Below we summarize some results

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2.2. The dynamical problem 79

which will be needed in Chapter 3. In doing so, we will concentrate on one- particle excitations, i.e. individual electrons and holes moving in the force field of all other electrons as well as in the force field of the atomic cores.

Correlation effects on one-particle excitations. Density functional theory.

Correlation effects on one-particle excitations may be treated by means of the Green's function theory of many particle-systems. The poles of the one particle Green's function in the complex energy plane represent oneparticle excitation energies (more strictly speaking, the real parts of these poles are the energy levels, and the imaginary parts are the lifetime broadening en- ergies of the one-particle excitations). The one-particle Green's function is governed by the Dyson equation, which contains correlation effects through the so-called mass or self-energy operator.

Another method which works well for oneparticle states involved in the ground state of the many-particle system is known as density functional theory. This method relies on a theorem, the Hohenberg-Kohn theorem, which ensures that the ground state energy Eo of an interacting electron system in an external potential Vc(x) is a functional Eo[n(x) ] of the total electron density n(x) of the ground state alone. This implies, first of all, that the total energy Eo[n(x) ] depends on the oneparticle wavefunctions only through the ground state density n,(x) and, moreover, that the density enters at every point x, through an integral over X . The oneparticle wavefunctions determine the ground state density by means of the equation

(2.62)

where cpvi(x) denote the one-particle states which, in the ground state of the N-electron system, are populated by electrons z = 1,2, ..., N. According to the variational principle of quantum mechanics, the wavefunctions cpv,(x) adjust so that, while keeping their norms (cpudlqv,) constant, the total en- vrgy Eo[n(x)] is minimized. This requires the vanishing of the variational derivative of the functional Eo[n(z) ] -x i E,(cp,Ip,) with respect to p:*(x), where the factors E , are variational parameters, therefore

(2.63)

In this functional derivative the value of cp:t(x) at a certain point x is taken as an independent variable, with respect to which the common derivative is taken.

The total energy functional Eo[n(x)] of the ground statr may be decom- posed into several energy contributions, namely, the kinetic energy E ~ & ( x ) ] , the external potential energy Ec[n(x ) ] , the Hartree energy Ew[n(x) ] and the

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80 Chapter 2. Electronic structure of ideal crystals

exchange and correlation energies which are usually summed in the exchange- correlation energy E x c [ n ( x ) ] . Thus

Eo[n(x)] = Ekin[n(x)l + Ec[n(x)l + &dn(x)l + Exc[n(x)l. (2.64)

The functionals Ec[n(x) ] and E ~ [ n ( x ) ] are easily obtained as

2 n(x') . n ( x ) R R Ix' - XI '

E ~ [ n ( x ) ] = / / d3x'd3x

(2.65)

(2.66)

The functional E x c [ n ( x ) ] is less obvious. It is usually taken in a local ap- proximation called the local density approximation (LDA). The LDA starts with the homogeneous electron system without any external potential. In this case the density n ( x ) is a constant n in space, and E x c ( n ) reduces to an ordinary function of n. Dividing E x c ( n ) by the total number nR of electrons yields the exchange-correlation energy E X C ( ~ ) of the free electron gas, per electron. The total exchangecorrelation energy Exc(n) of a weakly inho- mogeneous electron gas of density n ( x ) should then be given approximately by the expression

(2.67)

Finally, the kinetic energy fuIictiona1 E:k.tn[n(x)] has to be specified. definition, we have

By

(2.68)

Although this expression does not look like a functional of n ( x ) il is in- deed possible l o transform it into such a form because all other terms in the total energy functional Eo[n(x)] of (2.64) are functionals of n ( x ) , and the Hohenberg-Kohn theorem enforces this for E & t ( x ) ] . For the waluation of the variational equation (2.63) we do not, however, need the explicit func- tional form of EkznIn(x)l; expression (2.68) suffices. Its variational derivative with respect to p;,(x) iu given by

For the rcmaining functional derivatives, it follows that

(2.69)

(2.70)

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2.2. The dynamicd problem 81

where

(2.71)

(2.72)

(2.73)

denotes the ~xcliangPcnrrelation potential. The latter can be determined if the exchange-correlatiou energy Exc(n) of the homogeneous electron gas is known as fniictio~i of n. This dependence can hr obtained by calcrilat- ing E>yc(n) numerically for tfifkrent values of n and then lilting the data to appropriate explicit functions. TJsing this procedure. various Pxchang+ correlation potentials have been proposed, for exaniylr

1/3 Lkc(x) - - (:) e2n1j3[x) [I + 0.7734 z In

with z - rS/2l, where rS = and CI.B the Bohr radius (Hedin, Lundqvist, 1971). Suhstitiiting i n h quat ion (2.63) the frlnctionrtl drriva- tives obtained above, one arrives at

(2.75)

with v(x) = VC(X) I Vrr(X) + i<Tc:(x) (2.76)

as an effective one-elcrtron potential. Th? electron ind~x a has heen omitted here bwauw thP equation is the bame for all electrons.

Relation (2.75), with the potential V(x) given by (2.76), is known as Kohn-Sham equatzon. ,4s rompnred to the oneeleclron potential V(X) of the Hartrw ur Hartret-Fock equations, that of the Kohn-Sham equation additionally accounts €or correlation effects. The physical significance of tlir solutions of the Kohn-Sham equation is, however, less direct than that of the solutions of thP llartree or Hartrw-Fock equations. Generally, the eigcnvalucs of the Kohn-Sham equation cannot be understood in the sense of oneparticle excitation energies of the N-electron system. as it is possible for the eigenvalues of the IIartree or HartrePFock quations according to Koapman's theoiern. A misinterpretation of this kind may lead to large errors. This applies, in particular, to electron-hole excitation energies in senlimnductor crystals, defining the energy gap. The resulting erroneous gaps are about 50% smaller than the experimental values. However, the

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82 Chapter 2. Electronic structure of ideal crystals

eigenvalues and eigenfunctions of the Kohn-Sham equation can be properly used to calculate the total energy of the ground state of the N-electron system by means of the total energy functional (2.64). The dependence of the total energy on external parameters as, for example, on the positions of atomic cores, also can be obtained in this way. Minimizing the total energy with respect to the core positions yields the atomic structure of the crystal. The oneparticle excitation energies may be obtained as the differences of the total energies of the ( N t 1)- or ( N - 1)-electron systems, on the one hand, and the N-electron system, on the other hand, where the ( N + 1)-electron system applies for electlan excitations and the ( N - 1)-electron system for hole excitations.

2.3 General properties of stationary one-electron states in a crystal

The oneelectron Schrodinger equation (2.53) is specified by the form of the potential V(x) which is different for crystals of different chemical composi- tion and atomic strurlure. However, there are certain general properties of V(x) which do no1 depend on the particular material nature of the crystal. As we already know, a crystal remains invariant under a transformation by an element of i t s space group. Rotations, reflections and rotation-reflections of the point group of directions transform a given crystal direction into a physically equivalent one. The symmetry of the crystal is transferred di- rectly to the potential V(x). Consequently, the Schrsdinger equation (2.53) is endowed with corresponding symmetry properties. We shall first describe these and then explore their implications for the stationary oneelectron states cpu(x). In doing so, we initially neglect electron spin.

2.3.1 Symmetry properties of the one-electron Schrodinger equation

For simplicity, we restrict our considerations here to crystals their space groups are symmorphic. These are groups which contain solely translations, rotations, reflections as well as rotation-reflections, while screw rotations and glide-reflections are excluded. The general case of non-symmorphic space groups, and with it the particularly iniyortant case of the diamond structure, i s treated in Appendix A. It turns out that the results derived for symmorphic space groups arc also valid, with minor modifications, for non-symmorphic ones.

We use the symbol t R to denote the lattice translation operator which causes a translation of all points x through a lattice vector R,

t R x = x + R. (2.77)

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2.3. General properties o f stationary oneelectron states in B crystal 83

The set of all lattice translation operators forms the translation group, which we already encountered in Chapter 1. Since the lattice translation operators are symmetry dements of the crystal, the translation gioup is B subgroup of the space group which, by definition: contains all symmetry elements of the crystal. As in the case of translations, we also assign operators to rotations, reflections and rotation-reflmtions of the point group of directions. These operators also art on the spatial vectors x, and WP denote them by the symbol a and call them point symmetry operataorw. For the symrnorphic space groups considered here, each element y may be thought of as a product 0 . t ~ or t ~ r r of a translcttion t~ and a point syrnrnetrg operation a. Since an arbitrary position vector x may be represented in a basis spanned by the three primitive lattice vectors al. a 2 and a3, it suffices to s p e d y the &ect of a on these. We define (see Appendix A}

(2.78)

with ov as real coefficimts. By means of this relation each opeiator a is uniquely associakd with a corresponding matrix aV. If the thee primitive lattice vectors a, are orthogonal and of the same length, the uaJ form orthog- onal niatriw$, i.e. their inverse matrices are the same as their transposed ones. However, this holds only for the primitive cubic lattice, and is not true fur all other 13 Binvais btticrs. Thus, the utJ arc not iu general orthogoual matrices. The effect of a on an arbitrary position vector x may easily be determind by ~neans of i t s decomposition

x - c z j a 3 t

(2.79)

with respect to the three primitive lattice vectors aj. The xj are the com- ponents of x with respect to these vectors. Applying a,

j

we may rewrite this relation in the form

(2.80)

(2.81)

from which it follows that the transformation a , which was ariginaIly defined as a transformation of the basis vectors ai with fixed coordinates zz, may also be understood as a counter-tra~sformation of thc coordinates with fixed basis vectors. Indeecl, lh coordinate counter-transformation takes place with the

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84 Chapter 2. Electronic structure of ideal crystals

transpose matrix a j i of aij, in contradistinction to the transformation (2.78) of the basis vectors. This is t o say that

(2.82)

If the crystal remains invarianl under the transformations t R and a, then this mist also hold for the potential V(x) with which the tramformed crystal acts on an electron at position x. To express this fact formally, we &fine the operation of t~ and n on an arbitrary position dependent scab1 crystal function S(x) as follows:

tf$(x) = S ( t R ' X ) , (2.83)

a S ( x ) = S ( a - l x ) . (2.84)

It is striking that the operators t~ and a act on S(x) in such a way that x ifi replaced by tR1x or m - l x . but not by k ~ x or c k x . as one might hme expectd. The rhosen deftnition strms from the rrcognition that the trans- formed property is that of the transfarmrcl crystal at the original position x. This is the Sam?, howewr. as the proprrty o f the original crystal at the inverse transformd position. Formally the definitions (2.83) and (2.84) giiarantee the correct multiplication order of two no11 commuting operators under the fiinction symbol, because ( q c q ) - ' - rtl a2 . Applying these definitions to the potential V ( x ) and simultaneously requiring crystal sym- metry, w r have

-1 -1

tnv(x) = V(t,lX) - v(x), U V ( X ) = V ( d X ) = v(xj.

(2.85)

(2.86)

Further application of these opwatm relations to a wavefunction p(x) one obtains

and since ~ ( x ) may be chosen arbitrarily, the relations

(2.89)

(2.W)

follow, with IA, B] = A B - B A as abbreviation for the commutator of two operators A and 3. One says that t~ and 1y commute with the potential V ( x ) . Such cornmutivity also holds for the uthcr contribution of the Hamil- tonian, the kinetic enerw operator ?' = p2/2m. For t~ this follows directly

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2.3. General properties of s ia ihnary oneelectron s&atw in a crystal 85

from the relations p = - ihV , and V, = V x + ~ , while for a it follows from the fact that p2 is the square of the length of the momentum vector, as well as from the observation that the length of a vector is not changed by a translation, rotation or reflection. Thus.

[tR, TI = [a, 2-1 = 0. (2.91)

Taken together with ('2.89) and (2.90), these relations yield

[ t R , HI = 0, (2.92)

]u, H ] = 0. (2.93)

Since the elements g of a symmorphic space group may be written as prod- ucts of t~ and C Y , one also has

The latter relation expresses in h a 1 form the implication of crystal symmetry for the Schrodinger equation. We will use it extensively in the analysis of symmetry properties of stationary oneelectron st,ates to follow.

2.3.2 Bloch theorem

Let p~ be an cigenfunction of H having eigenvalue E . Then the stationary Schrodinger equation (2.53) holds, whrrc the state index v has been replaced by E .

H V E W = E P E ( 4 . (2.95)

In addition, one has the periodicity condition with respect to a periodicity region

VE(X -k Gaj) = (PE(X) 1 J - 2 , 3, (2.96)

and the normalization condition

Through the periodicity condition (2.961, the symmetry group, which at the outset includes an infinite number of lattice translations, is reduccd to the finite subgraup containing only those translations t R which do not fall outside the periodicity region. The commutivity of t~ and a with H has the consequence that, along with cp~(x), also t ~ p ~ ( x ) , w,uE(x) and g p ~ ( x ) are eigenfunctions of H having ihe same cigenvalue E . Thus, one has

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86 Chapter 2. Electronic structure of id& crystah

(2.100)

Consider now the eigcnfunction g i p p If the symmetry operation g ranges over the whole space goup, the totality of vectors g p ~ spans a subspace of the Hilbert space w l i o s ~ dimension d i s in general larger than 1. This is to say that the eignvalue E, for symmetry reasons, is d-fold degenerate. It can be shown that, apart from spmial cases, d equals the number of dif- ferent elements m of the point group, and then the d basis functions of the subspace may be chosen in the form with Q ranging over the entire point group. This result will not. howevpr, be usrd in what follows and the basis funrtions will be written in the general form p ~ l , p ~ 2 , . . . , IpE& They span that subspace of the Hilbert space which contains the eigenfunctions of the Hamiltonian with the eigenvalue E . According to i t s construction this space is invariant under the operation of an arbitrary element g of the space group. In terms of the concept of rmducible represenfatema of groups (an introduction is provided in Appendix A), we may alternatively express this observation by saying that the eigenfunctions of H for a partic- ular eigenvalue E give rise to a &dimensional irreducible representation of the space group. The same statement also follows for each subgroup of this group, particularly for the subgroup of all translations. The space spanned by r p ~ l , P E T , . . . , ( p ~ d : thus also provides B representation of the translation group. However, this representation is, in general, no longer irreducible. That means that the original basis p ~ l , p ~ 2 , . . . , ( P E ~ may be transloorrued

(d-1) x I into a new basis (p~,(pk,'plh;, in such a way that the represen- tation matrices of all translation operators t~ written in the new basis are constructed from lower dimensional matIices, odered along the diagonal. It is of special importance that the translation group is a group of Abehan type: The resiilt of two translations t~~ and t~~ does not depend on the sequence in which they are executed. Formally this is expressed by the equation

.,pE

(2,101)

As is demonstrated in Appendix A, all irreducible representations of Abelian groups are 1 dimensional. This means that the lower dimensional matri- ces along the diagonals of the tH-representation matrices are 1-dimensional. Each of the basis functions &, cp;, . . . , ' pE therefore forms a repre- srtntation space of the translation group by itself. Thus, su functions tRpE, with t~ an arbitrary element of the translation group, are linearly dependent on p~ and dl functions t ~ & are linearly dependent on & etc. 'I'his means that thp t p p ~ itself may be written in the form

(d-1) x I

tRcPE(x) = c(R)rFE(x) (2.102)

where r(R) is a complex coefficient. Analogous equations hold for all otha functions t~lpb, tR&., , . .. This is equivalent to the statement that the cho-

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2.3. General properties of stationary oneelectron states in a crystal 87

sen ( P E , 'pk, &, . . . are eigenfunctions of the translation operator. This result forms the content of the

Bloch theorem:

The eigenfunctions p~ of the Hamiltonian H of a crystal can be chosen such that they are simultaneously eigenfunctions of the lattice translation operators of the crystal

The particular energy eigenfunctions whose existence is stated by this the- orem are termed Bloch functions. The proof of the Bloch theorem sketched above relies in an essential way on features of group representations. Al- though it forms the most appropriate proof, one may, however, also proceed without the tools of group theory, using a mathematical theorem which eo- sues that two commuting Hermitian or unitary operators have a common set of eigenfunctions.

The Bloch theorem is of such great importance that several remarks are appropriate. In the first place, it has to be emphasized that this theorem is an immediate consequence of the commutivity of translations. For the sym- metry operations a of the point group, for example, which in general do not commute, it does not hold, which means that in general the eigenfunctions of H cannot be chosen to be simultaneous eigenfunctions of the operators u

of the point group. Secondly, the theorem does not say that every conceiv- able eigenfunction of W is also necessarily an eigenfunction of tR . This holds only for specially chosen c p ~ . The Bloch theorem insures that such a choice is always possible. Thirdly, this theorem also does not imply that only one eigenfunction exists for particular eigenvalues E and c(R) of, respectively, H and tR . In reality there are alwtLys several. The pair of eigenvalues El c ( R ) is therefore not sufficient to uniquely characterize the eigenstates of H and of the group of operators t R . Quantities which provide such unique charac- terization have yet to be identified. It, turns out that there are three real numbers k l , E z , k3 which allow one to distinguish the different irreducible representations of the translation group of the crystal.

To understand this we must examine these representations in greater detail. They are defined by the eigenvalue equation (2.102) of the translation operators. To start, we will show that the eigenvalues c k l k 2 k 3 ( R ) of this equation may be written in the form

(2.1 03)

P k 1 k 2 k 3 ( R ) = ( -2r)(hri + ~ Y z + h ' 3 ) . (2.104)

Here, the E l , kz, k3 are the above mentioned real numbers which determine the representations uniquely. The factor L 2 ~ 1 is introduced to simplify ex- pressions which later will arise. To prove equation (2.103) we first show

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88 Chapter 2. Electronic structure of ideal crystals

(2.105)

(the subscript indices k l , k2, Ic3 on c(R) and P(R) will be suppressed tem- porarily). The proof is based on the normalization of p~ according to equa- tion (2.97), which leads to

(tR(PE I t R P E ) = ( P E I (PE). (2.106)

This holds because the translation t~ of the integration variable through R in ( t ~ p I t ~ p ) may be absorbed by a change of variables jointly with an application of the periodicity of the wavefunction p(x) with respect to the periodicity region. On the other hand, it follows from relation (2.102) that

(tRLPE I tR(PE) -I c(R) l2 (PaE I P E ) . (2.107)

Considering (2.106) and (2.107) together, (2.103) is verified at once. Sec- ondly, we show that

c(R1+ R2) = c(R1) . c(R2) (2.108)

must hold. To prove this, we use the following obvious relations:

Comparison of the last two relations immediately shows that (2.108) is also true. Employing (2.103) for c(R), equation (2.108) now yields

exp[iP(Rl+ Rdl = exp[iP(R~)I . ~xP[WWI, (2.112)

whence P(Ri t R2) = P(R.1) t P(R2). (2.113)

This means that P(R) is a homogeneous linear function of the components TI, 7-2, r3 of R. As such P(R) must have the form (2.104).

We now proceed to the eigenfunctions of the translation operator. Em- ploying ( ~ ~ k ~ k ~ k ~ (x) to denote the eigenfunction having eigenvalue Ck1kzk3 (R) of equation (2.103), we claim that pEklkzks(x) can be written in the form

(2.114)

where u ~ k ~ k ~ k ~ ( x ) denotes a lattice-periodic function, such that, for any lattice vector R,

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2.3. General properties of stationary oneelectron states in a crystal 89

UEklkzk3(X - R) = UEk1kzk3(X)' (2.115)

The factor l/& with 0 = G300 is introduced in (2.114) in order that the normalization integral of UEklkzk3 (x) with respect to a primitive unit cell be 1. The proof of (2.114) may be carried out by verifying that functions of the form (2.114) obey equation (2.102). This results in

which we know to he true. The functions p ~ k ~ k ~ k ~ ( x ) of (2.114) are referred to as Bloch functions and the factors u ~ k ~ k + ~ ( x ) as Bloch factors. Recalling that the p ~ k ~ k ~ k ~ ( ~ ) are also simultaneously eigenfunctions of the Hamilto- nian, we may express the Bloch theorem in a somewhat more specific form than we did above:

The ezgenfinctions of the one-electron Hamiltonaan of a crystal can be chosen in the f o r m (2.114).

The real numbers kl, k2, k3 characterizing the various Bloch functions may he understood as components of a vector. However, as we will soon see, this is not a vector in coordinate space, but one in a space which is reciprocal to coordinate space.

2.3.3

The starting point for understanding the nature of the components k l , k2, k3 is their transformation law under point symmetry transformations a in co- ordinate space. This will now be explored.

Reciprocal vector space and the reciprocal lattice

Transformation properties of kl, 122, k3

At the outset, it is clear from equation (2.99) that both p ~ k ~ k ~ k ~ ( x ) and a q E k l k z k 3 (X) are degenerate eigenfunctions of the Hamiltonian H with the same eigenvalue E. The question arises whether a p , g k l k Z k 3 ( x ) is also an eigenfunction of the translation operator and, if so, what eigenvalue of t R pertains to it. In order to answer this question, we form tRfffpEklkzk3 =

t ~ p ~ k ~ k ~ k ~ ( a - ' ~ ) and obtain the equation

= e x ~ l i P k , k ~ k - , ( ~ ~ ~ ~ R ) l ~ ~ ' ~ ~ k ~ k ~ k ~ ( ~ ) . (2.117)

The latter relation means that a p ~ k ~ k ~ k ~ ( ~ ) is indeed an eigenfunction of the translation operator t R with eigenvalue e ~ p [ i P k ~ k ~ k ~ ( a - ~ R ) ] . Evaluating ,f3klkZ~3(a-1R) explicitly using equations (2.104) and (2.82), we have

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90 Electronic structure of ideal C S ~ S ~ A I S

(2.118) j i

This expression is subject to an interpretation which differs from the previous one and which has important consequences for later developmentu. The new interpretation is that the transformation, which hitherto operated on the vector components of R, should now be understood to operate on the real numbers kl, k z , k~ inslead. Of course, t,he new interpretation cannot change the value of the expression (2.118). To assnre this, the kj must transform according t,o the transposed matrix a;' of a;' jnsted of the matrix aij ,

which applies in the case of the components ~j of R. This is to say that

(2.119)

must hold. One may therefore write

(2.121) 3

This equation means that the transformation a, which originally operated on the wavefunction p ~ k ~ l f a k ~ ( x ) , has been transposd to operate 021 the k,, which werp initially introduced as real numbers without any particular transformation behavior. The transformation properties of the kj thusly defined are similar, but not identical to, those of vector components. The difference liea in the fact that it ia not the matrix arj itself that multiplies the column vector of the components, but rather it is the transposed inverse matrix a;'. We will now prove that this is exactly the way the components of a vector transform in a space reciprocal to the ordinary space of position \-=tors x.

Definition of the reciprocal vector space

l h e space of position vectors x is defined through its basis al, a2, w. The corresponding reciprocal vect.or space is determined by a basis bl, b2, b3 said to be reciprocal to the original basis a1, az, a. The reciprocal basis is defined by the set of equations

EQ ' bj = 2 ~ 5 i j (2.122)

These equations state that the vectors of the T & P T Q C ~ basis me normal to, respectively, one of the three planes spanned by pairs of direct basis vectors.

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2.3. General properties of stationary oneelectron states in a crystal 91

,4mong themselves, the bj are in general non-orthogonal to the extent that the a, are mutually non-orthogonal. The lengths of the reciprocal basis vectors bj are determined by the three equations which follow from (2.122) for j = a. Altogether, the equation set (2.122) determines the reciprocal basis vectors bj uniquely, aa follows:

2 x 27r 2x 0 0 0 0 Qo

bl= -[a2 x q] , bz - x a11 , b3 : -[a1 x az] . (2.123)

Here Ro = a1 . [a2 x a31 is the volume of a primitive unit cell of the crystal lattice.

If one subjets the direct basis to a rotation or reflection, this induces the same rotation or reflection of the rwiprocal basis, as follows immediately from the defining equations (2.122) or (2.123). Because of these relations the 'reciprocal' basis (bl, bz, b3) is rigidy joined with the direct basis (al: a2, ag) and will transform when the latter is tranaformd However, since the recip- rocal basis is different from the direct one (only if (al, az,a3) are orthogonal unit vectors are the two basis sets the same), the matrix LY which describes the rotation or reflection of the ai is different from that which transforms the bj. Using equation (2.123) one may easily show that the bj are transformed with the transposed inverse of the transposed matrix u p which transforms the q, that is, with the inverse matrix a;'. In other words, the inverse matrix describes the same rotation OF reflection of the reciprocal basis as the transposed matrix docs fur the direct basis.

Consider now an arbitrary vector k of reciprocal vector space, which can be written as

(2.124) i

As we know, the vector components themselves transform in accordance with the transpose of the transformation of the basis vectors. Since the transfor- mation of the reciprocal basis calls for the inverse matrix, the components bi will transform with the transposed inverse matrix 05'. This verifies our earlier assertion that quation (2.121) describes the transformation of the components of a reciprocal vector. Using this result, we obtain

kkb, = cxk. i

(2.125)

The reciprocal vector k of (2.124) may be used to express the phase ,&kka(R) of the eigenvalue (2.103) of the translation operator in the more compact form

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92 Chapter 2. Electronic structure of ideal crystah

B k l k p l ~ 3 ( R ) - -k. R. (2.126)

Using this expression the eigenvalue of t,hP treiistntion operator (2.103) be- comes

which provides a more compact notation fm thp Bloch We may write

(2.128)

with u ~ k ( x ) as a short notatinii for the Bloch factor u ~ l ; ~ k ~ k ~ ( x ) . Because of equation (2.115) one has

~ g k ( X - R) U E k ( X ) . (2.129)

A further consideration illuminates the physical meaning of the reciprocal vector k. For this p u r p o s ~ we first assume the potential V(X) to be a constant independent of x. appropriate to a free electron. All the abovementioned r e s u l t s hold. of roursc, in this caw. Rut thrrp is mom In this caw translations through arbitrary vectors r and rotations about any axis and through arbi- trary angles are symiwtry opcrations, for they do not change the potential and, consequently, also leave the Hamilt onian innriant. Therefore.

" E k b - 4 = " E k W (2.130)

for arbitrary vectors r. This mcans that the Bloch factor ugk(x) is a con- s twt , and t,he eigenfunctious of the HamiItonian are of the form

(2.131)

This is just the well-known result that the stationary states of a €ree electron may be taken as plane waves of a given wavwwtor k. If the potentid is not completely constant but. as happens in a crystal, it remains constant only under translations through lattice vectors, the meaning of k as wavevector is largely prwervd. It is thrrefore called a qaasa-wavevector: Of course, the dimension of the quasi-wsvevector is also that of a reciprocal length. The Bloch functions y ~ k ( x ) of (2.128) may be understood as travelling waves. modulated spdiallJ by thr latticeperiodic Bloch factor U E ~ ( X ) . That the stationary oneelectron states may be chosen in this form, is the content of B l o c h ~ thwrpm. This theorem d o e riot say, howevrr, that these states are necessarily modulated tmvellzng plane waves, just as the stationary states

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2.3. General properti= of stationary one-electron states in a crystd 93

Figure 2.4: Mesh of allowed points in k-space due to the periodic boundary conditions in a 2-dimensional model.

of a free electron need not be travelling plane waves. One ran also h a m standing plane waves, sphrriral waves d c .

Discretization of k-space

'The periodicity condition (2.96) for the eigenstates of the Hamiltonian has largely been ignored so far. Fortunately, it can be satisfied very simply. The Ebch fuucfions V E ~ ( X ) obey the relation

if and only if the components k j of the k-vector are of the form

1 k,, - - l j , j = 1! 2 , 3 , G

(2.1 33)

with 4 BS arbitrary integers, to assure satisfaction of the requirement

The k-vectors defined by (2.133) form a finely meshed net in k-space (see Figure 2.4). The only permissible k-vectors must be points of this net since the Bloch functions are periodic with respect to the Periodicity region. The k-space thus has a discrete structure; its set of points is countable. The distance betwren different k-vectors is, however, so small because of the large size of C that k can pract,ically be treat4 as it continuously varying quantity despite its discrete character. This approximation will be used later extensively.

Reciprocal lattice

Just t t h in direct spaw, one may also consider a point lattice in reciprocal space. In this context, the so-called mciprocal lattice i s especially useful. It is dehed by taking the reciprocal basis vectors bl, bz, b3 of (2.123) aa its

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94 Chapter 2. Electronic structure of ideal crystals

primitive lattice vectors. A reciprocal lattice vector K is then given by the relation

K = C K j b j 3

(2.135)

with arbitrary integers Kj. As in coordinate space, one also has 14 different lattice types in reciprocal space, namely the 14 Bravais lattices. The recip- rocal lattices bear a close relation to the corresponding direct lattices. The following properties can be verified easily: (i) A reciprocal lattice has the same point symmetry as the direct lattice with which it is associated. (ii) The reciprocal lattice of a reciprocal lattice is the direct lattice. (iii) If the direct lattice is primitive, then the same holds for the correspond- ing reciprocal lattice. The reciprocal lattice of a centered direct lattice is also centered.

The Bravais types of primitive direct lattices are the same as those of their reciprocal lattices, while the Bravais types of centered direct lattices and of their reciprocal lattices differ in certain cases, For example, a body centered cubic reciprocal lattice corresponds to a face centered cubic direct lattice, and, conversely, a face centered cubic reciprocal lattice corresponds to a body centered cubic direct lattice.

2.3,4 Relation between energy eigenvalues and quasi-wave- vector

We have not as yet directed attention to the question if there may be a connection between the quasi-wavevector k and the energy eigenvalue E: and what form it might take. In fact such B connection does e x i s t , and it will now be explored in some detail.

That a Bloch function p ~ k for a given wavevector k cannot be art eigen- function of I1 for an arbitrary energy eigenvalue E may be yeen as follows, If one substitutes i p ~ k ( x ) (2.128) into the Schriidinger equation (2.95), one obtains the following eigenvalue equation for U E ~ ( X ) :

1 1 -(P t rzk)2 -1- v(x) UE&) = E U E k ( X ) .

Izm (2.136)

The k-dependence of the operator in this equation transfers to the eigenvalue E, i.e. E becomes H function E = E(k) of the quasi-wavevector k. In the case of a free particle with V(x) = 0, one has E(k) = (H2/2m)k2. For a crystal one expects, of course, more compljcated dependencies than this. The specific form of the function E(k) is determined by the particular shape of the periodic potential V(x). Some general properties of E(k) follow,

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2.3. General properties of stationary one-electron states in a crystal 95

however, just from the space and time symmetry of V(x). We will discuss this now, beginning with point symmetry, whose implications are relatively easy to examine.

Point symmetry

We consider the Schrodinger equation (2.95), on the one hand, for the quasi- wavevector crk, where TY i s taken to be an arbitrary element of the point group of the crystal:

(2.137)

and, on the other hand, for the quasi-wavevector k, but with an application of the transformation (Y as follows

HWEdX) E(kbPEk(X). (2.138)

The relations (2.120) and (2.124), derived in the preceding subsection, mean that U p E k is an eigenfunction of H for quasi-wavevector ak. If there is only orbe eigenfunction for a given wavevector, which we initially take to be the case, a c p ~ k ( x ) must, be identical with ( P E ~ ~ ( x ) , because ( P E ~ ~ ( x ) is by definition the eigenfunction for ak. Thus,

and a comparison of (2.137) and (2.138) then yields

E(k) = E ( a k ) . (2.140)

The assumption that there is only one eigenfunction of the Hamiltonian for the quasi-wavevector k holds for almost all k. There is an exception for those special k values for which ak does not differ from k or from a vector k + K equivalent to k for all a. One refers to such vectors as symmetrical k- vectors. They will be considered in section 2.5, but we exclude symmetrical k-vectors at this point. For non-symmetrical k-vectors, the fact that there is only one eigenfunction of H , follows from the onedimensionality of the irreducible representations of the translation group (see Appendix A).

Time reversal symmetry

A property of the Hamiltonian H which we have not employed thus far is its invariance under time reversal. In order for the Schrodinger equation to remain unchanged under time reversal, the time reversed wavefunction must be defined as the complex conjugate of the original wavefunction. Then the

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96 Chapter 2. Electronic structure of ideal crystals

time reversed Bloch function of wavevector k has wavevector -k. Since both Bloch functions have the same eigenvalue, one obtains

E(k) - E(-k). (2.141)

This relation has additional significance, beyond that of relation (2.140) obtained by means of spatial symmetry, only then, if the point group of the crystal does not contain the inversion.

Translation symmetry: Extended and reduced zone schemes

The lattice translation invariance of H has a remarkable consequence for the eigenvalue function E(k). It arises in conjunction with the degeneracy of the eigenvalues ck(R) of the translation operator tR. Specifically, all wavefunctions (PEk+K with K, any reciprocal lattice vector, have the same eigenvalue of thc translation operator. Independently of the actual value of K, the latter is

(2.142)

which follows directly from the relation

-2k.R C k + K ( R ) = e

K . R - (2.143) 1

with K , and T % as integers. Therefore we obtain the full spwtrum of eigen- values of t R if k varies within a particular primitive unit cell of k-space. The corresponding k-vectors will br denoted by kl. The k-vectors of any other primitive unit cell differ from these by a reciprocal lattice vector K and thus do not lead to new eigenvalues of the translation operator. However, the en- ergy cigenvalues g ( k ) do differ, so that E(k1) f h’(k1-t- K) holds in general. Table 2.3 illustrates these connections.

This asymmetry with respect to the k-dependence of rk(R) and E(k) is the reason for the introduction of two different representation schemes for the energy eigenvalue function E(k). In the representation scheme employed thus far, k varies over the entire reciprocal space, and to each vector k we associate only one k-dependent energy function E(k). With respect to the eigenvalues of the translation operator, this description i s highly redundant, since one encounters the same eigenvaliie an infinite number of times. This disadvantage is eliminated if the selection of the region over which k varies is not fitted to the energy E(k) but rather to the ck(R) eigenvalues of the translation operator, i.e., if k is allowed to vary only over a primitive unit cell rather than over the whole k-space. Then we must accept, however, that an infinite number of different energy eigenvalues are assigned to each k-vector, namely all those which follow frotn E(k) by means of the prescription

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2.3. Gcnmd propertic3 of stationary oneelectron states in a crystal

~ X P [-ik. R] ~ exp [-ikl . R]

97

E(k + K)

Table 2.3: Representation of band structure in the cxtcnded and reduced zonc schemes.

Wavevector

k from Infinite Space

k = kl from Unit Cell

I Eigenvalue of t R Eigenvalue of H

I

1

'

Represent at ion

General

Extended Zone Scheme

Reduced Zone Scheme

whme kl is a vector of the primitive unit cell at K ~ 0. In this way the singlevalued function E(k) over all k-space is transformed to an infinitely multi-valued function EK(k1) over a primitive unit cell. An analogous state- ment holds for the eigenfunctions of the Hamiltonian. Each Bloch function p,yk over all k-space corresponds to an infinite number of different Bloch functions pEKkl(x) - pqkl+Iqkl+I{(x). One cells the representation in- volving the entire k-space the extended zone scheme, and that involving the primitive unit call at K = 0 the w d u d zonw schpme. For most problems in semiconductor physics the reduced scheme is more convenient than the extended scheme.

Later, we will demonstrate that the energy E(k), as a function over all k-space, is not continuous everywhere but has discontinuities at certain k- planes. Anticipating this, one may then inquire whether one can choose the primitive unit cell of the reduced scheme at K ~ 0 in such a way that the discontinuities of the branches EK(k) do not occur in the interior, but only on the surface of the primitive unit cell. If such a special primitive unit cell exists at all, it must have at least the symmetry of the point group of direclionb, as followb from the symmetry reletion (2.140). This nieans that only the Wigner-Seitz cell of the reciprocal lattice is a possible candidate for siich a primitive unit cell. Of course, its point group is that of the crystal lattice and, theIefore, larger than the requisite point group of directions.

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98 Chapter 2. Electronic structure of ideal crystals

We know, however, that lattices have fewer point groups than there are point groups of equivalent crystal directions. The next-lower lattice point group would not contain the needed point group of directions. The issue in question thus revolves upon whether the function branches EK(k) are continuous in the interior of the Wigner-Seitz cell of the reciprocal lattice at K = 0. We will prove that this is indeed the case for the branch Eo(kl) in the next section. Moreover, it will be shown that the other branches EK(k1) of the infinitely multi-valued function E(k) may be redefined in terms of new branches such that each single-valued branch is also continuous over the Wigner-Seitz cell at K = 0. One calls these continuous branches energy bands and the Wigner-Seitz cell of the reciprocal lattice at K = 0 is the f i r s t Brillouin zone.

Having started from general considerations and obvious conjectures, we thus arrive at the important conclusion that the spectrum of allowed energy values of crystal electrons will have the form of energy bands, and that these energy bands arise by varying the quasi-wavevector k over the first Brillouin zone. We will discuss the concept of ‘Brillouin zones’ along with the still outstanding proof of the continuity of E(k) over these zones more fully in the next section.

2.4 Schr6dinger equation solution in the nearly- free-electron approximation

It is well known that the Schrodinger equation can be solved approximately if the potential, or a part of it, represents a small effect which may be treated by means of perturbation theory. To apply this procedure, the solution of the unperturbed problem must be known. At this point, we will treat the entire periodic potential by means of perturbation theory. Even if it is not to be expected that this will yield results which are quantitatively accurate, some qualitative understanding can be gained in this way.

If one takes the periodic potential as a perturbation, then the unper- turbed problem is that of a completely free electron. The corresponding Hamiltonian Ho is simply the kinetic energy of the electron, i.e.

P2 H o = -. 2m

(2.145)

The perturbing operator H I , which together with Ho forms the total Hamil- tonian

H = Ho + Hi, (2.146)

is then the periodic potential

HI = V(x). (2.147)

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2.4. Schriidinger equation solution in the nearly-freeelectron approximation 99

In perturbation theory, the exact eigenfunctions Cpk and eigenvalues E(k) are expanded in power series,

E(k) = Eo(k) + 6E1(k) + 6E2(k) + . . . (2.149)

with respect to the perturbation potential, here with respect to the periodic potential V(x). In this discussion, we suppress the energy index E of the eigenfunctions for the sake of brevity. This leads to no ambiguity since we work in the extended zone scheme, where the energy is a unique function of the quasi-wavevectors k. The zero-th order terms obey the equation

Hop; = Eo(k)Cpg. (2.150)

In order that the perturbation theory be applicable, the zero-th order solu- tions must form a complete orthonormal set of functions, such that

and -yCpp(x’)Cp;(x) = 6(x’ - x) k

(2.152)

must hold. The k-summation of (2.152) is extended over all points of the infinite finely meshed net of Figure 2.4. The two relations (2.152) and (2.153) are in fact valid for the solutions

(2.153)

of equation (2.150). The validity of the orthonormality relation (2.151) may be easily verified by direct calculation, and the completeness relation (2.152) is proved in the theory of Fourier series. The energy eigenvalue Eo(k) of zero-th order for pg is

0 Ti2 E (k) = -k2

2m (2.154)

We will assume at the outset that this eigenvalue is not degenerate, more exactly, that this holds for k-values for which the perturbation operator V has non-vanishing matrix elements (pi! I V I ~ p g ) with any vector k’. It is not possible to exclude degeneracy completely because wavevectors k of the same length lead to the same energy eigenvalue (2.154).

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100 Chapter 2. Electronic structurc of ideal crystals

2.4.1 Non-degenerate perturbation theory

h accordance with the procedures of quantum mechanical perturbation t h e ory, we have

(2.155)

(2.156)

The energy correction 6E1(k) of the first order i s the average value (cpg I V I pi) of the periodic potential. This value may be set equal to zcro. The off-diagonal matrix element of the potential involved in (2.1551, (2.156) are given by

(z. 157)

Because of the lattice periodicity of V(x) we can transform the integral over the periodicity region in (2.157) into a sum of integrals over the primitive unit cell Qo(0) at R = 0:

Noling that

(2.159)

(2.160)

we arrive at the result

Using this result, the correction 6 ~ : to the wavefunction takes the form

(2.162)

and the total wavvcfunclion pk = ‘p! t 6pk is given in this approximation as

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2.4. SchrMingw equation solution in the nearly-fieet3-electron approximation 101

One recognizes that this has the expected Bloch form (2.128) b r energy eigenfunctions of the crystal Hamiltonian. Moreover, (2.163) yields the Bloch factor as

(2.16 4)

The second order energy correction E2(k) may be determined using (2.161) as

(2.165)

The corrections to the eigenfunctions (2.162) and the enerffi eigmvtrlues (2.165) will become large if the denommator in these expressions becomes srnall. Actually. the denominator may even vanish when

E"(k) - Eo(k I K) - 0. (2 166)

In such a case, I ~ P basic premise for validity of non-degenerate perturbation theory is violated, namely that there be no degeneracy among the unper- turbed energy cigenvalues for those k-values fur which the matrix elements of V ( x ) do not vanish. However, just such a degeneracy exists if equation (2.166) holds, with the states 9: and cp:-K degenerate with each other. Below, we will accommodate this case and discuss the corresponding degen- erate perturbation theory, but first we analyze the k-values for which the condition (2.166) is fulfilled. Using (2.154) for E o ( k ) , we get from (2.151)

1 k . K + - I K 1 2 = 0.

2 (2.16 7)

This equation can be solved geometrically. It is obvious from Figure 2.5 that the tIps of the corresponding k-vectors lie on a plane perpendicular to the vectors K or -K, with this plane intersecting the vector -K at its half-length. Such planes are familiar from the theory of X-ray diffraction by crystal lattices. They aTe called Bmgg reflectam planes.

The non-degenerate perturbation-theoretic results discussed above may be interpreted quite clearly if one regards the problem of the calculation of the eigenfunctions in the periodic potential as a scattering problem. In this, the unperturbed eigenfunctions p: are thought of as incoming waves and the perturbed eigenfunctions 9 k as outgoing waves, 6 ~ : corresponding to the scattered part of the latter. That the perturbation theory according to formula (2.162). yields only minor corrections to the unperturbed eigenfunc- tions is related in this picture to the fact that the incoming and scattered plane waves in\-olve different wavelengths, since 1 k If jk + KI, and cor- respondingly they are not capable of significant constructive interference.

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102 Chapter 2. Electronic structure of ideal crystals

Figure 2.5: Construction of Bragg reflection planes.

-&

-K \ f

Figure 2.6: Illustration of perturba tion theory with respect to the peri- odic crystal potential in the vicinity of a Bragg reflection plans where the zero- th order energy levels are degenerate.

For incoming waves with k-vectors close to the Bragg reflection plane of the reciprocal vector -K, the scattered part dpk contains a par t idar ly large plane wave component of wavevector k + K. This is due to the fact that, in this case, the incoming plane wave and the Ecaltered wave of wavevec- tor k + K, have almost the same wavelengths. They are thus rapable of constructive interference which strongly enhances the amplitude of this par- ticular scattered wave. If k is located exactly on the Bragg reflection plane, one may understand the scattered wave with wavevector k + K as the plane wave reflected at this plane in the sense of geometrical optics. This follows immediately from the construction of the k h g g reflection plancs in Figure 2.5. The scattered wave of wavevector k + K is strengthened by inter- ference, whch makes it the refkected wave (bear in mind that in the wave picture reflection represents an interference phenomenon). This interpreta- tion can be universally applied to the propagtltion of plane waves of any kind in a system of scattering centers periodically ordered on a lattice, including X-rays. Actually, the above results for elwtron waves were discoverd iisinE X-ray radiation.

The amplitudes of plane waves reflected at Bragg planes, according to formula (2.162), become idnitely large formally What is expectd. how- ever. is that the amplitudes of the refiected and incoming waves should be comparable. This contradiction arises from the misuse of non-degenerate perturbation theov, w h h is no longer applicable under the conditions of reflection. For k-vectors which obey the Bragg condition (2.167), one must apply degenerate perturbation theory.

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2.4. Schrdinger equation solution in tho nearly-free-electron approximation 103

2.4.2 Degenerate perturbation theory

Let ko be a point on a Bragg reflection plane, so that

(2.168) 1

k o . K + - I K 1 2 = 0 2

holds for a particular reciprocal lattice vector K. We denote a small deviation from ko by Ak where 'small' means that I Ak I<<] K I. A vector k of the form

k = k o + A k (2.169)

is then a vector close the Bragg plane involved. The same holds for the vector k + K, as it lies close to the parallel plane in Figure 2.6.

According to quantum mechanical perturbation theory for the case of degenerate zero-th order states, one proceeds as follows. The perturbed eigenfunctions cpk are sought as linear combinations of the two (almost) degenerate eigenfunctions cpi and (p& in zero-th order, writing

(PIFo(x) = cocp:(x) + C K d + K ( X ) (2.170)

with co and CK coefficients to be determined. In the lowest non-vanishing order of perturbation theory, the Schriidinger equation leads to the following set of equations;

V(K)* (2.171) Eo(k) - E ( V(K) Eo(k$ K ) - E For the set to have a non-trivial solution, its determinant must vanish, whence

Eo(k) - E V(K) V(K)* Eo(k+ K) - E

must hold. The two solutions are E = Ek(k) with

= 0 (2.172)

1 1 2 2

Ek(k) -- ,[Eo(k) + Eo(k + K)] f -\/[Eo(k) - Eo(k + K)]* + 4 I V(K) I*. (2.173)

For further evaluation of this expression, we assume that Ak is directed parallel to K (see Figure 2 .6 ) . Expanding the square root in powers of I Ak I and terminating the series with the second order, we find

jpK2 E*(k) = Eo(ko)f I V(K) I 4 1 f 2m (2.174)

li2 [ I V(K) I ] Ak2.

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104 Chapter 2. Electronic structure of ideal crystals

Figure 2.7: The degeneracy of the two energy parabolas E(k + Ak) and E(k + K + Ak) at the Bragg reflection plane Ak = 0 (left part of figure) is removed by the periodic crystal potential. In the formerly continuous energy spectrum a gap arises at this plane (right part of the figure).

The remarkable feature of this result is that the 2-fold degenerate energy eigenvalue Eo(ko) = Eo(ko + K) at the Bragg reflection plane, i.e. for Ak = 0, splits into two different eigenvalues which are shifted by IV(K)I to higher and lower energies, respectively. Between them there is a region of width 2 I V(k) I where no allowed energy values may exist (see Figure 2.7). This means that, in the formerly continuous energy spectrum of the free electron. an energy gap appears as a result of the periodic potential. This occurs at all Bragg reflection planes, provided that the corresponding Fourier components I'(K) of the periodic potential do not vanish. Except for the Bragg planes, the function E ( k ) and the energy spectrum remain continuous. The result derived here by means of perturbation theory is also valid well beyond the framework of validity of this approximation. The following statement remains true also in the general case.

The energy f u n c h o n E(k) LS continuous everywhere in k-space, with the exceptzon of the Bragg reflectLon planes where, 271 generat, discontinuities occur and the energy spectrum exhihts gaps.

In regard to the shape of the function E*(k) close to the Bragg planes, the relation (2.174) provides the following picture (see Figure 2.7). The two branches E+(k) and E-(k) approach Ak = 0 with horizontal tangents to the limiting values Eo(k) + IV(K)I and Eo(k) - ll'(K)I, the upper branch with positive curvature, and the lower with a negative one. The latter ob- servation follonrs from (2.174) since the applicability of perturbation theory is restricted in validity by 1 V(k) I << h2K2/2m, which means that the sign

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2.5. Band structure 105

of the angular bracket in (2.174) is always determined by the second term.

2.5 Band structure

The perturbation-theory calculations of the preceding section were carried out in the extended zone scheme for E(k}. Alternatively, there is also the reduced scheme introduced in section 2.3, in which the quasi-wavevector k varies ody over a primitive unit cell of reciprocal space. In this , each of an infinite number of choices for this cell can be used. Here, we will determine the choice of primitive unit cell of the reciprocal lattice in such a way that the desired property of the function E(k} discussed above, namely that it have discontinuities only at Bragg reflection planes and otherwise be continuous, is described as simply as possible.

Simplicity of the description calls for the discontinuities of E ( k ) to occur only on the boundaries of the unit cell, with E(k) continuous in the interior. The question of whether there is a primitive unit cell which guarantees that. is equivalent to the question of whether there exists a primitive unit cell which is bounded by Bragg planes, but has no other such planes in its inte- rior. The prescription for the construction of the Bragg planes (see section 2.4) makes it immediately clear that the Wigner-Seitz cell of the reciprocal lattice at K = 0 has the desired property - it is bounded completely by planes which obey equation (2.1671, namely Bragg planes, and in its interior it is devoid of such planes. This means that within the Wiper-Seitz cell of the reciprocal lattice at K = 0, the function E(k) is continuous. However, it remains to be clarified how the rest of k-space can be reduced t o the Wigner- Seitz cell at K = 0 in such a way that the resulting new function branches for E(k) are also continuous over this celL It is clear that this cannot be done by simply dividing k-space into Wiper-Seitz cells and then translating all cells not containing the origin through reciprocal lattice vectors back to the cell at K = 0. The reason for this is that the non-central Wigner-Seitz cells are cut by Bragg planes. The correct procedure goes back to Brillouin and will be described below.

2.5.1 Brillouin zones

Definition

Consider, in k-space, centro-symmetric bodies which contain the origin and are entirely bounded by parts of Bragg reflection planes. These bodies are arranged and enumerated according to their volume. The smallest will be the Wigner-Seitz cell at K = 0. The second, next-largest body, has the volume of two Wigner-Seitz cells, as one may easily demonstrate. It contains the

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106 Chapter 2. Electronic structure of ideal crystals

ZONE 1 2 3 4 5 6 7 8 9 1 0

Figme 2.8: The first, 10 Rrillouin zones or H sqimre plane 1al.tic:e. (After Brillouin, 1953).

first body jointly with the Bragg planes bonnding i t . If one removes the first body from the second, then one obtains a hollow body of the volume of one Wigner-Seitz cell, which is bounded by Bragg planes inside and outside, but contains no siirh planes in i t s interior. This conhtruction may be carried further. Removing the v-th body from the (v + 1)-th one again obtains a hollow body having the volume of the primitive unit cell in k-space. It i s boiindd by Bragg planes, and has no such planes in its interior. These difference bodies are called Brdloum zones (abbreviated 8 8 s ) . The v-th body i s called the v-th R%, the Wigner-Seitz cell at K = 0 is, accordingly, the first B Z . In Figure 2.8 the first 10 Brillouin zones are shown for a 2-dimensional square reciprocal lattice. The first B Z for two important 3- dimensional lattices, i.e. the frr lattice and the hexagonal lattice, are shown in Figure 2.9. We will derive the shape of the first of them below.

Focusing on a point k, of the v-th R Z , a vetor K(k,) of the reciprocal lattice is attached k, such that k,- K(k,) represents a vector kl of the first B Z ,

kl = k, - K(k,). (2.175)

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2.5. Band structure 107

Figure 2.9: The first Brillouin zones of the two Bravais lattices which apply to many semiconductors: (left) the face centered cubic lattice of diamond, zincblende and rocksalt structures; (right) the hexagonal lattice of wurtzite and selenium structures.

We omit the formal proof for this important result here. For the square lattice it follows immediately from Figures 2.8 and 2.10. Its validity in the general 3-dimensional case is clearly plausible. If, instead of k,, another point kh of the v-th B Z is chosen, another reciprocal lattice vector K(kh) may be necessary in order that kh - K(kl ) shall be a vector of the first B Z . The set of reciprocal lattice vectors K(k,), K(kL), . . . which arises if k, varies over the whole v-th B Z , is just the set of K-vectors which defines the internal boundary planes of the v-th BZ. The correspondence of the points of the v-th B Z to points of the first B Z given by equation (2.175) is unambiguous in both directions - each k, corresponds exactly to one kl, and each kl exactly to one k,. One terms this correspondence ‘folding’ of the v-th B Z into the first. The term ‘displacing’ would probably provide a better description. During such folding or displacing, the external or internal boundary planes of v-th B Z may be translated to the interior of the first B Z . There, they border on other boundary planes of the v-th BZ. This means that original kl-vectors which lie immediately to the right and to the left of such planes will involve different reciprocal lattice vectors.

The folding operation (2.175) provides the mechanism for changing from the extended description of the energy function E(k) over all infinite k-space to the reduced description within the first B Z . The transfer of the function values is, defined above by relation (2.144). It states that the value E(k) at a point k = k, of the v-th B Z is assigned to that particular point of the first B Z which arises from k, by folding the v-th BZ into the first. By this assignment a multi-valued function E,(kl) is defined within the first BZ. The unique function E(k) over the whole k-space is mapped into the multi- valued function E,(kl) within the first BZ. Formally, this is expressed by

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108 Chapter 2. Electronic structure of ideal crystals

Figure 2.10: Folding of the second through ninth Rrillouiri zones into the first R Z for the plane square lattice of Figure 2.9. (After Brillouin, 1953)).

the equation

Ev(ki) - E ( k i + K(k,)) E(kv). (2.176)

We maintain that the branches E,(kl) of the multi-valued function are con- tinuous within their entire range of ddinition, i.e. within the fust B Z . For points kl which correspond to points k, of the interior of the v th B Z , the validity of this assertion is obvious. For those psrticiilar kl-planes sris- ing from a bo~iirlary plane of the v-th B Z , continuity is not immediately evident, for in penetrating such a kl-plane the original vector k,, of the Y-

th R Z jumps by the negative of the rcciprocal latticc vector. which defines this boundary plane. However, this jump does not affect the energy eigen- value E(k,) in the extended zone scheme as may be seen by means of the degenerate perturbation tbwry in swtion 2.4. Thus the functions E,(k) are also continuous on the particular planes in question.

Each of the function branches F;,(kl) defined above encompasses some finite interval of values on the energy scale. The term energy hnnd for the set of E,(kl)-values is thus obvious; I / is the so-called band andm Between the various function branches, or energy bands, energy regions may exist in which no energy eigenvalues occur. These regions are called forbidden zone8 or energy gaps. l'he whole s r t of functions R,(kl) is referred to collectively as thr band structure.

In Figure 2.11 we illustrate the band structurc using the model of a 1-dimensional lattice. The solid curves correspond l o a rornplelely free elec- tron, and the dashed ones to an elwtzon in a weak periodic potential. One recognizes that energy gaps orcur not only on the boundaries of the first BZ

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2.5. Band striictiire 109

Figure 2.11: Rcduction of the energy parabola of a free electron into the first B Z of a 1-dimensional lattice. The changes caused by a weak, periodic lattice potential are indicated by dashed lines.

but also at its center at k = 0. This is due to the fact that, during folding, surface points of the first B Z are displaced to the B Z center, and the energy gaps from those points also appear at the center.

The folding procedure into the first B Z must also be carried out for the eigenfunctions of the Harniltonian. Let p v k l denote the Bloch function of energy E,(kl) and of reduced wavevector kl. In the extended scheme the same wavefunction reads q q k , ) k v ( x ) . Thus,

The representation of E(k) as a multi-valued function E,(kl) over the first B Z is by far the most common and frequently used description of the energy spectrum of a crystal. The first Brillouin zone is, therefore, of outstanding importance in solid state physics. Unless otherwise specified, the quasi- wavevector will henceforth always be understood to lie in the first BZ. Thus we will write k instead of kl, which means, in particular, E,(k) instead of E,(kl). The Schrodinger equation for an electron in a crystal then takes the form

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110 Chapter 2. Electronic structure of ideal crystals

f f ~ v / c ( x ) = &(k)pvk(x) . (2.178)

The first RZ is the fully symmetric primitive unit cell of the reciprocal lattice. The relation between the reciprocal and the direct lattices was dis- cussed in section 2.3, where it was seen that each of the 14 direct Bravais lattices corresponds to a particular reciprocal lattice of the same symmetry, although not necessarily of the same Rravais type. Accordingly, there are 14 different reciprocal Bravais lattices, which ah0 means 14 different first BZs . Two of them, the two that are most important for semiconductors, are shown in Figure 2.9.

Empty-lattice band st ructure

In Figure 2.11 the energy E ( k ) of a free electron moving in 1 dimension has been represented over the first B Z of R I-dimensional lattice. Similar representations are possible in 3 dimensions. Below we demonstrate this for the face centered cubic lattice which applies to crystals of the diamond and zincblende structures, According to Table 1.2, the primitive vectors of this lattice may be chosen in the form

(2.179)

The corresponding basis vectors of the reciprocal lattice follow from equation (2.123), with the result

a a a

2 2 2 a1 = - ( O , 1, I ) , a2 = - ( l , O , I), a 3 = -(I, I, 0) .

27r 27r - 2n - - bl = -(I, T, I), b2 = -(l, 1, T) , b3 = -(I, a 1,l). (2.180)

The first BZ is bounded by the 14 planes perpendicular to the following reciprocal lattice vectors:

(1) 8 lattice vectors having the length of the primitive lattice vectors given above, which means the smallest possible length overall. These vectors read:

2n 27r - 2x - - 2n -(i,i, T ) , -(I, 1, T) , -(I, 1, I), - (T,T,T) , a a a a

(2.181) 27r - 2 s 2n 27r -(1, 1,1), -(I, T, I), -(I, 1, T ) , -(I, 1,i).

(2) 6 lattice vectors having the next-smallest possible length:

27r 27r 2n -4% a 0, O ) , -(0,2,0), a - ( O , a 0,Z). (2.182)

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2.5. Band structure 111

Figure 2.12: Symmetry points and irreducible parts of the first BZ of the fcc lattice (left) and the hexagonal lattice (right).

The first B Z of the fcc lattice obtained by means of these reciprocal lattice vectors is shown in Figure 2.9. Perpendicular to the three cubic axes it is bounded by squares, and normal to the four space diagonals by hexagons. It is common to denote the symmetry points of the first BZ by capital letters. Greek letters are assigned to symmetry points in the interior of the first BZ, and latin letters to symmetry points on its surface (see Figure 2.12) . The center of the B Z , for example, is I?, the point at which a cubic axis cuts the B Z surface is X , and the connecting line between I? and X is A.

We will now reduce the energy parabola E(k) = (fi2/2m)k2 of a free electron along the A-line, i.e. for points

(2.183) 2R 2R 2?r a a a

r = -(o, o,o) A = -(o, o , ~ ) ; x = -(o, 0, 1)

with 0 < C < 1. In the reduced zone scheme we have

E,(k) = -(k h2 + K ( ~ Y ) ) ~ . 2m (2.184)

Using the above listed 14 reciprocal lattice vectors K(k,) in this relation, the second and third BZ's are folded into the first. For convenience we introduce Eo = ( T ~ ' / 2 m ) ( 2 7 r / a ) ~ as an energy unit, and set

E,(k) 71 Eu. ~v(k). (2.185)

With lattice constant a of 5 A, a value of 5.9 eV follows for Eo. For ty(k) we have

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112 Chapter 2. Electronic structure of ideal crystals

(2.186)

where the K,, Kvy, K,, are the components of the vectors K, in units of 2nla. In Table 2.4 we show the reduced energy functions ty(<) for the 14 vectors K, together with those for 0. In some cases the same value of the function E , ( C ) is obtained for 4 different K,’s, meaning that the correspond- ing energy bands are 4-fold degenerate. In Figure 2.13 the content of Table 2.4 is illustrated graphically, along with the band structure parallel to the A-line between 0 and L not given in Table 2.4. Again, one recognizes that there are k-points associated with the same energy value several times.

The band structure shown in Figure 2.13 is that of an ‘empty’ fcc lattice, i.e. a fcc lattice whose lattice points are not occupied by atoms, since the latter would create a non-zero periodic potential V(x) if they were present. For a vanishing potential V(x), no definite lattice exists because of its arbi- trary translation symmetry. Nevertheless, a definite lattice, the fcc lattice, and a definite lattice constant, were chosen in the above procedure. This was an arbitrary choice, in the sense that we could have also chosen any other of the 14 Bravais lattices, and any other lattice constants. However, if we want the empty lattice band structure to resemble the band structure of a really existing crystal, then we cannot take any empty lattice band struc- ture but must chose the one for the lattice applying to the crystal under consideration. Later we will verify that the band structures of real crys- tals and the pertinent empty lattice band structures have in fact common features. In Figure 2.11 such features are, for example, the appearance of the lowest energy eigenvalue at the I?-point and the development of several non-degenerate energy bands from this level along the A-line, one of them crossing another band at the X-point. It is also consistent that the energet- ically high-lying bands display a relatively high degeneracy at the I?-point, although for real fcc crystals the degeneracy at maximum can be only 3-fold, and not 8-fold as for the empty fcc lattice at the energy E = 3&, and not &fold as for the fcc lattice at E = 4Eo. A degeneracy higher than %fold turns out be accidental, i.e. not caused by symmetry, but by the particular values of the potential, which is identically zero in this case. The width of the lowest energy band in Figure 2.11 is about Eo, i.e. 6 eV. Also, this result is not too far from the actual widths in fcc crystals, as we will verify below.

As with the reduction along A- and A-lines, one can plot the band struc- ture along other symmetry lines in the interior and on the boundary of the first B Z . A complete representation of the functions E,(k) over the points of 3-dimensional k-space would require a Cdimensional space. In the 3-dimensional space available to us it is only possible to represent bands E,(k) over planes in k-space. Such representations are, however, uncom-

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2.5. Band structure

(KV&V,K,)

113

c y ( c ) Notation Degree of Degeneracy

( i i i ) (iii) (111) (iii)

(zoo) (200) (020) (020)

2 + (6 + I)* c 2 4

4 I c z F3 4

mon. -4s a rulc, one plots the energy against lines in k-space, as in Figure 2.13.

The question of what valws the energy band functions E,(k) take out- side the first B Z does not arise or is meaningless, since the E,(k) were

" L A r A X Wavevector -

Figure 2.13: Band structure of an empty fcc lattice.

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114 Chapter 2. Electronic structure of ideal crystals

just defined by transferring the function values from other B Z s to the first one. For this reason, the E,(k) already encompass the entire spectrum of eigenvalues as k varies over the first B Z . If one wishes to also consider the functions E,(k) outside of the first B Z , one has to define them there. The most natural manner to do this is the periodic continuation.

Periodic continuation

In this context, the values of E,(k) in Wigner-Seitz cells having centers at K # 0 are defined by the relation

(2.187)

where k is a point of the first B Z . By periodic continuation, one can of course extend each function originally defined only within the first B Z to the entire k-space. The question is what analytic properties does this function have? If one considers an arbitrary function, discontinuities will occur on the boundaries of the Wigner-Seitz cells, i.e. the periodically continued function will not be continuous in the entire k-space. In order to have continuity, E,(k) must satisfy certain conditions on the boundary of the first B Z . These conditions follow from the fact that the boundary of the first B Z consists of pairs of equivalent parallel planes (see Figure 2.6). We denote a point on one plane of such a pair by ko, and kb denotes the equivalent point on the other plane of the pair. If ko belongs to the first BZ, as we will assume, then kb cannot also belong to i t , because a primitive unit cell contains only non- equivalent points. The value of E,(kb) is defined by the periodic continuation and, as such, it is equal to E,(ko). In order for the periodically continued func%iorr t o be continuous, thix value must coincide with the limiting value of E,(k) if one approaches the point kb coming from the interior of the first B Z . Thus, a necessary condition for continuity of the periodically continued function is that the original function, defined only over the first B Z , obey the boundary condition

lim E,(k) = E,(ko), (2.188) k- k&

where k is in the interior of the first B8. This condition is also siifFicient, since the continuity of the periodically continued function E,(k) implies that equation (2.188) holds. Therefore, the boundary condition (‘2.188) for the function E,(k) defined over the first R E , and the continuity of the periodically continued function E,(k) are just different descriptions of the same property. Tising continuity of the function &(k) defined over all k- space, this property may be expressed in a more convenient way. This is the reason why one continues the ene ra bend fiinctions R,(k), which actually

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2.5. nand structure 115

have meaning only within the first BZ, periodically over the entire k-space. One could also forego this, and use the condition (2.188) directly.

We will now prove that the energy bands E,(k) actually obey the bound- ary condition (2.188), and thus, are continuous everywhere as periodically continued functions. To this end, we use the Schrodinger equation (2.178) for the eigenvalues E,(k). The corresponding Bloch functions p,k(x) may be expanded in plane waves with wavevectors k + K, where K is an arbitrary reciprocal lattice vector, as follows:

Using this result, the Schrodinger equation (2.178) takes the form

(2.189)

(2.190)

The E,(k) are the eigenvalues of the matrix of coefficients of this system of equations and their k-dependence i s determined by that of the matrix. The latter is manifpstly continiioiis. That it also exhibits the periodicity of the reciprocal lattice, one may verify as follows. Consider the coefficient matrix of (2.190) at the point k + K” with K” an arbitrary reciprocal lattice vector. If one replaces the column and row indices K‘ and K, respcctively, by K’+K” and K + K”, using the relation (K’ + K”IVJK + K”) - (K’IVIK), one obtains the matrix at the original k-vertor. This implies that the eigenvalues Ev(k t K”) and E,(k) are identical, as stated.

The above proof employs only the lattice translation symmetry of the potential V(X). The continuity of the periodically rontinued function E,(k) in the entire reciprocal space is therefore an immediate consequence of the periodicity of V(x) in the direct lattice. The continuity of the periodically repeated energy band functions E,(k) is often very useful. It ensures, on the one hand, that the E,(k) may be npproximctted fairly closely by a Fourier series with relatively few terms. On the other hand, mathematical theorems dealing with the types and numbers of extrema of periodic analytic functions may also be applied. This i s particularly important for the theory of the optical proprrties of semiconductors.

We return now to a question which was previously explored in section 2.3, namely the implications of point symmetry for band structure. This was discussed above in the extended zone scheme, and symmetric k-points were excluded. NOW we USP the reduced zone scheme description and concentrate on symmetric k-vectors.

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116 Chapter 2. Electronic structure of ideal crystals

2.5.2 Degeneracy of energy bands

The first B Z , as the Wiper-Seitx cell of the reciprocal lattice, also cxhibits the point synmelry of this lattice which, for its part, is the same as the point bymmetry of the corresponding direct lattice. ‘The latter statement follows directly from the definition (2.123) of the reciprocal basis bl, bp, b3. The first B Z of the fcc lattice has, therefore, the point symmetry of a cube, and its point group is Oh(m3m). The energy bands likewise have a particular point symmetry, namely that of the equivalent directions of the crystal. This is expressed by quation (2.140), which underlies the extended zone scheme. It may, however, be transferred immediately to the reduced scheme, so that we may also write

&(k) = Ev(ak) , (2.19 1)

where a is an element of the point group of equivalent directions of the crystal. This group is generally smaller than the point symmetry group of the lattice, but in special circumstances it can also be the same. The latter rase occurs for the diamond structure, where the point group of equivalent crystal directions is likewise o h .

In section 2.3, symmetric k-vectors, i.e. vectors which are transformed by at least one element of the point group 1’ of equivalent directions, into themselves or into an equivalent vector, were excluded. Now we also admit such k-vectors to consideration. In rqard to their effect on symmetric k- vectors, the elements cr of P split into two subsets. The Erst contains those elements which transform k into itself or, at points on the surface of the first B Z , inlo a vector crk equivalent to k which differs from k only by a reciprocal lattice vector K. This set forms a subgroup pk of P , and is called the sn~a l l poznt group of k. The second subset contains all those elements of P which transform k neither into itself nor into a vector equivalent to k. The set of all different and non-equivalent vectors a k is called the star ofk. We denote the number of elements of the point group P by p , and the number of different star points of k by S k . Since each star point has the same point symmetry, the number pk of elements of the small point gToup Pk is given by the relation p k 7 p l s k . A general k-vector bas no point symmetry, whence p k = 1 and A k = p . For k-vectors on symmetry lines or planes, pk has a value between 1 and p . The center of the first BZ is special: all elements of P arc also dements of Z’k, so that pk = p and S k = 1. A wavevector whose small point group does not consist of only the unity element, is a symmetry point. The points of the first B Z in Figure 2.12 marked with Iargr greek or latin letters are among them.

Using the terminology above, the point symmetry of band structure as given by equation (2.191) may also be described by saying that the energy of a particular band has the same value at all points of the star of a wavevector,

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2.5. Band structure 117

Onc therefore also calls this symmetry the Y ~ U T degunrrucy OJ e n e r g y bands. Bmause of star degeneracy, the energy band functions E,(k) need to be calculatcd only for a scction of the first BZ which covers the region between adjacent star points (Figure 2.12). One calls such a section an zrrediiczble part of the first R E . The energy eigmvulues for the remainder of the first BZ are obtained by means of the symmetric continuation (2.191) of the values of the irreducible part. The irreducible part is the p-th part of the first BZ. If one considers that, in the case of diamond structure, the number p of elements of the point group 01, is 48, one can appreciate how greatly the description of band structure is simplified by exploiting the point symmetry of crystals.

In section 2.3 it was shown that for any eigenfunction (p,k(x), acp,k(x) = cpVk(cr-’x) is also an eigenfunction of the Haniiltonian H with the same eigcnvalue B,(k), provided (1 belongs to the symmetry group of 11. Those elements a of this group which also do not change the wavevector k, or change it only by a reciprocal lattice vector, will thus transform an eigen- fiinction pvk(x) of a particular Land v and k-vector, into an eigenfunction cp,k(~u-~x) of the same band and the same k-vector. Thus, lor the case in which not all wp,k(x) are linearly dependent, t h e are several linearly in- dependent eigenfunctions for a given band index 11 and wavevector k. Let their number be d,k. This means that the energy band E,(k) i s d,k-fold de- generate at the point k. One calls this kind of degeneracy band degeneracy.

We employ q l k ( x ) , 1 - 1,2 , . . , duk, to denote a basis set in the subspace of eigenfunctions or H having the eigenvalue E,(k). The functions crcplk(x) are then likewise eigenfunctions with the eigenvalue E,(k). As such, they can be expressed as linear combint\tions of the basis functions pVk(x), whence

(2.192)

with Dpl(a) as expansion coefficients. Through relation (2.192) each element a of the sniall point group is associated with t~ matrix Dpl(a). In Appendix A we explain that the matrices Dpl(a) form a r e p r e s e n t a t z o n of the small point group of the wavevector k. If the clegeneiacy of the energy eigenvalues i s only due to the symmetry of the Hamiltonian, then this representation is arreducihle. We may say that the eigcnfimctions of the Hamiltonian for a particular k-vector and band-index v span a subspace in Hilbert space which gives rise to an irreducible representation of the small point group. The dimension of this reprcsentation, i.r. the dimension of its matrices, corresponds to the degree of degeneracy of the energy band for the k-vrctor mider consideration. If one knows all irreducible representations of the small point group Pk, then one also knows what degrees of degeneracy of the energy bands are possible at k. This relation between degeneracy and symmetry

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118 Chapter 2. Electronic structure of ideal crystals

Table 2.5: Irreducible representations (notations and dmensions) of the smaLl point groups of symmetry points and lines of the fist B Z of the fcc lattice for crystals with the diamond structure. For the point X, the projective irreducible representations are given in the crystallographic factor system. (See Append& A ) .

- A

- X

- c K

is quite remarkable. It holds not only for the energy bands of crystals, but for the eigenvalues of any Hsmiltonisn in quantum mechanics. It is a good example how a relatively abstract mathematical theory - the theory of groups ~ has immediate physical consequences.

The number of distinct irreducible representations of an arbitrary finite group is, b i t e , as are the dimensions of the representations. For the point groups of equivalent directions, the irreducible representations can be only 1-, 2- and 3-dimensional; l-dimensional for small groups, 2-dimensional only for groups that are 'not too small', and 3-dimcnsional only for the largest point groups, specifically for Oh(m3m), 0(43m), Th and Td(332) of the cubic crystal system.

To illustrate these somewhat abstract statements and to prepare for the discussion of actual band structures in section 2.8, we consider some s p a i d k-vectors of the first BZ of the fcc lattice. At the center r, the small point group is identical with the full point group P for crystals of diamond structure, i.e. In the context of energy band theory, the irre- ducible representations of the small point groups are denuted by the same capital greek or latin letters which stand for the k-vectors, supplemented by subscripts. For the I?-point and diamond structure, the irreducible repre- sentations are denoted by rl, Tz, I112, r25. r15, and I?;, I?$, This somewhat strange indexing refers back to the so-call& compatibility rela-

with o h .

I&,

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2.5. Band structure 119

tions between different representations (see Appendix A for more detail). The primes indicate that the involved representations differ from the un- primed ones only by a minus sign for the inversion matrix. In Table 2.5, all irreducible representations of the point group o h , which is the small point group of r for crystals with the diamond structure, are listed. In addition to the symbols of the representations, their dimensions are given. The irre- ducible representations of the small point groups Pk of the symmetry points A, X , A and L for crystals having the diamond structure are also indicated in Table 2.5. According to this table, one has both 1-, 2- and %fold degen- erate bands at the center of the first BZ, while at all other points only 1- and 2-fold degeneracies occur for crystals which have diamond structure. At X there are 2-fold degenerate bands exclusively. This peculiarity is due to the fact that the space group of the diamond structure is non-symmorphic. In this case, the irreducible representations of the small point group of X involve a factor system, which means that multiplication of two representa- tion matrices yields the matrix of the product element, save only for a scalar factor (see Appendix A). In such circumstances, it is understandable that 1-dimensional representations may not be possible.

The degeneracy at I' may be compared with that of a free atom. In the latter case the degeneracy is likewise a consequence of rotation and reflec- tion symmetries. However, arbitrary rotations and reflections are possible in this case, i.e. the point group contains an infinite number of elements. The dimensions of the irreducible representations of this group are determined by the angular momentum quantum number I , where 1 can take the values 0,1,2,. . . , 00. Amounting to 21 + 1, all odd numbers are possible as dimen- sions of irreducible representations, and hence also as the multiplicities of energy eigenvalues. In the case of the hydrogen atom, the symmetry-related degeneracy is still not the full degeneracy. In addition, one has an acciden- tal degeneracy caused by the particular shape of the Coulomb potential. In this case the energy eigenvalues differ only for different principal quantum numbers n. Thus all eigenstates corresponding to a given n, with their vari- ous 1-values 0, 1 , 2 , . . . , n - 1, have the same energy. The degeneracy is thus n2-fold for the hydrogen atom.

2.5.3

The degeneracy of energy bands at symmetry points of the first BZ which we treated in the preceding subsection, is a direct consequence of the symmetry of these points. We proceed now to another consequence of this symmetry, which will lead us to the concept of critical points and effective masses.

If a particular symmetry point ko of the first BZ is transformed into itself under the action of a point symmetry operation a, then points ko + 6k in the vicinity of ko are necessarily transformed into points close to ko. If,

Critical points and effective masses

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120 Chapter 2. Electronic structure of ideal crystals

for example, a i s a reflection with respect to u plane normal to the x-axis, then points of the form ko + 6k e, with ex as unit vector in x-direction, will transform into ko - 6k . ex. Because of the symmetry relation (2.191) for energy bands, E,(k) will have the same value before and afler the reflection. From this it follows that the first derivative of the function EV(k) with respect to k, must vanish at the mirror plane. If ko also simultazlcously lies on a second mirror plane perpendicular to the first, e.g., one perpendicular to the y-axis, thrrr the b,-derivativp of f!!,(k) will also be zero. If ko i s even more symmetric and involves a third mirror plane normal to the z-axis, then the derivative with respcrt to k, also has to vanish. What is being demonstrated here using the example of mirror planes has a general significance: some or all of the first derivatives of the energy band functions E,(k) with respect to the three wavevwtor components vanish at symmetry points, depending on the degree of symmetry. With some ambiguity of expression, we may say that at symmetry centers all three fiist clerivativcs are zero, at symmetry lines two vanish, namely those with respect to the normal plane, and on symmetry planes one derivative vanishes, nainrly the one in the normal direction. On a symmetry line there are also often special points, wheie, for reasons that have nothing to do with symmetry, the remaining first derivative parallel to thv line also vanishes.

The first derivatives of the function Ey(k) with respect to k have a direct physical meaning. One can easily prove (and we will do so in section 3.3) that they are, apart from a constant factor, equal to the expectation value of the momentum operator p in a Bloch Btate, thus

(2.193)

If VkE,(k) vanishes, then the average momentum in the corresponding Bloc11 state is also zero.

These are occasions which warrant special consideration and examination of points the first B Z in which all first derivatives of E,(k) vanish simulta- neously. These points are called crztical points. We will later see that the so-called denszty of +da tw possesses singilnrities at energies whose constant energy surfaces contain critical points. This also explains the name of these points. The critical points h a w a close connection with symmetry points, without, however, being generally identical to them. Critical points k, are defined formally by the equation

(2.194)

With the requirement that E,(k) be a regular function at k, one may approximate it in the neighborhood of k, by a Taylor expansion to second

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2.5. Band structure 121

order

Here, we introduce the components M;:g of a second rank tensor M L 1 defined by the equation

(2.196)

IJsing this tensor, the expansion (2.195) of E,(k) may be written in the form

(2.197)

Thc tensor components cnter this expression in a manner similar to the way the reciprocal mass rn-l enters the energy dispersion relation of a free electron. Actually one can obtain the expression (2.154) for the energy of a free electron from (2.197) by Erst taking the energy zero at Eu(kc) which is unimportant, and then by substituting

(2.198)

The components of the invrrsr tensor M , of M;’ accordingly have the di- mension of a mass. One therefore calls Mu the effectzve mas8 tensor. The term ‘effective’ refers to the fact that the electrons of a crystal are not free elcctrons but arc subject to the influencc of thc periodic potential. The presence of this potential mandates that the wavevector of the free electron be repIared by the quasi-wavevwtot k, and that the quadratic dppendenre of the energy on k become a more complicated dependence. Only in the vicinity of critical points does the quadratic drprndenre persist, but with coefficients M;$, which differ from those of the flee electton. They are eflectztw coefficients which involvc the effrct of the periodical potmtial.

In contrast to the free electron case, where the coeficients form a multiple rn-l of the unit tensor 6,p, i.e. a scalar quantity, for the case of an electron in a crystal they represent u tensor ML$ with non-vanishing off-diagonal elements and general diagonal elements. An analogous statement holds for the inverse, as the cffective mass tensor M,. Thus, in general, the effective mass is a tensorial quantily. Therein the crystal potential manifests itself, generally generating different mass values in different spatial directions. In special c‘ase‘s one ran approximate the effertive mass by a scalar quantity rn;. Then, the difference between the mass of the free electron and that of the crystal electron reduces to the different magnitudes of the two masses.

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z 22 Chapter 2. Electronic structure of id& crystals

The mathematical description of effective mass can be simpl%d by exploit- ing the symmetry of the tensor Mv:, with rcsppct to an exchange of its indices a and 3, i.e. by using the relation

(2.199)

This propat,y follows directly from the defining equation (2 .196) for M&. As is well-known from linear algebra, such a real symmetric tensor cau be brought lo diagonal form by a coordinate transformation, in our case of the components of the k-vector, to the principal axis system. With ms' being the principal diagonal elements of M l l in this system, one has

The m;;' are real. Ifn general, they can take both positive and negative values. The same applies to thc inverse quantities, the effective masses rnk themselves. Taking k, as componenls of k in the principal axis system, we have

(2.201)

If all three effective masses are positive, the energy function Ew(k) has a minimum at the critical point kc, and if all three are negative it is a maxi- mum. If the three effective masses do not all have the same sign, then E, (k ) has a saddle point.

lies on a 4- fold symmetry axis of the first 32. and that this is simultaneously one of the principal axes. Then, for symmetry reasons, the two principal tensor elements normal to this axis must be equal. With LY = 3 for the symmetrical principal axis we thus have

Consider the particular case in which the critical point

mC1 = mt2 (2.202)

The corresponding element for the symmetrical principal axis m:3 E m:,, in general differs from mZI. If k,, however, lies on a symmetry center of the first B Z , then all three elements are in fact identical, i.e.

(2 203)

For the frequentIy observed case of a symmetry point at the center of the first BZ! i.e. with k, = 0, one gets the simple dispersion law

F12

2m; E,(k) = Ev(0) + -k2. (2.204)

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2.5. Band structure 123

This differs from that of a free electron only in that the effective mass rn; appears instead of the free mass and that the energy at k = 0 assumes a value E,(O) which depends on the band index and, therefore, cannot be set equal to zero for all bands by changing the zero of energy.

In the following subsection we will again deal with band structure in its general form E,(k). We will introduce the so called density of states that encompasses essential information on band structure in just one function of energy. Often, a knowledge of the density of states is enough to calculate important macroscopic properties of semiconductor crystals.

2.5.4 Density of states

Definition

We first consider an arbitrary infinite oneelectron system. Its Hamiltonian will be denoted by H , its eigenstates (disregarding spin) by li), and its energy eigenvalues by Ei, i = 1 ,2 , . . . , 00. Periodicity of the eigenstates is assumed with respect to a periodicity region of volume 51. The quantity

2 p ( E ) = - C 6 ( E - Ei).

O i (2.205)

is called the density of states (DOS) of the system. We will show that this designation is justified, i.e., that p(E) does represent the number of one particle states of energy E per unit energy and volume. To this end we integrate equation (2.205) over a small energy interval E < E' < E + A E , obtaining

2 E+AE P(E)AE = FL dE'6(E' - Ei) . (2.206)

The integral on the right hand side yields

1 for E < E i < E + A E

0 otherwise . (2.207) dE'6(E' - Ei) = JEE+AE

Thus, each state i whose energy lies between E and E + A E contributes '1' to the sum on i, with no contributions from all other states. The sum in (2.206) equals, therefore, the number Az(E) of states having energy in this interval. Considering spin, one has

(2.208) 2 51

~ ( E ) A E = -Az(E) ,

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124 Chapter 2. Electronic structure of idcal crystals

confirming the designation of p(E) as DOS. The L)OS expression (‘2.205) may br written in a more compact form

which will be used in Chapter 3 in the derivation of important general results. Noting that the 6-function 6 ( & ) is the imaginary part of -(1/7r)l/(E + ie) with t a small positive number, one can easily prove that p ( E ) of (2.205), except for a factor R, equals

p ( E ) = --Im Tr 7T

(2.209)

where trace sytxho1 ‘Tr’ denotes the sum over all diagonal elements with respect to any basis set in Hihert space. If this set is identified with thr eigmstatcs 12) of ff, expression (2.209) transforms immediately into (2.205).

We now consider the DOS p ( E ) defined by (2.205) for an ideal crystal. The cigenstates li) are Rloch states Ivk), in this case, and the DOS p ( F ) therefore reads

2 p ( E ) - - C 6 ( E - ’ uk (2.210)

The k-sum is over all points of the finely meshed net of Figure 2.4 which lie within the first B Z . Because of the fineness of this network, the result of the summation is almost identical to that of an integration. The substitution of the s m by an integral is done in accordance with the prescription

c...= -1 R d3k ... k 8 T 3

(2.211)

The factor f l / 8 m 3 in front of the integral ocrurs because the volume of a mesh element is the G3-th part of the volume 8n3/Slo of a primitive unit cell of the reciproral lattice, whirh is 87r3//n. One must divide by this volume in writing the sum as an integral. The density of states follows as

(2.212)

which is independent of the volume R of the system. The density of states restricted to the u-th energy band, p u ( E ) , is defined by the expression

(2.213)

and it differs from zero only for energy values for which the argument of the 6-function may brrome zero, i.r. for energies in the v-th band. Summing over d l bands one again obtains the entire density of states

(2.2 14)

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2.5. Rand structure 125

Expression (2.213) for p , ( E ) may be transformed into a surface integral over the constant energy surface &(k) -= E using the rules of calculation for the &function, thus obtaining

(2.215)

with df as element of the constant energy surface E,(k) -= E. The inteErand l/IVkF,’,(k)l is, apart from a factor TL, the reciprocal absolute value of the group velocity of an electron in the Bloch state Ivkq). The smaller this velocity is at a particular k-point on the constant energy surface or, in other words, the longer an electron stays at it, the larger is its contribution to the density of states. If (VkE,(k)l - 0, i.e. at a critical point, the contribution is, formally, infinitely large. From this it follows that for energy values E whose iso-energy surfaces contain critic81 points, singularities of the density of states as a function of energy are to be expected. They are called van Hove singularitiee.

Parabolic energy bands

We aasunie an isotropic parabolic energy band, i.e. we suppose that the function E,(k) is given by the expression (2.201) with the sainc effective mass rn; along all three main axes. The critical point k, is taken as the origin 0. If we further replace E,(k,) by E N , then the expression for &(k) becomes

(2.2 16)

We use this to calculate the density of states p , ( E ) from equation (‘2.213). The effective mass rn; is taken to be positive, i.e., E,(k) should have a minimum at k = 0. The use of the dispersion law (2.216) i s only justified within its limits of validity, i.e. for energy values E,(k) close to the minimum E d . Only for these E-values can the density of states as calculated by means of (2.216), be relied upon. For such energies it so happens that the value of p , (E) does not depend on the upper limit of integration. We transfer this limit, for simplicity, from the edge of lhe first B Z to infinity, obtaining

(2.2 17)

In this integral, we transform to new variables x, y, z To start we, execute the integration over z using the rule

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126 Chapter 2. Electronic structure of ideal crystals

1 2zo

E - EA - x 2 - y2 - z2) = - [S(z - ZO) + 6 ( z + ZO)] (2.2

with

Zo = ,/E - E , , ~ - x 2 - y2 f o r E - E& > 0. (2.2

It follows that

where

1 f o r E > Euo

0 f o r E < Euo B(E - Euo) = (2.221)

is the Heaviside unit step function. If one considers, instead of an isotropic parabolic band, an anisotropic one with three different effective masses m:l, m:2, m:3 corresponding to the principal axes of the effective mass ten- sor, then the density of states is again given by expression (2.220), but m: must be replaced as follows

(2.222)

by the so-called density-of-states-mass m b . This may be seen immediately if one reviews the calculation for the isotropic case. The anisotropic effec- tive mass involves a change only upon substitution of the variables for the components of the k-vector - the factors for the 3 components are different and lead to the modification indicated in equation (2.222).

According to expression (2.220) the density of states of a parabolic band with m*, > 0 exhibits square root-like behavior at the lower band edge, and continues following a square root law up to infinitely large energies. The latter property is a consequence of our untenable assumption of parabolic dispersion up to the upper band edge and the replacement of this edge by the point at infinity. In reality the band edge lies at finite energies, and the density of states again falls off to 0 there. It has, qualitatively, the shape shown by the dashed curve of Figure 2.14. The sudden square root-like increase of the DOS at the band edge reflects the fact that the function p(E) has singularities at these energies - the first derivative with respect to E is 0 if one approaches the edge from the forbidden zone, and it is +CXJ if one approaches it from the interior of the band. This is one of the already mentioned van Hove singularities of the DOS. These singularities occur not only at energies where a particular band E,(k) has a minimum, as in the

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2.5. Band structure 127

Figure 2.14: DOS of a parabolic 5 - I I I I

imum (solid curve). The dashed curve isotropic band in the vicinity of its min-

shows the DOS of a more realistic band. - 4 -

< c = 3- I

vl 0 0

2 -

1- -

0 -1 0 1 2 3 4

Energy lorb. unlts)

case considered here, but at all energies corresponding to critical points of the energy bands, thus also at maxima and saddle points.

Counting states: 3D, 2D, and 1D density of states

The square root enerw dependence of the density of states in the case of a parabolic dispersion law may also be demonstrated in the following, more vivid way (also see Figure 2.15). For simplicity we set E,o = 0. and omit factors which do not depend on k or E. The n m b r r AZ of bmd states with energy between E and E .f A E is the number of different k-points of the finely meshed net which yield energy values in t,his interval. These k- values lie in a thin spherical shell in k-space [see Figure 2.15) which, because E o(. k2, has radius k~ 6: &. and thickness Ak which, since Ah' CK k&k, is given by

(2.223) 1 Ak 0; -AE. fi

For the volume AV of this shell, it follows that

AV m k L A k cc A A E . (2.224)

Since the density of the finely meshed net of k-points of the first BZ i s the same everywhere, the number of k-points in the spherical shell is propor- tional to its volume AV. Hence AZ cy v%AE and

p ( E ) o(. 4%. (2 2 2 5)

Such considerations can easily be applied to energy bands in 2-dimensiouel (2D) and 1-dimensional (1D) k-spaces. Such k-Bpaces have physical (as well as mathematical) meaning, in particular for electron systems whose free

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128 Chapter 2. Electronic structure of ideal crystals

b C

Figure 2.15: Counting states in (a) 3D, (b) 2D, and (c) 1D k-space.

motion is confined to one or two dimensions of 3-dimensional space. We wiU encounter examples of such systems in Chapter 3 - surfaces and quantum wells for the 2D-case, and electrons in a homogeneous magnetic fieId for the 1D-case. For 2 0 k-space, the volume of the spherical shell is replaced by the area AF of a circular ring, such that A F a k E A k LX A E and

p ( E ) 3: const. (2.226)

The density of states of a 2D-electron system is therefore independent of E. In the case of a ID k-space: AV is replaced by the length Ak of the k-interval itself. Thus,

1 p ( E ) DC -AE. f i

(2.227)

2.5.5 Spin

Thus far, spin has been omitted from our discussion of the general properties of stationary oneelectron states of crystals. It turns out. however, that spin and spin-orbit interaction play an important role in most semiconductors. Therefore, the question arises as to how the results derived above for scalar wavefunctions change when electron states are no longer scalar but spinor functions and the spin-orbit interaction H , is taken into account.

The starting point to address this question is the one-electron Schrodinger equation (2.58) with spin, which may be written in the following short form

(2.228)

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2.5. Band structure 129

The first point we will consider is that of spatial symmetry in the presence of spin.

Spatial symmetry

Inspecting the explicit form of the spin-orbit interaction operator ( b e y for- mula ( 2 . 5 6 ) ) , one readily recognizes that TKso has thc full symrwtry of the crystal. Thus it colrimutes with all elements 9 of the space group. Since the same holds for the spin free part H of the total Hamiltonian !i + one has

I f f + H,. g] = 0 . (2.229)

To exploit this symmetry property of thv total Hamiltonian w e mu61 know how thc components p(x, s) of a spinor transforin i d e r thr operations g of the space group. Considering translations first. we have

In t.he absence of spin, the cornmulivity of the HanliltoIlittn with translations gave rise to the Bloch theorrm. In t.he same way that this theorem was proved without spin above. its validity may be also demonstratd here - in the pwsrnrr of spin. It. stdates that the solutions {p~k(x, 4). y ~ ( x , f)} of t.hr Schrodingeer equation (2.228) for the eigenvalue E A ( ~ ) can be chosen in the form of Bloch type spinor functions with a particiilar quasi-wavevecbor k as

where I ~ X ~ [ X . zti) are the spin-dependent Bloch factors. If k is restricted to the first B Z . then E x ( k ) rrprwents a continuous function of k. Thc band index X here refers to both the state of the orbital motion and also to spin state.

Second, we consider point symmetry operations u, whose action in tram- forming t,hc components of a spinor is discussed in Appendix -4. The results of Appendix A are applied here in the form

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130 Chapter 2. Electronic structure of ideal crystals

with

In this, $, B and p are the Euler angles of the orthogonal transformation a. If the spinor is spatially constant, the transformation just reduces to multi- plication by the matrix D r ( a ) . The set of all orthogonal transforniations cy.

forins an (infinite) group. Through relation (2.233) this group is assigned a matrix group D i ( a ) . A peculiarity of thm mapping is that it is not unique

- a change of the angles cp or $J in (2.233) by 27r leads to a change of the sign of the associated matrix, even though this signifies the identity trans- formation. One may say that under a rotation through 27r a spinor does not transform into itself, but into its negative. This gives rise to multiplication rules for the representation matrices which deviate from those of the group itself. Thus the representations of the full orthogonal group in the space of two-component spinors are not representations in the ordinary sense, but in a generalized sense. They form projective representat iom with a special factor system (see Appendix A), or, in short, spinor representatiom. By means of the so-called double group, which includes each element, of the fill1 orthogonal group twice, once in the original form, and once multiplied with a rotation through 27r, one may trace back the spinor representations to or- dinary representations. The spinor representations are those representations of the double group which do not occur among the ordinary representations of the single g~oup (for the derivation of this result see Appendix A).

Up to this point we have only given attention to the transformations of spinors in spin space. The states of electrons are described by spinor fields having positional dependence. These give rise to spinor representations of the (full) orthogonal group which differ, in general, from D i , The total- ity of spinor representations may be obtained from the ordinary irreducible representations V, and the particular spinor representation D1 of the full orthogonal group. Indeed, if a spinor field (P,(x, s), with s = fk, transforms according to a certain ordinary representation D, in coordinete space, then its transformation in coordinate ~ spin space is governed by the product representation Vi x Vv. If, here, D, encompasses all vcctor representations, then all spinor representations are obtained.

Using the concept of spinor representations, the already discussed con- nection between the eigenfunctions of the crystal Hamiltonian for a given eigenvalue and the irreducible representations of its symmetry group may be generalized in the following way:

The spinor eigenfunctions of the crystal Hamil tonian W + H , having the same energy eigenvalue span a representatdon space of a n irreducible spinor

2

1

2

2

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2.5. Band structure 13 1

Table 2.6: Irreducible spinar representations (notations and dimensions) of the small point groups of symmetry points and lines of the firat B Z of the f ix lattice for crystala with the diamond structure.

representation of the full symmetry group of H + H,, (namely, the space group of the crystal).

The product representations Dr. x Vv, taken as representations of the sman point groups of a particular wavevector rather than for the full orthogonal group, are generally reducible. This means that bands which are degener- ate without spin by reason of spatial symmetry, may split if spin is taken into account. The magnitude of the energy splitting depends on the spin- orbit interaction, without this interaction, degeneracy persists, but as an accidental rather than a symmetry induced degeneracy. The following ex- ample illustrates the removal of degeneracy. We consider the valence band

of zincblende type semiconductors which is triply degenerate at I'. The product representation in this case is Di x 1-15. It decomposes into the two irreducible spinor representations ra and r7 of the point group T d of I', where rs is of dimension 4 and r7 of dimensiun 2. This means that, due to the spin-orbit interaction, the triply degenerate r15-valence band without spin splits into the $-fold degenerate rs-band and the 2-fold degenerate r7- band, accounting for the effects of spin. In Table 2.6 the irreducible spinor representations are shown for some symmetry paints of the first BZ of dia- mond type semiconductors. A systematic treatment of these representations is given in Appendix A.

a

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1 32 Chapter 2. Electronic structure of ideal crystals

Time reversal symmetry

As the consequences of spatial symmetry are altered by the phenomenology of spin, so is the action of time reversal symmetry changed by spin. First,we will show that the operator K which transforms a spinor {cpx(x, $), cp~(x, $)} into the spinor K{cpx(x, i), cpx(x, i)} at reversed time, is given by the rela- tion

-

(2.234)

To prove this assertion, one executes the time reversal operation on the spin-orbit interaction operator H,, of equation (2.56) explicitly. The vector a' then transforms into -a*, and p into -p, so that the net change in Hso is the replacement of d by a'*. Employing the operator K , on the other hand, the effect of timc reversal on H , , may be expressed in terms of the similarity transformed operator KH,,K-'. For the two expressions to be identical, K must have the form given above in equation (2.234), apart from a phase - factor which remains undetermined. The wavefunction {cpx(x, i), cpx(x, $)} must be replaced by K{cpx(x, i), cpx(x, $)} for the Schrodinger equation (2.180) to remain valid.

The question, under what conditions time reversal symmetry includes degeneracy of eigenfunctions not accounted for by spatial symmetry alone, is treated in Appendix A in full generality. Here we deal only with a specjal case. We consider a non-symmetric wavevector k. Let {cpxk(x, i), cp~(x, i)} be an eigenfunction with energy Ex(k). If the point group of directions contains the inversion, then the eigenfunction {cpx-k(x, i), cpx-k(x, i)} cor- responds to the same eigenvalue Ex(k) = Ex(-k). The time reverse of the Bloch function {cpx-k(x, i), cpx-k(x, $)} likewise has energy Ex(k) and wavevector k. It is linearly independent of {(Px~(x, i), cpxk(x, $)}, since

-

-

- -

as one may easily show by evaluating the scalar product (here, one also has to sum over the spin coordinate s ) . Thus, two linearly independent eigenstates of the same energy have the wavevector k. Since k was chosen arbitrarily, it follows that, due to time reversal symmetry, all bands of inversion-symmetric crystals are at least twofold degenerate.

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2.5. Band structure 133

2.5.6 Calculational methods for band structure determina- tion

There are many methods, quasi-analytic and numerical, for calculating the band structure of crystals. In the following we will give an overview.

In band structure calculations one is confronted with two problems, firstly with the formulation of the one-elcctron Schrodingcr equation, i.c. with the determination of the effective periodic oneelectron potential V(x) of the crystal, and secondly with finding the eigenvalues and eigenfunctions of that equation. The various methods of band structure calculation differ in the manner in which thcsc two problcms are resolved. Hcre we will deal mainly with the second problem, i.e. with the solution of the Schrodinger equation whose potential is known. The first problem, the determination of thc cffectivc oncclcctron potential V(x) , has in principle brcn solved in section 2.2, where we dealt with the one-electron approximation for the many-electron system of a crystal. Here, we only address some further dctails of this problem.

Determination of the effective one-electron potential

The simplcst way of dealing with thc oneelectron potential V(x) is to treat its matrix elements with respect to a particular basis set as empirical param- eters rather than integrals to be evaluated from a knowledge of the particular profile of V(x) . Having done that, one has an empzmcal methodof band struc- ture calculation. We will provide examples below. If one wants to forego, however, the aid of empirical data and calculate the band structure from first pnnczples, this evasion is unacceptable. The effective one-electron potential V(x) of the crystal must then be known as a function of x. Methods which proceed in this way are referred to as ab znztzo methods.

Within the frozen core approximation, V(x) describes three interactions of a valence electron. The f is t is the Coulomb interaction with the atomic nuclei - this part is not problematic. The second is interaction with the (frozen) core electrons - the energy levels and eigenfunctions of the core states of the free atoms have to be determined in advance for us to be able to write down this potential contribution explicitly. It includes the Hartree and exchange potentials as well as the correlation potential of the valence electron-core electron interaction. These potentials are described by ex- pressions which are completely analogous to those for the valence electron- valence electron interaction derived in section 2.2. The third part of the periodic crystal potential accounts for the effect of the remaining valence electrons on the valence electron specifically considered. It depends on the very same eigenfunctions that are to be calculated. Because of this, the calculation of the band structure must be done self-consistently. In many

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134 Chapter 2. Electronic structure of ideal crystaIs

cases, the Hartree and HartreeFock approximations fail to give satisfying results, so that correlation effects must also be inchided. As indicated in section 2.2. one way this can be done is by means of the dematy finctional theory in its local approximation (local density approximation or LDA). The LDA-method yields very good results as far as valence bands are concerned, but it fails if applied to the conduction bands. In particular, as already mentioned in section 2.2, it does not reproduce the correct d u e of the fun- damental energy gap. A procedure which avoids this failure is the Gmen’s functzon method mentioned in section 2.2. This approach is now being ap- plied more frequently under the name puma-particle method (see, e.g., Bech- stedt, 1992). Within this framework the self-eneqy operator is often taken in the so-called GW -approximation [‘G’ stands for ‘Green’s function’ and ‘W’ for the Coulomb potential).

We add two further remarks, related to the potential of the atomic cores. Firstly, only for relatively light atoms such as C or Si, may spin and the spin-orbit interaction of the valence electrons be ignored. For the heavier atoms, such as Ge, this interaction is essential and must be incorporated in the effective one-electron potential. Secondly, the decomposition of the total electron system into valence and core electrons need not to be made in the literal sense of these terms. What counts is which electrons are frozen in their atomic states and, therefore, need not be treated self-consistently, and which electrons must be. The latter are ‘valence electrons’ in a more general sense. In 111-V semiconductors, for example, they can also include d-electrons which, of course, do not belong to the valence shell of one of the two elements involved. In the extreme case, all electrons are treated as ‘valence electrons’. Then one has the so-called call electron problem. The solution of this problem involves an extraordinary large numerical effort, so that such all-electron band structure calculations have been performed in ody a few cases to date. From the physical point of view, they are the most satisfying. With increasing computing power they will become ever more important.

Solution of the Schrodinger equation

The solution procedures can be divided into three groups, firstly! the matrix methods, secondly, the cell methods, and thirdly, the muffin-tin methods.

Matrix methods

In the application of matrix methods one represents Bloch type eigenfunc- tions of a given quasi-wavevector k in a particular basis set consisting of a finite number of functions of the Bloch type. The Hamiltonian of the crystal is thereby represented by a k-dependent complex Hermitian matrix which

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2.5. Rand structure 135

has as many columns and rows as the basis set has functions per k-vector. The calculation of band structure is thus traced back to the determination of the eigenvalues and eigenvectors of a k-dependent Hamiltonian matrix.

The basis sets employed differ by the number and kinds of functions they contain. One would like to manage with the fewest possible functions, to minimize the numerical effort. The price for this is a loss of precision, because with fewer basis vectors the eigenfunctions are necessarily approxi- mated more crudely than with a larger number - in the extreme, one needs an inh i t e number of functions. For a fixed number of basis functions the level of precision achieved is higher for basis functions which are better ad- justed to represent the eigenstates. The following basis functions prove to be of practical use:

-Plane waves Iki-K) with k being a vector of the first BZ and K a reciprocal lattice vector. This constitutes a generalization of the nearly free electron approximation.

~ Bloch sums of atomic orbitals, also called LCAO's (Lznear Combznatzons of Atomzc Orbatuls), or of other localized functions. This includes the tight binding met hod.

~ Bloch functions with Bloch factors for a special wavevector, referred to as Luttznger-Kohn funetzons. The so-called k . p-method uses these functions.

In some circumstances, the tight binding and k pmethods may be used to derive analytic expressions for the k-dispersion of energy bands. Such expressions are extremely useful to achieve a physical understanding of band structure. Therefore, we treat these two methods in greater detail below (sections 2.6 and 2.7).

- Orthogonalized plane waves, called OY W's. An OPW-function IOPW~+K) is obtained from n plane wave Ik+K) by subtracting a certain linear combi- nation of core band eigenfunctions Irk) of the crystal Hamiltonian. Within the frozen core approximation, the core band eigenfunctions (ck) may be taken as Bloch sums of the core states of single atoms. The linear combi- nations to be subtracted are chosen such that the OPW's are orthogonal to all core eigenstates, so that

(2.236)

holds. The OPW's must have the form

lOl 'Wk+~) = Ik. + K) - C(cklk + K ) Ick) (2.237)

in order to satisfy the orthogonality condition (2.236). Making the expan- sion functions orthogonal to the core eigenstates accounts for the fact that

C

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136 Chapter 2. Electronic structure of ideal crystals

eigenstates of the Hamiltonian for different energies are always mutually orthogonal. This means that the sought-after eigenstates of the valence electrons of the crystal for a given quasi-wavevector k must be orthogonal to the core eigenstates having the same quasi-wavevector k. For this reason the OPW's are much better adjusted to represent the eigenfunctions of the valence electrons than are pure plane waves; correspondingly one needs fewer OPW's than plane waves to accurately represent the valence eigenfunctions. A further development of the OPW-method is the pseudopotential method.

Pseudopotential method

The idea underlying the pseudopotential method is to transfer the core state orthogonality term in equation (2.237) from the OPW's to the one-electron Hamiltonian H of the crystal. This transfer is done as follows. Consider the core electrons. which we have hitherto taken jointly with the atomic nuclei to form the cores, to be independent particles, just like the valence electrons. This means that the potential created by the core electrons is no longer included in the core potential Vc, but is added to the eEective o n e electron potential 1'~ + 15-c of the electron-electron interaction. Because of this reinterpretation of the oneelectron Hamiltonian H , its eigenstates now also include core states, for which we have

H i c k ) = E,lck). (2.238)

The valence band eigenvalues E and eigenfunctions ~a similarly satisfy the eigenvalue equation

The expansion of ~ $ m with respect to the OPW's reads

Removal of the core states from the expansion functions mandates changing from the eigenfunctions $a to other functions v p i that have the same expansion coefficients, but plane waves as expansion functions:

(2.241)

Surprisingly, the artificial wavefunctions ~ $ g ~ are in fact eigenfunct ions of a particular Hamiltonian Hps , having the same eigenvalue E to which the eigenfunctions $a of H correspond. In fact, by applying of H to $I% one immediately Ends that

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2.5. Band structure 137

(2.242)

H P S = H + E ( E - E,)lck)(ckl . (2.243) C

The Hamiltonian HPS is a well-defined linear operator, although it is non- local and depends on the eigenvalue E itself. It is called a pseudo-Hamilto- nian The $gk are termed pseudo-wavefunctions. Each eigenvalue of HP" is simultaneously an eigenvalue of H , but the reverse statement does not hold. While core levels E, are eigenvalues of H , they are not also eigenvalues of HPS since the pseudo-wavefunctions of the core states vanish.

The pseudo-Hamiltonian HPS may be written in a form which clarifies its meaning. To begin with one revokes the re-interpretation of the Hamiltonian H , i.e. considers the core electrons no longer as independent particles which enter the effective one-electron potential VH + Vxc of the electron-electron interaction, but includes them again into the atomic cores making them contributors to the core potential V,. Then HPS takes the form

H P s = - + v p2 H + Vxc + Vc + C ( E - E,)lck)(ckl . 2m C

(2.244)

The last two terms in this expression jointly constitute the so-called pseu- dopotential V,p"

(2.245)

The second term on the right is significant only in the core regions. There, it is preferentially positive, indicating that it repels valence and conduction band electrons away from the cores. One can show that it largely compen- sates the variation of the core potential V, in these regions. Because of this, the pseudopotential V,p" is relatively smooth throughout the cores. It can he made even smoother if one exploits a property of V,p" which has not yet heen discussed. We refer to the non-uniqueness of the repulsive part of V,p" in equation (2.245). If there the bra-vectors (E - E,)(ckl in (2.245) are r e placed by completely arbitrary functions (while keeping the ket-vectors Ick) unchanged), the eigenvalues of the pseudo-Schrodinger equation (2.242) re- main the same, only the pseudo-wavefunctions change. This freedom may be used to make the pseudopotential still smoother, and also fulfill other require- ments, such as, for example, norm-conservation of the pseudo-wavefunctions. The smoothness of the pseudoptential makes it possible to restrict the plane- wave expansion (2.241) of the pseudo-wavefunction to terms having small reciprocal lattice vectors K. Consequently, the representation matrix of the pseudo-Hamiltonian HPS is small and easy to diagonalize. This is the reason

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138 Chapter 2. F k t m n i c structure of ideal crystals

that the pseudopotential method is very helpful in calculating valence and conduction band structures of semiconductors.

In order to apply this method, the psendopotential must be known explic- itly, of course, In principle it can be determined from the defining equation (2.245), with core levels and wavefunctions taken from atomic calculations, and with bra-vectors (ckl substituted by appropriate functions. In practical applications, the pspudopotential V,p" i s replaced by approximate expressions which range from empirical local pseudopotentials with adjustable param- eters up to non-local pseudoyotentiah including core states of s-! p- and d-symmetr y.

The pseudopotent,ial method is generally successful if the true valence band eigenfunctions are also sufficiently smooth outside ol the core region. This is the c a ~ e as long as the valence band states are composed mainly of atomic 3- and p-orbitals. If d-orbitals cont,ribute to these states in an essential manner, i.e. if the d-electrons of the atoms are sigmficantly involved in t,he chemical bonding of the cryshl, then the pseudopotential method becomes problematic, because it,s main advantage in having the pseudo- eigenfunctions built up from a relatively small number of plane waves, no longer applies. Thus may occur in trhe rase of TII-V and 11-VI wmpound semiconductors whose cations have flat d-levels as in the case of Zn, for exmaple (see Table 2.2).

A completely different approach to solving the oneelectron Schrodinger equation is taken in the so-called cell methods.

Cell methods

These methods are 3-dimensional generalizations of the method of match- ing conditions usually employed in solving thr Schrodinger equation for the square well potential and other 1-dimensional potentials. In cell methods? one first determines linearly independent solutiona of the Schrodinger equa- tion within a primitive unit cell for arbitrary energies. Forming Bloch sums with them, one constructs the solution for the total crystal. The require ments that these functions, and their first derivatives, be continuous at the boundaries of the unit cells determines the energy eigenvalues and eigenfunc- tiuns. This method suffers from the arbitrariness of t,lie choice of the unit cells and the difficulty of satiseying the boundary conditions over the whole surface, i.s. at an infinite number of points. The most natural choice of unit cell is the Wigner-Seitz cell. A further development, of cell methods lies, in a sense, in the mufin-tin methods.

Muf&n-tin methods

In the present method one delimits spheres around the atoms, and leaves some empty space around them. The crystal then looks something like

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2.5. Band structure 139

a muffin-tin, whence thc method’s narnp. Within the spheres one uses the spherimlly syuunelric potentials of the atoms, trnd in the surrounding regions one takes the potential to be uniform.

In a special mufin-tin method, t,he augmented plane wave (AP W)-method, the solutions of the Schriidinger equation within the spheres are expanded with respect to angular momentum eigsnfunctians. The radial parts of the expansion are determined by the radial Schrijdinger equation. This is solved for the various angular momentum quantum numbers numerically. Then, the still unknown expansion coefficients are determined by the requirement that t,he soliit,ions within the spheres mist join continuausly with the solutions outside. The latter are plane waves of some wavevector k + K. The func- tions constructed in this way are called augmented plane wawes {APW’s). They are, of course not eigenfunctions of the oneelectron Schrdinger equa- tion of the crystal, but may be taken as a basis for them. In contradistinction to bhe basis functions used in matrix methods, the APW’Y still depend on the unknown energy eigenvalues. An rigenfunction expansion with respect to APW’s leads, as in matrix methods, l o B homogenwus set of equations €or the expansion coefficients. but the matrix elements are, unlike the Hamil- tonian in matrix nielhoads, funcbiom of energy. However, the A PW-matrices are in general smaller, as a consequence of the use of better adjusted ba- sis functions. Often, a linearized energy dependence of the matrix element,s yields useful results in the linearized APW o r LAPW method. If, in con- structing the APW’s, Gaussian functions are used instead of the angular momentum eigenfunctions, one speaks of rnuff-tin orbitals (MTO ’s), upon which the MTO and LMTO methods rest.

The different APW’s are indexed by reciprocal lattice vectors K, in addition to angular momentum quantum numbers. In another muffin-tin method, called the Komnga-Kohn-Rostoker (KKR)-metho$, which takes ad- vantage of the formal scattering theory of quantum mechanics in the Green‘s function formulation, the expansion functions are also angular momentum eigenfunctions between the spheres.

In band structure calculations for semiconductors, some of the methods listed above are used more frequently than others. Among the ab initio procedures, the pseudopotentid method combined with density functional theory in its local approximation, and lately with the Greens’s function method, is particularly important. In addition, also the APW and LMTO methods are used. mainly in their linearized forms. Of practical importance among the empirical procedures are, above dl , the empirical versions of the tight binding and of the pseudopotential methods.

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140 Chapter 2. Electronic structure of ideal crystals

2.6 Tight binding approximation

In the nearly-free-electron approximation, the eigenstates of the one-electron Hamiltonian of a crystal are represented by a superposition of plane waves. The non-diagonal matrix elements of the periodic potential with respect to these functions are treated as a small perturbation. Thus the eigenstates are weakly disturbed plane waves in which lhe electrons are spread out almost uniformly over the whole crystal. Such a distribution might be valid for drctrons of tho conduction band, biit it does not correspond to the reality one should expect for valence electrons if one considers the crystal to be formed from previously isolated atoms. In free atoms, the valence electrons are l o c a l i d at their respeclive atomic cores. Although this localization must be partially breached in the crystal, in order for chemical bonding to take place, an almost complete deloralization such as is assumed in the nearly- free-electron approximation is not to be expected. This suggests a more appropriate approximation which takcs atomic wavefunctions as the basis set and treats the non-diagonal elements with respect to these functions as small perturbations. This approximation is callcd a tzght b m d m g (TB) appro.mmntzon The errors of this approximation are expected to be small if the valence electrons of the crystal are well localized at the atoms. The approximation of nearly free electrons will work very poorly in this case, while it gives good results if the electrons are weakly localized, i.e. when the tight binding approximation is not applicable. In this sense the two approximations are complementary. Results which are obtained from both these approximations may be considered to be independent of any particular approximation, i.e. to be exact. The term ‘exact’ here means within the framework of the simplifications made earlier. One of these simplifications, the approximation of frozen cores introduced in section 2.1, is particularly important because it allows us to deal with only the valence electrons of the free atoms, while the core electrons are incorporated in the atomic cores.

In this section we will develop the basic principles of the TB approxima- tion. These principles will be applied to semiconductors of the diamond and zincblende type, and it will be shown that the TB approximation not only is capable of explaining the valence band structure of these crystals, but it also provides insight into their chemical bonding and atomic structures.

2.6.1 Fundamentals

Atomic Orbitals

The basis functions of the TB approximation are the one-particle eigenfunc- tions of the valence electrons of the free atoms, more strictly, of the atoms composing the crystal under consideration. These eigenfunctions are called

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2.6. Tight binding approximation 141

atomic oibatnls. The spinor character of the orbitals may be taken into ac- count, but we will omit it here brevity. The mobt important property of the orhit&, which turns out to be decisive for the TB approximation, is their spatial symmetry. The latter i s determined by the symmetry of the Hartree oi HartrecFock potentials of the atomic cores. These potentials are isotropic if all core shells are fully populated by electrons. In the case of atoms forming diamond and zincblende type semiconductors, this condition is always satisfirul. Thus we may assume isotropic core potentials, and the atomic orbitals of given energy eigcnvalues form basis sets of irrediicible rep- resentations of the full orthogonal symmetry group. These representations are characterizd by an angular momentum quantum number 1 which may assume all non-negative integral values. The irreducible representation of a given quantum number 1 is (21 + 1)-fold degenerate. The 21 + 1 basis func- tions are distinguished by the magnetic quantum niimbcr 7n which takes all integei values between - 1 and + l . The energy spectrum of the free atom is degenerate with respect to m. Foi the hydrogen atom there is also a d c generacy with respect to 1. This is not due to the spatial symmetry of the potcntial but to its purr Coulomb form. Here this additional degeneracy may not be assumed. For each value ol lhe quantum number 1 one has an infinite set of different energy eigenvalues. These are distinguished by the mdin quantum number n which may take all intcgcr values horn 1 + 1 to 00.

In this way the energy levels E,l of an atom depend on thc two yuanturn numbers r~ and 1 , and the corresponding eigenfunctions dnlm(x) depend on thrw quantum numbers r ~ , I , m. 'l'ht. eigcnfunctions &,jrn(x) can be written as products of the radial wavefunctions &(I x I) and spherical harmonics Km(x l I x I),

(2.246)

Here, it is assumed that the tttorriir core is located iit the coordinate origin x = 0. A wavefunction with the quantum number I = 0 is called an s-orbital, one with 1 = 1 a p-orbital, with 1 - 2 a d-orbital etc.

In order to represent the eigenstates of the valence electronb: of a crystal, one needs, rigorously speaking, nll orbitals of the cores of its atoms since only the totality of all orbitals forms a complete basis set in Hilbert space. However, not all of these contribute in an essential manner. 'l'he largest contributions are to be expccted from orbitals forming the valence shells of the free atoms. Within the TB approximation one takes only these orbitals into account, This corresponds to a perturbation-theoretic treatment of the Hamiltonian matrix with respect to the atomic orbital basis; only matrix clcmerits between valence orbitals are considered while those involving other orbitals are neglected. For the elemental semiconductors of the fourth group of the periodic table the valence shell orbitals are formed by the four ns-

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142 Chapter 2. Electronic structure of ided crystals

and np-states with n = 2 for C,n = 3 for Si, n = 4 for Ge, and n = 5 for a-Sn. In the case of semiconductors composed of different elements, the valrnce shell orbitals of the various atoms must be considered. For GaAs that means the one 4s-state and the three 4p-states of Ga, and the one 4s- and t h e Q-states of As. For GaP one bas, besides the abovementioned 4s- and lip-states of Ga, the one 3s-state and the three Spstates of P.

The valence shell orbitals used as basis functions need not, of course, be populated by electrons in the case of the free atoms. For Si, for example, two of the three porbitals are empty. Similarly, the eigenstates of the crystal, which will be calculated later by means of the TB approximation, will also not be completely populated. As this applies to quantum mechanics in general, the eigenstates are candidates for posstble population. Whether they are popdated or not. depends on the macroscopic state of the system, e.g., on the temperature of the crystal

In the examples considered above the valence shells of the atoms are formed by s- and p-states. This is the typical case for tetrahedral semi- conductors composed of elements of the main groups 11, IV, and VI of the periodic table, but it is by no means the only possibility, especially if one also includes other material classes. For body-centered cubic metals such as Cu and Ni, for example, the valence shells are formed by 3d- and 4s- states. In section 2.1 it was already mention4 that d-states may contribute to the valence shell of 11-VI semiconductors with heavy metal atoms such as Zn. Here, we wiU exclusively consider semiconductor materials for which only s- and p-states need to be taken as basis functions. The corresponding spherical harmonics X,(X/ I x 1) are

Using these harmonics, eigenfunctions &oo, $n.cll, &lo, &i-i may be formed according to equation (2.246). Instead of drill and d ~ ~ 1 - 1 , one may use the linear combinations

The latter are also energy eigenfunctions because of the degeneracy of qbn1l aiid 0~1-1 with respect to the magnetic quantum number m. For the sake of uniformity, one sets

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2.6. Tight binding approximation 143

Figure 2.16: Polar diagrams of the atomic s-and p- orbitals in Cartesian represen- tation.

The eigenfunctions of equations (2.246) to (2.248) will be referred to as spherical orbitals, and those of (2.249), (2.250) as Carte3ian orbitak. The latter are visualized in Figure 2.16. The quantum numbers nlm of the spher- ical orbitals, as well as n, 6, z, y, z of the Cartesian, will be abbreviated by a general index a. The orbitals considered above are orthonormalized, i.e. one has

($a’ I $fZ) ‘af,. (2.251)

If the atomic core is not located at the coordinate origin, as has been assumed thus far, but at a particular lattice position R+G, the corresponding orbitals will be denoted by &j~(x). They may be traced back to the orbitals &(x) of atoms located at the origin by shifting their arguments in accordance with

Two different orbitals & f j f R ‘ ( X ) and & ~ R ( x ) with identical values of R’ and R as well 8s of j ’ and j , but different values of a‘ and a, are also orthogonal to each other. For R’ f R or j’ # j , i.e. for orbitals at different cen- ters, no such orthogonality exists. Although the two orbitals are localized in different spatial regions, and the integral over the product of the two, the so-callcd ovcrlap integral, turns out to be relatively small, it may not be ne glected because its influence on the energy eigenvalues is of the same order of magnitude as the matrix elements of the Hamiltonian between orbitals at different centers. The latter elements are essential because they are r e sponsible for the bonding between atoms in a crystal and for the splitting of the atomic energy levels into bands. The non-orthogonality overlap integrals must therefore also be taken into account. This may be done directly, by writing down and solving the eigenvalue problem for the crystal Hamilto- nian in the non-orthogonal basis set of the atomic orbitals. This procedure

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144 Chapter 2. Ektranic structure of ideal crystds

is, Iiowcvcr, quite inconvenient becaiise the matrix of overlap integrals has to be calculated explicitly and diagonalized together with the Hamiltonian matrix. It is more useful to employ a set of orthoganalizd orbitals by form- ing suitable h e a r combinations of the q a j ~ ( x ) . Here, 'suitable' means that the new orbitals should have t,he same spatial symmetries as thc original atomic nrhitrals Q U j ~ ( x ) , because these symmetries are the essential proper- ties that allow the matrix elements of the Ilamiltonian to be reduced to a few const.ants. Consequently, the new orbitals must, likewise form basis sets of irreducible representations of the full orthogonal group, each set heing char- a c t w i d by a particular angular morrieriturri quantum miniber 1. That there are indeed linear combinations of atomic orbitals possessing these properties, forms the content of Liimdin,'<s thcowm (we, e.g., Slater, Koster, 1954). We will iise this theorem below. The orbitals m U j ~ ( x ) ! which thus far have been taken as ordinary atomic orbitals, will henceforth be understood as atomic orbitals in the sense of Ldwdin's theorem. The orbitals qaj~(x ) are now, of course, no longer given by the expressions [2.246) to (2.250), but their indices n E nlm will retain the meaning they previously had for the ordi- nary atomic orbitals. hlthough explicit expressions for the Gwdin orbitals & j ~ ( x ) can in principle be providd, they will not be given here since thcy are nut, nmdd if one proceeds in thc manner to be discussed below. The Lowdin orbitals arc, by definition, aIso orthogonal for different centers, such that

(&!jrR( I O a j R ) = bo'abj'$R'R. (2.253)

With Ro as an arbitrary lattice vector, the d U 3 ~ ( x ) obey the relation

(2.254)

Bloch sums. Schrodinger equation in matrix representation

Consider, again, the Schrodinger equation (2.75) of the crystal. As before, the eigenfunctions of this equation are taken in the form of Bloch functions pyk(x). To represent the Bloch type eigenfunctions by means of atomic orbitals o a J ~ ( x ) . it is convenient to transform the latter into Bloch type orbitals puJk(x). This is done by means of the k-dependent unitary trans- format ion

(2.255)

The sum over R extends over all lattice points of the periodicity region. By means of (2.254) one may readily verify that

(2.2 56)

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2.6. Tight binding approximation 145

which identifies the & j k ( x ) as Bloch functions of quasfwavevector k. The 4 , j k ( X ) are called Bloch sums of atomic orbitals. If one replaces k in &jk by a wavevector which differs from k by a reciprocal lattice vector K then, because of the relation exp[iK. R] = 1, the original wavefunction is reproduced expect for an unimportant phase factor. Therefore, it suffices to take k in the first B Z . Obviously, the use of the Bloch sums (2.255) automatically puts us in the reduced zone scheme. The orthogonality of the Lowdin orbitals & ~ R ( x ) results in the orthogonality of their Bloch sums 4 a j k ( X ) , such that

(‘#‘alk’j’ I ‘#‘ajk) 6 a ’ a 6 j 1 j 6 k 1 k . (2.257)

In the TB approximat,ion, the eigenfunctions C p v k ( x ) of the Schrodinger equa- tion are written as linear combinations

cPvlc(x) = x(’Jjk I c P v k ) c P a j k ( x ) (2.2 58)

of Bloch sums q 5 a j k ( x ) . Employing this representation in the Schrodinger equation (2.75), we obtain

j Q

x ( a j k I H I a’j’k)(a’j’k I V v k ) = Ev(k)(ajk I cPvk), (2.259)

where the matrix elements of the Hamiltonian are given by the expressions

j ’a ‘

with

( a j 0 I H I a’j’R’) = d 3 x ~!J:(x - ? ~ ) H ~ , I ( x - R’ - 5,). (2.261) s In deriving equation (2.260), the lattice translational symmetry of H has been used. For a given wavevector k, the (ajk I H I a’j’k) form a square matrix with a finite number of rows and columns, the same as the number of different orbitals per primitive unit cell which were used for the represen- tation of the eigenfunctions, i.e. ‘number J of atoms per primitive unit cell times number A of orbitals per atom’. The integrals ( a j 0 I H I a’j’R’) in (2.261) describe the interaction of electrons in orbitals a and a’, where one of the orbitals belongs to the atom at ?’ and the other to the atom at R’+ ?,I.

These integrals substantially decrease with increasing distance between the atoms, so that it suffices in most cases to extend the sum on R’ in (2.260) over the nearest, or if need be, the second nearest neighbor atoms only. Once the integrah ( a j 0 I H I a’j’R’) and, through them, also the matrix elements (ajk 1 H 1 a’j’k) are known, the energy eigenvalues and eigenfunctions of

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146 Chapter 2. Electronic structure of ideal crystals

the Schriidinger equation may be obtained by calculating the eigenvalues and eigenvectors of the ( J x A ) x ( J x A)-dimensional IIamiltonian matrix of equation (2.260). This is an easily solvable task which occasionally can be treated analytically, but, can in any case, be done numerically. For each k one has J x A eigenvalues and eigenvectors. They will be partially degen- erate if symmetrical k-vectors are considered. If k varies over the first B Z , the J x A energy eigenvalues form J x A energy bands.

Simple example: cubic crystal composed of s-atoms

We will illustrate the above discussion with a simple model. Consider a crystal having a primitive cubic lattice and 1 atom per primitive unit cell ( j - 1). The atoms are placed at the lattice points R, with - 0. The set of atomic orbitals will be restricted to one ns-orbital only, such that a = ns. With J = 1 and A - 1, a matrix of size 1 x 1 = 1 is obtained. This the solution of thP eigenvalw problem is, in fact, trivial. In evalu- ating the matrix element (nslk IH I nslk), we will terminate the R’-sum in expression (2.260) at the nearest neighbor atoms. The latter are located at R1,z = (fa,O,O),R3,4 = (O,fa,O),Rs,s = (O,O,fa). For the matrix elements occurring in equation (2.260), we use the notation

(nslO I H I nslO) = ens, (2.262)

(nslO I H I nslRt) = pns, t = 1 , 2 , . . . 6 . (2.263)

Here, we have employed the fact that, for reasons of symmetry, all six neigh- bor atoms give rise to the same value of the integral (2.261). The value of ,BnS depends on the overlap of the ns-orbitals localized at adjacent atoms. With increasing distance between the atoms, &, approaches 0. The value of E,,,

in a crude approximation, may be identified with the energy of the ns-level of the free atom. In terms of E,, and Pns, the matrix element (nslk IH I nslk) of (2.260) is given by

6 (nslk I H I nslk) = ens + &sxeik’Rt. (2.264)

It follows that the energy eigenvalues Ens(k) of the Schrodinger equation are

t=l

Ens(k) = ern + pns2 [cos(k,a) + cos(kya) + cos(k,a)] . (2.265)

It is of interest to further discuss the eigenvalues e,,(k), to gain insight into the formation of energy bands. In this context, the main quantum number n will be allowed to take not just one value, as previously assumed,

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2.6. Tight binding approximation

Figure 2.17: Energy bands in Tight-Binding ayproxirnat,ion for the simple s-atom crystal described in the text.

ks

147

P3s

P 2s

P l S

l-r - a

Wavevector (O,O,k,) - -IT - 0 D

but several. This means that the valence electrons in the free atom do not occupy only one 5 level, but several s-levels ens differing in the value of n. To justify the application of the results obtained above in the present case, the matrix elements of H between s-orbitals having different values of n must be negligibly small. We assume this to be true. In addition, we suppose that ens is negative for all n, and that the sign of pns alternates, such that for n = 1 it is taken to be negative, for n - 2 positive, for n - 3 negative etc. This behavior reflects the differing numbers of nodes of thc atomic wavefiinctions for different values of n. The absolute values of Pns are expected to increase with growing n, corresponding to the larger values which the ns-orbitals with larger n have at the nearest neighbor aloms. The separation between adjacent energy levels cn8 should, however, always be larger than 4Pns.

If the above conditions are met, the energy band dispersion along the line (0, 0, kz) of the first B Z of the simple cubic lattice under consideration, has the form shown in Figurr 2.17. Each level ens of the free atom gives rise to an energy band of the crystal. The width of these bands amounts to 4&., thus increasing with increasing n. The bands are separated by forbidden energy regions. Their widths decrease with growing n. If the distance be- tween nearest neighbor atoms increaees, then the parameters fins of equation (2.263) decrease because of decreasing ovrrlap of the orbitals, as has been

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148 Chapter 2. Electronic structure of ideal crystals

pointed out above. The same holds for the widths of the energy bands, they become narrower if the distance between neighboring atoms grows. They approach the discrete levels ern of free atoms as this distance becomes in- finitely large, corresponding to ,&, = 0. In the latter case. electron states with different values of the wavevector component k, have the same energy, ens. These levels are, therefore, highly degenerate. If the infinitely remote atoms again approach each other, then the ,&,-values become finite. because of the onset of overlap of neighboring orbitals. Correspondingly, the degen- eracy of electron states with different kz is removed. and the discrete levels of the free atoms spread into bands. The TB approximation quite naturally explains, in this way, how discrete energy levels of the free atoms transform into energy bands of the crystal. The gaps between the bands occur nat- urally in this approach, since the discrete atomic levels are separated by energy gaps from the outset. This stands in contrast to the approximation of nearly free electrons, where the occurrence of energy gaps calls for an explanation. On the other hand, 'bandwidth', in the form of the infinitely broad energy continuum of the free electron. is present at the outset in the latter approach. The gaps induced into the energy continuum were seen to arise because of the strong perturbation of plane wave states by the peri- odic lattice array of atomic coTes fox wavevectors on the Bragg reflection planes. Comparison of the two approximation procedures reveals the differ- ence between the underlying concepts - the TB approximation emphasizes the atoms and the short-range ordered complexes of the crystal. while the approximation of nearly free electrons focuses on the crystal as a whole and the long-range ordering of the atoms. Such comparison also shows that the short-range and long-range order concepts are equivalent in the sense that they result in the same characteristic features of the electronic structure of crystals -both concepts predict the existence of energy bands separated by gaps. Figure 2.18 depicts the manner in which the bands and gaps arise in the two approximations.

2.6.2 TB theory of diamond and zincblende type semicon- ductors

Semiconductors of the diamond and zincblende type are tetrahedrally coor- dinate cubic crystals with two atoms per primitive unit cell and four valence shell orbitals per atom, among them one s-orbital([ = 0) and three porbitals ( I = 1). The application of the TB method to this specid case is of particu- lar importance. It will be developed in the present subsection. Only nearest neighbor interactions will be taken into account because this introduces con- siderable simplification and nevertheless gives results of reasonable accuracy. For the p-orbitah. we choose the Cartesian form, so that the orbital index a takes the values ns, nz,ny and nz . The Cartesian components I , yrz refer

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2.6. Tight binding approximation 149

Figure 2.18: Illustrat,ion of the origin of energy bands and gaps in the enerw spectrum of a crystal.

to the cubic crystal axes. The main quantum number n will be suppressed below because it may assume only one value here for each of the t,wo atoms, although not necessarily the same. As a first step we determine the 8 x 8- Hamiltonian matrix for diamond type crystals. Later: it will be generalized to zincblende type structures.

Hamiltonian matrix

We start with matrix elements bet,ween orbitals at equivalent atoms.

Matrix elements between orbitals at equivalent atoms

These are elements of the general form ( a j k 1 H 1 a'jk). In the case of diamond type crystals. the two atoms J = 1 and J = 2 of the primitive unit cell are of the same chemical nature, thus their matrix elements will be identical. Since we are restricting ourselves to nearest neighbor interactions, only the term with R' = 0 needs to be considered in the R'-sum of formula (2.260). Thereby, ( a j k 1 H I a'jk) becomes independent of k. The eigen- functions I s l O j and I zlO), 1 ylO), \ 210) transform, respectively, according to the unit representation rl and the vector representation I'15 of the cubic point group Oh. Since oh is the symmetry group of H , the matrix element ( s lk I H 1 s j k ) likewise belongs to the unity representation. According to Appendix A this means that its value, in general, is non-zero, and we denote

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150 Chapter 2. Electronic structure of ideal crystals

it by E ~ . In a crude approximation, es is the energy of the s-orbital of the free atom.

The matrix elements between s- and p-orbitals at the same atom trans- form according to the representation rl x I'l x r 1 5 = r15, which does not con- tain the unit representation. Then, according to Appendix A, these elements must vanish. The matrix elements between p- and p-orbitals belong to the representation r 1 5 x I ' l x r 1 5 = X'1+r12+I'i5+I'b5 (see Table A.27). Here, the unit representation occurs exactly once, which means that the p - p-matrix contains exactly one independent constant. A more detailed analysis shows that this constant corresponds to the three non-vanishing and mutually iden- tical elements (210 I H I s10) = (y10 I H I y10) = (210 I H I 210) = E,,. Again, as in the case of c8, the value of cp is roughly the energy of the corre- sponding p-orbital of the free atom. The non-diagonal p -p-matrix elements must be zero according to the above symmetry analysis. Summarizing, we have

(2.266)

Matrix elements between orbitals at non-equivalent atoms

In evaluating the matrix elements between orbitals at non-equivalent atoms, i.e. elements of the general form (ujk 1 H I a'j'k) with j # j ' , the R'- sum in (2.260) may be restricted to the 4 lattice points &, t = 1,2,3,4, whose primitive unit cells host the 4 nearest neighbor atoms. With this simplification the Hamiltonian matrix of (2.260) becomes

4

(ujk I H I a'j'k) = x e i k ' ( R t f 5 f - % ; . ) ( ~ j 0 IH I U ' j ' R t ) . (2.267) t=1

The values of j and j' are complementary to each other because the nearest neighbor atoms lie in the other respective sublattice. For j = 1 one has j' = 2, and for j = 2 then j ' = 1. We will restrict ourselves to the first case, i.e. j = 1,j' = 2, because the second may determined from the first with minor changes. The four nearest neighbors of a 1-atom are located in the primitive unit cells at

R1 = 0, R2 = -al, R 3 = -a2, R 4 = -a3, (2.268)

(see Figure 2.19). The calculation of the matrix elements (a10 I H I a'2Rt) between the different Cartesian orbitals and the four different values of Rt is

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2.6. Tight binding appmxhation 15 1

24

#’

Figure 2.19: Atom of sublattice 1 and its four nearat neighbors in sublattice 2.

somewhat laborious. It will be carried out in several (five) steps. In the first step we determine the matrices (a10 I H I a’2Rt) between spherical orbitals.

i) Spherical orbitals in the pair-related coordinate system

The spherical orbitals are related to a spherical coordinate system whose z-axis is in the direction of the line connecting the central 1-atom and its nearest neighbor atom 2t in the unit cell at & (see Figure 2.19). To dis- tinguish between this z-axis and the crystal-related cubic z-axis, we denote the former by 2. For each of the four next neighbor atoms, I has B different direction. The matrix elements (a10 I H I a‘2Rt) between the so-defined or- bitals are equal for all four nearest neighbors. The orbitals differ, however, because they refer to different pair-related coordinate systems. Consider, in particular, the neighbor atom 21 which lies in the same unit cell as the central 1-atom. Here, R1 = 0. The corresponding Z-axis represents a %fold symmetry axis of the crystal which contains three mirror symmetry planes. In evaluating the matrix elements (Im 10 I H I I’rn’20) one may therefore use the fact that the Hamiltonian exhibits the symmetry of the point group C h with respect to the Z-axis. This yields diagonality of these elements with respect to the magnetic quantum numbers rn, m’, so that

(Lm1O I H I I’m’20) = & , , , ( h l O 1 N 1 l‘m20) BmmrV~ppn 12 , (2.269)

The Cg,-symmetry of the Hamiltonian H holds with respect to the nearest

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152 Chapter 2. Eiectronic structure of ideal crystals

neighbor dirwtions. For the second-nearest neighbor directions, the symme- try of H is smaller. Rwauw of that, the matrix elements (alk I H I a’2k) of equation (2.260) are, in general, non-diagonal with respect to m, m‘ if the R’-sum is extended to the swond-warest or more remote atoms. The diago- nality holds, however, in an approximate sense, which may be seen as foHows. The periodic potential V(x) in H represents a sum &,,,, vji ,(x - R” - i y ) of potential contributions of all the individual atomic cores of the crystal. The matrix elements of H between orbitals at different renterrs jR and J’R‘ thereby decompose into bums over all centers JI’R’‘. The largest contribu- tions will arise from t a m s where the center index j“R” coincides either with 1R or with J’R‘. If one considers only such twecenter terms and neglects all threecenter contributions, then the integrand of the matrix elements (Im10 I H I l’m’20) has the full axial symmetry, so that these dements become diagonal with respect to m, m’.

The hermiticily of the Hamiltonian and the particular form of the wave- functions in (2.252) lcsd t o the relation

(lrn10 I H I I‘m20) = (l‘m20 1 H I lm10). (2.270)

The three matrix elements

(0010

(0010

(1010

(2.271)

are independent of each other: and the two elements

(1110 I H I 1120) = ( i i i o I H I 1720) = v& (2.272)

are identicaL These elements are illustrated in Figure 2.20. This figure also shows how the elements (In110 I H 1 l’m20) behave if the two atoms 1 and 2 are interchanged. One has

(2.2 73)

Since the atomic orbitals under consideration are those of bound states, and since the eigenvalues of the Hamiltonian for bound orbitals are generally negative, one may expect negative values for k<3n, Vm, and positive ones for Ifaw, IT&,. Taking account of the strength of the overlap of the orbitals

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2.6. Tight binding approximation 153

12 v s s u

= vuG = vssu

21

-

21 = V p p u

= V p p c r -

21, = V P P T

Figure 2.20: Illwtratiori of tight binding matrix elements.

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154 Chapter 2. Electronic structure of ideal q y s t d s

in Figure 2.20, one can conclude that the absolute values of these elements should obey the relations

I VPPO I ’ I v,, I ’ I VSSU I >I V,, I . (2.274)

These expectations are in fact valid in most cases.

ii) Cartesian orbitals with respect to the pair-related coordinate system

The matrix elements (lm10 I H I I’m’flO) of H with respect to spherical or- bitals calculated above will be used to derive, in the second step, the matrix elements of this operator with respect to the Cartesian orbitals. The z-axis of the Cartesian coordinate system is taken to be the same as that of the spherical coordinate system used above. This means that the z-axis points in the direction of the connecting line between atom 10 and atom 21. The corresponding 2- and y-axes lie in the plane normal to the z-axis, apart from an irrelevant rotation about this axis. The pair-related Cartesian coordinate system thus defined differs from the formerly introduced crystal-related sys- tem which is given by the three cubic crystal axes. The coordinates in the pair-related Bystem will be denoted by 2 , 5, 2. The pair-related ;-orbital co- incides with the 8- orbital with respect to the cubic-axes system. In terms of this notation, the matrix elements of H between Cartesian orbitals in the pair-related system read (310 I H I 3.20),(3.10 I H I 120),(3.10 I H I 520), (3.10 I H I .520), (110 I H I 120), (110 I H I 520) etc. Since the Car te sian orbitals are defined in terms of spherical orbitals by equations (2.249), (2.250), the matrix elements between Cartesian orbitals can also be related to those between spherical orbitals. The corresponding relations are given below. Elements which are complex conjugates due to hermiticity of the Hamiltonian, such as (310 I H I 220) and (120 1 H I ;lo), are listed only once. The relations read

(310 I H I 3.20) = VSScr,

(3.10 I H I 520) = (3.10 H I 520) = 0, (3.10 I H I 220) = VSF,

(110 I H 1220) = (510 I H 1520) = Vw,

(510 I H 17/20) = (g10 I H 1.520) = (210 I H 1120) = 0,

(El0 I H 1.220) = Vppu. (2.275)

To develop the representation of the Hamiltonian matrix (2.267), we need the elements (a10 I H 1 a’2Rt) of H between Cartesian orbitals which refer to the three cubic crystal axes rather than to the pair-related ones. We determine these elements in the third step.

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2.6. Tight binding approximation 155

iii) Cartesian orbitals in the crystal-related system

To this end, the Cartesian orbitals 2, y, z related to the cubic-axes system must be expressed in terms of the pair-related Cartesian orbitals ?,c ,Z . To determine this relation, we consider a rotation which transforms the crystal-related axes system into the pair-related one. The transformation is characterized by Euler angles $, 6 and 'p. Since the orientation of the pair- related system is defined only up to an arbitrary rotation about the Z-axis, the Euler angle 'p may, without any loss of generality, be set equal to zero. As noted in Appendix A, the rotation matrix A of equation (A.31) transforms the coordinates before rotation into the rotated one. The basis vectors are transformed by the transposed matrix, which, in the present case of rotation, is also the inverse matrix. Since the 2, y, z and ? ,y , Z are understood here in the sense of basis vectors, we have

cos @ - cos 6 sin $ sin 0 sin $ (:) = (sin@ c o ~ e c o ~ ~ -sinecos$) (i). (2.276) 0 sin 0 cos e

Below, we will see that the direction cosines (PI, q1, 71) of the pair-related 2 -axis with respect to the crystal-related cubic z-axis play an important role. These are the elements of the third column of the rotation matrix in equation (2.276), i.e.

p l = sin 0 sin $, q1 = - sin 6 cos $, r1 = cos 0. (2.277)

Using equations (2.275) to (2.277), the matrix elements in the crystal-related system are evaluated as One obtains

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156 Chapter 2. Electronic structure of ideal crystals

These relations are valid for matrix elements involving the two nearest neigh- bor atoms belonging to the same unit cell at R = 0. From these elements we may determine the elements with the nearest neighbor atoms of different unit cells. This will be done in the next step.

iv) fourth step.

The vectors pointing from the central 1-atom to the four nearest neighbor atoms will be denoted by dt, t = 1,2,3,4, such that dt = Rt + - TI. For the diamond structure, the two sublattices are displaced with respect to each other by the vector (a/4)(1,1,1). Using equation (2.268) for Rt and the explicit form of the primitive lattice vectors of the face-centered cubic lattice, we obtain

U a - a - - dl = a(l, 1, l), d2 = -(1,1,1), d3 = -(l, 1, I), d4 = -(l, 1,l). (2.279)

4 4 4 4

These relations determine the direction cosines ( p t , qt, rt ) of the connecting lines between the central 1-atom and the nearest neighbor atoms 2t. In par-

unknown matrix elements (a10 I H I a‘2Rt) for t = 2,3,4 follow from the elements (a10 I H 1 a’2R1) in equation (2.278) by replacing the direction

ticular, (PI, a, 7.1) = ( l / f i ) ( L 1,1), ( P Z , 42,72) = (l/fi)(LI,l) etc. The

cosines (PI, m, ri) by ( p t , qt, r t ) in them.

v) fifth step

The elements (a10 I H I a’2Rt) determined above are used to calculate the k-dependent matrix elements (alk I H I a’2k) between Bloch sums. For complex conjugate elements, again, only one relation will be given. With the help of equation (2.267), we obtain

4 4

(s lk I H 1 s2k) = ~ e i k - d f ( s 1 0 I H I s2Rt) = V,,,xeik.dt, (2.280) t=l t=l

4 4

(s lk 1 H I 22k) = xeik .d t ( s10 I H 1 22Rt) = Vswxeik-dtpt. t=l t=1

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2.6. Tight binding approximation 157

4 4

4

The combinations of matrix e1ement.s in these expressions are commonly abbreviated by

The seven different t-sums which enter t,he matrix elements (2.280) may be reduced to just four because of the obvious relations ptpt = r t , p t r t = qt. and qtrt = p t . The four independent sums are

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158 Chapter 2. Electronic structure of ideal crystals

9l(k) = eikdi + ,ik.dz + ,ik.d3 + ,ik.d4,

g2(k) = eik.dl + .ik+d2 - .ik.d3 - .ik.d4

9 3 0 4 = .ik.d~ I .&.dz + .ik.ds - ,ik.dn:

9 4 0 4 = - - e

(2.282)

.ik.dl ik-dz - ,ik.ds + ,ik.d4

Finally, we can write down the Hamiltonian matrix (ajk I H I a’j’k) between Bloch sums in explicit form. Arranging the eight basis functions 1 ajk) in the sequence I s l k ) , I z lk ) , I ylk) , I z lk) , I s2k), I 22k), I y2k), I z2k), this matrix is given by

Band structure. Empirical TB method

To obtain the band structure of diamond type crystals, one has to calcu- late the eight eigenvalues El(k), Ez(k), . . . , Es(k) of the Hamiltonian matrix (2.283) at the various points k of the first BZ. The k-dependence of the eigenvalues stems from the factors 91, 92, g3,g4 in (2.282), which are univer- sal functions of k. The constants r,, e p , E,,, E,, E,, and E,, are related to material properties. In principle, these constants can be calculated from the defining equations (2.280) and (2.281) and the Hamiltonian matrix elements (2.261) between atomic orbitals. In quantum chemistry one often proceeds in this way. However, one also may forego the calculation of these Hamil- tonian matrix elements and considers them as empirical parameters. Their values may obtained as follows. First of all, one calculates the eigenvalues of the Hamiltonian matrix (ajk I H I a’j’k), at at least 6 special k-points as functions of the unknown parameters cg, epr E,,, E,, E,, and E,. Then one measures the band energies at these points 01 calculates them by some other

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2.6. Tight binding approximation 159

Table 2.7: Tight binding matrix elements for diamond type semiconductors (in eV). (After Harrason, f9RO. ) 1 ~i 1 6s 1 EP 1 VsSa 1 V v 1 vma 1 1;

Ge -14.38 -6.36 -1.79 2.36 4.15 -1.04

-17.52 -8.97 -4.50 5.91 10.41 -2.60

-13.55 -6.52 -1.93 2.54 4.47

Table 2.8: Universal inter-atomic TB parameters. (After Majewski and Vogl, 1987.) mi

-1.38 1.92 1.68 2.20 -0.55

method. Equating the TB eigenvalues with those measured or known from other calculations, one obtains a system of equations which determines the unknown parameters uniquely. There are other variants of this procedure. For instance, the intra-atomic matrix elements t3 and tp may be identified with the atomic s- and p-energies, leaving only the inter-atomic constants E,,, ESP, Epp and Epp for the fitting procedure. Instead of EsS, E,, E, and E,, one often takes the inter-atomic matrix elements V,,,, Vspo, V,, and V,, as independent parameters. Together with c3 and e p , they are referred to as tight binding parameters. (see Table 2.7) . Once these are determined, one is able to calculate the band structure at all k-points. The whole pro- cedure may therefore be considered to be an interpolation of band energies between special points of the first BZ. It is called the empirical tight bind- ing (ETB)-method. This method can also be applied to the calculation of the electronic structure of perturbed semiconductors, which will be treated in Chapter 3. In this case it serves as an extrapolation method, because it helps to extrapolate from the electronic structure of the ideal crystal to that of the perturbed one.

Inspecting the inter-atomic TB parameters listed for the various dia- mond type semiconductors in Table 2.7, it may be seen that, within an appropriate error limit, each of these parameters may be expressed in terms of the material-dependent nearest neighbor distance do of the crystal, and a

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160 Chapter 2. Electronic structure of ideal crystals

Table 2.9: Intra-atomic TB parameters (in eV) to be used in conjunction with the universal inter-atomic TB parameters of Table 2.8. (Source same as in Table 2.8.)

universal material-independent factor q l t m . This approximate relation is

(2.284)

signifying that the inter-atomic matrix elements of the Hamiltonian scale with the inversesquare of the nearest neighbor distance. The factor ( f i 2 / m ) = 7.62 eVA2 in equation (2.284) has been jntroducd in order to make the universal factors qzpm dimensionless. The nrarest neighbor distances dg fol- low from the cubic lattice constants a in Table 1.2 by ineans of the rela- tion do --- (fi/4a). Ultimately, the empirical do'-dependence in equation (2.284) originates from the kinetic energy operator of the crystal Harnilto- nian (li'royen, Harrison. 1979). The eigenstates of this operator are plane waves and its eigenvalues are proportional to the inversesquare of the wave length of i t s eigenstates. As a unnsequmce of this, the empty lattice band structure scales with the inverse square of the latlice constmt. The TB band structure follows by diagonalizing the T H Hamiltonian (2.283), resulting in c l o d analytical expressionh for the energy hand lev& at special k-points of the first BZ. If these expressions are identified with the empty lattice band levels at, the 5ame k-points, respectively, linear eqiiations for the TB parameters ctre obtained. Their solutions stale with the inverse-square of the lattice constant a because the empty lattice band levels do so. hlormver, numerical valiies for the universal TB parameters q ~ ' ~ follow from these equations. They are close t o the values listed in Table 2.8 which have been derived from more precise band structure data (the meaning of f i g p a wiU be explained below in connection with zincblende type semiconductors).

The intra-atomic matrix elements e g and c p corresponding to the set of universal inter-atomic TB parameters ql+, in 'lable 2.8 are shown in Table 2.9 for a series of atoms forming diamond and zincblende type semiconduc-

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2.6. Tight binding approximation 161

tors. They represent atomic s- and p-level energies which deviate somewhat horn the energy levels given in Table 2.1.

Here, the eigenvalues and eigenfunctions of the Hamilton matrix (2.283) may br o b tained ~ I I closed analytic form. The results me the following 8 eigenvalues &(a). i = 1,2,. . . , 8 .

A k-point of prtictilar interest is the €32 center k = 0.

The components of the corresponding eigenfunctions I EiO) read as follows:

1 E10) = (l/&)(l, o ,o , 0,l .O. 0, O), 1 E z O j = ( I / f i ) ( O , 1,0, o.o,T, o,oj ,

1 . ~ ~ 0 ) - ( i / ~ ) ( 0 , o , i , 0 , o , O , i , o ) , 1 E40) - (i/&j[o,o,o, i ,o .o ,o , i j ,

I E ~ O ) - (1/&)(0,0, LO, o,o, i ,o j , I E ~ O ) - ( i / f i ) { o , o , 0, I, 0.0, o,i) . I R5O) = (1/&)(1,0, o,o, 1,0, 0, 01, I EGO) (1/&)(0, 1,0,0,0. 1, o,o),

(2.286)

Since cP is negative and EZz positive, the triply degenerate level 6 2 ( 0 ) =

E3(0) = E4(0) lies below the triply degcneratc level Rs[0) = Fe(O) = E7(0) . The constants E~ and E,, both have negative values, therefore the El(O)-hvel is lower than the Es(O)-level. Owing to the fact that the atomic s-energy eS lies below thp pencrgy cp7 the pigPnvahx9 E l ( 0 ) is also smaller than the deeper of the two Zriyly degenerate levels. This means that El(0) is the deepest of the foul levels. Thm+nrP, the energetic ordering of the lwels is determined by the relatiom

El@) < ER(O), (2.2~17)

E l ( 0 ) < E z ( 0 ) - E3(0) = E4(0) < E5(0) Es(0 ) - &;7(0).

For the pmitinn of the &(O) lcvcl, there are still three pussibilitks (impor- tant malerials to whirl1 the three possible cases apply are listed alongside the cases), namely :

i ) &(0] > &(O) > Es(0 ) - for C and Si,

ii) E3(0) < &(0) < &(0) - Ge,

iii) &(O) < &(0) < &(o) ~ ru-sn.

These relations Inem that the E'c;(O)-level moves down with respect to the other two levels as the size of the atoms increases.

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162 Chapter 2. Electronic structure of ideal crystals

It turns out that the ordering of the eight energy bands at k = 0 remains the same over the entire first BZ. This is important because the positions of the energy bands relative to each other determine the likelihood of their population by electrons. As already mentioned at the beginning of this section, not all of the bands are expected to be populated, just as the s- and p-levels of the free atom whose orbitals were used as basis functions were not completely filled. Electron population of the ground state of the crystal, i.e. at temperature T = 0 , may be obtained as follows:

For a simple band, each k-value corresponds to 2 eigenstates of opposite spin. For a periodicity region of volume Q = G3Ro, the first BZ contains, as does every primitive unit cell of reciprocal space, G3 allowed k -values (see section 2.3). A simple energy band therefore has 2 x G3 states. Four such bands are necessary to host the (2 x 4) G3 = 8G3 valence electrons of a periodicity region. In the ground state of the crystal, therefore, the four lowest bands are populated, and the four highest bands are empty. This means that in the case of C , Si and Ge, El(k) , E2(k), E3(k), E4(k) are the populated valence bands, and Es(k), &(k), E7(k), Es(k) are the empty conduction bands. For a - Sn, El(k) and &(k) form valence bands, together with two of the three bands E2(k), E3(k), E4(k). The remaining band of the three is a conduction band. Since it is degenerate at k = 0 with the highest valence band, the energy gap of a-Sn vanishes.

The above band assignment allows us to determine the symmetry of the valence and conduction band states at r.

Symmetry of valence a n d conduction band states at r We know that degenerate eigenst ate8 of the crystal Hamiltonian having the same energy value form a set of basis functions for an irreducible repre- sentation of the small point group for the wavevector k. For k = 0 this group coincide8 with the full point group of cquivalent crystal directions, whch, here, is oh, The dimensions of the irreducible representations are the same as the degrres of degeneracy of the corresponding energy lev- els. Therefore, the eigenfunctions for &(0) and &(O) each belong to 1- dimensional representations, and those of E2(0) = &(o) = E4(0) and E s ( 0 ) = E s ( 0 ) = E r ( 0 ) each belong to 3-dimensional representations. Ac- cording to equation (2.286), the deepest valence bend level Ei(0) ha5 the eigenfunction (l/fi)[I s10)+ I s20)]. In order to determine its transfor- mation properties under the operations of the point group Oh, it is useful to decompose oh into two parts, firstly, the tetrahedron subgroup ‘I> con- taining only elements which are not involved with an exchange of the two sublattices 1 and 2, and secondly, the remainder of oh which is composed of all elemfxttb of T d multiplied by the inversion. Each of the elements of the second part of 01, exchanges the two sublattices. For brevity, we will term

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2.6. Tight binding approximation 163

the latter elements 'exchanging', and the former ones 'non-exchanging'. The eigenfunction (l/&)[l s 1 0 ) + 1 sZO)] for El(0) transforms into itself under the action of both types of elements, thus it belongs to the unit representa- tion rl. For the eigenfunction (l/fi){l sl0)- I s 2 0 ) ] of the Es(O)-level, the transformation into itself occurs only under the action of non-exchanging el- ements, while a factor -1 is generated in the case of exchanging elements. It follows from the character table of the irreducible representations of Oh given in Appendix A, that this transformation corresponds to the representation

The upper valence band level E z ( 0 ) = E3(O) = E4(0) possesses the three eigenstates (~/fi)[ l 210)- 1 220)], (l / f i){l Yloj- I Y~o) ] . (1/&)[1 ~ 1 0 ) - I d o ) ] . Under the action of the non-exchanging elements of oh.

these functions transform like vector components. Inversion, which is part of the exchanging elements. reverses the sign of the vector components, and the exchange of the two sublattices also reverses the sign of the whole eigen- functions. The three eigenfunctions therefore transform as they wodd under the action of the corresponding non-exchanging elements. The character ta- ble of the irreducible representations of Oh in Appendix A shows that this transformation is characteristic of the representation J&. Similarly, one finds that the eigenstates (1/&)[1 x10)+ I d o ) ] , (1/&)[1 y10)+ I Y ~ O ) ] , ( 1 / f i ) [ I zl0j-t I ZZO) ] belonging to the eigenvalues Es(0 ) = Ee(0) - E7(O), trans- form according to the irrcduciblc representation rl;.

We summarize the results of our TI3 band structure calculations as fol- lows: For crystals which have the diamond structure, i.e. for C, Si, Ge, as a rule, the highest valence band is 3-fold degenerate and belangs to the irreducible representation r&, of the point group oh. The lowest conduction band at r exhibits either a similar %fold degeneracy, in which case it be- longs to the repraentation f 1 5 (C, Si), or i t is non-degenerate and belongs to the representation I'i (Ge). For LY - Sn, the F2-band lies below the I'g -band, which in this way, is partially a conduction band. These results are illustrated graphically iu Figure 2.21.

r;.

Extension t o semiconductors of zincblcnde type

Band structures of diianioand type semiconductors calculated by means of the empirical TB method reflect the essential features of the real valence bands of these materials quite well. In order to apply the TB approximation to semiconductors having the zincblende structure, the Hamiltonian matrix [Z.ZS3) must be modified as follows. Firstly, it has to be recognized that the matrix elements E~ and between orbitals at the mme center depend on whether the center is an atom of chemical species 1 or 2. This means that two different s- and p-energies have to be inserted into the two 4 x 4 diagonal blocks of the matrix (2.283); ti, t i in the block at the upper left, and e ; , t:

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164 Chapter 2. Elect,ronic structure of ideal crystals

6, - r,’ -

r,’ - 6, -

diamond structure

r;, - conduct ion

r, - bands

valence r;, - 6 - bands

zinblende structure

Figure 2.21: Ordering of energy bands at the center 1’ of the fint B Z for semicon- ductors of diamond and zincblende type.

in that at the lower right. Secondly, the relation V$$ -V:$ G -V3w of equation (2.273) cannot be used in the zincblende case because it rests on the chemical identity of lhe two atoms of A unit cell. For the matrix element

l2 , the s-state belongs to a 1 atom, and the p-state to a 2-atom, while for ;z the s-state belongs to a 2-atom, and the p-state to a 1-atom. Therefore, thr matrix element, V& is given by an independent constant -espc rather than by -V:$, (the parameters iSw in Table 2.8 are associated with csF by meany of equation (2.284). The other excliangr relations in (2.284) remain valid because they refer to matrix elements whose orbitals at the two different atoms belong to the same state. With these two changes, in equation (2.283) the TB Hamiltonian matrix for zincblende type semiconductors becomes

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2.6. Tight binding approximation 165

with ESP = (l/fi)c&,. TTsing this matrix, the band structure and the eigenstates of zincblende type semiconductors may be calculated. YIP spa- tial symmetry of the eigcnstatcs a t thr RZ rentpr is similar to thtlf, which was found for diamond type crystals above, except that the point group Ott has to he r ~ p l a r d by the tetrahedron group T d . The wprPsPntaliun ri, of Oh thereby becomes the representation r15 of T d , I'i is replaced by rl, and r1.5 remains ri5. In contrast to diamond type crystals, one has practically only one energetic ordering of the conduction bands here - the rl-band being the deepest.

2.6.3 sp3-hybrids, total energy and chemical bonding

Once the bald stnicture is known, the total energy of the valence electrons ran be calculated. Formula (2.54) of section 2 . 1 indicates how this may be done for the ground state of the crystal. One hw to sum the energy levels of all valence electrons! and subsrquently remove the doubly counted electron- electron Coulomb interaction energy from this sum. The total energy of the vakncr elwtrons of the crystal or, strictly speaking, its deviation from the total energy of the valence electrons of the free atoms which were brought together to form the crystal, defines the energy gain due to chemical bonding. known as coh.esion energy of the crystal. This definition is reasonable because the valence elect,rons are the only parts of the atoms whose stat= change when the crystal is formed.

In order to obtain the total e n e r a in closed analytic form, one needs explicit mathematical expressions for the valence band energies st. all paints k of the f i d BZ. However, the TB approximation in the form developed above produces such expressions only at particular symmetry points. To find them everywhere, one must introduce further simplifications. A starting point for this is a formulation of the 'I'B approximation which employs certain linear combinations of 8- and p orbitals as basis functions, rather than the atomic orhihals of definite angular momentum quantum numbers I ! which were used above. These linear combinations are callcd sp'--hgrhr%d orbitals. In contrast to the atomic orbitals, they are not energy eigenfunctions of the atoms.

sp3-hybrid orbitals

The four hybrid orbitals I h~lR), I hzlR) , I hslR), I hslR) of a 1-atom in the unit cell at R are defind by the equations

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166 Chapter 2. Electronic structure of ideal crystals

23 23

23 23

Figure 2.22: Illustration of the wavefunctions involved in chemical bonding in tetrahedrally coordinated semiconductors: sp3-hybrid orbitals (a, b), bonding or- bitals (c) and anti-bonding orbitals (d).

1 I h31R) = 5 [dslR(X) - # z l r t ( X ) -t b y d x ) - $zlR(X) 1 1

I hdR) = 5 [ $ s ~ R ( x ) - 4 z d X ) - &lR(X) + d'.zlR(X) 1

In Figure 2.22 the probability distributions of the four sp3-hybrid orbitals are shown in the form of polar diagrams. The orbitals resemble clubs pointing to one of the nearest neighbor atoms in the sublattice 2, e.g., I hllR) to atom 21, 1 hzlR) to atom 22 etc.

1

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2.6. Tight binding approximation 167

Similarly, the four sp3-hybrid orbitals I h12R), I h22R), I h32R), I h42R) at a 2-atom in the unit cell at R, are defined by the relations

These orbitals point to an atom of type 1. The following four orbitals are directed to the 1-atom in the unit cell at R: orbital 1 h12Rl+ R) at atom 21, I h22R2f R) at atom 22, I h32R3+ R) at atom 23, and I h42&+ R) at atom 24. Hybrid orbitals at the same center are orthonormalized with respect to each other, as the s- and porbitals from which they are constructed. If the latter are understood in the sense of Liiwdin, orthogonality also holds for hybrid orbitals at different centers. For each unit cell there are 8 associated hybrid orbitals 1 &jR) with t = 1,2,3,4 and j = 1, 2, as there are 8 atomic orbitals for the two free atoms of a unit cell. From the hybrid orbitals of a given sublattice one may form Bloch sums I htjk) by means of the relation

(2.291)

in complete analogy to the Bloch sums q b ~ k involving atomic orbitals in equa- tion (2.255). Due to the definitions (2.289) and (2.290) of the hybrid orbitals in terms of atomic orbitals, the Bloch sums I htjk) may be thought to arise from the Bloch sums # a 3 ~ of atomic orbitals by means of a unitary transfor- mation. The same holds for the Hamiltonian matrix (h t jk I H I htrg'k) with respect to the basis I htj k); this matrix may be understood as arising horn the above mentioned unitary transformation of the known Hamiltonian ma- trix ( a j k I H I a'j'k) with respect to the basis daJk. As such, it has the same eigenvalues and eigenfunctions as the original matrix. The implementation of the TB method by means of hybrid orbitals instead of atomic orbitals having a definite angular momentum quantum number is therefore nothing but the solution of the same eigenvalue problem in another representation. This statement holds, of course, only as long as equivalent approximations are made in the two representations. However, the hybrid orbital represen- tation is well s u i t e d for further approximations, beyond those already made earlier. These approximations facilitate the derivation of simple analytical

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168 Chapter 2. Electronic structure of ideal crystals

expressions for the eigenvalues and eigenvectors which may be used to ex- plicitly determine the total energy of the valence electrons of a crystal in closed analytical form.

Hamiltonian in hybrid-orbital representation

Here, we describe the most important additional approximation available in hybrid orbital representation, which utilizes the fact that those particular hybrid orbitals at nearest neighbor atoms which point toward one another will also overlap one another more strongly than all others. In consequence of this, the Hamiltonian matrix elements (htl R I H I ht2 Rt + R) between these orbitals will be the largest. Approximately, one need only consider these elements, while all others may be neglected. The former elements do not depend on t and R, as can be seen from the explicit expressions for the hybrid orbitals (2.289), (2.290). Their common value, denoted by V2, may be determined from (2.289), (2.290) and equations (2.271) to (2.273) as

The corresponding matrix elements (h+lk I H I ht2k) between the Bloch sums of hybrid-orbitals follow from VJ by multiplying this quantity with the factor

et ~ eak.dt (2 293)

In the casc of Hemiltonian matrix elements between hybrid orbitals at the same center, one has to distinguish between diagonal and non-diagonal ele- ments. Thc non-diagonal elements ( ~ Q R 1 If 1 h t , ~ R ) have thc samc value for all R and orbital quantum numbers t , t'. For the common value V1 one finds by means of (2.289), (2.290) the expression

(2.294) 1 4

with e3 and ep defined in equation (2.266). Since only nearest neighbor atoms are considered, the rrvult is the same

as for the matrix element ( h y k I ZI I ht f j k) between the corresponding Rlocli sums. An analogous result holds for the diagonal elements (&jR 1 H 1 ht3R) at the same center. Their common value ctt is

1 4

Vl 5 ( k t j R 1 H 1 ht , jR) - - ( c S - E ~ ) , t # t' .

(2.295)

Again, this result is the same as for the matrix elements (huk 1 H 1 h u k ) between the corresponding Uloch sums. Numerical values for th, along with values of Vi and VL, are listed in Table 2.10.

~h ( h t j R I H I h y R ) = -(~g -t- k p ) .

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2.6. Tight binding approximation 169

In writing down the Bamiltonian matrix (htjk I H I hrj’k) in the hybrid- orbital representation, we arrange the eight basis functions in the sequence

the matrix (h t jk I H I httj’k) takes the form I hl lk) , I hzlk), I h s W , I h 4 W , I h l W , I h Z W , I h32k), I h 4 W . Then

. (2.296)

This matrix is also known as the Weare-l‘horpe Uarniltonian. Remarkably, its 8 eigenvalues can be obtained in closed analytical form. Denoting them by E,b, E:, i = 1 ,2 ,3 ,4 , we havve

Table 2.10: Hybrid matrix elements calculated from the TB parameters in Table 2.7 (in eV).

-8.27 -1.76 -2.98

Ge -8.37 -2.01 -2.76

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170 Chapter 2. Electronic structure of ideal crystals

r X wovevector

Figure 2.23: Evolution OF the energy bards of semiconductors with the diamond slructura within the TB approximation The right-hand part shows the band struc- t,ure of Si calculated by meam of equations (2.297).

Here gl(k) is the structuredependent factor d e b 4 in equation (2.282). The actual positions and k-dispersions of the bands (2.297) are deter-

mined by the parameters th, 1’1 and Vz. Using the values for S i given in Table 2-10, one obtains t h ~ bnrrd structure shown in trhf right hmd part of Figure 2.23. In this regard. the 4 bands indicakl by b lie below the energy gap, arid thP 4 bands indicated by u lie above it. This means that in the ground state of the crystal. the &lands are fully populated by electrons, i.c. they form the valence bands, and the a-bands are complekly empty, thus they are the conduction bands. Essential features of these relationships are the negative sign of V2, and the validity of the magnitude relation I V1 1 > 1 “2 I between the absolute values of and Vz. Figure 2.23 also illustrates how the different bands emerge from the atomic s-and p-1evrlb due to the two interactions V1 and V z . Roughly speaking, Vz determines the distance be- tween the cpnters of gravity of the valence and conduction band complexes, and V1 the width of thpse bands.

Now wc return to the main goal of this subsection. the ralcuIatjon of the total energy of the crystal. To accomplish this, the energy values of the four valence bands E,b(k) determined above must be summed over all i and k. It

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2.0'. Yi'ghht binding approximation 171

turns out that this task may even be carried out analytically if a suitable additional approximation, the so-called bond orbital approsirnation. is made.

Bond orbital approximation

To introduce this approximation, we review the already solved problem of diagonalizing the Hamiltonian (2.296). but procwd in a sompwhat different way, The diagonalization will now bt. carried out in two steps. In the first step, the eigenvalues and rigenvectors of the matrix (2.296) are calculated without taking account of the VI-terms, i.e. temporarily setting Vl to zero. As may he scen from formula (2.297). this leads to the reduction of the enerffi bands to the two dispersionless levels givw by

5k) belonging t o q, have the components

bik) (l/fi)[l, D,O,O,e;,O, O , O ) , (2.299)

(htlk 1 b4k) = ( l / J z ) ( O , 0: O , l , O , O , O , e: ) ,

and the 4 eigenfunctions I atk) belonging to E , have the components

(ht2k I agk) = ( l / & ) ( O , O ; 1,0,0,0: -ez, 0 ) ,

(ht2k I aqk) = ( l / & ) ( O , O , O , 1.0,0,0, -.;I.

Each of the eigenfunctions I btk) and 1 atk) is a linear combination of Bloch sums of two hybrid orbitals pointing toward one another, or, equivalently, a Bloch sum of the linear combinations of these orbitals. In the case of &states the hybrid orbitals are added, and the corresponding linear combinations I btR) are given by

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1 72 Chapter 2. Electronic structure of ideal crystah

(2.301) 1 I kR) = - 11 htl R)+ I h2Rt + R)] , t - 1,2 ,3 ,4 . 1/2

Thc exponential factors eF of the eigenvectors (2.299) and (2.300) were com- pensated by the et-factors of the Bloch sums. In the case of a-states the hybrid orbitals are subtracted, and the corresponding linear combinations I a , R ) are given by

The polar diagrams of these functions are shown in Figure 2.22. For reasons which will be clarified later, one refers to the orbitals I h t R ) as bondiny and to the orbitals 1 arR) as anhi-bonding orbitals. With the help of bonding and anti-bonding orbitals the eigenfunctions of the Hamiltonian matrix (2.296) with b'1 -- 0 may be written in the form

1 I btk) - eik'R I b t R ) , 1 - 1 ,2 ,3 ,4 , a R (2.303)

(2.304)

In the second step of the diagonalization procedure the Vl-terms are in- cluded. The Hamilton matrix (2.296) is transformed iuto the basis set of the previously calculated eigenvectors 1 btk) and 1 atk). This results in the (8 x 8)-matrix

(2.305)

with the (4 x 4) -matrices

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2.6. Tight binding approximation 173

{ 2.3 07)

as block elements. The expression for If, follows from that €or H & if in the latter q, is replaced by E,. The structure factors g t p and &) in (2.306) and (2.307) are defined as follows:

The (4 x 4)-matrix H M couples the various bonding states, and H,, couples the various anti-bop.ding states. The non-diagonal matrix Hab describes the interaction between the two types of states. It gives rise to corrections to the eigenvalues of relative order of magnitude I 1’1 I /2 I V2 I. Considering the actual values of V1 and V2. these corrections are rather small. This suggests treating them as perturbations. The zero-th approximation, i.e. the complete neglect of the interaction between bonding and anti-bonding states, is referred to as the b o d ovbital approximation. Within this approximation the valence and conduction bands follow from separate eigenvalue equations, the former by diagonalizing the matrix Hbbq the latter by diagonalizing Hw. To calculate the total energy of the crystal one needs the total energy of all valence electrons. The latter may be calculated by means of formnla (2.54) which expresses the total energy of an interacting electron system by means of its one-particle energies. One has

Er$&f’ = 2 E:(k) - E c d (2.309)

where E d means the Coulomb energy of the interacting valence electrons which is counted twice in summing upon all band states. The factor 2 accounts for the two spin states. Within the bond-orbital approximation, the i-sum in (2.309) may be carried out in closed form, with the result

k z

(2.310)

To pro>*e this relation, we write the eigenvalues @(k) in (2.310) as diag- onal elements of H M between eigenstates, and note that within the bond orbital approximation the eigenstates for a given wavevector k are linear

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1 74 Chapter 2. Electronic structure of ideal crystals

combinations of the bonding states I bik) only. In this matter, the linear combinations are generated by the unitary transformation which diagonal- izes the Hermitian matrix H a of (2.306). If one sums the eigenvalues @(k) over all i , and takes advantage of unitarity, then the diagonal elements of Hw with respect to the eigenvectors become the diagonal elements of this matrix with respect to the bonding states, i.e. q,, as is stated in relation (2.310). One may also prove this relation in another way, using Vieta's the- orem, which states that the sum of all zeros of a polynomial of degree n equals the negative of the coefficient of the (n - 1) - th degree term. In the case of the characteristic polynomial of a matrix, this coefficient repre sents the negative of the sum of all diagonal elements, here, therefore, -4q, confirming the validity of equation (2.310).

Total energy and covalent bonding

We proceed on the assumption that the atoms are arranged, as above, in the form of a diamond type crystal. The distance d between the nearest neighbor atoms will now be chosen, however, to have a value different from the do-value of the actual crystal. The total energy of this fictional diamond type crystal represents a function EEfAt"l(d) of d. We will demonstrate that E Z Z t a Z ( d ) reaches its absolute minimum at the finite distance d = do. This result constitutes a theoretical proof that the atoms bind themselves into the form of a crystal, and that the nearest neighbor distance will have the experimentally observed value. This does not prove the correctness of the di- amond structure of the crystal, for that was assumed a priori. To verify this, one must also show that no other crystal structure can yield a lower total energy minimum. We will not address this question, but rely on experience, which indicates that elements of the fourth group of the periodic table, under normal conditions, crystallize into the diamond structure. Our considera- tions here have the sole purpose of understanding why, in this structure, the total energy reaches a minimum at a finite distance d = do. In other words, we want to understand why chemical bonding should occur at all between atoms of group IV.

The total energy E;f$a'(d) of the crystal is composed of the energy of the valence electron system in the field of the atomic cores, as well as the energy of the atomic cores and their mutual electrostatic interaction energy. For the total energy EE$;f'(d) of the valence electrons of a crystal with G3 unit cells, one obtains from the relations (2.309), (2.310) and (2.298) the value

(2.3 11)

To get the total energy of the crystal, the energy of the atomic cores must be

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2.6. Tight binding approximation 175

added to the energy value of equation (2.311). In doing so, one may again use the fact that the core states of the crystal do not differ from those of the free atoms. This means that only the mutual electrostatic interaction of the cores results in a structuredependent energy contribution, while the internal core energies sum to a constant Eo. The core-core interaction energy has, approximately, the same value as the electron-electron interaction energy E d between the valence electrons on different atoms. This is true because the valence electron charge of an atom equals its core charge for the crystals considered here. The corecore interaction energy approximately cancels, therefore, against the negative Coulomb energy -E,1 of valence electrons in expression (2.311). Finally, the total energy of the crystal is given by

(2.3 12)

The d-dependence of this energy is due to the fact that both Eh and Vz depend on d - the hybrid energies Eh have d-dependence as they are defined by the diagonal matrix elements of the Hamiltonian H between Lowdin orbitals which contain overlap integrals between s- and p- orbitals of adjacent atoms, and V2 because this quantity is the matrix element of H between hybrid orbitals at nearest neighbor atoms. V2 decreases with decreasing distance d, corresponding to an attractive force between the atoms. The hybrid energy q, increases as d decreases, corresponding to a repulsive force. For large d, the attraction dominates over the repulsion, and for small d, the repulsion dominates over the attraction. Overall, the total energy E z l b l ( d ) of the crystal varies with d as shown in Figure 2.24 schematically. At the equilibrium distance do, it takes its absolute minimum value. This means that the initially free atoms will not remain free but form a diamond type crystal with nearest neighbor distance do. They experience what is called covalent chemical bonding.

In order to provide a better physical understanding of the nature of covalent chemical bonding, we compare the total energy E g t p l ( d ) of the crystal with the total energy E f $ T of 2G3 free atoms. For the elements of the fourth group of the periodic table with their two electrons in atomic €,-levels and two in atomic cp-levels, one has

(2.313)

where Eo, again, accounts for the energy of the atomic cores. The negative difference of the two energies (2.312) and (2.313) represents the cohesion eriergy of the crystal. It is given by the expression

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1 76 Chapter 2. Elecbronic structure of ideal crystals

Figure 2.24: Dependence of energy difference Eza?''' ( d ) -E::+Ts on f,he inter-atomic distance d (schematically).

crystal atom Etotol -Eta.ral

Formally, the occurrence of a positive cohesion energy is due to the fact that the matrix element V2 o l H bctween hybrid oxbitals at adjacent atoms pointing toward one another is negative, and that I 4v2 I exceeds the energy differericc (cP - cs). The latter difference may he understood as the energy increase of an atom if one of itb two .+electrons i s lifted into a p-state or, equivalently, if its four valence electrons a x put into four sp'-kybrid orbitals rathe1 than into two s- and two p-orbitals. One calls this population the pro- moted configuration of the atom. In sp'--hybrid states, the electrons of adja- cent atoms are capable of pronounced interference. This can be constructive or destructive, depending on whether bonding or anti-bonding states are con- sidered. In the casc of constructive interference, the probability amplitude becomes relatively large in the region between the two atoms and the two electrons of the interfering sp3-4ybrjds undergo a delocalization (see Figure 2.22). In this process, the potential energy of the Coulomb interaction of the two electrons among themselves and with the atomic cores remains al- most unchanged. However, their kinetic energy decreases considerably. This may be imderstood in terms of the Heisenberg TTncertainty Principle which tells us that a weaker localization, i.e. a larger positional uncertainty, cor- responds to a smaller momentum uncertainty and, therefore, to a smaller kinetic ene~gy. Altogether, the energy of the two electrons decreases, b e cause of constructive interference, in a bonding state. The energy gain per atom amounts to 4 I Vz 1. If it exceeds the energy necessary for promoting an atom into its sp'-state, i.e. if the condition 4 I V2 I> ( e p - F ~ ) holds, it is energetically favorable for covalent chemical bonding to occur. As we have seen, quanturti rtieclianical phenonienology is essential in the interpretation of this behavior. Unlike the bonding of electrically diflerent charged ions, covalent bonding between neutral atoms Lannot bc understood in terms of classical physics.

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The c:ondition necessary for thc occiirrcnce of bonding eigenstates able to host all vrtlprrcp electrons is the ordering of the newest, neigIi1,ors of an atom on the comers of a tetrahedron, i.e. the diamond structure of the crystal. In this way thc above consideration also justifies focming on the tetrtlhedral crystal structure of diamond type crystals, which was merely assumed at the outset. The atomic structure follows, so to speak, from the electronic structure.

Ionic bonding

The ionic contributions to chemical bonding will now be calculated for mat,+ rials having the zincblende structure. As is well-known, a scries of 111-V and 11-VI compound semiconductors form crystals of this type. For the Ilamil- tonian matrix (2.305), the transit,ion from the diamond to the zincblendc structure means that ch in the upper left (4 x 4)-block has to be replaced by the hybrid energy EL of the 1-atom? and in the lower (4 x 4 )-block by the hybrid energy e i of the 2-atom. With this replacement, the bonding and anti-bonding energy levels become

where !+, = c i - ~ 8 . Thr energy separation betwren the two levels is larger than that of diamond type crystals. This results in an enlargement of the energy gap betwcm the valence and conduclion hands. The bonding and anti-bonding oibilals arp given by the expressions

where we set ap - Vs/dV; + bp. T h e f d o r s {1/2)(1 -ap) and (1/2)(1+ctp) in ('2.316) and (2.317) represent the probabiMes of finding an elrctron in the bonding state at atom 1 or 2, respectively. One calls aP the polarity uj bonding orbitals or simply the polardty of bonding. If f f f is deeper than c i , ! then I!?, and also up, art' positive. The electron prpferpntially stays at atom 2. In this way the polarity of bonding orbitals is such that, in the ground state, whcre the electrons occupy only boxrding orbitals, the previously electrically neutral atoms heconie charged. Atom I becomes the positive cation, and atom 2 is the negative anion. The charge of the cation is given by cZ* with Z' = (Z, - 4 t $ap), where 21 is the number of valence electrons at

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178 Chapter 2. Electronic structure of ideal crystals

thc free 1-atom. The anion charge is -e (Zz - 4 - 4ap) = - e Z * , i.e. the unit cell is neutral. Owing to this redistribution of electron charge, the electron-electron interaction energy to be subtracted from the sum of one- particle energies, because of double counting, takes a different, value. It is, therefore, no longer completely compensated by the electrostatic interaction energy between atomic cores. This leads to an additional contribution to the total energy of the crystal which may be interpreted as the electrostatic interaction cnergy between anions and cations. Onc calls it the Madelony energy EM&- The general expression for E M a d iu

where the sum extends over a periodicity region. With E M a d , the total energy of the crystal is

The Madelung energy is negative, i.e. it strengthens chemical bonding. Since the bonding is then pdrtially due l o attractive forces between ions, OTW refers to it as partially zonzc bondzng. The absolute value of the Madelung energy is, on the one hand, proportional to the number GT of unit cells, and on the other hand, inversely proportional to the distance d between two adjacent ions. One therefore sets

(2.320)

with cy as the so-called Madelung constant. The latter depends on crys- tal structure and can easily be calculated numerically. In Table 2.11, the n-values are listed for crystal structures which are observed in materials composed of group IV elements as well as 111-V, 11-VI and I-VII compounds. The value for the wurtzite structiirp in Table 2.11 corresponds to the ideal tetrahedral case with an equivalent cubic lattice constant &a (see Chap- ter 1). The contribution of the Madelung energy to the total cnergy of a given compound will be larger for larger effwtive charge number Z* of the compound. This results in a tendency of compounds with larger 8* valiirs to crystallize in structures with Madelung constants larger than that of the zincblende structure. Therefore, in passing from the 111-V through the II- VI to the I-VII compounds, one observes a transition from the zincblende structure through the wurtzite to the rocksalt and cesium chloride struc- tures. The cesium chloride structure follows from the rocksalt structure by replacing the two facecentered cubic sublattices by two primitive cubic sub- lattices, shifted in the same way with respect to each other as in the rocksalt

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2.7. k . p -method 179

Table 2.11: Madelung constants for several crystal structures.

Zincblende 1.6381 U’urtzite 1.6410 Rocksalt 1.7476

structure, i.e. by ( a / 2 , a /2 , a/2). With growing polarity of the bonding, the energy gap becomes larger, as mentioned above. This explains the transition from the semiconducting properties of the group IV crystals to the insulating nature of the I-VII compound crystals.

In the case of the I-VII compounds, the absolute values of V3 are so large in comparison with V2 that the bonding polarity op is approximately unity. This implies that almost all valence electrons of the compound stay at the anion. Then the crystal consists of positive ions of the group I atoms, which have lost all their valence electrons, and negative ions of the group VII atoms whose valence shells are completely filled. One refers to such crystals as tonac crystals. In this case, the energy gain due to the transfer of electrons from cations to anions, which represents an essentia1 part of the bonding energy and forms the driving force for the formation of ions, no longer depends on the crystal structure. This structure is determined by the hladelung energy only. Therefore, ionic crystals exhibit structures with particularly large hladelung energies, i.e. rocksalt and cesium chloride structures.

2.7 k. p -method

2.7.1 Fundamentals

Luttinger-Kohn functions

The k. p-method rests on a particular property of the BIoch type eigen- functions pV,+(x) of the crystal Hamiltonian H . As we know these functions (which will be denoted below by (xluk) instead of q y k ( x ) ) are the product of an exponential factor exp(ik.x) and the latticeperiodic BIoch factor uvk(x). If one replaces the wavevector k in uYk(x) by a constant ko, while retaining k in the exponential factor, then the resulting functions

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1x0

[2.321)

are no longer eigenfunctions of H ~ of course, but they do form a complete orthonarmalizPc1 basis set in Hilbert space, as wcll as the Bloch functions. whence

(v’k’ko(vkko) - 6 v l u 6 k ~ k , (2 322)

~ ( x ’ l v k k o ) ( v k k o l x ) -- b(x‘ - x). (2.323) uk

Thp vdidily of these relations f d o w b directly from the rorripletenyss and orthonormality of the Bloch functions. The (xlvkko) are referred tu as LuftmgGgP7 - K u h fiLa~t7onu. T h y arp determined by the Blorh factors uVk(x)

for the special wavevector ko in contrast to the Bloch functiom which require full knowledge u l u,k(x) lor all wavevectors k. The k . p-method takw advantage of this properky of the Luttinga-Kohn functions.

In this method, one represents the Srhriidinger quation for R crystal electron in terms of the complete orthonormaIized set of these functions. The rpsulting matrix elcments of H can be expressed, as we will s e e later, by the matrur elements of H between the Bloch factors Uyk(X) for k = ko. These elements arc, of course, just as little known as the Bloch factors them- selves. However, one may take them as empirical parameters. If one does so and inserts values for the parameters. then the Hamiltonian matrix in the Luttinger-Kohn basis is completely determined. Uiagonalizing this matrix yields the eigenvalues and eigenfunctions of the crystal Hamiltonian H for all valiies of k. Tllis means that the k p-method allows 011r to calrulate, from the Bloch matrix elements at only one point kol the eigenr-alues and cigenfunctions over the entirc first €32, i.e. to extrapolate from thc particu- lar point ko to the entire first BZ. ()€ten one is only interested in solutions in the vicinity of a critical point k, e.g. in the vicinity of the valence band ninxinium or the coridurtion band minimimi. Then it is expedient. although not necessary, to icienliljr ko with k,. If k, hes. for example, at the center of the 6rst BZ, as often occurs. one has ko = 0. This choice will be used later. At the outset, ko should still be considered an arbitrary point of the first HZ.

In order to accomplish the ahove program. we expand thP Bloch functions (xjuk} with respect to Luttinger Kohn functions (xjpk’ko). On\y terms with k’ = k occur in this expansion because of the lattice translation symmetry of both functions, whence

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2.7. k . p -method 18 1

With this expansion, the Schriidinger equation (2.178) in the Luttinger-Kohn representation bmomes

k . p - Hamiltonian

‘l’he matrix elements (pkko I H I p’k‘ko) of the llarniltonian bptween T,uttingpr Kohu functions can he t r a d back to matrix elements (pko I p j p’ko) ol the momentum operator p between Bloch functions. if one uses the easily provcn commutation relation

[p2, eik-x] = eik-x (p2 f 2fik. p + h2k2) , (2.326)

which yields

where we have set

(2.328) E:(k) = E,(ko) + -(k - ko)2.

The matriv on the right-hand side of (2.327) allows for an important rewrit- ing. If one defines

Ti2

2m

(2.329) h

Hk.p(k) = Ho(k) + -(k - ko) . p! na

with

(PkkolfflP’kko) = (PkoIHk.,(k)lP’ko). (2.3 3 1)

The latter relation means that the actual Hamiltonian matrix W in the k- dependent Luttinger-Kohn basis Ipkko) equals the representative matrix of a fictional k-dependent Harniltonian Hk.p(k) in the k-independent partial Bloch basis lpkoko) = jpko) for the wavevector k = ko. The k-dependence of the Luttinger-Kohn basis on the left hand-side of equation (2.327) has been transferred to the new Hamiltonian Hk.p(k) on the right-hand side. The SchrGdinger equation (2.325), with this new Hamilt onian, reads

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182 Chapter 2. Eiectronic structure of ideal crystals

The components of the eigenvectors Ivk) in (2.332) refer to the Luttinger- Kohn basis Ipkko), although the operator Hk.p(k) is represented in the Bloch basis Ipko).

Solution of the Schrodinger equation (2.332) involves the diagonalization of the matrix (pkolHkE,(k)lp'ko). For k = ko this matrix is automatically diagonal, by virtue of the fact that Bloch functions Ipko) are eigenfunctions of the Hamiltonian Hk.p(kO) = Ho. For k # ko, the Ipko) states are no longer eigenfunctions of Hk.&), so that the matrix (pkolHk,(k)Ip'ko) has off-diagonal elements with respect to the band indices. Formally. one may interpret these non-vanishing elements as arising horn an interaction between different bands. Since this interaction results from the (k - ko) . p-term in Hk.p(k), one calls it the k . pinteractian. In this, the bands which are mutually coupled, are not bands in the sense of the eigenvalues of the actual crystal Hamiltonian H - the latter are uncoupled by definition - they are fictional bands E:(k) defined by equation (2.328).

As the point k in Hk.,,(k) approaches ko, the k . p-interaction tends to zero. For k-vectors sufficiently close to ICO, one can treat this interaction with the help of quantum mechanical perturbation theory. Apart fkom the square term in (k - ko) already present in E:(k), this entails a power series expansion of the energy bands E,(k) with respect to (k - ko) about the point ko. The form of the perturbation theoretical expansion depends on whet her the unperturbed bands, i.e. the eigenvalues E,(ko), are degenerate or not. We will first consider the simpler case of non-degenerate bands.

Application to non-degenerate bands. Effective masses

In first, order perturbation theory the eigenvalue E$(k) arising from EE(k) is given by the relation

(2.333)

and the Bloch function Ivk)' arising from 1.k)' E Ivkko) by the relation

Since the f m t derivatives VkE,(k) of the exact band energies E, (k ) at ko depend only on linear expansion terms of E,(k) in k - ko, no approximation is needed to obtain the relation OkE,(k)lk, = VkEb(k)lb. Considering

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2.7. k . p method 183

(2.3331, this exact relation yields V&v(k)lb = ( h / r n ) ( v b l p l v h ) . This holds the same content as equation (2.193) used above without proof, because ko may be an arbitrary point of the first BZ. In particular, if ko is a critical point kc, i.e. if V&,(k)\k,, = 0 holds, then the k. p-correction vanishes in first order perturbation theory. One must proceed to the second order to get a non-vanishing contribution from the k p-perturbation. The result reads

where, for brevity, we set (I& I p I pkc) = (v 1 p I p). We rewrite this expres- sion in the form

Generalizing the terminology introduced in section 2.6, we call ML1 the effective mass tensor at the critical point k,. For the diagonal elements m z ' of At;' with respect to the principal axis system, one obtains from (2.337) the relation

This relation connects the effective masses with the matrix elements of the momentum operator between different bands and with the energy separation of bands at the critical point. The tendency indicated is that the absolute values of the effective masses become larger for smaller momentum matrix elements and larger band separations. One expects small effective masses for large momentum matrix elements and small band separations. As far as the band separations are concerned (only for them can one make an easy estimate), we will later find conhmation of this tendency in all concrete cases. For pairs of bands which are closer to each other than to all other bands and, therefore, whose mutual interaction is stronger than that with all other bands, relation (2.338) allows one to also draw a conclusion about the signs of the effective masses. According to it, the energetically higher of the two bands should have a large positive effective mass, and the energetically lower a mass of the same large absolute value but of negative sign. This

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184 Chapter 2. Electronic structure of ideal C F Z . S ~ ~ ~ S

conclusion also proves to be valid in all cases in which the assumptions of this calculation apply.

Band degeneracy

Critical points are often symmetry centers or lie on symmetry lines, and at these symmetry points, degeneracy of the energy bands often occurs. If this happens, one must carry out second order k. pperturbation theory €or degenerate bands. In quantum mechanics, perturbation theory for degener- ate energy levels is c o m o n l y of first order - the matrix of the perturbing Hamiltonian operator between the degenerate states has to be diagonalized (we remind the reader of the nearly free electron approximation in section 2.4). This procedure does not apply here because the perturbation matrix at critical points vanishes in first order. One must therefore choose a variant of perturbation theory for degenerate energies which works in second order. To this end, one constructs the matrix of the perturbation operator not between degenerate unperturbed eigenstates, as is commonly done. but between the (also degenerate) eigenstates of first order of perturbation theory. By diago- nalizing this matrix one obtains the eigenvalues in second order perturbation theory. These are, in general, no longer degenerate.

An important case in which the k . p-perturbation matrix between the degenerate unperturbed states vanishes, is the valence band maximum of semiconductors with diamond structure. This case will now be investigated. In doing so, we initially neglect the spin-oIbit interaction. This approxima- tion is valid for semiconductor materials composed of light elements only, including, for example, Si. For other materials this procedure serves as a zero-order approximation which can be used to proceed further (as we will do below).

2.7.2 Valence bands of diamond structure semiconductors without spin-orbit interaction

As we know from section 2.3, the valence band maximum of diamond type semiconductors is located at the center r' of the first B Z . Therefore, we set & = 0. The maximum is 3-fold degenerate. We denote the three pertinent Bloch functions by IvmO), where m can assume the values 2, g, 2. According to section 2.6, these eigenfunctions belong to the irreducible represent ation l?b5 of the cubic group oh. As indicated in Appendix -4. a basis of this rep- resentation is formed by the products yz, zx, z y of the components 2, y, z of position vector x. Therefore, with regard to their transformation properties under the action of elements of o h , we may identify IvzO) with yz, IvyO) with zz, and IvzO) with zy. The vector components 2, y, z of the position vector itself transform in accordance with the irreducible representation I'15 of Oh.

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2.7. k . p -method 185

For the subgroup T d of oh the two representations F15 and r h 5 coincide. For semiconductors having the zincblende structure, the three degenerate states ( v I O ) , IuyO), IuzO) of the valence band maximum may therefore be associ- ated with I, y, z insofar as their transformation behavior is concerned. In the case of the diamond structure, 2, y, z are merely a short hand notation.

The vanishing of the matrix (vmOlplum'0) of the momentum operator between valence band states at I?, anticipated above, may easily be demon- strated using the pertinent criterion for such vanishing given in Appendix A: The operator p transforms according to the irreducible representation I'15

of oh. The matrix (umOlplvm'0) therefore belongs to the reducible repre- sentation x r 1 5 x rl,, = I?;, x (rh + ri2 + I'15 + r25) , wherein the identity representation does not occur. According to Appendix A this means that the matrix (umOlplvm'0) must vanish. One can also obtain this result by means of inversion symmetry alone. We have chosen the somewhat more troublesome method of proof because it may also be applied in other, less obvious cases, as we will see immediately below.

k . p -perturbation theory to second order with degeneracy

In order to apply degenerate second order perturbation theory, the solu- tions of Schrodingers equation (2.332) are needed to first order in the k . p- perturbation. For the orthonormalieed Rloch valence band eigenstates (vmk)l one finds

(2.339)

where we set E,(O) = for brevity, and the degenerate valence band energy EJO) is denoted by Ev. The third term in (2.339) guarantees the normal- ization. (:onsidering the sum on p , tlir value 11 = 71m does not need to be specifically excluded because the matrix elements ( p I p I vm') for p = 117n vanish anyway. Expressions of tlic form (2.339) also hold for the approxi- mate Bloch functions Ipk)' of the remaining bands p with p f urn, but we omit an explicit presentation of them here. The states Ivmk)' and Ipk)' with p # t m will now be used as a basis set to represent the IIamiltonian H . The resulting matrix is 'almost' diagonal, because the basis functions are 'almost' eigenfunctions. In particular, the submatrix of the three velence bands is coupled to the remainder of the matrix only by elements of second order in the k . p-perturbation. These elements give rise to corrections of the valence band energies which are only of third order and can be neglected.

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186 Chapter 2. Electronic structure of ideal crystals

In second order perturbation theory, the valence band energies Ev and For- responding Rloch states I&,) therefore follow from an rigenvalue equation which is decoupled from the remaining bands, namely

'(vmklH!vm'k)l l(wn'kl&:y) = E ' , '(vrnklE,). (2.340)

The initial occurrence of interaction betwren the valence band and the r e maining bands is incorporated in the matrix (u>m k I H I wn'k) in first order perturbation theory.

rn,

Harniltonian matrix

The (3 x 3)-Hamiltonian matrix '(vmklHlvm'k)' of equation (2.340) can be obtained by means of expression (2.339) for the perturbed states Ivrnk)'. A short calculation yields

where

is a fourth-rank tensor. Since the states I p ) = 1 P O } are eigenfunctions of H with eigenvalue EEl = E,(O), one may write (2.342) in the more compact form

With respect to the indices arPl the tensor D z i , is symmetric, and with respect to the indices m , m' it is IIermitiaa From equation ('2.345) one can see that D Z k , transforms under symmetry operations of the cubic group Oh according to the $-fold product representation [rb5 x ri5]a x [1'1~ x l?1~],

where the index s denotes the fiyrnmetrical part of the product. According to Appendix A, L?zk, then contains as many independent elements as the number of times the identity representation occurs in the product [I?& x riE;lrn x [I'Is x I115Is. Using Appendix A , one finds [I'15 x rl5Is = [I& x r&lS =

3c, -k 5T'k5 The tensor D z L , therefore has three independent components. one can show that these correspond t o the three types of non-vanishing

r1 + r12 + ra,, which yields [rk5 x r& x [rI5 x rl& = 3r1 + rz + 4r12 +

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2.7. k. p -method 187

matrix elements D z z , Ll;& and D Z . We introduce the abbreviations L - D g , hl = D$$, and N = D$ f D g . The elements L, M and can be calculated if the Bloch factors are known. In the absence of this informalion, however, we consider L , M and N to IIP empirical parameters (as indicated at the outset) and use their connection with the Bloch factors only to identify some general properti=, such as the fact that they can be chosen real. Since all remaining matrix elements Dmm, vanish, the Hamiltmian matrix of the valence band has the form

QB

Method of invariants

The Hamiltonian matrix (2.344) can also be derived in a somewhat different way, whirh leads to the goal more quickly, but is formally more dernand- ing. One uses the fact that the Hamiltonian matrix l(vmklHlum'k)l can be represented as a linear combination of the 9 matrices of a basis in the product space 1 vmO) (iirn'O 1 which transforms according to the repreaenta- tion [I& x r:,] of the point group oh. This representation is reducible. By decomposing it into i t s irmliicible parts, m e obtains a basis which consists of subbases, each of which belongs to a particular irreducible representation of Oh. Such a matrix basis can easily be constructed by means of the 3- dimrnsionel angular niomenturn matrices Iz, Iv, Iz (considered in Appendix A ) and their products, since it is known how these matrices transform, namely according to the pseudovector representation I':s. In the product spacc k,f-fi of the components of the vwtor k, one proceeds in a similar way. One determimes a basis from subbases which transform according to the ir- reducible parts of the representation [I'15 x r15Iy. The Hamiltonian matrix reprewrits an element in the product space of Ihe two spaces which is invari- ant under transformations of the point group Oh. Such invariant elements of the product space can be produced by forming scalar pruducts of subbases of the two spaces whi& transform according to the same irreducible repre sentation. As seen in Appendix A, the corresponding scalar products belong to the- identity representation, that is to say, they are invariant.

To find the most general Hamiltonian matrix compatible with the sym- metry oh, one has to determine all invariants of the product space. I.€ one then multiplies each by a real scalar factor and sums them all, one obtains

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188 Chapter 2. Electronic structure of ideal crystals

the most general invariant of the product space and thus the most general Hamiltonian matrix compatible with Oh symmetry. This process is called the method of invariants. It is applicable to arbitrary symmetry groups and degrees of degeneracy, and it quickly leads to the goal if one considers spin and spin-orbit interaction. It also allows one to determine the matrices for perturbing Hamiltonians other than that of the k . p-interaction, such as the interaction between the angular momentum of Bloch electrons and an external magnetic field (see section 3.9) or the interaction with mechanical strain. In this book, we will only use the method of invariants occasionally. A comprehensive outline of the method with several applications i s given by Bir and Pikus (1974).

Valence band s t ruc ture

The eigenvalues of the matrix (2.344) form three valence bands E,l(k), E,n(k), E,3(k). For the three symmetric k-directions [loo], [lll] and [110] the dispersion curves are determined by simple analytical expressions as fol- lows:

E,lp(k) = M k 2 , E,3(k) = L k 2 , (2.345)

1 1 3 3 E,lp(k) = - [L + 2M - N]k2, E,3(k) = - [L + 2M + 2 N ] k 2 , (2.346)

E,l(k) = M k 2 , (2.347)

1 1 E,2(k) = - [L + M + N]k2, E,3(k) = - [L + M - N ] k 2 . (2.348)

2 2

Along the two directions [loo] and [lll], the valence band, being triply degenerate at r, splits into two bands, one 2-fold degenerate and one non- degenerate (see Figure 2.25). In the [llO]-direction and also for all more asymmetric k-vectors, no degeneracy remains. This indicates a %fold split- ting of the valence band for such k. All bands are parabolic, but evidently, in general, not isotropic. One speaks of a warping of energy bands. In the case of Si, one has L = -5.64, M = -3.60, N = -8.68 in units of (h2/2m). Using these values, the two degenerate bands E,l/a(k) of equations (2.345)

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2.7. k . p method 189

Figure 2.25: Valence band dispersion for diamond type semiconductors in the vicinity of the I'i5-maximum for different k-directions.

or (2.346) have smaller curvatures than the third band E,s(k) in these equa- tions. Thus the first two bands correspond to the heavy holes and the third band to the light holes of Si. Isotropy exists only if N = 0 and L = M . Then, there also is no longer any distinction between light and heavy holes. Conversely, anisotropy grows stronger as the difference between the masses of the two types of holes becomes larger.

The results discussed above were obtained without consideration o l spin- orbit interaction. However, for most of the diamond and zincblende type semiconductors, the valence band structure is significantly influenced by this interaction (in the case of Si it is small, hut often not negligible). We now proceed to consider the effects of spin-orbit interaction.

2.7.3 Lut t inger-K o hn model

Including of spin-orbit interaction

To include spin-orbit interaction the electrons must be treated as particles with spin. The electron states then depend on both the position coordinate x, and on the spin-coordinate 3, and the quantum numbers of the electron states contain provision for both the motion in coordinate and spin state u describing the spin motion. A set of basis functions for representation of the electron states in coordinatespin space may be formed lrom the Bloch functions (xlpk) or the Luttingw-Kohn functions (xlpkO) without spin, aug- mented by multiplication with the eigenfunctions ( s l u ) of the z-component of the spin operator. Here u means the spin quantum number which can have the two values u -r (spin up) and cr =I (spin down). We denote the product functions by (sxlpk) or (sx(pkO), respectively. Then, by definition,

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190 Chapter 2. Electronic structure of ideal crystah

we have

The total Hamiltonian of the system is obtained by adding the spin-orbit interaction H, to the Hamiltonian H in its absence, where Hgo is given by equation (2.56) as

(2.3501

One has to be aware that the (sxlvuk) are eigenfunctions of H , but not of H + H,. Correspondingly, the (xlvuko) signify the spin-dependent Luttinger-Kohn functions of H , but not of fi + f lW. The matrix repre senlation of the Schrtdinger equation with respect to the spin-dependent Luttinger-Kahn basis reads

x ( p r k O t H + H,, Ip’v‘kO) (p’u’kO I El , ) - E , ( pu kO I E,) . p W

(2.351)

In calcuhting the matrix of H t H,, of (2.351)) an additional k-dependent term appears in comparison with the spin-less case, as a consequence of the fact that H , contains the momentum operator. This term has the same form as the (A/rn)k. p-term arising from H . except that the p-operator is replaced by the operator p+(1/4rnc2)[5 x OV(x)]. The additional term can be taken into account by replacing the operator Ht.p of equation (2.329) (for ko = 0) by the operator

Tl Hk.x = H o ( k ) + -k . a, (2.352)

m with

1 4m c=

ii = p + ---[a x VV(X)]. (2.353)

With this the Luttinger-Kohn representation of H + H,, becomes

(pukOlH + H80/p’a’kO) = ( ~ u O I H ~ . ~ + Hso{p‘u‘O),

and the Schrodinger equation (2.351)) takes the form

(2.35 4)

C ( ~ ( T O I H ~ . ~ + HsoIp‘u’O)(p’u‘kOIEu) = E , ( p n k O l E , ) . P’U’

(2.355)

Up to this point, we have kept the discussion general. Now we wish to ex- plore the particular consequences of spin-orbit interaction for the previously

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2.7. k p method 191

considered valence band states. To this end, we need the matrix ele- ments (vrnuO~Hso~wm’u’O) of H,, between the spin-dependent Bloch states I vm 0 .0 )~ namely,

F,

4m c ( u ~ u O I H , , I ~ ~ ’ U ’ O ) = ~ ( v v ~ O I [VV(X) x p] Ium‘O) x (aldld). (2.356)

To evaluate the matrix element (vmOl[OV(x) x p]]vm’Oj in coordinate space we make use of crystal symmetry, as was done before, in the calculation of the matrix elements of the momentum operator p. The operator [VV(x) x pi is a pseudovector and transforms according to the irreducible representation ri5 of oh. The entire third-rank tensor (vrnOI[VV(x) x p]lvrn’O) therefore belongs to the reducible representation I’b5 x Ti, x P2, = rl,, x (r2 + r12 + T’i5+I7L5), in which the identity representation occurs exactly once. The ten- sor (vmO ! [VV(x) x p] I vm’O) consequently contains one independent con- stant. This constant coincides with the matrix elements (vyOl[VV(x) x pIz I v z O j = (vzO I [VV(xj x plX I vyO) = (vzO I [VV(x) x p]ylvzO), as well as

where (vxOl[VV(x) x plz IuyO) = -(uyO I [VV(x) x p],lvzO) holds. Because the Bloch factors are real? these elements are pure imaginary. We denote the value of (vyOl[VV(x) x p],jvzO) by (4m2c2/h)(i/3)A, i.e. we set

(~.Ol[V(X)XPl* lvy0) = (WYOI [ w x ~ x P l . / ~ ~ o ) = (VZOl PV(XjXp1ytvrO).

h A 4m c 3

,(uyO~[VV(x) x plrlvzO) = i - . (2.357)

Below, we will see that the constant A is the energy splitting of the valence band at J? due to spin-orbit interaction. Applying equation (2.357) and the explicit form of the spin matrices given in (2.57), the matrix (avmOlH,I a’w m ‘0 j becomes

A 3

(vmaOp,,Ivm.’u’O) = -

0 - i 0 0 0 1

i 0 0 0 0 - 4 0 0 0 - 1 i 0 0 0 - 1 o i 0 0 0 - i - i o 0

1 i 0 0 0 0

(2.358)

Here the rows end columns are associated with the basis functions in the sequence Ivz T O j = Iz t) , Ivy 7 0 ) = ly t j , . . ., It)z 1 0 ) 3 1s 1). Spin- orbit interaction couples orbital states and spin states to each other. At k = 0 the expressions E:(k) and k ii are both zero. The eigenenergies and eigenfunctions of the total Hamiltonian Hk.a + H , are therefore also those

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192 Chapter 2. Electsonic structure of ideal crystals

of H,, alone, so thsl the Schrfidinger equation (2.355) at k - 0 becomes

c (vmaO~H,,lvm'a'O)(vm'a'O~E) = E(wamO(E). (2.359) u'm'

Eigenfunctions at r. Angular momcntum basis.

The matrix (2.355) has a 4-fold degenerate eigenvalue

1 Ev1/2/3/4 = p

IE ) ~ T ( 1 , i, o,o, O,O), IEv2) -

with the four corresponding eigenstates, respectively,

1 v1 - Jz

(2.360)

1 i IEv3) = z(l, - i , O , 0 ,0 ,2 ) , IEv4) = -(O, O,O, 1, - i , O ) , (2.361) Jz

and a 2-fold degenerate eigerivslue

(2.3 62) 2 3

Ev5/6 - --A

with the two corresponding eigenstates

I E v 5 ) - -(o, 0, 1, 1, i, o), i (2.363)

1 1E ) - ----(--I, i, o,o, 0, 1). fi w6 - d3

The components of the eigenvectors given by (2.361) to (2.363) refer l o the basis functions IvmaO). If we abbreviate these by Ima) IwmaO), using Iz T), Jy I), 12 I), Iz J), ly I ) , Iz i), the cigcnvcctors take the form

The eigenvectors lEvt), i - 1,2, . . . , 6 of (2.361) have a simple meaning. The lEvl), IEvz), IEv3), IEv4) are basis functions of the irreducible representation

of the cubic group o h . Acrording to Appendix A, these representations

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2.7. k p -method 193

emerge from the representation 213 of the full rotation group if, D D ~ is taken as a representation of the subgroup oh with +1 for inversion. It has also been shown that the basis functions of this representation are the simultaneous eigenfunctions of the angular-momentum-squared 5' €or the eigenvalue j ( J + 1) (in units R ), and of the z-component J z of J for the eigenvalues m j =

of (2.364) by

2 - -

2 , 3 1 . - 1 . = L 2 , - and -$ 4. One therefore denotes the first four eigenfunctions

(2.365) 3 3 1 3 1 i I--) = -1rz + iy t), I--) = -"-2Iz t) + la: + iY 1 1 1 7 2 2 4 2 2 f i

3T 1 3 3 i I--) = -[I. - iy T ) + 2 / 2 111% I--) = -1. - i Y I), 2 2 & 2 2 Jz

The lEvs), IE,s) are basis functions of the irreducible representation I'y of o h . These representations do not arise from any representation V3 of the full orthogonal group. in particular not from a representation for J = 4 (this happens with r t in the case of Oh or r:) in the case of T d . but the expectation values of J2 and J z are the same as those in the Dl basis. Therefore one also uses the angular momentum notation €or the last two eigenfunctiona of (2.3641, i.e. one sets

Each of the dgenvectors (2.366 and (2.366) is determined only up to a phase factor, which is chosen heie such that the states with negative total angular momentum, I?:), I$:), 1 follow, respectively: from the states with positive total angular mornenturn, I$$), I;;), 1;;) by means of time reversal, ie . by forming the comphx conjugate of the original eigenvector and subsequently multiplying it by rP.

We refer to the functions Ijmj) of (2.365) and (2.366) henceforth as the angular momentum basis. According to (2.360) and (2.3621, eigenstates hav- ing the same eiggmvalue of the the angular momentum squared, J2, also have the same energy eigenvalue, while the rncrgy eigenvalues differ if states with different eigcnvalues of J' are consitirrrd. The valence band: being B-fold degenerate at the r-point if spin is not taken into account, therefore splits into two bands, one with j : 3, and one with j = 4, if the spin-orbit in- teraction is considcrccl. That such a .ynin,-o)rhit qditting must occiir? one can recognize just by means of a grnnp theoretical analysis of the problem. The six valence hand states at r transform in accordance with the &dimensional representation Dl x I';, of Oh. Tlicse representations are reducible, accord-

ing to Appendix A , as Di x = Y$ + I';. The size of the splitting is given T

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194 Chapter 2. Electronic structure of ideal crystaJs

by the constant A, which determines the strength of the spin-orbit interac- tion. Therefore A is called spin-orbit splitting energy. One may interpret A as the difference of the spin-orbit interaction energy between the states with j = 3 and those with j = i. As one should expect, the states with larger angular momentum lie energetically above the states with smalIer angular momentum. States with different m j , i.e. with different projections Jz of total angular momentum on the z-axis, but the same J2, have the same spin- orbit interaction energies. Therefore the degeneracy of these states remains.

3

Valence band structure off r The above statements refer to valence band states at the center r of the first B Z , where the k p-interaction vanishes. Off r, this interaction is no longer zero and must be taken into account in addition to the spin-orbit interaction. We have seen how this can be done approximately in the preceding section, without consideration of spin. The method used there indicates the following procedure in the presence of spin and spin-orbit interaction: One determines the functions Ipok)' which diagonalhe the operator H k q of (2.352) in first order perturbation theory. In analogy to equation (2.339), one finds for the valence band states Ivmcrk)' the expression

where we use the same abbreviations as in (2.339). Analogous relations hold for the states of the other bands. The functions Ipgk)' form a complete orthonormal set in t a m s of which the Hamiltonian H may be represented. The submatrix with respect to the valence band states (vmmk)' is decou- pled from the remainder of the matrix in second order perturbation theory. Since H,, also only couples the valence band states among themselves, but not to states from other bands, the Schrdinger equation (2.355) in this representation reads

The spin-dependent term of k. 7i in the eigenstates Ivmcrk)' of (2.367) d e scribes the change of the k. p-interaction due to spin-orbit interaction. '5' ince t,he two interactious are supposed to be weak, this change i s second order small. It will be omitted below. Then we have, approximately,

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2.7. k 9 p -nrt.th<>d 195

(hjlff,olj’m;) =

(vmakIH,,lv7n’a’k)’ (vmaOl H,qolvm’a’O).

Now we use the fact that H,, is diagonal in the angular momentum basis Ijmj) of (2.365) and (2.366). It is clear that this basis follows from Ivma0) E Imu) by a unitary transformation

o $ o o 0 0

o o + o 0 0

o o o g 0 0 ’

0 0 0 0 - 9 0

(2.372)

~ j m j ) CUr rm jm j Ima). (2.3 70) ma

The corresponding unitary transformation matrix UmjmJ can be readily obtained from the rclations (2.365) and (2.366). One has

(2.371)

If one applies this transformation to the Schrodinger equation (2.355), then the matrix (vmaOlH,,(vm’a’O) takes the diagonal form

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196 Chapter 2. Electronic structure of ideal crystds

The sum of the two matrices (2.372) and (2.373) is the new Hamiltonian matrix. It has the same eigenvalues as the original matrix, even though its form deviates from that of the original. The difference in form is, above all, that the new matrix is already diagonal at k = 0. Kon-diagonal elements occur for k # 0. Among them, the elements between basis vectors ] j m 3 ) and [ j ' rn$) with m3 # m i , , but = 3' play a different role than the ones with .f j'. While the influence of the 2-diagonal elements on the eigenvalues is

independent of the size of the spin-orbit splitting A, it does depend on it for the j off-diagonal elements. The magnitude of the latter can be estimated as the larger of the two terms Nlki2 or IL - Mllkl'. If one assumes that l L f a ~ ( ~ V , IL - AIl}lk12 << A holds, then a perturbation theoretical treatment is possible. It yields an energy correction of order of magnitude [ M a r { N , IL- M1}]21k14/A. Under the assumptions made, this is small compared to A. That means that the 3-off-diagonal elements of the transformed Hamiltonian matrix can be neglected if the k-vectors are sufficiently close to r. We will assume below that this is the case, although the Luttinger-Kohn model also covers the general case of a (6 x 6 ) -Hamiltonian matrix. Neglecting ]-off- diagonal elements the Hamltonian decomposes into two blocks, one (4 x 4) -block conesponding to the basis vectors of the representation I'i, and one (2 x 2)-block for the basis vectors of the representation I't. The rsf-matrix reads

Here, therows a n d c o l u r r ~ u a r e o r d e r e d i n t , ~ ~ ~ ~ u ~ c e I;:), I;;], I::), I::), thc 1 multiplying A/3 is thc (4 x 4) unity matrix, and the quantities &? R, S,T stand for the fobwing expressions;

9

The Hamiltonian matrix of thr rz-valence band can also be derivcd using the method of iiivarinnts, which was discuss& at an earlier stagc. To do this, one needs the angular momentum matricrs for spin J = 4, as well aa their products with earh other. These matrices arp given in Appendix A. The

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2.7. k . p +method 197

resulting Bamiltonian matrix agrees exactly with that of equation (2.374). This means that the neglect of the coupling terms between the spin-orbit and k p-interactions has no influence on the general form of the r$-Hamiltonian matrix. Even with this additional approximation one obtains the most gen- eral I'8+-Hamiltonian compatible with the symmetry of the crystal. Only the explicit expressions for the constants L , M , N in the matrix DZ:, of (2.343) are affected. Since these arc understood as empirical parameters, this also does not play an important role.

The matrix (2.374) has the two 2-fold degenerate eigenvalues E&(k),

(2.376) 2

Using the explicit expressions for Q, T, R , S, this yields

1 1 1 3 3 3

A = - (L + 2M), B = - (L - Ad), C 2 7 - [ N 2 - ( L - M ) ' ) ] . (2.378)

Thc T': (2 x 2)-matrix block of the full (6 x 6)-matrix is already diagonal. Its 2-fold degenerate eigenvalue Er7(k) is given by

The energy level Eo1/2/3/4 Bra(0) in equation (2.360) is 4fold degen- erate at I?. Correspondingly, two 2-fold degenerate bands Erf,(k) arise off of I?, and starting from the 2-fold ckgenerdte level Ew5/6 = Ey7(0) at r in (2.362), which lies at an energy separation A below, a similar %fold degen- cratc band Ep,(k) (see Figure 2.26) evolves off of r. The 2-fold degeneracy of ixll band6 is a consequence of time reversal symmetry jointly with spatial inversion symmetry (see Appendix A). The E&-band has weaker curvature, it corresponds to the band of heavy holas. For k-, = k-, = 0, the pertinent eigenfunctions are I$$) and 1:;). 'l'hc EFa-band is that of the Zzght holes.

The eigenfunctions for k, - k, - 0 read 1;;) and 1;;). For arbitrary k- directions, the heavy and light hole states are linear combinations of all four

The described structure of the valence band around r represents what is called the Lallznger-Kohn modal. The particular form of the bands is determined by the three consttlnts A , B, C. Instead of the latter, one can use the dimensionless parameters 7 1 , 7 2 , 7 3 called Luttznger parameters, defined by :

-

basis functions I;:), I:;), 122) 3T and 1;;).

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198 Chapter 2. Electronic structure of ideal crystals

I s; II [loo] 5; II [Ill]

I G + l

I

Figure 2.26: Valence band structure of diamond and zincblende type semicon- ductors in the Luttinger-Kohn model. The dispersion for the two k-directions is different (band warping).

(2 3 8 0 ) h2 ha h2

A -----TI, B = - , 2 ~ , C2 = ---12(7,2 - 7:). 2m Lni 2m

Values of the Luttinger parameters are listed in Table 2.12. The constants L , M , N , which were originally used, may be expressed in terms of the Lut- tinger parameters by the relations:

Both energy band functions E&(k) and Er.,(k) depend on the square of Ikl, i.e. they are parabolic. This would not have resulted if the interaction between the rBf-states, and the spin-orbit-split r,f states, had not been neglected as it in fact was. Concerning the dependencies on the direction of k, the spin-orbit-split band Er7(k) is isotropic, while the heavy and light hole bands Er8(k) are not. In their case one again has a warping of energy bands as discussed above (see Figure 2.26). The constant C measures the strength of warping. In the case C = 0, the warping vanishes. For C # 0, the point k = 0 is a singularity of the energy band functions Erf,(k) in that the second derivatives with respect to the components of k depend on the direction from which one approaches the point k = 0. The effective mass tensor, as given by equation (2.196), is not defined in such circumstances. Instead, one can define an anisotropic effective mass, by differentiating in equation (2.377) not with respect to the components of k, but with respect to Ikl. If the warping of energy bands is ignored, for instance, by averaging

f

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2.7. k. p -method

Material

C

Si

Ge

cz - Sn

GaAs

GaSb InSb

ZnSe

HgTe

199

71 yz 7 3 E i A Ep 0.94 0.23 0.23 5.48 0

4.26 0.38 1.56 3.4 0.044

13.35 4.25 5.69 0.90 0.29 26.3

19.2 13.2 16.3 -0.4 0.8 39

6.85 2.1 2.9 1.52 0.34 25.7

11.8 4.03 5.26 0.70 0.8 22.4

35.08 15.64 16.91 0.18 0.98 21.2

4.8 0.67 1.53 2.67 0.42

12.8 10.6 8.8 -0.30 1.08 17.5

Table 2.12: k . p-band parameters for selected diamond and zincblende type serni- conductors. E i , A, and E, in eV. Ep = ( 2 m / h 2 ) P 2 is a measure of constant P in Kane's band model. The values for 71, yz, 7 3 are adjusted to the Luttinger-Kohn model. Temperature below 70 K . (After Landoldt-Bomstein, 1982.)

over all directions, this yields the ordinary isotropic heavy and light hole masses but in the sense of an average.

The Luttinger-Kohn model was described above for the case of diamond type semiconductors. Formally, for materials having the zincblende struc- ture, the model does not apply because the matrix elements of the momen- tum operator p between the triply degenerate valence band states without spin. IvrnO), are in general non-zero: These states transform according to the vector representation r15 of the tetrahedral group T d r and the matrix elements (vrnOlplvm'0) belong to the product representation I'15 x I'15 x r15. The latter contains the unity representation, as distinguished from diamond type semiconductors, where the unity representation does not occur in the corresponding I'i, x r15 x product. The reason for this is the absence of inversion symmetry in the zincblende structure. The non-vanishing matrix elements (omOlplvrn'0) give rise to terms linear with respect to k in the I'15 valence band Hamiltonian, besides the quadratic terms which are are al- ready present in the diamond case (see equation (2.344)). However, as a rule, the k-linear terms are small, and the Luttinger-Kohn model also applies to zincblende type materials, provided the other requirements which underlie this model are satisfied. Table 2.12 therefore also contains Luttinger-Kohn parameters for semiconductors of the zincblende type.

The most important requirement for the Luttinger-Kohn model to be valid is

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200 Chapter 2. Electrunic structrrre of ideal crystals

the validity of the assumption that the k p-interaction of the valence band with thc deepest conduction band is weak enough to be treated by means of perturbation theory. This is justified as long as the encrgy separation EF from the lowest condiiclion band and the I&-valenre band (not to be confused with the fimdamental cnergy gap) is siifficiently large. One expects deviations from the Luttinger-Kohn model to become noticeable if E i is sinall Table 2.12 shows thal the EF-value for InSb, for example, clearly lies below those for C, Si and Ge. In the case of a-Sn and HgTe, Ef; even becomes negative. Simultaneously, the spin-orbit splitting energy A becomes relatively large in some of the zincblende type materials, so that even A > /i$ holds. Describing of l he valence band of such semiconductors by means o l the Luttingrr-Kohn model would entail treating the effect of the remote spin-orbit-split band exactly, while taking iuto account only the energetically closer conduction band by means of perturbation theory. Such a procedure is not meaningful and one must seek a different, more appropriate description.

A model which is tailored exactly to such circumstances is the Kana model, which we will now discuss. In this matler, we asbume that the point group of equivalent directions is the tetrahedral group ‘rh, and no longer the cubic group o h as above, thercby encompessing both typeb of semiconduc- lois, those of zinrldende type EM well as those of diamond type. In the latter case, inversion symmetry still has to be added. This involves a spwialization of the results, which may be casily done, should thc need arise.

2.7.4 Kane model

The Kane model is based on the following assumptions. P’zrstly, it is assumed that the k. p-interaction of the valence band with the deqest conduction band at 1’ is so strong that it must bc treated exactly. Secondly, at r the spiuless valcnce band should have the symmetry T15, and the spinlesu conduction band should have the symmetry rl. This assump- tion corresponds to the situation which actually exists in semiconductors of Llncblendc type. Thirdly, the interaction of the valence and conduction bands with all re- maining bands (referrcd to as ~ e m o t ~ ) is assumed to be srrrall, so that it rriay be treated by perturbation theory, similar l o the Lultinger-Kohn model in which the interaction of the valence band with all other bands was treated in this way. Here, we will simplify further and neglwt this interaction COHI- plelely. Tn addition, we will also neglect the direct k. p-inleiaction among the three r15-valence bands which, as mentioned above, does not rigorously vanish for zincblenrle type crystals, giving risr to k-linmr terms in the Hamil- tonian. It turns out that the latter approximation is valid in most cases.

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2.7. k. p -methad 201

Neglect of interaction with remote bands

We analyzr the generally valid Ychrbdinger equation (2.351) using the as- sumptions and approximations discussed, above. considering spin md spin- orbit interaction horn the outset. We again denote the thrw valence band indices by urn, ni - I. u , z. The conduction band index c will be augmented by 3. idirating the 3- or rl-synlm&-y of the conduction band state at r (the coincidence of this notation s with that of the s p h variable 9 is unfortu- nate, but unavoidable, and the reader should keep the dxerent meaning of s dearly in mirid to avoid confusion). The pertinent spin-dependent Bloch functions at k - 0 are IvmuO) and IcsvO), rmpectively. The matrix elements of the term Ho(k] of f f k X are given by

(vmaO)Ho(k)jvm'dO) = bmm~6umi-k ri2 2 , (2.382)

(2.383)

(wmOlHo(k)IcsO) = 0. (2.384)

Iii the otkw o p ~ a t o r term of A V ~ . ~ , namely {Fa/m)k-.?i, WP may neglect thc spin-dqwndcnt part by virtue of the same arguments as in the Luttinger- kohn model. The three needed matrix elements of this operatw may then be determined as

2m

(csdlHo(k)lcsdO) = 6,,)-k , h2 2

2m

(2.385)

The matrix elements (vmOtplzlm'0) of p between the valence band states are wglectctcd in accordance with the assumptions madr above. The diagonal rlemenl (csOlplcs0) of p in the conduction band state IcsO) vanishes exactly. The matrix elements {cs01plvmO) between valence and conduction band states tranuform according to the produrt representation r1 x I115 x r15 = Fl + trls i l725. Since the unity representation is contained in it exactly once, the matrix does not haw to vanish; it contains exactly one independed rlcnwnt. As such, one may chose ( c s O l p , l v d ) and set it equal t o z (m/h)P . The nun-vanishing matrix elements of p are then given by the relation

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202 Chapter 2. Electronic strr1c:ture of ideal crystals

(2.386) m F,

~ c s o ~ p , ~ 7 ~ r o ~ = ~ r 8 ~ ~ p v ~ I v y ~ ) = ( c so /p ,~~~t .o ) = i - - ~

The factor 2 giiarantws that P is real, if the Bloch factors are real as we assume. The other factor (m/Ta) was introduced for convenience in the final dispersion rclations. With the moinentiim matrix elements of (2.3861, thp Hamiltonian matrix (pu01 Hk ,(k)lp’u’O) takes the form

E c

--iPk,

-iPkg

- i P k ,

0

0

0

0

, iPk , i Y k ,

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0

0

0

0

E C

- i P k ,

- i P k ,

- i P k ,

0 0 0

0 0 0

0 0 0

0 0 0

iPk, iPk , i P k ,

0 0 0

0 0 0

0 0 I]

(2.387)

where thr lines and columns arp ordprcd in the sequence / s I), la I), Ig I),

Finally ihe matrix demerits (puO1 Hsolp‘m‘O) nl t h e spiri-orbit interaction operator ITrn haw to lie detrrminpd. For the Flj-vaknrc band rlernents (wmcrOlH,,lwm’a’O), one can adopt the results which were formerly derived for the valence band. because l’i5 coincides with I’15 for the tetrahedral group. There are new matrix elPrnents (vmuOlH,lrsa’O) and (csuQIH,oj cscr‘0) involving the conduction band states. The coordinatedependent fac- tor of the first matrix element transforms according to the representation r15 x l?25 x I’l - I’z+ I’15 + r25. The unity representation does not oc cur here, thus this factor vanishes and with it the whole matrix element (umaOIH,,I csa‘O), whence

I2 th I. 1). lz 1), I Y 11, 12 1).

(um.aOlHs”Icsrr’0) = 0. (2.388)

The caordinatedeyendrnt factor of (csuOlH,olcsdO) belongs to the product representation rl x r25 x rl : rztj arid must therefore likewise wnish,

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2.7. k . p -method 203

Thus, the total Hamiltonian Hk.T + H,, is gken by the following matrix representation

, Ec

-iPk,

- iPkg

-iPk,

0

0

0

0 \

iPk , iPk , iPk , 0 -ig A 0

i$ o o 0 0 0

0 0 0

0 0 -- A 3 . A 0 0 -2z

3 2 5 O - a . A

0

0

0

0

Ec

--iPk,

-iPkY

-iPk,

0 0 0 - 0 O ?

0 0 - 2 7

-- $ i+ o . A

iPkz iPk , iPk,

0 i+ o -+ 0 0

0 0 0

Here, the order of rows and columns is the same as in (2.387). For k = 0 and vanishing Ec, this matrix reduces to the spin-orbit interaction opcrator WSw If one rearranges the rows and columns of this matrix in such a way that those relating to the conduction band states 1s 1) and 1s J) occur in the left upper corner, side by side, then the matrix decomposes into a (2 x 2)- block for the conduction bt~ncl, and B (6 x 6)-block for the valence band. The eigenfunctioiis of the two blocks arc simultaneously also e i g e h c t i o n s of the total matrix. T h {2 x 2)-co,nduction band block is already diagonal, i.c. 1s t ) and 1s J] are eigenhnctions of the Hamiltonian matrix (2.390) at k - 0. The (6 x G)-vdencr band block is identical with the matrix of the spin-orbit interaction operator H,, of (2.350) for the Luttinger-Kohn model. The eigenfunctions at k - 0 here are therefore also the vectors 1$ms) and i$rn 1) of ihe ctngpubr momentum basis (2.366).

The latter basis should be particularly suitable for solution of the eigen- value problem for the Hamiltonian matrix (2.490) at k # 0. The matrix (2.390) is, however, so simple, indeed, that one can also obtain the secular equation directly. We will do t h s . before we h r t h w consider the angular momentum basis. To diminate the free elmtron part (h2/2m)k2 from the eigenvalues Elk), we write them in the form

T

(2.391) h2

E(k) : E’(k) -I- -k2 2 m

where E’(k) satisfies the following

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204 Chapter 2. Electronic structure of ideal crystals

Secular equation

(2.392)

The fact that the two factors in round brackets appear squared, signifies an at least 2-fold degeneracy of all eigenvalues. The reason for this is. again, time reversal symmetry jointly with spatial inversion symmetry (we remind the reader that the term of the Hamiltonian which can break inversion symmetry in the case of Td-symmetry has been neglected). Accordingly, one has in general four 2-fold degenerate bands Ei(k). Ei(k), E$(k), Ei(k). It is also noteworthy that in the secular equation (2.392). k enters only in the form of k2. This means that all four bands are isotropic, in contrast to the Luttinger- Kohn model where a warping of the valence bands occurs. In the case of k = 0, the energy levels of (2.392) are given by

One may draw conclusions from these expressions in regard to the meaning of the four energy bands E,(k): El(k) is the J?s-conduction band. Ez(k ) 2nd Es(k) are the two upper degenerate rs-valence bands at r. and E4(k) corresponds to the spin-orbit-split rT-valence band. The energy separation EL of the r6-conduction band and the I'g-valence bend at. r is obtained as

(2.394)

As long as E: is positive, it represents the energy gap E , at I?. The case of negative E,' is discussed below. For one of the two upper valence bands - the one which arises from the vanishing of the first factor of the secular equation (2.392) and which is denoted by i = 2 - the energy El(k) does not depend on k. For E$(k). a k-dependence follows with finite negative curvature, as we will soon see. Thus E;(k) corresponds to a band of (infinitely) heavy holes, and Ei(k) to a band of light holes. If one adds the (Ti2/2m)k2-term, then the band Ez(k) displays a positive curvature. It is relatively small because of the large free electron mass, but the positive sign contradicts what is to be expected for a valence band. This unexpected prediction for E z ( k ) results from the fact that the interaction of the valence band with all remaining bands, except with the deepest conduction band, was completely neglected. In order to treat the heavy hole band correctly, the interaction with remote bands must also be considered at least by perturbation theory as in the Luttinger-Kohn model. This will be done below.

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205

Interaction with remote bands

According to the assumptions made at the outset. the interaction with r e mote bands is weak and may be taken into account by means of second order k. p perturbation theory. The 8 x 8 Hamiltonian matrix of cquatiori (2,390) for the conriuclion-valence band complex contains two 4 x 4 blocks of def- inite spin with rows and columns referring to the conduction band s state and the three 2--, y--, z-valence band statrs without spin, respectively. To include the inteTactjori with remote bands. RII additional 4 x 4 matrix o€ second order in k has to be added to each of these 4 x 4 blocks. Since the interaction bctween conduct ion and valrncc ba ld states contributes alrpady in first order, second order corrections occurring at $2--, sy-, s z - , and zs--, ys-, zs-positions may be omitted. For the ss-element and the 3 x 3 valence band submatrix, second order corrections beconie import ant. Their genpral forms follow from symmetry arguments as above. The correction of the ss-element may be written in the form Ack2, with A, a constant. The perturbation correction to the 3 x 3 valence band submatrix has the general form of the 3 x 3 matrix in qiiation (2.344) with parameters L , M. N &fin-1 like the matrix elements D E , D g in equation (2.341), however, with the conduction band excluded from the summation over { t bewuse this band is not reinotc. The 4 x 4 matrix block thus determined is added to carti of the two 4 x 4 diagonal blocks already present in the 8 x 8 matrix (2.390). Finally, the whole 8 x 8 matrix is subjected to a unitary transformation into a basis set in which the spin-orbit interaction part of the Hamiltonian becomes diagonal. The latter requirement is evidently satisfied by a basis which, as in the Luttinger-Kohn model. contains the 6 angular momentum eigenfunctions I$:), I$$), I:!), I;:), I%+). I;$), and in addition the two con- duction band states 1s 1) and 1s 1). This corresponds to a 8 x 8 unitary transformation matrix composed of a 2 x 2 unity matrix block for the two conduction band states, and the 6 x 6 matrix block from equation (2.371) for the six valence band states.

Carrying out the unitary transformation one obtains the general 8 x 8 Kane Hamiltonian which applies to any diamond or zincblende type materi- als, including those which are already well described by the Luttinger-Kohn model. However, even in these cases the Kane model is more precise than the Luttinger-Kohn model, because the valence-conduction band interaction is treated exactly rather than approximately like in the Luttinger-Kohn model. If one uses the Kane model in cases in which the Luttinger-Kohn model al- ready works well. one has to be aware that generally the parameters 71, 7 2 , 73 have different Values in the two models since those of the Luttinger-Kohn model contain the valenceconduction band interaction while those of the Kane model do not so. The general 8 x 8 Kane Hamiltonian is given by rather lengthy expres-

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206 Chapter 2. EIwtronic structure of ideal crystals

sions. To avoid these below, the 4 x 4 block matrix of the remote band interaction will be reduced to a special case before proceeding further. We put L = M -- A,, and N = 0, which means physically that the remote bands affect heavy and light holes in the same way, and do not disturb the isotropy of the bands. -Ordering rows and columns in the sequence 1s t), 1s l), I:;), I;$), . . . , I$$), the transformed 8 x 8 Hamiltonian matrix with simplified remote band interaction becomes

U 0 a P,

0 U D

-iP- 0 V

- f i P - 0

-i& -i$&P, 0

0 0 0

0 0 D

v 0 0

0 V 0

0 0 W

0 0 0

-& 0

0

O

0

0

U' (2.3

where the notations l'h = (1/&)P(kz f ik,), Pz = Pk,, U = Ec + A&', V = (1/3)A + A,k2, and W = -(2/3)A + A,k2 are used. The eigenvalues of this matrix follow from the secular equation

2 E'(k) + -A - A,k2 E'(k) +

i 3 I - i 3

This equation only differs from equation (2.392) in that the factors whose vanishing define the conduction and valence bands contain, respectively, the additional terms A,k2 and A,k2.

Solution of the secular equation in limiting cases

The zeros of the h s t factor in (2.396) yield, as seen previously, the I's-band of heavy holes (i = 2). However, the dispersion relation for it now reads

A ii2 En(k) = - + A,k2 + -k2 3 2m

(2.397)

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2.7. k. p -metbod 207

By choosing a negative value for of appropriate magnitude A,, the dispersion for heavy holes can be brought into agreement with experimental findings. The zeros of the second factor in (2.396) determine the dispersion of the Fa-conduction band ( i = l), the rs-band of light holes (i = 3), and the spin-orbit-split I'T-band (i = 4). For the conduction band, the dispersion is changed due to the A,k2-term, and for the two valence bands due to the A,k2-term. But, here, these corrections are added to already existing strong dispersion terms. We will therefore neglect them in the following, as we neglected the weak dispersion duc to the free electron term (li2/2m)k2 earlier. Then the eigenvahe equation for the three bands i = I, 3,4 reads

- [E'i(k) + $1 P2k2 = 0. (2.398)

This equation will be solved approximately in three limiting cases with re sped to the order of magnitude relations between the energy gap E i and spin-orbit splittiiig energy A, as w ~ l l as with respect to the s i g n of E,'. namely firstly for EF >> A, Eg > 0, secondly for EF -CK A, E; > 0, and thirdly for IEiI < A, E i < 0. The significance of a negative value of E i will be discussed while treating the third case. All three cases actually oc- cur in zincblende type semiconductors, as a look at Table 2.12 immediately shows.

The first case corresponds to materials with wide energy gaps whose va- lence band complex could be described just as well by means of the Luttinger- Kohn model; the second case refers to semiconductors with narrow energy gaps; and the third to materials whose energy gaps vanish.

Case 1: E: z E, >> A, E F > o

We consider energy values Ei(k) in the various bands with energy separations IEi(k) - &(O)( from the respective band exbrema which are small compared with EF. For such energies, the conduction band El(k) approximately obeys the equation

[El(k) - E,]E, - P2k2 = 0. (2.399)

and for the two valence bands E3(k) and E4(k) we have

b i ( k ) - $1 b i ( k ) + TA E, + &(k) + - P2k2 = 0, i = 3,4 . (2.400) 2 1 [ :I From (2.399) it follows immediately that

Page 222: Fundamentals of Semiconductor Physics and Devices

208 Chapter 2. Electronic structure of ideal crystals

(2.401)

and from (2.400) we obtain

8 2 1 A P2 A P2

E3/4(k) = -- 2 [ 3 - + -k E, 2]*ij[?f-Kk2] +m' (2.402)

Under the condition (P2/Eg)k2 << A, a parabolic approximation for the otherwise non-parabolic valence band E3/4(k) is possible, namely

(2.403) - (2 13) (P2/ E,) k2

- (2/3)A - (1/ 3) ( P2/ E, ) k2. E3/4(k) =

The structure of the four bands E1/2/3/4(k) is illustrated in Figure 2.27a.

Case 2: Ef = E, << A, Ef > 0

The energy values Ei(k) now have energy separations IEi(k) - Ei(O)I from the respective band extrema which are small compared to A. This means they can be comparable to E i . For the conduction band and the light hole band one then gets from (2.398), approximately,

= 0, i = 1,3, (2.404)

from which

follows. In general, the dispersion laws for the electrons and light holes are again non-parabolic. Only for energy separations from the band extrema which are small compared to E,, more exactly for P2k2 << (8/3)E,, a k2- dependence emerges, namely

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2.7. k . p -method

r, 7A 2

209

\i 4

b)

3

I 4 4 I Figure 2.27: Valence band structure of zincblende type semiconductors in the Kane model for limiting cases: (a) E, >> A, ( b ) E g << A, E: > 0, ( c ) E , << A, E; < 0.

Since the energy region of width Eg above the band extrema is relatively narrow for the narrow gap semiconductors considered here, one has in these materials, even at relatively small separations from the band edges, non- parabolicities in the dispersion laws of the electrons and light holes. As far as these particles are concerned, small energy gaps and non-parabolicities occur together.

For the spin-orbit-split band, one obtains, without further approxima- t ions,

2 P Z 2 E4(k) = --A - -k .

3 3A (2.407)

The dispersion curves of all four bands in the limiting case considered here are depicted in Figure 2.27b.

Case 3: IEil << A, E i < 0

According to equation (2.394), negative values of EF mean that the r6-

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2 10 Chapter 2. Electronic structure of ideal crystals

conduction band level lies below the rglevel. The relation lEil << A guarantees that the spin-orbit-split r;.-level is found further below it (see Figure 2.27~) . If. again, only energy values are considered with separations IE,(k) - E,(O)I from the respective band extrema which are small compared to A, one obtains dispersion relations having the same form as in the previ- ously considered case EF << A, E, > 0 (see equation (2.406). For electrons and light holes they yield. under the condition (P2/A)k2 << IEil, the ap- proximate parabolic dispersion laws

Because of the negative sign of E:, band 1 now lies energetically lower and exhibits a negative effective mass, and band 3 lies higher and has a positive effective mass. For the spin-orbit-split band E4(k), equation (2.407) holds unchanged, and for the band E i ( k ) of the heavy holes, relation (2.397) also remains the same. Thus, the band E 3 ( k ) is, among all four bands, the highest energetically with the exception of I', where it degenerates with the band of the heavy holes. Since the 8 valence electrons of a zincblende type semicon- ductor are only enough to occupy 6 of the 8 bands of the I?;.--. FS-. ??&band complex - 2 electrons per unit cell are necessary to fill the deepest I's -valence band, omitted from consideration here - the E3[k) band remains empty. It becomes the conduction band, which is also in accord with its positive effec- tive mass. At r, its separation from the uppermost valence band, the Ez(k) band of heavy holes, is zero. This means that the energy gap vanishes in this case. Negative d u e s of the parameter E i . which signifies the energy gap when it is positive, cause EF to lose this significance, and the real energy gap becomes zero. Materials with vanishing energy gap are called zero-gap semtconductors. Examples are HgTe as a zincblende type semiconductor, and D - SR as a semiconductor of the diamond type.

In Table 2.13. the effective masses m t are listed for the three limiting cases considered above. Generally, one has 1m.I I 5 I rn31 < IrnZl. The ef- fective masses of the electrons and light holes are proportional to / E g l / P 2 throughout. The rule discussed above for degenerate bands is thereby con- firmed to be valid also for non-degenerate bands wherein mass decreases as lEgl decreases and P increases. For IE,l << A the effective masses of the electrons and light holes are almost identical, for E, >> A the electrons are lighter than the light holes. For the spin-orbit-split band, the proportion- ality of the effective masses to IE,l/P2 exists only in the case E, >> A; for

<< A the mass mS becomes independent of Es and proportional to A.

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2.8. Rand structure of important seniiconductors

Band

rdl) rs(3)

rd4)

211

E F > > A , E ; > O E L < A , h ' , r > O EL << A, EF < 0

1 . ( F , / P 2 ) (3/2) . (E,/P2) -(3/2) ' ( I 8; I / P 2 )

-3 ' (A/ I q I) -(3/2) . (Eg/P' ) -(3/2) . ( E S / f " ) (3/2) . (I -Ei I / p 2 )

-3 ' ( E , / P 2 ) -3 ' (A/Fs,)

2.8 Band structure of important semiconductors

In this section we discuss the band structures of some important semicon- ductors. In all cases, the results presented are based on both theoretical and experiment a1 investigations.

obtained by means of optical reflectance spectroscopy. It turns out that char- acteristic structures of the reflectance spectra, like peaks or shoulders, are directly related to optical transitions at critical points of the energy difference between the initial and final bands involved. The frequencies of these struc- tures are experimental measures of the energy separations between initial and final bands at critical points, To enhance the charactelistic spectral features, changes of the reflectance spectra are measured due to external perturba- tions, as, for example mechanical strain, clmtiic and magnetic fields, light, or heat. By modulating the perturbations periodically in time with frequen- cies in the kHz range, these spectral changes can be measured very precisely by means of frequency and phase scnsitive techniques. Exttniple5 of this so- called modulataon spectroscopy are electrorepectance (ER), pzezorejlectanw, thermarejlectance, and photoreflectance (PR) (for morc on elect roreflwtance see section 3.7). Details of the band structure at critical points, like effective masses of free carriers, may also be extracted from transport meaxmemenits. Again, external perturbations, in particular, magnetic fields, are applied to induce changes. Magnetotransport phenomena, like rnngnetoreuzutanc.r and Shubnzkov-dp-Haas effect are examples. In cyclotron rmonance one mea- sures the absorption of microwave radiation by a semiconductor sample in the presence of a magnetic field to obtain the effective masses of electrons and holes (section 3.8).

None of the experimental methods is capable of revealing the entire band strurture of a given semiconductor material at all poiutv of the first BZ. To

Experimental data concerning band struct ure of semiconductors arc mainly

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212 Chapter 2. Electronic structure of ideal crystds

obtain this. one is nhliged to carry out band structure calculations. Expai- meutal data enter thrse calculatiolis in various ways. Tlus is obvious if em- pirical methods are cmployed their results have to be fitted to experimental data, as, for example, to energy separations between bands at critical points obtained from modulation spectroscopy. Less obvious. but nevertheless ex- isting, is the need of experimental data for ab-initio calculations. Although these methods are free of fitting parameters, various approximations are in- volved which call for experimental confirmation or even corrections of the results. as, for example, in the case of the erroneous fundamental energy gap in the local density approximation.

Below we represent the results of of uumprical hand structure calcula- tions using one or mother of the methods described in section 2.5. We will not specify which particular method was applied since that is not of interest here. Our main concern is with the qualitative features of the energy bands. We will demonstratp that thew may be understood. at least partially, just by means of the general results derived in the preceding sections and in Ap- pendix A. This is particularly true for features irivolving the degree of band degrnrracy at symmetry points of the first B Z , which follow from the i r re dudble representations of the space group of the crystal under consideration (see Appendix A). The band structure models derived by the empty lattice approach. the k . p method, and the tight binding method in sections 2.4, 2.6 and 2.7, respectively. will also be helpful. We begin our discussion with silicon.

2.8J Silicon

In Figure 2.28 the band structure of Si is shown. Spin-orbit interaction plays only a minor role for the ovcwJl behavior of energy bands in Si, 80 the energy levels may be classified by means of the ordinary irreducible representations of the small point groups of the wavevectors k. l'hesr are subgroups of the full point group 01, of eqiiivalent dirrctions. According to what we already know about the dimensions of these representations at the various symmetry points of the first B Z (see Table 2.5 and Appendix A), one expcrts at most %fold degenerate levels at the symmetry center I', only 2-fold a t the the symmetry point X. and at most 2-fold on thp symmetry lines A, A and at L. This expectation is codinned by the band structure shown in Figure 2.28. The deepest energy level of the entire band structure occurs at r, i s non-degenerate. and belongs to the irreducible representation r'1. The same result was previously obtained in the band structure analysis of diamond type crystals by means of t h e empty lattice altd tight binding approaches in earlipr swtions (see, Figwrs 2.13 and 2.2L rcspectively). Also, off of l?, the numerical calculation of the PI-band of Figure 2.28 exhibits the behavior prdictfd by these simple approaches. DiEmmrw between the numerical

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2.8. Band structure of important semiconductors 2 13

Figure 2.28: Rmid structure of Si. The energy unit i eV. (After Chelikowskp and Cohen, 1974.j

and empty-lattice band structures occur at the second level at r as may be seen from Figure 2.29. The second empty lattice has an 8-fold degeneracy which exceeds what is compatible with the Oh-symmetry of r. In a real diamond type crystal this degeneracy is removed. As indicated in Figure 2.29, a splitting into levels of rl-, I'b-, I'b5-, and r l5-symetry will occur. In this way the corresponding rk5-, rls-, and I'k-levels in Figure 2.28 could have emerged from the 8-fold degenerate empty lattice level. The tight binding analysis of the band structure of diamond type semiconductors in section 2.5 has already shown that the second level (from below) at I? has symmetry were possible, according to this analysis. 4 s Figure 2.28 shows, r15 applies in the case of Si, while the I'b-level is the fourth.

For the third level at r, the representations r15 or

A410ng the A- and A-lines, the two 3-fold degenerate let~ls I'L and r15 must split since only 1- and 2-dimensional irreducible representations are possible there. Both a %fold splitting into 3 simple bands as well its a 2-fold splitting in a 2-fold degenerate and a simple band are conceivable. From Figure 2.28 one sees that the latter case holds, both along the A and A- lines. and for the I?&- and the I'ls-levels. Since there are only 2-dimensional representations at X (for an explanation see Appendix A), two simple bands along the A-line must merge at this point, as actually happens in Figure 2.28.

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214 Chapter 2. Electronic structure of ideal crystals

t lo Figure 2.29: Band structure of the empty fcc lattice. The energy

A unit is the same aa in Figure 2.28. The irreducible representations of - 8 the energy bands are also indicated. For comparison, the band structure of Si is shown in the same figure by dotted lines.

0 w m cu C

L

w 6

r, 62 5; r; 5' 55 6;

r A X Wavevector - r1

A 2-fold degenerate band along A cannot merge with another band at X , but miist terminate in a doubly degenerate level at X. From this it follows, for example, that the upper of the two bands arising from the I'k5-level along the A-line must be 2-fold degenerate, and the lower band must be simple. A look at Table 2.4 shows that the only 2-dimensional representation of the small point group of A is A5. At X this representation is compatible either with X 1 or X 3 (the compatibility relations between irreducible representations are derived I n Appendix A). Figure 2.28 shows that X 1 is correct in this case. In a similar way one may conclude that the representation of the lower level at X, arising from the I'i,-level, must be X z . A similar analysis for the splitting of the rls-level along the A-line shows that the lower level is non- degenerate and belongs to the irreducible representation Al, and the upper level is 2-fold degenerate and belongs to .As. The intersection between the Ah-band emerging from the r!+vel, with the As-band emerging from the l'ls-level, is not due to symmetry, but reflects an accidental degeneracy.

Consider next the two bands arising from the l'$s-level on the A-line. The upper is the 2-fold degenerate As-baud, and the lower is the simple A1-band. The simple band does not merge with another simple band at L , but remains separated from it by a finite gap, in contrast to the band behavior at X. This is possible because at L there arc also 1-dimensional representations, L1 and L i , which give rise to the two lowest levels at L. The splitting of the rls-level along the A-line is quite similar to that of the l?~5-level; the lower band is non-degenerate and belongs to A l , and the upper

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2.8. Band structure of important semiconductors 2 15

is 2-fold degenerate and belongs t o h ~ . The occupation of the energy bands of Si and other diamond type crystals has already been discussed in section 2.6, using the band structure which follows in tight binding approximation. In the ground state of the crystal, the band associated with the 1-fold I'&vel, and the bands arising from the %fold l?h5-1evd, are completely occupkd, while the above-lying r16- and rb- bands are completely empty. Thus the rl- and r&-bands form the valence bands of Si. and the rls- and rb-bands. as well as all higher bands form the conduction hands. The two groups of bands are separated by k-dependent forbidden energy regions, marked by dashes in Figure 2.28. Moreover, there are forbidden energy values which occur at all k-vectors. This means that the bami structme calcidations for Si yield a finite energy gap, and, inded, they explain the semiconductor character of Si.

While the absolute maximum of the uppermost valence band lies at r, the absolute minimum of the lowest conduction band is located on the A-line close to the edge of the h s t B Z , and its irreducible representation is AL- Semiconductors with the absolute extrema of the valence and conduction bands located at different points of the first B Z , are called ipadiwct. Silicon is, therefore, an indirect aernicondurtor. If the extrema occur at the same point, the- semiconductor i s called dawct. The property of a semiconductor material in being direct or indirect (Table 2.14 contains information about this) has important physical and technological consequences. For example, indirect materials are in general not suitable for manufacturing light-emitting devices, unless one takes special measures such as a particular doping.

Following the general lines of section 2.5, we will now examine the e€fec- tive mass tensors of Si. We will restrict oursehes to the two critical points mentioned above, in which the conduction band has its absolute minimum and the valence band has its absolute maximum - the vicinities of these points are the regions of the first 32 which. under thermodynamic equi- librium conditions, host most of the free eIectrons and holes. The effective masses at these points are therefore the effective masses of the electrons and holes of silicon.

Owing to the cubic symmetry of the band structure of Si, a minimum of the conduction band on a particular A-line is automatically accompanied by 5 other minima on the star of A-lines. This means that there are, altogether, 6 minima OT valleys, a term which is often used for the vicinities of the min- ima. Since Si has several valleys. one calls it a many-valley semiconductor. The valleys are centered at the points

with kc = 0.85(2n/a). The cubic axes are, simultaneously, the principal

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2 16 Chapter 2. Electronic striictnre of ideal crystals

axe8 of the effective mass tensors of the various valleys. Each principal axis represents a Cfold rotation symmetry axis. To p r o r d further. we select, arbitrarily, the valley centered at (O,O, kJ. The band structure of this valley is given by the expressions (2.201) and (2.2@2), which are applicable here, as their conditions of validity are satisfied. Setting the band index v equal to c, which refers to thp coiiductioii band, the dispersion rdation EJk) of this band becomes

(2.409)

If t,hc zero point of the energy scale is put at. thr valence band maximum, as we do here. then E, is t,he fundamental energy gap.

In the vicinity of the valencc band maximum, the general results of the k . p-method of section 2.7 are applicable. Without spin-orbit interaction (see cqiiations (2.345) to (2.348)) one has two valence bands for each of the two symmetry directions A and A, an upper E,l,z(k) which is 2-fold degenerate, and a lower &(k) which i s non-degenerate. Along less sym- mdricnl k-directions E,l/z(k) split 5 into two bands. However, spin-orbit interaction cannot be neglected; although it has little effect on the overall band structure of Si, it inftuences the valence band structure considerably in an energy interval of several kI' below the rnaxiniurn at r, which is the energy region where most of the holes Bre located. Spin-orbit interaction makes the uppermost valence band level at r, which has a 6;-fold degener- acy in Si if spin i s considered, split into an uppm &fold degenerat.e I?$-level and a lower 2-fold degenerate I';-lcvel. The splitting rncrgy amounts to 44 m e V . Away from I?: the upper ra-level decomposes into the two bands Efp(k) of heavy and light holes according to equation (2.377). The lower r$-levd givw rise to the spin-orbit-split r;-band. The heavy and light hole bands are strongly warped and each exhibits %€old spin-degeneracy, which is due to timc reversal syrrirriet,ry in coinbination wihh spatial inversion (see Appendix A). If warping i s neglected the two bands Era (k) can he described by isotropic effective mass# m$. 'I'hey are negative because of the maximum at I'. Extracting the negative sign, we define the positive effective masses rnEh and m.G1 of, respectively, heavy and light holeb;: setting m:,, = -m; , and ntl = - m l . For the heavy and light hole valence bands EF8(k) G E,h(k) and E&(k) = E,,,(k): we haw

(2.410)

Numerical values for the effective hole masses of Si are given in Table 2.14. A vivid view of the conduction and valence band structure in the vicinity

of the band edges is provided by the corresponding iso-enerm surfaces. These

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2.8. B i d structure dimportant semiconductors 217

Figure 2.30: Iso-energy surfaces of the electrons (on the left) and holes (on the right) for Si.

I

are obtained by k i n g a specific energy and then drawing all k-vectors for which the bands of equations (2.409) and (2.411) yield this energy value (see Figure 2.30). For the conduction band, the iso-energy surfaces are ellipsoids of revolution, pointing in the direction of the symmetry axis. Each of the six star points is the center of such an ellipsoid. For the valence bands, within the isotropic approximation, the iso-energy surfaces are concentric spheres centered at I?. The inner sphere corresponds to the light hole band, and the outer to the heavy hole band. In reality the valence bands are not isotropic but have only cubic symmetry. Thus their iso-energy surfaces are warped, as shown in Figure 2.30. If the conduction band is populated by electrons up to a given energy, then the k-vectors of these electrons lie within the ellipsoid of revolution corresponding to this energy. Accordingly, the k-values of the holes lie within one of the two spheres or the two bodies bounded by the warped surfaces.

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218 Chapter 2. Electronic structure of ideal crystals

Figure 2.31: Band structure of Ge. The energy unit is eV. (After Chelikowsky and Cohen, 1.974.)

2.8.2 Germanium

In Figure 2.31 the band structure of germanium is depicted. It is similar to that of silicon, owing to the fact that both materials have diamond struc- ture. The differences between the two band structures are, apart from other reasom, due to the fact that the spin-orbit interaction is considerably larger in the case of Ge as compared to Si (see Table 2.14). This results from the larger orbital velocity of the valence electrons of Ge because of the larger atomic nucleus of this element. With the regard of spin-orbit interaction the valencp band maximum at r is formed by an upper 4-fold degenerate rsf-level and a lowcr 2-fold degenerate I';-level, separated from the upper level by 340 meV. Away from I', the upper I',f-level decompoaes into the two 2-fold degenerate bands of heavy and light holes of, respectively, A6 and A7 symmetry, As in the case of Si, the bands are warped but often considered in an isotropic approximation.

The lowest conduction band level r, at !2 arises from the ri-level without spin, which was the second conduction band level in the case of Si. In Ge this level i s shifted down considerably so that it becomes lower than the r, and I?, doublet which arises from the I'15-level without spin. The most important

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2.8. Band structure of important semiconductors 219

0.24

0.07

Table 2.14: Characteristic data of the barid structure of selected semiconductors. Energies in eV, effective masses in free electron masses. Temperature 300 K. For Si, Ge, GaAs, Gap, CdTe, and IIgTe read i = u h , v l (heavy, light holes), and for CdS (hcxagonal phasc), PbTe, Te, and Se read i ~ 1 1 , I (parallel, perpendicular to syniniebry axis). (After Candoldt-Blirnstein, 1982.)

0.028

0.1

- - Milate-

rial

Si

GI2

GaAS

GaP

CdTe

i_.

HPn: CdY

PbTe

‘k

Se - -

C-Blmd-

Mimirnurn

A

L

r X

r r r L

H

Z(?)

Energy

Gap

1.1 1

0.66

1.43

2.27

1.43

0.00

2.50

0.30

0.33

1.8

0.044

0.29

0.34

0.08

0.7

0.30

0.064

- -

h l II I_

0.54

0.3

0.45

0.67

0.4

0.3

5

0.31

0.11

- -

0.15

0.044

0.087

0.17

0.1

0.03

0.7

0.022

0.24

- -

difference between the conduction bands of Ge and Si is the location of the respective absolute minima. In Ce it does not occur on a A-line, but at L , being the BZ boundary point of the A-line. Therefore, the conduction band of Ge has 8 half-valleys instead of 6 full valleys in the case of Si. The minima at L are L;-levels arising from the &-band on the A-line which enters the r7f-level at I‘.

2.8.3 111-V Semiconductors

For the two 111-V compound semiconductors GaAs and Gap, the band struc- tures are shown in Figure 2.32. Their first B Z s are the same as that of Si and Ge, since they have the same Bravais lattice. The full point group of equivalent directions is T d for both materials. Spin-orbit interaction is im- portant for the overall band structure of GaAs, while for GaP it may be neglected. Thus the band structure of GaAs is described in terms of spinor representations, and that of GaP in terms of ordinary representations. For both materials, the valence band maximum lies at r. Without spin and

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220 Chapter 2. Electronic structure of ideal crystals

Figure 2.32: Band structure of 111-V semiconductors GaAs (top) and GaP (hot tom). The energy unit is el’. (AfLer Chelikousky rand Gohen, 1974.j

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2.8. Band structure of important serniconductors 22 1

I 1 I i r x U,K r -61 I I

L A r A X U , K E r

Figure 2.33: Band structure of IT-VI sernimnductom Cd‘k (left) and IIgTe (right). The energy unit is el.‘. (Af ter Chadi, Walter, Cohen, Patroff and Balkanski, 2972.)

spin-orbit interaction (the case of Gap), it belongs to the irreducible repre sentation I’l5 of the point group Ta which arises from the representation I?;, of Oh if the latter is taken as a representation of Td. With spin-orbit interac- tion (the caw of GtrAs), the r15 vulence band maximum splits into an upper r8-level, and a lower I’7-levd In the upper l’g-level, the two 2-fold degener- ate heavy and light hole bands mcrgr together as in the case of Ge and Si. The lower l”;.-level gives rise to the 8-fold degenerale spin-orbit-split band. The %fold degenpracy of the valence hands at X, observed in the case of Si and Ge, splits in CaAs and Gap. The reason €or this is that the t,wn atoms of the primitive unit cell are no longer identical, which means that the point symmdry of equivalent directions is rduccd from Of, t o Td. Therefore, one also has 2-dimensional spinor representations ( X s , X:) instead of only one 4-dimensional ( X , ) in the case of Si and Ge, and also 1-dimensional ordinary representations (XI X2, X3, Xq) instead of only 2-dimensional ones ( X i , X z ! x3> Xq) in the case of Si and Ge (see Appendix A).

The conduction bend minirnnrn of GaAs occurs at the r-point and be- longs to the spinor representation r7. Thus, CaAs has a dirwt energy gap. One of the peculiarities of its condiiction band structure is the relative min- imum at the L-point only about 0.4 eV above the absolute minimum atr r. In the case of Gap, the conduction band minimum occurs at X and belongs to the representation XI. Since the valence band maximum resides at I’, GaP is an indirect semiconductor.

2.8.4 11-VI semiconductors

The band structures of two typical IT-VI semiconductors with zincblcnde structure, CdTe and HgTe, are shown in Figure 2.33. The band structure of

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222 Chapter 2. Electronic structure of i d 4 crystah

8

6

L

2

a

- 2

CdTe is similar to that of GaAs, in particular the conduction and valence band edges are located at the r-point, as also occurs in GaAs. The same holds for HgTe, but as mentioned in section 2.7, the two bands touch each other at r, so that the energy gap vanishes. The reason for this is easy to understand, and was also already discussed in section 2.7: The r6-level which lies above the I‘s-level in CdTe, is found below it in HgTe. The 8 valence electrons per primitive unit cell therefore suffice to occupy only one of the two rs-bands merging at I?. The second band remains unoccupied and becomes

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2.8. Band structure of important semiconductors 223

lxc, I

L r X K.U I4 r Figure 2.35: Band structure of PbTe. The energy unit is eV. (After Martinez, Schliiter and Cohen, 1975.)

the conduction band. Since it is degenerate with the uppermost valence band at I', the fundamental energy gap is zero. According to the general definitions of Chapter 1, this means that HgTe is no longer a semiconductor, because in semiconductors the valence and conduction bands must be separated by a finite energy gap. It is more fitting to term HgTe a semimetal.

As an example of a 11-VI semiconductor which does not exhibit the zincblende but rather the wurtzite structure, we chose the hexagonal CdS (see Figure 2.34). The first BZ is that of the hexagonal Bravais lattice, which is shown along with its symmetry points in Figure 2.12. CdS has a direct energy gap at r. Furthermore, the valence band is additionally split off at r as compared to the cubic materials of diamond and zincblende struc- ture, because of the hexagonal deformation of CdS. Thus, the valence band is composed of 3 bands, namely r7-band which is separated from the other two by spin-orbit interaction, and the two bands r6 and I?7 into which the rg-band decomposes under the hexagonal deformation. The latter effect is called crystal field splitting (see the right part of Figure 2.34).

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224 Chapter 2. Electronic structure of idea[ crystals

12

8

4

0

-4

-8

r z H K r 1 - z H K r Figure 2.36: Band structure of Te (left) and Se (right). The energy unit is eV. (After Maurhke, 1 g71; Stuff, Maachke and Laude, i973.)

2.8.5 IV-VI semiconductors

The band structure of a typical IV-VI semiconductor, PbTe, is depicted in Figure 2.35. It crystallizes into the rocksalt structure. Its Bravais lattice is, thus, again the fcc lattice and its first B Z is the same as that of the diamond and zincblende type crystals. The conduction band minimum and the valence band maximum both lie at the edge-point L of the first B Z . This means that PbTe is a direct gap semiconductor with many valleys, a special feature which is hardly realized in any other semiconductor families apart from the IV-W compounds. The energy gap of PbTe is relatively small, it amounts to about 0.2 e l F . Like InSb of the 111-V-compounds, PbTe belongs to the group of narrow gap semiconductors.

2.8.6 Tellurium and Selenium

The band structures of tellurium and selenium are shown in Figure 2.36. Both materials have hexagonal Bravais lattices. Therefore, they have the same first B Z s as the hexagonal CdS. In the case of Te. both the valence band maximum and the conduction band minimum occur at the H-point of the first B Z . The material is therefore direct. Se has an indirect gap, the two edges both lie outside of I-, that of the conduction band is probably at 2, and that of the valence band is probably at H.

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22 5

Chapter 3

Electronic structure of semiconductor crystals with perturbations

Real semiconductor crystals are imperfect. They differ from ideal crystals in their atomic structure as they contain structural defects, and their chemi- cal cornpodmn is not exactly lhal of the correspondiug ideal crystals. It is, however, not the atomic structure which provides real semiconductor crystals with special scientific and technological importance. but their peculiar elec- tronic structure that is manifested in pronounced macroscopic effmts such as. for example, the increase of electrical conductirity by orders of magni- tude. With some justification one can even say that the particular scientific and technological importance of semiconductors rests on the peculiarities of the electronic structure of amperfeet semiconductor crystals. Nevertheless, one must also deal with the atomic structure of imperfect semiconductor crystals since that is, ultimately, the source of the particular nature of their electronic structure. This will be done in section 3.1. The one-electron Schrodinger equation of imperfect semiconductor crystals will be derived in section 3.2. - 4 ~ important theoretical method €or its solution, the so-called effective mass theory, will be developed in section 3.3. This equation allows one to determine the effects of perturbations whose interaction potentials with electrons are slowly varying over the atomic length scale. Interaction potentials of this kind apply to many perturbations of atomic structure, in particular to the so-called shallow centers which will be treated in section 3.4. There are. however. also other perturbations. like the so-called deep centers, to be dealt with in section 3.5, in which potentials exhibit consider- able change over the atomic length scale. They cannot be treated by means of effective mass theory, but require other methods. This is also true for clean semiconductor surfaces considered in section 3.6.

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226 Chapter 3. Electronic s-tnictirre of semiconductor crystals with perturbations

Besidc the study of semironductor crystals having a perturbed atomic struc- tiire, 11 is also of interest to study semiconductors which are exposed to macroscopic external perturbations such as dectrir OT magnetic fields, me- chanical strain, or arbifmid atomic superstructures. The electronic structure of semiconductors in the presence of such macroscopic external perturbations is treated in the second part of this chapter. This will include superlattices (section 3.7) and semiconductors subject to an electric field {section 3.8) or a magnetic fidd (section 3.9). In all cases, again, the effective mass theory plays a major part in our theoretical understanding. This theory applies because the macroscopic perturbation potentials are naturally smooth over the atomic length scale.

3.1 Atomic structure of real semiconductor crys- t als

The reference system for the description of the atomic structure of a real crystal is the ideal crystal. As discussed in Chapter I, the latter may be characterized as follows: All regular sites

Ri 7 rial + r2a2 + 7'3a3 + 6 defined by the crystal structure, are occupied by atoms of the 'correct' chem- ical species, and other sites are empty. In a real crystal, this characterization holds for the vast majority of regular and irregular sites - providing the jus- tification to speak of a crystal at all, albeit a perturbed one. However, there is deviation from the ideal occupation at some of the regular and irregular sites. In this sense. one then refers to the crystal as either perturbed or real.

(3.1)

3.1.1 Classification of p e r t u r b a t i o n s

The perturbations to be treated can be classified in accordance with several points of view. On the one hand, one can distinguish them on the basis of whether they are of a purely chemical nature, or purely structural, or mixed type. In the first case, only regular sites of the crystal are occupied, but. not with chemically 'correct' atoms throughout. One refers to this as chemical or cornpositzonal dzsorder. In the case of elemental crystals, this kind of disorder necessarily means the presence of impuaty atoms. In this context the perturbed crystal, which hosts the impurity atom. is called the host crystal. A crystal formed from a chemical compound may also exhibit compositional disorder without impurity atoms, namely because of a per- turbed stoichiornetrical composition. In the case of structural perturbations, only chemically 'correct' atoms are present, but these are not always posi- tioned on regular crystal sites but also on irregular ones. Furthermore, not

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3.1. Atomic structure of red semicoriductor crystals 227

all regular siles are occupied by atoms, some sites remain empty. Structural perturbations are also called strurtural & f w h or simply dpferts. The third case is the most general one - regular or irregular sites of the crystal are partially occupied by rhrrnirnlly ‘wrong’ atoms, end also the ‘correct’ atoms partially occupy ‘wrong’ sites.

Deviations from an ideal crystal may, on the other hand, be distinguished according to the macroscopic extension of the perturbations. A perturbation that is limited to one or a few neighboring regular or irregular crystal sites is called 0-dimensional or a poznt perturbatzon. If the perturbation extends over sites located on a line or a planc, it is referred to as a 1-dimensional or lzne perturbatton and a 2-dimensional or planP perturbatton, rpspert ively, Combining the two classification schemes, one may refer to structural point or line perturbations, compositional point perturbations etc. The dimen- sions in the second classification scheme apply to the microscopic core of the perturbation. There may be smaller perturbations induced by that core which extend in three dimensions. Examples are charged impurity atoms, which, due to their long-range Coulomb forces, change the potential energy of an electron even over distances large compared to the lattice constant.

Below we will characterize the various perturbations in more detail, start- ing with point perturbations.

3.1.2 Point perturbations

Types of point perturbations

In this subsection, we describe the most important compositional and struc- tural point perturbations of semiconductor crystals. An illustration is given in Figure 3.1.

(1) We begin with an impurity atom on a regular crystal site. Since the impurity atom substitutes an atom of the host crystal (see Figure 3.lb)’ it is referred to as a substztutzonal vrnpuraty. Examples of substitutional impurities are a phosphorus atom in Si on a Si-site, or a sulphur atom in GaAs on a As-site. To avoid a somewhat cumbersornc description, in the first case one uses the symhol S I : P , and in the second, the ~ynibol GaAs ; S A ~ . As a rule, impurity atoms which are chemically similar to atoms of the host crystal, are incorporated substitutionally. For this reason, many elements of the mein group of the periodic table, if added l o group-IV elemental semiconductors as well as binary 111-V and 11-VI-compound semiconductors, form substitutional impurities. The substitutional incorporation, in most cases, occurs on that lattice site which corresponds to the chemically most similar of the two atonis in the binary compound semiconductor. ‘l’herefore, the doping of GaAs with S leads to the above mentioned point perturbation

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228 Chapter 3. Elcctronic structure of semiconductor crystah with perturbations

CaAs : S A ~ (and not GaAs : S G ~ ) , and the doping of GaAs with Si leads to GaAs : Sica (and not GaAs : S ~ A ~ ) .

(2) An unoccupied regular crystal site is called a vncuncy, as depicted in Figure 3. Ic. Tn semiconductors made of binary chemical compoiinds, one has to distinguish between cation and anion vacancies, as shown Figure 3 . 1 ~ . Vacancies occur in all important semiconductor crystals; the general symbol is V. The vacancy in Si is denoted by Si : c': the cation vacancy in C h 4 s by GaAs : T;&, and the anion vacancy by GaAs : VA*.

(3) If the impurity atom does not occupy a repular crystal site but a site between regdar ones, one has an interstitial ampurity atom (Figure 3.16). In order for an impurity atom to stay at an interstfitrial site, it must have sufficiently low energy there. It is quite clear that this will be satisfied for interstitial sites which either haw high local symmetry or which lie on a bond between two at'orns. In thc latter case t,he crystal has bond centered interstitiah. The high symmetry interstitial sites in t e t r ahda l seniiconduc- tors may be such with tetrahcdral local symmetry in the neighborhood of the cation or of the anion (for group IV elemental semiconductors the latter distinction is void, of course). Moreover, there are high symmetry sites with hcxagonal symmetry. One refers to these7 respectively, as f e t v u h c h l and hexagonal interatitiak The incorporation of impurity atoms on interstitial crystal sites is especially likely when the impurity atom deviates relatively strongly from the atoms of the host crystal as! for example, in t,he case of transition metal atoms in semiconductors composrd of elements of the main groups. The general symbol for an interstitial is I . An interstitid Fe atom in Si on a tetrahedral site is denoted by Si : I;,.

(4) If a chemically 'correct' atom of the crystal occupics an interstitial site rather than a regular one, one has a self-interstitial (as shown in Figure 3.le). In order for such a structural point defect to develop in a crystal, there must he enough space between the host ntmns, i.e., the crystal should not be packed too densely. This happens, for example, in the case of tatra- hedral semiconductors, particularly in Si and Ge which have purely covalent b ondinE.

( 5 ) If,a crystal consists of two different chemical elements, then an atom of the h s t may occupy a regular site of the sccond, and vice vcrsa Such point perturbations are called antisite defects, as i lhs t ra td in Figure 3 . 1 ~ In the case of CiaAs, for example, a Ga atom may be located ttt an h-s i te ~ this is called B I ~ As-antisite defect,, and tla As atom may occupy a Cia-site - this is called a Ga-antisite defect. The symbols are As& for the As-antisit>e defect, and G U A ~ for the Ga-antisite defect.

Interstitials, vacancies and antisite defects are structural point perlur- bations, or point defects. The compositional point perturbations, i.e. the

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3 . 1 . Atomic structure of real semiconductor crystals

Ideal crystal

Substitutional impurity (S) SA

Vacancy (V)

Interstitial ( I 1

Interst. impurlty -___ A,-Antisite

A -Atom

0 B-Atom Impurity atom

S B

229

4

B,-Antisite

e)

Figure 3.1: Illustration of the most important point perturbations in semiconduc- tors using the example of a crystal with two atoms per unit cell of the same chemi- cal element (left-hand side) and different chemical elements (middle and right-hand side).

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230 Chapter 3. Electronic structure of semiconductor cxystals with perturbations

Table 3.1: Electron corifiguratiori of main group elements. In the rightrnost column the respective closed shells are indicated.

impurity atoms, need to be further specified. This will be done next.

Classification of impurity atoms

The division of elements into groups, which is commonly used in chemistry, also proves to be helpful for the classification of impurity atoms in semi- conductors. This is not surprising because the incorporation of an impurity atom in a crystal indicates a more or less strong chemical bonding. We summarize this group division of chemical elements below.

The periodic table consists of two types of groups of elements, the main groups, and the transition groups. Of the first 98 elements, 50 belong to the main groups and 48 to the transition groups. The elements of the main groups in Table 3.1 have in common the feature that electron shells with angular momentum quantum numbers 1 2 2 either do not occur at all or, if they exist, they are completely filled or completely empty, i.e. no partially filled shells of this kind occur. The energetically highest, and thus in general, only partially filled shells of these elements either have 1 = 0 or 1 = 1. Therefore, they are s- and p-shells. Because of the relatively large spatial extension of s- and p-shells in comparison with d- and f-shells, the former are simultaneously also the outer shells of the atoms which are responsible for chemical bonding. One thus also speaks of sp-bonding elements. The rare gas elements are special cases, in which the s- and pshells are also completely occupied.

For the elements of the transition groups presented in Tables 3.2 and 3.3, the shells with 1 2 2 are energetically the highest and, thus, in gen-

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3.1. Atomic structure of real semiconductor cry&&

ziSc 3d4~3~

231.

=Ti 23V z4Cr wMn mF'e z7cO Ni 3d248' 3d34s2 3d5& 3d"4s2 3d'4s2 3d74s2 3d64aa 3a23p6

Table 3.2: Electron configuration o€ transition dernent,s. In the rightmost column the respective closed shells are shown.

n Iron m o w

era1 they are the not completely occupied oms. Because of the relation R 2 I + 1 between the main quantum number ?a and the angular mornen- turn quantum numbe1 E. the d-shells (I - 2) are possible only for n >_ 3. the f-shells ( 1 - 3) only for n 2 4, etc. Accordingly, one has the shells 36, &, 4f , 5d, Sf, ;1g, 6 4 Sf, 69, fih ptc. Since among the first 98 elemmts of

Table 3.3: Electron configuration of rare earths and actinides. In the right column the respective closed shells are indicated.

Actinides

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232 Chaptcr 3. Electronic slructure of semiconductor cr,ystals with perturbations

the periodic tablc, however, the 5g-shell already remains unoccupied, only d- and f-shells are to be considered, namely the d-shells 3d , 44 5d , 6 d , and the f-shells 4f and 5 f . The filling of the 3d-, 4d- and 5d-shells takes place in the series of transztzon metals (together with the Elling of the 4s-, 5s- and 6s-shells). Among the transition metals, one distinguishrs the iron groixp in which the 3d- and 3s-shells are being filled, the palladium group in which the samc happens with the 4d- and 4.s-shells, and the platinum group where the 5d- and 5s-shells are being Elled. The 4f-shells are being filled in the rare earth elements, and the 5 f-shells in the actinides. In comparison with the s- and p-orbitals, the d- and f-orbitals have a smaller extension in spacc, they lie mostly within the s- and p-shells of the same main quantum number 7 ~ . 'l'herefore, mainly s- and p-electrons are involved in chemical bonding. This explains the remarkable chemical similarity of thc rare earth elements with each other, and a certain similarity of these elements with the elements of the main groups.

Complexes of point perturbations; associatos

Just as atoms arc bound in molecules when it is energeticaqy advantageous, point perturbations also associate if the result is a state with lower energy. They are called poznt perturbatzoa complezeu or u8souutP8. Bonding can occur between various point perturbations: between chemically identical or different impurity atoms, between impurity atoms and point defectu, and among point defects themsclvcs. The associates may consist of two or of several constituents. The diversity of these associates is comparably large to that of molcculcs in chemistry. We give some examples below.

Donor-acceptor pairs

In the case of an ionized donor and an ionized acceptor, the lowering of total energy through the formation of a bound complex of associtlteb is par- ticularly obvious the two point perturbations are differently charged and attract each other through electrostatic forces. This leads to the forma- tion of donor-acceptor paws, in which the donor and acceptor atoms occupy neighboring sites in the crystal. In general, the pairs are stable at several possible distances, whirh gives rise to a variety of different donor-acceptor pair complexes.

Di- and multi-vacancies

When there are two vacancies, the mechanism for the formation of bound pairs can also be easily undertitood the (internal) surfacc of thc crystal i s reduced if two previously isolated vacancies move together to occupy neigh-

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3.1. Atomic structure of red semiconductor crystah 233

boring cryYtal sites. One caIls this associate a ddvacancy. Analogous atate-

ments holds for the association of more than two vacancies, which are called multrvarunczes.

Frenkel defects

If, in a crystal, an atom moves froin a regular site to an interstitial site, then it h v e x behind a vacancy which attracts the interstitial. Thus a defect pair is formed in this process which consists of a self-interstitial and a vacancy. It is called a Freibel defprt.

There arr important puint perturbation complexes which occur only in a specific material or matrrial group. We now consider some examples for si snd GAS.

Point perturbation complexes in Si

Wr various reasons, hydrogen is often present in Si. In p-type Si, H atoms undergo chemical bonding with the availablc wreptoi atoms. Thereby the electron of the H atom is captured by the acceptor atom: which becomes singly negatively charged. One may also conclude that the acceptor expends i t h hound hole to the H atom ralher than to the valence band because it has lower energy there. In B-doped Si, for example, it negatively charged B-ion and ti positively charged H-ion are formed in this way. The two ions attract each other by- Coulomb forces, which results in the formation of a neutral ( H . B) pair. Of course, the pair will not be able t o arcept an clcrtron which means that the B atom has lost its ability to act as an acceptor.

Chalcogen atoms like S, Se, and Te are incorporated in S i not only as single atoms, but tlBo as two-atom molecules. Oxygen in Si enters into bonding with a vacancy, forming a pair which probably constitutes the so called A-renter known from caparity 111ea~iremmtfi. Oxygpn is also involved in a wries of other defect complexes in Si, among others the so-called thermal damrs. which are thusly named hPraiise of their origin in thwmal treatment.

Point perturbation complexes in GaAs

A prominent defect associate in GaAs is the so-called RX-center, which acts as a donor. It is found in GaAs and also in (Ga, A1)As mixed crystals under appropriate conditions (e.g., high pressure, for more see section 3.5). Originally, the DX-center was attributed to a donor atom, like S&, bound to another point perturbation whose nature was unhiowii at t!hHt time and, therefow, was denoted by X. Currently, the DX-center is thought to be due to the donor atom alone, more strictly; to a donor atom which is incorpo- rated interstitially but not substitutionally, as commonly happens. Another

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234 Chapter 3. Hectronic structure of semiconductor crystals with perturhations

typical defect associate in GaAs and other Ill-V semiconductors is the com- plex formed by an As-antisite defect (a Ga atom on an &-site) associated with an As-interstitial. Now, this complex is believed to form the care of the so-called b;L!Lcentei- in GaAs. This center manifests itself as deep level hav- ing strong influence OR the electrical and optical properties of GaAs (more detail on the EL2-rcnter may br found in section 3.5).

Point perturbation aggregates

I€ the complexes formed by point perturbations bmome larger and reach mesoscopic size, one refers to them as aggw.qates. Complexes of macroscopic size such as, for example, oxygen or heavy metals in Si cryshls, are called precipitates.

Latticc relaxation

The forces on atoms in the irnmediatc vicinity of a point perturbation dif- fer frorri those in lhr ideal crystal. They are non-xr?rol in general, at. the ideal crystal sites. Thus the atoms are forced t.o move t.u new equilibrium sitcs. This is known as lattice relamtion (in Figure 3.1 this effect is omit- ted). The new rqiiilihriiim sites are initially tinknown. In principle, they can be determined by means uf atomic structure calculatious for t he per- turbed crystal. These have to be performed sirnultanmusly with calculations of the elect,ronic structure, just as is done in self-consistmi. calcidations of the electronic and atomic st.ructwes of ideal crystals described in Chapter 2, section 2.2. HoweveT, there is an impwtant difference between the two cases. For ideal crystals, the calculation of at,omic structure may be avoided since, for t,he latter, complete and reliable experimental data are available. However, in regard t,o the atomic structure of a crystal in the vicinity of a point perturbdion, in many cases, hardly more t,hari t.he symmebry is known from experiment. Thus the self-consktent calculation of the electronic and at.omir: sbriictures must actually be carried out. if lattice relaxat,ion becomes important.

Experimental information concerning the symmetry of lattice relaxation may be derivrd from observation of t,he Jahn-TelEev efleet. This eflect results in splitting of the degenerate energy levels of a point perturbation due to a symmetry-lowcring latt.ice relaxation. In the case of a varaiicy: for exam- ple, the lattice rehxation can reduce the original tetrahedral symmetry T d to tetragonal symmetv D2d. Such spontaneous symmetry breaking occurs when it leads to lowering of the total energy of the crystal. This mag; happen when, in the unrelaxed state, i.he point perturbation has a c1rgenerat.e level which is only parhially occupied. First. of all, the degenerate level will split off due to the symmetry lowering relaxation. According to perturbation theory,

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3.1. Atomic structure of real semiconductor crystals 235

this splitting proceeds such that the center of gravity of the levels remains unchanged. Thus, along with levels shifted up, there are also levels which are shifted down. If only the latter are predominantly occupied, then the energy of the electrons localized at the point perturbation decreases. This energy reduction can compensate the increase in total energy due to the removal of atoms from their equilibrium sites. If this happens the relaxation is energetically favorable and will take place spontaneously.

In the case of the Jahn-Teller effect, the displacements of atoms are of the order of magnitude of one tenth of an Angstrom, i.e. they are relatively small. Larger displacements, of the order of magnitude of one Angstrom, are observed at point perturbations for which different atomic structures are stable, depending on external conditions as, for example, the position of the Fermi level. This phenomenon is observed at the D X centers in GaAs and (Ga, A1)As mentioned above (see section 3.5 for more detail).

3.1.3

Formation of structural defects

Formation of point perturbations and their movement

All of the above mentioned defects of ideal crystal structure may in fact exist in real semiconductors. There are various reasons for their occurrence, the most important and general being the second law of thermodynamics. According to this law, the thcrrnodynamic cqiiilibrium state of a crystal at temperature T and pressure p , is characterized by a minimum of the Gibbs free energy G = H - TS. Here H is thc enthalpy and S the entropy of the system. The entropy of a macroscopic state is proportional to the logarithm of its microscopic realization probability (or ‘thermodynamic probability’) and the proportionality factor is given by Boltzmann’s constant k. The system considered here is the totality of the atoms of the crystal. Lets assume that there arc only chemically ‘correct’ atoms, and that these are randomly distributcd in space. Then the idedl crystal is formed wherein the atoms move to the regular crystal sites. This corresponds to a very special state of the system. It is extremely improbable compared to the large number of states in which deviations from the ideal configuration of atoms appear as described above. The entropy S of the ideel crystalline state i s smaller, therefore, than that of the imperfect crystalline states.

The minimum of enthalpy II i s reached in the ideal crystalline state. Since the entropy S takes larger values in other states, the minimum of H is not necessarily coincident with a minimum of the Gibhs free energy G. Depending on temperature, a minimum of G = H - T S can also be adjusted for a state of the crystal which is less advantageous with respect to enthalpy, but more advantageous with respect to entropy, i.e. for a b t a k with a non- vanishing concentration of structural defects. As a concrete examplcs we

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236 Chapter 3. Efectronic structure of semiconductor crystals with perturbations

consider a vacancy in an elemental semiconductor. In an ideal crystal. the periodicity region contains a number J of identical

atorris. A vacancy is created w h m one of t h e atoms is removed from the crystal. Altogether, there are J different possibilities for such a removal, as many as thme are atorns. For each realization of the irlcal crystal one has, therefore, J realizations of the crystal with B vacancy. II n independent vacancies exist, the number o f realizations is .J(J- I ) . . ( J - r L + l)/nL The gain of entropy AS compared to the ideal cryatd t hrrefore mntnints to

A s = k l n [ J ! 1 , n!(J - I t ) !

In a more rigorous treatment, the entropy of lattice oscillations has yet to bP considered in AS, more accurately, the change of this entropy because of the alteration of the spectrum of lattice oscillations in the presence oI the vacancies. We will neglect this effect in what follows. What must, however, he taken into account is the enthalpy Rf necessary for the formation of n

vacancy (at constant pressure). For R independent vacancies the enthalpy requirement is n H f . Altogether, the gain of Gibbs free energy AG due to the formation of n. vacancies becomes

This expression Ims t o be minimized with respect to TL for a hed value of T and p. According to of Stirling's formula, h { n ! ) N n In(n) - n, the minimum coiidition may be written as

Since n << d can be assumed, the ilctivalion 1.m~

follows from this relation. The enthalpy of formatiult HJ €or B vacancy typ- i d l y amounts to several c l ' . Assuming H f = 2 5 el/ and T - 1400 K (roughly the crystalhdation temperature of Si), one gels the value n = J x 5 x 10-l' from expression (3.5). Because J - s x cm-' (for Si), a varancy conrmtration of about 1013 crriL3 follows. Similar concentrations are ohtnincd for other point deferts. Thew v a h s refer. hy thcir derivation, to the high temperatures assumed above, at which the cqs td is grown. Cooling down to room temperature often does not, however, substantially change the defect concentrations The defects are frozrii in. From this one may conclude that the presence of a considerable number of point defects in

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3.1. Atomic struclore of real semiconductor crystals 237

crystals is essentially unavoidable. The Si singlecrystal bars used in micro- electronics its a material base, in fact, contain vacancies and interstitials in the above estimated concentration of about ~ r n - ~ .

Incorporation of impiirity a t o m s

Chemical point perturbations are not as unavoidable as structural defects. Of course, the laws of thermodynamics in this case also act in a direction which leads away fro111 the absolute chemical purity of the ideal crystal. This tendency, however, can only be effective to the extent that chemically 'wrong' atoms are available in the raw materials and the chamber in which the crystal is grown. The state of a mure or less homogeneous distribution of impurity atoms in the whole chamber, i.e. the state in which the growing crystal also contains impurity atoms, has smaller Gibbs free energy than that of the chemically absolutely pure crystal with all the impurity atoms confined to the remainder of the chamber. However, by cleaning the raw materials and the growth chamber ~ in thermodynamic terms by an extrac- tion of entropy - the number of chemically 'wrong' atoms can be reduced. Theoretically, no limit exists for the degree of purity achievable, practically such td limit is set by the cleaning expenses, of course. Therefore, one re- duces the concentration of impurity atoms only to a level which is absolutely necessary for the application iinder consideration.

A boundary for the achievable concent,ration of impurity atoms in a crys- tal exists, as a rule, in the form of an upper limit. Under thermodynamic equilibriiim conditions it is given by the solubility of the corresponding el- ements in the semiconductor crystal. The solubility increases with rising temperature acrording to a law which is similar to that for the vacancy con- centration of equation (3.5), providcd Hf is underst,ood as the formation enthalpy of the impurity. The latter depends on the chemical nature of the impurity atom and host crystal, and it also differs for incorporations at differ- ciit crystal sites like substitutional or interstitial ones. High solubility values are achieved, as a rule, if the element of lhe impurity atom is chemically similar to at least one of the elements of the crystal. Incorporation at sub- stitutional sites is preferred in this case. If the crystal consists of elements of tho main groups, which i s the case for the majority of known semiconductor materials, one has relatively high solubility, especially for impurity atoms of these groups themselves. In the extreme rase, an alloy will be formed with t,he impurity atoms. The dissolving of A1 in GaAs, for example, results in a (Gtt, Al)As alloy. In such cases, t,he solubility equals the concentration of the host crystal atoms, i.e. crn-'. For P in Si or Ge in G A S , solubility values of lo2' c n ~ - ~ are reached, which lie just a little below. For transi- tion metal elements, which d i k r st,rongly chemically from the main group elements, t>he solubilities in semiconductors made of such elements are sub-

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238 Chapter 3. Electronic structure of semiconductor crystals with perturbations

stantially smaller. They lie in the region between lOI4 cm-3 in the case of the elemental semiconductors, where the transition metal atoms can be incorporated both substitutionally and interstitially, and 1017 ~ r n - ~ in the case of thc compound semiconductors where, as a rule, substitutional incor- poration is preferred. Generally, solubilities for interstitial impurities depend strongly on the sizes of the impurity atoms. Small atoms, like H, may be interstitially incorporated in higher concentrations than larger atoms, such as atoms of the transition metals. It is remarkable that rare earth atoms are predominantly incorporated substitutionally in tetrahedral semiconductors of the main group elements. This may be understood by means of the fact that the not-yet-completed 4f-shells are smaller than the already filled 5p- and 6s-shells, so that 3- and pshells are involved in chemical bonding, just as in crystals of main group elements. The solubilities of the rare earths are nevertheless relatively small, typical values being of the order of magnitude I 014 cm-3.

There are various procedures to introduce impurity atoms into crystals. The easiest one exploits the growth of the crystal from the melt: the impuri- ties are added to the melt of the host material, &hen the melt is rooled down to cause crystallization. Another method proceeds in the solid state: the irn- purity atoms are ‘diffused-in’ from an outer source. To speed up diffusion, the crystal is heated. Besides the two processes mentioned, which introduce impurity atoms under equilibrium conditions, there are also non-equilibrium procedures to reach this goal, among them the so-called i o n i m p l a n t a t z o n In the latter, the impurity atoms, after they have initially been ionized and then accelerated by an electric field, penetrate into the crystal where they are ‘implanted’. This implantation process must be followed by heating of the crystal in order to heal out the defects which arise during implantation (annealzng). The heating also makes it possible for the impurity atoms, which were initially placed at sites relatively indiscriminately, to reach their equilibrium sites. If the concentration of implanted impurity atoms should be larger than the equilibrium solubility, the surplus atoms are later ex- cluded from the crystal to a certain extent (precipitation). In many cases, a relatively large number of surplus atoms remains in the crystal, which then is in a ‘frozen-in’ non-equilibrium state.

Migration and diffusion of point perturbations

Point perturbations interact with the atoms of the crystal surrounding them. Both the point perturbations themselves as well as the environment atoms are in permanent motion because of the lattice oscillations. This results in more or less chaotic forces on the point perturbations. Due to the effects of these forces the perturbations move over to equivalent crystal sites in adjacent primitive unit cells. This is referred to as a migra t ion of point

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3.1. Atomic structure of real semiconductor crystals 239

perturbations. The migration of a vacancy proceeds such that an equivalent neighboring atom moves to the vacancy site and leaves a vacancy behind at its former position. The latter vacancy is displaced with respect to the original vacancy. A substitutional impurity atom migrates mainly with the aid of a vacancy, namdy by filling a neighboring vacancy site. In contrast to this, an interstitial impurity atom does not need the help of a vacancy for its migration, it can move directly to an adjacent equivalent crystal site.

The initial and final states of an elementary migration act are relatively stable states of the crystal. Between them, intermediate states with larger total energies occur. This means that ttn energy barrim has to be overcome in an elementary migration act. One calls it the magration barrier Em. In order for migration to occur, the lattice oscillations must provide the energy Em or the enthalpy Hm = Em + p V , with the pressure p held constant rather than the volume V, as comnionly OCCUTS. In this way, the migration rate becomes proportional to the activation factor e x p ( - H , / t T f . For the migration af a vacancy in Si and Ge, H, amounts la about 1 eV, and for the migration of a self-interstitial in these materials, it is smaller than 0.25 el/.

Migration constitutes the elementary microscopic event underlying the mtacroscopic process of diffusion. The latter occurs then when the distribu- tion of the migrating point perturbations is spatially inhomogeneous. For not-too-large gradients of the defect concentrations, the first Fick's law may be applied in many cases. Conespondmgly, the diffusion of a particular kind of point perturbation may be traced back to just one material con- stant, namely the diffusion coefficient D. The temperature dppendence of the latter also exhibits an activation behavior.

where Do is the limit of U at high temperatures. In the case of diffusion through interstitial sites, the activation enthalpy Q equals the correspond- ing migration enthalpy H,. For diffusion through vacancies, Q is given by the sum of the migration mthalpy Hm and the formation enthalpy H f for a vacancy. Substitutional impurity atoms preferentially diffuse through vacan- cies. In Table 3.4 the Q-values for substitutional and interstitial diffusion of some impurity atoms in Si are listed. For P in Si, which diffuses substitution- ally, the diffusion coeficient at 1500 K is about lo-'' cm2 sec-'. At1 atoms in Si preferentially diffuse through interstitial sites. The d i f f ~ s i ~ ~ coefficient at 1500 K of about 2 x c m 2 scc-l exceeds that for P in Si by 6 orders of mappitude. The reason for this hiigp d i k e n c e is thp siibstantially smaller activation enthalpy for interstitial diffusion as compand to substitutional.

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240 Chapter 3. Electronic structure of semiconductor crystals with perturbations

impurity atom

activation enthalpy (eV)

B Ga P H Au Cu

3.7 3.9 3.7 0.5 0.39 0.43

3.1.4 Line and planar defects

With an increase in the number of point perturbations, a tendency arises for these perturbations to arrange themselves on lines or planes of macro- scopic size. Then one has 1- and, respectively, 2-dimensional perturbations in contrast to the 0-dimensional ones considered above. In practice, order- ing on lines and planes occurs only for structural point perturbations. The most important representatives of a 1-dimensional perturbation are step- and screw dislocations. At a step dislocation, the occupation of a particular lattice plane breaks down along a line of lattice points - the plane is occupied on one side of this line (representing the step dislocation line), but not on the other side (see Figure 3.2). If one calculates the line integral over the path shown in Figure 3.2 (left-hand side), then the result will not be equal to zero, as would be the case for an ideal crystal, but equal to a non-zero vector perpendicular to the dislocation line. It is called the Burger’s vector. For a screw dislocation one has to view the crystal as being cut by a semi-infinite lattice plane, which is bounded by a line of lattice points representing the screw line. Then the lattice planes left and right of the cutting plane are shifted parallel to the screw dislocation line by a lattice vector. After that one reconnects the two crystal halves again (see Figure 3.2, right-hand side). The line integral about the screw dislocation line yields a vector parallel to this line. In a dislocation, deviations from crystal structure are, indeed, not limited to a line as one could initially assume. Slight atomic displacements (strain) occur in a finite macroscopic environment. The microscopic core of the perturbation is limited, however, to the dislocation line.

The two most important examples of 2-dimensional perturbations are stacking faults and grain boundaries. In a crystal with stacking faults, the various lattice planes carrying the atoms are not stacked in the same way as in the ideal crystal, but certain lattice planes are twisted by an angle. Grain boundaries occur in crystals which in fact consist of two differently oriented half crystals - the grain boundary is the lattice plane at which these two half crystals meet. In a sense, surfaces of crystals may also be considered as 2-dimensional perturbations.

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3.2. One-electron Schriidinger equation for point pcrturbations 24 1

Figure 3.2: Illustration of a step dislocation {left-hand side) and a screw dislocation (right-hand side). ( A j L w f35w.. 1990.)

Having surveyed of the atomic structure of real semiconductor crystals we will now proceed to the electronic structure of such crystals. In this, we restrict, our considerations to crystals with compositional or structural point perturbations, therefore, we excliide 1- and 2-dimensional perturbations. 'rhis is done in recognition of the fact that only point perturbations can play an important positive role in semiconductor devices, while perturbations of higher dimensions are generally disruptive. Fortunately, the latter can also be more easily avoided than 0-dimensional perturbations, because they are less effective in increasing entropy. The silicon bars used in microelectronics are now grown practically free of dislocations and grain boundaries.

In the following sect'ion we will formulat'e the Schrijdinger equation for the oneelectron states of a crystal with a point perturbation.

3.2 One-electron SchrGdinger equation for point perturbations

In regard to its geiwral form, thr ontLclectrou Schrodinger quat ion for a crystal with point perturbations does not substantially differ from the one electron Schriidinger equation for an ideal crystal. The reason for this is that the derivation of the latter equation in section 2.2 never actually involved the periodicity of the ideal lattice. Only the oue-elwtron potentid l imrt(x) of the peTtriirbecl rrystal diffws from the rarresponding potential V ( x ) of the ideal crystal given in equation (2.76). However, just like the latter, also the potential V ~ " ( x ) of the perturbed crystal is the sum of three contributions, the potential V ' c ? 7 L ( ~ ) caused by thf atomic cores of the perturbed crystal, the Hartret. yotentid I/Frt(x) of the valence electrons, and the exchange- correlation potential VF:(x) of these electrons, In analogy to equation (2.76) we therefore have

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242 Chapter 3. Electronic structure of semiconductor crystals with perturbations

VWTt(X) = VL?'t(X) f V,"'"x) + Vg;t(X). (3.7)

With this, thc one-electron Schrodinger equation for the wave function $lV(x)

of an electron of the perturbed crystal reads

3.2.1 Elect ron-core interact ion

First of all, we discuss the potential VcPTt(x) of the electron-core interaction. We decompose it into a sum of the core potential Vc(x) for the ideal crystal, and a potential V,'(x) which describes the change of the core potential due to the point perturbation. Thus,

VC~'"X) = K ( X ) I V,(x). (3.9)

In further clisrussion, we assume that only one point perturbation exists in the crystal. The center of this perturbation will be taken as the origin of our Cartesian coordinatc systcm, and we procecd on the assumption that the potential VJx) falls off with increasing distance form the origin and finally approaches zero:

V,'(x) -+ 0 f o r I x I + 00. (3.10)

The point perturbations listed in section 3.1 differ as to how fast this decay proceeds.

We consider, initially, the case of a substitutional impurity atom whose core contains almost the same numbers of protons and neutrons as the core of the host atom. With this rqiiirement, the two atomic cores differ mainly through their different charges. One refers to them as isocoric impurity atoms. This case occurs, for example, if a P atom with only one additional proton and one addit,ional neutron substitutes a host atom of a Si-crystal (as illustrated in Figure 3.3). We denote the number of (positive) elementary charges of the core of the impurity atom by Zr , and the number of (positive) elementary charges of the core of the host atom by Z H . Because of charge neutrality of the individual atoms, ZI and ZH are simultaneously also the numbers of valence electrons of these atoms. The potential energy of a valence electron of an impurity atom of the type described differs, above all, by the change of Coulomb potential from the potential energy of an electron of the host atom. If one considers the cores to be point-like and neglects spatial dispersion of dielectric screening, then one has, approximately,

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3.2. Oneelectron Schrcdinger equation for pint perturbations 243

(3.11)

where E is the static dielectric constant of the semiconductor material. Devi- ations from this perturbation potential are to be expected in close proximity to the impurity atom, on the one hand, bwansc there the spatial dispersion of dielwtric screening cannot be neglected, and. on the other hand, because the core of thp impurity atom diffcrs not only by its charge number from that of the host atom, but also in other respects. In fact each core has a iinite spatial extension because of its spatially extended r o w electrons (the nucleus of the core may be treated as a point charge). In close proximity to the center, the core electrons give rise to additional forces, heside the elerlrostatic point charge forces already counted in [3.11). These additional forces are caused by higher moments of the core electron charge distribution, and also by exchange and correlation effects. If the cores of the host and impurity atoms differ, whch always happens iI the atoms arc not identical, the additional core forces will also differ. Both effects, the spatial dispersion of screening and the differpiirps of additioiiat rorc forces, are jointly termed central cell correcfzonu.

The perturbation potential (3.11) is distinguished in that. with the ex- clusion of a close environment of the impurity atom, its variation over a primitive unit cell is relatively small. which implies that it remains signifi- cant over a relatively large distances from the impurity atom compared to the lattice constant. In this context, it is called a long-range potential. We may state. therefore, that isocoric impurity atoms are approximately described by smooth or long-range perturbation potentials.

The potentials which apply for non-isocoric impurity atoms are different. Consider first the case in which the charge eZf of the impurity atom core equals the charge eZH of the host atom core. This does not necessarily mean that the nuclei of the two atoms must have the same numbers of core electrons, because different numbers of protons of the atomic nuclei may compensate the difference of core electron charge. The requirement of equal numbers of core charges implies, however, that the numbers of valence electrons of the two atoms must coincide - as described by the term asovalent impuritr atoms. Examples for non-isocoric isovalent impurity atoms are C atoms in Si- or Ge-crystals, or N atoms substituting P atoms in Gap. The potential energy of an electron at a non-isocoric. isovalent impurity atom differs from the potential energy at the host atom not just by the screened Coulomb potential (3.10). but by the different core electron shells in the two cases. The perturbation potential b:(x) accordingly contains only electrostatic contributions of higher moments of the core electron charge distribution difference, as well as exchange and correlation contributions due to this difference. Both kinds of contributions decay more rapidly with

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244 Chapter 3. Electronic structure of semiconductor crystals with perturbations

pert v; = v, -v,

v

Si Si Si Si Si Si Si - _ _ _ _ _ _

Figure 3.3: Illustration of thc origin of long-range (left-hand side) and short-range (right-haritl side) core perturbation potentials.

increasing distance fiorn the center than does tlic Coulomb potential of a point charge, they are therefore described as short runqe. These arc also the potential contributions, whicb in the context of the ibocoric impurity atoms abovc, gave rise to central cell corrections.

The exact &termination of the pertrirhatiori potential is a problem which, like the determination of the periodic core potential of an ideal crystal, can ultimately be solved only by numerical calculations. These show, in fact, that the pet turbation potentials of isovdlenl impurity atoms differ horn zero substantially only over a distance of a few lattice constants. Consequently, we have

v'(x) 0 f o r I x I 5 a f e u lattice constants (3.12) (0 for all other I x 1

v(;(x)

Short-range perturbation potmtials apply not only to isovelcnt, non-isocoiic substitutional impurities, but also to structural defects such a8 vacancies or interstitials. l n thc case of a vacancy, the occurrence of a short-range perturbation potential is particiilatly obvioiis. The remuval of a Si at<nrn from the chain of S i atoms in Figure 3.3a yields the potential profile of VJx) + V,'(x) as depicted in Figure 3.3b, and the perturbation potential has the form shown in Figure 3 . 3 ~ . The latter approximately represents the ncgative potential v,'(x) of the missing atom.

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3.2. One-electron Schrljdinges equation for point perturbations 245

Finally we foiego the requirement that the core charges of the two non- isocoric atoms be the same. Examples of this most general case are Cd atoms with their 2-fold core charge in a crystal of S i atoms with their 4- fold core charge, or Sn atoms with their Cfold core charge on a G&site in GaAs, thereby substituting a triply positively charged host atom core. In these cases, the perturbation potential represents a superpobition of the screened Coulomb potential (3.11), which accounts for the different core charges, and a short-range potential which takes accoimt of all remaining differences bctween the cores d5 well as aspwts of the spatial dispersion of screening beyond those accounted for in equation (3.11). ‘The effects of the two potential contributions are not independent of each other. The same bhort-range potential has a greater effect if the Coulomb potential, due to a larger core charge difference between the impurity and host atoms, is stronger. This comes about because the stronger Coulomb potential pulls the electrons closer to the core, where the the short range potential is essential.

Just like the elwtron-core inteiaction potential, the Hartree potential V,””(x) and the exchangecorrelation potential V,”,’‘(x) also undergo chan- ges in the prrsence of point perturbationb. Below, we describe these changes for the Hartrre potential VIyt ( (x) and the exchange potential V,””((x) of the liartree-Fock approximation. Similar results can be derived for the exc~iaiig~rorrt~ation potential vzt(x) of the LDA mettiod.

3.2.2 Elect ron-electron interaction

Hartree potential

The Hartree potential V r t ( x z ) of the r-th electron is, as before (see relation 2.49), given by the expression

(3.13)

However, the oneparticle states $,(x2) here are not those of the ideal crystal, but those of the pertiirbed crystal and the summation runs over the one particle states u i occupied in the ground state of the pcrturbwl crystal.

From the physical point of view it is quite clear that the prrtiirbed rrys- tal ~ H S two kinds of stationary one-particle states; first, those with energy eigenvalues which were already allowed for the ideal crystal, i.e. those within the bands and, secondly, those with F I I C ‘ I ~ ~ eigenvalues within the energy gap between the valence and conduction bands. The states of the first kind rep- resent pure Bloch states only in zero-order approximation with respect to the perturbation potential, while, in higher approximations, superpositions

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246 Chapter 3. Electronic structure of semiconductor crystals with perturbations

of other Bloch states have to be added due to scattering by the perturba- tion potential. The pure Bloch state property of being infinitely extended over the entire crystal also applies to these scattered states for most of the energies. The states of the second kind, which belong to energies in the for- bidden zone of the energy spectrum of the ideal crystal, are localized, unlike the Bloch states, at the position of the point perturbation (formal proofs of this will be given in sections 3.4 and 3.5). We identify the extended states by quantum numbers v " ~ , and the localized states by quantum numbers doc. Furthermore we assume that, in the ground state, n of the total number N of electrons of the perturbed crystal are found in localized states v;OC O, and the remaining N - n electrons are in extended states viztd '. Accordingly, we decompose the Hartree potential of the perturbed crystal into two parts, the part V Z t d ( x i ) stemming from electrons in extended states vrtd O, and the part VAOC(xi) due to electrons in localized states v;OC ',

with

extd V g t d ( x i ) = e2 / d3x'

k J

(3.14)

and

(3.16)

Here we assume that the electron i on which the potential acts is located at the center. This assumption will also be maintained below.

The localized part VhOC(xi) of the Hartree potential depends on the num- ber n of electrons at the center. There are two sources of this dependence. First, the number of electrons which enter changes with n, Vfi"(xi) becomes more repulsive if n is large, and less so if n is small. Second, the wave functions of localized occupied states vLOC ' over which the sum in (3.16) is extended, depend on n. They become more localized if electrons are removed from the center, and less localized if electrons are added. One says that the wavefunctions relax.

For reasons similar to those causing the localized part VhOC(xi) of the Hartree potential of the perturbed crystal to depend on the number n of electrons at the center, the extended part V;&(xi) also differs from that of the ideal crystal, firstly, because the number N of extended electrons is decreased by the localized ones and, secondly, because the states of the extended electrons relax. The first change causes a relative potential cor- rection of order of magnitude 1/N and thus can be omitted (a comparable

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3.2. One-electron Schriidinger eqmtim for point perturbations 247

approximation was employed previously in the derivations of the Hartree potential and of Koopman’s theorem in section 2.2). The change of the extended states is essential, however: because all N - n occupied states are affected. In this connection, the probability amplitude of the extended states is reduced close to the localized electrons of the center because of Coulomb repulsion. Hence, a positive excess charge arises in the vicinity of the local- ized electrons which screens the Coulomb potential of these electrons. This change may be accounted for by replacing the localized part VAm(xi) of the perturbed Hartree potential by a screened potential Vh(%), and simulta- neously substituting the extended part V F d ( x i ) of the perturbed Hartree potential by the Hartree potential V ~ ( x i ) of the ideal crystal. Accordingly, we set Vkm(-) --t VA<xi) and V r t d ( q ) .+ V~(xi). Then it follows

(3.17)

where, by definition, VA(xi) is given by the expression

where e - ‘ ( q , x‘) is the inhomogeneous position-dependent screening func- tion of the crystaL

Exchange potential

As we know from section 2.2, the exchange potential describes the Coulomb interaction with the exchange hole which occurs because two electrons of the same spin cannot reside at the same position. For the localized electrons of the perturbed crystal, the positional uncertainty is smaller than that of the extended ones. Accordingly, their exchange interaction will be stronger than that of the extended electrons. It is therefore again expedient to decompose the entire exchange potential V,P”t(x,), which an electron a localized at the center feels, into the two parts V,..td(&) and l r ~ ( x ~ ) of, respectively, the (N - n) extended and n localized electrons. We replace the extended part by the exchange potential V x ( x i ) of the ideal crystal and simultaneously substitute the localized part by an effective exchange potential Vi(xz). With this replacement, we have

v,p”t(X,f = VX(Xi) + Vfi(X2). (3.19)

Using relation (2.61), it follows that

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248 Chapter 3. Electronic structure of semiconductor crystals with perturbations

(3.20) where the summation over k inrlirdes only particles whose spin W k equals the spin LT? of particle i.

Substituting trhe three potential parts (3.91, (3.17) and (3.19) into the oneelcctron Schrbdinger equation of the perturbed crystal, the various terms may bc arranged such that the effective oneelectron potential E'(x} of the ideal crystal (2.60) occurs. The equation reads

The three perturbativc potential parts in this equation have different ef fwts. In gcnpral, the core perturbation potential l<'(xz) is the strangest. The occiirrenre of localized states at the center. the central feature of our considcrations above, is maidy due to this potential contribution. If only one electron is localird at the center, tlip two othw potential parts L$(x%) and Vdk (x,) due to electron-electron interactions vanish completely. <;enerafly, they a e non zero and lead to corrections of the localized eigensolutions of ihe Schriictinger PqiiBtion (3.21) which rontaiiis L:(x) as the only potential. 'I'hcse corrections will now be estimated by means of perturbation theory.

Pert rirbation theory

The Srhrodingrr equation of zcro th order reads

[ H f q x ) ] ls'v(x) - E U T h ( X ) , (3.22)

whcrt. I1 - p2/2na + V(x), as bcfore. signifies the oneelectron Hamiltonian operator of the ideal ciystal. The index i is omitted because all electrons feel the same core perturbation potential. If only one particle is localized at the center, equation (3.22) is exact.

Hartree energy

The energy correction due to the Hartrcc potential for an electron z in local- ized stateqVt i s given in first order perturbation theory b - ~ the rxpectation valri~ ($V, 1 VAw I vv,). [!ring (3.181, this expression may be written as

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3.2. One-electron Schrodinger equation for point perturbations 249

(3.23) We extract the factor (E-'(x, x')/ I x-x' 1 ) from the x'-integral by evaluating it at an average position X in place of x'. The remaining x'-integral yields 1 because of the normalization of the wavefunction @,,(x'). The integral over X, multiplied by e2, will be denoted by Uy,, i.e. we set

(3.24)

With this, the expectation value of the perturbation of the Hartree potential becomes

The energy Up' of (3 .24 ) is called the Hubbard energy.

Exchange energy

Next we will calculate the corresponding energy correction due to the pertur- bation of exchange potential V%{x) of (3.19). The latter depends on the spin of the electron because. contrary to the ideal crystal case where the num- bers of electrons with 'spin-up' and 'spin-down' are equal in ground state, the corresponding numbers n~ and nl of electrons localized at the center can differ. In terms of the total number n = n~ +nl and the total spin projection MS = (1/2)(nT - "1) of the localized electrons, the two partial numbers ny and nl may be written in the form

"1 = - ( n + 2 h l s ) , 1 1 2 2

nl = -(n - 211iIS) . (3 .26)

For the expectation value (&,t 1 15 I iu,) of the perturbation of the exchange potential, equation (3.20) yields the expression

( ~ v , I ~'i I +v,) -(no, - I ) J , f o r not 2 1. (3 .27)

where g, is the spin quantum number of the state v,, and

(3.28)

is so-called the exchange integrcab The dependence of this integral on the orbital state v' = Uk has been ignored in expression (3.28).

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250 Chapter 3. Electronic structure of semiconductor crystals with perturbations

To solve the Schrodinger equation (3.21) explicitly) the number R of electrons at the center must be known. This question will now be addressed.

Number of electrons at the center

To say that an electron is at the center means occupying a state which is IocaIized there. The number n. of electrons at the center equals, therefore, the number of electrons in localized states. This number is initially unknown. It can only be determined self-consistenbly because the Hamihonian which determines the states depends on it. In thermodynamic equilibrium and at. temperature T = 0 K the number of electrons at the center may be found by self-consistently counting the number of localized states which belong to energy eigenvalues below the Fermi level. Since the position of the Fermi level depends on the thermodynamic state of t,he semiconductor, the number n. of electrons at. the center is also a state dependent quantity.

Below, we introduce some common not,ations used in this context and estimate n for particular impurity atoms whose chemical bonding to the host crystal is known. We assume the point perturbation to be elechically neutral, which means that in generating it, the same number of positive and negative elementary charges were added to the ideal crystal (as in the case of an impurity atom on an interstitial site): or removed from it (as in the case of a vacancy) : or removed and added (as in the case of a substitutional impurity atom). Depending on i t s environment in the crystal, the paint perturbation enters into a more 01 less strong chemical bonding with the surroundiug atoms. A phosphorus atom in a Si-crystal, for example, is chemically bound like a Si atom, i.e. the .Y- and p-states of the P atom are involved in the formation of the valence band states of the crystal, and 4 of its 5 a2p3-valence electrons are hosted by these states. Therefore, only 1 of the 5 valence electrons of the P atom i s available to occupy bcal izd states in the energy gap. This means that n = 1 holds for the neutral phosphorus substitutional impurity in Si. If a Si atom in a Si-crystal is substiluld by a boron atom, one has TI = - 1. Therefore, 1 bole is available to occupy the states localized at the center. Similarly, n 7 -1 emerges if a P atom in P ZnS-cryst81 replaces an S atom. The number V of electrons of an impurity atom X involved in its bonding to the crystal and which, thus, have energies in the valence barid and not in the gap, is generally referred to as the oddation state of the atom. The latter is denoted by X”+. A phosphorus atom, for example: which is installed in a si- c r y s t d on a regular crystal-site, has the oxidation state P4+, This notation, which looks like the number of e1ernent.at-y charges at the atom without being such, originates from its prior use for crystals and impurity atoms with pureIy ionic bonding. In the latter, the valence electrons in fact pass from the impurity atom to the surrounding crystal and the impurity atom is left behind as an Xv+-ion. This is no

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3.2. One-electron Schrodinger equation for point perturbations 25 1

longer true if the bonding is covalent or partially covalent. Then some of the bonding electrons remain at the impurity atom, and the true number of its elementary charges differs from V. The n valence electrons of the impurity atom, which remain after the departure of the V electrons into the bonds with the crystal, and are available for occupation of the localized levels in the energy gap, are sometimes called active electrons. In the case of an impurity atom with Vi valence electrons one has n = Vi - V active electrons.

So far, the charge state of the impurity atom prior to its incorporation into the crystal was taken to be neutral. However, positively or negatively charged ions can also be introduced into the crystal, and atoms introduced in a neutral charge state can have electrons removed or added within the crystal. Similar statements hold for structural defects. If a particular per- turbation center X is not neutral for the lack of Q electrons, in the sense just specified, one says that the charge state of X is Q and writes X ( Q ) (where negative Q mean surplus electrons). The charge state should not be confused with the oxidation state. In the purely ionic case, the distinction between the two can be expressed most easily: the charge state counts the elementary charges of the atom outside the crystal, and the oxidation state counts its elementary charges inside. Generally, the oxidation state of a cen- ter in charge state Q will be X('+Q)+ if the oxidation state of the neutral center is V . The reason is that, in this case, V + Q valence electrons are not available for occupation of localized states. The number n of electrons which are available, i.e. the number of active electrons, amounts to Vi - (V + Q ) if V,, as before, denotes the number of valence electrons of the neutral im- purity atom. The notations for the oxidation state and the charge state are summed up in the common symbol X('+Q)+(Q+).

A simply ionized sulphur atom in a Si-crystal, for example, is denoted by S5+(l+). Of the 6 valence electrons of the S atom, 6 - 5 = 1 are available for occupation of localized energy levels in the gap, instead of 2 electrons in the case of the neutral center S4+(O+). Transition metal (TM) atoms can be installed in Si- and other tetrahedral semiconductor crystals both sub- stitutionally as well as interstitially. The oxidation states V and, therefore also the numbers n of electrons at the impurity atom, are different in the two cases. For substitutional TM atoms, V equals the number of electrons which are left at the atom after it is bound to the crystal. Interstitial im- purity atoms are only weakly bound, i.e. the number V of electrons of the TM atom which occupy bonding valence states, is almost zero. The oxida- tion state is therefore T M o f . Oxidation and charge states coincide in this case, and the number n of electrons at a TM atom equals the number of its valence electrons. Interstitial Fe atoms in Si are found in the oxidation state F e o f , substitutional in the oxidation state Fe4+. In the first case, n = 8 (six electrons in 3d-orbitals and two in 4s-orbitals), and in the second case the value of n = 4.

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With the background analysis set forth above, we are now sufFiciently p r e pared to address the solution of the one-electron Schriidinger equation for the crystal with a point pcrturbation. Which solution methods can be applied with succcss depends decisively on whethe1 lhe core yrrt iulmlion potential is long- or short-range. Of course, thr lattice translation symmetry, which drastically simplifirs the bolution of t h ~ Schriidinger eqiiation hi the casr of an ideal crystal is perturbed by both kinds of potentials. but only for short-range potentials in an e s s e a t d mannet-. For long-range potentials the dec iations from lattice trauslation symmetry are relatively weak. In this case the Scfirodinger equation, in a certain sense, may be decomposed into two quations, one for the periodic potential of the ideal crystal, which was solved previously, and one for the perturbation potential. The latter is called the LPffective mass equation’, which we will derive in the following section.

3.3 Effective mass equation

ure start with the Schrodinger equation (3.8) of the perturbed crystal in the form (3.21). The most, iiriport.ant assumpt.ion we will make relat.es to Ihe core part V ~ ( X ) of the pert,urhatiozt potential VATt(x) in this equation. This potential part is supposed to be smooth on the atomic length scale. In conjunction wit,h t,his wxmption, point pcrtiirhatioiis with shos-t-range core pert.iirbation pot.entials Vf(x) are ruled out from the very beginning, because these caiinot be considered to I c smooth. Point perturbations with Coihm- bic core perturbation potentials are still allowed, although the Coilonib form of this potential is not necessary for the derivation of t.he effective mass equa- tion. The other parts of thr t o l d perturbation potential V&.t(xj of eqiiation (3.21)! i.~. the pdurbations of the Hartree and exchange potentials l7L(x) and ViCx), arc automat,ically smooth if V&(x) has this property because these potential parts are dctcrmined by t.he solutions qV of Ychrodinger equa- tion (3.21) which are smooth if L’;(x) is also. The wwelunction dependence of Vh(x) and L’i(x) requires self-consistent solutions of the Schriidinger equation. In this section we do not intend to solve this quabion explicitly hut iclther to t,ransform it into another equation, the cffective mass equa- tion? a l i i th can be solved more easily. Althoiigh the self-consistency demand is transferred to the effective mass equation. it does not iulerfere with its derivation. For the latter? Vh(x) and Vk(x) may be treated as smooth ex- ternal potdialii which, together with V,‘(x)! add lip lo form a smooth total perturbation potential i’’(x). Far hhe derivation which follows, the PO- t.ential need not be t.he smooth perturbation potential VieVt(x) of a point perturbelion, any smo0t.h potential U ( x ) is allowed. This is important inns- much as it becomes possible in this way to utilize the effective mass equalion not only €01 point perturbations with Coulombir core perturbation poten-

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3.3. Effective ma95 equation 253

tials, hilt also for macroscopic perturbations, such 8s that associatd with an external electric Geld, which are smooth on the atomic length scale by definition.

The goal of the following consideralions is lo simplify the oneelectron Schriidingcr eyiiation

(3.29)

for t,he perturbed crystal in stages, so that., ultimately: a mare easily solvable equation, namely the effective mass equation, results. In this mat,ter, we will employ t,he fact that, in the vicinity of critical points of a certain band v , the energy of an electron of an ideal crystal depends quadratically on quasi-wavevector k, i.e. in the same way that the energy of a free electron depends on its momentum. However, the free electron mass 7n is replaced by the effective mms rnc of the particular band and critical point chosen. The e&ct.ivc mass includes the effects of interaction of the electron with the periodic potential of crystal. Therefore, it is generally a tensor, and if il can be reduced to a scalar maw, the value of the lat.ter generally differs horn the mass of a free electron. Having in mind the effective mass description of the band energy versus quasi-wavevector k relat,ion, one may suspect that the influence of a perturbation potential V(X) on an &&on of the cryisla1 can be calculated in an appr0ximat.e way as follows: One eliminates the periodic potential of t h e oiic cltrtron Schrdinge~ equation for the perturbed crystal! while simultaneously replacing the free mass m in the kinetic energ); operator by the effective mass m:. The resulting Schrodinger equation represents the one-bend e#ective mass eyuatian in its simplest form It can be solved much more easily than the original Schrodinger equation? which includes the periodic crystal potential ill explicit form. For a siibstitutioiial P atom in Si, for example. t.he effective mass equation is twsentinlly the same a8 the Schrodiuger equation for the hydrogen atom whose solutions are already known.

The procedure described above needs, of COZITSC, further justificat.ion. To provide this, quest,ions have to bc addressed which have been left, open so far, for example, how the wavefunction of the e f l d i v e mass equation reiates to the wavefunction of the original Schrodinger equation, and the matter of what conditions must be placed on the perturbation potential V(x) and the wavefunction 7;j(x) in order for bhe effectivp mass equation to be applicable.

3.3.1

To address these questions and derive the effective mass equation for a single band, we rewrite the Schrodinger equdion (3.29) in the Bloch representation,

Effective mass equation for a single band

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254 Chapter 3. Electronic structure of semiconductor crystals with perturbations

expanding ~ ( x ) with respect to the complete set of Bloch functions ‘pvk(x),

whence

(3.30)

with the expansion coefficients given by

(vk I = d 3 x l p X x ) ~ ( x ) . (3.31)

Since the (pYk(x) are eigenfunctions of H with eigenvalue E,(k), the Schrodinger equation (3.29) takes the following form:

[ E y ( k ) 6 v f y f 6 k k ~ + (vklUjv’k‘)] (Jk’lV) = E(vkl$) . (3.32) v‘k’

To transform this equation into the effective mass equation, the assumption made at the outset that U ( x ) should be a smooth potential, must be specified further. This is done by assuming that the change of U ( x ) over a primitive unit cell is small in comparison with the change of the periodic crystal po- tential t ’ ( ~ ) over such a cell. To formulate the condition for smoothness in this sense quantitatively, we decompose U(x) in a Fourier series, using the same notation introduced previously in Chapter 2. We have

with Fourier coefficients

(3.33)

(3.34)

The sum in (3.33) is extended over the entire infinite k-space, meaning over all Brillouin zones. Smooth functions U ( x ) in the above sense have Fourier coefficients (klU) which, for large k-vectors, more strictly speaking, for k- vectors outside of the first BZ, are small compared to the Fourier coefficients (kll.’) of the periodic crysta1 potential V ( x ) . The latter components were calculated in section 2.4. According to formula (2.161), they are non-zero only if k is a reciprocal lattice vector K differing from zero. This means that the smoothness condition for U ( x ) may be expressed its

essentially non - zero f o r Ikl within first 32

<< I(KlV)l, K # 0, f o r Ikl outside f irs t BZ (3.35)

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3.3. Effective mass equation 255

The left-hand xlde of the inequality (3.36) is of the order of magnitude of the change of the perturbation potential U ( X ) over a primitive unit cell of the direct lattice, that is, a1VUI with a as lattice constant. The right-hand side of (3.35) can be estimated by a characteristic band structure energy, like the fundamental energy gap E,q between the valence and conduction bands. With this, the inequillity (3.35) takes the form

alVUl << Eg. (3.36)

It states that, for a perturbation potential U ( x ) to be smooth and our first condition to be fulfilled, the changes of U(x) over a primitive unit cell must be small compared to the energy gap. Secondly, we assume that the wavefunction $(x) can be set up exclusively from Rloch functions of only one band YO, i.e., that $(x) should be of the form

(3.37)

with F , ( k ) as a function which has yet to be determined. It turns out that this requirement has no contradictions if the perturbation potential is smooth.

The thzrd requirement refers to the function F,(k). It is assumed that Fvo(k) differs from zero only for small k-vectors in the sense used in equation

Fourthly, it will be assumed that for small k-vectors, which according to (3.35) have to be considered exclusively, the Bloch factors uWk(x) in the Bloch functions p,k(x) can be approximately replaced by their values at k = 0 :

(3.35).

U v o k ( X ) ZS .mo(x). (3.38)

In the following subsection 3.3.2 we will see that the two last requirements can be justified when eigenstates of the perturbed crystal exist having energy eigenvalues just above and below the edge of band vo, and if only these eigenstates are considered. If, additionally, the band edge lies at k : 0, the small energy deviations also correspond to small k-values, and relation (3.38) holds approximately. The restriction of the location of the band edge to k = 0 can be omitted, and the derivation procedure can also be applied to band extrema other than the center of the first B Z , the only modification being the replacement, of the R X center k = 0 by the non-central critical point k, as well BY of k by k - k, in the corresponding equations.

The above four conditions will now be used to simplify the Schrodinger equation (3.32). Applying relation (3.38) the wavefunction (3.37) may be written as

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256 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Tihe k-sum is the Fourier transform F,(x) of the function F,(k), whence

1 F,(x) = - CFvo(k)eik.x. (3.40)

& k

Therefore $ ( x ) may bc written as

$44 = ~ , O ( X ) ~ Y l ( X ) . (3.41)

The function F,(x) is termed the envelope functzon or, in short, the enve- lope. The envelope function, by definition, i s a bmooth function. Equation (3.41) means that the truc wavefunction + ( x ) is obtained by enveloping the rapidly oscillating Bloch lactor uvgo(x) with the smooth cnvelope function

The two ronditions conccrning the smoothness of U(x) and Fvo(x) make it possible to represent the matrix element (uk(l/(v’k’) in the Schrodinger equation (3.32) in a substantially simpler form. We rewrite this element using thc Fourier representation (3.33) of U(x) and the product form of the Bloch functions p v k ( x ) , obtaining

Fuo ( X ) .

Furthermore, we transform the integral over the periodicity region into a sum of integrals over the unit cells of this region. A lattice point R is associated with each unit cell, so that the sum runs over all lattice points of the periodicity region. Because of the lattice periodicity of the Bloch factors, only terms of the form exp [i(k’ + k” - k) . R] remain under the lattice sum. The summation over them can be executed easily. With K as an arbitrary reciprocal lattice vector, it follows that

C .i(k’+k”-k)-R = G3 S k l f k ’ l - k , K . (3.43) R K

With this, expression (3.42) takes the form

1 (vklUlv’k’) = - C ( k - k’ + KIU)Ci$’(K), (3.44)

f i K

wherein the notation for the so-called Bloch integral

(3.45)

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3.3. Effective mass equation 257

has been introduced (0 = G3Ro). The Bloch components of the wavefunc- tion @(x) in the Schrodinger equation (3.32), (vkl$) and (v’k’lg), differ from zero only for small k and k’ because of the smoothness of the envelope. Therefore we need the matrix elements (vklUlv’k’) only for small k and k’. With regard to the smoothness requirement for the potential U(x), the terms in expression (3.44) for (vklUlv’k’) yield no significant contributions to the Schrodinger equation (3.32) if K # 0, so that

1 (vklUlv’k’) N -(k - k’lU)C$t’(O). a (3.46)

Approximating the Bloch factors u,k(x) and U v l k t ( X ) by, respectively, ud(x) and uU,o(x), and applying the orthogonality of the Bloch functions, the Bloch integrals of equation (3.45) c:~:(o) follow as

c,,, kk’ (0) = 6,,t. (3.47)

Substituting this relation in expression (3.44) we obtain

(v kl U Iv’k’) w (kl U 1 k’)6,,, . (3.48)

With this relation, the decisive step in simplifying the Schrodinger equation (3.32) for the perturbed crystal has been done - the coupling between the different energy bands caused by the perturbation potential has been elim- inated. The Schrodinger equation now decomposes into separate equations for the various individual bands, and these equations can in fact be solved by a one-band ‘Ansatz’ of the form (3.37). The coupling between different wavevectors remains in place. This can be processed relatively easily. We employ all results achieved up to now in the Schrodinger equation (3.32), obtaining the following relation for F,(k):

This equation will be transformed from k-space into coordinate space. To this end we multiply by (1/a) exp ( ik . x) = (xlk) and sum over k. The first term on the left-hand side thereby becomes

C &,, (k) Fuo (k) (XI k) = Ev0 (-iVz) Fvo (x) . (3.50)

One can easily prove that this transformation is correct by expanding E,(k) in a power series and by using the identity

k

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258 Chapter 3. Electronic structure of semiconductor crystals with perturbations

The second term on the left-hand side of equation (3.49) can be written in the form

C ( x l k ) (klU l k’Pm (k’) U(x)Fm(x), (3.52)

and the right-hand side of (3.49) inluiediately becomw EF,(x) . With this, one finally gets the following equation for F,,(x)

kk‘

If one further replaces Evu(-iVz), in accordance with equation (2.204), by a quadratic function of -iVz7 which is justified because of the smoothness of F,(x) , and assumes an isotropic effective mass m+,, then it follows that

h2 E,(-iV,) = E,(O) + -(-iVx)2,

2mg (3.54)

and

(3.55) 1 [ L . d O ) - -a2 2m& I U ( x ) F,(x) = E F , ( x ) . h2

With the derivation of this equation, initially conjectured and now verified, it, is quite clear that the influence of the perturbation potential can be ap- proximakly determined from an equation in which the periodic potential no longer appears and the effective mass replaces the free electron mass. Equation (3.55) is therefore the desired effective muss equation The eigen- function Fvo(x) of this equation plays the role of an envelope function for the Uloch factor uvoo(x) in equation (3.41) for the true wavefunction $J(x). Equations (3.53) or (3.55) are also called envelope function equatioas in t,his context .

The essential requirements involved in the derivation of the effective mass equation were the smoothness of the perturbation potential, the smoothness of the wavefunction and its composition of Bloch functions from only one band, as well as the k-independence of the Bloch factors. These assumptions are decisive, and are often not fulfilled. Short-range perturbation potentials, for example, are not smooth. Nevertheless, the effective mass equation (3.55) is a suitable instrument for the solution of a series of important problems of solid state physics. Point perturbations with smooth potentials are only one example of this. Other problems which can be solved with the help of the effective mass equation include artificial superstructures in a crystal and external electric fields. These will be treated later in section 3.7 and 3.8, respectively, Also, the Coulomb attraction between electrons and holes - which results in the formation of excitons as pointed out in Chapter 2 -

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3.3. Effective maw equation 259

may be treated by means of this equation. To account for external magnetic fields, the effective maw equation has to be modified in a way which will be indicated later in section 3.9.

3.3.2 Multiband effective mass equation

The oneband effective mass equation in its general form (3.53), derived in the preceding subsection, completely solves the eigenvalue problem for an electron in a crystal in the presence of a smooth perturbation potential. The practical use of this equation depmds, however, on the dispersion law E,(k) of the band under consideration. In the vicinity of the minima or maxima of non-degenerate bands of cubic crystals one has a purely parabolic and isotropic k-dependence, and the effective mass equation is, as we have seen, no more difficult to solve than an ordinary Schrodinger equation with an external potential. This picture changes for bands which display degeneracy in the extreme. As demonstrated in section 2.7 on k p-theory, one then in general has non-parabolic and non-isotropic dispersion laws. This does not involve a major difference if the one-band effective mash equation can be solved in k-space, i.e. in the form of equation (3.49). IIowever, for various reasons it can be necessary to transform the effective mass equation into co- ordinate space and to solve it there. This applies if basis functions other than plane waves are better adjusted to the symmetry of the perturbation poten- tial as, for example, the Coulomb potential of an impurity atom, or if no perturbation potential is present, but the perturbation is introduced through boundary conditions, such as in the case of artificial superstructures like su- perlattices and quantum wells. Because of the non-parabolic and anisotropic dispersion laws in the case of degenerate bands, the effective mass equation (3.53) in coordinate space is more complicated, in particular higher order differentia1 operators occur which make solution of the eigenvalue problem practically impossible.

A resolution of this situation is offered by k . p-perturbation theory. As we have seen, the parabolic and anisotropic dispersion laws of degenerate bands occur in this theory through diagonalization of the matrix of the Hamiltonian with respect to a basis set of Bloch functions Ivk)', which are exact up to the first order in the k . p-perturbation (see formula (2.339)). The elements of this matrix are relatively simple linear and quadratic func- tions of the components of k. In the case of degenerate bands this sug- gests, therefore, not to take a onecomponent effective mass equation as starting point, but a multi-component one which is obtained by writing the Schrodinger equation for the perturbed crystal in the approximate Bloch basis 1vk)l.

We will undertake this program now. Regarding the perturbation poten- tial U ( x ) and the envelope function F ( x ) , we pose the same requirements

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260 Chapter 3. Electronic structure of semiconductor crystals with perturbations

as in the previous subsection - both should be smooth in the sense that they change much less over a primitive unit cell than the periodic crystal potential V ( x ) does. For the wavefunction $(x) of the perturbed crystal, the expansion with respect to the approximate Bloch functions reads

?clW = c ‘(.kl$)(xtvk)’, (3.56) uk

and the Schrodinger equation (3.29) in this representation takes the form

‘(vklH + Ulv’k’)’ ‘(v’k’l$) = E’(vk($). (3.57) uk

The matrix ’(vklHIv’k’)’ of the unperturbed Hamiltonian H is diagonal with respect to k, k’, and block-diagonal with respect to the band indices v, v’ with the blocks referring to degenerate bands. The diagonal elements ’(vk(U(vk’)’ of the perturbation potential U may be calculated by means of relation (3.48) which gives the elements of U with respect to approximate Bloch functions their Bloch factors u v k were replaced by u d . In the terminol- ogy of section 2.7, these are Bloch functions in zero-order k . p-perturbation theory or Luttinger-Kohn functions. Using relation (2.339), which expresses the first order Bloch functions Ivk)’ in terms of Luttinger-Kohn functions, it follows that the diagonal elements ’(vklUlvk’)’ arc the Fourier transforms (klUJk’), just as in relation (3.48) of subsection 3.3.1. However, unlike this ielation, thc matrix ’(vklU lv’k’)’ has also lion-vanishing off-diagonal ele- ments with respect to I / , 11’. This is due to the k-dependence of the Bloch fac- tors in )vk)l which has been omitted in subsection 3.3.1. The non-vanishing off-diagonal elerrienls ‘(vklU)v’k‘)’ may he calculated in the same manner as the diagonal elements before. One obtains

The factor at (klVWlk’) in equation (3.58) has the order of rnagiitudr of the lattice constant a, as can easily be seen replacing p by x by means of Heisenberg’s quation of motion. This implies that the off-diagonal elements of U ( x ) are of the order of magnitude of the relative change of U ( x ) over a primitive unit cell. Terms of this order of magnitude are to be neglected within the framework of effective mass thpory. Thus the matrix elements of // with respect to first order Bloch functions Ivk)’ are approximately given by the relation

(vkl U Idk’)’ = (kl U I k’)buu, , (3.59)

which corresponds to relation (3.48) of subsection 3.3.1. Note lhat equation (3.59) is valid independent of whether the two bands v and Y’ arc dcgeneratc

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3.3. Mective mass equation 26 1

or not; even degenerate bands are not coupled by perturbation potentials U if these potentials are smooth.

Before simplifying the Schrodinger equation (3.57) still further for the actually interesting case of degenerate bands, we will examine the case of non-degenerate bands.

Non-degenerate bands

WTe assume that p(x) can be expressed in terms of the approximate Bloch functions of only one band vg? so that

'(vkl@) = b,,F,(k).

With the Ansatz (3.60), vj(x) may be written as

(3.60)

$(x) = CFw(k) (x lvok) l . (3.61)

Employing expression (2.334) for the approximate Bloch functions jvok)', it follows that

k

where &(x) is the Fourier transform of F,(k) in coordinate space. The second term of @(x) of equation (3.62) does not occur in the corresponding equation (3.41) for +(x). This term is again due t o the k-dependence of the Bloch factor in Ivk)', which was neglected in subsection 3.3.1. In fact, its relative magnitude with respect to the first term is of the order of the relative change of the envelope function F,(x) over a unit cell. Again terms of this order of magnitude are again to be neglected within the framework of effective mass theory. We therefore obtain the same expression V(x) as in equation (3.41) above.

The Hamiltonian matrix '(vklHlv'k')' in (3.57) is approximately diag- onal with respect to the band indices, and the diagonal elements are the eigenvalues Ez(k) of H calculated in section 2.7 in second order k. p per- turbation theory. Employing expression (2.336) for E?(k), we obtain

where hi&8 are the elements of the reciprocal effective mass tensor accord- ing to formula (2.337). Using the Ansatz (3.59) for '(vklp) and expression (3.631 for l(vk)H]vk)', the Schrodinger equation (3.57) yields

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262 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Fourier-transforming this equation into coordinate space, we have

This equation is in agreement with the effective mass equation (3.53) if the latter has E,,,(--zV) replaced by a parabolic expression in the components of -iV, refraining, however, from the assumption of isotropy of the band struc- ture. If the effective mass tensor reduces to a scalar quantity ( l / m ~ & ) 6 ~ 0 , then (3.65) yields the effective mass equation in the form (3.55). The deriva- tion of the effective mass equation based on k . p perturbation theory thus produces this equation automatically in parabolic approximation. In the ear- lier derivation of subsection 3.3.1, an effective mass equation was obtained that did not yet contain this approximation, which was invoked only later. The advantage of the derivation of the effective mass equation within the framework of k * p-perturbation theory is clearly manifested when degener- ate bands are considered, which we will address next.

Degenerate bands

We direct our attention to the valence band maximum of semiconductors with diamond type structure (see section 2.7). Accordingly, we assume that at k = 0 a degenerate band level E,(O) exists with either symmetry rhs or r: depending on whether spin is omitted or not. First we consider the case without spin. The three degenerate T'b, Rloch states of energy Ew(0) arc- distinguished, as before, by an integer index m. The Bloch functions at k = 0 arc therefore written as )omO) . If we are interested in eigcnstates $(x) of the Schrodinger equation of the perturbed crystal whose energy eigenvalues are expected to be near the valence band edge, we can represent $(x) as a superposition of the Bloch functions Ivrnk)' in first order k . p perturbation theory. Thus, we set

(3.66)

As above, in the case of non-degenerate bands, it approximately follows that

(3.67)

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3.3. EEective mass equation 263

with F,(x) as the Fourier transform of F,(k) in coordinate representation. Now, expression (3.67) for '(ukl$) is substituted into the Schrodinger equa- t.ion (3.57). The subspace of valence band states is approximately decoupled from the remainder of the Hilbert space. Using equation (2.341), the matrix elements '(vklHlvm'k')' may be written as

where the coefficients D;:, are defined in equation (2.342). The matrix ele- ments '(vklUlvrn'k')' of the perturbation potential U follow from equation (3.59). They vanish if v # wm'. For v = urn' they differ only from zero if m = m'. With this, the Schrodinger equation (3.57) takes the form

Transforming to coordinate space and taking account of relation (3.51), the equation

+ Ci(X)h,,! Fm'(X) = EFm(x) . (3.70) 1 +$ @ ($

* follows. Using the notation D for the fourth-rank tensor ~ ~ l ~ = D:fm, I for the unit matrix 6,i, and F (x) for the vector (F,(x), F,(x). F,(x)), t,his equation may be written in matrix form as

(3.71)

In the case of the I& valence band the Hamiltonian matrix is given by rela- t,ion (2.344). With t,his, the envelope function equation then reads, explicitly,

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264 Chapter 3 . Electronic structure of semiconductor cry.staLs with perturbations

Here, derivatives of the type 62/6z2 are denoted by symbols like a;, and those of type a 2 / d r d y by symbols like a&.

Consideration of spin leads to two changes. Firstly. one has to use spin- dependent Bloch functions (sxlvmak0)' and, therefore, also spin-dependent envelope functions Frrw(x, s ) , and secondly the spin-orbit interaction oper- ator H,, is to be added to the Hamiltonian operator H . Thereby, Hk.p

becomes Hk.=, which has the consequence for D that the matrix elements of p are replaced by those of ii from equation (2.353). If interference terms be- tween the k ' p-interaction and the spin-orbit interaction H , are neglected, the representation of the latter in the approximate Bloch basis leads to a

k-independent tensor Hs,. The envelope function equation then reads

U

*

It is advantageous to first transform this equation into the angular momen- tum basis l jmg) , in which the operator fiso is diagonal, before one proceeds to solve it. Using the arguments of section 2.7, and assuming a sufficiently large spin-orbit splitting energy A, the coupling between the I'i and r$ valence bands may be neglected. For the T'i valence band, i.e. with j = $, one arrives at the following $-component envelope function equation:

U

The quantities d, R, $, here are differential operators defined as follows:

(3.75) 1 2

Q = --(L + nf)(a: + a;) - Ma:,

+ = --(L + 5n/I)(d2 + a;) - - (2L + M p : , (3.76) 1 1 6 3

(3.77) * N s = i-(& - ia&,

v%

[3.78) (L - M ) ( a , 2 2 - a,) - 2iNa,B,] .

The energy origin in (3.74) was moved to the valence band maximum, which was previously located at A/3. The envelope equation (3.74) makes it pos- sible to calculate acceptor states and hole bands in superlattices of the dia- mond and zincblende type semiconductors. One can derive a similar equation within the Kane model of the rs-valence-band-rs-conduction band complex of zincblende type semiconductors.

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3.4. Shallow levels. Donor and acceptor states 265

3.4 Shallow levels. Donor and acceptor states

We will now explore stationary states of a semiconductor crystal caused by point perturbations which givv rise to smooth long-range perturbation potentials V'(x). The most important point perturbations of this kind are isocoric substitutional impurity atoms, i.e. atoms from rows and columns of the periodic table which are close to the row and column of the host atom which is substituted. Below we will concentrate on such point perturbations. We denote the difference of the core charge numbers of the impurity and host atoms by

AZ = Z I - XI?, (3.79)

and assume that it is non-zero. The sign of AZ can be both positive as well as negative. In this sense we can speak of positive and negative point perturbations the point perturbation is positive if the core charge number of the impurity atom is larger than that of the host atom, negative, if it is smaller. For the pcrtiubation V,'(x) of the core potential, we may use expression (3.11) which here we write as

(3.80)

In general, V,'(x) does not yet represent the entire perturbation potential in the one-particle Schrodinger equation (3.21) for the perturbed crystal. One still must add the Hartree potential Vh(x) and the exchange potential Vfr(x) caused by other electrons localized at the center. For impurity atoms with only one valence electron more or less than the host atom, these potentials, as a rule, have the effect that only one electron or hole can be bound at the center. For this one electron or hole, the Hartree and the exchange parts of the perturbation potentials vanish, i.e. V,'(X) is itself the entire perturbation potential, and the one-particle Schrodinger equation has the form (3.22). For impurity atoms with lAZl > 1, Vb(x) and Vfr(x) do not vanish. The effect of a non-vanishing VA(x) will be discussed later.

To solve the one-electron Schrodinger equation (3.22) with the potential V,'(x) of (3.80), we may use the effective mass theory derived in section 3.3. Here, effective mass equations must be written down for those critical points of energy bands where changes of the spectrum of energy eigenvalues dae to the perturbation potential are to be expected. In Chapter 1, it was pointed out that such changes occur at the bottom of the conduction band and the top of the valence band - for impurity atoms with A 2 > 0 discrete levels appear in the energy gap just underneath the conduction band edge,

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266 Chapter 3. Electronic structure of semiconductor crystaIs with perturbations

and for impurity atoms with A 2 < 0 one has such levels in the energy gap just above the valence band edge. Effective mass equations are needed, therefore, both for the conduction band edge as well as for the valence band edge. In writing down these equations we start with a simple twc-band model. The minimum of the conduction band (vo = c ) , and the maximum of the valence band (v, = w) are assumed to be non-degenerate and to lie at k = 0. Furthermore, the k-dispersion of the two bands is assumed to be isotropic and parabolic in the vicinity of k = 0, whence we have

Ti2

2m; &(k) = -----k2,

(3.81)

(3.82)

with rn; and ml, as effective masses of electrons and holes, respectively. In reality, for many semiconductors such as Si and Ge. the conduction band edge does not lie at k = 0. The valence band edge zs found at that point for many semiconductors including the ones mentioned; but there is degeneracy between heavy and light hole bands at k = 0. Nevertheless, we will initially proceed with the above simplifying assumptions. The relevance and neces- sary improvements of this idealized model are yet to be discussed separately. For reasons which will become clear below, the model is often referred to as the hydrogen model

3.4.1 Hydrogen model

For the hydrogen model we have the two effective mass equations

(3.83)

(3.84)

In the usual form of a Schrodinger equation they read

(3.86)

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3.4. Shallow leveb. Donor and acceptor states 267

Apart from notation, signs, and constant factors these are the Schrodinger equations for charged particleu in the potential of a point charge, just like in the case of the hydrogen atom. From experience in quantum mechanics, we know that h r an dttrtlclite potential, there is a continuum of positive eiierpy eigmvalues and, in addition, there ale discrete energy levels which occur for negative energies. Fur repulbive potentials there are only pusitixe energy eigenvalues, but no bound statrs at negative energies. From this it follows that for a positive point perturbation (upper sign in equations (3.851, (3.86)), discrete enera levels are to be expected just below the conduction band edge, and for negative point pertiirhations such levels are exppctpd just above the valence band edge. Thew dibcrele levels lie in the emrgy gap between the conduction and valence bands where, as we know, no energy eigenvalues can appear in an ideal crystal. We consider first the cwe of the conduction band.

Conduction band

Transferring results from the quantum mechanical treatment of the hydro- gen atom to our present situation, we get the discrete energy levels & of principal quantum niimher 71, n = I , 2 , . . ., as ( E , - E ~ ) = - ~ * / n ~ or

with

(3.87)

(3.88)

where Ef - (e'm//2ha) denotes the binding or Rydberg energy of the hydro- gen atom. The m~av~fin~rtioii FTdm(x) for principal quantum number n = 1, angular mumedurn quantum number I ~ 0 and magnetic quantum number m - 0 reads

with

(3.89)

(3.90)

where ag = (h2 /e2m) denotes the Bohr radius of the hydrogen atom. The orders of magnitude of the energy Eg and the length ag can be estimated easily. With E g FY 13.6 eV, a: = 0.5 A , (mz/m) = 0.2, c = 11.4 and = 1, it follows lhat

(3.91)

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268 Chapter 3. Electronic structure of semiconductor crystals with perturbations

11.4 a g --" 0.5 (=) A M 29 A. (3.92)

Compared with the width of the energy gap which is of order of magnitude 1 el/, the discrete energy level El lies closely below the conduction band edge (see Figure 3.4). Even closer are the states with n = 2 , 3 , . . .. The des- ignation shallow levels is used in this situation. The wavefunction for energy level El decays exponentially with distance from the core of the impurity atom with the characteristic decay length ag . In contrast to band states, which are evenly spread out over the entire crystal, we have, therefore, lo- calized electron states (see Figure 3.4). The localization radius a g is clearly larger than the distance between two crystal atoms, which is approximately 2.5 A. This means that the localization of an electron at an attracting impurity atom is weak if considered on the interatomic length scale of the crystal. Exactly this is to be expected considering the smoothness require- ment made at the outset, which states that only Fourier components of small wavevectors k should contribute to the wavefunction. We can also examine the validity of this requirement quantitatively. Omitting a k-independent factor, the Fourier transform of Fcloo(x) is given by the expression

(4.93)

At the edges of the first B Z one has k FY (../a) and a g k M w ( a g / a ) FY 10. Between the center of the first B Z and its edge, Fcloo(k) therefore falls off by about 4 powers of ten. The smoothness requirement is therefore fulfilled very well. In Figure 3.4 the localization of the wavefunction in k-space is also indicated.

Recalling the theory of the hydrogen atom, it is known that the effect of an attractive Coulomb potential is not restricted to the formation of bound states with energy levels in the previously forbidden negative energy region. Changes also occur in the continuum of positive energy eigenvalues which are allowed even without Coulomb potential. They are manifested in the fact that the eigenstates, which were previously spherical waves, are scattered by the Coulomb potential. Their amplitude at the positive center becomes larger, and further away from the center it is smaller. This leads to a change of the density of states in the continuous part of the energy spectrum (the definition of the density of states is given in section 2.5). According to Levinson's theorem, which we will discuss in the next section in somewhat greater detail, this change takes place such that the total number of all states in the presence of the perturbating potential, including the bound states at negative energies, remains the same as the total number of states without the perturbating potential. For each bound state of negative energy a state of positive energy is therefore excluded from the energy spectrum.

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3.4. Shdaw lev& Donor and acceptor fitates 269

AZ a 0

L1 O k

dZ.0

r\ O k

Figure 3.4: Illmtration of shallow localized States in coordina1.e spat:e and in k- space far mibslitutional impurity atoms with either posit.ive or negative differences A 2 beOween the impuril,y core charge nunthers and the h o d core charge numhers (upper part of figure). The occupation of the shallow levels by eI&rons is presented in the lower part of the figure. The levels act as donors of elwtctruns for A 2 > 0 and ss acceptors of electrons for A 2 < 0.

The abovr-Inmtionfd result,^ for shallow impurity levels appended to an isotropic parabolic conduction band can be immediately iransferrd to an isotropic parabolic valence band. Below, we will writs them down without further derival ion.

Valence band

For a negative point perturbation there appear discrete energy lpvels in the energy gap closely above the valence band dgr . The energies ol thew levels are given by the exprcssiorrs

(3.94) EB En - - ,z '

(3.95)

The pcrtinent wavefunctions are localized at the perturbation center. The wavefunction of the ground state, n = 1 and 1 = m = 0, has the form (3.89)

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270 Chapter 3. Electronic structure of semiconductor crystals with perturbations

with

(3.96)

as the effective Bohr radius. In Figure 3.4 (lower part) the energy levels and localization regions are shown schematically.

Although the population of energy levels by electrons will not be treated systematically until the next chapter, we wish to deal now with a special case, namely with the occupation of the shallow impurity levels discussed above at absolute zero temperaturc, 7’ = 0 K . This consideration will lead us to a better understanding of the nature of these impurity states. First of all, wc assume the case of a positive point perturbation, i.e., the core charge number 21 of the impurity atom is taken to be larger than the core charge number Z H of the host atom. For simplicity, we assume 21 = ZH + 1. Then the impurity atom also has one valence electron more than the host atom. To be specific, we can imagine it as a P atom in a Si-crystal. According to Levinson’s theorem, which mandates that the total number of states must remain the same both with and without perturbation, one can proceed on the assumption that nothing will change due to the impurity atom regarding the number of states in the valence band. Therefore, all the valence electrons of the host atoms and all those of the impurity atom, except for the additional electron, can be placed in the valence band. The additional impurity electron has to reside in the lowest unoccupied energy level. That is the shallow n = 1-level just below the conduction band edge which, according to the above results, arises from the impurity atom. The resulting occupation is shown in Figure 3.4. If the temperature is increased this electron can easily be excited from the shallow level to the conduction band. There, it is no longer localized but spread out uniformly over the whole crystal. representing a freely mobile negative charge carrier. The shallow energy level therefore functions as a donor of a free carrier. One calls it a donor level and the impurity atom itself as a donor. One can also say that a donor atom (with AZ = 1) has one surplus valence electron which is bound relatively weakly and can be transferred easily by thermal excitation to the conduction band. The impurity atom remains in the single positively charged state, i.e., it is singly ionized. For A 2 2 2 one has doubly or multiply ionizable donors. We will discuss this later more exactly.

Next, we consider an impurity atom with ZI = ZH - I, i.e. with one positive elementary charge in the core and, with this, also one less valence electron than in the host atom. Due to the substitution of a host atom by such an impurity atom, the number of states in the valence band decreases - according to Levinson’s theorem by one if only the n = 1-level is counted. With this, the number of valence band states is still large enough to host all

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3.4. Shallow Jevels. Donor and acceptor states 271

valence electrons, but, also, there remain no unoccupied states in the valence band. The n = 1-level in the energy gap remains empty at T = 0 X. If the temperature is increased, an electron from the valence band can easily be excited into the impurity level. The level accepts an electron, it functions as an acceptor. In the valence band itself, a hole remains which, as we know from Chapter 1, behaves like a positive freely mobile charge carrier. Consequently, we can also say that, under thermal excitation, the acceptor transfers a hole to the valence band, whereas at T = 0 K the hole was bound to the acceptor. This picture corresponds to the model of a negatively charged point perturbation which binds a positive hole and transfers it to the valence band under excitation. The analogy to the positively charged donor center which binds an electron and donates it to the conduction band is obvious. Similar interpretations are available for doubly and multiply ionizable acceptors.

3.4.2

As already indicated, the above hydrogen model of shallow impurities, for which the band extrema lie at the I? point and the bands are non-degenerate at r as well as isotropic and parabolic in its neighborhood, cannot be directly applied to many semiconductor materials including Si. Moreover, as we know from section 3.2, the perturbation potential V’(x) has, in general. a short-range part as well as an electron-eIectron interaction part which are not considered in the hydrogen model. Below, we treat corrections to the hydrogen model which can be traced back to the cited effects. Some of these corrections appear only at donors. others only at acceptors. Therefore, we treat the two kinds of impurities separately.

Improvements upon the hydrogen model

Donors

The conduction band of Si and other indirect semiconductor differs in three ways from the simple isotropic hand model. First, the minimum lies out- side of the center of the first BZ; second, for symmetry reasons, one min- imum carries the implication of several equivalent minima or valleys; and third, the band structure in the neighborhood of each individual minimum is anisotropic. While the off-center location of band minima has no direct ef- fect on the donor binding enerm, the many-valley feature and the anisotropy of the band striiccture rln have such an effect. Due to the existence of several valleys (dist.inguished by a valley index i ) , the wavefunction 01 the donor state $(XI has to be expanded with respect. to the appr0ximat.c Bloch func- tions (xlclri + [k - kill1 for all mutu& degenerate minima ki, rather than with respect to the approximate Rloch function (xlck, + [k - kc])‘ of one minimum k, only Because of this, the one-component envelope function

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272 Chapter 3.

F,(k) becomes

Electronic structure of semiconductor crystah with perturbations

a multi-component envelope function Fi(k), and we have

The envelope function equation in k-space reads

h 2 rx - [ (k - kilMC’lk’ - ki) 6i j6kk l + (klV’lk’)] Fj(k’ - kj) = 2 k’ j

= EFi(k - ki), (3.98)

where Mi is the effective mass tensor of valley i. The wavevectors k - ki and k‘ - kj vary in a small neighborhood about the zero point, i.e. k and k’ are vectors near the respective minima ki and kj. For i # j , the matrix elements (kJV’Jk’) of the perturbation potential V’(x) can approximately be replaced by the k- and k’-independent expres- sions (kilV’lkj). The latter can easily be transformed into coordinate space where they result in an additional 6-function-like potential. With i = j , the matrix elements (kJV’Jk’) result in the perturbation potential V’(x) which is already present in the hydrogen model. The effective mass equation (3.98) in coordinate space thus reads

= EFi (X) . (3.99)

If V’(x) is the screened Coulomb potential of equation (3.80), as was as- sumed, the absolute values of the inter-valley Fourier components (kilV’1kj) are negligibly small in comparison with the intra-valley Fourier components, because of the large wavevectors ki - k j occurring in the former combined with the (l/lkl’)-dependence of (klV’) on k. However, as we know from sec- tion 3.2, the true perturbation potential V’(x) differs from the pure Coulomb potential, because the screening of a point charge by the semiconductor is wavevector dependent, and on the other hand, because V’(x) contains a short-range part, which cannot be traced back to the potential of a point charge in any way. Both modifications can cause the inter-valley matrix elements (hlV‘Jkj) of V’(x) to take larger values. Nevertheless, they still remain so small that they can be considered by means of perturbation the- ory. In zero-order approximation they may be omitted completely. Thereby, one obtains a separate effective mass equation for each valley from (3.99). In case of k, = ( O , O , k z ) , it reads, in coordinate space

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3.4. Shallow levels. Donor and acceptor states 273

Analogous equations hold for the other valleys. This means that, neglecting inter-valley coupling, all valleys give rise to the same donor level.

Anisotropic effective masties

In contrast to the hydrogen model, the effective mass of the envelope q u a - tion (3.100) depends on direction in k-space. In this circumstance, analytical solutions arc not available, and one mist resort to approximations and nu- merical calculations. One possible approach is the application of a variational procedure, in which the eigenfunctions are represented as linear combinations of niembers of a set of auxiliary functions whose general shapes are adjusted as well as possible to the real solution, but which still contain free parame- ters. Using this representation one calculates the energy expectation value of the Ilamiltonian and varies the parameters until its absolute minimum is reached. This minimiim value then yields an approximation for the ground state energy, which is better as the linear combination of auxiliary functions approximates the actual eigcnfunction. The success of this approach thus depends in an essential way on the auxiliary functions used. They should, in particular) correctly reflect the symmetry of the actual eigenfunctions, in the case considered here the symmetry of the ground state wavefunction. One can also employ the variational process for the calculation of excited states. In this case the wavefunction of the first excited state must, in ad- dition, be orthogonal to the ground state wavefunction, that of the second excited state must be orthogonal to both previously calculated states, etc. Numerical results obtained by means of this procedure for the energy levels of the P-donor in Si are reproduced in Figure 3.5. For a simple estimate of the anisotropy effect, one can replace the reciprocal effective mass rn: of the hydrogen model by the reciprocal effective mass (l/m:ll + 2 / m ; ) / 3 obtained by averaging the direction-dependent reciprocal effective masses of (3.100) over all directions in k-space. With this, one obtains energy levels for the simply ionizable donor in Si, which are shown in Figure 3.5a. The binding energy EB is represented as 29 meV in this approximation.

Inter-valley coupling

As a second corrective step, the inter-valley coupling in the effective mass equation (3.99) will be considered. In perturbation theory, the calcula- tion of eigenvalue corrections involves the diagonalization of the matrix O(kilV’lkj)lFj(0)12. Using the example of Si, this calculation will now be

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274 Chapter 3. Electronic: structure of semiconductor crystals with perturbations

Figure 3.5: Ground state and first excited states of a phos- phorus donor in Si: a) hydro- gen model; b) including effcc- tive mass anisotropy; c) includ- ing central cell corrections, effer- live mass anisotropy arid inter- valley coupling; d) exxperimen- tal results. ( A f t w Rlakemore, 1985.)

- T 0'0° > 2! -0,Ol

CI, w w

I

-0,02

- 0.03

-0,04

-0.05

O b b c b b b O b b c b b b 0 b b c

c b b O b b b c b b O b b b c b b o

performed explicitly. In this case one has 6 minima, which lie on the cubic axes k,, k,, k , and k,, k,, k , close to the respective X-points. From symme- try considerations it follows that fl(k,lV'lk,)II;k(0)12 has the general form

_ - ~

7 (3.101)

b = fi(k~lv'lky)l~z(0)12, r = ~t(kzIv'lkz)l.)IFz(0)12 (3.102)

have been used. Lines and columns in (3.101) are written in the sequence k,, k,, kzr k,, k,, k,. The matrix (3.101) has the three eigenvalues

- _ _

A&I = 4b + C, 1 - fold

AEz = -2b + C , 2 fold

AEg - -c, 3 - f o l d (3.103)

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3.4. Shallow lwels. Donor and acceptor states 275

with the degrees of degeneracy indicated. Thus, inter-valley coupling par- tially removes the 6-fold degeneracy among the valleys.

The sphtting of the ground state donor lrvel in Si due to inter-valley coupling can also be predicted jus t oti the basis of group theory. One con- siders the representation of the symmetry group of the envelope equehn (3.99) for si, i.e. of o h ) in the space of the 6 reciprocal vector compo- nents k,, kg, k,, k,, k,,, k,. This representation is reducible. To demonstrate this one may employ the transformation rules of the vector components x, y, z , a, 7 j , Z under the action of the clcments of o h (summarized in Table A.7 of Apprndix A) , realizing that these component? transform in the same way as thc reciprocal vector components considered here. Then one easily finds that the irreducible parts of this representation arc rl ( A , ) . r12 ( E ) and I’z5(Tz). ‘The notations A , E , Tz are commonly used in the theor,- of l o c a l i d states, and WP will also employ thcm here. With this. it is clear that the symmetry of the AEI-state is A l , that of the two AETstatcs is E , and that of the three AE3 states is 12.

The size of the splitting depends on the inter-valley nitrtrix elemen1 of the perturbation yolmtial. If one considers only thc first of thc two above- mentioned effects which result in matrix elernentv ol considerable size. i.e. the wavevector dependence of screening, and assumes that screening has become fully ineffective at large wavevectors k, - kj, therefore replaring the screened Coulomb potential V’(x) of equation (3.80) by the bare Coulomb potential -(e2/IxI) in (ktlV’lkj), then it follows that

_ _ _

(3.104)

With this, one obtains b as -2.36 mel’ and c as -1.2 met‘. This yields AEl = -10.8 meV. AEz = 3.6 met‘ and AE3 = 1.2 meV.

A 3-fold splitting of the P-donor ground state level in Si has, in fact, been observed experimentally (Aggrawal and Ramdas, 1965). The simple theoretical estimate presented above agrees remarkably well with the ex- perimental splittings of 11.6 meV between the Al- and T2-levels, and of 1.3 meV between the E- and T2-levels (not resolved in Figure 3.5). The results of a numerical treatment of inter-valley coupling shown in Figure 3.5 are even closer to the experimental values. However, the agreement is not as good for the absolute position of the ground state level. The addition of A E l , i.e. of 10.8 meV to the binding energy of 29 meV without inter-valley interaction yields a corrected binding energy of 39.8 meL’ which is closer to the experimental value of 45 meV, for Si:P but still clearly below.

Chemical shifts

In Table 3.5 we list experimental binding energies for a number of singly

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276 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Table 3.5: Experimental binding energies of several shallow donors and acceptors in Si, Ge and GaAs. (After Landoldt-Bornstein, 1982.)

Material

Si

Ge

GaAs

Sb Ga In 153

P 13 As 14 Sb 10 Bi 13

B 11 A1 11 Ga 11 In 12

ionizable donor and acceptor atoms in Si, Ge and GaAs. Not only are the absolute values of these energies striking, but also the fact that they are not equal for donor or acceptor atoms of different chemicals stands out. Contrary to the prediction of the above theory, there are dependencies of binding energies on the chemical nature of the impurity atoms. One refers to these as chemical shifts.

The absence of chemical shifts in the calculated binding energies is a consequence of the approximation of the perturbation potential V’(x) as a purely Coulombic potential in the hydrogen model. As we know from section 3.2, the correct perturbation potential of an impurity atom also contains a short-range part, to which all central cell corrections contribute. Among these, there are also contributions which depend on the chemical nature of the donor. The latter are the main reason for the experimentally measured chemical shifts of the donor binding energies. The effects of central ceE corrections are particularly large in the case of the 1s-ground state, as one can recognize from Figure 3.5 for the special case of the P-donor in Si. A weaker influence, and, accordingly, better agreement between experiment and the predictions of a theory omitting central cell effects occurs for the excited states with n = 2 , 3 , . . .. This is to be expected since the excited state wavefunctions have maxima that do not lie at the perturbation center x = 0

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3.4. ShalIow levels. Donor and acceptor states 277

(which does happen in the case of the ground state) but further outside. Electrons in the excited states are therefore less affected by changes of the potential in the central cell than are electrons in the ground state.

Generally, the occurrence of pronounced chemical shifts means that the perturbation center is no longer shallow and the effective mass theory is no longer applicable for its theoretical treatment (see section 3.5 for further discussion).

Acceptors

Corrections to the simple hydrogen model are also necessary for acceptors. Of course, the maximum of the valence band lies in the center r of the first B Z for all diamond and zincblende type semiconductors, as is assumed in the hydrogen model. However. this maximum is degenerate: depending on whether spin-orbit interaction is important or not one has, respectively. the %fold degeneracies of the representations l'b5 (diamond) or (zincblende) in the space of scalar functions. or the 4-fold degeneracies of the representa- tions I?$ (diamond) or Ts (zincblende) in the space of twc-component spinor functions. In the vicinity of r, this degeneracy splits off and one has ihree or two anisotropic valence bands. In section 2.7 these were approximated by two isotropic parabolic bands. one for heavy holes and one for light holes. If one also uses this approximation here, then there are two hydrogen-like series of acceptor le\-els, one for each sort of holes. Calculating acceptor bind- ing energies by means of expression (3.95) in the case of Si yields 50 met7 for heavy holes which are expected to form the ground state. In the case of Ge. the result is 17 meV for heavy holes. One can hardy expect these values to be in agreement with experiment. In fact, they differ apprecia- bly from the measured ground state binding energies shown in Table 3.5. Evidently. non-parabolicity and anisotropy of the band structure play an essential role and must be taken into account in realistic calculations. This may be done by means of the multfband effective mass theory developed in the preceding section. Below. we will explain the application of this theory to simply ionizable acceptors in Si. Owing to the small spin-orbit splitting energy A of 44 meV and the relatively large acceptor binding energies EB (68 me\' for the isocoric impurity atom Al). spin-orbit interaction initially will be neglected. Later this approximation needs to be corrected since E g is not much larger than A. Without spin and spin-orbit interaction the wavefunction ~ A ( x ) is a linear combination of the three Bloch functions (xlumO), m = x. y, z of the representation Ti,, which are enveloped by the components F,(x) of the envelope function vector F (x). Thus

*

(3.105)

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278 Chapter 3. Electronic structure of semiconductor crystals with perturbations

The corresponding effective mass equation of the I’b5-valence band is given by relation (3.72). If, therein. U ( x ) is identified as the Coulomb potential. (3.80), one obtains, in the case A 2 = -1,

The eiigenvalues and eigeuvectors of this complicated set of equations cannot be obtained, of course, in closed analytical form. One has to use spproxinia tions and numerical calculations. The variational method again represents a reasonable way to proceed. In this approach one starts with a choice of suitable auxiliary functions for the expansion of the multi-component enve-

lope fundion F (x). This function should have symmetry properties which fit the symmetry of the acceptor state @ A ( X ) to be calculated. The mean- ing of this will be explained below. We consider a particular acceptor level E n , which in general will be degenerate. It belongs to an irrdiicible reprc sentation r A of the point moup oh. The corresponding aigenhnctions are denoted by $A~(x) , @m(x), . . ., and the pertinent multi-component envelope function by ~ , . q l (x), FA? ( x ) , . W e have to find the representation DA of O h to which the set of vectors F A ~ (x), F A ~ , . . . must belong in order that the wavefunctions T , ~ A ~ ( x ) , $~z(x):. , . formed from these vectors by meam of equation (3.105), actually transform according to the representation r ~ . The FA^ (x). FA^ (x), . . . belong to a space which differs from the ordinary Hilbert syaccr in that it is riot spanned hy oue-comyonent basis funrlious, but by three-component basis functions. A representation in this space is given by the product of two represt?nt,atinns, one of which corresponds to the 3D rqresentation according to which the components of each of the three- component, fiinctions FA^, FA^ transform separately, and the other is the representation according to which the threecomponent functions transform among each other. One can easily show that the latter representation must be r A , so that $,.ql(x), +,.q2(x), . . ., forincd according to relation (3.105), b e long to the representation r A , as has been supposed. For the representation D A of FA^ (x): FA^ (x), . . . it follows that

=$

* * =$-

3 *

= $ *

* *

(3.107)

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3.4. Shallow levels. Donor and acceptor states 279

This representation is reducible. The irreducible components determine the symmetries of the three-component basis functions to be used for the r e p resentation of the threecomponent envelope function of the acceptor state under consideration.

Equation (3.107) yields a remarkable conclusion concerning the symme- try of acceptor states. An envelope function basis vector which transforms according to the unity-representation occurs in the expansion of the accep- tor eigenfunctions $ A ~ [ X ) , @ ~ ( X ) ~ . . . only when the symmetry r-4 of these functions is I'h5. The basis vector belonging to the unity-representation is the only one which does not vanish at the central site x = 0. Since the wave function of the ground state will also be non-zero at x = 0: DA must contain the unity-representation if A is the ground state. According to (3.107) this is only possible if the ground state has the symmetry Fh5. Acceptor levels of other symmetry are necessarily excited states. The symmetries of all three component basis functions involved in the construction of the ground state envelope function are given by the relation

(3.108)

shown in Table A.27 of Appendix -4. Relation (3.108) determines the symme- tries of the three-component auxiliary functions under rotations, therefore the angular dependencies of these functions. In order to also get their ra- diaI dependencies one expands the rl-, r12-, and ri5 -functions with respect to angular momentum eigenfunctions with the various quantum num- bers 6, rn and multiplies the expansion coefficients of different values of I by corresponding radial wavefunctions r1 exp(-T/q). These are formed in anal- ogy to the eigenfunctions for the Coulomb potential, but with &dependent localization radii rl. The latter are treated as variational parameters, just like the Coefficients of the auxiliary functions belonging to different irre- ducible representations. Applying this procedure to the acceptor ground state of Si, a binding energy of 31 me&' is obtained (Schechter, 1962). More recent calculations, taking spin-orbit interaction into account, have resulted in a value of 44 mel/ (Baldereschi and Lipari. 1977), which is very close to the experimental value of 45 meV for 3 in Table 3.5 but, of course, does not account for the pronounced chemical shifts seen in this table.

Calculations of acceptor binding energies have also been performed for materials like Ge whose spin-orbit splitting energies are large compared to the acceptor binding energies. Spin has to be taken into account under such circumstances, i.e. the T'& valence band has to be replaced by the two rg- and r$-bands. Owing to the large spin-orbit-splitting energy, however, the spin-orbit-split I'F-band can be omitted. For the expansion of the 4- component envelope function of the remaining rg-band one needs auxiliary functions of + 2r15 + 2r25 symmetry (see Table x I'i = ri + fk +

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280 Chapter 3. Electronic structure of semiconductor crystals with perturbations

(A.28)). In the calculations for Ge, an acceptor binding energy of 10 m e v is obtained, close to the experimental values of 12 nieV for In and 11 meV for all other elements listed in Table 3.5. Obviously, chemical shifts are very small in the case of Ge, unlikr Si where these shifts were found to be very pronounced. The different behavior of acceptors in Ge and Si is undmstandable if one looks at the absolute magnitudes of their binding energies EB: in Ge, ED is siihstantially srrialler than in Si, reflecting the fact that the effective heavy hole mass in Ge is smaller than in Si. This implies that the acceptor wavefunctions of Ge have smaller amplitudes in the central cell than those of Si. Thus the point charge Coulomb potential which neglects central cell corrections giving rise to chemical shifts is expected to work much better for Ge than for Si, as it dow in fact. For the same reason, excited acceptor states in both Ge and Si are well described by the multi-band effective mass theory using a point charge perturbation potential (Balsderesrhi and Lipari, 1978).

Multiply ionizable donors and acceptors

Additional changes arc to be expected for substitutional impurity atoms whose core charge numbers do not differ by & A 2 - 1 from those of the host atom, as we assumed above, but differ by 2, 3 or more. Then 2, 3 or more electrons or holes are weakly bound to the impurity atom at T - 0 K , and 1, 2, 3 or more electrons can be thermally excited into the conduction or va- lence bands, leaving behind a 1-, 2-, 3- or morefold ionized impurity atom. The interaction between the carriers bound at the center is described by the Hubbard energy of equation (3.25). Without this interaction the effective mass equat,ion for a non-degenerate isotropic band yields a hydrogenic s e ries of energy levels with binding energy increased by the factor lA21' in comparison with the corrcsponding energy for a simply ionizable donor or acceptor. The Hubbard correction (3.25) results in a shiit of the energy lev- els towards higher energies by an amount which depends on the number 'IL

of electrons or holes bound at the center. We illustrate these remarks using thc example of the S-donor in Si. The

binding enera without H a r t r e potential is four times larger than that of the P-donor in this material. In the ground state, 2 electrons are bound at the S-donor, both of which can be placed in the 1.9-level because of 2-fold spin- degeneracy. If the Hubbard correction, which in this case is approximately given by

is taken into account, then the 1s-level is shifted up by just this energy Us. By exciting one electron into the conduction band, the neutral S-donor

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3.8. Deep lewls 28 1

S(D+) bccarnes a single positively charged S(1+) donor. For the Is-lad 01 h e S ( l 1 ) ioii thr shift in energy by Us do= not occur. Thus this level i s shifted down by I/, in comparison with the 1s-level of tbc S(0 +-) atom. Exprrirneniully, one finds t,hat the ionk-ation energy of the neutral S(O+) donor i s 0.31 eV, and the ionization energy of the S(1t) donor js 0.69 eV (see Figure 3.6). From thefie values a Hubbard energy I i , of 0.28 eV may be deduced. IIowever, both ionization energies are suhstantially larger than the result. 4 x 0.03 el7 = 0.12 eb’ which follows from the hydrogen model. Evident.ly, the central cell short-range poteniid cont,ribution is ewedial in the case of the S-donor in Si. ‘ l l is donor is a deep center rather than a shallow me.

3.5 Deep levels

3+5.1

As we know from previous sc3ctions, hhc potential of a point perturbat.ion, generally, consists of a long-range (Coulornbic] part and a short-rangp part.. In regard to their ability to bind states, the two potential contributions differ substantially. In the case of the long-range Conlomb potential, hound $Pates exist, for all possible potential strengths, i.e. for arbitrary magnitudes of the point charge and dielect,rir: cobstant. Short-range potentials, on t,he contrary, must have a minimum st.rengt.h to he able to bind a stale. For example, for a 3-dimensional potential box of depth Vo and radius a, the condition Vo > h2/(8w>.uli) must he fulfilled for a b a u d state to exist (see, e g . ? Schiff, 1968). In this, Heisenberg’s uncertainty principle manifests itself, in that localization due to the potential leads to a non-vanishing expectation value of momentum and 1,hus also of kinelic energy of the particle. Only if the dept8h of the potential box exceeds the expectation value of this kinetic energy! can t.hc potential prevent the partick from escaping the center. (Not.e, however, that in the case of a 1-dimensional potential box, bound states exist for arbitrarily small m d l depths sincc the 1ocalizat.ion and: with this, the average kinetic energy decreases sufficiently- fast with decreasing well dept.h. This indicates already at t,he outset that 1-dimensional models of deep centers are ralher poor).

According to wction 3.4, shallow levek ocrur for statps which are bound by a Coulomb potential, i.e., by the long-range contribution of the total perturbation potential. The concornit,ant short-range potential part does not suffice in this case for binding, it only leads to corrections of the binding energy and the eigenfunction of the ground state, k1loa.n from section 3.4 as c e n l m l tell correcfioru. The spatial extension of the eigenfunction is essentially given by the Bohr radius. If t.he strength of the short-range

General characterization of deep levels

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282 Chapter 3. Electrunin structiire of Bemiconductor crystals w i t h perturbations

part of the potential increases, then a point will be reached at which the short range potential is itself able to bind states. Provided the pertinent binding energy i s Iargcr than that due l o the Coulomb potential, binding to the center will be dominated by the short-range potential contribution, and the chararter of the bound state will change. The spatial cxtension of the eigenfunctjon of the ground state is then no longer dekrrninml by the Bohr radius, but by thr lattice constant. The Coulomb potential only leads to corrections to t h r binding energy and the cigmfunctions, which are mainly determined by the short-range potential alone.

States which are primarily bound by the short range potential part, are termed d w p . ‘ h e rorresponding la-& arc called deep levels, and the point perturbations at which they occur, are called as deep centers. Literally, the term ‘deep level’ refers to an energy eigenvalue that lies deep in the energy gap, far away from the two band edges, in contrast to shallow levels which lie close to one of these edges. Actually. the location of deep levels deep in thr energy gap is only a particularly important special case. as there are also yet other case. - derp levels can also be close t o or even wzthm one of the bands. The essential features of deep levels are their binding by a short-range potential and, in combination with this, their localization radius bring restricted to magnitude of the lattice constant. Because of their strong localization, deep states resolve the spatially periodic fluctuations of the rrystal potential while the eigenfunctions of shallow levels average them out. The shallow levels can therefore be treated by means of an effective mass equation. in the case of deep levels the requirements of effective mass theory, namely smoothness of the perturbation potmtial and of the corrpsponding wavefunction, are not fulfilled. Contrary to the assumptions of PEective mass theory, these states cannot be s p t h e s i z d by Bbch functions of k-vectors drawn from a small vicinity of a rritical point, and also not from Bbch functions of only one-band. In a theory of deep states, therefore, neither the effective mass is meaningful, nor can such a theory be based on a oneband equation, not even a degenerate multi-band equation. Attempts t o retain the effective mass concept and to consider solely ihe multi-band character of deep level theory met with little success. In this context one can say that skidow levels are eigenvalues of point perturbations which are capable of description by means of effective mass theory (the concomitant short-range potential contribution can be treated by rncans of perturbation thcorg in this rase), while deep levels are eigcnvaliies whose treatment by effective mass theory is impossible.

The discussion above is amply intuitive. it anticipates, in part, results which are i n d d plausible, but which have yet to be proven rigorously. This applies, in particular, to the fundamental feature which distinguishes between shallow and deep levels, namely, that short-range potentials can dominate binding only if they exceed a minimum threahold potential strength

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283

2 1 0 -2 -4 -6

.3

Acceptors 2 .4 - - -_ F:

a .-

.5

.6 (eV)

2 1 -2 -4 -G -8 -1OleV) Atomic Electronegativity

Figure 3.6: Ionization energy of substitutional impurity atoms in varioum tedra- hedrally coordinate host semironductors as B function of the strength of the pcr- turbation potential, measured in terms of valence shell s-level differences between impurity and host atoms in the case of donors, and valence shell p-level differences in the caae of acceptors. Donor energies are reIative to the conduction band edge, and acceptor energies to the valence band edge. (After Vogl, 1981. Reproduced from Boer, 1990.

- in the discussion above, the periodic potential of the crystal was omitted from consideration. An experimental proof of the threshdd behavior of short-range potentials in crystals may be taken from Figure 3.6. There: the ionization emrgies of a series of substitutional impurity a tom are plotted as a function of the difference! between the atomic valence shdl energy levels of the host and impurity atoms. This difkrence can serve as a measure of the strength of the short-range potential. The fact that for donor states

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284 Chapf,er 3. Electronic structure of semiconductor crystals with perturbations

the diffrrence of 8-levels is taken, and for acccptor states the differenre of p-levels, is not important foi the present discussion (we will return to this point later). With increasing horizontal scparation of the impurity atom from the host atom in the periodic table, the potential strength increases more or less continuously. The ionization energies initially retain their small valucs characteristic of shallow lcvels, Starting at a threshold distanre, they suddenly become substantially larger, a manifestation of the fact that the short-range potential has takm over binding and the level has become deep,

.Just like shallow levels, decp levels can also act as donors or acceptors (sometimes even as both of them). Due to their mostly larger separation from the hand edges they are, however, generally less effective than shallow levels in enhancing the concentrations of free charge carriers. They exhibit their greatest efiectiveness in just thp opposite process, the lowering of free carrier concentrations for which, in turn, shallow levels are not very effective. If, in addition to the neutral center, the single negatively charged center also forms a deep level in the gap with sufficient separation from the conduction band edge, the neutral center will capture electrons from the conduction land, which are available there eithrr as equilibrium charge carriers due to n-doping, or ~ L Y non-eyujlibiium cariiers due to optical or other excitation of the semiconductor. In the first case, one has a compe~wut ion of the donors by the deep center (for more 5ee Chapter 4), and in the second case, a cup- ture of non-equilibrium electrons by the center (see Chapter 5). A center which forms deep levels in the gap both in the neutral and simply posi- tively charged state, plays an analogous role in regard to the compensation of acceptors and the capture of non-equilibrium holes. Centers which can capture both electrons as well as holes act as catalysts for the radiation- less recombination of electron-hole pairs (see Chapter 5). If one wants to enhance radiative recombination as strongly as possible, such as in semi- conductor light emitting diodes or laser diodes, non-radiative recombination processes have to be avoided, which involves the elimination of the corre- sponding deep centers. On the other hand, if one is interested in having a short lifetime for non-equilibrium charge carriers, as in the case of fast tran- sistors and photodetectors, one should intentionally introduce deep centers in order to enhance non-radiative recombination. In general, deep centers play a role of similar importance to that of shallow ones for semiconductor devices, although in a completely different way.

Having discussed the general character and importance of deep centers, we now will treat their electronic structure. In subsection 3.5.2 we will de- velop a simple model of a deep center, the so-called defect-molecule model. In subsection 3.5.3 we treat some methods of solution of the one-electron Schrodinger equation for deep centers. The consequences of many-electron effects in the electronic structure of deep centers will be discussed in subsec- tion 3.5.4. In section 3.5.5 we display experimental and theoretical results

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3.5. Deep levels 285

for selected deep centers, among them the vacancy in Si. the substitutional impurity at,oms of the main groups of the periodic table. the group of 3d- transition metals. and the group of rare earths. Also, we discuss the D X - center and the EL2-center in GaAs.

3.5.2 Defect molecule model

The 'Tight 3inding' (TB) method developed in section 2.6 represents one of the various procedures for calculating band structures of ideal crystals. Unlike other methods it uses basis functions to represent the Hamiltonian which are localized on the atomic length scale. Since the perturbation poten- tials of deep centers are localized on the same scale, the TB method should be particulady well suited for such centers. Of course, one must chose the Hamiltonian matrix elements empirically in order to arrive at useful practi- cal results, and the results also cannot be expected to be very accurate in a quantitative sense. However, the method should be suited to the deriva- tion of simple models that exhibit the essential physical features of real deep centers. The simplest among these models is the so-called defect-molecule model, which we will introduce below. In doing so, many particle effects and lattice relaxation will be ignored. We will mainly use the model to demonstrate the existence of deep levels and to explore the symmetry of the pertinent eigenfunctions.

At the outset we have to clarify which of the various tight binding basis sets should be used for the representation of the Hamiltonian in the case of a deep center. En the ideal crystal case considered in section 2.6, the atomic orbitals or hybrid orbitals IhtRj) were not used directly, but rather. we employed wavevector-dependent Bloch sums lukj) or ihtkj) formed from them. This was advantageous because the translation symmetry of the crys- tal could be exploited in this way. The latter symmetry no longer exists in a crystal with a deep center. so that localized basis functions, i.e. atomic or hybrid orbitals. can also be used without any loss. We select hybrid orbitals because these produce drastic simplifications of the Hamiltonian matrix which, like the Bloch sums of hybrid orbitals in the case of an ideal crystal, allow the eigenvahes of the Hamiltonian to be calculated in closed analytical form.

We introduce the defect molecule model using the vacancy as an example. Later, we will apply this model to substitutional impurities from the main groups of the periodic table. As the host crystal we take, in all cases, an elemental semiconductor of group IV, like Si.

Vacancy

Figure 3.7 shows part of a Si crystal containing a vacancy. Symmetry consid-

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286 Chapter 3. Electronic structure of semiconductor crystals with perturbations

erations make clear that the origin of the vacancy is not important, whether it originated by removal of a Si atom from sublattice 1 or from sublattice 2. Here, we consider the removal of an atom from sublattice 1, and to be still more specific, from the primitive unit cell at R = 0. The perturba- tion potential V’(x) is the negative of the potential produced by this atom in the crystal. Because of the removal of the 1-atom, the hybrid orbitals lht2Rt), t = 1,2 ,3 ,4 , of the four surrounding 2-atoms in the unit cells Rt, pointing inwards, no longer have a hybrid orbital of a 1-atom to which they can bind. They are called danglzng hybmds. The three other hybrid orbitals at a surrounding 2-atom interact with inwardly directed hybrid orbitals at atoms lying still further away (these atoms are not shown in Figure 3.7). The hybrid orbitals at a pal ticular siirrounding 2-atom also interact among themselves, including the one dangling hybrid at this atom. This means that the dangling hybrids are coupled to the entire crystal through nearest neigh- bor interactions. If interactions between hybrid orbitals at the same atom are omitted, i.e. if the matrix element V1 introduced in section 2.6 is set to zero, which means - cp, then the dangling hybrids are decouplcd from the remainder of the crystal. They still interact only among themselves. Since the atoms at which these hybrids are located are second-nearest neighbors, this interaction is not within the framework of Erst-nearest neighbor inter- action which we have been exclusively considering in the treatment of an ideal crystal in section 2.6, but, rather, it ha6 the sense of second-nearest neighbor interaction. The latter must be taken into account here in order to arrive at non-trivial results.

Based on the approximations made above, the crystal with a vacancy decomposes into two partial systems which do not interact with each other, ~ first, the partial system of the four interacting dangling hybrid orbitals, one at each of the four atoms surrounding the vacancy, and second, the yarlial system of all remaining hybrid orbitals of the crystal with the vacancy, i.e. the hybrid orbitals of all atoms which are not directly adjacent to the vacancy, and the three hybrids at each one of the four adjacent atoms which &re not alrpady included in the first ptrrtial system. With some ambiguity

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3.5. Deep levels 287

we can refer to the first partial system as a vacancy molecule: and to the second t t s the ‘rest-of-crystal’. This designation also encompasses h e term defect molecule model for the tight binding approximation described above.

For each hybrid of the rest-of-crystal another hybrid exists in the rest- of-crystal which points to it. The Hamiltonian matrix elements between the various pairs have the same value, namely the value given above in equation (2.292) defining matrix element T/2. The energy spertriim of the rest-of- crystal is therefore identical with t ,hd of the infinite idea1 cryslal in bhe simplest TB approximation? consisting of the bonding level t b = th - IVzl, and the anti-bonding level E, = E,+ + I&I. The splitting of these highly degenerate levels into bands remains incomplete because of the neglect of Ihe interaction between hybrids at. the same &om? i.e. the neglect. of r/l.

In calculating the energy spectrum of the first partial system, i.c., the vacancy-molecule, we need the matrix elements of the pertarbed Hamilta- nian H + V’ of the crystal with vacancy. The diagonal elements (ht2RtlH + V‘lht2Rt) are simply the hybrid energies ~h since the elements (h,t2Rt(V‘ lht2Rt) of the vacancy potential V‘ between hybrid orbitals localized off the %mancy sit.e are small. One therefore has

In order to obtain the non-diitlgounel (second-nettrest neighbor) matrix el- ements (h t2Rt lH f V”lht32FQ) between different dangling hybrid orbitals t , t‘ f t , one has to recaIl the symmetry of the perturbed Hamiltonian H +V‘. Since, by creating a vacancy in sublattice 1) the two sublattices are no longer eqiiivalent, the symmetry of the crystal is no longer given by thr full cubic point group Oh, but rather by the tetrahedral group y d . Nevertheless, this means that all non-diagonal elements are equal for symmetry reasons, such that

wit,b W as second-nearest neighbor interaction energy. Because of the pre- dominantly negative values of the operator fI + V’ acting on hybrid orbitals and the predominantly positive values of the hybrid orbital products. W is expect.ed to be negative. ‘I’he absolute value of W must be determined empirically.

the matrix The energy levels of the vacancy-molecule are obtained by diagonaliziug

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288 Chapter 3. Electronic structurc of scmiconductor crystals with perturbalions

Its eigenvalues Eyzz,4 read

(3.113)

(3.1 14)

I E Y ) = [IIZL~RI) - IhaZRz)] , (3.116) Jz

v 5

45

(3.117)

(3.118)

One can easily demonstrate that IEI") belongs to the irreducible represen- tation A1 of the symmetry group Td of the vacancy, and the three functions IE$uc), IEY'), IEiWc) to the irreducible representation T2 of this group (see Table A.6 of Bppendix 4) . The eigenfunction of the A1-level resembles an atomic s-orbital of a Si atom. and the three Tz-eigenfunctions are similar to the three p-orbitals of such an atom. Evidently, the sp3-hybridization of the atomic orbitals in a Si crystal is removed at a vacancy. The states are more atom-like. in consequence of the fact that the crystal potential is no longer fully effective in the vicinity of a vacancy.

In Figure 3.8. the e n e r e spectrum of the defect molecule model of a vacancy is shown along with that of the rest-of-crystal. The A1-level lies below the Tz-level because of the negative value of Vt-. Whether it lies below or above the bonding level of the crystal depends on whether we have 31W1 < l\Jzl or 31WI > lF'21. The question of whether it is found in the valence band or in the energy gap, cannot be answered within the defect molecule model because. therein, the valence band has shrunk to the bonding energy level. It is just as difficult to decide whether the Tz-level lia in the conduction bend or in the energy gap. Experiment and more exact calculations, which we will discuss later in more detail. show that the -41- level lies in the valence band. and the 7'2-level in the energy gap. With this,

1 VaC - IE, ) ~ - [ I ~ I ~ R I ) - Jh2R3)] I

1 138"") = - [lh12R1) - lh42R9)].

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3.5. Deep levels 289

Figure 3.8: Energy spectrum of the defect molecule model of ~ 1 . Si-vacancy along with that. of the rest-of-crystal. The distribution of thc 4 electtona of the defect molecule over the energy levels is also shown.

the T2-level is the actual deep level of the vacancy. Of the four electrons of lhe neutral vacancy - each of the four dangling bonds yields one ~ two must he hosted by this lcvcl while the other two occupy the A1-level in the valence band. IJsing the terminology introduced above, we may say that the oxidation state of the neutral vacancy is V2+.

The defect moleciile model of the vacancy reflects the actual relation- ships remarkably well. In any case it, provides a qualitatively correct phys- ical picture of the electronic structure of a vacancy in group-IV elemental semicondiictors. Refinements of this picture will be discussed further b e low. Here we treat a second example to illustrate the defect molecule model which emerges from the v.mancy by occupying its empty lattice site with an impurity atom.

Substitutional impurity a t o m s wi th sp3-bonding

We consider a substitutional impurity alom in an elemental semiconductor of ~ o u p IV, which we can again imagine as a Si rrystal (see Figure 3.9). Let the siibstituted Si atom be that of sublattice 1 in the primitive unit cell R = 0. Like Si, the impurity atom should belong to one of the main groups 11,111, IV, V, or VI of the periodic table so that the valence shell is formcd by s- and p-orbitals, which all lie energetically higher than any other occupied orbitals (in contrast to rare earth atoms). The perturbation potentials of these impurity atoms in an elemental semiconductor of group I V possess, as we know, both a short-range part and also a long-range Coulomb part. The

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290 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Figure 3.9: Defect molecule model of a sub- stitutional main group &bonding impurity atom in a tetrahedrally coordinated semicon- ductor.

L

latter will be omitted from consideration below. This approximation does not affect the answer to the question of whether a particular impurity atom forms a deep level or not. although it influences the actual position of this level in the energy gap. The latter cannot be determined within the defect molecule model anyway.

The four sp3-hybrid orbitals of the impurity atom will be denoted by IhtiO), and the four hybrid orbitals of the host crystal pointing in the direction of this atom will be denoted by IhtzR,), t = 1,2.3,4. Xeglect- ing interactions between the hybrid orbitals at the host atoms, the crystal with a substitutional impurity decomposes, just as before in the vacancy case, into two partial systems, first. the defect molecule with the 8 orbitals jh t iO) , jht2Rt), t = 1.2,3.4, and second. the rest-of-crystal with all remain- ing orbitals. The energy spectrum of the rest-of-crystal again coincides with that of the ideal crystal within the framework of the approximations used here. The Hamiltonian matrix of the defect molecule is composed of elements of the general form

fhtjRtIH + 17’lht~j’R~~)~ (3.11 9)

where j and j ’ take the values i and 2 independently of each other. We consider only the most important elements of this matrix, namely

In this, E ; and c i signify, respectively, the hybrid energies of the impu- rity and host atoms, V2 corresponds to the matrix element of H of equation (2.292) between hybrid orbitals at nearest-neighbor atoms of the ideal crystal

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3.5. Deep levels 291

pointing toward each other, and W describes, as in the vacancy case, the sec- ond nearest neighbor interaction between differing host atom hybrid orbitals pointing toward the impurity atom. All elements (h t jRt lH + V’lht‘J’Rt,) not listed above are neglected, among them also the Vl-like elements (htzRt(H + V’lht,i&,) between different hybrid orbitals at the impurity atom, i.e. with t’ f t. With these approximations, the Hamiltonian matrix of the defect molecule may now be written down in explicit form. In doing so the rows and columns are ordered in the sequence IhliRl), lh2iR2), lh3iR3), lh4iR4), lh12R1)) lh22R2), lh32R3), I@&), and the Hamiltonian matrix is

L i 0 0 0 1 7 2 0 0 0

0 € e , O 0 0 v z o 0

0 0 6 i 0 0 0 v 2 0

o a o a o v2 v 2 a o o € ; W W W

0 v 2 o 0 W E k W W

0 0 v 2 0 W W E k W

( 0 0 0 I$ w w w e;.

(3.124)

This matrix has four distinct eigenvalues, two simple and two triply degen- erate. The eigenfunctions of the two simple levels (we distinguish them by indices b and a ) belong to the 1-dimensional irreducible representation A1 of the tetrahedral group T d , and the two triply degenerate (again distinguished by indices b and a ) belong to the 3-dimensional irreducible representation T2 of T d . The corresponding energy levels are, respectively, denoted as E i T / b and E&Y:,b, whence

In Figure 3.10 these levels are plotted as functions of the difference ( c i - E;) between the hybrid energies of impurity and host atoms. This difference rep- resents a measure of the strength of the short-range perturbation potential of the impurity atom. In order to better understand the physical meaning of the energy levels derived above, it is helpful to consider two limiting cases.

First, we assume the impurity atom to coincide with the host atom, which means t i = c;. If one also neglects the second nearest-neighbor interaction energy W , then the energy spectrum of the perturbed crystal of equations (3.125) and (3.126) must coincide with that of the ideal host crystal in the

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292 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Figure 3.10: Energy levels Eimp of a substitutional main group sp3-bonding impurity atom in a tetrahedral semiconductor as function of the hybrid energy dif- ference ( t i - e i ) . Horizontal lines indicate the bonding level t b and the anti-bonding level of the host crystal.

simplest TB approximation, i.e. a bonding energy level q = ~ h - IVzI, and an anti-bonding energy level E, = € h + IV21 must emerge. This is in fact the case. Thus it is also clear that the two energy levels EiT,, of the perturbed crystal correspond to bonding and anti-bonding states of A1-symmetry, and the two energy levels E;Y$. to bonding and anti-bonding states of Tysymmetry. The two undisturbed levels €h f IV21 in Figure 3.10 encompass the energy gap of the host crystal.

Second, we consider the limiting case of ( E ; - tk) tending toward $00

or -00 which, for a given host crystal, also means t i -+ +m or E ; -+ -00,

respectively. In the limit

Ei + $00

equations (3.125) and (3.126) yield

and in the limit

E i 4 --oo

we have E i T = tk + 3W,

(3.127)

(3.128)

(3.129)

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3.5. Deep levels 293

(3.130) EAlb Tzb - + --O0.

In the limiting case t i -+ +m, the two anti-bonding levels EiT and EkYZ of (3.127) tend, along with t i , toward 00, i.e. leaving the energy spectrum on its high energy side, while the two bonding levels E i Y and EkYl limit toward, respectively, the two Al- and Tz-levels of the vacancy of equations (3.113) and (3.114). This is understandable because the limiting case EL ---f +m means that no impurity hybrid energy level exists at finite energy and, therefore, there is also effectively no impurity atom. In other words, there is a vacancy.

With E ; -+ -m the two bonding energy levels E i Y and E$T leave the energy spectrum at its low energy side, while the anti-bonding levels limit toward the two A l - and T2-levels of the vacancy of equations (3.113) and (3.114). This occurs for the same reason as above in the limit t i -+ +m, namely because t i + -00 means that there is a vacancy at the impurity site.

For finite positive values of (EL - E;), the bonding Al- and Tz-levels can occur in the energy gap, forming deep levels there, and for finite negative values of ( t L - ~ k ) the same can happen with the two anti-bonding Al- and Tz- levels. A look at Figure 3.10 shows that the most favorable candidate for an impurity level to be in the energy gap is the bonding Tz-level if ( ~ i - 6;) > 0 holds, and the anti-bonding A1-level if ( 6 ; - 6 ; ) < 0 holds. More rigorous calculations confirm this conjecture in that they find exactly these levels to be in the gap. Below, we will describe such calculations and discuss methods for solving the one-electron Schrodinger equation (3.8) for the crystal with a point perturbation.

zmp = Ezmp ~

3.5.3 Solution methods for the one-electron Schrodinger equa. tion of a crystal with a point perturbation

The solution of the Schrodinger equation for a crystal with a point pertur- bation begins with the determination of the effective one-electron potential VPrt(x). In section 3.2 this task already was addressed in general terms. Two remarkable differences were found in relation to an ideal crystal: First, the effective one-electron potential VPert(x) depends on the electron popu- lation of (localized) one-particle states, and second the atomic structure of the perturbed crystal, which participates in the determination of the po- tential Vwrt(x), is in many cases initially unknown and it must be self- consistently calculated jointly with the electronic structure. Below, we will eliminate such additional difficulties of electronic structure calculations for point perturbations by assuming that the population of the center and the atomic structure of the perturbed crystal are known. Thus the potential Vwrt(x) of the center can also be considered to be known. In regard to the

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294 Chapter 3. Electronic structure of semiconductor crystals with perturbations

electron-electron interaction contribution to VPd(x) , this only means that the functional dependence of this contribution to VpeTt(x) on the eigenfunc- tions of the one-electron Schrodinger equation is known, but not its actual value. The form of this dependence is governed by the particular oneelectron approximation used in the calculations - the possibilities include Hartree, Hartree-Fock, Local-Density-Functional theory or the quasi-particle method. Because of the dependence of the potential on the eigenfunctions of the Schrodinger equation, any solution procedure must be iterated repeatedly until self-consistency is achieved.

The three most important procedures are the cluster method, the super- cell method and the Green's function method. As a rule, the replacement of the real potential by a pseudopotential - a technique which has been \-cry

successful for ideal crystals (see section 2.5) - is also applied in the supercell and Green's function methods.

Cluster method

Employing the cluster method, the infinite crystal with a point perturbation is replaced by a finite part which contains the point perturbation at its cen- ter and which. on the one hand, is large enough that the band structure of the infinite crystal is almost completely devploprd and, on the other hand, is small enough that the Schrodinger equation for it can be solved easily. This finite part of the crystal is called a rlwter. The cluster with the point pprturbatinn at its center is clearly just a large molecule. Its atomic and elec- tronic struclureb can be calculated ubing the methods of quantum chemistry for the determination of the structure o f molecules. In contrast to infinite Lrystals, clusters have a surface. The latter also acts as a perturbation of the periodic potential and can gi%e rise to bound slates in the gap. Such statrs must be remowd in a suitable way in order for the cluster method to be applicable. To this end, different procedures haw been proposed such as, far example, the saturation of the dangling bonds at the surface by hydro- gen atoms. Correspondingly, the surface levels are lowered deep below the valence band edge and raised high above the conduction band edge, where they cause no further difficulty.

Supercell method

At the boundary of a properly shaped ciuster, one can add an additional cluster of the same kind. If one does this repeatedly and continues the whole proress ad infiniturn, one finally arrives at an infinite crystal. The primitive unit cell of this crystal is now, however, no longer that of the crystal hosting the point perturbation. but it is the cluster, which in this context is called a supeveell Thp crystal is referred to as a supercrystal. The

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3.5. Deep levels 295

periodic repetition of the point perturbation in the supercell method causes the discrete dwp levels to become k-dependent bands of h i t e widths. If the supercell is made large enough, these widths are negligibly small and it suffices to calculate the band structure of the supercrystal for one k-point only, e.g., for k = 0. The appealing feature of the supercell method is that the whole apparatus of band structure calculations for ideal crystals can be exploited for the electronic and atomic structure determinations of deep centers.

In the cluster and supercell methods one obtains the deep levels of the perturbed crystal by a process in which one numerically calculates the energy spectrum of a model system, i.e., of the cluster in the first method and of the supercrystal in the second. Results for the band structure of the ideal crystal are neither necessary nor useful in either method. Also, the decomposition of the entire effective oneelectron potential into that of an ideal crystal and a perturbation potential need not to be made. In the third method for calculating the electronic structure of perturbed crystals, the Green's function method, this decomposition is essential, and the band structure of the ideal crystal is required.

Green's function method

The Green's function method employs techniques and insights of quantum mechanical scattering theory. It provides results not only about bound states with energy levels in the gap of the ideal crystal, but also on scattering states having energies in the allowed energy spectrum of the ideal crystal. First we consider the bound states.

Bound states: Kostcr-Slater method

To explain this method we write down the Schrdinger (3.21) equation for the perturbed crystal once again in a somewhat modified form,

[E - HI+ - L"@. (3.131)

Since we are interested in bound states. the eigenvalues E which WP seek from this equation lie outside of the bands, to be mare precise. in the iundamenlal energy gap between the highest valence band and the lowest conduction band. As tl preliminary we note that equation (3.131) can be formally solved for @ by multiplying both sides by the inverse [E - H1-l of the operator [ E - HI. The inverse operator [ F - H1-l stands in close relation t o the Green's function of the unperturbed crystal. The retarded Green's finctaon O " ( E ) is defined by the equation

(3.132) 1

E t i 6 - H ' GO@) =

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296 Chapter 3. Electronic structure of semiconductor crystals with perturbations

where 6 is an infinitesimal positive imaginary part which is set to zero after serving to remove singularities at the energy eigenvalues of the unperturbed crystal. This procedure assures that the wavefunction response of the un- perturbed crystal conforms to the causality principle, in that the response occurs only after the system has been perturbed. As our present interest is in deep levels in the forbidden part of the energy spectrum of the ideal crystal, not in changes of wavefunctions with energies in the already existing continuous part, no singularities occur in G"(E) of equation (3.132) in the range of E of interest to us, and we may ignore 6. Using G " ( E ) , equation (3.131) may be formally rewritten as

[@(E)V' - I]?) 1 0. (3.133)

Non-trivial solutions me possible only for energies E for which the deter- minant of the matrix of the operator [C0(E)V' - 11, in an appropriate or- thonormalized basis set, vanishps, i.r. if

Uet[G"(E)V' - 11 = 0 (3.134)

holds. The energy eigenvalues E satisfying this equation are deep levels. These levels ran thus be determined by calculating the Green's function G o ( E ) of the unperturbed crystal and solving equation (3.134). This p r o w dure is rcfrrred to as Koster Slater method, and equation (3.131) as Koster- Slater equation

The Green's operator @ ( E ) can be cdriilattul if the band structure E,(k) and the Bloch functions of the ideal crystal are known. The matrix represenlation of d ' ( E ) in the Bloch basis reads

(3.135)

With this we only need to know the matrix elements (uk(V'1u'k') of the perturbation potential V' in Bloch representation in order to determine the deep levels using equation (3.134). However, i t would be more expcdicnt for the matrix representation of the perturbation potential to use localized wavefunctions as) for example, atomic orbitals. If one does so, then the matrix representation of the (heen's opertrtoi is not as simple as in the Bloch basis. A particular compromise in choosing a basis set for the representation ol the eigenvalue equation (3.134) is the use of so-called Wunnzer finctzons.

Koster-Slater method in Wannier representation

Wannirr functions are linear combinations of all Bloch functions for a given band index v of the form

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3.5. Deep levels 297

(3.136)

Here the summation is over the whole first BZ, G“ is the number of primitive unit, rdls in a ppriodicity rcgioa. and ‘ t k are certain phase a~glru. If the latter are chosen properly, the Wunnier h c t i o n s IvRJ turn out to be well localized in the unit cell at the lattice point FL If the eigenvalue equation (3.134) i s

written in terms of this basis, it reads

The matrix elements of the Green’s operator G o ( E ) may be obtained by means of (3.143) as

The matrix elements of the potential (vRIV’jv’R’1 are particularly large if R and R‘ are identical, and both are equal to the lattice vector of the cell which hosts the point perturbation. As before, we assume & = 0. Neglecting all other elements we find

Employing (3.138) and (3.139)) equation (3.137) becomes

For the particular lattice point R = 0 it reads

(3.140)

{ 3.141)

Equation 13.141) forms a closed set of equations for the central cell compo- nents ( ~ O ~ I J ) of $ only. If the latter are known, all other components (vRI$) follow at once from equation (3.140). For a non-trivial solution of the ho- mogeneous system (3.141) to exist, its determinant has to vanish separately, i.e.

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298 Chapter 3. Electronic structure of semiconductor crystals with perturbations

must hold. The diagonal elements (vOIGo(E)IvO) = G!(E) of the Green's operator in equation (3.142) may be obtained from (3.138) as

1 1 G ; ( E ) (vOlG*(E)IvO) = 3

G Ic E + i r - E E , ( k ) ' (3.143)

The dominant bands in equation (3.142) are those which form the gap, i.e. the uppermost valence band u and the lowest conduction band c. If we consider only them and neglect all others equation (3.142) becomes

or more explicitly

To solve this equation, the host Green's functions G:(E) and G : ( E ) , as well the perturbation potential matrix elements, have t o be known. Below we calculate the host Green's functions for isotropic parabolic bands of finite widths. In expression (3.143) for G:(E) we rqlace the k-sum by an integraL This yields

' d%i v = c, v. (3.146) 1

E -I- i t - E,(k) ' 0 G J E ) = --

G 3 8x3 , 1 s t ~ ~

with st beiug the volume of a periodicity region. Introducing the identity operator Jrm dE'd(b" - E ) into this integral, G:(E:) may be written as

whrre p v ( E ) is the density of states per unit energy and spin state

R p , ( E ) : - d3kd(E - &(k]) , v = c,w,

h3 1.232

(3.147)

(3.148)

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3.5. Deep levels 299

which differs from the DOS in equation (2.213) by a factor (0/2). In eval- uating p,(E), we approximate the true valence and conduction bands by isotropic parabolic ones, however, taking their band widths A E , to be the same as those of the original bands. This corresponds to the use of an effec- tive mass for each band averaged over the whole first B Z . The bandwidths are introduced by putting the total number of states of the approximate isotropic parabolic bands equal to G3. Then it follows that

(3.149)

3 G3 p c ( E ) = ~ ~ ~ ~ [ O ( E - Eg) - B(E - E, - AE,)] . (3.150)

where B(E) is the unit step function. We substitute these expressions into the Green’s function G ; ( E ) of equation (3.147). The E‘-integral is readily done. The imaginary part ofG;(E) equals (-TI times pv(E). The real parts ReG:(E) and ReG:(E) are given by

-I.;TI’F] (3.152)

Although complex numbers appear in these expressions, they Eare in fact real. This would hP obvious if we rcplaced the In-function by an arctan-function. We avoid this because it is more convenient to handle the many-valued char- acter of the In-function rather than that of the arctarz-function. Altogether, the host Green’s functions of OUT model depend on three parameters. the energy gap EB, and the two band widths A& and AE,. The latter are measure3 of the awrage kinetic energies of electrons in the vakncce or con- duction bands ~ large bandwidths mean large average kinetic energies (or small eEective masses).

Later, the Grm’s function method in Wannicr representation will be used to address the question of whether or not main group impurity atoms in tetrahedral semiconductors form deep levels in the gay.

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300 Chapter 3. E1wtmnic Btructure of serniconductar crystals with perturbations

Scattering states

The perturbation potential also gives rise to changes within the energy bands of the ideal rrystal. Of course, the wergia of t hew bands are still allowed quantum mechanically in the presence of the perturbation, so that in this rwpeet there is 110 change. However, change within the energy hands of the ideal rrysttll IS indured by the perturbation in the form of a modified density nf allowed energy levels pcr unit mnergy, i.e. in the form of a modifid density ol statvs. To calculat? this change it is expedient to introduce the Green's fiirirtion of the perturbed crystal,

(3.153)

According to formula (2.209). the imaginary part Im T r [ G ( E ) ] of the trace represents (apart from a factor -1/ir) the density of states p ( E ) of the syst,em, here thai of the crystal with the point perturbation. The Green's function of the per turb4 crystal oLrys the equation

1

G ( E ) - I; 4- i6 - [H + V ' ] .

G ( E ) = G o ( E ) t G'(E)V 'C(E) . (3.154)

which follows at once from the definition (3.1533 of G'(E). In quantum field tbwry. this relation is known as the D p o n equatzon Using thk equation and performing some simple calculations, the DOS expression (2.209) may be brought into the form

1 d A d E

p ( E ) - I m - In Det[G(E)] . (3.155)

In an analogous way? the DOS P O ( ! ? ) of the unperturbed crystal may be expressed in terms of the unperturbed Green's function Co(E). We seek the change

M E ) = P ( E ) - P O ( W (3.156)

ol t h r nos clue lo the point perturbation. This change may be obtained from (3.155) and (3.156) as

(3 157) 1 d x d E Ap(E) = --Irrt-lnDDet[l- G'(E)V'I.

This relation can be used to ralculatc the total change of the DOS in the entire enerRy range between -oo and too. Integrating A p ( E ) over this interval, and considering the fact that G'( f i ) vanishes for li: -+ f m , yields

lm ~ E A ~ ( E ) =. 0. (3.158) M

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3.5. Deep levels 30 1

This relation signiEes that under the action of the perturbation potential, the total number of states remains unchanged. This result already was used in section 3.4 in lhe context of shallow levels, and there it was referred to as Levznson’s theorem. Here, this theorem means that for each new state of the crystal created by the point perturbation, a state which existed without perturbation must cease to exist. The deep states in the energy gap occur therefore at the expense of band states. For each state occurring in the gap, R stale is lost fiom a band,

dEAp(h’:) - - d B A p ( E ) . (3.159) Lap. In the above derivat,ion of Levinson’s theorem, no assumptions about the spatial variation of the perturbation potential were made. This theorem, therefore, also applies for purely long-range potenlials. Thus, we have proven what was anticipated earlier in section 3.4, namely, that for each shallow level in the gap, a state is lost from a band. In Chapter 4, where we will calculate the electron population distribution over the energy levels in the gap and the bands in thermodynamic equilibrium, this result will play an important role.

At certain energy values in the bands, the DOS of the perturbed crys- tal can display maxima or minima. One speaks of these as resonance and antz-resonance state.9. These states emcrge when the short-range pertur- bation polential can bind or ‘anti-bind’ states at band energies. Since the corresponding localized level is degenrrate with the band energy continuum, the localization of resonance and anti-resonance states differs from that of deep level states in the gap - with increasing distance from the center, the eigenfunctions do not decay exponentially but oscillate with an amplitude decaying to zero arcording to a power law.

3.5.4 Correlation effects

Correlation effects, as discussed in section 2.1, are important for the N - electron system of a crystal with oneparticle states localized at a deep centel. One of thcse e f k t s is based on tlie configuration dependence of the Hartlee and exchange potentials. Another results from the fact that Slater determinants, even thosc calrulatcd by means of configuration de- pendent Hartrw and exchange potentials, are not exact eigenstates of the N-electron Hamiltonian. The exact eigenstates are linear coinbinations of different Sl&r determinants, an rffwt which is referred to as cor~fignralzon znteraetzon (see section 2.1). The two correlation effects, ‘configuration de- pendenre’ and ‘configuration interact ion’, will be discussed helow for deep centers. In regard to ‘configuration dependence’, we continue the general discussion of section 3.2 here.

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302 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Configuration dependence

For an ideal crystal, the energy eigenvalues E, of the onepartick Schrijdin- ger equation have direct physical meaning. Apart from their sign, they are the ionization energies I,, of the corresponding eigenststes Y , i.e. E , -I,. In this regard, according to section 2.2, the ionization energy Iv is definned as difference

(3.160)

of the total energy R t o ~ ( { v ) ” ) of the N-electron system in the ionized stair {v}”, and the total energy E t o d ( { v } ’ ] in theground state {v}’. Theionized state (v)” differs from the ground state (v)” in that a particle, which in the ground state of the system occupies a oneparticle state of energy E,, is transferred to a oneparticle state of energy 0 corresponding to the vacuum level. Like in section 2.1 one says that a31 electron is removed from the system. This expression has to he used with care, however, for if taken literally i t misrepresents the charge neutrality of the system. The total energy of the ground state follows, according to formula (2.541, by summing all occupied oneparticle energies, followed by subtraction of the electrostatic interaction energy of the electrons because the latter is doublecounted in forming the sum of one-particle energies.

The equation E , = - I , is the content of Koopman’s theorem, which was explained in section 2.1. The essential requirement for the validity of this theorem is the approximate population independence of the Hartree and exchange potentials or, more generally, of the effective one-particle potential. This requirement is not satisfied for a crystal with a point perturbation. The potentials depend on the number n of electrons occupying oneparticle states localized at the center, and so do the energy eigenvalues E, of the one-particle Schrodinger equation of the crystal with a point perturbation. We will denote these oneparticle energies by E:) henceforth to emphasize this dependence explicitly. Of course, the definition (3.160) of the ionization energy is also valid in this case, but it is no longer true that I,, represents the

That this cannot hold is immediately clear if one recognizes that the eigenvalues IT?) and E P - l ) of the one- particle Schrodinger equation with. respectively, n and n - 1 electrons at the center differ from each other - the level EP-” is deeper than the level E p ) because the removal of an electron results in the positive core being less strongly screened, so that the remaining electrons are more strongly attracted. In this situation, we say that the electrons at the center relax on the removal of an electron from the center. If one further considers that the total energies of both systems enter in the definition (3.160) of I,, that of the relaxed system with (n- 1) electrons at the center, and that of the unrelaxed

negative oneparticle energy -E$. (

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3.5. Deep levels 303

system with n electrons at the center, there is no way to explain why the energy difference E b t d ( { v } v ) - E ~ M ( { V } ' ) should be equal to the negative

eigenvalue -EP-') of the relaxed system. In reality, the ionization energy Ebtal ( {v}v) - E ~ M ( { v } ' ) is not equal to either one, but lies somewhere in between the two. It can be shown that it is approximately given by the negative of the one-particle energy eigenvalue of the N-electron system with the fictitious number (n - 1/2) of electrons at the center. This is plausible because the electron, during its removal, feels, so to speak, the potential with n localized electrons half of the time, and feels the potential with (n - 1) localized electrons there also half of the time.

To carry out an exact calculation of the ionization energy of a center which, in its ground state, has n localized electrons, the one-particle energy eigenvalues E P ) are not sufficient. Applying equation (2.54), these one- particle energies provide the total energy E b ~ ( { v } ' ) of the ground state, but they cannot be used to calculate the total energy Etotal({v}v) of the ionized state with n - 1 localized electrons at the center. To obtain the latter, one also needs the one-particle energies EP-') of the N-electron system with n - 1 localized electrons. In density functional theory, the two total energies follow more directly: one determines the eigenfunctions of the Kohn-Sham equation for the centers with n and n - 1 electrons, then forms the corresponding ground state densities, and evaluates the total energy functional (2.64) using these densities.

Ionization, i.e. exciting an electron to the vacuum level, is only one of the various possible one-particle excitation processes of the N-electron system of a crystal having a perturbation center. Generally, one may examine an excited state { v } ~ ' ~ with an electron in a formerly unoccupied one-particle state Y' of energy below the vacuum level, and a hole in a formerly occupied one-particle state v. The corresponding excitation energy Iuiv is given by the total energy difference between the excited state { Y } ~ ' ~ and the ground state {v}',

eigenvalue - E Z ) ( of the unrelaxed system and not equal to the negative

(3.16 1)

As in the case of ionization, the excitation energy Iulv is not equal to the one-particle energy difference E,i - E,. Here, we are interested in excitation processes involving changes of the populations of one-particle states local- ized at the deep center. There are different types of such processes. Firstly, in the final state, all electrons may still be localized at the center, but with one electron having changed its localized one-particle state. Such excitations are called internal transitions of the center. Secondly, an electron originally localized at the center may undergo a transition to the bottom of the con- duction band. This is referred to as a donor transi t ion Thirdly, an electron

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304 Chapler ,7. Electronic structure of semiconductor crystals with perturbations

Figure 3.11: Reletion between donor and acceptor ionazation levels.

from the t,op of thevalencr band may be transferred into a previously empt,y one-particle state Iocalized at the center. This type of excitation is called an acceptor transition. The excitation energy for an electron from the t,op of the valence band to the bottom of the rnnduct,ion band defines the en- ergy gap. As excitation energies of the N-electron syst.em of the perturbed crystal, all transition energies discussed above may be plotted in the same energy scheme, just as if they- were oneelectron levels. But. they are not: they are one-particle ezcitation levels. More strictly speaking, one has donor excitation levels E$V - E, - and acceptor excitation lcvels ~2~ = 1 ~ 1 ~ .

The donor excitation level for a center D in the neutral charge state D(O) is denoted by n(O/+), and the mceplor excit,ation level for a center A in its neutral charge state A(0) is denoted by A(O/ - ) . If the donor excitation level D ( + / 2 + ) at t.he single positively chargcd donor center D ( + ) also lies in the gap, one has a dovhly i u n i m b k , OT douhk donor. If only the excitation level D(O/+) lies in the gap, the donor D ( 0 ) is simply ionizable, or a single donor. In the general case, t,here are m d t i p k donors. An analogous terminology i s

used for acceptors. Viewing the hand edges as ‘int.erna1 vacuum Icvcls’ one 0ft.m refers to donor and acceptor transitions as donor and acceptor ion- izut,iuab, and to & curraponding excitation energies as ionization lez~elx.. Below, we will use this terminology

Acceptor ionization lcvcls may be traced back to donor ionization levels. In fact, ionizing a neutral acceptor center A(0) leaves this center in singly negative charge state A ( - ) , and a hole appear8 at the valcncc hand edge. If one subjects the centex A(-) E D ( - ) to a donor transition D ( - / O ) then the center returns t.o its neutral charge slate D(0) A(O), and an electron appears at the conduction band edge. As a resdt of this t,wo-step ioniza- tion process of the center A(O), an electron-bolc pair is excited while the state of the center has not changed. The sum of the two ionizalion energies A ( O / - ) and D ( - / O ) equals, therefore, the (minimum) excitation energy of an electron-hole pair, namely, the gap energy E, (see Figure 3.11). Gener- ally, for an x:cept.nr A ( Q ) in charge state &, one has

(3.162)

If one measures thc acceptor ionization level relative to the conduction b u d

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3.5. Deep levels 305

edge instead of the valence band edge (as is done in equation (3.162)), thus setting A’(&/(& - 1)) = Eg - A(Q/(& - l)), then it follows from (3.162) that

This unified description of donor and acceptor excitation levels makes it possible to decide in simple way whether a given center X represents a donor, an acceptor, both a donor and an acceptor or none of them. One has

(a) a pure donor, when the ionization level X(Q/(Q + 1)) lies in the gap and, simultaneously, the ionization level X((Q - 1)/Q) does not;

(b) a pure acceptor, if X((Q - l ) /Q) lie in the gap and, simultaneously, X(Q/(Q + 1)) does not;

(c) if both levels X((Q - l ) / Q ) and X(Q/(Q + 1)) are found in the gap, the center is both a donor and an acceptor. One then calls it amphoterzc;

- if neither of the two levels X((Q - 1)/Q) and X(Q/(Q + 1)) lies in the gap the center is neither a donor nor an acceptor.

The difference between the acceptor ionization energy X((Q - 1)/Q) and the donor ionization energy X(Q/(Q + 1)) of an amphoteric center X(Q) is, by definition, the Hubbard energy 1.J. One therefore has

Since, coninionly, the Hubbard energy U is positive, the acceptor ionization level commonly lies highcr in thc gap then the donor ionization level.

The number of electrons bound at a center in thermodynamic equilibrium depends on the position of the Fermi level with respect to the ionization levels. From the outset it i s clear that the ionization level X(Q/(Q + 1)) also marks that position of the Fermi level at which the charge state of the cenlei changes: if E p lies just above X(Q/(Q I l)), the charge state Q is realized, if EF lies just below X(Q/(Q + l)), the charge state (Q + 1) is realized. h o r n this observation one may conclude that the charge state Q occurs when the Fernii Pnergy lies above the ionization level X ( y / ( y t 1)) and simultaneously below the ionization level X((Q- l ) /Q), i.e. between the two levels X(Q/(Q + 1)) and X((Q - l ) /Q). Of course, for this conclusion to be valid, both levels have to be located in the gap.

Configuration interaction

If there is degeneracy among the various oneparticle states of an N-electron system, then there is also degeneracy between the Slater determinants formed

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306 Chapter 3. Electronic structure of semiconductor crystals with perturbations

from one-particle states of this type. Here, we concentrate on those degener- ate one-particle states which are localized at the deep center. The existence of such degenerate states was demonstrated in subsection 3.5.2 - the vacancy in Si exhibits a %fold degenerate state transforming according to the irre- ducible representation T2 of the symmetry group Td of the vacancy, beside a non-degenerate state belonging to the irreducible representation A l .

To analyze the consequences of configuration interaction, we use a sim- ple model: a deep center with 3 degenerate localized one-particle states of symmetry Tz, occupied by 2 of the N valence electrons of the crystal; the remaining N - 2 electrons are in extended valence band states. The extended electrons contribute to configuration interaction only little; we dis- regard them in our further discussion. Neglecting spin-orbit interaction, the Slater determinants for the two electrons of the deep center are prod- ucts of coordinatedependent and spin-dependent wavefunctions. Since a Slater determinant itself is antisymmetric with the respect to the exchange of two electrons, there are two possibilities for the behavior of the two fac- tors in regard to particle exchange: either the spin-dependent wavefunction is antisymmetric and the coordinate-dependent wavefunction symmetric, or the spin-dependent wavefunction is symmetric and the coordinate-dependent wavefunction antisymmetric. From quantum mechanics it is known that, for a system with two electrons, the first case applies if the total spin S is 0, and the second case if S = 1 (other values of the total spin are not allowed). Using the three wavefunctions Iz), ly), 12) of the 7'2-state, one can combine 6 symmetric two-particle wavefunctions, namely

and 3 antisymmetric two-particle wavefunctions, namely

These are, altogether, 3' = 9, as one should expect. If spin and exchange interaction are considered, the symmetric and an-

tisymmetric wavefunctions have slightly different energies because the ex- change energy depends on total spin S , which differs for the two groups of states. Within each group there is degeneracy, however, at least within the framework of the one-particle approximation. If the configuration in- teraction is taken into account, this degeneracy is removed. The result of this removil can be derived by means of group theory, strictly speaking,

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3.5. Deep levels 307

by means of the decomposition of symmetric and antisymmetric product representations into irreducible parts (see Appendix A). For the symmetric product one obtains (T2 x T Z } ~ = A1 + E2 + T2, and for the antisymmetric product (2'2 x T2}a = T I . In these relations, the factors on the left-hand sides are representations in oneparticle Hilbert space, while on the right- hand side one has representations in two-particle Hilbert space. This means that every state on the left-hand side can host 1 electron, and every state on the right-hand side 2 electrons. To avoid confusion, the representations in oneparticle space are denoted by lower-case letters in this context, i.e. by a l , a2, e, tl, t2 etc., instead by upper-case letters A l , A2, E , T I , Tz etc., which are common in group theory. Here, upper-case letters are used for two- (or, generally, multi-) particle representations. This is chosen in accordance with the notation for free atoms, where s, p, d , f represent oneparticle states, and S, P, D , F represent many-particle states. Below, we will employ the distinc- tion between lower- and upper-case representations wherever an ambiguity might occur. As in the free atom case, the total spin S of the many-electron state is indicated by the spin-multiplicity 2s + 1, appended to the represen- tation letter at its upper left. Using the notations introduced above, we may conclude the analysis of our model deep center by stating that configuration interaction will split its {t;}-configuration into the 4 different two-electron energy levels 'A1, ' E , 'T2, and 3T1.

If more than two electrons are localized at the center, and if oneparticle states of different irreducible representations are involved as, for example, in the ground state configuration {aSt;} of the neutralvacancy with 4 electrons, then the corresponding many-electron levels can be obtained in the same way as above. The only difference is that the symmetric and antisymmetric products to be decomposed into irreducible parts have more than two factors and the factors are not necessarily the same.

The splitting of the many-particle levels of deep centers of crystals has its counterpart in the fine structure of the many-particle energy levels of free atoms with more than one electron. For such atoms, many-particle states, formed from one-particle states of given total spin S and orbital angular mo- mentum L , having different total angular momenta J , give rise to slightly different energy levels. (Recall that the irreducible representations of the full rotation group, into which the products of the irreducible one-particle repre- sentations decompose, are distinguished by J . ) This fine structure splitting has its largest effect for electron shells which are strongly localized, i.e. for d- and f-shells (as opposed to s- and p-shells). One may expect, therefore, that impurity atoms with unoccupied d- and f-shells, i.e. atoms of tran- sition groups of the periodic table, should result in deep levels exhibiting pronounced fine structure splittings. This is in fact the case, as we will see below.

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308 Chapter 3. Electronic structure of semiconductor crystak with perturbations

3.5.5 Results for selected deep centers

Below we discuss the structure of several deep centers which are important either from the scientific or technological point of view. Knowledge about these centers is, in every, case the product of combined experimental and theoretical investigations. Since we have thus far treated only theoretical methods, we first present a short overview of the experimental methods.

Experimental methods

One can divide the experimental methods for investigation of deep centers into two groups, on the one hand, methods which measure ground state properties of the centers, and on the other hand, methods which give exper- imental data on center properties in thermally or optically excited states.

Among the methods of the first kind are measurements of magnetic properties, like Electron-Paramagnetic Resonance (EPR); Electron-Nuclear- Double Resonance (ENDOR) and magnetic susceptibility. These methods provide data concerning the total spin S of the centers and, if anisotropy effects are measured, also spatial symmetries. The chemical identity of the centers can be determined (in addition to other methods) by means of mass spectroscopy or of Rutherford Backscattering (RBS). Measurements of the Extended-State X-ray Absorption Fine Structure (EXSAFS) provide data about the geometrical ordering of the atoms in the vicinity of a point per- turbation.

To investigate the excitation properties of deep centers, optical and elec- trical methods are available. Ionization energies can be determined by means of optical absorption spectroscopy and photoconductivity measurements, mainly in the infrared spectral region. The cross-sections of deep centers for emission of free charge carriers can be determined by means of t ime resolved current or capacitance measurements at pn-junctions or at other depletion layers. By suddenly applying a reverse bias at such a junction, the deep levels are lifted relative to the Fermi level The new equilibrium state of the junction corresponds to fewer electrons in the deep levels than previ- ously. This state does not occur suddenly, however. it adjusts exponentially through emission of electrons from the deep levels into the conduction band (we assume deep donor levels here, the case of deep acceptor levels may be treated analogously). In the conduction band, the electrons are freely mobile carriers and are immediately sucked up by the positive electrode at the n- region. This results in an exponentially decaying current, which for its part leads to an increasing positive charging and. thus. an increasing capacitance of the junction. The decay time of the current and the rise time of the asso- ciated capacitance change are determined by the emission probability from

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3.5. I?wp let& 309

the dttep centers. Measuring the capacitance rise time yields experimental values for the emission probability. Of particular importatice is the so-called Deep Level Trawaerat Spectroscopy (DLTS), wherein a reverse biased pn- junction is exposed to periodically repeted voltage pulses of forward {deep level filling) polarity. The recovery time for the capacitance change after a filling pulse has been switched off, is measured as a fimrtion of temperature. This fiinction exhibits maxima which, under certain conditions, can be used to determine the ionization energies of the deep centers. Ionization energy values from LILTS measurements and other thermal quililrrium technique are, as a rule, smaller than optically measured values: the lattice has time to rrlax if ionization proceeds thermally, and this lowers thc enerm of the finel stat?. Optical ionization occurs instantaneously, hence lattice relaxation is is not possible.

we initiate our discussion of particular deep centers with it. Perhaps the best understood point perturbation is the vacancy in Si, so

Vacaricy in Si

The defwt molerule model of the vscancy was treated in section 3.2. It predicts the existence of two bound one-particle states, a non-degenerate al-state, and a tripiy degenerate t2-state. More rigorous calculations within one-particle approximation (3araff, Schliiter, 1980; Bernholc, Pantelides. 1980) show that the al-level can be excluded as a deep state in the gap because it lies in the valence band (see Figure 3.12a). As such, it is fully occupied having two electrons. The t 2 -level can lie in the gap depend- ing on how many electrons it hosts. ,4t maximum, this can be 6, and at minimum 0. Thus there exist 7 charge states of the vacancy, namely t7(2+), I,’(+), i ’ (O), I,’(-), V(2-), Ir(3-) and V(4-). The one-particle en- ergies of these centers differ by the Hubbard energy U. A simple estimate yields 0.3 el’ for C3. Thus, many-body effects, more strictly speaking. con- figuration dependencies of one-particle energies: should be important in the case of the Si-vacancy. Other many-body effects, including configuration mixing, are small, and a description of the vacancy in terms of one-particle states, albeit configuration dependent ones. is approximately justified.

Owing to the value of about 0.3 e V for U , it may be expected that three or four ionization levels codd fit in the Si gap of about 1.1 eV. A4ctualIy, the donor levels 5’(+/2+), t ’ (O/+) , and the acceptor levels V(- /0 ) and V ( 2 - /-1 are found there (as above the acceptor levels are counted relative to the conduction band edge). In Figure 3.12 (part b). calculated positions of these levels are shown. These are not yet final positions because lattice relaxation has not yet been considered. The latter leads to changes which are discussed below (see Figure 3.13). In charge state V(Z+) , no electrons are available

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310 Chapter 3. Electronic structure of semiconductor crystals with perturbations

for the population of the t 2 level, therefore no Jahn-Teller distortion of the vacancy occurs. In charge state V ( + ) , the tz-level is occupied by 1 electron. Its energy can be degraded through a tetragonal Jahn-Teller distortion. The symmetry after the distortion is DM. For this group, the 3-dimensional irreducible representation t 2 splits into the 1-dimensional representation 62 and the 2-dimensional representation c. The bz-level lies energetically below the e-level and is single occupied. In charge state V ( 0 ) the additional electron can also be hosted by the bz-level. In order to gain additional energy, the tetragonal Jahn-Teller distortion is strengthened. In charge state V (-), population of the e-level begins. Through a further distortion of the vacancy, which reduces its symmetry from DM to CzVi the e-level splits into two levels and the additional electron is placed in the deeper of the two. This level can also still host the additional electron of the charge state V(2-), with an increase of the C2,-distortion. If one takes account of the energy shifts diie to Jahn-Teller distortion, the resulting level positions are as shown in Figure 3 .12~ . 'I'he exchange interaction evidently plays only a minor role, so that practically no spin-splitting of the vacancy levels occurs. Considering that the wavefunction of the deep vacancy levels extend as far as the nearest neighbor atoms, this is understandable.

A surprising result concerning the level positions in Figure 3.12~ is that the donor level V(O/+) lies below the donor Ievel V ( + / 2 + ) . The usual ordering of the ionization level of the more negative charge state above the ionization level of the less negative charge state is thus reversed. Formally. it seems as if the Hubbard energy U would be negative instead of positive for the transition V ( + / 2 + ) . One therefore also calls the vacancy in Si a negatzwe-U center. Of course, the interaction energy between two electrons at the center does not really change its sign, but the increase of ionization energy due to the Jahn-Teller effect on the V ( + / 2 + ) transition amounts to only about halfof that for the V(O/l+) transition since the Jab-Teller effect is absent at the V ( 2 + ) center. If the vacancy was initially in the neutral state, i.e. with Fermi energy lying above the V(O/+) donor level and below the V(- /O) acceptor level, with subsequent lowering of the Fermi energy below the V(O/+) level, then the vacancy will initially capture a hole from the valence band and pass into charge state IT(+) , and from there, without further change of the Fermi energy, i.e. spontaneously, it will capture another hole passing into charge state V(2+).

The occurrence of an effectively negatke iJ at the vacancy in Si was first predicted theoretically (BarrafF. Schliiter, 1980) and later found experimen- tally in correlated EPR and DLTS measurements. This phenomenon has now been observed at a number of other deep centers.

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3.5. Deep levels 311

Figure 3.12: Deep levels of avacsncy in Si: a) Calculated ionization levels without Hubbard corredions and lattice relaxation. b) Levels of a) with Hubbard correc- tions. c) Fsxperimetital ionization levels which, in addition, include Jahn-Teller shifts. The numbers give the level distances from the valence band edge in eV. (After Watkins, l S S 4 . )

Main group impurity atoms in tetrahedral semiconductors

The elements of the main groups of the periodic table are distinguished by the fact that their valence shells arise born 3- and p-orbitals. The group- IV elements, as well as thc IV-IV, 111-VI. and 11-VI binary compounds, form crystals with, respectively, diamond and zincblende structure in most caws. Relow we refer to them as 'tetrahedral semiconductors'. If main group elements appear as impurity atoms in such crystals, then they are chemically bound in a manner similar to that of the hust crystal atoms which they replace, i.e. through sp'-hybrids. That implies that the incorporation of impurity atoms should be mainly substitntional in such crystals. In the case of a compound semiconductor, the lattice site which is prpferred is the one that belongs to the chemically most similar atom of the host crystal, therefore the anion site, if the impurity atom is a nan-metal atom. and. the cation site, if it is a metal atom. The solubility of the main group elements in tetrahedral senlicntiductors is, accordingly, high. It lies between 10" for chemically very similar systems such as GaAs.Al, and lOI5 ~ r n - ~ for

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312 Chapter 3. Electronic structure of semiconductor crystals with perturbations

cl v’

d ) v -

D2d - Dzd

- o + d - b t c

Figurr 3.13: Defect molecule model of a vacancy in Si. Different charge states of the vacancy are shown, taking into account the torresponding Jtthn-Teller distortions. On the right-hand side, the basis functions of the irreducible representations of the deep k v e l ~ are indicated (a, b, c Bnd d bband for the dangling hybrids of the 4 surrounding atom). (After Watktns, l U U 4 . )

relatively different systems as, for inst ance, Si:Hg.

The perturbation polrntial of a main group impurity atom in a tetra- hedral semiconductor contains, besides the short-range part, as a rule, also a long-range Coulomb part. For isocoric impurity atoms, i.e. atoms whose cores do not deviate too strongly from that of the host atom, the Coulomb potential is in general the main contribution. As shown in section 3.4, this potential leads to shallow donor or acceptor levels. These are the levels principally involved if main group elements are used as doping atoms for

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3.5. Deep ler7ek 313

?r

Be B C N 0 - Ec - 0.14 -

h k A1 Vac. P S - EC - 0.43 - Ec - 0.31

Zn Ga Ge As Se

- - E, - 0.3 E1, + 0.32 -

Cd In Sn Sb Te E, + 0.55 E, t 0.15 - - Ec - 0.20

Hg T1 Pb Bi Po Ec - 0.31 E , + 0.25 ~ -

1

Table 3.6: Experimental deep l e d positions of neutral main group substitutional impurity atoms in Si (in eV). "Dash" indicates that no deep level occurs in the gap €or this particular substitutional impurity, meaning either that no localized state exists at all (C, Ge, Sn, Pb), or that such states exist but are shallow (B, Al, Ga, P, As, Sb. Bi). An empty space means that the corresponding impurity atom is either not incorporated substitutionally, or that there is no unambiguous experimental data. The neutral vacancy (Vac.) is shown for comparison. [Data compiled from Landoldt-Barnstein, 1982.)

tetrahedral semiconductors. For non-isocoric elements of the main groups, however, the short-range potential dominates in general, and, in particular cases. gives rise to deep levels in the gap (see Table 3.6). If an impurity atom belongs to the same column of the periodic table as the host atom, one says that the two atoms are isovcalent. In this case. the Coulomb potential vanishes completely and the short-range potential remains as the only po- tential contribution. Isovalent substitutional impurity atoms therefore lead either to deep levels (this occurs if the isovalent host and impurity atoms are chemically dissimilar, as in the case of GaP:P; or ZnTe:O), or no localized levels OCCLU at all (this takes place for chemically very similar isovalent host and impurity atoms as in the case of Si:Ge. which forms an alloy).

An important theoretical problem which has yet to be solved for main- group impurity atoms in tetrahedral semiconductors is to understand why certain elements cause deep levels in the gap while others do not. The impu- rity problem mentioned was treated above within the defect molecule modeL Although group-IV elemental semiconductors were assumed as host crystals, the general results obtained there also apply to compound semiconductors.

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3 14 Chapter 3. Electronic structure of semiconductor crystah with perturbations

Accordingly, a substitutional sp3-bonding impurity atom in a sp3-bonding host crystal will introduce two bonding levels EL? and E t T of, respec- tively, a1 and t 2 symmetry, and two anti-bonding levels E:T and E;:’ of these symmetries (see Figure 3.10). Which of these levels are located in the gap, and which are not, cannot be decided by means of the defect molecule model. On the assumption that there is a deep level in the gap. one can guess its symmetry as folIows: The perturbation potential V’ = e t - E: has negative values (or is attracting) if the hybrid energy of the impurity atom is lower than that of the substituted host atom, i.e. if the group number of the impurity atom is higher than that of the host atom. Furthermore, V’ has positive values (or is repelling) if the hybrid energy of the impurity atom is larger than that of the host atom, i.e. for impurity atoms with lower group numbers than the host atom. If, by varying the impurity atom, the per- turbation potential V’ = ER - E: takes increasingly negative values starting from zero, then a deep level in the gap will evolve from the conduction band below a certain negative threshold value. If the perturbation potential V‘ takes increasingly positive values starting from zero, then one expects a deep level to arise from the valence band above a certain positive threshold value. Using this observation, a look at. Figure 3.10 indicates that for negative (at- tracting) perturbation potentials, the deep level should be the anti-bonding u1-leve1, and for positive (repelling) perturbation potentials, the deep level in the gap should be the bonding ta-level.

The above conclusions are essentially confirmed by the more accurate Green’s function tight binding calculations performed by Hjalmarson, Vogl, Wolford, and Dow (1980). Some results of these authors. concerning al- levels, are depicted in Figure 3.14. Since the a1-levels arise mainly from atomic s-orbitals, they are plotted against the s-orbital energy of the impu- rity atoms in Figure 3.14. Concerning the question of whether or not a main group impurity atom gives rise to a deep level, the indications of Figure 3.14 agree surprisingly well with experiment. For the particular case of Si this can be seen by means of comparison of Figure 3.14 with Table 3.6, which summarizes experimental results for this host crystal. The data from both sources agree that no deep levels exist in the case of the isovalent group-IV atoms C, Ge, Sn, Pb. Among the group-V substitutional impurity atoms N, P, ils, Sb, Bi, only IY gives rise to a deep level while all others result in shallow levels (even those do not exist for the isovalent group-IV atoms). In the case of group-\’I atoms, Figure 3.14 indicates deep levels for substi- tutional 0, S, and Se. while experimentally one also finds a deep level for substitutional Te. For impurity elements left of column IV of the periodic table, like Ga or Zn. the perturbation potential is positive, and the deep level in the gap is expected to be tz-like rather than al-like. Such levels are not well described in the TB calculations quoted above.

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3.5. Dcep levels 315

>" QI

W Y

0

-1

-2 L I I I I I I 0

A -1 > w Q) u

-2

-3

11 I I!

t Eandedge -

- Cation Site

I I I 1 I I A

0 -30 -20 -1 0

s - Orbital Energy (ev)

Figure 3.14: Deep levels for main group substitutional impurity atoms of various tetrahedral semiconductors. The anti-bonding levels of al-symmetry are shown as a function of the s-orbital energy of the impurity atoms. For further discussion, see the main text. (After Hjalmarson, Vogl, Wolford, and Dow, 1980)

The fact that the perturbation potential is negative (attracting) for the anti- bonding al-level and positive (repelling) for the bonding &level allows an important conclusion on the nature of the deep impurity states: in both cases, the corresponding wavefunct ions have larger amplitudes at the sur- rounding host atoms than at the impurity atom. In this sense these deep impurity states are more host-like than impurity-like.

Quantitatively, the al-level positions in Figure 3.14 differ from the experi- mental ones. This is to be expected because of the great simplifications made in the calculations. many-body effects, particularly, configuration dependen- cies of the one-electron levels, have been neglected, and lattice relaxation has not been taken into account (Scherz and Scheffler, 1993). To get an idea how these effects would modify the results, we show in Figure 3.15 the expected occupancy of the deep oneelectron levels in the particular case of Si as host crystal, using the level ordering suggested by the above-quoted calculations

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316 Chapter 3. Electronic structure of semiconductor crystals with perturbations

f I el d )

Figure 3.15: Energy levels and their populations for sp3-bonding main group im- purity atoms in elemental semiconductors of group IV, within the defect molecule model. The partial illustrations a) to h) correspond to impurity a tom of groups I to VII as follows: a - V, b - \’I, c - VII, d - HI, e - 11; f - I.

and the defect molecule model. Of the 9 electrons of a Sirgroup-1’ atom molecule (remember that only N results in a deep level in this case), 8 are hosted by the four bonding states and the remaining electron occupies the anti-bonding al-state in the gap. In a 5’t:groupVI atom molecule with 10 electrons, 2 occupy the al-level, and in the 3t:groty-VI atom molecule with 11 electrons, in addition 1 electron has to be placed into the anti-bonding tl-level. This state of the molecule will certainly not be stable. As in the va- cancy case, a Jahn-Teller distortion will occur which removes the degeneracy of the ta-level and allows for a lower total energy by occupying levels shifted downwards. The defect molecules of group-111. -11 and -I atoms have. respec- tively, 7, 6 and 5 electrons. Of them 2 electrons are hosted by the bonding al-level in the valence band. There are, respectively. 5, 4, and 3 electrons to be placed in the bonding tz-state which forms the deep le\Tel in this case. One may also say that this level hosts, respectively, 1. 2, and 3 holes. Again, a Jab-Teller distortion will occur which lowers the total energy. Qualitative

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3.5. Deep levels 317

differences from this model occui for elements of the first main group which have no occupied p-states but relatively shallow closed d-shells (Cu, Ag) or d- and f-shells (Au). Foi these atoms, d-electrons participate in chemical bonding with the host crystal. We will discuss this problem in more detail in the context of the transition metal impuiity atoms.

tutional impurity atom, ran he treated arralytzrally, employing the Wannier representalion of the Green’s function derived above. As lhis calculation provides further physical insight into the formation of deep levels, we will discuss it hclow, agdin taking Si as the host crystal. The Chen’s function in Wannier representation has already been determined above (see equa- tions (3.158) and (3.159)). For the evaluation of the Koster-Slatcr equation (3.152) we need the matrix elemeuts V,,,, V,, V , of the perturbation poten- tial V‘ in this representation. To calculate them we use the results of the TR approximation of scction 2.5. In the simplest veision of this upproxi- mation, the ILamiltonian of the ideal crystal is diagonalized by the bonding and anti-bonding Bloch functions Ibtk) and In&), t = 1 , 2 , 3 , 4 , of equations (2.303) and (2.304). Thus, in an approximate sense, the bonding and anti- bonding orbitals IbtR) and latR) are the Wannier functions of the eigenvalue equation (3.152). We will use thmn to grt explicit expressions for V,,,,, V,,, and V,. Fiist, we write down the matrix representation of the unperturbed Hamiltonian H between bonding orbitals. It is given ’by

The question of whether deep levels exist or not for a particular substi

(btRIHI4R’) fhGRR‘, h (3.165) where E; is the hybrid energy of a host atom. For the perturbed Hamiltonian EZ fV’, one has the same matrix elements for all R and R’ with the exception of R R’ = 0. The latter elements are

(3.166) (btOlH i- V’lbtO) - S(E; + EL), where E L is the hybrid energy of the impurity atom. Siiicc none of the matrix elements of (3.165) and (3.166) depend on t , one may identify IbtO) with the Wannier function of the uppermost valence band. Then, taking the difference of equations (3.166) and (3.165), an expression for (vO1V’(wo) G V,, follows

1

S S

vlf, - v o , vu s ( 6 h 1 t - Eh). h (3.16 7)

The factor in (3.166) arises because the Wannier function is equally spread out over the two atomic sites of a unit cell, while the substitution of an atom occurs at only one site.

Analogoiisly, matrix elementu of fI and IT I V’ between anti-bonding orbitals may be used to obtain an expression for (cO(V‘lc0) I_ v,,, and matrix

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318 Chapter 3. Electronic structure of semiconductor crystals with perturbations

elements between bonding and anti-bonding orbitals yield the expression for (wO1V'lcO) V,. It turns out that all elements are the same, i.e.

v, = v, = v,, = LG. (3.168)

We seek deep level solutions Et of the Koster-Slater equation (3.152) located in the gap, i.e. with 0 < Et < EQ. For such energies, the imaginary parts of G:(E'} and G,!(E) are zero: and equation (3.152) transforms into

Solutions of equation (3.169) within the gap do not exist for all possible values of the 4 parameters entering, i.e. the gap energy Eg, the two band widths AE,, A&, and the perturbation potential constant Vo. This is evi- dent if one considers the particular case AE,,v >> Eg, which can be realized. The arguments of the logarithmic functions in G:(E) and G:(E) are close to 1 in thia case, and the logarithms themselves become negligibly small. If one assumes, in addition, that A& = AEw, then

Re[G:(E) + G:fE)] - (F) --& [,,/=- fi] . (3.170)

With this exprwsion! the deep level condition (3.169) takes the form

(3.171)

It has solutions Et only in the energy gap 0 < Et < E,, within which Et has to be restricted anyway because the deep level condition (3.170) is valid only there. For negative Vo, one necessarily has Ef > Eg/2, and for positive Vo, Et < E,/2. For Vi -t +m, El approaches the value E,/2, i.e. the deep level is pinned at the midgap position. This corresponds to the pinning of & to the vacancy level discussed in subsection 3.5.2. In order for a solution Et of equation (3.171) to exist at all, lV01 must exceed a minimum value J V O ) ~ ~ ~ given by

(3.172)

This minimum value of (Vo( follows from equation (3.171) by identifying Et with the lowest possible deep level position in the gap namely Et = 0 in the case of positive Vo, and by identifying Et with the highest possible value of Et namely E, in the case of negative VO. Considering equation 13.1721, the solution of equation (3.171) may be written in the closed form

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3.5. Deep levels 319

Table 3.7: Perturbation potential matrix elements Vo = (1/2)(~: - $) (in eV) for substitutional impurity atoms of main groups in Si, calculated by means of Herman- Skillman s- and p-orbital energies. The latter are partially reproduced in Table 2.2. (After Herman and Skillman, 1963.)

B e B C N 0 F 1.57 0.08 -1.41 -3.04 -4.8 -6.71 Mg A1 Vac P s c l 2.18 1.05 8.28 -1.12 -2.31 -3.56 Zn Ga Ge As Se Br 1.82 0.88 -0.04 -0.99 -1.97 -2.98 Cd In Sn Sb Te I 1.91 1.14 0.35 -0.42 -1.22 -2.03 Hg TI Pb Bi Po At 1.87 1.17 0.47 -0.24 -0.96 -1.68

According to condition (3.172) for a deep level to be formed, the absolute value of the potential and the energy gap must be relatively large, and the bandwidth relatively small. That lVol needs to be large enough is obvious. The bandwidth must be sufficiently small because it corresponds to the av- erage kinetic energy, which militates against binding. The gap is the energy range where the deep level has t o be placed, so it should not be too small. EnIarging the gap and lowering the bandwidth increases the likelihood of forming a deep leveL

The condition lVol > IV0lmifi may even serve as a quantitative guide, as the following numerical example for impurity atoms in Si demonstrates. In this case one has Eg = 1.1 eV, and AE, = A E , = 3.3 eV is a reasonable choice. With these values, IVolmipl is 1.21 eV. The perturbation potentials V-0 for impurity atoms of the main groups II to VI of the periodic table, calculated by means of equation (3.167), are listed in Table 3.7. According to our model, a deep level should exist for lV0l > 1.21 eV, and should not exist for IVoi < 1.21 eV. Comparison with experimental results €or deep levels of substitutional impurity atoms in Si shown in Table 3.6 reveals that

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320 Chapter 3. Electronic struetrue of semiconductor crystals with perturbations

the rritprion is correct in all caws. pxrept for In and TI, which are close to the border line of our model but still on thp side where no deep level6 exist. while experiment ally surh 1wrls art- found.

Transition metal impurity atoms

The transition metal ('YM)-impurity atoms of the iron group, i.e. those with closed ad-shells (see Table 3.2), are incorporated in Si crystals prdom- inantly on interstitial sites of tetrahedral symmetry, while the 5d-TM atoms prefer substitutional cation sites. The position of the 4d-TM atoms in Si lies between the two, i.e. both interstitial and substitutional incvrpuretions are observed. In III-V and 11-Vl semiconductors, all three groups of TM atoms prefer substitutional incorporation on cation sites. The solubility of most transition metal elements in Si is relatively small, lying in the range of

rm-3. Higher values in the range of 101frm.3 arc? reached in tetrahedral compound semiconductors, and for M n in GaAs the solubility reaches as high as 10" ~ r n - ~ . R;In also is a transition metal which gives rise to alloys with cerhizl 11-VI compound semicondudors, such as, for example, (Zn, b1n)Te or (Cd.hln)Te. Semiconduct,or alloys exist also with Fe and Co.

In tetrehdral semiconductors, most of the TM atmns form deep levels. Among them, the levels of 'I'M-elements with closed 3d-shells (the so-called 3d-TM cleinents) are by far the bestst known. Concerning 4d- and 5d-TM imyurily atoms, it is understood that. deep levels also exist in their rasp (Beeler, Andersm. Sdief€lcr, 1985 and 1990; Alves, Leihe, 1986). This is not surprising since the Thl elements are all chemically very similar. Here, wc restrict our considerations to the 3d-TM atoms. Them are the elements Sc! Ti, V, Cr, Mn, Fe, Go and Xi. Sometimes, the neighbors of Ni to the right in the periodic table, CU and Zn? are also included, although, owing t.o their respective 3d1'4s- and 3d1'4s2-confi~~ations. these are not in fact transition metals. In tetrahedral semiconductor crystals, however, they behave simi- larly. Special interest in the investigation of 3d-Thl atoms results mainly from t,he fact that their deep levels play an import.ant part in electronic de- vices. They can serve as recorrilrizlation centers through which the lifetime of non-equilibrium carriers is shortened, or as capture centers for h e carriers which partially compensate the effect of dopant atoms. These effects are imclesirable in most caws, thus one must. try to minimize the contamination of the devicw by 3d-'I'M atoms. Occasionally, 3d-TM impurities can elso be useful. 'I'his happens, for example, in the case of Cr in GaAs which, due to its compensating effect! makes the crystals serui-insulitting.

in tehahedral semiconductors (interested readers are referr4 to the book by Fkurov and Kikoin, 1994). Beside donor and acceptor transitions alsn in- ternal transitions of the 3d-TM imp1irit.y atoms are observed. Below we

There are numcrouq experimental investigations on 3d-'l'h.I impurit.y &oms

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3.5. Deep levels 32 1

concentrate on the donor and acceptor transitions. Figure 3.16 contains data related to this, as well as showing the ionization energies of the free 3d-TM ions. The latter are larger than the ionization energies of the 3d-Thl atoms hosted by semiconductor crystals by more than one order of magni- tude. The case of free atoms is also striking in regard to the large Iyariations of ionization energies between different ionization states. In semiconductor crystals these variations are almost two orders of magnitude smaller so that, as in the case of the Iacancy, ionization levels of several charge states may be found in the enera gap. Considering the substantially stronger localization of d-electrons and the much weaker screening of their interaction in compar- ison with the valence electrons this is surprising and needs explanation. We will return to this point later.

Theoretically. the donor and acceptor ionization levels of substitutional 3d-TM impurity atoms are now well understood. A simple oneparticle model which is supported by ab initio calculations, is a defect molecule with the 3d-TM atom on a substitutional cation site at its center (see Figure 3.17). In this model, the TM atom is represented by the five 3d-orbitals and the one 4s-orbital of its valence shell, and the representation of the crystal is embodied in the four sp3-hybrid orbitals of the four surrounding cation atoms pointing towards the TM atom. The energies of these orbitals will be denoted by Ed, eS and eh, respectively. Considering the tetrahedral sgmme try of the impurity center, we decompose the four sp3-hybrid orbitals of the four surrounding cation atonis pointing lowdrdb the ThI atom into a state with A1-symmetry and three states Itah) with T2-symmetry. The s-orbits1 of tlir TM atom is deformed by the crystal into an orbital of A1-symmetry, and hhe five d-states decompose into two states 1.) with E symmetry and th rw states It%) with Tz-symetry. The two 01-orbitals interact with each othrr tand give rise to a bonding state dwp in the valenct. band and an anti- bonding state deep in the conduction band. The two estates remain without bonding, their energy levels are likely to be found either in the gap or in the valence band. The two triply drgeiicratc 12 states Itah) and 1t2d) do interact mutually. 'rhe corresponding interaction matrix elements are of the type rj' and bym {see section 2.6). Neglecting bbP-type matrix elements, one triply degrnertlte bonding level Eb nnd one triply degenerate anti-bonding level Ea arise. For these two levels, the same formulas apply 8s those which wmt' derived in section 2.6 for interacting hybrids at nearest neighbor atoms of an ideal crystal. Denoting the Vpp,-type matrix elemen1 by Vhd, we hwe

(3.174)

where

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322 Chapter 3. Electronic structure of semiconductor crystah with perturbations

Figure 3.16: Experimental ionizatjon energjes of free 3d- Tzvl ions (lower part), as well as donor and acceptor ioniza- tion energiea of 3d-TM impu- rity atoms and ions in various host semiconductors (upper five parts). (After Zernger, 1986.j

d'/do d%' d?d2 dyd' d%l' d?d5 d?da d%' d%ln d'o/d9 I I I I I I I ' 1 1 1

Go As to/- 2.4

The corresponding thecomponent wavefunctions It2b) and Itza) of, respec- tively, energy Eb and E,, are

where we set

1 s 1 s 2 A a2 = - ( I - - a)' a 2 = - ( l+-) . (3.177)

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3.5. Deep levels

c -band

32 3

t 2 a \ t 2 4 s 3 d

v - band

Figure 3.17: Energy level diagram of the defect molecule model for a substitutional 3d-TM impurity atom in a tetrahedral semiconductor. Explanations are given in the text.

Since the two bonding and anti-bonding eigenvectors Itzb) and Itza) of equa- tion (3.175) are linear combinations of basis functions with tz-symmetry, they also have t2-symmetry (as already indicated by the notation). It turns out that the bonding tzb-level lies in the valence band, the non-bonding e- level in the valence band or the gay, and the anti-bonding tk-level is in the gap above the e-level or in the conduction band. In regard to deep levels of the 3d-TM atom i s in the energy gap, therefore, the 2-fold degenerate e-level and the abovelying triply degenerate anti-bonding tp,-level are candidates (see Figure 3.17). The eigenfunctions of the e-level are linear combinations of the &orbitals of the TM atom and are therefore strongly localized at the latter. Considering localization of the t a b and tn,-leveh, the position of the two orbital energies F d and th relative to each other is decisive. If Ed lies deeper than Ch, then a! > ,8 holds, meaning that the bonding tzb-level is mainly formed from the d-ststes of the TM atom, while the anti-bonding tza-level, which forms the deep level, arises to a considerable extent from the tz-components of the four sp3-hybrid orbitalb of the surrounding host crys- tal atoms. Exactly this picture emerges from the above-mentioned Green’s function ab initio calculations (Zunger, Lindfeldt, 1983). In Figure 3.18 the calculated oneparticle energy levels are shown for some 3d-TM atoms in Gap.

The most striking feature of the above description of the electronic struc- ture of substitutional 3d-TM impurities is the existence of deep levels whose wavefunctions are spread out over the surrounding host atoms. These levels

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324 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Figure 3.18: Calculated one- electron energy levels of substi- tutional 3d-TM impurity atoms in Gap. (After Zunger and Lindt- felt, 1983.)

3 4ae *\ 1-

-3 -2 -jFq are host-like to a certain extent rather than purely TM atom-like. This pic- ture stands in remarkable contrast to the so-called ionic model, which was long believed to be correct for substitutional TM atoms also. In this model, the deep level is strongly localized at the TM atom. Today there exists clear experimental evidence that this is not the case. The ionic model is based on the so-called ligand field theory. This entails an approach to the deep level problem for impurity atoms which differs fundamentally from the one used in this book. Within ligand field theory, it is assumed that an impurity atom X in the crystal has essentially the same electronic structure as the free X"+-ion, where V + means the oxidation state of the impurity atom in the crystal introduced above. The energy levels and wavefunctions of this ion are weakly disturbed by the crystal field at the impurity site, an effect referred to as crystal field splitting. While this model applies relatively well in the case of ionic crystals (where ch lies deeper than c d ) , it evidently fails in the case of the covalent or partially covalent tetrahedral semiconductor crystals. The oxidation states of ligand field theory are, however, also r e produced in the approach taken here. To demonstrat,e this we consider the example of a Co atom substituting the metal atom in a 111-V compound. There are 9 electrons in the 3d74s2 valence shell of Co, and 5 in the valence shell of the group V atom to which the Co atom binds. 8 of these 14 electrons are hosted by bonding states of the deep center (2 by the bonding al-state, and 6 by the bonding t 2 state). S i x electrons remain for the population of the deep impurity states, i.e. the non-bonding e-states and the anti-bonding t2-states. Since 9 electrons are expected at a neutral Co atom, the oxidation state of Co in a 111-V compound is Co3+. Substitutional Co in a 11-VI com- pound has the oxidation state Co2+ since one additional valence electron is provided by the group VI atom of the host crystal.

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3.5. Deep levels 325

3.0

I

2 55 - 2.0

LT W z w

1 .o

0.0

Figure 3.19: Calculated inultiplet structure for substitutional 3d impurity atoms in ZnS. [After Faatio, Caldas, and Zungcr, 2984)

Ligand field theory accounts for the fine structure of the deep impurity lev- els, which is not explainrd by the approach taken above. Although this approach provides qualitative understanding of deep 3d-TM centers in tetra- hedral semiconductors, the excitation energies it yields are not correct quan- titatively, particularly not for the internal transitions of the 3d-TM centers. To be quantitatively correct, this approach must be refined by including many-body effects, in particular, configuration interaction. While the latter effect does not play an important role for main group impurity atoms, as we have scen above, it becomes important for T M impurity atoms with their strongly localized open d-shells (as was specultited in section 3.5.4 using the analogy with frcc I'M atoms). Figure 3.19 demonstrates this in thc case of TM atoms in ZnS.

The differences between the donor or acceptor ioniaation energies of a particular TM atom in different host crystals exhibit interesting behavior. Experimentally one finds that these diffeiences are almost independent of the atom considered. Therefore the ' ionization-encrgy-versus-'I'M atom curves' of different semiconductors are parallel to each other and ran be translated to overlay by rigid displacements along the energy axis (Caldas, Fazzio, Zunger, 1984; Langer, Heinrich, 1985). In Figure 3.20 this fact is shown for acceptor

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326 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Figure 3.20: (a) Experimental acceptor ionization energies of 3d-TM impurities in 111-V semiconductors. (b) Experimental donor (open symbols) and acceptor (filled symbols) ionization energies of 3d-TM atoms in IFVI semicoIiducton. (After Langer and Heinrich, 1985.)

and donor ionization levels of a 3d-TM impurity atom in a series of I11 V and 11-VT compound semiconductors. The rigid displacement along the energy axis has an important physical meaning as we will discuss further below. Here, we will provide a simple explanation for the similarity of the 'ioni- zation-energy-versus-TM atom' curves in different semiconductors (Tersoff, Harrison, 1986). In this discussion we will use the defect molecule model of 3d-TM impurity atoms presented above, in addition, however, the depen- dence of the energy levels f d and Eh on their population will be taken into account in a self-consistent way. Let n d , n h and n:, n: be the electron num- bers in, respectively, the atomic d- and sp3-hybrid orbitals, the former after charge transfer between the impurity atom and the crystal has taken place, and the latter before. The dependence of the levels Ed and ~h on population may, in the linear approximation of equation (3.25), be written as

Ed = td 0 + U d ( n d - r i d ) , 0 Eh = €1, 0 tJh(nh - nh). 0 (3.178)

Here c:, E: denote the energy levels without charge transfer, and [Jd, Uh are, respectively, the Hubbsrd energies of the d- and the h-states of the sp3- hybrids. The electron numbers in the bonding and anti-bonding ta-states will, respectively, be denoted by rib and n,, and the electron number for the estates by ne. Of the total number of electrons of the defect molecule,

n d = nbu! 2 + nap2 i- n, (3.179)

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3.5. Deep levels

are in d-states, and

327

n h = n b P 2 +,,a2 (3.180)

are in h-states. The bonding T2-level Eb lies in the valence band, its three states are therefore completely occupied and it holds n b = 6. The difference 6 = (Ed - € h ) / 2 between the d- and h-states adjusts self-consistently: if Eh

increases, then 6 becomes small and, with this, also ,B and n h are small. As a consequence of this, n d increases which, according to (3.178), results in an increase of € d and, hence, an increase of 6. Self-consistency is then achieved if the two equations (3.178) for the level positions t d and Eh are satisfied with values n d and n h from equations (3.179) and (3.180), respectively, and values a2, P2 from (3.177). For the level difference 6 = ( € d - ~ h ) / 2 this condition yields

1 6 + u h ,(nu + 6 ) - -(nu 2 - 6)- A - n:)] " (3.18 1)

where 260 = (E! - c i ) denotes the level distance without charge transfer. The crucial point at this juncture is that [ I d , because of strong localization of the &electrons and wcak scrcening of their interaction, has very large values (on the order of magnitude 10 e V ) . Therefore, the factor multiplying U d in equation (3.181) must practically vanish itself in order to satisfy this equation, leading to the approximate rclation

(3.182) 1 1 6 0 i ( ~ a + 6 ) + s(nu - 6 ) - A 1 ne - n d w 0,

which uniquely determines 6. According to this relation, the d-level adjusts in the crystal in such a way that the charge transfer between the crystal and the d-shell practically vanishes (of course, it cannot vanish completely be- cause adjustrrierii of the levels requiies a bit of rhargc to bc transferred). This adjustment is called the 'neutrality level'. According to equation (3.181), the self-consistent value of h and, thus, also the energy separation tId-€h = 26 be- tween the d- and < h level, is indepmrlent of Ch. Since t h is the only quantity which changes substantially in the series of the 111-V and 11-VI-cornpound semiconductors, the separation between Ed and Eh is the same in different semiconductors. This also holds for the anti-bonding tza-level dnd for thc non-bonding e-level, i.e. for the two levels which are candidatev fol deep levels in the gap - also, their clihtance from c h is independent of Eh. For the ta,-level this follows from equation (3.174), and for the e-level from the fact that its position is determined by the location of the d-level in the c ~ y s - tal. We thus arrive at the conclusion that the deep levels of 'I'M atoms in

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328 Chapter 3. Electronic structure of semiconductor crystals wilh perturbations

the gap are ‘pinned’ at the hybrid level F ? ~ of the surrounding host anions. If the coupling hetwccn the anion-hybrids and the rest-of-crystal is taken into account, the anion-hybrid energy Eh has to be replaced by the average cation-anion hybrid energy of the crystal.

The rebUlt8 above explain lhe experimental finding that the ‘deep level- versus-Thl atom-curves’ for two different semiconductors ran be tranbleted to overlay each other by rigid displacements. Moreover. they associate the magnitude of the displa~ements with the difference between the average hy- brid energies of the two semiconductors. Thls quantity has a close relation to the valence band discontinuity at a semiconductor heterostnicture. as will be discussed iu more delail in section 3.8

The large Hubbard energies of the atomic d-states of the ‘I’M impurity atoms and thFir pinning at lhc dangling hybrid lev& of the surrounding atoms are also contributo~y to the striking phenomenology noted above. in that the deep levels of TM atom8 depend only relatively weakly on their population, so that levels of several charge states can fit in the gap. In fact, if an additional electron is put in the deep tzo-lwel, and, with this, also part ofit IS added to the TM atom, the d-level of the latter will be shifted up and the bondmg tB-level will be depolarized ( a becoming smaller and /3 larger in equtliiori (3.177)). This means that electron charge density will flow from the Thl atom to the surrounding host atoms. in almost the same measure as was added t o the TM atom when the additional electron was placed in the deep ant i-bonding &-level. Since the Hubbard energies of the s- and p-shells of the host atoms are substantially smaller than those of the &orbitals of the Thl atoms, the deep TM-levels are shifted up only by several tenths of an e V rather than 10 el, for the case of the free TM-ion if one more electron is p la~ed at the center (Haldan, Anderson, 1976).

Cu, Ag, Au in Si

These three elements (abbreviated below as ‘Nhl‘, for noble metal) play an important part in silicon device technology. As impurity atoms, they constitute effective capture centers for non-equilibrium charge carriers. Their high solubility in Si is remarkable (in the region of ~ r n - ~ at melting temperature), as is their fast diffusion. Their incorporation in the host crystal is predominantly substitutional. The elements tend to form complexes with other elements. for example, Au with 0. Fe or 3d-TM’s. This makes it somewhat difficult to identity the pure substitutional NM centers.

Although the elements Cu, Ag and Au belong to the fast main group of the periodic table, their behavior its impurity atoms in tetrahedral semicun- ductors resembles that of transition metals. This is understandable if one considers that for these elements only the outer s-shell, but not the outer p-shell, is occupied, and that the closed d-sheIl (Cu, Ag) or d- and f-shells

cmd3 to

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3.5. Deep levels 32 9

(Auj lie energetically relatively shallow below the s- and p-shells. According to calculations by Fazzio, Caldas and Zunger (1985). the anti-bonding t 2 - state lies in the energy gap, just as in the case of transition metals, and as in the latter case, this state arises from the interaction of the impurity-derived d-states (more exactly their tg-component) with the tg-components of the dangling bonds of the surrounding host atoms. Because of the relatively weak localization of this deep impurity state, its Hubbard energy is small so that. again. several ionization levels can fit within the gap. For all three K M atoms, the calculations result in a Nh/l(O/+) donor level, and a NM(-/O) acceptor level. In the case of Ag and Au, the amphoteric character has also been conhmed experimentally.

Rare earth atoms

The rare earth (RE) atoms are among the following elements: Ce, Pr, Nd, Pm, Sm, Eu. Ac. Th. Pa, U. Np, Pu, Am, and Cm. The valence shell config- uration of the majority of these atoms may be described as 4fn6s2 with the number n of f-electrons varying from 2 for Ce to 14 for Yb (see Table 3.3). Exceptions are Gd and Tb which have one 5d valence electron in addition. The investigation of RE impurities in Si as well as in the 111-V and 11-VI semiconductors has received new stimulus very recently. The reason for this is the luminescence of RE impurity atoms in these crystals in the technologi- cally interesting visible and infrared spectral regions. In equilibrium the RE atoms are installed both substitutionally as well as interstitially. The equi- librium solubility of the RE atoms is, however; rather small. For practical applications, non-equilibrium incorporation techniques like ion implant ation must be used, although, generally, only parts of the implanted atoms are optically or electrically active. For Erbium (Er), impurity concentrations of about lo1’ ~ r n - ~ have been reported in Si, and of about 10l8 C W I - ~ in Ga4s.

The technologically interesting luminescence discussed above is caused by internal transitions within the 4f shell of the RE atoms, more strictly speaking, between the various levels arising from the many-particle levels of the 4f-shell under the influence of the crystal field. The 4f-orbitals remain, in fact. almost unchanged by installation of the RE atoms in the crystal. This may be traced back to the fact that these orbitals are strongly localized and, moreover, strongly shielded by the 6s-electrons shutting out influences of the surrounding crystal. Furthermore, for RE elements heavier than Nd, there is also strong shielding by the completely occupied 5s- and 5p-shells lying outside the 4f-shells for these elements. While internal electron transitions within the 4f-shell have been studied already for a long time, transitions to energy levels outside of the 4f-shell, including donor and acceptor transitions into conduction and valence band states of the crystal, have been subjected to more detailed investigation only recently.

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330 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Ce Pr Nd F'm Sm Eu Gd Tb Dy Ho Er Tm Yb 8 ! ' I ' I j 1 1 1 I " ,

6 4

2

5 0

P -2 g - 4

v

-6

-8

-10 -12

k E(3+/4+) I

1 I I

, I ! I I / I I , ! I I Ce Pr Nd Rn Sm Eu Gd Tb Dy Ho Er Tm Yb

Figure 3.21: Donor ionization levels of substitutional rare earth impurity atoms in different semiconductors. (After Delerue and Lannoo, 1994.)

A rough but qualitatively correct picture of the electronic structure of sub- stitutional RE impurity atoms in tetrahedral semiconductors may again be obtained by means of a defect molecule model (Delerue and Lannoo, 1991). In this model, the 4f-orbitals neither interact with the 5d- and 6s-orbitals of the RE atom nor do they couple with the orbitals of the surrounding host atoms. The RE atom is represented by its 5d- and 6s-orbitals only. The 4f- orbitals, nevertheless, are involved in forming deep states because electrons are transferred to or from them. The host crystal is described by the four sp3-hybrid orbitals at the four neighboring atoms pointing towards the RE atom. The model is therefore largely analogous to that of 3d-TM atoms de- scribed above: The five 5d-orbitals decompose into two e-orbitals and three tz-orbitals, and the 6s-orbital becomes a al-orbital under the influence of the tetrahedrally symmetric crystal field. The four sp3-hybrid orbitals at the neighboring atoms pointing toward the RE atom split into one al- and three t2-orbitals. The interaction between the two al-orbitals results in a bonding and an anti-bonding al-state lying deep in the valence and conduc- tion bands, respectively. The e-states remain without bonding to the host crystal, while the host and RE derived t2-states couple to each other forming bonding and anti-bonding tz-states. The differences between 3d-TM and RE impurities are essentially of a quantitative nature. The 5d-levels of the RE atoms are higher in energy and therefore closer to the hybrid levels at the neighboring atoms than are the 3d-levels in the 3d-TM case. Because of this, the e-level and the anti-bonding tz-level of the RE atom (which may lie in the gap in the case of 3d-TM atoms) are lifted into the conduction band.

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3.5. Deep levels 33 1

The position of the If-shell with respect to the above levels still has to be determined. In the model by Delerue and Lannoo this is done assuming that the distance between the 4f-shell and the average 5d-level in the crystal is the same as that between the 4f-shell and the 5d-shell in the free RE atom.

The ionization levels of the 4J-shell calculated by means of the above defect molecule model are shown in Figure 3.21 for various RE atoms and host crystals. Two diffcrcnt oxidation states of the RE atom, RE3+ and RE2+, have been assumed. One recognizes that, for InP, the oxidation state RE3+ is stabk in most rasrs, because the RE(3-t /4 t ) ionization levels are still below the Fermi level, while the RE(2 t /3+) ionization levels are above it. For CdTe, the oxidation state RE2+ is found to be stable in most rases. With a few exceptions, these predictions are confirmed by experimental in- vestigations. Oxidation states can also be derived in a more direct way from the above defect molecule model. In a 111-V compound host crystal, the RE defect molecule has 2 + n + 5 electrons, 2 + n of them from the HE atom, and 5 of them from the host crystal anion. These electrons orrupy either states in the 4f-shell, or are in bonds between the RE atom and the crystal. The bonding al- and t2-states host 2 f 6 = 8 electrons provided that they are energetically lower than the f-shell, which in fact turns out to be the case. Thus, 2 + n + 5 - 8 = JZ - 1 electrons remain for the f-shell. The oxidation state of a neutral RE atom is RE3+ in such circumstances, because 3 elec- trons are missing at the RE atom, the one f -electron and the two s-electrons in bonds with the host crystal. In 11-VI compounds, the oxidation state of a neutral RE atom should be RE2+ according to this consideration, since the host crystal anion delivers 6 electrons instead of 5.

A striking feature of the ‘ionization-energy-versus-RE-atom-curves’ for different semiconductors in Figure 3.21 is that they can be brought to over- lie each other by rigid relative displacements. This is reminiscent of the ionization energies of the 3d-TM atoms where the same feature is observed. The reason is similar to that of the 3d-TM case: one the one hand, the 4 f -levels are pinned electrostatically to the average 5d-levels by means of the self-consistent charge exchange between the d- and the f-shell with its large Hubbard energy, and on the other hand, the 5d-levels are pinned at the average hybrid energy levels.

As-antisite defect in GaAs (GaAs: AsG~)

The interest in this defect arises mainly from the fact that it is closely con- nected with the so-called EL2-center which is one of the most common point perturbations in GaAs. The EL2-center is a double donor and plays an im- portant role in GaAs electronics. Through deliberate use of its properties, p-type GaAs, which normally arises in crystal growth, can be transformed into semi-insulating GaAs which is required for the manufacture of GaAs

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332 Chapter 3. Electronic structure of semiconductor crystals with perturbations

MESFET’s and other electronic devices based on GaAs. However, in regard to materials for light emitting optoelectronic devices, the concentration of EL2-centers has to be kept as small as possible because EL2 acts to quench luminescence. Whether the As-antisite defect GaAs: ASG, is really identical with the EL2-center, or if it is only part of a complex which represents the EL2-center, is still somewhat controversial today. Nevertheless, important properties of the EL2-center , in particular its structural metastability, can be explained in terms of the simple GaAs:Asc, antisite defect alone (Chadi, 1992, Dabrowski, Scheffler, 1992).

The GaAs:AsG, antisite defect may be viewed as a special case of a substitutional main group impurity atom with sp3-bonding in a tetrahedral semiconductor. For the latter point perturbation, the defect molecule model was developed in subsection 3.5.3. Accordingly, one has a bonding and an anti-bonding A1-level, and a bonding and an anti-bonding T2-level. From ab initio calculations it is known that the bonding levels lie within the valence band, the anti-bonding A1-level in the gap, and the anti-bonding T2-level in the conduction band. Of the 10 electrons of the defect molecule, 8 can be placed into the two bonding levels within the valence band. The remaining two electrons just suffice to fill the anti-bonding A1-level in the gap. These are the two electrons which, by exciting the crystal, are transferred to the conduction band giving rise to the doubledonor nature of the center. How- ever, the population of the two anti-bonding states is energetically costly. Therefore it is not surprising that for the As atom other positions than the substitutional one may be more favorable for minimization of the total en ergy. In the above mentioned ab initio calculations, it was shown that a displacement of the As atom by about I A in [Ill]-direction, a displacement away of one of the 4 nearest neighbor A s atoms towards an interstitial po- sition about 0.2 A below the plane spanned by the other three As atoms (see Figure 3.22), wsiilts in a relative totdl energy minirrium which lies only 0.25 eV above the absolute minimum of the substitutional As-site. In ad- dition to the stable substitutional state of the defect, there exists, there fore, a metastable interstitial state. Thc stable state ia separated from the metastable by an energy barrier of about 0.8 eV, and in the reverse direction the barrier amounts to about 0.34 eV. If onr of the two donor electrons at the substitutional center is optically excited, the barrier decreases siibstan- tially and a thermal transition into the metastable state is possible. In this state the center is not capable of capturing the electron whir11 was previously optically excited into the conduction baud. This electron remains there re- sulting in the experimentally observed persistent photoconductivity. Only by heating the sample above toom tempwature can the As atom return to Its stable substitutional site.

The main reason for the relatively small energy difference between the substitutional and interstitial locations of the As atom iu that, between an

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3.5. Deep levels 333

Direction

Figure 3.22: Geometry of the stable and metastable state of the As-antisitc defect in GaAs. (After Dabrowski and SchrJ’ler, 1992.)

interstitial As atom and Ohe three adjacent, A s atoms, st,rong sy2-bonds can be formed similar to the bonds in the graphite structure of carbon and that of g a y As. The non-bonding p-orbital of the A s atom, which lics rcla- tively high enmgetically, does not increase t,he total energy of thc interstif,ial s h t e because it remains largely unoccupied, the two remaining electrons are hosted by the dangling bond of t,hc fourth remotc lying As atom.

DX-centers

A metastability similar to that of the C:aAs:AsG,, ailtisite defect is also ob- served at ot,hcr point perturbations, among them at t,he so-called L)X-cent,ers in GaAs and (Ga,Al)As mixed crystals (Lang, Logan, 1977). The micro- scopic nature of these centers was a puzzle for a long time. It was only clear that they were related to impurity atoms or the main groups lV and VI of the periodic table which normally are incorporaled subst,itiitionally on cation and anion sitcs, respectively, and form shallow donors. lTndcr cer- tain conditions hydrostatic pressure or strong Ti-type doping in the case of GaAs, and AlAs mole fractions larger than 22% in the case of (Al, Ga)As - deep levels emerge from these shallow donors. Originally this transition was attributed to the formation of a defect complex involving the donor atom (a) and ti11 iinknown point, perturballon (X). Now, it is clear t,hat the D X - center j u s t represents another state ol 1he shallow donor. In contrast to the lat,t,er, the donor atom of a DX-ccntcr is not neutral, but singly negatively charged, and its installation does not occur at a substitutional site but at an interstitial site. If the special Conditions mentioned above are not fuliilled, t,hen the shallow donor represents a stable state of the impurity atom, the DX-state is metastbble. However, as for the EL2-center, thc rclalive total energy minimum of the DX-state lies only slightly (about 0.2 e V ) above thc absolute minimum of the shallow donor state. The reason for this en- ergy balance is the sainc as in the E L 2 case, namely the energetically costly populalion of the anti-bonding A1-level at the shallow donor. In the singly negatively charged state which the donor takes under high doping, (and ev-

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334 Chapter 3. Electronic structure of semiconductor crystals with perturbations

idently also under the other above-cited conditions), 2 electrons must be hosted by this level. The energy costs thereby become so high that the DX- state is energetically more favorable for the impurity atom than the shallow donor state. The donor atom - to be concrete we will identify it below with a Si atom - moves into the interstitial position shown in Figure 3.22, where it can enter into sp2-bonding with the 3 surrounding As atoms. The non- bonding porbital of the Si atom remains largely unoccupied, instead the more deeply lying dangling hybrid orbital of the fourth more remote lying As atom becomes occupied (Scherz, Scheffler, 1993).

This model explains in a natural way why the DX-center exhibits persis- tent photoconductivity. By optically exciting an electron at the DX-center into the conduction band, the center passes into its neutral charge state. This means that the substitutional shallow donor is a stable state of the point perturbation. Therefore, optical excitation transfers the DX-center into a shallow donor in its ground state, and the donor is not able to cap- ture the excited electron from the conduction band. It remains there for a longer time and gives rise to persistent photoconducting. Only after thermal excitation does the center return to its original metastable state.

The structural metastability observed at the DX- and EL2-centers is not restricted to these point perturbations, but represents a relatively com- mon phenomenon in tetrahedral semiconductors. It arises, evidently, from the fact that (depending on the number of electrons to be placed at the center), either the sp3- or the sp2-hybridization of the s- and p-orbitals of the impurity atom allow for a lower total energy of the perturbation center, jointly with the fact that the two kinds of hybridizations lead to different atomic structures - the sp3-hybrids to the tetrahedral diamond structure, and the sp2- orbitals to the graphite structure. Whether or not a metastable state actually exists for a particular point perturbation can, however, only be decided by ab initio calculations.

3.6 Clean semiconductor surfaces

3.6.1

Every solid is bounded by a surface. Nonetheless, the model of an infinite solid which neglects the presence of a surface works very well in many cases. Why is this possible? The reason is, first, that in many cases one deals with properties, such as transport, optical, magnetic, mechanical or thermal properties, to which all the atoms of the solid contribute more or less to the same extent, and, secondly, that there are many, many more atoms in the bulk of a solid than at its surface, provided the solid is of macroscopic size. In the case of a silicon cube of 1 cm x 1 cm x 1 cm, for example,

The concept of clean surfaces

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3.6. Clean semiconductor surfaces 335

one has 5 x loz2 bulk atoms and only 4 x 1015 surface atoms. Beside the above mentioned bulk properties, there are other properties or processes, like crystal growth, oxidation, etching or catalysis, which are determined by the surface atoms only. In these rases, the model of an infinite solid does not apply, of course. Moreover, in semiconductors, bulk properties like transport are often not controlled by all atoms but only by dopant atoms. Then the number Ns of surface atoms per cni’ i s to be compared with the number of dopant atoms. For a semiconductor sample of thickness d , area 1 cm2, and doping concentration N D , the number N o x d of dopant atoms per c7n2

can easily become as small as the number IV, of surface atoms. This means that the transport properties of such a semiconductor sample will depend on its surface. In the history of semiconductor physics, this was recognized at a very early stage. Examples which demonstrate this are the electric rectification at a semiconductor-metal contact (discovered by F. Braun in 1874), and the unsuccessful attempts to build a field effect transistor in the late thirties of this century. The failure was caused by electron states localized at the surface which captured all the electrons induced by the extcrnal voltage.

‘l’he surfaces used in the early field effect experiments were far from perfect. They were made by cutting or cleaving a semiconductor sample in air. If at all, they were clean and smooth in a macroscopic sense, but not so microscopically. They exhibited surface roughness on the 100 nm scale, structural defwts on the 1 nm scale, and impurity atoms at and below the surface. The surface states responsible for the difficulties of the early field effect transistor were due to these imperfections. In the present section we will deal with surfaces which are free of such imperfections. Perfect surfaces of semiconductor crystals in this sense necessarily represent a particular lattice plane occupied only by chemically ‘correct’ atoms at regular sites. No impurity atoms are allowed above or below the surface. The surface is reduced to what it means ideally, the termination of the crystal.

Perfwt surfaces in this sense cannot, of course, be realized in practice. One may only try to approximate them so closely that the existing imper- fections do not essentially change the properties of the surface as compared to the properties which a perfect surface would have, if it really could be made. One calls such almost perfect surfaces clean. Although this term refers only to chemical composition, it also implies structural perfection or atonzc smoothness. There are essentially th rw ways to manufactwe clean surfaces. All thrw need ultrahigh vacuum (UHV), i.e. pressures below Torr:

(i) Treatment of imperfect surfaces by ion bombardment and thermal an- nealing (generally in several cycles).

(ii) Cleavage under UHV conditions (only surfaces which are cleavagc planes

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336 Chapter 3. Electronic structure of semiconductor crystals wilh perturbations

of the crystal can be made in this way, of course).

(iii) Epitaxial growth of crystal layers by means of molecular beam epitaxy (MUE, see section 3.7).

Although most surfaces used in practice are not clean but imperfect, studies of clean surfaces are also of practical importance. This is due to the fact that imperfect surfaces are such complex subjects that it is very difficult to approach them directly. One first has to reduce the level of COEApleXity by considering clean surfaces for investigation. Later, the complexities are introduced step by step and examined for changes. In this way, real imperfect surfaces may also be undeistood. Clean surfaces are, howevei, important themselves. It was already pointed out that these are the surfaces upon which epitaxial crystal growth proceeds in MUE.

In this section, we will deal with the atomic and electronic structure of clean semiconductor surfaces. The basic principles will be treated in subsection 3.6.2 for atomic structure and in subsection 3.6.3 for electronic structure. Particular suxfuces are discussed in subsection 3.6.4 taking Si and GaAs as examples.

3.6.2

A geometrical construction which is of special significance in describing crys- tal surfaces is that of lattice planes, which we will now describe.

Atomic structure of clean surfaces

Lattice planes of 3D crystals

Such planes are usually denoted by Miller indices ( h k l ) where h, k , 1 are the integer reciprocal axis intervals given by the intersections of the lattice planes with the three crystallographic axes. The symbol (loo), for example, dcnotes lattice planes perpendicular to the cubic x-axis, (111) means lattice plunes perpendicular to the space diagonal in the first octant of the cubic unit cell, and (110) denotes lattice planes perpendicular to the face diagonal in the first quadrant of the ry-planc of the cubic unit cell. In the case of trigonal and hexagonal lattices, four crystallographic axes are considered (three instead of two perpendicular to the c-axis). The lattice planes then are characterized by four indices ( h k i l ) instead of three. The first three, however, are not independent of each other, since i + h + k = 0. The (hki l ) are sometimes termed Bravais ~ T L ~ ~ C C R .

A particular geometrical plane can also be characterized by its normal direction. To define lattice planes in this manner, it is convenient to write the normal direction as a linear combination of the primitive vectors bl, b2, b3 of the reciprocal lattice introduced in scction 2.4, with integer coefficients h l , I k 2 , jL3,

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3.6. Clean semiconductor surfaces 337

n - h l b l + h 2 b z + h ~ b 3 . (3.182)

Hrre, reciprocal lattice vectors are understood without the factor 27~ in def- inition (2.122). If the normal direction n is given, the roefkients ( h l h ~ h 3 ) are only determined up to a common integer factor. We chose this factor such that h l , h2 , hs have no common divisor. Then the coeffirients hi, h2, h 3 are the Miller indices of the lattice plane under consideration, however, r e ferring to the primitive lattice vectors al, a 2 , a 3 as coordinate axes rather than to the crystallographic axes. The latter are parallel to the piimitive lattice vectors only in the case of primitive Bravais lattices. For centered laltires, like the face rentrred cubic one, they have different directions, and the coefficienls ( h l h ~ h 3 ) differ from the corrirnon Miller indices. Tf necessary, one can rasily switrh from one representation to the other. The ( h l h 2 h 3 ) are, apart from a co~linion factor, obtained by multiplying the ( h k l ) with the matrix which transforms the three nun-primitive crystallographic axis vec- tors into thc three primitive lattice vectors. Using the above characterization of lattice planes by their normal direction n, the lattice points

Ro = rloa1-k r20az + '7'30ag (3.183)

of the lattice plane perpendicular to n which contains the zero point, may be defined by the equation

n & E h17.10 + h2r20 + h3r30 = 0. (3.184)

The unique character of this equation lies in the fact that only integer so- lutions r10,r2O1r3o are allomwl. In mathematics it is called a Dzophantin equation.

Whereas the Miller indices definr an infinite family of parallel lattice planes, equation (3.184) yields only a single plane, namely that member of the infinite family which contains the zero point. One can show that all lattice planes

Ri = wal+ v z a 2 + w a ~ (3.185)

of the infinite family of parallel plancs are obtained by replacing the right hand side of quation (3.181) by arbitrary integers 1 ,

(3.186)

The points of the lattice plane defined by equation (3.184) form a 2D lattice. The primitive vectors of this lattice will be denoted by fi and f2. This i s

done in such a way that the three vectors f1, f2, n form a right hand coor- dinate system. The f1 , f2 may be expressed in terms of the primitive lattice

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338 Chapter 3. Electronic structure of semiconductor cryst& with perturbations

Table 3.8: Primitive surface lattice \rectors and stacking vectors of low index sur- faces of semiconductor crystals.

n Crpst'al Structures

I Diamond, Zincblende, Rocksalt. I1

WurtBite, Selenium 1 vectors al ,az,q of the crystal under consideration. In order to determine the coefficients of this representation, the Diophantin equation (3.184) has to he so lvd , which ran be done by Incans of the Euclidean algorithm (set, Bechstrdt and Enderlein, 1988j. The piirriitive vectors fi, fz obtained in this way are shown in Table [3.8) for several Iow index lattice planes of the 5 common scrniconductor crystal structures. IJsing thc lattice vectors fi . f2

and ailitrary integers ~ 1 ~ 5 2 , the lattice plane given by relations (3.183) and (3.184) can bc represented as

Ro = S l f i + sgh. (3.187)

The lattice planes Rr defined by equations (3.185 and (3.186) can be written in the form

where f.3 is a vcctor complementing F1 and f 2 to form a s ~ t of primitive lattice vrctors Fi , f2, F3 of the 313 bulk lattice of the crystal. The vector €3 can br detcrinined from thr Diophantin equation (3.186) for 1 ~ 1 in just the same way, i . ~ . by means of the Euclidean algorithm. BS the vectors Fi,fid were obtained above. The choice of $3 is not uniquc. of course, and any vector fi which differs from f 3 hy a vector aithiii the latticr plan? can also be used. We call f 3 the stacking vector because it determines how the lattice planes arc stacked in the crystal.

The considerations above show that the primitive unit cell of a crystal may be chasm as a parallelrpiyed with one of its pair6 of parallel laws par- allel to a gwen lattice plane. This implies that the structure of a crystal

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3.6. Clem semiconductor surfaces 339

Figure 3.23; Construrtion of a crystal from its lattice planes. The lattice points of a given plane are occupicd by identical atoms. Different planes may host different atoms if the crystal has a basis. In the figure, a crystal with o~ily one atom per primitive unit cell is shown.

may be characterized as consisting or parallel lattice planes which are dis- placed with respect to each other and which are each occi ipid hy atoms of a particular species (see Figure 3.23). 'I'o be more specific, the lattice plane Ro is occupied by atoms of species 1, the next plane, displaced by 6 with respect to the first onc, is occupied by atoms of species 2, etc., and the plane displaced by FJ is occupied by atoms of type J . It may happen that two OT more atoms of the basis are located at the same lattice plane. In that case an ntomic layer consists of two or more basis atoms. The lattice plane Ro + 7j completev the constsuction of a crystal slab which, in the vertical direction, encompasses exactly one yrimitivc bulk unit cell. This slab is callcd a p r i m i t i v e crystal slab. A lattice plane occupied by atoms is referred to nb: a t o m i c layer. The second primitive crystal slab begins with a lattice plane occupied by atoms of species 1 and is displaced with respect to the zero-th plane of the first layer by the stacking vector f3 , followed by a plane with atoms of species 2 which is displaced by f3 t ?2 etc., the last plane of the second primitive crystal slab being occupied by atoms of species J and displaced by f3 + T'J, The second primitive crystal slab is followed by another J planes dlvplaced with respect to the first slab by 2f.j instead of f3. The crystal can therefore be thought of as consisting of successive primitive crystal slabs sitiihtcd one above the other, each of which consists of atomic layers containing difkrmt types of atoms and which are laterally displaced with respect to onc another. The location of an individual atom can bc specified by the number 1 of the primitive crystal slab, the numbers j of the atomic sublattice and the integer coordinates s1, s2 witahin thc lattice plane. The position R(j, I , s1, s2) of an atom j in crystal slab 1 can therefore be written as

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340 Chapter 3. Electronic structure of semiconductor crystals with perturbations

(3.189)

The complete set of locations of atoms in an infinite 3D crystal can be obtained by assigning all possible integer values from --oo to +m for 1 , s 1 , s 2 ,

and all integer values from 1 to J for j . The normal direction n, upon which the whole construction of lattice planes is based, is arbitrary in the case of an infinite 3D crystal. Any choice of n yields the same crystal.

We can immediately employ the above representation of an infinite crys- tal in describing a crystal with a surface. The normal direction n is then, however, fixed by the direction perpendicular to the surface. We start with the description of a crystal having an ideal surface. The exact meaning of ‘ideal’ in this context is explained immediately below.

Ideal crystal surfaces

Regular crystal sites

We consider a crystal surface given by a lattice plane perpendicular to n as defined by equation (3.184). A crystal having such a surface may be generated from an infinite crystal of infinite extension by removing all atomic layers above the surface and retaining those below. Since the forces acting on atoms situated beneath a lattice plane in an infinite crystal are also partially due to the atoms located above the plane, we can, in general, expect that the forces acting on atoms in a crystal with a surface should differ from those acting in an infinite crystal. The deviation from the infinite case, however, diminishes with increasing distance of atoms from the surface and we can thus assume that the forces acting on, and hence the positions of, atoms deep inside the crystal bulk are, to a good approximation, the same as those in an infinite crystal. This is, however, not true for atoms situated near the surface, and the forces acting on them are appreciably different, resulting in displacements of atomic positions with respect to those of the infinite crystal. We will discuss these displacements below. Here, we assume that they are not present and, correspondingly, the atoms at and immediately below the surface have the same positions as they would in an infinite crystal. The term ideal surface is used to refer to this configuration. The atoms of a crystal having an ideal surface are thus located at the positions R(j, 1, ~ 1 , s ~ ) given by equation (3.189), however, only positions below the surface plane are occupied. These obey the relation

n . R(j, 1 , s 1 , 5 2 ) = I + n . Tj 5 0. (3.190)

The surface or first atomic layer is obtained if the left hand side of this relation is taken to be zero. A solution of (3.190) is 1 = 0 and = 0.

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3.6. Clean semiconductor surfaces 34 1

The latter condition signifies j = 1 since 71 = 0 is assumed. Thus, the first atomic layer corresponds to the particular lattice plane perpendicular to n which goes through zero and whose lattice points are occupied by basis atoms of species 1. There may be other ?j beside ?I which, although not being zero themselves, have a zero projection n .3 with respect to n. Then the basis atoms of this species j are also located in the first atomic layer. They are shifted with respect to the atoms of species 1 by a vector Tj parallel to the surface. Such multiple-species occupancy of an atomic layer occurs, for instance, in the case of (110) surfaces of diamond and zincblende type crystals. In this case one has two atoms in each primitive unit cell of the 2D lattice of a lattice plane, in the case of Si, for example, Si atoms, and in the case of GaAs, 1 Ga atom and 1 As atom.

The distinguished role of atomic species 1 in the above considerations results from the choice of the origin - it has been placed at an atomic site of species 1. Of course, the coordinate system may be shifted in such a way that its origin coincides with the location of any other basis atom. In each case a different surface is obtained. Even if the basis atoms are chemically identical the surfaces may differ from each other in a topological sense. In the case of diamond type crystals, for instance, two topologically different (111) surfaces exist, one with three nearest neighbors above and one below the surface and another with one above and three below. The latter surface is more stable than the former one and the latter is meant if one refers to a (111) surface. For other crystal structures and surfaces the situation is similar. If topologically different surfaces exist for a given set of Miller indices, generally, one of them is more stable than any other, and that is the one which is commonly realized in experiment and studied theoretically. We first consider the translation symmetry of an ideal surface and the cor- responding lattice.

Translation symmetry and lattices of ideal crystal surfaces

The translation symmetry of a crystal with an ideal surface can be derived from the following observations: (i) Only translations within the surface lattice plane perpendicular to n are admissible. Any translation leading out of this plane would alter the spatial location of the surface and thus would not transform the system ‘crystal with surface’ into itself. The symmetry group of translations and, corre- spondingly, also the lattice of a crystal possessing a surface are, therefore, only 2-dimensional, although the crystal with surface is 3-dimensional in extent. (ii) Only those translations are admissible which transform each atomic layer into itself. The construction of a crystal, layer by layer, as described above implies that if a particular translation transforms the first atomic layer into

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342 Chapter 3. Electronic structure of semiconductor crystals with perturbations

itself, this also holds for any other layer. It immediately follows that the group of translations of a crystal with a surface is identical to the translation group of the first atomic layer. The lattice vectors r of a crystal possessing an ideal surface perpendicular to the normal direction n are therefore those of equation (3.187), so that

r = s l f 1 + s 2 f 2 (3.191)

holds. The vectors f1, f 2 are the primitive lattice vectors of the 2D lattice of the ideal surface. The primitive unit cell of the crystal with surface has the form of a prism bounded above by the parallelogram spanned by f1, f2

and extending to minus infinity in the direction of -f3. For the low-index surfaces of the common semiconductor crystal structures, the three vectors f1, f2, f3 are listed in Table 3.8.

As in the 3D case, 2D lattices may be divided into crystal systems and Bravais lattices according to their symmetry with respect to rotations and reflections. We now consider the possible plane crystal systems and plane Bravais lattices. Their point symmetry elements are necessarily rotations about axes which are perpendicular to the 2D lattice plane, and reflections at lines within the 2D lattice plane. Thus, the point groups are either C, (n) or C,, (nm, nmm). Since the rotation through 180' is always a sym- metry element of a plane lattice, only even n are allowed. The highest value of n may be readily obtained from the derivation of the possible rotation symmetry axes of 3D crystals in Chapter 1. There, n 5 6 was found. Thus, the possible multiplicities of rotation symmetry axis of plane lattices are n = 2,4, and 6.

A lattice which only contains a 2-fold symmetry axis is either a com- pletely general oblique lattice or a rectangular lattice. The point groups of these lattices are, respectively, C 2 (2) and C2, (2mm). Lattices with a 4-fold symmetry axis also possess 4 reflection lines which are rotated through 45' with respect to each other. The point group of such a lattice is therefore C4v

(4mm). Similarly, lattices with a 6-fold axis have 6 reflection lines which meet at an angle of 30'. In this case the point group is c 6 v (6mm). There are thus 4 different plane crystal systems - the oblique with holohedral point group 2, the rectangular with holohedral point group 2mm, the quadratic with holohedral point group 4mm and the hexagonal with holohedral point group 6mm (see Figure 3.24).

The possible plane Bravais lattices are obtained as follows. First, one takes the four primitive lattices with unit cells which are parallelograms having either no particular symmetry, or that of a rectangle, a square or an equilateral hexagon. Then one adds additional points to each of the unit cells of these lattices in such a way that their point symmetries are not lowered. Only one new lattice is obtained in this way, namely the 'body' centered

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3.6. Clean semiconductor surfaces 343

oblique

P

rectangular

P

quadratic

P

hexagonal

P

C

Figure 3.24: The 5 plane Bravais lattices of the 4 plane crystal systems.

rectangular lattice. This lattice cannot be transformed by continuous and symmetry preserving transformations in the primitive rectangular lattice, nor in any other primitive lattice. So it forms an additional Bravais lattice. In all other cases, the addition of points either leads back to the primitive lattice or results in no lattice at all (i.e. it creates a crystal with a basis). We conclude that five plane Bravais lattices can be realized within the framework of the 4 plane crystal systems: only the primitive lattice in the oblique case, the primitive and the centered in the rectangular case, and again only the primitive in the quadratic and hexagonal cases (see Figure 3.24).

With the last statements we have completed the symmetry classification of the plane lattices of crystal surfaces. The lattice types of the ideal low index surfaces of the common semiconductor structures are summarized in Table 3.9. We now turn our attention to the symmetries of crystals with surface as a whole.

Point and space group symmetries of ideal crystal surfaces

We wish to establish, on the one hand, point groups which transfer equivalent directions of a crystal with surface into one another and, on the other hand, space groups which transform the crystal with a surface into itself. In the latter case this implies that not only does the 2D lattice transform into itself, but so do equivalent atoms occupying the primitive unit cells. These atoms are located at positions R(j, I, sir 92) given by equation (3.189). To express the 2D nature of the translation symmetry of a crystal with a surface explicitly, we denote the atomic positions by Rjl(s1, s2) R(j, I , ~ 1 , s ~ ) and write

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344 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Table 3.9: Structural properties of ideal low index surfaces of the five common semiconductor crystal structures. Column 6 and 7 give, respectively, the number of atomic layers of an irreducible crystal slab and the number of basis atoms per layer.

(ioio)

(ioio)

(1120)

where

Bravais Lattice Point Group Space Group Irr. Slab Basis

hexagonal 3m p3ml 6 1

square 2mm p2mm 4 I

p-rectangular 2mm p2mg 2 2

hexagonal 3m p3m1 6 1

square 2mm p2mm 4 I

p-rectangular m p l m l 2 2

hexagonal 3m p3ml 6 1

square 4mm P h m 2 2

prectangular 2mm p2mm 2 2

hexagonal 3m p3ml 4 1

prectangular m p lml 4 2

prectangular m P W 4 4

hexagonal 1 Pl 3 1

p-rectangular 1 Pl 6 1

p-rectangular 2 P21 4 1

represents the basis of the crystal with surface. This basis contains an in- finite number of vectors corresponding to the infinite number of atoms in a primitive unit cell. The point and space group elements of a crystal with surface must leave the surface invariant, i.e. only 2D symmetry groups need to be taken into consideration. Here, as in the case of translation symmetry, we therefore also have the situation that, although a crystal bounded by a surface extends in three dimensions, its point and space groups are only 2-dimensional.

The point groups of directions of crystals with plane surfaces are nec- essarily subgroups of the holohedral point groups of the plane lattices, i.e. subgroups of the point groups 2,2mm, 4mm and 6mm. There are exactly

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3.6. Clean semiconductor surfaces 345

a 0 1

2 2 mm

c3 m

@ ..

(Ii .4 mm

4

@ . . a @ 3m

(I>

6mm

6

3

Figure 3.25: The 10 point groups of equivalent directions of crystals with plane surface.

10 such groups. Their stereograms are shown in Figure 3.25. This implies the existence of 10 different crystal classes which are associated with corre- sponding crystal systems as indicated in Figure 3.25.

The possible plane space groups can be found as follows (see Figure 3.26) . To start with, it is evident that each of the 10 point groups of equivalent directions combined with the corresponding associated lattice gives rise to a space group. The space groups p l , p 2 1 1 , p l m l , p 2 m m , p 4 , p 4 m m , p 3 , p 3 m l , p 6 and p 6 m m originate in this manner. Since the point groups of the rectan- gular crystal system are each associated with two Bravais lattices, primitive and centered, we find two further space groups, c l m l and c 2 m m . In the case of the point group 3 m , there are two different possibilities of positioning the reflection lines relative to the hexagonal lattice vectors, either through the vertices of the equilateral hexagon of the Wigner-Seitz cell, as assumed in the case of p 3 m 1 , or such that ,hey bisect its edges. In the latter case one

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346 Chapter 3. Electronic structure of semiconductor crystals with perturbations

c r y s t a l B r a v a i s o o i n t soace 1 svstem

oblique 2

-

rec- t angular 2 m n

l a t t i c e

LLZ

p-rec- tang ul o r

c-rec- t a n g u l a r

6

I p'g' I

Ip- I

Figure 3.26: The 17 space groups of crystals with plane surface.

has the group p31m as a 13-th space group. One should further note that the point group of directions of a crystal remains unchanged if, in its space group, a glide reflection line is substituted for an ordinary reflection line. One must therefore examine the 13 space groups already established to determine whether the substitution of a reflection line m by a glide reflection line g (i.e. a reflection in m in conjunction with a translation 7' by half of the shortest lattice vector parallel to m ) leads to a new space group. One easily finds that this is not the case for the hexagonal crystal system. In the quadratic crystal system it is possible to substitute a system of glide reflection line for one but not both of the non-equivalent reflection line system. This yields the additional space group p4mg. The additional space groups in the case of the

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3.6. Clean semiconductor surfaces 347

p-square

,-hexa- gonal

2mn

4

- 4mn

- 3

c2mn I p4 I p4mn I

p3m1 I

Figure 3.26: Continuation: The 17 space groups of crystals with plane surface.

primitive rectangular crystal system are p l g l (from plrnl) and p2rng,p2gg (from p2mrn). The centered rectangular and the oblique crystal systems do not give rise to additional space groups. In the case of crystals with plane surface, there are therefore a total of 17 different space groups. Four or them involve glide reflections. i.e. they are non-symmorphic.

The 2D point and space groups of the various surfaces of a given crystal may be derived from the 3D point and space groups of the infinite crys- tal under consideration. They are, in fact, the subgroups of the 3D groups which contain only those symmetry elements which transform lattice planes

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348 Chapter 3. Electronic structure of semiconductor crystak with perturbations

situated parallel to the surface into themselves. More precisely, they con- tain elements which transform directions in a lattice plane into equivalent directions in that plane as far as point groups are concerned, and transform atomic layers located parallel to the surface into themselves if space groups are under consideration. The 2D point and space groups of the three low- est index surfaces of the five common crystal structures are summarized in Table 3.9, columns 4 and 5. The point and space group symmetry of an ideal crystal surface may also be derived from the projection of the crystal onto its surface. In Figure 3.27 such projections are shown for the three low index surfaces of diamond and zincblende type crystals. Atoms from different layers below the surface are illustrated differently with sizes as in- dicated. After some number of atomic layers (6 for (lll), 4 for ( loo) , and 2 doubly occupied layers for ( l lo)) , the projections repeat themselves. A crystal slab which contains the minimum number of atomic layers necessary for completing the projection, is called an irreducible crystal slab. The point and space group symmetries of a crystal with an ideal surface are those of its irreducible crystal slab.

Relaxed and reconstructed surfaces

Surface-induced atomic displacements

In the preceding section the atomic structure of crystal surfaces was con- sidered under the assumption that the atoms of the crystal bounded by a surface occupy the same positions Rjl(s1, s2) as they did in the infinite 3D crystal, if the former is generated from the latter by removing one half of it. As already noted, this assumption is actually not valid. The atoms of the surface layer experience different forces than those acting in the bulk of the crystal, and are thus subjected to displacements from their original sites in the crystal bulk. Since the forces acting on the atoms of the second layer are in part determined by the positions of the atoms in the first, these forces are also subjected to changes accompanied by displacements in the second layer and so on for each successive layer. All one can assume is that the displacements decrease from one layer to the next and vanish altogether at a depth that is relatively far from the surface. Here, we present a more detailed description of the surfaceinduced displacements of atoms and dis- cuss the resulting altered symmetries as compared to those of ideal crystal surfaces.

Translation symmetry of relaxed and reconstructed surfaces

We denote the displacements of atoms due to the formation of the surface by 6Rjl(sl, 4, and the new positions of atoms in the crystal with surface

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3.6. Clean semiconductor surfaces 349

(1111

(1101 coiii

(111) (100) (110)

0 1 A 0 1 A 0 1 A O 2 B 0 2 6 0 1 6 0 3 A 0 3 A 2 A

Figure 3.27: Projections of a diamond or zincblende type crystal onto its low index surfaces.

by R’j~(s1, sz). Then

6Rjl(sl, s2) 4 0 f o r I -+ -m. (3.195)

The displacements 6Rjl may be divided into two classes with regard to their effect on translation symmetry. If the latter is not affected, the displacements are termed r-e-eluzation. In this case equivalent atoms in different primitive unit cells are displaced in the same way, i.e.

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350 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Figure 3.28: Surface relaxation (a) and surface reconstruction (b) extending up to the second atomic layer. A 2 x 2 reconstruction is shown in part (b).

BRjl(s1, s2) = 6Tji‘jl for all 51, s2. (3.196)

Only the vectors 61 of the basis are altered, the lattice vectors remain un- changed in this case (see Figure 3.28a). If, on the other hand, the translation symmetry is altered, the displacements are termed reconstruction. In this case equivalent atoms in different unit cells are not all displaced in the same manner, i.e. bRjl(s1, s2) depends on sl, s2. Both the basis and the lattice are changed. We first consider the changed lattice translation symmetry of reconstructed surfaces (see Figure 3.28b).

In describing reconstruction it is useful to divide the crystal with a surface into two slabs parallel to the surface, an upper slab containing the atomic layers with displaced atoms, and a lower slab encompassing all other layers, i.e. layers with non-displaced atoms. The upper slab is sometimes called a ‘selvedge’, here we use the terms ‘surface slab’ and ‘bulk slab’ (or simply ‘bulk’) for the upper and lower slabs, respectively. By Ts and Tb we denote, respectively, the plane translation symmetry groups of the surface and bulk slabs. The translations which transform the crystal with surface into itself must belong to both groups of translations, Ts as well as Tb. The translation group T of the whole crystal with surface is thus the intersection

T = Ts fl Tb (3.19 7)

of Ts and Tb. Alternatively, one can say that T is the largest common subgroup of both groups Ts and Tb. There are two possibilities, either T only consists of the identity translation, which means that the lattices defined by Ts and Tb are non-commensurate and the crystal with surface does not possess any lattice translation symmetry, or T contains more elements than just the identity, in which case one says that the two lattices derived from Ts and Tb are commensurate. The lattice associated with the T is called the coincidence lattice. If, in particular, Ts is a subgroup of Tb then T is equal

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3.6. Clean semiconductor surfaces 35 1

to T,, i.e. the coincidence lattice is identical to the lattice T, of the surface slab. If T, is not a subgroup of Tb then T cannot be equal to T, and is necessarily a proper subgroup of T,, i.e. it is smaller than T,. One can thus distinguish between the following three cases with regard to the translation symmetry of crystals having a reconstructed surface:

(i) no translation symmetry,

(ii) translation symmetry exists but is smaller than that of the surface slab,

(iii) translation symmetry exists and is the same as that of the surface slab.

If one assumes that, among the various conceivable surface reconstructions with a given degree of translation symmetry, that particular reconstruction will take place which allows for maximum translation symmetry of the crystal with surface (the system ‘surface slab plus bulk slab’), then only the third of the above possibilities can be realized. A formal proof of this assumption does not exist, and it is probably not valid without exceptions, however, as a rule, it generally yields correct results.

Using the above conclusions we are able to determine the primitive lattice vectors of the reconstructed surface in terms of the primitive lattice vectors f1 , f 2 of the ideal surface. The latter are, by definition, also the primitive lattice vectors of the bulk slab of the crystal with surface. Let be fi and fi the primitive lattice vectors of the reconstructed surface slab. They may be linearly composed of f1 , f 2 according to

f: = Qllfl + 912f2

f; = Q2lfl + 922f2.

(3.198)

(3.199)

Here, the coefficients Q i k , i, k = 1,2, of the transformation matrix Q are, at the outset, arbitrary real numbers. They need to be rational if and only if the two lattices derived from f i , f 2 and f ’ l , f ’2 have a coincidence lattice, i.e. in case (ii) above. If the coincidence lattice is identical to the lattice derived from fi, fi, i.e. in the particularly interesting case (iii), it follows that the Qik must have integer values. In this case, the lattice derived from f : , f i

is simultaneously the lattice of the reconstructed surface. We thus arrive at the important conclusion that in the only case of practical interest (iii) above, surface reconstruction can be described by a 2 x 2 matrix with integer elements.

The two most common forms of this type of reconstruction have a special notation (Wood notation): (1) The non-diagonal elements vanish, i.e. f: and fi are parallel to f 1 , f 2 , respectively, and their lengths are integer multiples of the respective lengths of the latter. We thus have

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352 Chapter 3. Electronic structure of semiconductor crystak with perturbations

f: = nfi, f& = mf2 , (3.200)

with n and m being integers. The primitive unit cell of the surface slab con- tains n x m primitive unit cells of the bulk slab. This is said to constitute an n x m reconstruction An n x m reconstructed crystal surface of a particular material C parallel to a lattice plane with Miller indices (hlcl) (or ( h k i l ) in the case of hexagonal symmetry) is characterized by the symbolic notation

C(hlc1) n x m. (3.201)

(2) The off-diagonal elements of Q are not equal zero, i.e. fi is not parallel to fi and/or fi is not parallel to f2. The angles L(f!, fi) and L(f& fz), which are in general not equal to each other, are assumed to be equal in the case under consideration. This means that the two vectors f1, f2 (with tails joined at the same point) can be transformed into the two corresponding vectors f;, fi by a rotation through the same angle L(f;, fi) = L(f4, f2) = a about an axis which is perpendicular to the surface, with a subsequent rescaling of f1, f2 by the factors lfiI/fll and ~ f ~ ~ / ~ f ~ ~ , respectively. A (hlcl) surface of a particular material C reconstructed in this way is characterized by the symbolic notation

lfil 141 C(hk l ) - x - - a. lfil If21

(3.202)

The factors lf~l/lf~l and lfil/lf21 are in general irrational in contrast to the qik, which in the case considered here are integers. Examples of both of the special reconstruction forms discussed, as well as for the general reconstruc- tion form in case (iii), are shown in Figure 3.29.

Sometimes, in the notation (3.201), n x m is replaced by p - n x m or c - n x m. The lattice vectors fi = nfi and fi = mfz are then not necessarily primitive as originally assumed in the notation (3.201), and in addition to primitive ( p ) reconstructed surface lattices, also centered ( c ) ones are possible. This can only take place, however, for rectangular surface lattices. Thus, the modified notation applies only to this case, although it is also sometimes used (formally incorrectly) for square lattices. In the rectangular case the notation c - n x m describes a type of reconstruction which is not covered by one of the two notations (3.201), (3.202), and which can otherwise only be characterized by the 2 x 2 matrix Q itself. For square reconstructed lattices the c - n x m notation is just a simpler description of a reconstruction of type (3.202).

The point symmetry of a reconstructed surface lattice is generally lower than that of the ideal surface lattice from which it is derived. This im- plies that the crystal with reconstructed surface belongs to a different plane

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3.6. Clean semiconductor surfaces 353

Figure 3.29: Three different types of surface reconstructions: (a) 1 x 2, (b) 3h x 31 -30°, (c) general type, the Wood notation does not apply in this case, the matrix notation does with 411 = 5, 412 = -1, 421 = 2, 422 = 2.

crystal system than the crystal with ideal surface. For example, the 2 x 1 reconstruction of a square lattice leads to a rectangular lattice. The same holds for the 2 x 1 reconstruction of a hexagonal lattice which also results in a rectangular lattice.

A further item is worth mentioning, concerning the surface reconstruc- tion itself. It follows from the point symmetry of the crystal with ideal surface itself. If the latter belongs to the square crystal system, i.e. if it has a square lattice and one of the two point symmetry groups 4 m m or 4, the directions of the two primitive lattice vectors are symmetrically equiv- alent. A particular reconstruction which increases the surface unit cell in the direction of fl by a factor n and in the direction of f2 by a factor m, is equivalent to another reconstruction which does the same for, respectively, the symmetrically equivalent vectors f2 and fi (Figure 3.30). An analogous statement holds for an ideal surface of the hexagonal crystal system, having a hexagonal lattice and one of the point groups 6mm, 6 , 3 m or 3. In this case, three symmetrically equivalent direction exist (Figure 3.30). If there is no physical reason which makes one of the different symmetrically equivalent reconstructions more likely than another, they will take place simultaneously in different regions of the surface. The result is the formation of domains of otherwise identical, but differently oriented, reconstructed unit cells. Due to the domain structure, the overall translation symmetry of the surface is destroyed. Structural imperfections of a more local nature occur where the boundaries of such domains meet.

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354 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Figure 3.30: Symmetrically equivalent 2 x 1 reconstructions in the case of ideal surfaces belonging to the square (left) and hexagonal crystal systems.

Point and space symmetries

The point and space symmetries of relaxed or reconstructed surfaces are gen- erally of lower degree than those of the corresponding ideal surfaces. They do not only depend on the point and space symmetries of the irreducible crystal slab as in the case of an ideal surface, but also on the point and space symmetries of the relaxed or reconstructed surface slab. If the surface slab consists of more layers than the irreducible crystal slab, then its point and space groups can be taken to determine the point and space groups of the whole crystal with relaxed or reconstructed surface. If the surface slab contains fewer layers than the irreducible crystal slab it is expedient to add further atomic layers (which then do not contain displaced atoms) to make up the difference. The point and space group elements of the thus extended surface slab which are simultaneously point and space group elements of the irreducible crystal slab form, respectively, the point and space groups of the whole crystal with relaxed or reconstructed surface. Similarly like this was done for the translation symmetry group above, one may argue that the point and space groups of the relaxed or reconstructed surface slab should be subgroups of the point and space groups of the corresponding ideal surface. If this is the case, the point and space groups of the relaxed or reconstructed surface slab are, respectively, the point and space groups of whole crystal with relaxed or reconstructed surface.

3.6.3 Electronic structure of crystals with a surface

The electrons and cores of a crystal with a surface undergo the same interac- tions as the electrons and cores of an infinite bulk crystal, the only difference being that the cores and electrons of the removed semi-infinite crystal above the surface are missing, and the positions of the cores just below the surface

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3.6. Clean semiconductor surfaces 355

differ from those in the infinite bulk. The two basic approximations of the theory of the interacting electron-core system of an infinite crystal, namely the adiabatic approximation and the one-electron approximation, are not influenced by these modifications, so they may also be used in the presence of a surface.

In the electronic structure calculations of infinite bulk semiconductor crystals, the core positions commonly are taken as input data. This is pos- sible because these positions are crystal sites of high symmetry which are well-known from X-ray diffraction experiments. For crystals with a surface, this can no longer be assumed. The positions of atoms in the surface layers of relaxed or reconstructed surfaces are crystal sites of lower symmetry. In many cases, they are not known or only incompletely known from experimen- tal investigations, for these investigations are more difficult and less precise than X-ray diffraction in the case of bulk crystals (a short review of these methods will be given further below). X-ray diffraction cannot be applied to surfaces because it lacks surface sensitivity. In such circumstances, the positions of atomic cores at surfaces are to be treated as output, rather than input, data of the electronic structure calculations. The way that this can be accomplished was discussed in section 2.2 in general terms, and in section 3.5 with respect to point defects. It involves, first, the calculation of the total energy of the electron-core system for a variety of different sets of core positions and, second, the minimization of the total energy with respect to these sets. The minimum set gives the core positions which really apply. The most critical part of the total energy is the energy of the electron system. In order to obtain it, the one-electron energies of the crystal with surface have to be calculated for assumed core positions. Formally, this involves the same task as in the case of bulk crystals, namely, the calculation of station- ary one-electron states for given core positions. Below, we will demonstrate how this problem can be solved in the case of crystals with surfaces. The remaining parts of the procedure for determining the atomic and electronic structures of surfaces, the calculation of the entire total energy including the core-core interaction energy, and the minimization of the total energy with respect to the core positions, will not be treated here because it is mainly a numerical task.

The assumption of a priori known core positions is valid if ideal surfaces are considered. In this case they are the infinite bulk positions. The elec- tronic structures of ideal surfaces are important as reference data for the electronic structures of relaxed and reconstructed surfaces. Thus we will also deal with them.

One-electron Schrodinger equation

The one-particle Schrodinger equation for the wavefunction $E(x) of an

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356 Chapter 3. Electronic structure of semiconductor crystals with perturbations

electron in a crystal with surface has the same general form

(3.203)

as in the case of an infinite bulk crystal (see equation 2.53). The oneelectron potential V(x) may be written as the sum

of the electron-core interaction potential Vc(x), and an effective one-electron potential Ve(x) due to electron-electron interaction. For Ve(x), any of the one-electron approximations introduced in section 2.1 may be used. The core potential Vc(x) follows from the corresponding expression for an infinite crystal if the summation is restricted to cores within and below the surface. Using the surface adapted notation 5 + If3 + r for the position of the j-th basis atom of the primitive bulk unit cell at the bulk lattice point r + If3,

j = 1 , 2 , . . . , J , 1 = 0, -1,. . . , -00, this leads to

-W J (3.205)

where q(x) is the potential of a core of species j located at R = 0. The core potential Vc(x) and, hence, also Ve(x) and V(x) have the translation symmetry of the 2D surface lattice, so that

v(x) = V(x + r) . (3.206)

As in the 3D bulk case, this symmetry can be used to derive the Bloch theorem.

Bloch theorem

This theorem states that the energy eigenfunctions $E(x) of the Schrodinger equation (3.203) may be chosen simultaneously as eigenfunctions of the sur- face lattice translation operators t,.. This allows one to write these functions in the form of Bloch functions @Q(x) with a 2D quasi-wavevector

4 = 91g1 + 9282, (3.207)

where q1,92 are arbitrary real numbers, and gl and gz are the primitive lattice vectors of the reciprocal surface lattice, defined by the relations

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3.6. Clean semiconductor surfaces 35 7

fi . g k = 27T6&, i, k = 1,2,

The latter equations are solved by the vectors

(3.208)

g1 = N-lf2 x [fl x f2], g2 = N-lfl x [f2 x fl] (3.209)

with N = (1/2n)[(f1 . fl)(fZ. fz) - (fi . f2)’] as normalization constant. The two vectors gl, gz of (3.209) lie in the plane spanned by the two primitive surface lattice vectors f1, fz, i.e., within the surface. The same statement applies to the 2D wavevector q.

As in the 3D bulk case it is convenient to introduce a region of macro- scopic size with respect to which the eigenfunctions $J,E~(x) can be assumed to be periodic. In the case of a crystal with surface this region forms a paral- lelogram spanned by the edge vectors Gfl, Gf2, with G being a large integer. The area RII of the periodicity region is G21f1 x f2l. The Bloch functions normalized to it may be written as

1 Q,Eq(x) = -e2qx U E q ( X ) , fi (3.210)

where U Q ( X ) is the Bloch factor, which has the periodicity U E ~ ( X ) = U E ~ ( X +

r) of the surface lattice and is normalized with respect to a primitive surface unit cell. To guarantee the periodicity of the Bloch functions $~~(x) of (3.210) with respect to the periodicity parallelogram, the wavevectors q must have the form

(3.211)

with p1,pz as integers. This means that the q-vectors must belong to a finely-meshed lattice similar to that of Figure 2.4. For the macroscopically large values of G which we assume, q is practically continuous, although the number of different q-values within a given region of q-space is finite.

Surface Brillouin zones and surface energy bands

The energy eigenvalues E of a particular Bloch type eigenfunction of equa- tion (3.203) depend on q. If g varies over the whole infinite space, as we assume here, E represents a unique function E ( q ) of q. This description corresponds to the extended zone scheme in the case of an infinite bulk crys- tal considered in section 2.4. As in the latter case, one may switch from the extended to the reduced zone scheme in which q varies only over a primitive unit cell of the reciprocal surface lattice. Any other point of the inh i t e q- space may be written as q + g where g is a surface reciprocal lattice vector.

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358 Chapter 3. Electronic structure of semiconductor crystals with perturbations

In the reduced scheme, the energy function E(q+ g) over the whole q-space is replaced by the manifold of functions E,(q) E(q + g ) defined only over a primitive unit cell. As in the bulk case, there is a distinguished choice of the primitive unit cell, as we will see below.

Using first order perturbation theory with respect to the periodic poten- tial V(x), one may demonstrate that, in the extended scheme, E(q) repre- sents a continuous function of q everywhere, except for the q-vectors with lql = lq + gl or

q . g + - 1 2 g = 0, (3.212) 2

where g is an arbitrary surface reciprocal lattice vector. Equation (3.212) defines lines in the 2D q-space which are analogous to Bragg reflection planes in the 3D k-space of a bulk crystal. On Bragg reflection lines, the energy function E(q) has discontinuities. These lines may be used to define 2D Brillouin zones in just the same way as was done in the 3D case in section 2.4. One speaks of surface Brillouin zones (BZs). The surface B Z s of the square lattice have, in fact, already been used in section 2.4 as an illustration for the 3D case. Of particular importance is the f i rs t surface B Z . It may also be defined as the Wigner-Seitz cell of the reciprocal surface lattice.

Since there are 5 different plane Bravais lattices and, hence, 5 different reciprocal surface lattices, there are also 5 different first surface BZs. They are shown in Figure 3.31. Their shapes are the same as those of the Wigner- Seitz cells of the corresponding direct lattices since the Bravais types of the direct and reciprocal surface lattices always coincide.

The first surface B Z s are, by definition, free of Bragg reflection lines. Thus the energy function E(q) is continuous within these zones. Further- more, any higher surface B Z of order p may be reduced to the first surface B Z , and the energy function E(q) in the p-th surface B Z can be folded back to the first surface B Z . There, it forms a continuous function E,(q) which is called a surface energy band. Thus we may state that the energy eigenvalues of a crystal with surface form energy bands over the first surface B Z .

There are surface bands of different types, regarding their relations to the energy bands of 3D bulk crystals without surface. Below we characterize these differences in a qualitative way.

Types of eigenstates

Bulk states

Consider an infinite bulk crystal, and cleave it into two semi-infinite crystals with a surface parallel to a particular lattice plane. The spectra of energy

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3.6. Clesn semiconductor surfaces

t'

359

tY

Figure 3.31: Surface B Z s of the 5 plane lattices: (a) oblique, (b) p-rectangular, (c) c-rectangular, (d) square, (e) hexagonal. Symmetry lines and points are also shown, and their notations are introduced.

eigenvalues of the two semi-infinite crystals will contain all energy levels which were already eigenvalues of the infinite bulk crystal before cleaving. This implies that each crystal with surface possesses an energy eigenvalue spectrum which partially is made up from energy eigenvalues of the infinite bulk crystals from which it is derived. The eigenfunctions of the crystal with surface corresponding to these eigenvalues, if examined at positions outside of the crystal, will decay exponentially with increasing distance from the surface, while inside they will practically be the same as those of the infinite crystal, i.e. they will exhibit undamped oscillations throughout the whole semi-infinite crystal, as do the eigenfunctions of the infinite bulk crystal (see Figure 3.32). Eigenstates of a crystal with surface exhibiting such properties are called bulk states. The corresponding energy eigenvalues form bulk state surface energy bands Ep(q). These bands can be obtained by projecting the bulk bands E?"(k) of the infinite crystal onto the first surface B Z . Here, 'projecting' means that one assigns to a particular point q of the first surface B Z all bulk band energies EF"(k) corresponding to k-vectors of the first

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360 Chapter 3. Electronic structure of semiconductor crystals with perturbations

f C 0 x 0 C W 3 C

w + W

.- -w

I P

surface state

9- 0 x-

Figure 3.32: The three different types of electron energy eigenstates of a crystal with surface below the vacuum level (schematically).

bulk B Z having the same projection q on the surface B Z plane, but with various components kl perpendicular to it. Formally, this can be expressed by the relations

The bulk state surface band index p in (3.213) replaces the bulk quantum numbers v k l . As kl varies continuously, @ also does. This means that a particular band of the infinite bulk crystal gives rise to a continuum of surface bands. In this way, the infinitely large number of atoms in a primitive unit cell of the crystal with surface manifests itself.

We illustrate the projection of bulk bands onto the surface B Z using the (100) surface of a diamond type crystal as an example. The band structure is taken from the empty lattice model introduced in section 2.4. As a first step, the bulk B Z is to be projected onto the plane of the first surface B Z (see Figure 3.32). In doing so, one notes that part of the projected bulk B Z lies in the second surface B Z . This has to be folded back to the first surface B Z , together with its energy values. In this way we obtain the projections of the lowest three empty lattice bands shown in Figure 3.33. In addition to q-vectors for which all energy values are allowed, there are also q-vectors occurring for which certain energy values are forbidden. This peculiarity is found in other, more realistic cases as well, it represents a general feature of the projected bulk band structure. In the forbidden energy regions states may occur which are localized at the surface.

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3.6. Clean semiconductor surfaces 36 1

Figure 3.33: First (100) surface B Z (shaded area) together with the projected first bulk B Z for a crystal with fcc Bravais lattice.

‘I

Surface states

Such states are to be expected for the same reason that localized states are observed in the case of point or 0-dimensional perturbations. In contrast to the latter, surfaces constitute 2-dimensional perturbations. The localization occurs, therefore, only in 3 - 2 = 1 dimension, namely that perpendicular to the surface. If the energy lies in the forbidden region of the projected bulk band structure, the decay of the eigenfunction towards the bulk proceeds exponentially (see Figure 3.32). The states are then called bound surface states and the corresponding energy bands E,(q) are bound surface bands. Besides these, one has surface resonances and antiresonances. The latter occur at energies in the allowed region of the projected bulk band structure and give rise to resonant or antiresonant surface bands Ep(q). They mani- fest themselves in an increase (resonance) or decrease (antiresonance) of the density of states. The eigenfunctions at these energies are also localized at the surface, but decay less rapidly towards the bulk (according to a power law) than the exponentially decaying bound surface states. Antiresonances are necessary in order to satisfy Levinson’s theorem, which holds for surfaces as well as for point perturbations. If bound surface states exist, the antires- onances must compensate the increase of the total DOS in the previously forbidden energy region.

Implications of symmetry for surface band structure

The spatial symmetry of a crystal with surface has implications for the pos- sible degrees of degeneracy of surface energy bands Ep(q), p = p, (T, p, at a

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362 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Figure 3.34: Projection of the empty lattice band structure of a fcc bulk crystal onto the first - (100) surface B Z . Three bulk 5 bands with reciprocal lattice vec- n

2 tors K = 0 ( 1 = 0), K = - 7

E? (2n /a ) ( l l l ) ( I = 1) and K =

ered.

1 9 ul

%

a, C W

(2n/a) (m) ( I = -1) are consid-

5

3

1

r K r

given wavevector q. It also results in symmetry relations between the values of EF(q) for different q values, and it determines the spatial symmetries of the eigenfunctions $i)w(~). The key for such conclusions are, in analogy to the infinite bulk case, the irreducible representations of the space group of the given crystal with surface. This is based on the fact that the eigenfunc- tions for a particular energy eigenvalue form a basis set of an irreducible representation of this group. Such a representation may be characterized by the star {q} of the wavevector q and the irreducible representations of the small point group of q with the factor system of equation (A. 157). The dimensions of the irreducible representations determine the possible degrees of degeneracy of an energy band E,(q) at the point q. Moreover, at all points of the star {q}, E,(q) has the same value. Since the 10 possible point groups of equivalent directions have at most 12 elements (this happens in the case of C6v(6mm)), and since the small point groups of the symmetric q-points are in general even smaller than the corresponding point groups of equivalent directions, only irreducible representations with dimensions equal to 1 or 2 will appear. 2D representations are likely to occur in cases where the small point groups are, on the one hand, large enough, and on the other hand, the corresponding space groups contain glide reflections. Then non- trivial factor systems arise for points q on the boundary of the first BZ,

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3.6. Clean semiconductor surfaces 363

pig1 r, r, - A;-?;--Z;- ~ r ~ r + - ~ 2 - - - ~ - ~ ; - X ; - - ~ ; - M 2 - Z ; - I

- A\-x:-z;--M,--z,- x,-A,- ~--A~;-&-z;-M,-zF- I I I

x2

x1

Figure 3.35: Symmetry and degeneracy of the surface energy bands for the 5 space groups of the p-rectangular Bravais lattice. [After Terzibaschian and Enderlein, 1986.)

and 1D representations might not be possible at all. This happens for 2 of the 5 space groups of the primitive rectangular Bravais lattice, which below will be studied as an example. In Figure 3.35, the possible types of band structures are shown on certain symmetry lines of the p-rectangular surface BZ. In the case of the space group p2mg, which applies for (110) surfaces of diamond type crystals, only 2D representations exist at M and X . The two 1D representations on the 2-line connecting these points belong to the same energy eigenvalue because of time reversal symmetry. For the space group p2gg, one has 2D representations only at X and X'. The representations on the connecting line X - 2 - M - 2' - X ' are lD, but the corresponding en- ergy eigenvalues are degenerate because of time reversal symmetry. For the three remaining space groups p2mm, p l m l , and p lg l , the representations at all symmetry points are 1D.

Numerical methods for calculation of the electronic structure of surfaces

A 3D crystal with surface may be characterized as a crystal with a 2D lattice and primitive unit cells extending infinitely in the direction perpendicular

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364 Chapter 3. Electronic structure of semiconductor crystals with perturbations

to the surface. This point of view gives rise to the so-called slab method for calculating the electronic structure of crystals with surfaces.

Slab method

The electronic structure of a crystal with a 2D lattice may be calculated by means of any of the methods known from band structure calculations of 3D crystals, with the exception that the dimensionality of the lattice has to be changed from 3 to 2. In the tight binding method, for example, one has to use Bloch sums of atomic orbitals &(x) over all points r of the 2D surface lattice, rather than over all points R of the 3D bulk lattice. With the surface adapted notation, r+G+Zf3, for a regular crystal site, the orbital a localized at such a site is given by &(x - r - 3 - 1f3) E &jl,.(x), The corresponding Bloch sums &jlp(x) are defined as

(3.2 14)

The number of different Bloch sums is infinitely large even if a finite number of orbitals is used per atom, because the integer 1 counting the primitive crystal slabs, runs from 0 to -co. Consequently the Hamiltonian is given by an infinite matrix in the atomic orbital representation &jl,.(x). Some specific approximation is necessary to transform this to a finite matrix. One possibility is to consider a slab of the crystal, i.e. to cut off the semi-infinite crystal at a particular lattice plane parallel to the surface and discard the remainder. One may say that, in the direction perpendicular to the surface, the true semi-infinite crystal is replaced by a cluster (see Figure 3.36). The latter has two plane surfaces, one of them being real (that of the semi-infinite crystal), and the other one (obtained by cutting the semi-infinite crystal) not. This differs from the cluster method in the case of point defects discussed in section 3.5 where the whole cluster surface is artificial. The slab method may be combined with any band structure calculational method for infinite bulk crystals. Its combination with the tight binding method will be illustrated below by means of an example.

We consider the ideal (111) surface of a diamond type crystal which according to subsection 3.6.2, has a hexagonal Bravais lattice. Instead of the one s- and three p-orbitals per atom we use the four hybrid-orbitals, i.e. we replace a in the Bloch sum (3.214) by ht, t = 1 ,2 ,3 ,4 . Only nearest neighbor interactions are taken into account. An illustration of this model is given in Figure 3.37. To get the matrix elements of the Hamiltonian H between Bloch sums, we first need these elements between the localized hybrid orbitals +htjlT Ihtjlr). The diagonal elements are

(3.2 15)

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3.6. Clean semiconductor surfaces 365

Figure 3.36: Illustration of the slab method for surface band structure calculations.

Figure 3 . 3 7 Defect molecule model of the ideal (111) surface of a crystal with diamond structure.

and the non-diagonal elements between different hybrids at the same atom j = j ’ are

(htllrlH(htJ1lr) = VI , t # t’ . (3.216)

The two types of matrix elements in (3.215) and (3.216) are equal for atoms at the surface and in the bulk. This is not true for elements between hybrids located at different atoms. Let j = 1,1= 0, designate the surface atom layer and consider the elements (ht10rJH]ht20rt) between the hybrid ht located at the surface atom 10r and the hybrid lht20x-t) at its nearest neighbor 20rt below the surface, pointing toward (htlOr). Since no nearest neighbor hybrid exists for the hybrid )hllOr) pointing out of the surface, all nearest neighbor elements involving IhllOr) vanish. The other three elements with t = 2,3 ,4 are equal to the parameter V2 introduced in equation (2.292),

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366 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Using equations (3.215), (3.216) and (3.217), the matrix elements between the corresponding Bloch sums (3.214) may be formed. They are

(htlOq(Hlht,lOq) = v1 , t # t’ . (3.2 19)

(htIOqlHlht20q) = etV2, t = 2 , 3 , 4 , (3.220)

where et is given by equation (2.240), with dt being a vector which points from the surface atom 10r toward its nearest neighbor atom 20rt in the direction of the hybrid t. We have

(3.221)

The hybrids lht201-i) at the nearest neighbor atoms pointing td the surface atom lor, are coupled to the other hybrids at these atoms, and these hybrids interact with hybrids at more remote atoms. Thus an infinite matrix would occur if we would not restrict consideration to a slab, as we in fact do. Here, we will go one step further and neglect all couplings between the hybrids at the nearest neighbor atoms. Then the 7 Bloch sums IhtlOq), t = 1,2,3,4, and (ht20q), t = 2 ,3 ,4 , are completely decoupled from the rest of the Bloch sums (see Figure 3.67). In this way we arrive at a simplified model of the crystal with surface which effectively reduces it to the two first atomic layers and treats the atoms of the second layer only in an approximate way. This model represents an analog of the defect molecule model in the case of a point perturbation. If the basis functions are arranged in the order IhllOq), IhzlOq), h l o q ) , h l o q ) , lh220q), lh320q), lh420q), then the Hamiltonian matrix is

a a - - d2 = - ( l T T ) , d3 = -(lll), d4 = a(TT1). 2 2 2

(3.222)

The eigenvalues of this matrix are plotted in Figure 3.38 for different sym- metry lines of the first surface B Z of the hexagonal lattice of Figure 3.31.

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3.6. Clean semiconductor surfaces 367

r M K r W avevec tor

Figure 3.38: Band structure of the ideal (111) surface of a diamond crystal within the defect molecule model. The TB parameters are V1 = 2.13 eV, V2 = 6.98 eV. The hybrid energy Eh is used as the energy origin.

The three lowest and three highest bands are bulk state surface bands. They arise from the three hybrids of the surface atom, which bind this atom back to its nearest neighbors in the second layer. More strictly, they correspond to the 3 back bonding and antibonding states. The band in the gap is due to the dangling hybrid of the surface atom. It forms a bound surface band. If we were to consider more than 2 atomic layers, the region where the bulk bands in Figure 3.38 occur would be covered by more bulk state surface bands, while the bound surface band in the gap would hardly change.

Supercell method

The slab considered in the preceding subsection may be repeated periodi- cally in the direction perpendicular to the surface, simultaneously inserting several layers of vacancies between two neighboring slabs (see Figure 3.39). This structure may be considered to be an infinite repetition of the origi- nal crystal with surface, each repetition being approximated by a finite slab of several atomic layers embedded between vacancy layers. This arrange- ment represents a 3D supercrystal composed of 1D supercells (bear in mind that in the case of point defects in section 3.5, we similarly considered a

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368 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Figure 3.39: Supercrystal obtained by a periodic repetition of supercells. The latter are composed of crystal slabs embedded between vacancy layers.

3D supercrystal composed of 3D supercells). The band structure of the su- percrystal is approximately the same as that of the original crystal with surface, provided the vacancy slabs are thick enough to suppress coupling between neighboring crystal slabs, and the latter slabs are thick enough to simulate a semi-infinite crystal. The bands Ep(q) of the crystal with surface are obtained from the bulk bands EF(k) of the supercrystal by plotting the latter on the surface B Z . In this, k l may be chosen arbitrarily because any dispersion of EF(k) = EF(q , k l ) with respect to k l would indicate a coupling between the slabs, which has been excluded. The band structure of the supercrystal may be obtained by means of any 3 D band structure calculation method without modification. This makes the supercell method particularly appealing. Combined with the pseudopotential method, as well as the local density functional or quasi-particle approximations, it represents the most important calculational method for electronic and atomic structure determinations of surfaces.

Defect model and Green’s function method

A crystal with surface may also be viewed as a 3 D crystal with a 2D pertur- bation. The significance of this characterization is the following: Consider first an ideal infinite 3 D crystal, then remove some number of neighboring atomic layers parallel to the surface under consideration or, equivalently,

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3.6. Clean semiconductor surfaces 369

Figure 3.40: Illustration of the defect method for calculating surface energy band structure.

create the same number of vacancy layers (see Figure 3.40). What remains are two identical semi-infinite crystals which are only displaced with respect to each other. They do not interact provided the number of vacancy layers is large enough, which we will assume. The electronic structures of the two semi-infinite crystals coincide, and are identical with that of the considered crystal with surface. As in the case of a single vacancy, the states bound at the defect may be obtained by means of the Greens function Go(E) of the unperturbed 3D crystal, more strictly, by the vanishing of the determinant of [Go(E)V’ - 11 (see equation (3.134)). Here, the perturbation potential V’(x) represents the difference of the potential energy of an electron in the perturbed and unperturbed crystals. It is the negative of the sum of the potentials of the removed atoms or the sum of all vacancy potentials. It has the 2D lattice symmetry of the surface so that

~ ’ ( x ) = V’(X + r). (3.223)

If only nearest neighbor interactions are taken into account, a single layer of vacancies is enough to decouple the two semi-infinite crystals. For the analysis of the bound state condition (3.74), one has to use a particular basis set, just as in the case of a point perturbation. Wannier functions are again a possible choice, but here they involve localization only in one direction, namely that perpendicular to the surface. We denote these functions by IvqRl) where RI is the component of a 3D lattice vector R perpendicular to the surface, such that

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370 Chapter 3. Electronic structure of semiconductor crystals with perturbations

(3.224)

The matrix representation of the Green’s function Go(E) with respect to this basis set is

For the perturbation potential V1(x) we take that of a single layer of vacan- cies located at RI = 0. Then the matrix representation of V1(x) is

where VL,,(q) = (vq01V’Jv’qO) has been used. The relevant matrix elements (vqOIGoV1 - 1lvlqO) of GoV1 - 1 take the form

with (3.228)

According to equation (3.134), the determinant of the matrix (3.227) must vanish for energy eigenvalues E in the gap of the ideal crystal, i.e.

Det [G:(E, q)VL,,(q) - 6,,,] = 0. (3.229)

If one compares this equation with the corresponding relation (3.94) for deep centers, it will be noted that the q-dependence which occurs here was absent there. This dependence in the case of surfaces causes the eigenvalues in the gap to form ‘deep’ bands rather than deep levels (which was the case for point perturbat ions).

Transfer matrix method

Finally, still another method for surface band structure calculations should be mentioned. It is based on the transfer matrix concept of quantum mechan- ics. The transfer matrix M ( E ) is formed from solutions of the Schrodinger equation upon one unit cell of the bulk crystal for particular boundary condi- tions in the direction perpendicular to the surface. The energy E is arbitrary first of all. Transferring the wave function from the surface unit cell to the

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3.6. Clean semiconductor surfaces 371

n - th unit cell below the surface can be done by applying the n - th power W ( E ) of M ( E ) . Bound surface states are obtained for such energy values E for which M n ( E ) decays exponentially with n. The practical use of this method seems to be limited, however.

3.6.4 Atomic and electronic structure of particular surfaces

In this subsection we will deal with the atomic and electronic structures of some important semiconductor surfaces, including the various reconstruction states of the Si (111) surface, the Si (100) surface, the (110) surface of GaAs and other 111-V compound semiconductors as well as the (111) and (100) surfaces of GaAs. It is advisable to treat the atomic structure together with the electronic structure because of the close relation between the two kinds of structures, as was pointed out earlier. We begin with a short introduction to the experimental methods of surface structure analysis, again referring to both the atomic and electronic aspects.

Experimental methods for surface s t ruc ture analysis

Experimental methods for determining the atomic structure of bulk crystals are all based on the interaction of waves with the atomic cores and valence electrons of the crystal. If the wavelength is of the order of the distance between the atoms, i.e. of the order of magnitude of 1 A, the crystal consti- tutes a 3D diffraction lattice and diffraction maxima will occur in prescribed directions in space. The crystal structure may be determined from the po- sitions and intensities of these maxima. The various experimental methods differ primarily in the nature of the waves employed. X-rays are by far the most important for determining the structure of bulk crystals. A wavelength in the region of 1 corresponds to a photon energy in the range of 10 k e V . Electron and neutron waves are also diffracted by bulk crystals. Electron energies in the region of 100 eV and neutron energies of 0.1 e V are required for wavelengths in the d region. Since they are neutral particles, X-ray pho- tons and neutrons interact only relatively weakly with the crystal. They can pass through crystals of macroscopic thickness and be backscattered from them from macroscopic depths within them. X-rays and neutrons thus yield information on all atomic layers of a crystal, including those at the surface. Since the number of surface layers is extremely small in comparison to the total number of layers, the diffraction patterns are dominated by the bulk of the crystals.

The interaction of electrons with the atomic cores and the valence elec- trons of a crystal is significantly stronger than that of photons and neutrons. Electrons having energy less than 100 k e V can not pass through a crystal of

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372 Chapter 3. Electronic structure of semiconductor crystals with perturbations

macroscopic thickness. Experimentally, one therefore has only the backscat- tering available and then only elastically backscattered electrons can be em- ployed in forming diffraction patterns. These originate at a depth which, on average, is equal to the inelastic mean free path of the electrons. This varies relatively independently of the particular crystal under consideration from 4 A to 10 for energies of 20 eV to 300 eV. Electron diffraction in this energy region is thus hardly suitable for examining bulk crystals but it can be readily employed in studying crystal surfaces. The diffraction of low energy electrons in the region of 100 eV is in fact the most intensively used method for surface structure determination. It is referred to as LEED (Low Energy Electron Diffract ion).

The principle of LEED may be explained as follows. We consider an incident electron wave with the wavevector ki. The interaction with the crystal generates scattered waves with wavevectors k,. The scattering po- tential has the translation symmetry of the surface lattice, thus its Fourier components differ from zero only for vectors g of the reciprocal surface lat- tice. This means that only those scattered wavevectors k, can occur whose components k,ll parallel to the surface differ from the parallel component kill of ki by a reciprocal surface lattice vector g, hence

(3.230)

There is no relation between the components of k, and ki perpendicular to the surface because there is no translation symmetry of the scattering potential in this direction. In writing down equation (3.230) we have implic- itly assumed that only one scattering event takes place. This relation also applies, however, to multiple scattering processes. This is important be- cause electrons scattered back from the surface have, as a rule, experienced many scattering events, in contrast to X-ray photons which typically have been scattered only once. This difference is due to the above mentioned fact that electrons interact with the crystal much more strongly than do X-ray photons. The electrons measured in LEED are elastically scattered. One therefore has

Ik,/= lkzl . (3,231)

A solution of the two equations (3.230), (3.231) always exists for given vec- tors ki and g (this is in remarkable contrast to coherent scattering of elec- trons from 3D bulk crystal, which can only occur if k, lies on a Bragg re- flection plane). The solution of equations (3.230) and (3.231) can be readily carried out using the construction shown in Figure 3.41. The points at which the vertical lines passing through the reciprocal lattice points g intersect the

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3.6. Clean semiconductor surfaces 373

Figure 3.41: LEED maxima.

Construction of

sphere Jk,] = lkzl, determine the directions in which diffraction maxima oc- cur. There is exactly one maximum for each reciprocal lattice point g. The reciprocal surface lattice can thus be read immediately from the distribution of the diffraction maxima on the registration screen. The direct surface lat- tice is the reciprocal of the reciprocal surface lattice. Some typical LEED images are shown in Figure 3.42. The bright points correspond to the re- ciprocal lattice of the ideal surface, and the less bright points to the finer reciprocal lattice of the reconstructed surface. In this way it is relatively easy to determine the surface lattice by means of LEED. To obtain the ac- tual positions of atoms is more diacult. One needs additional experimental and theoretical information about the intensity of the diffraction maxima as a function of the energy of the incident electrons (dynamical LEED).

Besides LEED, there are other methods for surface structure analysis which, although they are not a substitute for LEED, can supplement it. These methods include diffraction of energetic electrons in the region of some 10 k e V , known as ‘Re5ection High Energy Electron Diffraction’ (MEED), diffraction of X-rays incident almost parallel to the surface, and diffrac- tion of slow Helium atoms (of M I00 meV). Scattering of energetic ions (M 1 M e V ) is used in techniques like ‘Rutherford backscattering’ (RBS) and ‘ion channeling’. Imaging procedures of significance are transmission elec- tron microscopy (TEM), scanning tunneling microscopy (STM) and atomic force microscopy (AFM). The latter two methods have become particularly important.

Like for atomic structure determinations, a variety of methods exist to study the electronic structure of surfaces, in particular the bound surface states in the energy gap of the bulk crystal. The most powerful and uni- versal method is photoemission spectroscopy (PES). This method relies on the external photoeffect in which an electron is emitted from the crystal by

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374 Chapter 3. Electronic structure of semiconductor crystals with perturbations

x6

Figure 3.42: LEED pictures of six differently prepared GaAs (100) surfaces. (After Drathen, Ranke and Jacobi, 1978.)

absorbing a photon of sufficiently high energy. The emitted photoelectrons are spectrally decomposed with respect to their kinetic energies. The thus obtained energy spectrum of photoelectrons maps the density of states of occupied electron levels of the crystal. To enhance surface states and dis- criminate bulk states, photoelectrons with kinetic energies around 50 eV are used whose inelastic mean free path is only about 5 A and which can therefore only come from this depth below the surface. These electron ener- gies correspond to photon energies which are not substantially larger, i.e. in the far ultraviolet region. The term UPS (Ultraviolet Photoemission Spec- troscopy) is used in this context. The only practically suitable radiation source in this energy region is the electron synchrotron. By measuring an- gular resolved photoemission spectra (ARUPS), the wavevector dispersion of the bound surface energy bands can be determined. To study moccu-

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3.6. Clean semiconductor surfaces 375

11101 hi01

Figure 3.43: Geometry of the ideal Si (111) surface (left) and of the Si (111) 2 x 1 surface according to the buckling model (right).

pied surface states one may use inverse PES in which electrons captured by such states emit photons. Beside photoemission, a variety of other tech- niques exists which can provide data on surface states. In principle, any experimental technique which probes the electronic structure of bulk crys- tals can be employed for surface electronic structure investigations, provided it can be made surface-sensitive. This applies to optical reflectivity, elec- trical transport, photoconductivity, and capacity measurements, as well as electron energy loss spectroscopy (EELS). Experimental techniques like field effect measurements fulfill this requirement from the very beginning. Con- trolling the energy of tunneling electrons in scanning tunneling microscopy, surface states can be resolved spatially and energetically (scanning tunneling spectroscopy).

Experimental techniques which primarily measure the electronic struc- ture, can also provide data on the atomic structure. The solid state shifts of core levels (see section 2.1) are an example. These shifts differ for atoms in the bulk and at the surface because of the altered atomic structure at the surface. The difference (typically some tenths of an e v ) can be mea- sured by means of PES and UPS. On the other hand, they can be calculated on the basis of a particular surface structure model. By comparing theory and experiment one can evaluate the feasibility of various models of surface structure.

The calculation of the total energy is a purely theoretical test of the validity of a particular surface structure model, and it may be used to deter- mine the parameters which can be varied in such a model. If the model has optimized parameters and results in a lower total energy than other models it may be given preference over them.

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376 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Figure 3.44: Surface band structure of an ideal Si (111) sur- I ' O r face.

-5 - * .. .. .. ,-I I

r

!ii

iii

::I

I l l ...

t i *!:!

..... .*....

I K r 2D wave vector

Si surfaces

(1 11) surface

The geometry of the ideal (111) surface of diamond type crystals is illus- trated in Figure 3.43 (left). The surface lattice is hexagonal, and the two primitive lattice vectors are fi, f2 of Table 3.8. There is one surface atom per primitive unit cell, and one dangling bond per surface atom. The (111) sur- face is the cleavage plane of diamond type crystals. By cleaving a Si crystal in UHV at room temperature, one obtains a 2 x 1 reconstructed (111) sur- face. After annealing it at 500 C , a 7 x 7 reconstruction state evolves, which remains stable at room temperature. The occurrence of a 2 x 1 reconstruc- tion immediately after cleavage is to be expected if one examines the band structure of the ideal (111) surface (see Figure 3.44) and considers, in partic- ular, the electron occupancy of the bound surface band in the fundamental gap. This band arises from dangling sp3-hybrids of surface atoms, and can host 2 electrons per surface unit cell. Since 3 of the 4 valence electrons of a surface atom are in bonds with second-layer atoms, only 1 electron per surface unit cell is left for the bound surface band. Thus this band remains only half-filled. The ideal Si (111) surface is metallic.

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3.6. Clean semiconductor surfaces 377

This state is unlikely to be stable, however, i.e. surface reconstruction is likely to take place. Below we discuss a particularly simple reconstruction model, the so-called buckling model (see Figure 3.43, right-hand side) which in the early days of clean semiconductor surface physics was believed to be correct. Later, it was realized that buckling is energetically advantageous only for 111-V compound semiconductor surfaces, while it is not advanta- geous for group-IV semiconductor surfaces including the (111) surface of Si. To introduce the buckling model we consider doubling of the primitive unit cell in the direction of primitive lattice vector fi, which according to Table 3.8 points in the direction [ l i O ] . The hexagonal lattice with doubled primitive unit cell forms a rectangular lattice with primitive lattice vectors 2f1 -t f2 and f2 The short side of the rectangular primitive unit cell, shown in Figure 3.43 right, is parallel to [Olq, and its long side parallel to [2m. The corresponding first surface BZ is also a rectangle (see Figure 3.31) with its long side, i.e. its r - X-direction, parallel to [Olq, and its short side, i.e its X - M-direction, parallel to [2ii]. The rectangular BZ is half as big as the original hexagonal BZ, and each band of the latter gives rise two band in the former, a direct and a back-folded one. There is no gap between these two bands because they arise from the same band of the larger BZ. The surface is still metallic.

A gap arises if the so far formal 2 x 1 reconstruction is made real. This can be done by a buckling of the surface, i.e. by alternately raising and lowering atoms in rows parallel to f1 above the surface and below it (see Fig- ure 3.43, right). In this, the three back-bonding hybrids of a raised atoms becomes more p-like, and the dangling hybrid at this atom more s-like si- multaneously lowering its energy, while the three back-bonding hybrids at a lowered atom become more sp2-like and the dangling hybrid at this atom more plike simultaneously raising its energy. The two bound surface bands derived from these s- and p-like dangling hybrids are just the bands below and above the gap discussed before. The lower s-like band can host all elec- trons of the dangling hybrids, while no electrons remain for the population of the upper p-like band. If the total energy of this state were in fact lower than that of the ideal surface, buckling would take place spontaneously, i.e the translation symmetry of the surface would spontaneously be lowered, A similar spontaneous symmetry breaking, the Jahn-Teller effect, was dis- cussed in the context of point perturbations in section 3.5. There, the point symmetry was broken, while no translation symmetry was involved. If the translation symmetry is broken, as in the case of surface reconstruction, one speaks of a Peierls instability or a Peierls t rans i t i on

However, as has been indicated at the outset, buckling turns out to be energetically not favorable in the case of Si (111) surfaces. Populating the lower s-like band with two electrons per primitive surface unit cell means transferring charge from the atoms lowered below the surface to the atoms

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378 Chapter 3. Electronic structure of semiconductor crystals with perturbations

r?

'P r- U

[1?01 [I101

Side view

a ) b )

Figure 3.45: a-bonded chain model of the Si (111) 2 x 1 surface (After Pandey, 1982). Part (a) shows the unreconstructed surface in top and side views. The top view in the second row has been rotated with respect to the top view in the first row in order to allow for the side view below. Part (b) shows the same views of the surface as in part (a), but after reconstruction has taken place.

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3.6. Clean semiconductor surfaces 379

raised above. This implies the creation of an electric dipole which is too costly in energy to actually take place. Using the terminology of section 2.2 we may say that correlation effects of electron electron interaction, more strictly speaking, the configuration dependence of oneelectron states, pre- vents the buckled Si (111) surface to be lower in energy than the ideal one.

The reconstruction model which actually applies to the Si (111) surface is the so-called s-bonded chain model, illustrated in Figure 3.45. In this model, second layer atoms in rows parallel to fi + f2, i.e. along the [lo3 direction in Figure 3.45a (including atom number 2) are raised into the first layer as shown in Figure 3.45b, breaking their bonds with atoms in the third layer (for example, the 2-5 bond). The dangling bond of the new surface atom (say atom 2) is used to establish bonds with atoms of the first layer (the 2-1 bond in this case). These can only be s-bonds (indicated by double lines in Figure 3.45b) because the dangling bonds are perpendicular to the surface. In this way s-bonded chains occur along the [ l O q direction (Pandey, 1982). The dangling bonds left at the third layer atoms (for example, atom 5) are saturated by hybrids of the first layer atoms which have been lowered down to the second layer (for example, atom 3). The surface is in fact 2 x 1 reconstructed. This may be seen by taking the primitive lattice vectors of the ideal surface to be f 1 + f 2 and -f2. Doubling -f2 yields the rectangular lattice indicated in Figure 3.45b by dashed lines. Its primitive lattice vectors are f1 + f 2 and -2f2+ (fi +f2) = f 1 - f 2 so that the short side of the rectangle is parallel to the chain direction [lOT], and the long side perpendicular to it (parallel to [121]). A peculiarity of the n-bonded chain model is that it has a different bonding topology in comparison with the ideal (111) surface and also with respect to the buckling model. While the latter exhibit rings with 6 mutually bonded atoms (see Figure 3.45a), the former shows alternating rings with 5 and 7 bonded atoms (Figure 3.45b). This is due to the fact that bonds existing at the ideal and buckled surfaces are broken and new bonds are established in the s-bonded chain model.

The total energy of this model is clearly below that of the ideal surface (about 0.5 eV per surface atom). Thus it represents a good candidate for the reconstruction of the (111) Si surface. Further evidence is provided by ARUPS and optical measurements. Figure 3.46 shows the wavevector dis- persion of the two bound surface bands as obtained from ARUPS measure- ments together with the calculated dispersion of these bands. The agreement is quite satisfying. The strong dispersion of the bound surface band on the r-X-line and the weak dispersion on the X-M-line of the rectangular sur- face B Z is easily understandable: the long r-X-side of the rectangular unit cell in q-space corresponds to the short side of the rectangular unit cell in coordinate space, which is also the direction of the s-bonded chains. One expects strong dispersion along the chains and weak for the perpendicular direction, exactly what is seen in Figure 3.46.

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380 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Figure 3.46: Dispersion of the bonding (B) and antibonding t 2 (A) bound surface state bands along the r - X - M line for the Si (1 11) 2 x 1 surface. Curves are

chain model, points are obtained d 0

by means from ARUPS measure-

centi, and Hansson, 1985.) Yl

- 2

3 1

- >

W calculated within the 7r-bonding

Q, a x ments. (After Martensson, Cri-

g o W

-1

r X nf 20 wave vector

Figure 3.47: Differential reflectiv- ity spectrum of the Si (111) 2 x 1 surface (After Chiarotti, Nannarone, Pastore and Chiaradia, 1971.)

0 93 0.A $5 0,s [ 7

Energy (ev) -

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3.6. Clean semiconductor surfaces 38 1

Figure 3.48: Polarization dependence of the differential reflectivity spectrum of the Si (111) 2 x 1 surface of Figure 3.45 (taken at its maximum). The solid curve is calculated using the x-bonded chain model, the dashed curve using the buckling model, and the points are experimental data (After Del Sole and Selloni, 1984.)

Support for the r-bonded chain model comes also from optical measure- ments. The differential reflectivity spectrum of the 2 x 1 reconstructed (111) surface of Si is shown in Figure 3.47. There is no doubt that the observed peak at 0.5 eV is due to optical transitions between occupied and unoccu- pied bound surface bands. Such bands exist both in the buckling model as well as in the 7r-bonding chain model (in the latter one has 7r-bonding and x-antibonding bound surface bands). However, the two models differ in regard to their predictions on polarization dependence of optical reflectiv- ity. According to the 7r-bonded chain model, transitions with light polarized parallel to the chains, i.e. parallel to [lOq, should be allowed and transi- tions for light polarized parallel to the perpendicular direction [121] should be forbidden, while this should be reversed for the buckling model. The ex- perimentally observed polarization dependence shown in Figure 3.48 is that predicted by .I-bonding chain model. It rules out the buckling model.

Besides the 2 x 1 reconstruction, there are other reconstruction states of the Si (111) surface. The 7 x 7 reconstructed surface is the most stable one. The complicated structure of this surface has finally been resolved by combining the results of various experimental methods including STM (see Figure 3.49). The model which accounts for all experimental data utilizes three structural disturbances of the ideal surface, these being dimers (D), adatoms (A) and stacking faults (S). It is referred to as the DAS model

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382 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Figure 3.49: Scanning tunneling microscopy (STM) image of the Si (111) 7 x 7 surface. (After Quate, 1986.)

(Takayanagi, 1984). The DAS model of the 7 x 7 reconstructed (111) Si surface is shown in Figure 3.50.

(100) surface

The (100) surface is the surface of choice for electronic device applications of Si. The geometry of the ideal (100) surface is shown in Figure 3.51. The surface lattice is a square one, with primitive lattice vectors f1 , f2 given in Table 3.8. The primitive unit cell has one surface atom, and each atom has two dangling bonds which point out of the surface like ‘rabbit ears’ (see Figure 3.51a).

Each dangling bond is only half-filled as in the case of the (111) surface considered above. Thus, the ideal surface is metallic, and this state is un- likely to be stable. A state of lower total energy can be established by a 2 x 1 reconstruction as follows: The atoms of two neighboring rows parallel to [ O l i ] (or [Oll]) move slightly towards each other in order to allow bonding between two of their four dangling hybrids. This dimerization of the sur- face gives rise to bonding and antibonding bound surface states, the lower bonding state being completely filled and the upper antibonding state being completely empty. The primitive lattice vector in the direction of f1 doubles, thus a 2 x 1 reconstruction takes place, and the surface lattice becomes rect- angular. The remaining two dangling bonds of a dimer are still half-filled, however, so that the surface is not yet stable. It is stabilized by buckling,

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3.6. Clean semiconductor surfaces

a1

383

Figure 3.50: Dimer-Adatom-Stacking-Fault (DAS) model of the Si (111) 7 x 7 surface. The side view (a) is shown to identify the atoms: large shaded circles are adatoms, open circles are surface atoms of the first (large circles) and second (smaller circles) monolayer, solid circles are bulk atoms which do not undergo re- construction. The top view (b) shows a 7 x 7 surface unit cell and its surroundings. The small circles within shaded circles represent second layer atoms vertically below the adatoms. (After Takayanagi, 1984.)

which turns out to be energetically favorable in this case. One of the two atoms of a dimer moves above the surface and one below. The dangling hybrid at the lowered atom, being p-like, takes a higher energy and is corre- spondingly empty, while the dangling hybrid at the raised atom, being s-like, takes a lower energy and is filled. Rigorous structure calculations essentially confirm this simple tight binding picture. They only add displacements of the dimer atoms parallel to the surface in addition to the perpendicular ones. Due to the two kinds of displacements, the dimers become asymmetric. The described 2 x 1 reconstruction of the Si (100) surface is therefore called the

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384 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Top view

I-- 0'-

0

--0 10113

Side view

(100)

n

I t - -?

0 U

U

a ) b )

Figure 3.51: Geometry of the ideal Si (100) surface (a), and of the asymmetric dimer model of this surface.

asymmetric dimer model. Although the 2 x 1 reconstruction is the most common superstructure of the Si (100) surface, other reconstructions are also observed, for example, 2 x 2, c - 2 x 2, c - 4 x 2. Most of these structures may be traced back to asymmetric dimers as building blocks.

GaAs and other 111-V compounds

(110) surfaces

The (110) surface represents the cleavage plane of zincblende type crystals. The best understood (110) surface of all zincblende type 111-V semiconduc- tors is the (110) surface of GaAs. Its surface geometry is shown in Figure

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3.6. Clean semiconductor surfaces 385

Top view

I I I I I I 1 I I

Side view

a )

(110)

I?

5 0 Y

[?I0 1

- 0 c- 7 U

[?I01

Ga o As 0

I 1

I I I I I

I I

I I

b)

Figure 3.52: Geometry of the ideal GaAs (110) surface (a), and of the same surface after relaxation (b).

3.52. The surface lattice is p-rectangular. The primitive lattice vectors are given in Table 3.8. There are two surface atoms in a primitive unit cell, one Ga and one As atom. Two bonds of each surface atom lie within the surface, one is directed back and one is dangling. The two dangling hybrids per unit cell have different energies since they belong to either a Ga- or an As atom. Thus one expects two bound surface bands, one Ga-like and one As-like. These are in fact seen in the band structure of the ideal (110) GaAs surface depicted in Figure 3.53. The lower As-like bound surface band is completely occupied, and the higher Ga-like band is completely empty. Thus the ideal surface is semiconducting. Nevertheless, it does not yet represent the stable state, as the gap between the two bound surface states is too small. It can be enlarged by moving the As atom above the surface, rendering its dangling hybrid more s-like and lowering its energy, and moving the Ga atom below

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386 Chapter 3. Electronic structure of semiconductor crystals with perturbations

(111)

Ga As (1101 (1001

-10

r M K r r x ' M X r

2 D wave vector

r i ~ l r

Figure 3.53: Band structure of the ideal GaAs (110) surface. (After Talwar and Ting, 1992.)

the surface, rendering its dangling hybrid plike and raising its energy (see Figure 3.52). These displacements do not change the translation symmetry of the surface because the two atoms belong to the same unit cell. One therefore has a relaxation instead of a reconstruction of the (110) surface. This is consistent with LEED measurements, which do not show spots other than those due to the ideal prectangular lattice. The experimental value for the rotation angle w of a Ga-As bond with respect to the [ l i O ] direction is close to 30°, in good agreement with structure calculations. Experiment and theory also agree in regard to an essential feature of the band structure of the relaxed (110) surface. As indicated in Figure 3.54, relaxation moves the bound surface bands completely out of the fundamental gap, the As-like band merges into the valence band, and the Ga-like band merges into the conduction band. This implies that perturbations of the surface such as, for example, coverage by an insulator, can easily create surface states in the gap. Such states are in fact present at GaAs/insulator interfaces. They pin the Fermi level and preclude the possibility of making GaAs-based field ef- fect transistors in the same way as the Si-based MISFET (see Chapter 7 for further discussion).

Relaxations similar to that of the (110) surface of GaAs are also observed at the (110) surfaces of other 111-V compounds. For all materials except Gap, relaxation moves the bound surface states out of the gap (see Figure 3.54).

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3.6. Clean semiconductor surfaces

- 2 h - - 3 - - 0 5 - 4 -

9 -5- g

E - 7 -

3 u

3 -6- e x

w

387

A

(110) cleavage face GaP GaAs GaSb In P In As In Sb

U n

Figure 3.54: Energy level diagrams of bound surface states and bulk states of var- ious 111-V compound semiconductors below the vacuum level. Solid lines represent experimental results, and dashed regions represent dangling bond surface bands. F'ramed undashed regions indicate bulk bands. (After Bertoni, Bisi, Calandra, and Manghi, 1978.)

Other surfaces

Besides the (110) cleavage surface, investigations have mainly been focused on the low index (111) and (100) surfaces of GaAs. (100) is the preferred sur- face orientation of GaAs wafers used in device fabrication (partially because the (011) plane perpendicular to this surface represents the cleavage plane of GaAs). The geometrical structures of the ideal surfaces are the same as those of the corresponding Si surfaces shown, respectively, in Figures 3.43 and 3.51. The (111) and (100) surfaces of GaAs differ from the (110) surface of GaAs mainly in regard to the fact that two different surface terminations are possible in their case, one by Ga atoms and another by As atoms. One says that these surfaces are polar, in contrast to the (110) surface, which is said to be non-polar. In forming a polar surface, an electric dipole is created between a Ga-layer and an As-layer which is costly in energy. This explains why for GaAs and other 111-V compound semiconductors, the cleavage plane is neither ( l l l ) , like in the case of group-IV materials, nor (loo), but the non-polar (110) plane. In the latter case the 1:l ratio of Ga- and As atoms is strictly k e d by chemical stoichiometry. For the polar surfaces, there are, however, no stoichiometrical reasons which cause a surface termination by

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388 Chapter 3. Electronic structure of semiconductor crystals with perturbations

only Ga- or only As atoms. Both kinds of terminations can occur simulta- neously at a given surface, and in order to define the surface uniquely one has to specify the percentage of Ga and As it contains. A rough distinction is that between Ga-rich and As-rich surfaces.

The structure of a particular polar GaAs surface depends decisively on its termination. For the Ga-rich (111) surface one finds a 2 x 2 reconstruction, and for the As-rich case there are f i x 8 - 30° and a x - 23.4'' reconstructions in addition. For the Ga-terminated 2 x 2 (111) surface, a model has been proposed with a quarter of the surface atoms missing. The remaining first-layer Ga atoms and the second layer As atoms undergo a buckling, which raises the As atoms close to the surface.

The structure of the GaAs (100) surface exhibits an even greater vari- ety, depending on surface termination, surface treatment and temperature. Some of the possible reconstruction states can be seen in the LEED pic- tures of Figure 3.42. The c - 2 x 8 structure is found for an As-stabilized surface, which is important for MBE-growth because this commonly begins and ends with As-rich conditions. For the Ga-stabilized surface, the c - 8 x 2 reconstruction is found to be stable.

3.7 Semiconductor microstructures

Semiconductors with a clean planar surface, considered in the section above, may be thought of as units of two infinite half spaces, one filled with the semiconductor material, and the other being empty. If the vacuum is re- placed by a semiconductor material different from the first, then one obtains a semiconductor heterojunction or single semiconductor heterostructure. Fig- ure 3.55 shows an example. Below the plane at z = 0 one has semiconductor material 1, say GaAs, and above it is the material 2, say AlAs. The plane at z = 0 is called the interface. The microstructures to be considered in this subsection are composed of semiconductor heterojunctions. Thus, before addressing microstructures we must deal with het ero j unct ions.

3.7.1 Heterojunctions

Below we describe the electronic structure of semiconductor heterojunctions. The two semiconductor materials are taken to be undoped. Free carrier ef- fects on the electronic structure can be neglected in these circumstances. In the case of heterostructures formed from doped semiconductors, such ef- fects might be important. They are treated in Chapter 6 in a systematic way. Below, we discuss the various stationary one-electron states of undoped heterostructures, omitting free carrier effects.

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3.7. Semiconductor microstructures 389

Stationary one-electron states

As in the case of a crystal with a clean surface, heterojunctions of the kind described above possess a 2-dimensional rather than a 3-dimensional lattice translation symmetry. Generally, their one-electron states cp(x) are Bloch states (Pk,,(xI(, z ) in regard to their dependence on the position vector com- ponent XII parallel to the interface, with 2-dimensional quasi-wavevectors kll. The corresponding energy eigenvalues form bands in the 2-dimensional first B Z of the heterostructure. Just as in the surface case, almost all energy eigenvalues of the two infinite bulk materials, i.e. the Bloch bands E,l(k) and E,a(k) of these materials, are also energy eigenvalues of the heterojunc- tion. What may change are the eigenfunctions of these energy bands. Let us fix a particular quasi-wavevector ko. If an energy level E,n(ko) of material 2 does not coincide with any of the allowed energy levels E,l(ko) at ko of ma- terial 1, then the eigenfunction belonging to this level and quasi-wavevector will be localized in material 2. There, its wavefunction is spread out uni- formly over the whole semi-infinite crystal from z = 0 to z = +co; it forms a bulk state of material 2. This is illustrated in Figure 3.56a by representing energy levels of this kind by lines extending from z = 0 to z = +co. Vice versa, if, at ko, an energy level E,l(ko) of material 1 does not coincide with any of the allowed energy levels E,z(ko) of material 2, it forms a bulk state of material 1, and may be represented by a line extending from z = -co to 0. If there are energy levels at ko which are identical in both materials, i.e. with E,l(ko) = EA(ko), the corresponding wavefunctions will extend over both materials (Figure 3.56b). In the case of identical energy levels E,l(ko) and E,a(kb) at dzflerent wavevectors ko and kb, it is not a priori clear what will happen. If a matching of the corresponding two eigenfunctions and their derivatives at the interface turns out to be possible, then states extending over the whole heterojunction from -cc to f m will exist (Figure 3.56~). If

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390 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Material 1 I Material 2

P a, L.

Figure 3.56: Various bulk states of a semiconductor heterojunction.

no matching is possible, then the two eigenfunctions will remain localized in their respective material regions. One has a situation similar to that in the case of electromagnetic waves propagating between two semi-infinite dielec- tric media: for certain wavevectors they may propagate from one medium into the other, and for others they are internally reflected, as indicated in Figure 3 . 5 6 ~ .

Besides bulk states, there may be stationary states with energy eigenval- ues which are allowed in none of the two materials. The wavefunctions of such states are localized at the interface between the two materials, just like the bound surface states in the case of clean surfaces. They are called bound interjuce states. In analogy to the clean surface case, interface resonances

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3.7. Semiconductor microstructures 39 1

may also occur, their energy levels lie in one of the bands and their wave functions are weakly localized at the interface. If the two materials forming a heterojunction are composed of chemically similar atoms as in the case of GaAs and AlAs and in many other cases of practical importance, then no bound interface states will be possible, because the perturbation at the interface is too weak.

Valence band discontinuity

In many semiconductors, the maximum energy of the valence band occurs at the center r of the first BZ. Below we will restrict ourselves to materials of this kind. Of course, the maximum energy of the valence band itself depends on the material under consideration. This statement may seem to contradict the results of Chapter 2, where this energy was set to zero for all materials. However, this was done in the context of dealing with the band structure of only one infinite bulk semiconductor at a time. For such a single semiconductor, the energy origin could be chosen arbitrarily, and we took it to be at the valence band maximum. In the heterojunction of Figure 3.55, two semiconductor materials are involved, but the energy origin of the heterojunction can be fmed only once. If the valence band maximum is taken as zero for one material, it will, in general, differ from zero for the other material. Here, instead of setting it to zero for any material, we select the vacuum level as the common energy origin. This level may be defined as the minimum energy which an electron in an infinite semiconductor sample must have in order to escape. We will denote the valence band edge of a particular material i , i = 1,2, referred to the vacuum level as origin, by E$. Then the minimum energy which must be expended to remove a valence electron from material i , is given by --E$ If the electron should escape by absorbing a photon, then -Eti is the minimum photon energy required (photo-t hreshold energy).

For the actual positions of the valence band edges Evl and Ev2 at a heterojunction, the infinite bulk values Etl and Et2 have only an indirect meaning. In fact, for each of the two semiconductors of a heterojunction the adjacent material is foreign and represents an external perturbation. Even if no free charge carriers are available, as we assume, each of the semiconduc- tors reacts to this perturbation by redistributing its electrons, in this case its bound valence band electrons. How this occurs is quite clear physically, and it can also be analyzed in a more rigorous treatment: valence electron charge will flow from the material with the higher valence band edge into the material with the lower valence band edge, thus lowering the total energy of the heterojunction. In this way, an electric dipole layer develops at the interface, having a thickness of several atomic monolayers. Macroscopically, the spatial extension of the dipole layer is zero, but the associated dipole

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392 Chapter 3. Electronic structure of semiconductor crystals with perturbations

moment has a non-zero macroscopic magnitude. As is well-known from elec- trostatics, the potential cp exhibits a discontinuity in passing through such a dipole layer. We denote the left boundary potential value by cp1, and the right one by 9 2 . Then the actual positions of the valence band edges E,1 and E,2 at the heterojunction are given by the expressions

(3.232)

The difference AE, zz [E,1 - E,2] between the two valence band edges is called valence band discontinuity or valence band offset of the ‘material l/material 2’ heterojunction. By means of equation (3.232) the valence band discontinuity AE, is expressed as

(3.233)

Beside the pure bulk contribution [E$ - E&], the band discontinuity also contains the dipole contribution -e[cpi-pg]. The latter generally depends on the properties of the interface. If one were to ignore this dipole contribution, then AE, would obey the so-called ‘transitivity rule’ which states that the AE,-values for a succession of two heterojunctions ‘1/2‘ and ‘213‘ should add up to the AE,-value of the combination ‘1/3’. In practice this ‘rule’ is rarely fulfilled, which points up the importance of the dipole contribution to A E,.

The dipole contribution -e[cpl - cpz] may be estimated by means of the tight binding method developed in section 2.6. The bonding energy level € b considered there forms a rough measure for the average valence band energy. In formula (2.315)’ € b is given for a zincblende type semiconductor. The corresponding bonding orbitals IbtR) of equation (2.316) describes the charge transfer from the cation c (there denoted by an upper index ‘1’) to the anion a (there denoted by an upper index ‘2’) in terms of the hybrid energy difference EL - €2. If the hybrid energies were equal, no charge transfer would occur. A semiconductor material i composed of cations ci and anions ai, forms a big molecule of the average hybrid energy 2; = (€2 + &/2. Thus the charge transfer between material 1 and material 2 is governed by the average hybrid energy difference S i - 2;. If this difference is non-zero, valence electron charge will be transferred into the material with the lower value of 2;. As pointed out above, the charge transfer will result in an electrostatic potential difference at the interface. The potentials on either side of the interface will add to the average hybrid energies and diminish their difference. Charge will flow until the average hybrid energy difference has been equilibrated by the potential jump. This is described by the relation

(3.234) -1 ch - ecpi = F; - ecp2,

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3.7. Semiconductor microstructures 393

i.e. the average hybrid energy levels on the two sides of the heterojunction align, and the potential contribution -e[cpl- 1,721 to the valence band offset matches IS; - T i ] .

At this point it is advisable to recall a result concerning deep levels of transition metal (TM) atoms obtained in section 3.5. There, it was shown that the difference between the position E+M of a deep level of a particular TM atom in a semiconductor material 2, and the position E&M of the same deep level in a semiconductor material 1, equals the average hybrid energy difference between the two materials,

ETM 2 - E&M = 7; - Sk. (3.235)

Taking the deep level positions to be measured with respect to the valence band edges E& of the two infinite bulk materials, we denote these positions by E ) ~ , i = 1 ,2 , so that

E ) ~ = EkM - EVi. 0 (3.236)

Considering relations (3.234) to (3.236), the difference [ c $ ~ - t+M] between the positions of the deep level in the two materials is just the valence band discontinuity A E , of the two materials, hence

1 (3.237) 2 AEIJ = [ E T M - 6 ~ ~ 1 .

This relation reduces the experimental determination of valence band discon- tinuities to measurements of deep level positions of TM atoms with respect to the valence band edge. Other experimental methods rely on measurements of photoemission, optical or transport properties of heterojunctions. Ex- perimental values for valence band discontinuities of several heterojunctions are shown in Table 3.10. Often, different values are obtained by different methods or even by different authors using the same method. This points up the experimental difficulties in determining valence band discontinuities, and also to the dependence of the discontinuities on the preparation of the heterojunctions. In order to obtain theoretical values for valence band dis- continuities, electronic structure calculations are required with an accuracy of less than 0.1 eV. Such an accuracy is difficult to achieve so that, in many cases, calculated valence band discontinuities have also considerable uncer- tainty. To this day, a great deal of effort is devoted to the task of obtaining more reIiable experimental and theoretical data for AE,, even in the case of the most thoroughly explored heterojunction, that being between GaAs and (Ga,Al) As alloys.

Other band discontinuities

Knowing the lineup of the valence band edge at I? for a particular hetero- junction, the lineups of all other energy levels at I' and off I' can be deter- mined from the known bulk band structures of the two materials. Below we

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394 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Junction Ge/GaAs InAs/GaSb GaAs/ZnSe

A E , ( e V ) 0.54 -0.46 0.96

AE, (eV) 0.23 0.81 0.29

A E g ( e V ) 0.77 0.35 1.25

HgTe/CdTe InP/CdS

0.02 1.63

1.41 -0.39

1.43 1.24

demonstrate this taking the conduction band edges of a GaAs/Gal-,Al,As heterojunction as an example. In Figure 3.57 the valence and conduction band edges of the Gal-,Al,As alloy are plotted as a function of x. While GaAs and Gal-,Al,As with x < 0.42 are direct semiconductors with both the valence band maximum and the conduction band minimum at I‘, alloys with z > 0.42 and pure AIAs form indirect materials with the valence band maximum still at r, but the conduction band minimum at the BZ bound- ary point X . These two cases are also to be distinguished in aligning the conduction band edges of the heterojunction.

In the f is t case (x < 0.42, the conduction band edges Ec1,2 of the two materials are obtained from the relation

Ec1 = Ev1 + E ~ I , Ec2 = E v 2 + Eg2, (3.238)

where E,l,2 are the gap energies of the two materials. For the conduction band discontinuity AE, = E,2 - E,1 of the ‘material l/material 2’ hetero- junction, it follows that

AE, Ec2 - Ecl = -AEv + AEg, (3.239)

where AE, = E,2 - E,1 is the discontinuity of the energy gap. If AE, and AE, are known, AE, can be calculated by means of relation (3.239). The gap discontinuity AE, follows from the gap energies Egl, Eg2 of the two infinite bulk materials. The lineup thus obtained for the conduction band edge of a GaAs/Gal_,Al,As heterojunction with x < 0.42 is schematically plotted in Figure 3.58a, together with the lineup of the valence band edge. Often, the two band offsets AE, and AEc are expressed in percent of the gap discontinuity AE, by means of the ratios Qv = AE,/AE, and Qc = AE,/AE,. Because of AE, + AE, = AE,, one has Q , + Qc = 1.

In the second case, i.e. for GaAs/AlAs or GaAs/Gal-,Al,As hetero- junctions with alloys having an indirect gap (z > 0.42), the conduction band

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3.7. Semiconductor microstructures 395

f 2.0 Figure 3.57: Valence band edge at r together with the lowest conduction band levels at I? and X for Gal-xA1xAs

Fe(1 + /2+) acceptor level is used as energy origin following the discussion in the main text. (Af ter Langer and Hein- rich, 1985.)

c

3 c

alloys of varying composition x. The 1.6

Fe( 1+/2+) ---.- -.+ 0.4

Ec2

Ev2 I

X

r

r

Composition x -

r I t c 2

Ev2

Figure 3.58: Conduction and valence band lineups for a GaAs/Gal-,Al,As hetero- junction. In part (a) ( x < 0.42) the alloy gap is direct, and in part (b) (x > 0.4!2), the alloy gap is indirect. In the latter case, the r and X levels forming the con- duction band edges in, respectively, GaAs and Gal-,AI,As are also plotted in the respective other material where they do not form the conduction band edge.

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396 Chapter 3. Electronic structure of semiconductor crystals with perturbations

edges of the two materials occur at different k-points, that of GaAs at r, and that of the Gal-,Al,As alloy at X . Recalling the discussion about the stationary electron states of heterojunctions at the outset of this section, it becomes evident that it is no longer meaningful to plot the conduction band edge throughout the whole heterojunction as before. What may be plotted is the lineup of the r and X levels forming the band edges in one of the two materials. This is done in Figure 3.58b. The illustration indicates that for the conduction band states of GaAs with wavevector r, the Gal-,Al,As alloy region forms a barrier, and for the conduction band states of the alloy at X , the GaAs region does so.

Types of heterojunctions

Consider two direct gap semiconductors with the valence and conduction band edge at I?. There are several qualitatively different possibilities for the lineup of the two band edges E,i and Eci, i = 1,2 (see Figure 3.59). The conduction band edge of one material, say of materiai 1, may lie below the conduction band edge of material 2, while simultaneously the valence band edge of material 1 may lie above that of material 2. This case is referred to as heterojunction of type I. In this case the gap of material 1 is located entirely within the gap of material 2. If both the conduction and valence band edges of a particular material, say, again 1, are, respectively, below the two edges of material 2, one has a heterojunction of type II. The staggered type I I heterojunction occurs when we have Ec2 < E,1, and also E,1 < Ec2, E,1 < Eva holds, and the misaligned case applies if Ec2 > E,1. In the staggered case the heterojunction still has a gap, while in the misaligned case the conduction band of material 1 overlaps with the valence band of material 2 so that the gap of the heterojunction disappears. Sometimes, type ZZI heterojunctions are defined. These are heterojunctions of type I with zero energy gap in material 1. A look at Table 3.10 shows that the heterojunction Ge/GaAs is of type I, CdS/InP of type 11- staggered, InAs/GaSb of type 11-misaligned, and HgTe/CdTe is of type 111.

3.7.2 Microstructures: Fabrication, classifications, examples

Heterojunctions cannot be fabricated by simply putting together two sepa- rately made semiconductor samples with plane surfaces. If one would pro- ceed in such a way the result would be a completely rough, polluted interface, incapable of hosting electron and hole states which extend upon both mate- rials, a property which has been assumed above and which turns out to be crucial for the electronic behavior of a heterojunction. In actual practice, the second material must be placed on top of a crystal made from the first by continuing the crystal growth, a process referred to as epitaxial growth. In

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3.7. Semiconductor microstructures

>r

W C

P W

397

+

x

F 15

Ev2

A

Type staaaered EC2

t v i

misaligned d

0 2 0 2 material 1 material 2 material 1 material 2

Figure 3.59: Heterojunctions of type I, I1 and 111.

this kind of growth the underlying crystal, the so-called substrate, imposes its structure onto the growing layer, in contrast to ordinary deposition of an evaporated material which commonly results in a non-regularly structured layer. For epitaxial growth to be possible, the two materials must have sim- ilar crystallographic structures, and their lattice constants must be close to each other.

Epitaxial growth

The fabrication of heterojunctions by means of epitaxy suggests to proceed from simple structures consisting only of the substrate and the epitaxial layer, to more complex structures by growing a second epitaxial layer of an- other material on top of the first, a third layer on top of the second etc. If one does so, one obtains double and multiple heterostructares. The term su- perlattice is used if alternating layers of two materials are grown with equal thicknesses for layers of the same material (see Figure 3.60). Two epitaxial growth techniques are particularly important in this context, namely Molec- ular Beam Epitaxy (MBE) and Metal Organic Vapor Deposition (MOCVD).

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398 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Figure 3.60: Heterostructures as grown by epitaxy: single heterostructure (a), double heterostructure (b), multiple heterostructure (c), superlattice (d).

The principle of MBE is shown in Figure 3.61. The whole growth process takes place in a vacuum chamber under UHV conditions (typically about 2 x 10-l' T o w ) . In this chamber, the substrate is placed in front of a number of effusion cells which contain the various chemical elements to be deposited on the substrate. If one, for example, wants to grow a AlAs layer on top of a GaAs substrate, two cells are required, one for A1 and one for As, and their shutters must be open at the same time. Doping of the layer may be achieved by opening a third effusion cell containing the dopant atoms. If one wants to proceed with a GaAs layer on top of the AlAs layer, one needs a fourth cell with Ga. In order to grow a GaAs/AlAs superlattice, the shutters of the Ga and A1 cells have to be opened and closed alternatively while the As shutter has to be kept open all the time. There are many parameters which influence the growth process, in particular, the temperature of the substrate, the flux from the effusion cells, and the partial pressures of the various elements involved. The atoms or molecules from the effusion cells may have different sticking coefficients on the substrate. Arsenic atoms, for example, stick much less well on GaAs then Ga atoms. Thus the As partial pressure must be much larger than the Ga partial pressure in the epitaxy of GaAs on a GaAs substrate.

The fact that MBE proceeds entirely in UHV may be exploited in con- trolling the growth process. One may employ characterization techniques which require UHV, such as RHEED for studying surface perfection, or mass spectroscopy for analyzing the composition of the residual gas in the growth chamber (for more on MBE see, e.g., Herman and Sitter, 1989).

Unlike MBE, MOCVD takes place in a chemical reactor at atmospheric

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3.7. Semiconductor microstructures

LIQUID NITROGEN

399

RHEED GUN

COOLED SHROUDS / MAIN SHUTTER

TO VARIABLE SPEED MOTOR

ANDSUBSTRATE HEATER SUPPLY

FLUORESCENT SCREEN

Figure 3.61: Principle of MBE growth.[After Cho and Cheng, 1981.)

pressure or slightly below. The atoms to be deposited, say A1 and As, are provided by metal-organic gases, Al(CH3)3 (trimethyl aluminum) and AsH3 (Arsin) in this case. The A1 and As atoms are liberated from their compounds by means of a pyrolytic process which takes place on top of the heated substrate. After this the atoms are chemically bound to the substrate. In order to proceed with a layer of different chemical composition, say GaAs, Al(CH3)3 has to be replaced by Ga(CH3)3 (trimethyl gallium) in the growth reactor. MOCVD, unlike MBE, operates close to equilibrium conditions. The growth velocity in MOCVD is typically somewhat larger than in MBE because more atoms are provided to the growing layer in MOGVD than in MBE. This makes MOCVD particularly suitable for producing devices based on heterostructures. MBE is more universal and more responsive to control than MOCVD. The advantages of both methods are combined in MOMBE (Metal organic MBE): the growth process takes place in an UHV chamber, as in MBE, but the atoms to be deposited are provided by the pyrolysis of

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400 Chapter 3. Electronic structure of semiconductor crystals with perturbations

metal-organic compounds, as in MOCVD. By means of these and some other growth techniques it became possible to grow high-quality heterostructures of many elemental and compound semi- conductors, as well as from semiconducting alloys. The interfaces of these structures can be made almost abrupt, meaning that the transition from one material to the other occurs essentially within one atomic layer. Un- wanted impurities and structural defects can be excluded to a large extent, both at the interface as well as in the bulk. One can grow layers up to several millimeter thicknesses and beyond but, what is more important from the physical point of view, one can also grow very thin layers, down to the ultimate limit of one atomic monolayer.

It turns out that the electronic structures of multiple heterostructures differ from those of the constituent materials if the layer thicknesses reach the nanometer range. In subsection 3.7.4 we will prove this rigorously and describe the modified electronic structures in greater detail. Here, we will start with a qualitative discussion. We consider a double heterostructure formed by a GaAs layer embedded between two Gal_,Al,As layers with x < 0.42 (see Figure 3.62a). The electron states of this heterostructure with energies close to the conduction band bottom will be spatially confined to the GaAs layer. The latter is called a quantum well (QW) in this context, and the alloy layers are referred to as barriers (for a systematic introduction of these concepts see subsection 3.7.4). The confinement of electron states in the quantum well raises their energy and creates discrete levels, just like for a particle in a potential box. The same statement applies to holes in the case of the double heterostructure of Figure 3.62a. If, instead of type I GaAs/Gal-,Al,As double heterostructures, we consider those of type I1 (see Figure 3.62b,c), electrons and holes are no longer confined to layers of the same material, but to layers of different materials. The central layer in Figure 3.62b which forms a well for electrons is a barrier for holes, and the two outermost layers which are barriers for electrons are (semi-infinite) wells for holes. In the misaligned case of Figure 3.62c, an energy region exists where the stationary states of the double heterostructure are mixtures of the electron states of the two outermost layers and the hole states of the central layer.

Another example of multiple heterostructures with modified electronic properties is provided by superlattices composed of alternating (and suffi- ciently thin) layers of GaAs and Gal-,Al,As with z < 0.42 (see Figure 3.62d). In this case electrons from a GaAs well layer may tunnel through the neighboring Gal_,Al,As barrier layer reaching the next GaAs well, from which they may tunnel through the following Gal_,Al,As barrier layer, and so on. This leads to the formation of Bloch states and an additional en- ergy band structure superposed upon the bulk band structure. It is called a miniband structure. Qualitatively, the same behavior is expected for su-

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P Q,

5

Semiconductor microstructures 40 1

a ) d)

Type I I -staggered

Type II - misaligned

c b

2 2

Figure 3.62: Double heterostructure of type I acting as a quantum well both for electrons and holes (a), and of type I1 acting as a quantum well for electrons and a barrier for holes (b, c). In the misaligned case of type I1 (c) an energy region occurs where electrons and holes may coexist. Parts (d), (e) and ( f ) of the figure show the correponding superlattices giving rise to minibands on top of the bulk band structure. Further discussion is given in the main text.

perlattices of the staggered type I1 shown in Figure 3.62e; solely the gap becomes indirect in coordinate space in this case. For misaligned type I1 superlattices shown in part f of Figure 3.62, Bloch states are formed from the electron and hole states of the respective wells.

It is not surprising that the layer thicknesses must lie in the nanometer range for the confinement and tunneling effects discussed above to occur: 1 nm = 10 A is close to the distance between nearest neighbor atoms in a natural crystal (in GaAs the nearest neighbor distance is about 2.5 A). Het- erostructures with such thin layers possess, so to speak, an artificial atomic superstructure which is likely to result in a modified electronic structure.

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402 Chapter 3. Electronic structure of semiconductor crystals with perturbations

The term artificial semiconductor microstructures or just semiconductor mi - crostructures is therefore used in this context. The term nanostructures is also common. Besides epitaxially grown planar heterostructures and su- perlattices having a 1-dimensional artificial microstructure (in the growth direction), systems with a 2- or 3-dimensional microstructure are also in- vestigated. The number of non-microstructured dimensions determines the number of spatial degrees of freedom of electrons and holes - these systems are termed quasi %, 1-, and 0-dimensionaL Quasi 1-dimensional systems are also referred to as quantum wires, and quasi 0-dimensional as quan- t u m dots. Such systems may be fabricated by means of an additional lat- eral structuring of epitaxially grown thin layers. The more appealing nut- ural growth of quantum wires and dots forms an area of active research at present. One refers to this as self-organized growth (Leonard, Krishnamurthy, Reaves, Denbaars, and Petroff (1993); Christen and Bimberg (1990); Notzel, Ledentsov, Daweritz, Hohenstein, and Ploog (1991); Zrenner, Butov, H a p , Abstreiter, Bom, and Weiman (1994); Stutzmann (1995)). In this book we concentrate on planar microstructures.

The modified electronic structure of semiconductor microstructures r e sults in transport and optical properties which differ from those of the con- stituent bulk materials. Moreover, these properties may be tuned to a cer- tain extent by varying the layer thicknesses and chemical compositions of the materials. The possibility of tailoring their properties makes semiconductor microstructures extremely interesting subjects for micro- and optoelectron- ics. Devices may be created with performance data superior to those of conventional electronic components, or with functions not accessible at all to elements made of bulk materials. High Electron Mobility Transistors ( H E M T s ) and Quantum Well (QW) laser diodes are tangible examples of this that already exist. The concept of semiconductor microstructures was first introduced by Esaki and Tsu (1970). Today, investigations of artificial microstructures are the most active area of semiconductor physics.

Before we examine the electronic structure of these systems in more de- tail, we will present a short overview of the various kinds of planar semi- conductor microstructures. Besides heterostructure systems discussed ex- clusively hitherto, doping microstructures will also be covered. In these, the spatial variation of material composition is replaced by the spatial variation of doping. We start with microstructures composed of different materials, which are by far the most important ones.

Compositional microstructures

As a guide to finding material combinations from which compositional mi- crostructures with modified electronic structure may be formed, one can use a diagram which plots the energy gaps of the various semiconductors against

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3.7. Semiconductor microstructures 403

3.5

3.0

v 2 2.5 h

g 2.0 0

>r 1.5 P 2 1.0 w

0.5

' ZnS ' I \ I I I I I I I

MnSe _ . -

0.0

5.4 5.6 5.8 6.0 6.2 6.4 6.6 Lattice constant ( L I

Figure 3.63: Energy gap versus lattice constant for diamond and zincblende type semiconductors. Full lines represent alloys with direct gaps, and dashed lines rep- resent alloys with indirect gaps.

their lattice constants. Such a diagram is shown in Figure 3.63 for elemental and compound semiconductors of diamond and zincblende structure. Lines connecting two different materials indicate the gap energies of alloys made of these materials. The composition of an alloy is related to its lattice constant by means of Vegard's rule which linearly interpolates between the lattice con- stants of the two alloy components. For two direct gap materials 1 and 2, the gap difference AE, provides some hints about their band edge discontinu- ities. If the gap discontinuity AE, vanishes, then band edge discontinuities may, but need not, occur. If a gap discontinuity does exist, then at least one of the two band edges must also exhibit a discontinuity. In general, both the valence and conduction band discontinuities AE, and AE, differ from zero and their magnitudes reflect the gap discontinuity to a certain extent.

Lattice mismatch

As the materials shown in Figure 3.63 are all of zincblende type, their main

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404 Chapter 3. Electronic structure of semiconductor crystals with perturbations

structural differences lie in their different cubic lattice constants a. Ideally, in epitaxial growth, the lattice constant a1 of the substrate (material 1) should be equal to the lattice constant a2 of the layer (material 2). If they differ, the layer material will not grow with exactly the same lattice parameters as would a free standing bulk crystal of this material, but with a lattice constant parallel to the layers equal to that of the substrate. One says that the layer grows in a pseudomorphic phase. The adjustment of the parallel lattice constant creates a mechanical strain in the layer. The corresponding strain energy increases the total energy of the layer, and this increase grows larger as the layer becomes thicker. Past a critical thickness d,, the accommodation of the lattice mismatch by strain becomes energetically more costly than the formation of lattice defects, in particular of dislocations. The strain will then be released by the formation of dislocation lines (see section 3.2). As long as one stays below d,, only a few dislocations occur and the layer will have good structural perfection despite the strain. One speaks of strained layers. The critical thickness depends on the magnitude of the lattice mismatch. Suppose that a layer of a cubic material 2 is growing on a (100) surface of a layer of cubic material 1, and that the cubic lattice constants a l , a:! of the two materials are different. The relative deviation f = (a:! - a l ) / a l of these constants is termed lattice misfit and measured in percent. For a lattice misfit f of a few percent the critical thickness may reach values not much smaller than 100 A. If f is less than a few tenths of a percent, the lattice mismatch has only little effect and, in many cases, it can be completely neglected. One speaks of lattice matched heterostructures. Otherwise one has lattice mismatched heterostructures.

In order to grow lattice mismatched heterostructures it is important to know the distribution of strain between layers of different thicknesses. Such layers may be the substrate and the epitaxial layer, but also a buffer layer on top of the substrate which is used to accommodate part of the lattice mismatch strain. We suppose again a layer of cubic material 2 which is growing on a (100) surface of a layer of cubic material 1. The non-vanishing components of the stress tensors ~ ( i ) of the two layers i = 1 , 2 are a,(i) and ayy(i) with aee(i) = ayy(i), and the non-vanishing strain components are e x s ( i ) , eyY(i) and e z z ( i ) with e r X ( i ) = eyY(i) . The independent stress- strain relations for a particular layer read

“ 2 4 4 = c l l ( i )€zz( i ) + c12(i)[err(i) + €yy(i)], (3.240)

with cll(i) and clz(i) being the elastic stiffness constants of the cubic layer of material i. Because m t z ( i ) = 0, it follows from (3.240) that

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3.7. Semiconductor microstructures 405

For the parallel strain components e z Z ( i ) and e y 3 / ( i ) one has

a2 - ai a i

4 2 ) - &(1) = tyy(2) - E v y ( l ) = t, t = -.

(3.241)

(3.242)

The total strain E defined in equation (3.242) equals the lattice misfit f divided by 100%. Using the stress and strain components, the total elastic energy Eda of the two layers may be calculated, with the result

(3.243)

where

ci = [Cll(i) + (1 - Ki)Cl2(i)l. (3.244)

Minimizing Eela with respect to txZ(2) (or ~ ~ ~ ( 1 ) ) yields the conditions

(3.245)

According to these relations, the thin layer is more heavily strained than the thick one, provided the elastic constants of the two layer materials are comparable. This means, in particular, that the substrate will be almost unstrained and the epitaxial layer will accommodate almost the whole misfit strain. This was anticipated in the discussion above. Another general point to be mentioned in the context of lattice mismatched heterostructures concerns the effect of strain on the band structure. From theoretical considerations and experimental studies it is well known that such effects can be quite large (Bir, Pikus, 1974). To give an example, we consider a Si layer on top of a Ge substrate. The strain components in the Si layer are e X x = cyy = 0.04, corresponding to a lattice misfit of 4% (see below), and e Z z = -0.03, using the elastic stiffness constants c11 = 16.1 x 1011 dyn/crn2 and c12 = 6.4 x 1011 dynlcm’ of Si. The degenerate heavy-light hole valence band maximum of Si splits by 0.31 el/ under this strain. To produce the same splitting by means of uniaxial strain applied from outside, a pressure of 73 Kbar would be necessary. This example shows that considerable changes of the band structure are to be expected because of the lattice mismatch strain in heterostructures. In this context, energy levels in different materials or at different points of the first B Z can shift in dserent ways. One may take advantage of this to adjust the band discontinuities of a heterostructure to specified conditions. The strain becomes, so to speak, an additional degree of freedom for tailoring the properties of a heterostructure.

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406 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Particular compositional microstructures

GaAs/(Ga,Al)As

GaAs/Gal-,Al,As multiple heterostructures are the prototypes for lattice matched type I microstructures. The valence and conduction band edges of the Gal-,Al,As alloys have already been discussed in the context of Figure 3.57. Here, we add some quantitative estimates. The valence band discontinuity AEv of the GaAs/Gal-,Al,As heterojunction scales almost linearly with x according to the relation AE, = 0.45 x x eV. For the gap discontinuity A E , we have A E , = 1.25 x x eV, as long as we consider the direct region x < 0.42. This gives A E , = 0.85 x x eV for the conduction band discontinuity. With this, the valence band discontinuity takes the constant ratio Q v = (45/125)% = 36% of the gap discontinuity, and the conduction band discontinuity has the constant ratio Q, = 64%.

Beside providing model systems for basic research, GaAs/Gal-,Al,As microstructures are used in devices like HEMTs, heterojunction bipolar tran- sistors (HBPT), and QW laser diodes.

(In,Ga)As/Ga( Sb,As)

The Inl-,Ga,As/GaSbl-yAsy material system forms heterostructures of type 11. The two band edges are lower in Inl-,Ga,As than in GaSbl-,Asy for all compositions x and y. For small values of x and y, especially for InAs/GaSb with x = y = 0, one has a misaligned type I1 heterostructure. The conduction band edge of InAs is 0.14eV below the valence band edge of GaSb. Superlattices based on InAsfGaSb heterostructures have been the subject of intensive basic research. Under certain conditions, these SLs exhibit metallic behavior. For Inl-,Ga,As/GaSbl-yAsY heterostructures with larger x and y, including the trivial GaAs/GaAs homostructure, one has staggered type I1 heterostructures, for which the conduction band edges are above the valence band edges in both materials. Perfect lattice matching is achieved if y = 0.918~ + 0.082.

Si and (Si,Ge)

The lattice constants of Si and Ge are, respectively, 5.43 A and 5.65 A. This gives the lattice mismatch of 4 %, which has already been used above. The critical thickness d, amounts to about 30 A which is rather small. Therefore, one also considers heterostructures between Si as substrate and Sil-,Ge, alloys with low Ge content (x < 0.5) as epitaxial layers. In this case d, can be 100 A or larger depending on the actual value of x. (Si, Ge) alloys may also be used as substrates. Consider, for example, a Sil-,Gey substrate on which a Sil-,Ge, alloy layer is grown first, followed by a pure Si layer. Both layers are strained in this case. For 0 < y < x, the strain in the alloy layer

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3.7. Semiconductor microstructures 407

is compressive, and that in the Si layer tensile. If the same Si/Sil-,Ge, double layer is grown on Si substrate, the whole mismatch strain occurs in the alloy layer while the Si layer remains unstrained. Because of the different strain distributions, the band alignment of a Si/Sio.~Ge0.5 heterostructure on top of a Sio.75Geo.25 substrate differs from the band alignment of the same Si/Si0.5Ge0.5 heterostructure on top of a Si substrate. The latter is of type I, with both the highest valence band edge and the lowest conduction band edge in the alloy. The former still has its highest valence band edge in the alloy, but the conduction band edge of Si is shifted down below the conduction band edge of the alloy by the tensile strain in the Si layer. Thus the Si/Si0.5Ge0.5 heterostructure on Si0.75Ge0.25 substrate is of type 11. The type I1 Si/Sil-,Ge, heterostructures are better suited for applications in electronic devices (in particular HEMTs), because here the free electrons are hosted by the pure Si layer, where the mobility is much higher than in the alloy layer.

(Ga,In)(As,P)/InP

Quaternary alloys of composition Inl--rGa,AsyP1-y may be thought to be formed of the four binary compounds I d s , InP, GaAs and Gap, with com- position ratios given, respectively, by (1-z)y, (1-z)(l-y),zy, and z(1-9). Generalizing Vegards rule, the lattice constant of the alloy may be estimated as 6.058(1- z)y + 5.869(1- z)(1- y) + 5 . 6 5 3 ~ ~ + 5.451~(1- y). It matches with that of InP if the relation z = 0.189y/(0.418 - 0.013~) holds. Varying y between 0 and 1, the lattice matched alloy transforms from pure InP to an Inl-,Ga,As alloy with I = 0.47. Lattice matched Ino,53Gao,47As/InP microstructures may be used to fabricate HEMTs. The fundamental en- ergy gap of the lattice matched Inl-,Ga,As,P1-, alloy varies almost lin- early between 1.35 eV for InP and 0.85 eV for Ino,53Gao,47As, therefore, covering light emission wavelengths down to the technologically important near infrared region. Using Ino.53Gao.47As/InP based microstructures, laser diodes and photodetectors may be fabricated for optical fiber communica- tion at 1.55 pm wavelength, which is the wavelength with minimum losses in quartz-based fibers.

(Zn, Cd) (Se,S) /GaAs

ZnSe and GaAs form an almost unstrained type I heterostructure. If a Znl-,Cd, S,Sel-, alloy of a certain composition z,y is combined with a Znl-,tCd+! SY!Sel-,~ alloy of another composition x’,y’, one obtains a strained heterostructure of type I. Blue-green laser diodes have been fabri- cated from such structures, more strictly, from Zn(S,Se)/(Zn,Cd)Se/Zn (S, Se) double heterostructures, embedded between other 11-VI-compound lay- ers, deposited on top of a GaAs substrate.

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408 Chapter 3. Electronic structure of semiconductor crystab with perturbations

(Hg,Cd)Te/CdTe

HgTe and CdTe are lattice matched because the lattice misfit is smaller than 0.4 %. Being a zero gap material, HgTe forms a type I11 heterostructure with CdTe. The valence band offset turns out to be rather small (M 0.02 e v ) . In Hgl-,Cd,Te alloys the gap opens at 2 = 0.15. For 0.15 < 2 < 1 the alloys are direct gap materials with the valence and conduction band edges at I'. In this range, Hgl-,Cd,Te/CdTe forms type I heterostructures which, for sufficiently small 2, can be applied in infrared detectors.

(Pb,Sn)Te/PbTe

The lattice misfit between PbTe and SnTe is about 2 %, so that strain effects in Pbl-,Sn,Te/PbTe heterostructures are not negligible. Pbl-,Sn,Te alloys are direct gap materials with the conduction and valence band edges on the first B Z boundary at L. For PbTe, the L$ state forms the valence band edge, and the Lg-state forms the conduction band edge (see Figure 2.35). For SnTe, this level ordering is inverted. Thus, the gap of Pbl-,Sn,Te alloys must go through zero for some composition TO. At 77 K , one has zo M 0.4. Thus, Pbl-, Sn,Te/PbTe heterostructures are of type I for z < 20, of type I11 for z = zo, and again of type I for 2 > 20, however, with inverted band edges of the well material in this case. Depending on strain, type I1 situations also seem to be possible. Microstructures based on Pbl-, Sn,Te/PbTe have a potential for applications in infrared laser diodes.

Doping microstructures

nipi-structures

Semiconductor samples with alternating n- and p-type doped layers may ex- hibit properties similar to those of superlattices composed of two different materials (Esaki, Tsu, 1970; Dohler, 1972). Such doping superlattices may be thought of as periodic arrays of pn- and np-junctions with alternating neg- atively and positively charged intrinsic (i) regions between the neutral n- and p-layers. In this context they are sometimes referred to as nipi - structures. The oscillating space charge distribution gives rise to an oscillating electro- static potential which modulates the valence and conduction band edges as shown in Figure 3.64a. If the modulation period approaches the nanometer range, minibands arise just as in the compositional superlattices considered above. Doping superlattices are principally of type I1 because the two band edges, at a given position, are shifted by the same amount of energy, namely the electrostatic energy of an electron. Their periods, generally, cannot be made smaller than 10 nm because of the unavoidable diffusion of dopant atoms. The most common nipi-structures are those based on GaAs.

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3.7. Semiconductor microstructures 409

b) + + + + + + +

+ + + + a) m + - + - + - + + + + +

1- I v- I - I

z Z

Figure 3.64: Valence and conduction band lineup for doping microstructures in a direct gap material for a) a nipi-structure and b) a n-type &doping structure.

&doping structures

By means of MBE, spikes of dopant atoms of only a few nanometer width may be constructed in an otherwise undoped sample. One speaks of planar doping or 6-doping (see, e.g., Schubert, 1994). Like nipi-structures, 6-doping structures are always of type 11. The sheet of ionized dopant atoms forms an electrostatic potential well which binds the emitted free (majority) carriers, i.e. electrons in the case of n-type &doping (see Figure 3.64b), and holes in the case of p-type &doping. The energies of these carriers become quantized just as in the case of a quantum well. The potential well experienced by the majority carriers represents a barrier for the minority carriers. In GaAs, n-type &doping has been achieved, for example, by means of Si, and p-type 6-doping by means of Be.

3.7.3 Methods for electronic structure calculations

The theoretical methods which were developed to calculate the electronic structure of bulk crystals in Chapter 2 and of clean surfaces in Chapter 3 are also suitable for artificial semiconductor microstructures. Below, we will discuss this in the case of compositional superlattices (SLs).

Bulk methods

We consider SLs composed of two zincblende type materials as, e.g., GaAs and AlAs. The interface is taken to be parallel to a (001) lattice plane. A GaAs layer of the SL contains a number 2n of (001) lattice planes alterna- tively occupied by Ga and As atoms, and the AlAs layer contains a number 2m of (001) lattice planes alternatively occupied by Al- and As atoms. The notation (GaAs),/(AlAs), is used for such a SL. The primitive unit cell of

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410 Chapter 3. Electronic structure of semiconductor crystah with perturbations

[ I IO ] 0 Ga O A s 8 A t

Figure 3.65: Primitive unit cell of a (GaAs)3/(AlAs)3 SL.

the SL is spanned by the primitive lattice vectors of a (001) lattice plane, namely, A1 = (a/2)(ez + ey), A2 = (u/2)(eZ - ey), and another vector A3. Provided that (n+m) is an integer, A3 may be taken as A3 = (a/2)(n+m)ez. The pertinent primitive vectors of the reciprocal lattice are then given by

27r 4n B1 = -(ez - ey), B2 = -(ez a + ey), B~ =

27r a ez. (3.246)

(n + m)a

The volume of the first B Z of the SL is a (n + m)-th fraction of that of the first bulk B Z . In Figures 3.65 and 3.66 we show, respectively, the primitive unit cell and the first B Z of a (GaAs)g/(AlAs)~ SL as an example.

The stationary states of the SL are Bloch states with quasi-wavevectors k of the first SL B Z . The energy eigenvalues form bands in this B Z , re- ferred to as minibands in regard to k-dispersion parallel t o the SL axis, and as subbunds if k-dispersion parallel to the SL layers is considered. The mini- bands arise from bulk bands folded back upon the first SL B Z . As the bulk B Z encompasses (n + m ) first SL B Z s , one bulk band gives rise to (n + m) SL minibands. The minibands are separated by min igaps which occur at the Bragg reflection planes of the reciprocal SL lattice perpendicular to the z-axis. The minigaps become wider for stronger perturbations of the bulk crystals due to the superlattice structure, i.e. the larger the difference of the periodic potentials of the two bulk crystals, and the shorter the SL period is (as long as it does not become too short), the wider the minigaps open.

All methods for determining band structure of bulk crystals can also be used to calculate the band structure of SLs, the only difference being the larger size of the primitive unit cell. Figure 3.67 shows the results of such a calculation performed by means of the TB method. The valence band disper- sion of a (GaAs)a/(Ga0,~A10,3As)3 SL is plotted along the r - Z symmetry

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3.7. Semiconductor microstructures

1". 41 1

Figure 3.66: First B Z of the (GaAs)3/(AlAs)3 SL of Figure 3.65.

line of the first SL B Z and, to demonstrate the folding character of the SL band structure, also along the r - X line of the first bulk B Z . Important properties of the corresponding SL eigenfunctions are illustrated in Figure 3.68 for a (GaAs)7/(AlAs)T SL. The lowest valence and conduction band states are represented with respect to their spatial variations and symmetry characters. While the valence band states are well localized in the GaAs layers and are dominantly composed of GaAs-r-states, both AlAs-X-states and GaAs-r-states contribute to the conduction band states of the SL. This illustrates the general discussion of subsection 3.7.1 about the existence of eigenstates of heterojunctions which arise from bulk states with different k-vectors in the two materials .

Bulk methods may also be used to calculate the electronic structure of non-periodic microstructures such as GaAs/AlAs single heterostructures or (Ga, Al)As/GaAs/(Ga, A1)As double heterostructures. In these cases one may use a procedure similar to the slab method for surfaces: At the outset, one defines a slab with a sufficiently thick but finite GaAs layer and forms a single heterostructure with a sufficiently thick but finite AlAs layer, and then repeats this slab periodically. In the case of the (Ga, Al)As/GaAs/(Ga, A1)As double heterostructure, one embeds the GaAs layer between thick but finite (Ga, A1)As layers, and repeats the slab thusly formed periodically. In this way an SL is simulated whose electronic structure can be calculated by means of any of the bulk band structure calculation methods.

However, in many cases the application of bulk methods to artificial mi- crostructures is not appropriate. These methods yield the totality of energy bands of a microstructure in all parts of the first B Z , while in most cases only a small number of minibands are of interest because only these undergo changes in comparison with the back-folded bulk bands. Moreover, even the

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412 Chapter 3. Electronic structure of semiconductor crystals with perturbations

- t o 3 - -1 A ol i!

W

-2

-3

- 4

-F;

- €

Figure 3.67 Valence band structure of a (GaAs)3/(Gao.7Alo.3As)3 SL calculated by means of the TB method, and plotted upon the r - 2 symmetry line of the first SL B Z (left) and the I?-X line of the first bulk B Z (right). (Af ter Riicker, Hanke, Bechstedt, and Enderlein, 1986.)

changes of these few minibands are small. The bulk methods are often not accurate enough to reproduce them sufficiently well. The situation is similar to the case of the shallow levels of impurity atoms - if one would try to ob- tain these levels by means of a full bulk band structure calculation method the same difficulties would occur. Fortunately, another method, namely the effective mass theory, exists in the shallow level case. It is particularly well tailored to calculate the small changes of electronic structure occurring at the conduction band minimum and the valence band maximum, where the kinetic energy is small enough to allow the perturbation potential to produce measurable effects. The situation in the case of artificial microstructures is comparable: the perturbation potential is relatively weak, and changes of the band structure are expected only for selected bands and in the vicinity

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3.7. Semiconductor microstructures 413

GaAs A1 As r X

Figure 3.68: Localization of the lowest valence and conduction band states of the SL at the center r of the first SL B Z (left), and the symmetry character of these states (right), for a (GaAs)7/(A1As)7 SL. While the valence band states are well localized in the GaAs layers and are dominantly composed of GaAs-r-states, conduction band states extend over both layers, and are composed of both AlAs- X-states and GaAs-I'-states. hl = -0.11 eV,h2 = -0.19 e V , el = 1.75 eV,e2 =

1.77 eV, e3 = 1.88 eV.(After Rucker, 1985.)

of critical points. There are, however, also differences between the shallow level problem and the microstructure problem. The perturbation potential, i.e. the difference of the periodic oneelectron potentials of the two m a t e rials of a heterostructure, is far from being smooth on the atomic length scale. Even if one averages out the microscopic potential fluctuations over a unit cell, the average potential difference has still an abrupt change at an interface. For this reason one has to determine at the outset whether or not the effective mass theory developed in section 3.3 can in fact be applied to artificial microstructures. We will address this question below (for a detailed discussion see, e.g., Burt, 1992).

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414 Chapter 3. Electronic structure of semiconductor crystab with perturbations

Effective mass theory for microstructures

Consider a single heterostructure. There is no doubt that the effective mass theory may be applied to each of the two infinite half spaces of this structure filled with material 1 for z < 0 and with material 2 for z > 0, provided the conditions of validity of effective mass theory are satisfied in each of the two regions separately, which we will assume. Accordingly, we restrict ourselves to eigenstates of the heterojunction having energies in the vicinity of the band edges E,1 and E,2 of the two bulk materials. Both edges should occur at the centers I? of the respective bulk B Z s . The two band edges at are assumed to be non-degenerate and isotropic (the degenerate case will be treated separately). Furthermore, we suppose that, at critical points off r, all bulk band energy levels are far removed from the energies of the eigenstates under consideration. Then these eigenstates are composed of only bulk states with k-vectors in the vicinity of r, just as is assumed in effective mass theory (in fact this theory assumes composition of bulk states from the vicinity of a particular critical point which need not necessarily be f). If the eigenstates were composed of bulk states from different parts of the B Z , say from f and X as happens in the case of the conduction band states of the (GaAs)7/(AlAs)i. SL shown in Figure 3.68, then the effective mass theory could not be applied. If, however, an eigenstate of the heterostructure may be formed only from bulk states at an off-center critical point like X , then the effective mass theory is applicable as well.

Non-degenerate band edges

In the case under consideration, the eigenfunctions &l(x ) and $4~) of the two material regions 1 and 2 having energies, respectively, close to the band edges Evl and E,2 at f, may be written as

The two envelope functions F,l(x) and Fv2(x) obey the effective mass equa- tions

F,l(x) = E,F,i(x) , x in mater ia l 1, (3.248)

F V ~ ( X ) = E , F , ~ ( x ) , x in mater ia l 2. (3.249)

Formally, these equations may be written as one equation for the envelope function

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3.7. Semiconductor microstructures 415

(3.250)

of the whole heterostructure by introducing a z-dependent effective mass m * ( z ) and a z-dependent band edge E,(z) defined, respectively, by

We write this one equation as

(3.252)

The true wavefunction $,(x) of the heterostructure is given by the expression

(3.253)

(3.254)

is the Bloch factor of the heterostructure. Clearly, equation (3.252) holds only within the two material half spaces, but not at the interface. Thus it determines the possible envelope function solutions for z < 0 and z > 0. At z = 0, the solutions for the two half spaces must be connected in an appro- priate way to form an envelope eigenfunction for the entire heterostructure. To do so, matching conditions are required, which we will now discuss.

Firstly, we derive a condition for the change of the envelope function across the interface. To this end we consider the interface behavior of the Bloch factor ud(x) in equation (3.253) for the total wavefunction &,(x). If this factor was continuous at the interface, the envelope function F,(x) also would have to be continuous there since the total wavefunction @,(x) must be continuous everywhere. Thus the condition

Fv(z, Y, 0 - 6 ) = F,(z, Y, 0 + 6 ) (3.255)

should hold in the limit 6 --+ 0. Unfortunately, no general proof exists for the continuity of the Bloch factor at a heterostructure interface. The only case in which such a continuity is assured is that of a 'heterostructure' composed of two identical materials. Having this obvious result in mind, it is commonly

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416 Chapter 3. Electronic structure of semiconductor crystals with perturbations

argued that the Bloch factor should be continuous at an interface, at least in an approximate sense, if the two materials of the heterostructure are not too different from each other with respect to their energy band structures. This happens, for example, in the case of GaAs/Gal-,Al,As heterostruc- tures with sufficiently low z-values. It turns out, however, that effective mass calculations based on envelope function continuity, also yield correct results for heterostructures made of materials with considerably different band structures. Thus, similarity of band structures, while being sufficient, does not seems to be necessary for the applicability of effective mass the- ory to heterostructures. In fact, there have been various attempts in the literature to justify the continuity condition (3.255) without using the band structure similarity argument. In our opinion, it is essential to realize that the Bloch factor ul/o(x) in equation (3.252) occurs in an eigenfunction of the entire heterostructure rather than in an eigenfunction of the two infinite bulk crystals, as is tacitly assumed in the above discussion. For an infinite bulk crystal, the Bloch factor is that particular solution of equation (2.136) which obeys the lattice periodicity condition. Without demanding satisfac- tion of this condition, there exists a variety of solutions of equation (2.136). In a heterostructure, the two Bloch factors at the interface need not be pe- riodic with respect to z because the lattice periodicity in the z-direction is perturbed. This freedom may be used to satisfy the continuity condition for the Bloch factor at the interface.

Secondly, we derive a condition for matching the interface values of the first derivative of the envelope function with respect to t. To this end we use the physically obvious fact that the probability current density joined with the total wavefunction $ J ~ ( X ) must be the same on the left and right of the interface. This must also hold after averaging with respect to a primitive unit cell, i.e. with respect to a region where the envelope function is almost constant. Applying the well-known quantum mechanical expression for the current density one obtains two contributions. One is due to the gradient of the Bloch factor and it vanishes after averaging. The other contribution arises from the gradient of the envelope function. It is multiplied by the squared modulus of the Bloch factor, which yields 1 after averaging because of normalization. This means that the average current density in state $v(x) is obtained by applying the quantum mechanical expression to the envelope function Fv(x) alone rather than to the total wavefunction &,(x). Since the average current density must be continuous at the interface, and since the envelope function itself has this property because of relation (3.255), we arrive at the conclusion that the first derivative of the wavefunction must obey the relation

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3.7. Semiconductor microstructures 417

which is known as the BenDaniel-Duke boundary condition. This condit,ion follows automatically if the Schrodinger equation (3.252) is also applied at z = 0 after the kinetic energy term has been replaced in accordance with

v. h2 v2 -.+ --v.- 2m*(z) 2 m * ( z )

h2 _- (3.25 7)

With this replacement the effective mass equation (3.252) becomes

Integrating this equation with respect to z from -6 to +6 one obtains the boundary condition (3.256). The kinetic energy operator of the Schrodinger equation (3.257) is Hermi- tian, as any reasonable kinetic energy operator must be in order to avoid complex energy eigenvalues. The original form -[l/2m*(z)]V2 of the kinetic energy operator is not Hermitian, and must be rejected. Its replacement by the right hand side of relation (3.256) cannot be justified, however, by the hermiticity demand alone since there are also other ways to introduce her- miticity. In fact, one can easily demonstrate that any operator of the form -(F,2/4)[m*a(z)Vm*~(z).Vm*~(z)+m*~(z)Vm*~(z).Vm*a(z)] is Hermitian if a, p, y are real numbers obeying the relation (Y + ,B + y = -1. This implies that the boundary condition (3.256) involves more than just the hermiticity of the kinetic energy operator.

In the effective mass equation (3.258) the band edge plays the role of an external potential. Since it depends only on z , and since the effective mass does so also, the dependence of the envelope function on the component XI[ of x parallel to the layers may be taken in the form of a plane wave of parallel wavevector kll, therefore as

(3.259)

For Fu(z) the Schrodinger equation follows as

where EL is given by

h2 2m* , k i EL = E, - - (3.261)

with l/E; as the average inverse effective mass (m;:' + m;T1)/2.

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418 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Degenerate band edges

In the case of degenerate band edges E v ( z ) , the Hamiltonian’s H i of the two materials i = 1,2, are of the general form

@

(3.262)

derived in equation (3.73), and the components Fym(X) of the envelope func- tions F~ ( x ) obey the effective mass equations

*

[-~g$r(z)aaaat + ~ v ( z ) 6 m m i ] Fumt(x) = ~ v ~ v r n ( x ) , z # 0, m’ ad

(3.263) with DE$(z) = DYZAl for z < 0, and Dg$l(z) = D;ELl for z > 0. The solution of equation (3.263) may again be taken as a plane wave

(3.264)

=+ parallel to the layers. The boundary conditions for F v ( x ) at the interface follow in the same way as in the non-degenerate case. The z-dependent factor F~ (2) of F,, ( x ) in equation (3.264) must be continuous at z = 0, i.e.

F U (z) lz=-6 =Fu ( z ) I~=+6 (3.265)

must hold for 6 + 0. To determine the condition for the derivative with respect to z, one defines the Hermitian Hamiltonian matrix

* *

* *

(3.266)

($

which exists for any value of z including z = 0 and equals H 1 for z < 0 and H z for z > 0. Integrating the effective mass equation for this generalized Hamiltonian with respect to z over a small interval across the interface leads to the conclusion that the expressions

e

must be continuous at the interface z = 0. Below, these general results are specified for the fourfold degenerate rs va- lence band edge of diamond and zincblende type materials. Using the 4 x 4 Luttinger-Kohn Hamiltonian of equation (3.74), the effective mass equation (3.263) becomes

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3.7. Semiconductor microstructures 419

(3.268)

The quantities Q ( z ) , f i ( z ) , g ( z ) , ?(z ) here are the differential operators

k ( z ) = --a [y2(zj(k: - k i ) - 2i73(z)k,kg] . (3.2 72)

The constants M , L , N in equations (3.75) to (3.78) have been replaced by the Luttinger parameters y1,72,y3 using equations (2.381). From equation (3.267) one finds that the following combinations of the components Fm(z) and their derivatives FLm(z) with respect to z must be continuous at the interface z = 0:

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420 Chapter 3. Electronic structure of semiconductor crystals with perturbations

X J L

Figure 3.69: Geometry of the SL whose stationary electron and holes states are calculated in the text.

3.7.4 Elec t ronic structure of particular microstructures

Compositional superIat tices and q u a n t u m wells

We consider a SL composed of two zincblende type semiconductor materials 1 and 2 with the conduction band edges Ecl and Ec2 located at r. To be specific, we may associate ‘1’ with GaAs, and ‘2’ with a Gal-,Al,As alloy for z < 0.42. The thicknesses of the two material layers are denoted by d l and d 2 , and the lattice constant in z-direction by d , with d = d l + d 2 ( s e e Figure 3.69). We want to calculate the stationary electron states of this SL with energies in the vicinity of the conduction band edges Ecl and E,2 using the effective mass theory derived above. These states are the ones which would host the free electrons of the SL, if there were any. We suppose that there are none, as we did before.

Electron states

Effective mass equation and its solutions

For the zincblende type materials we are considering, the conduction bands are non-degenerate and isotropic in the vicinity of I’. The two effective electron masses m:l and mE2 generally differ, but we will initially assume that they are equal, denoting their common value by m;. In the case of SLs composed of GaAs and Gal_,Al,As this is a reasonable approximation. Later, we discuss the modifications which occur if m;l and m:2 are different. The effective mass equation for the electron envelope function Fc(x) of the

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3.7. Semiconductor microstructures 421

thus specified SL follows from equation (3.252) if we set v = c and

f o r Id < z < l d + d l

f o r Id + d l < z < ( 1 + l ) d . (3.274)

Egl + A E , E c ( z ) =

Here 1 is the integer index of the various SL unit cells, -co < 1 < 00.

The envelope function F,(x) may be taken in the form Fckl, (x) of equation (3.260) describing a plane wave of wavevector kll parallel to the layers. The z-dependent factor F,(z) of Fckl, (x) obeys the equation

where E; is given by

EA = E, -

In solving this equation, the boundary conditions

Fc(ld + d l - 0 ) = F,(ld + d l + 0 ) , d d

dz dz -Fc(ld - 0) = -F,(ld + 0 ) ,

d dz -F,(ld + d l -

(3.276)

(3.277)

have to be satisfied as well as the normalization condition with respect to the periodicity interval of length C2i = Ld ( L denotes the number of SL unit cells in a periodicity region of volume 0). Applying the Bloch theorem to the SL under consideration it follows that F,(z) may be written as a lattice-periodically modulated plane wave

F,(z) = Fck(z) = L e i k r U c k ( z ) , (3.278)

where k is the quasi-wavevector component parallel to z, and Uck(z) is the su- perlattice Bloch factor. The component k varies within the (1-dimensional) first B Z of the superlattice between - r / d and r l d . It must have the form (2a /Ld) x (0, f l , f 2 , . . .) in order to guaranty the periodicity of F&) with respect to the periodicity interval. The Bloch factor Uck(z) of the superlat- tice has to be distinguished from the Bloch factor uco(x) of the two bulk

f i

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422 Chapter 3. Electronic structure of semiconductor crystals with perturbations

materials defined in equation (3.254). The occurrence of two different Bloch factors reflects the fact that two periodic potentials are present in a super- lattice, one due to the natural crystal structure, and one due to the artificial superstructure. The total wavefunction &(x) &k(x) contains the product of both. It reads

(3.279)

The effective mass equation (3.275), the matching conditions (3.277), and the Bloch condition (3.278) define an energy eigenvalue problem which differs from the well-known Kronig-Penney problem only by notation. The latter problem constitutes an exercise of elementary quantum mechanics and is commonly used to demonstrate the existence of energy bands. In the case of SLs this exercise takes on real physical meaning. In the following we sketch its solution. We restrict ourselves to energy values 0 < E' < A E , or E,1 < E < E,1 + AE, + (7i2/2mr)ki, meaning energies below the lowest allowed energy in material 2 for a given value of kll.

According to classical mechanics, an electron with such an energy cannot penetrate within the superlattice layers made of material 2, it will be confined to layers consisting of material 1. If it hits the interface to material 2, then it will be reflected back to the interior of its material 1 layer. In quantum mechanics, the reflection is not complete, there is a certain probability for the electron to tunnel through a material 2 layer and reach the next neighboring material 1 layer. Although the probability for an electron to stay in material 1 is not unity, as in classical mechanics, but smaller, it is still much larger than that for material 2. As has already been mentioned, one uses the terms quantum wells for the layers of material 1, and bamiers for the layers of material 2. In the wells and barriers the wavefunction F,]F(z) is given by different expressions. For the wells 0 < z < d l , Schrodinger's equation (3.276) yields

F,]F(z) = ale iKZ + ble-iKz, 0 < z < d l , (3.280)

with K as a real number which determines the energy E' by means of the relation

I h2 E, = -K2 2 m ~ ' (3.281)

and al , bl as coefficients which are still to be determined. For the barrier d l < z < d one has

Fck(z) = a2enz + b2e-nz, d l < z < d , (3.282)

where we have set

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3.7. Semiconductor microstructures 423

2 K E . =

2mE Ti2

-(AE, - EL). (3.283)

The four coefficients a l , b l , a2, b2 are governed by a set of linear equations which follows from the 4 matching conditions (3.277). These equations read

For this set to have a non-trivial solution, the determinant of the matrix of exponentials must vanish. This yields the secular equation

tc2 - K~ 2Kn:

cos(Kd1) cosh(tid2) + - sin(Kd)l sinh(~d2) = cos(kd). (3.285)

Allowed K - and &-values must satisfy equation (3.285). The values of K and K are not independent of each other, both are determined by the energy EL through equations (3.281) and (3.283), respectively. Thus equation (3.285) represents a condition for EL. It coincides with that of the Kronig-Penney problem. As we know from the solution of the latter, there are discrete energy eigenvalues EL, n = 1,2,. . ., for a given quasi wavenumber k. They split into energy bands E&(k), separated by energy gaps, if k varies within the first B Z . To obtain the total energy eigenvalues E, = Ecn(k) of the superlattice, according to equation (3.276), one still has to add E,1 + (Ti2/2m*)kll to E L ( k ) , with the result

h2 2mr

Em(k) = E&(k) + Egi + -kll (3.286)

Unlike the Kronig-Penney problem, these energy bands exhibit an additional dispersion with respect to the wavevector component kll along the SL layers. Commonly, the energy bands and gaps for a fixed value of kll, but with varying values of k , are termed minibands and minigaps. Bands which arise for fixed IF and varying kll are termed subbands. The kll or subband dispersion is due to the natural crystal structure, while the k or miniband dispersion follows from the artificial microstructure. In Figure 3.70, central part, the sub- and minibands of a SL are schematically plotted. The effective masses of the minibands become negative at the boundaries of the superlattice BZ in the case of odd band numbers n, and at the center of the BZ for even n.

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424 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Figure 3.70: Mini- and subbands of a superlattice. Also shown are the two limiting cases of extremely wide and/or high barriers and of narrow and/or flat barriers. In the lower part of the figure the wavefunctions are illustrated.

Multiple quantum well versus superlattice behavior

Figure 3.70 also shows two enlightening limiting cases. If the thickness d2 of the barriers between the wells becomes large, and/or the barriers AE, are high, then the hyperbolic terms in equation (3.285) dominate over the cos(kd)-term on the right hand side which is responsible for the k-dispersion. Neglecting the latter, the minibands degenerate into discrete energy levels, termed sublevels. These levels are just the discrete energy eigenvalues of a single quantum well. Obviously, the superlattice decomposes into isolated quantum wells in the limiting case under consideration. The electron states are almost completely confined within a well, and practically zero within the barriers. One uses the term multiple quantum well for a superlattice in this limit.

For infinitely high potential walls the secular equation (3.285) becomes sinKd1 = 0. The solutions of this equation are K-values of the general form K = (7r/dl)n where n is an integer. They give rise to the energy eigenvalues

2 El = , n = 1 , 2 , . . . ,

2mE cn (3.287)

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3.7. Semiconductor microstructures 425

and Bloch factors

(3.288)

The confinement to the well region is complete in this limit. Assuming d l = 100 A and mE = 0.067 m, one gets a value of 57 meV for the first energy level ELl. This means that the ground state of the quantum well is shifted up by this energy amount as compared to the conduction band bottom of the infinite bulk crystal. This shift is a consequence of the localization of an electron within the quantum well which leads to a momentum uncertainty because of Heisenberg's uncertainty principle.

We consider next the opposite limiting case, that of extremely narrow and/or flat barriers (see Figure 3.70). In zero-th approximation the secular equation (3.285) takes the form k = K . In this limit the minigaps vanish completely, and the superstructure which distinguishes the superlattice from a natural crystal has no effect at all. In the first non-vanishing approximation with respect to the SL potential, gaps are opened at the Bragg reflection points k = ( n n / d ) of the 1-dimensional reciprocal lattice of the SL. For the energy splitting between the two bands E&+l(n/d) and E&(n/d) , the perturbation theory developed in section 2.4 yields

1 E&+1 (:) - E L (:) = -AEc 7rn sin ( y d g ) . (3.289)

According to this expression, the minigaps are proportional to the conduc- tion band discontinuity AE,. They decrease with band number n. Thus, for large n, there are no substantial energy gaps even if sufficiently wide and high barriers exist. One may say that for minigaps to occur the product of wavevector k = ( x n / d ) and superlattice period d should not be large com- pared to 1. In other words, the superlattice period d should not be large in comparison with the electron wavelength X = 2 r / k corresponding to k. This proves and specifies the expectation stated at the outset of this section, that substantial changes of electronic structure are to be expected if the superstructure occurs on a sufficiently small length scale. The length scale turns out to be that of the de Broglie wavelength X of an electron. For a conduction band electron of silicon, having an energy equal to the average thermal energy $kT at room temperature, one gets A =" 20 nm. Similar values are obtained for other semiconductors. Thus the layer thicknesses of superlattices have to be on the order of, or smaller than, 10 nm in or- der that superlattice effects on free carriers may become important at room temperature.

The limit of narrow and flat barriers is the true superlattice case, as distinguished by the coupling of quantum wells with one another. The en-

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426 Chapter 3. Electronic structure of semiconductor crystals with perturbations

ergy eigenfunctions also have non-vanishing values within the barriers, in contrast to multiple quantum wells where these values are almost zero. In a superlattice, an electron having a particular allowed energy, may tunnel from its quantum well through the barrier to the neighboring well. Since there it encounters the same allowed energy value from which it may tunnel back, the tunneling will be amplified, as in a Fabry-Perot resonator in the case of light. One has so called r e s o n a n t tunnel ing . It causes the discrete energy levels of isolated quantum wells to broaden into bands.

Electron density of states of SLs and QWs

The general expression (2.212) for the density of states (DOS) p ( E ) of a 3D crystal may be immediately applied to a SL. To this end one has to replace the energy band structure E,(k) in formula (2.212) by the expression given in equation (3.286). Then the DOS p s ~ ( E ) of the SL is obtained as

To evaluate this expression further, the miniband dispersion EA(k) has to be known. In the particular case of isolated QWs the dispersion vanishes completely, and the DOS of equation (3.291) takes the form of the DOS PMQW(E) of a MQW structure,

(3.291)

Each subband n gives rise to a step-like partial DOS, being zero below the band bottom E L + E g l , and non-zero but constant above. Such behavior of the DOS of an electron which is free to move only in two dimensions was already found in section 2.5. The summation in expression (3.291) for p ~ ~ w over all subbands results in a staircase-like DOS as schematically shown in Figure 3.71. The heights of the steps have the constant value ( m * / n 2 h 2 ) ( n / d ) , while the step widths depend on n. If the barriers of the QW are taken to be infinite, then the energy levels EAn are given by (h .2 /2mE)(n~/d1)2 , and the step widths of the staircase scale with ( n / d ~ ) ~ . For very wide QWs, i.e. in the limit d l -+ 00, the staircase-like DOS trans- forms into the smooth square-root-like DOS p 3 ~ ( E ) of an electron free to move in three dimensions. Equation (3.291) yields

(3.292)

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3.7. Semiconductor microstructures 427

4

7 3 3

0 1

3 0 2 Q

1

Figure 3.71: Electron density of states PQW of a QW as function of energy.

with f = limd,,,(dl/d) being the geometrical fraction of material 1 regions in the infinite MQW. The staircases of the DOS in Figure 3.71 are always below or at the limiting fractional 3D DOS f p j ~ ( E ) . The case when they are at the DOS occurs for energies E at the subband bottoms.

Refinements

In the above treatment of the electron states of a SL, the effective masses mzl and mE2 were assumed to be the same in the two materials. If this is not the case, the following modifications will occur. First, in the boundary con- ditions (3.277) for the derivative ( d F , / d z ) , different mass factors will appear on the two sides of the interface and they will not cancel. For the secular equation (3.285) this means that all K's except those which are factors in arguments of trigonometric functions, have to be replaced by (m&'mal)K. Second, in solving the secular equation, the relations (3.281) and (3.283) between K and Ei and, respectively, n and E' must be written with m; and mE2 instead with mE. Finally, the kll-dispersion of the total energy eigenvalue in relation (3.286) becomes different in the two material layers. This results in an additional contribution (h2/2)[mZ- ' ( z ) - E ~ - ' ] k ~ to the SL potential, as may be seen from equation (3.260). The additional term introduces a kll-dependence of the envelope function F& which otherwise depends only on the z-component k .

Another simplification made in the above treatment of the electron states of a SL, was the assumption that no electrons are available to occupy the SL

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428 Chapter 3. Electronic structure of semiconductor crystals with perturbations

0.5 x10’2cm-2 1.5 x10’2cm-2 6 xlO’*cm-* t

Figure 3.72: Self-consistently calculated potential profiles and energy levels for a Alo.aGao..rAs QW of 100 A width containing different numbers of electrons per cm-2.

bands. If there are such electrons present, the potential seen by an electron will change due to its interaction with other electrons in the SL bands. As in the case of bulk semiconductors, this potential change may be decomposed into a Hartree and an exchange-correlation part. The expressions derived for these potentials in Chapter 2 also apply here if the stationary states of the bulk crystal are replaced by the stationary states of the SL. As the latter states are only known after the effective mass equation has been solved, the inclusion of the Hartree and exchange-correlation potentials must be performed in a self-consistent way. The net effect of the two potential parts is repulsive, thus the SL potential well will flatten. For a given number of electrons in SL states, this flattening will be enhanced by greater localization of the electrons in the well regions. It will be particularly pronounced for isolated QWs. In Figure 3.72 we show self-consistently calculated potential profiles and energy levels for a Alo.3Gao.7As QW of 100 A width, containing different numbers ng of electrons per ~ r n - ~ . The well bottom bends up with rising ns, and the energy levels are shifted to higher energies.

Hole states

We consider a QW structure composed of zincblende type well and barrier materials which are both described within the Luttinger-Kohn model. The effective mass equation for the hole states of such a QW with energies close to the well bottom follows from equation (3.268) by installing there the valence band edge profle E,(z):

0 f o r O < z < d

A E , f o r z < 0 and z > d . E,(z) = (3.293)

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3.7. Semiconductor microstructures 429

Figure 3.73: Hole subbands of a 100 A wide Alo.sGao.TAs QW. ( A f t e r Altarelli, Eken- berg, and Fasolino, 1985.)

In this way, a set of second order differential equations for the four hole envelope function components arises. The solutions must be normalized and must guarantee for the continuity of the expressions (3.273) at the two interfaces. Solutions of this kind can be found in different ways. One can, for example, numerically calculate normalized solutions for various values of E and kll, valid in one of the three regions -m < 0, 0 < z < d and z < +m. Matching these solutions in such a way that the continuity of the expressions (3.273) is guaranteed for a given value of kll, will only be possible for certain discrete energy values E,,(kll). These are the subband energy eigenvalues which are sought. Subband structures calculated in this way are shown in Figure 3.73 for a A10.3Ga0.7As QW of 100 A width. The lowest subband ( h h l ) in the vicinity of kll = 0 arises mainly from hole states of GaAs with the heavy hole mass parallel to z , and the second level ( Ih l ) mainly from hole states with the light hole mass parallel to z. Perpendicular to z , the curvature of the hhl subband at starts stronger than that of the Ihl subband. Thus the two bands would cross, if this were allowed. In reality, the bands repel each other leading to the anticrossing behavior seen in Figure 3.73. The alternating heavy- and light-hole-like curvatures parallel and perpendicular to z are consequences of the Luttinger-Kohn Hamiltonian (2.3 74). If the subbands of the QW are partially filled by holes, then hole-hole inter-

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430 Chapter 3. Electronic structure of semiconductor crystals with perturbations :::I p-type &well

8.0 x 10” cm-*

-200 -100 0 100 200 -50

z (4

50 r-type &well

-250 -200 -100 0 100 200

z (A)

Figure 3.74: Potential pofiles and energy levels of an isolated p-type &doping layer (left hand side) and an isolated n-type-&doping layer (right hand side) of the same sheet dopant concentration 8 x 10l2 ~ r n - ~ . The Fermi energy is shown by dotted lines. (After Sipahi, Enderlein, Scolfaro, and Leite, 1996.)

action effects become important which, as in the case of electrons, require self-consistent calculations.

&doping structures

Whereas in compositional microstructures free carriers may, but need not, be present (they were omitted above), &doping microstructures do not exist without such carriers because the doping layer would be neutral if none of the doping atoms would be ionized. This implies that potential profiles and energy levels of doping microstructures cannot be calculated without taking carrier-carrier interaction into account. The main effect of this interaction turns out to be the screening of the Coulomb potential created by the ionized dopant atoms. In Figure 3.74 we show the resulting potential wells for an isolated p-type &doping layer (left hand side) and an isolated n-type 6- doping layer (right hand side), together with the sublevel and Fermi level energies. The dopant atoms are taken to be completely ionized. The lower

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3.7. Semiconductor microstructures 43 1

Figure 3.75: Calculated cur- 100 rent-voltage characteristics of va- rious GaAs/ AlAs SLs with the electric field parallel to the SL axes. Curves 1 to 4 correspond to SLs of different miniband widths A. (After Lea, Horing, and Cui,

80 - a > 60 0 = 40

1991.) 0 > 20

0 0 10 20 30 40 SO 60 70 I

E [kV/cml D

well depth in the p-type case as compared to the n-type case results mainly from the fact that heavy holes are more strongly localized at the ionized dopant sheet and thus are more effective in screening out the sheet potential then the lighter electrons.

Macroscopic manifestations of the electron and hole states of semi- conductor microstructures

The modifications of electron and hole states of semiconductor microstruc- tures manifest themselves in the macroscopic electronic properties of these structures. The more detailed treatment of these properties is beyond the scope of this volume. Here we will give only two characteristic examples, one concerning electric transport and another concerning optical properties. Figure 3.75 shows the calculated drift velocity Vd versus electric field char- acteristics of GaAs/ AlAs SLs with the electric field E directed along the SL axes. Above a critical field value at which the Vd peaks to its maximum value, the drift velocity decreases with increasing field strength, i.e, the dif- ferential conductivity becomes negative. Such a behavior, which originally was predicted by Esaki and Tsu (1970), has been confirmed in more rigor- ous calculations by Lei, Horing and Cui (1991). Experimentally, a sublinear increase of Vd with E has been found above a critical field value, as shown in Figure 3.76. If corrected for effects of the Ohmic contacts and macroscopic electric field inhomogeneities, the experimental Vd versus E characteristics clearly reveals the negative differential drift velocity predicted by Esaki and Tsu. The underlying physics can be understood easily.

Consider a single electron in the lowest SL miniband EAl(b). In the presence of an electric field parallel to the SL axis, the electron is accelerated

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432 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Figure 3.76: Exper- imental current-voltage 300K characteristics of various GaAs/AlAs SLs with the electric field parallel to >- the SL axes. (After 5 Sibille, Palmier, Wang, and Mollot, 1990.)

-

2 4 6 0 BIAS (VI

3

until it reaches the inflection point of Eil(k). Thereafter, its effective mass becomes negative, causing deceleration instead of acceleration. Since the electron is inevitably scattered by phonons and impurities, its velocity takes a stationary value in a dc field. The latter increases with increasing field strength E if the corresponding k-value is below the inflection point, and it decreases with increasing E if the k-value of the electron is above this point. However, a superlattice contains a gas of many electrons, populating the miniband E:(k). For low field strengths, most of the electrons have low k-momenta and positive effective masses. Thus, the total current density of the gas increases if E is further increased. Simultaneously, the electron gas is heated up by the electric field, which implies that more electrons populate k-points with a negative effective mass and less with a positive one. This reduces the current increase if E raises further because a larger subgroup of electrons is decelerated instead of being accelerated by the field increase. If the subgroup of electrons having negative effective mass is sufficiently large, further increase of the field strength will actually decrease the current. This explains the negative differential drift velocity seen in Figure 3.75.

An example of an optical experiment is shown in Figure 3.77. There, the absorption spectra of GaAs/Al,Gal-,As QW structures of different wen widths are shown. The staircase-like shape of these spectra reflects the DOSs of the electron and hole subbands of a QW which above have been shown to be staircase-like as well. Also other features of the QW band structure observed above, as the widening of the fundamental energy gap and the shift of the steps towards higher energies with decreasing well width, can be clearly seen in the spectra of Figure 3.77. The enhancement of absorption at the step edges, which correspond to optical transitions from the tops of hole subbands to the bottoms of electron subbands, is due to the Coulomb

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3.8. Macroscopic electric fields 433

Photon energy ( e v ) ---w

Figure 3.77: Absorption spectra of GaAs/Al,Gal-,As QWs of various well widths. (After Dingle, Wiegmann, and Henry, 1974.)

interaction between electrons and holes (excitonic effects). Beside absorption spectra, photoluminescene spectra are extensively used

for the experimental characterization of semiconductor microstructures. In photoluminescene, one observes mainly radiative transitions between the electron and hole ground state levels of an (undoped) QW because only these are considerably populated. Due to the enlarged energy separation of these levels, a QW emits light at shorter wavelength than the corresponding bulk material does. In the case of GaAs this effect is particularly striking - the bulk crystal emits radiation in the non-visible infrared region, while a GaAs-quantum well of 50 A width emits visible red light.

3.8 Macroscopic electric fields

Macroscopic electric fields in semiconductors are the central focus of many device applications and give rise to important physical phenomena in these materials. For instance, if one applies an external electric field to a semicon- ductor sample, the response yields information about the band structure, impurity states and other microscopic properties of the material. The elec- tric transport properties and electro-optic effects are particularly well suited

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434 Chapter 3. Electronic structure of semiconductor crystals with perturbations

to extract this information. Often, electric fields in semiconductors need not, however, be applied externally as they are already present internally, either because of spatially inhomogeneous doping such as in the case of a pn-junction, or because of other spatial inhomogeneities as surfaces or in- terfaces. In the following theoretical description, the source of the electric field will play a role only inasmuch as we assume that the field is spatially uniform. In practical terms, this means that the field should not change appreciably within a periodicity region, i.e. within a characteristic length of G x a. Since the limit of the infinitely extended semiconductor is effectively reached in good approximation with G M 100 the characteristic length need not to be larger than about 0.01 ,urn. Fields which change only little over this very small distance can be considered as homogeneous. In addition to excluding spatial inhomogeneities of the electric field, we will also exclude temporal changes in our discussion below. This only means that the fre- quencies of these changes should be small compared with the characteristic frequencies of the electrons of the semiconductor.

3.8.1 Effective mass equation and stationary electron states

A semiconductor in a homogeneous external electric field E represents a perturbed crystal in the sense of sections 3.1 and 3.2. The presence of such a field can be described by adding a perturbation potential V’(x) to the one-electron Hamiltonian H of the ideal crystal. This potential is defined as the difference of the energy of a crystal electron in the presence of the electric field and without it, and thus is given by

V’(x) = eE . x. (3.294)

As in the case of point perturbations considered in sections 3.4 and 3.5, the perturbation potential V’(x) of equation (3.294) does not possess lattice translation symmetry. Moreover, V’(x) diverges at infinity. The latter fact implies that, unlike to the case of point perturbations, an infinite crystal in a homogeneous electric field cannot be replaced by a perturbed supercell whose periodic repetition forms an unperturbed supercrystal. The crystal in an electric field has to be treated as what it in fact is, namely an infinite system with 2-dimensional lattice symmetry perpendicular to the field, and no lattice symmetry along the field direction.

If the field strength E is not too large then the perturbation potential (3.295) fulfills the smoothness condition of effective mass theory of section 3.3, which here reads

e I E I a << Eg. (3.295)

With this condition fulfilled, the effective mass equation (3.53) is applicable. Here it has the form

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3.8. Macroscopic electric fields 43 5

{ E y ( - i V ) +eE.x}FV~,(x) = EvFv~,(x). (3.296)

We consider this equation in the particular case of an isotropic and parabolic band. Assuming the z-axis of our Cartesian coordinate system parallel to the field direction, the electron motion in 2- and y-directions is that of a free particle. Thus the envelope function F,(x) can be written as

F~E,(x) E F~E,~,~(X) = e i k ~ ~ . x ~ x,, (2). (3.297)

where x l and k,l denote, respectively, the components of x and of the wavevector perpendicular to the field. The z-dependent envelope function factor X , , ( t ) obeys the equation

X,,(z) = E ~ X , ~ ~ ( Z ) , (3.298)

with l i 2

E, = E” + -k:l. 2m

(3.299)

For the conduction and valence bands, in particular, equation (3.298) reads

(3.300)

(3.301)

The second equation can be interpreted as the effective mass equation for a hole with (positive) mass Im:l and energy (-.,). As the equation shows, the charge of the hole is positive, i.e. fe, in contrast to the electron with charge -e.

The spectra of energy eigenvalues of equations (3.300) and (3.301) are continuous, and vary between -co and fco as a direct consequence of the unrestricted motion of the electron or hole in the field direction. This allows one to use the energies E , as continuously varying quantum numbers of the stationary states X V C v ( z ) , and to assume normalization in terms of Dirac’s &function with respect to these energies, thus

The solutions of the corresponding energy eigenvalue problems for electrons and holes read, respectively,

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436 Chapter 3. Electronic structure of semiconductor crystals with perturbations

with

(3.303)

(3.304)

(3.305)

as so-called electro-optic frequency, and with Ai as the Airy-function of first kind. Note that stationary states X,,(z) of conduction and valence band electrons exist at all energies ey between --oo and +m in the presence of an electric field rather than only at energies in the corresponding e n e r a bands without such a field.

Assuming E = lo4 Vlcm and m: = 0.1 m, one obtains for 0 , a value of about 1013 s - l , and for he, a value of 10 m e V . The profiles of the two envelope functions X E C ( z ) and X , , ( z ) are shown in Figure 3.78 for eC = eV = 0 and Bc = 0,. The electron function decays exponentially with increasing z for e E z > -Eg, and oscillates with decreasing z for e E z < -Eg. On the decaying side, the total energy Eg of an electron is smaller than its potential energy e E z . Thus, this region is classically forbidden, it represents a potential barrier for electrons. The envelope function is non- zero only because electrons tunnel into this barrier. The envelope function for holes behaves similarly, it decays within the hole barrier with e E z < 0, and oscillates in the classically allowed hole region with e E z > 0.

The Schrodinger equations (3 .300) , (3.301) and their solutions (3.303) and (3.304) describe, respectively, stationary electron and hole states in an electric field. Electrons or holes in such states do not carry an electric current, more strictly speaking, the expectation value of the current density operator in the field direction vanishes. Although this does not mean that a current flow cannot be described at all by means of stationary states (if the electron system is characterized by a statistical ensemble with respect to stationary states, then non-zero non-diagonal elements will occur in the expectation value of the current density operator), one may suspect, however, that in addition to stationary states there will be yet other states which are better suited to the condition of a non-vanishing current. Such states do in fact exist, they are not eigenstates of the Hamiltonian but rather are non-stationary states cp,(x, t ) which devebp in time not only by means of a timedependent phase factor. They correspond to the classical trajectories of an electron in an electric field, with momentum growing linearly in time.

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3.8. Macroscopic electric fields 437

0.4

0.3 C 0

(D

3 N

.- u g 0.2 rn

g 0.1 e - 0 P

Lc - c g 0.0 2 rt v) Y

W

-0.1

-0.2

-Eg/eE 0 Z

Figure 3.78: Envelope functions of stationary electron and hole states in an electric field, taken at energy cC = E, = 0. Oc = OU = 8.

3.8.2

The envelope functions F,(x,t) of these states are time-dependent, so one has

Non-st at ionary states. Bloch oscillations

$v(x, t ) = ~ V O ( X ) F V ( X , t ) . (3.306)

To determine F,(x, t ) , a time-dependent effective mass equation is required. This may be obtained from the corresponding time-independent equation (3.296), wherein the energy E, on the right hand side is replaced by the operator ib(a/at) . The resulting equation reads

(3.307) {&,(--iV) + eE . z} Fv(x, t ) = iTi-F,(x, t ) .

In solving this equation we assume that the electron was in a Bloch state (p,k(x) at t = 0. Then the initial condition for F,(x, t ) may be stated as

a at

.ik.x FJX, t = 0) s F,/C(X, t = 0) = - G ' (3.308)

Here, the envelope function at t = 0 has been chosen such that its normal- ization integral with respect to the infinite interval over which the crystal in field direction extends, is given by a &function with respect to k. The solution F,k(x, t ) of equations (3.307) and (3.308) is given by

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438 Chapter 3. Electronic structure of semiconductor crystals with perturbations

with kt = k - ZEt.

f i

(3.309)

(3.3 10)

This is readily verified by a straightforward calculation. Owing to the en- velope function (3.309) and the previously made approximation U& = U u k ,

the total non-stationary wavefunction & , k ( x , t ) evolving from a Bloch state (pvk(x) at t = 0 remains a Bloch state for t > 0 also, however, with a time-dependent quasi-wavevector kt,

(3.311)

This important result may also be established without approximating the Bloch factor U v k by u,o as is done in effective mass theory. The only ap- proximation needed is the neglect of interband transitions induced by the electric field.

Equation (3.311) states that the quasi-wavevector k of a Bloch state in the presence of an electric field changes with time in just the same way as does the ordinary wavevector of a free electron. Differing from the latter, the quasi-wavevector is restricted to the first B Z . However, relation (3.310) results in kt-values which can also lie outside the first BZ. This means that in the above derivation we inadvertently changed from the reduced to the extended zone scheme. According to the general considerations of Chapter 2, one may recover the description in terms of the first BZ by subtracting a suitable reciprocal lattice vector K(t) . For electric fields in symmetry directions of the first B Z , there will be a particular point in time t = T(E) at which, with increasing t , the reduced wavevector kT - K(T) equals the wavevector kt at t = 0 for the first time. This means that the time-dependent Bloch state (3.311) returns, at t = T , to the Bloch state at t = 0 (not counting a phase factor). The same holds for time points 2T,3T etc. The time development of Bloch states in an electric field is therefore periodic, hence the term Bloch oscillations. The period T of these oscillations may be obtained from (3.310) as

(3.3 12)

The Bloch oscillations affect not only the wavefunctions but also the perti- nent energies E,(kt), meaning that Bloch electrons execute periodic motion within their energy bands.

Oscillations also occur in the velocity of Bloch electrons, defined by the quantum mechanical expectation value < ( d x / d t ) >= (qJvhI(dX/dt)IVvk,) of

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3.8. Macroscopic electric fields 439

the velocity operator (dx ldt ) with respect to the timedependent Bloch func- tion Pvk,. using the identity (dxldt) = ( l /m)p, the velocity < ( d x / d t ) > follows by means of the previously derived relation (2.193) for the expecta- tion value ((p,hJpJ(p,h) of the momentum operator p. The timeaverage of the velocity < ( d x / d t ) > obtained in this way, taken over a Bloch period, turns out to be zero. The same result holds for the timeaveraged current of the electrons of a crystal in an external electric field, it also vanishes. That this is actually not the case is due to perturbations of the Bloch oscillations by collisions of electrons with phonons, impurities and other point perturba- tions. To prove this statement we consider a field strength of lo4 V / c m and a primitive reciprocal lattice vector K typical of semiconductors. The Bloch period T is of the order of magnitude lo-'' s in such circumstances. The time rv between two collisions is substantially smaller than T , typically of the order of magnitude s. Thus, an electron starting a Bloch oscilla- tion will soon be scattered and its momentum will be distributed randomly over all k-space. In other terms, no periodic motion can develop in the pres- ence of collisions. We conclude that collisions are not only responsible for the fact that the current of a free electrons, which would otherwise be in- finitely large, remains finite, but also for the fact that an otherwise vanishing average current of a Bloch electron does not actually go to zero.

There is still another reason that Bloch oscillations cannot be observed in ordinary semiconductor crystals. It is due to the fact that the time depen- dent Bloch states (3.311) decay in time by themselves since the exponential factor oscillates with increasing frequency. This may easily be seen for quasi- wavevectors k close to a critical point. For zero wavevector component k, in the field direction, the phase of the exponential contains a term (1/3)e:t3 (0, is the electro-optic frequency of equation 3.305). If (1/3)0:t3 is consider- ably larger than 1, the exponential oscillates so fast that the time dependent Bloch states (3.311) average out to zero almost completely. One may say that these states have a finite lifetime 0;' due to the electric field. For not- too-large field strengths, e;' is small compared with the period T of Bloch oscillations, just as the time T, between two collisions above was found to be small with respect to T . This implies that, in actual crystals and under nor- mal conditions, Bloch oscillations are not observable even without collisions, just due to the finite field induced lifetime.

For the artificial superlattices considered in the previous section 3.7, the primitive lattice vectors and thus the periods T are much shorter than those in natural crystals. In such circumstances T may become larger than T~ and OF1, and Bloch oscillations may in fact be observed. The negative differential electric conductivity of a superlattice parallel to its axis, reported in section 3.7, also may be understood as a manifestation of Bloch oscillations.

If one considers only k-vectors close to critical points of the band struc- ture, then the energy bands E,(k) can be taken in parabolic approximation.

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440 Chapter 3. Electronic structure of semiconductor crystals with perturbations

This yields, approximately,

(3.313)

With the neglect of collisions, an equation similar to Newton’s law of motion

mt- d 2 < x > = -eE, (3.3 14) d t 2

follows from (3.313). It differs from the ordinary Newton’s equation in that the free electron mass is replaced by the effective mass of the respective band v. For electrons from the valence band the effective mass is negative. The pertinent particles with positive mass, the holes, behave like particles with positive charge e.

3.8.3 Interband tunneling

In the effective mass equation (3.296), the possibility of electric field in- duced coupling between the valence and conduction bands is neglected. In reality such coupling exists due to the non-vanishing interband matrix ele- ments of the potential V’(x) = e E . x. Taking account of these off-diagonal elements yields a description of electrons undergoing quantum mechanical transitions between the valence and conduction bands. For reasons of energy and momentum conservation, the initial and final state energies E,, Ec and wavevector components k,l, k,l perpendicular to the field have to be equal in such a transition. According to equation (3.299) this means E, = cC. The quantum mechanical probability W , for field induced interband transitions is proportional to the overlap integral between the two envelope functions F,,vkvl(x) and FECkci(x) with identical values for E,,, tC and k w l , kc-. If, in particular, transitions of electrons between the two band edges, i.e. with E, = ec = 0 and k,l = k,l = 0, are considered, the overlap integral has to be taken with respect to the product Ai([E,+ eE~]/Ae , )Ai ([ -eEz] /Ae , ) . This integral is non-zero since valence and conduction band states having the same energies E, = eC = 0 differ from zero simultaneously for almost all values of z , as may be seen from Figure 3.106. The reason for the non-zero overlap is the tunneling of the valence and conduction band electrons into their respective barriers. This explains why field induced interband transi- tions are referred to as i n t e r b a n d t u n n e l i n g (the term Z e n e r t u n n e l i n g is also used).

Performing the overlap integral of Ai( [E , + eEz]/~e,)Ai([-eEz]/Ae,) with respect to z one arrives at an expression proportional to the square of the Airy-function Ai3(E,/he,) where 8, is a frequency analogous to the electro-optic frequencies e , , of equation (3.305), formed here, however with the reduced effective mass mT, = mZ,m:/(mz + m:) of electrons and holes

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3.8. Macroscopic electric fields 44 1

instead of their simple effective masses m,*,,. One thus obtains the interband transition probability W , as

W,, oc Ai2 (z) . (3.3 15)

For realistic field strengths E, is measurably larger than TLB,,. Therefore, the asymptotic approximation

(3.3 16)

for the Airy-function applies. To understand what this result means phys- ically we may argue as follows. Expression (3.315) for the transition prob- ability W , was derived using the stationary electron states in an electric field of subsection 3.8.2. The same expression follows if one uses the non- stationary states of subsection 3.8.3. In this, the electro-optic energy Ti@,, may be understood as the energy uncertainty of an electron-hole pair due to its finite lifetime in the presence of an electric field. Without such an uncertainty interband tunneling would not be allowed by reason of energy conservation: the final state in the conduction band would differ in energy from the initial state in the valence band by the gap energy E,. The energy uncertainty relieves the need to satisfy energy conservation and permits the interband transition probability W , to be nonzero. The non-stationary approach also provides a rough estimate of the proportionality factor in re- lation (3.315). The probability W,, for an electron to tunnel per second from the valence into the conduction band at k,l = kl = 0, may be ob- tained from Ai2(E,/7iB,,) by multiplying this expression with the frequency 1/T of Bloch oscillations. This is plausible since this frequency indicates how often a valence band electron, per second, reaches the upper band edge, from which it can tunnel into the conduction band. It is just the probabil- ity for this tunneling step which the Airy-function expression Ai2(Eg/W,) represents. The complete expression for W , thus reads

W , = -Ai2 1 (2). T

(3.317)

Although the interband matrix element (p,kJeE. xlp,k) of the perturbation potential V'(x) does not enter this expression explicitly, as one would ex- pect, it occurs implicitly through the lattice constant a in T , which has the same order of magnitude as (p,k[xIp,k) . According to expression (3.315), the tunneling probability W,, grows larger as the energy uncertainty hOCw approaches the gap energy E,. It is small if the condition

fit', << Eg (3.3 18)

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442 Chapter 3. Electronic structure of semiconductor crystals with perturbations

holds. In typical cases this condition is fulfilled sufficiently well for field strengths of the order of magnitude lo5 V / c m , while it is definitely not valid for field strengths close to the inner-atomic fields of 10' V / c m . At least for such fields interband transitions are important. By means of tunneling, holes are generated in the valence band and electrons in the conduction band. These diminish the resistivity of a semiconductor material and initialize other processes such as carrier heating and impact ionization of the valence band which may finally result in electric breakdown.

The inequality (3.318) transforms into the condition (3.295) for the ap- plicability of effective mass theory if the gap energy Eg is formally identified with (h2/2m5,)1/a2. The latter energy represents an average energy gap in effective mass approximation. It has the same order of magnitude as E,, so that the two conditions (3.318) and (3.295) are equivalent.

3.8.4 Photon assisted interband tunneling

As pointed out in Chapter 1, transitions of electrons from the valence into the conduction band can also be caused by photons provided their energy JLw exceeds the gap energy E, of the semiconductor. In the presence of an electric field this condition no longer needs to be fulfilled, owing to the tunneling of electrons and holes into their respective barriers, effectively low- ering the gap (see Figure 3.79). The optical interband transition probability W z t between the two band edges is proportional to the overlap integral of the envelope functions FVevkvl (x) and FCL,kCl (x) with identical values for k,l, kl, just as in the case of field induced interband transitions, but, un- like the latter case, with energies eC = e,+hw. This means that W s t follows from W , in equation (3.315) by replacing E, with E, - hw. One therefore has

(3.319)

In an external electric field, the optical transition probability and, hence, the optical absorption coefficient, is also non-zero for photon energies below the gap. It decays exponentially as hw decreases. This phenomenon is called the Franz-Keldysh effect (Franz, 1958; Keldysh, 1958; Boer, Hansch and Kummel, 1959). Above the gap, the absorption coefficient with electric field oscillates with photon energy about the absorption coefficient without field, and we have Franz-Keldysh oscillations (Tharmalingan, 1963).

Field induced changes occw not only in the absorption coefficient, but also in other optical constants including the real part of the complex refrac- tive index. Besides the changes at the fundamental absorption edge consid- ered above, one also has changes of optical constants at any higher van Hove singularity of the joint density of states of the interband energy Ec(k)-E,(k)

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3.9. Macrmcopic magnetic fields 443

- ( E,-ho)/eE 0 2

Figure 3.79: The same envelope functions as in Figure 3.78, but with the energy of the electron exceeding that of the hole by the photon energy iiw.

(Aspnes, 1967; Enderlein and Keiper, 1967). This fact is exploited in elec- trorefiectance spectroscopy, in which field induced changes of the reflectivity of a semiconductor sample are measured by means of a modulation tech- nique. Photon energies which give rise to strong electroreflectance signals correspond to optical transitions at critical points of the interband energy (see Figure 3.80). By measuring the electroreflectance spectra of semicon- ductors one obtains experimental data relating to their valenceconduction band separation at critical points. These can be used as input parameters for empirical band structure calculations (Cardona, 1969) as well as for the characterization of semiconductor heterostructures, including superlattices and quantum wells (Pollak, 1994; Enderlein, 1996).

3.9 Macroscopic magnetic fields

Like electric fields, macroscopic magnetic fields also give rise to effects in semiconductors which provide experimental data concerning effective masses and other microscopic properties of electrons and holes. Transport proper- ties, particularly galvanomagnetic phenomena such as the Hall effect, mag- netoresitivity and cyclotron resonance, as well as magneto-optic properties, are of particular importance in this context. The basis for theoretical un- derstanding of these phenomena are the stationary one-electron states of a semiconductor crystal in the presence of a magnetic field. For most purposes

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444 Chapter 3. Electronic structure of semiconductor crystals with perturbations

I I I I I I I

1.0 1,5 20 25 30 35 400 4s -121 ’ Energy ( e V ) -

Figure 3.80: Electroreflectance spectrum of Germanium. (After Seraphin and Hess, 1965.)

this field can be assumed to be homogeneous in space and time. Just as in the case of external electric fields, this again only means that the magnetic field is approximately constant on length and time scales which are large compared with corresponding microscopic length and time scales. In the case of cyclotron resonance, the microscopic time scale is given by the ro- tation period of an electron about the magnetic field axis which typically lies in the G H z range. Thus, temporal changes of the magnetic field in the M H z range are still admissible. Below, we consider spatially and temporally constant magnetic fields H exclusively. The vector of the pertinent magnetic induction vector will be denoted by B, as usual.

Since the interaction of an electron with a magnetic field cannot be char- acterized by a scalar perturbation potential, but must be described by the vector potential A of the magnetic induction B, the effective mass theory developed in section 3.3 is not directly applicable to this case. Whether an effective mass equation can be derived at all with a magnetic field, and what it will look like if it is feasible, must be explored separately. This matter will be addressed in the following subsection.

3.9.1

Initially, spin will be omitted from our considerations (it will be taken into account later). In these circumstances, the role of the magnetic field in the Hamiltonian is fully subsumed in the kinetic energy operator (p2/2m) with the canonical momentum operator p replaced by the kinetic momentum operator p + (e/c)A(x). The vector potential A(x) is determined by the magnetic induction vector B, save for the gradient of an arbitrary gauge

Effective mass equation in a magnetic field

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3.9. Macroscopic magnetic fields 445

function which we choose so that A(x) has the form

A(x) = A[B x X I . (3.320) 2

This choice of gauge guarantees that p commutes with A(x), so that the order in which p and A(x) are multiplied in the expression [p + (e/c)A(x)I2 is inconsequential. The strength of the magnetic field will, as in section 3.8 on electric fields, be limited by the condition that the associated vector potential changes only little over a unit cell in comparison with the energy gap. In this sense, the vector potential is presumed to be smooth.

The Schrodinger equation of a spinless crystal electron in a magnetic field reads

(3.321)

As in the derivation of the effective mass equation for perturbing scalar po- tentials in section 3.3, we use the k . p-perturbation theory, i.e. we represent the Schrodinger equation in the approximate Bloch basis Iuk)', whence

l 2 '(vk(H + L A ( x . p) +

mc (:) A2(x)IvAk')l '(vhk'lg) = E '(vkI$).

u'k' (3.322)

with H = (p2/2m) +V(x) as Hamiltonian of the crystal without a magnetic field. Initially, we consider a non-degenerate band extremum at k = 0, and assume the form (3.60) for '(vkl$). In order to satisfy the Schrodinger equation (3.322) with this Ansatz, the component with u = vo must obey the relation

2 1 l(vok(H + e A ( x ) . p +

mc f) A2(x)(uok')'Fw(k') = EF,,(k). k'

(3.323)

Components '(vkl$) with v # vo must vanish according to equation (3.322). To verify this we consider the matrix elements '(vklHlvok')' of the unper- turbed Hamiltonian H , which were already evaluated above. The expression (3.63) derived there is applicable here as well. Accordingly, t-he matrix ele- ments of H with v # vo vanish.

Because of the assumed smoothness of the vector potential A(x), its matrix representation '(vkJAlv'k')' is approximately diagonal with respect to band indices, whence

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446 Chapter 3. Electronic structure of semiconductor crystals with perturbations

The diagonality of the matrix for A extends to the matrix for A2, as well, and therefore one also has

’ (vkl A2 Iv’k’)’ M 6,1( klA21 k’) . (3.325)

What remains to be evaluated in the Schrodinger equation (3.323) now is only the matrix of the operator ( e / m c ) A . p. This operator has the same structure as the k . p-perturbation operator. Just as the matrix represen- tation of the latter with respect to the approximate Bloch basis (vk)’ is band-diagonal in second order perturbation theory, the same also holds for the matrix representation of the operator ( e / m c ) A . p. We have

(3.326)

This relation finally confirms the vanishing of the v # vo-components of the left hand side of Schrodinger’s equation (3.323) and, thus, of ‘(vklq) with v # vo.

The band-diagonal elements (vo kl e / m c A . plvok’)’ will now be rewritten in a more convenient form. We proceed in two steps. In the first step, we determine the matrix elements of the operator ( e / m c ) A . p between the Luttinger-Kohn functions (vokO) Ivok)’ which enter the first-order Bloch functions Ivok)’ of equation (2.334) as unperturbed parts. In the second step we consider the first-order corrections.

First step

Since (3.327)

1 2

A . p = - L . B

with L = [x x p] as angular momentum operator, we have

(vokOleA. pIvok’0) = pB(vokO(Ti-’LIvok’O) . B. (3.328)

Here p~ = (eTi/2mc) denotes the Bohr magneton. The matrix elements of L can be rewritten by means of Heisenberg’s equation of motion which approximately yields

m c

(3.329)

Here all terms depending on the magnetic field have been neglected since they only give rise to quadratic terms with respect to B in the matrix ele- ments (3.328). Substituting relation (3.329) in expression (3.328) for these elements, and using the notation L , = zap^ - ”@pa for angular momentum components (c.Pr signifies the ordered triplet zyz or a cyclical permutation of it), we obtain

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3.9. Macroscopic magnetic fields 447

(3.330) In this, we used the fact that the interband matrix elements of the momen- tum operator are diagonal with respect to k and k’. The matrix elements (vokO)L,lvokO) do not depend on k, so we may use the relation

The quantity L,, is an antisymmetric tensor or, in other terms, a pseudovec- tor L,. This antisymmetric tensor bears a close relation to the reciprocal effective mass tensor M&La of equation (2.337). Apart from a constant fac- tor, the two quantities may be understood as real and imaginary parts of the same complex tensor, namely

(3.332)

In section 3.3 we found that the symmetric tensor M,Lp describes the k- dependence of the band-index- and k-diagonal Bloch matrix elements of the Hamiltonian (see equation (3.63)). Here we find, that the antisymmetric tensor L,, determines the B-dependence of these elements. The latter evidently describe the interaction of the magnetic moment of the orbital motion of a Bloch electron with the magnetic induction vector B. In order that it differ from zero, the angular momentum of the Bloch state IvoO) must differ from zero, too. In principle, this is possible because the motion of a Bloch electron cannot in general be reduced to a translation, since it contains a rotational part about the atoms which form the crystal. The tensor L,, measures the average angular momentum of this rotation in the Bloch state 1~00). In cubic materials it vanishes for symmetry reasons, at least as long as only non-degenerate Bloch states IvoO) at the center of the first BZ are considered and spin and spin-orbit interaction are omitted from consideration, as we do here. The situation is comparable with that for s-states of free atoms.

Second step

Beside the k-diagonal term discussed above, which describes the coupling of the magnetic field to the rotational part of the motion of a Bloch electron, the

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448 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Hamiltonian matrix of (3.323) still contains terms which are non-diagonal with respect to k, k’ and give rise to a coupling of B to the translational part of this motion. They occur through the formation of the matrix elements l(vok)(e/rnc)A. p)lvok’)’, where the perturbed part [Ivok)’ - IvokO)] of the approximate Bloch functions Ivok)’ (which was previously omitted) is taken into account. The complete result is given by

6kk,h-1pBB. L, + - 1 [k,e(k(Aa(k’) + kp‘(kja,jk’)] . (3.334) 4 2m

Summing the three terms (3.63), (3.334) and (3.335) obtained in rewriting the Schrodinger equation (3.323), and simultaneously transforming from k- space to coordinate space, the following equation results for the envelope function Fvo (x) :

For a non-degenerate isotropic parabolic band the magnetic moment L, vanishes in the spinless case considered here, as already mentioned. Then equation (3.335) takes the form

(3.336)

This equation changes if spin and spin-orbit interaction are taken into ac- count. The assumption of a non-degenerate band vo will be adopted, initially.

Spin a n d spin-orbit interaction

Considering the role of spin and spin-orbit interaction, the Schrodinger equa- tion (3.321) may be written as

where g is the gyromagnetic ratio of a free electron (g = 2.0023). Using the same procedure employed in the spinless case, i.e. by first representing equa- tion (3.337) in the spin-dependent Luttinger-Kohn basis, then transforming

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3.9. Macroscopic magnetic fields 449

it into the spin-dependent approximate Bloch basis Ivak)' = Ivk)'lo), and finally transforming it back to coordinate space, the effective mass equation for an isotropic parabolic band follows as

= E F , ( x , s ) . (3.338)

In obtaining (3.338), we used the fact that the diagonal elements '(voaklH,, Ivoa'k)' of the spin-orbit interaction operator vanish at k = 0 for symmetry reasons, while the non-zero k-linear terms in these elements are negligible. The matrix elements L, of the angular momentum operator emerge from the corresponding expressions (3.330), (3.331) without spin-orbit interaction by replacing p in the latter with the spin-dependent operator ?i of equation (2.353), obtaining

In accordance with their definition, the L,, are matrices in the two-compo- nent spin space. In regard to their transformation properties in coordinate space, they are components of a pseudovector. Apart from a constant factor, there is only one such pseudovector in the case of systems with the symmetry group oh, namely the vector of Pauli's spin matrices a. This means that L, is a multiple g,Acr' of cr'. The constant gvo is determined by the matrix elements of (3.339). With 1 = guo + 29, we may write

1 h-lLm + 2g cr' = ig&d. (3.340)

Using this expression, the effective mass equation (3.338) takes the form

The factor gEo is called an effective g-factor. Just like the effective mass, it represents a characteristic of an energy band vo at a particular critical point. In Table 3.11 experimental g:,-values for the conduction band min- imum of several diamond and zincblende type semiconductors are listed. As one may see, all values gE are less than 2, the g-factor of the free elec- tron. The deviation from 2 is determined by the induced magnetic moment of the orbital-motion component of a Bloch electron. As in a free atom, this moment is always negative, in other words it results in a diamagnetic

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450 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Table 3.11: Effective g-factors gE of electrons and magnetic interaction constants IC and q of holes in diamond and zincblende type semiconductors. (After Landoldt- Bornstein, 1982.)

GaAs GaSb InSb CdTe ZnSe

-0.42 3.41 1.20

0.06 0.04 0.13 0.15

contribution to the total magnetic moment of the Bloch electron. For Si, the effective electron g-value is close to 2, indicating that the diamagnetic contribution is small in this case. A negative effective g-value and hence a larger diamagnetic contribution occurs in Ge. This is to be expected since the diamagnetic moment is determined by the average angular momentum L,, which according to expression (3.339), and similar to the effective elec- tron mass of expression (2.338), increases if the gap decreases. The same behavior of the effective g-factor for decreasing gap is observed among the 111-V compound semiconductors: InSb, the material with the smallest gap, has the largest negative value of g:.

To describe the top of the valence band of diamond and zincblende type materials, we need an effective mass equation for degenerate bands, and will address this below.

Degenerate bands

We consider the valence band of a diamond and zincblende type semicon- ductor in the vicinity of the BZ center at k = 0 . Spin and the spin-orbit interaction will be omitted initially. Then the degeneracy is %fold. The Bloch states at k = 0 are IvmO), m = 2, y, z , and the pertinent Luttinger- Kohn functions read (vmkO). The eigenfunction $(x) is given in terms of the envelope function vector F , (x) = (Fwz(x), Fw(x) , F,,(x)) by equation (3.67). The elements MGip of the effective mass tensor are 3 x 3 matrices 04 D with elements DZ:,. The same holds for the components L,, of the effective angular momentum of equation (3.333) which may be expressed as matrices L~ with elements

*

0

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3.9. Macroscopic magnetic fields 45 1

It follows that the envelope function vector of the degenerate valence band in a magnetic field satisfies the equation

where the abbreviation

(3.344)

has been used for the angular momentum term. Like the first tensor in equa- tion (3.343), the second tensor H M can also be determined, apart from cer- tain constants, by means of the point symmetry of the crystal. The method of invariants, which was formulated in section 2.7 for the k-dependent term of the Hamiltonian, is also well suited for the B-dependent operator H M . The only change is the replacement of the wavevector k by the magnetic induction B. To apply this method to the I?;, valence band of a diamond type semiconductor, one needs a complete set of 9 linearly independent basis vectors in the 3 x 3-matrix space formed from the three matrices Izr Iyr I z of angular momentum quantum number j = 1 and from the six products of these matrices which transform according to the irreducible parts of the product representation x I?;, = I?l +I?12+I':,+I';, (for the explicit form of the angular momentum matrices see equations (A.143) of Appendix A). Since the magnetic field B, as a pseudovector, belongs to the representation I?:, of the point group oh, only subsets of matrices transforming according to the irreducible representation I?:, of Oh are allowed to form an hvari- ant H M under the point group operations of oh. The subset of matrices

9

9

9 9 9

9

which transforms according to the representation I?:, is the pseudovector I= (Iz, Iyr IzLitself. Thus the scalar product (BI I ) emerges as the only 9 9 9 9 9

- invariant, and H M follows as

(3.345)

with K being a constant which has to be determined from band structure calculation or by means of experimental investigation. The same result for H M follows for the I'l5-valence band of zincblende type semiconductors.

The above consideration shows that, in the case of degenerate bands, there is coupling of the orbital motion to the magnetic field which does not occur for non-degenerate bands. The source of it is the non-vanishing of angular momentum matrix elements between the degenerate valence band states. Of course, the average angular momentum of all valence band states also vanishes in the degenerate case. However, the various angular mo- mentum states, averaging to zero in the absence of a magnetic field, are

9

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452 Chapter 3. Electronic structure of semiconductor crystals with perturbations

differently affected by the magnetic field so that a magnetic splitting of the r;, or r15 valence band occurs, comparable to that of a p-level in the case of a free atom.

If one adds the effects of spin and spin-orbit interaction, then the I'b, band of diamond type semiconductors develops into the r,f and I': bands. The effective mass equation for the I?$ band without a magnetic field is given by (3.74). Two changes are necessary in the presence of a magnetic field: First, in the operators Q, R, $, f of equation (3.74), the replacements 8, --.$

d ,+( i e /hc )A , , Q = 2, y, z , have to be made. Second, an angular momentum term H M has to be added. The latter can again be obtained by means of the method of invariants. To apply this method in the present case, one must find a complete set of 16 basis vectors of the 4 x 4-matrix space formed from the three angular momentum matrices I ~ , ly, I z of quantum number 4 of equation (A.144) and their products which transform in accordance with the irreducible parts of the product representation r$ x I?$ = rl+ rz + I'12 + 21'i5+I'k5 of oh. As in the case of the I';, band without spin, invariants linear in B are possible only with subsets matrices of I'i5-symmetry. Differing from the spinless case, there are now two such subsets, namely I = ( Ix, Iyr I z ) ,

030303 *3 and also the triplet (Ix, I,, I z ) . We denote it symbolically by I . The

additional term H M in the Luttinger-Kohn Hamiltonian which describes the coupling of the magnetic field to the magnetic moment of the I'$-valence band states thus reads

($

( $ @ @

9 u u *

*

(3.346)

where K and q are again constants which must be determined either from band structure calculations or experimentally. That the presence of spin involves not only K , but also a second constant q, is understandable because the orbital and spin motions have different gyromagnetic ratios. In most cases q is negligibly small (see Table 3.11). The result (3.346) for H M which has been derived here for diamond type semiconductors, holds unchanged for materials with zincblende structure also.

(z

3.9.2

We demonstrate the solution of the effective mass equations in a magnetic field in the simplest possible case, that of a non-degenerate band without spin. In this, we take the magnetic field in the z-direction and employ the vector potential in the form A = (--By, 0,O). This gauge differs from that in expression (3.320) used in deriving the effective mass equation (3.336). Unlike (3.320), the new gauge lacks axial symmetry with respect to the magnetic field. The effective mass equation (3.336) also holds, however,

Solution of the effective mass equation

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3.9. Macroscopic magnetic fields 45 3

because for this gauge the vector potential A = (--By, 0,O) also commutes with the momentum operator p. The same holds for the more complex effective mass equation (3.343) for degenerate bands without as well as with spin. Using the present gauge, the effective mass equation (3.336) for a non-degenerate band without spin takes the form

2- am:, [ (Pz + $Y)2 +Pp +pq F d X ) = ( E , - E,o)F,(x). (3.347)

Since the Hamiltonian in (3.347) commutes with the translation operators for arbitrary displacements in the x- and z-directions, its eigenfunctions can be chosen to also be simultaneously eigenfunctions of these operators, which leads to

(3.348) 1 '

27T with k , and k , as wavevector components in the x- and z-direction. For X,(y), equation (3.347) yields

F (x) = -ez(kSzfkZa)X v o ( Y ) , vo

where we have set

eB 2 l i C , yo = rCk,, r z = -.

wcvD = Im:,lc eB (3.350)

The total energy eigenvalue E , of equation (3.347) is a function Ew(k,) of k,. It follows from the partial eigenvalue ELo of equation (3.349) associated with motion in y-direction through the relation

(3.35 1)

Equation (3.349) describes a harmonic oscillator with center at yo, mass lm&1 and eigenfrequency wcw. It is well-known that in this case the eigenstates can be described by integer quantum numbers n = 0 , 1 , 2 . . . , 00, with the corresponding eigenenergies E L given by

EL, E & ~ = FAW,,, n + - . sgn(m&). (3.352)

These energies are called Landau levels. For the eigenfunctions Xvo(y), we have

( a>

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454 Chapter 3. Electronic structure of semiconductor crystals with perturbations

(3.3 53)

with H , a Hermitian polynomial of order n. Thus the electron executes an oscillation in y-direction with angular frequency wc, about its equilibrium position yo. The pertinent frequency is given by

(3.3 54)

For mC, = 0.2 m and B = 1 Tesla, this frequency amounts to 140 G H z , i.e. it lies in the microwave region. According to equations (3.352) and (3.353), the spectrum of allowed energy values forms a set of 1-dimensional energy bands, the Landau subbands,

which exhibit non-zero dispersion in the k-direction parallel to the magnetic field.

It is enlightening to consider the density of states (DOS) of the energy spectrum of equation (3.355), taking the conduction band vo = c as an ex- ample. This DOS can easily be calculated by means of the general definition of equation (2.205). One need only replace the quantum number i in (2.205) by the set of quantum numbers n, k,, k y , and replace Ei by the set Ecn(k,) of Landau subbands from equation (3.355). Then the DOS, p c ~ ( E ) , of the conduction band becomes, apart from a constant factor,

In Figure 3.81, p c ~ ( E ) is shown as a function of E. It becomes infinitely large at the uniformly spaced energies E , = E, + hLw,(n + i). Each singularity corresponds to the minimum of a particular 1-dimensional Landau subband. For energies slightly above E,, the DOS decays like l / d G . The inverse square root-behavior of the DOS of a single 1-dimensional parabolic band has already been discussed in section 2.5.

The physical meaning of the eigenstates (3.353) and eigenenergies (3.352) of a Bloch electron in a magnetic field may be understood as follows. In the z- and I-directions the electron propagates like a plane wave. Of the two wavevector components k , and k,, however, only k, enters the energy eigenvalue Euon. This unusual absence of k , is due to the fact that h k , represents the eigenvalue of the canonical momentum operator component

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h

v) (I .-

Y

% 4 Q-

2

0 0

I I I I 1 2 4 6 0

E - Eg(hwCc)

I I I I 1 2 4 6 0

E - Eg(hwCc)

455

Figure 3.81: DOS in a magnetic field.

p , rather than that of the kinetic momentum operator component (pz + :By). From the expression for yo in equation (3.350) it may be seen that k , determines the equilibrium position of the harmonic oscillations along the y- axis. Since the eigenenergies Evan(kz) of these oscillations do not depend on the equilibrium positions, a high degree of degeneracy exists with respect to k,, and every arbitrary linear combination of eigenstates having different k,- values is again an eigenstate. If the vector potential would have been chosen in the form (3.320) which is symmetric with respect to I and y, then such linear combinations would have resulted in eigenstates which also execute oscillations with frequency wcvo in the 2-direction, but phase shifted by 7rj2 with respect to the oscillations in the y-direction. This signifies circular motion in the plane perpendicular to the magnetic field, known as cyclot.ron m o t i o n Such circular motion is a consequence of the axial symmetry of the magnetic field. In classical mechanics, one also has circular orbits for the motion of an electron in a magnetic field, projected on I - y plane.

The radius of the cyclotron orbit depends on the quantum number n, larger values of n and, with this, also larger energies ELan mean larger cy- clotron radii. By absorbing an energy quantum of an electromagnetic radia- tion field, an electron can make a transition from the n-th to the (n + l)-th cyclotron orbit. Such transitions are involved in cyclotron resonance: At a fixed frequency, the absorption of microwave radiation is measured as a function of magnetic induction B. Varying the magnetic field, the cyclotron frequency can be adjusted to match the fixed frequency of the microwave ra-

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456 Chapter 3. Electronic structure of semiconductor crystals with perturbations

Magnetic Induction/ Tesla 4

Figure 3.82: Cyclotron resonance spectrum of Germanium at 4 K. (After Dexter, Zeiger, and Lax 1956.)

diation, whereupon the absorption becomes a maximum. Using the position of this maximum on the magnetic field scale, the effective mass can be deter- mined. Measurement of cyclotron resonance is an important experimental method for the determination of effective masses of electrons and holes in semiconductors and metals. In Figure 3.82, the first peaks of the cyclotron resonance spectrum of germanium are shown. Because of the many valley structure of the conduction band, the electrons give rise to several cyclotron resonance maxima. The holes show up in two well-resolved maxima in Fig- ure 3.82, one due to heavy holes and the other due to light holes. Each of the two hole peaks has a doublet structure which can be seen experimentally at higher resolution (in Figure (3.82) it is absent.

To understand the cyclotron resonance spectrum of holes, the energy eigenvalues of the rs-valence band in the presence of a magnetic field must be calculated from equation (3.343) with D from equation (3.74)) and H M from equation (3.346). One finds that the energy levels can always be char- acterized by an integer quantum number n, as in the case of the harmonic oscillator, but in contrast to this case, each value of n determines a group of four energy levels, two corresponding to heavy and two to light holes. B e cause of broken time reversal symmetry in the presence of a magnetic field, each of the doubly spin-degenerate heavy and light hole levels in the absence of a magnetic field splits into a doublet. By means of cyclotron resonance measurements these levels can be determined experimentally. Comparison with the theoretical results also yields values for the constants n and q. In Table 3.11 these values are listed for several semiconductors.

CJ *

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457

Chapter 4

Electron system in t herrnodynamic equilibrium

4.1 Fundamentals of the statistical description

In the two preceding chapters we discussed the possible stationary quantum states of the electron system of a semiconductor. We found that these states could be represented, approximately, by Slater determinants of stationary oneparticle states. In the case of ideal semiconductor crystals the o n e particle states are spatially extended Bloch states, and the corresponding energies form energy bands. For real semiconductors with shallow and deep centers there are, in addition, localized oneparticle states associated with energy levels in the gap between the valence and conduction bands. The energies of extended one-particle states were found to be independent of their occupation, i.e. independent of the specific configuration of the many- electron system, while configuration dependencies due to Coulomb repulsion were found for the energies of localized states. We also described calcula- tional procedures used to determine the stationary oneparticle states and the corresponding energy bands and localized levels, and, in limiting cases, these energies were evaluated explicitly.

However, we have not yet addressed the question of whether the electron system is actually in a stationary state, and if so, which of the various one- particle states are involved in the construction of the corresponding Slater determinant. This is equivalent to asking of which of the infinite number of possibilities will be realized for distributing the electrons of a periodicity region over the infinitely many one-particle states. We have yet to deal with this question, as it is by no means already answered or even addressed by the determination of all possible stationary states of the electron system. The situation is similar to that in classical mechanics wherein the totality of all possible paths of a particle system, i.e. the set of all possible solutions of the

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458 Chapter 4. Electron system in thermodynamic equilibrium

mechanical equations of motion of the system, has been determined. To find the actual path of the classical system, one needs additional information in the form of initial conditions or initial values of the constants of the motion. In the case of a quantum system, the initial value of its wavefunction is required. In any case, this information would be necessary if one were seeking for a deterministic description of the system. But this is not the case.

Macroscopic systems are generally not described in a deterministic, but a statistical way. This means that the system is understood to be supplanted by an ensemble of systems, all of which have the same Hamiltonian as the original one, but may be in different quantum states. The properties of this ensemble on average are taken to be representative of the macroscopic properties of the system. The state of the ensemble is described by a so- called statistical operator b. This operator has to be chosen in accordance with the macroscopic state of the system. If the latter is a thermodynamic equilibrium state, b may be assumed to be diagonal with respect to a ba- sis set of stationary quantum states, in our case of Slater determinants. A particular diagonal element of j indicates how many individual systems of the ensemble are in the stationary state corresponding to the diagonal site under consideration. The diagonal elements are specified using further in- formation on the macroscopic state of the system. In our case, the electron system may exchange energy with the atomic cores of the crystal. Further- more, an electron transfer to and from these cores, in particular to and from impurity atoms, may take place. In statistical mechanics, the presence of such a heat and particle bath mandates a description in terms of the grand canonical ensemble. This ensemble is characterized by two macroscopic state parameters, the temperature T , and the chemical potential p. Its statistical operator b is given by

* - ( H - p f i ) / k T p = e

where H and denote, respectively, the Hamiltonian and the total particle number operators of the many-electron system, in the so called occupation number representation This representation is well suited to describe Slater determinants in a more compact way than was done in Chapter 2. Before proceeding further, we will introduce this representation (for more detail see Appendix C). We number the possible one-particle states by an integer i which can take the values 1,2,. . . ,m. If a particular one-particle state i occurs in a Slater determinant one says that it is occupied or that its oc- cupation number Ni is 1. If the state i does not occur, then Ni is 0. A given one-particle state cannot occur more than once, otherwise the Slater determinant vanishes in conformance with the Pauli exclusion principle for electrons. The occupation numbers Ni can therefore only take the values 0 and 1. Using occupation numbers, the Slater determinants can be described

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4.1. Fundamentals of the statistical description 459

as vectors IN1, N 2 , . . . , N,) in the occupation number space. These vec- tors have, altogether, as many Ni different from zero as the total number of electrons in the system. The IN1, Nz, . . . , N,) may be understood as eigen- vectors of particle number operators i?i for the eigenvalue Ni (see Appendix C ) . Therefore,

fiiIN1, N27.. . j Nw) NilN1, N 2 , . . ' 7 Nw), (4.2)

and the total particle number operator k of the electron system follows by summing the f i i over all i,

00

N = p i . i=l

(4.3)

Moreover, (4.2) and (4.3) imply that

The occupation number vectors IN1, N z , . . . , N,) form a complete basis set in Hilbert space of the many-electron system. These vectors may be used to represent an arbitrary many-electron operator. If only extended oneparticle states are involved, like in the case of an ideal crystal, the occupation number representation of k reads

W

H = EiNi . i = l

(4.5)

where Ei are the configuration-independent oneparticle energies. If also localized oneparticle states exist, like in the case of a real crystal having shallow and deep centers, the energies Ei are, in general, configuration- dependent. The Slater determinants IN1, Nz , . . . , N,) are, of course, also eigenvectors of I? so that

Using equations (4.2) to (4.6) one may easily show that the occupation number space matrix representation of the equilibrium density matrix of equation (4.1) is in fact diagonal with respect to the basis set of Slater determinants "1, N z , . . . , N,) , with the diagonal elements given by

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460 Chapter 4. Electron system in thermodynamic equilibrium

The average value < fii > of the particle number operator fii, like the average of any other operator of an observable quantity of the many-electron system, is formed in accordance with the relation

Here, the symbol Tr[ ...I stands for 'Trace' and means the summation over all diagonal elements of the operator within the brackets with respect to any basis in the Hilbert space of the many-particle system. One may easily demonstrate that the value of the trace is independent of the particular basis chosen. Of course, it is advantageous to use the set of Slater determinants IN1, Nz , . . . , N,) as basis, since the matrices of 6 and Ni are known in this represent ation.

4.2 Calculation of average particle numbers

Below, we will calculate average particle numbers in equilibrium by means of equation (4.8). In doing so it is necessary to distinguish between configuration- independent extended one-particle states, and localized one-particle states which do depend on the configuration of the many-electron system because of Coulomb repulsion effects. We begin with configuration-independent one- particle states.

4.2.1 Configuration-independent one-particle states

As above, we describe them by an integer quantum number i. Each value of i represents only one state, with different spin orientations corresponding to different values of i. The two trace expressions in the numerator and denominator of equation (4.8) become, respectively,

where we have set p = l / k T for brevity. The two expressions (4.9) and (4.10) can be written as products of sums over particle numbers for a particular state j , as indicated in equation (4.10). In the quotient, all factors cancel except the two with j = i. These remain and give rise to the expression

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4.2. Calculation of average particle numbers 46 1

Since the particle number Ni may take only the values 0 and 1, it follows that

(4.12)

The average particle number in the one-particle state i thus depends only on the energy of the state. If one identifies the chemical potential p with the Fermi energy E F , and uses the notation

(4.13)

for the Fermi distribution function previously introduced in Chapter 1, we obtain

(4.14)

We conclude that the average particle number for configuration-independent one-particle state i is given by the Fermi distribution, just like for an ideal gas. The distribution function of classical statistics, the Bol t zmann distri- bution

(4.15)

follows from the Fermi distribution for energies E with E - E F >> k T . This also means f(E) << 1. Under this condition the average particle number in the one-particle state is small compared to 1, and thus the state is almost unoccupied. Substantial deviations from the Boltzmann distribution occur when E - EF is no longer large in comparison with k T and, therefore, f (E) no longer small compared to 1. In this case of substantial occupancy the limited capacity of a one-particle state to host electrons (remember the Pauli exclusion principle) becomes important, and the average particle number in it deviates from that in classical statistics in which the hosting capacity of states is not limited.

The deviation of the average particle number in a oneparticle state from the value given by the Boltzmann distribution is called (statistical) degen- eracy. A particle system is referred to as non-degenerate if it is described by the Boltzmann distribution, otherwise it is called degenerate. The ap- pearance of statistical degeneracy is a purely quantum mechanical effect - it reflects the quantization of energy levels and the indistinguishability of elec- trons. Statistical degeneracy applies to other elementary particles as well. Indistinguishability refers to the fact that, in the exchange of two identi- cal elementary particles, the total wavefunction of the many-particle system transforms either into its negative (this holds in the case of electrons and leads to the Pauli exclusion principle and with it to the Fermi statistics),

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462 Chapter 4. Electron system in thermodynamic equilibrium

or the total wavefunction transforms into itself (then the population of the oneparticle states is not limited, but the quantization of energy yields a distribution function which also deviates from the Boltzmann distribution). While the Fermi distribution emerges in the former case, in the latter case it is the Bose distribution function, which may be formally obtained from the Fermi distribution function (4.13) by setting the chemical potential zero and replacing ‘+l’ by ‘-1’ in the denominator.

In the particular case of band states, the quantum number i is the pair vk of band index v and quasi-wavevector k. In order that the wavefunction luk) really represent only one quantum state, which is the case here, u must also describe the spin state, or, alternatively, we must augment v to include a spin quantum number CT. We use the latter description, i.e. we will write the band states, henceforth, in the form lvko). With this, we assume that both the band and spin quantum numbers I/ and u are compatible and meaningful simultaneously. This is only justified if the spin-orbit interaction plays no role and the two which we take to yields

and the average wavevector k follows as

states (uk + $) and lvk - g). have equal energy E,(k), be the case here. In such circumstances, equation (4.14 )

(4.16)

particle number < NUk+& + NUk-+ > for band u and

1 2

< fiuk6 > = f(E,(k)), ff = f-,

(4.17)

The Fermi distribution (4.14) solves the problem of determining the average equilibrium particle number < I?i > for one-particle states li) having energies which are independent of the configuration of the many-electron system. This distribution cannot be applied to configuration-dependent one-particle states. In the latter case the above calculation of average particle numbers has to repeated, taking account of the configuration dependencies of o n e particle energies because of Coulomb repulsion effects.

4.2.2 Configuration-dependent one-particle states

We consider oneparticle states i localized at shallow or deep centers. As in Chapter 3, a center with n electron ionization levels within the energy gap is referred to as an n-multiply ionizable donor, and a center with n hole ionization levels in the gap is referred to as an n-multiply ionizable acceptor. The simplest case in which localized states occur is that of a simply ioniz- able donor or acceptor center. However, the peculiarities of the occupancy statistics of localized states can be understood more easily in the case of double donors or acceptors. Therefore, we start with them. Their states

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4.2. Calculation of average particle numbers 463

are described in oneparticle approximation with a configuration-dependent Hartree potential (see section 3.2). The simply ionizable donor and acceptor centers may be treated as limiting cases of double donors and acceptors, as will be demonstrated below.

Double donor and acceptor centers

We assume a double donor. To be specific, we can think in terms of a substitutional sulfur atom in a silicon crystal. The one-particle states i are specified as follows. For simplicity, we first consider only the two 2-fold spin- degenerate oneparticle ground states ID f ;)' = 1D)'I f $) of the 1-fold occupied center, and the two spin-degenerate oneparticle ground states IDf

states. The pertinent configuration-dependent one-particle energy levels are deno'ied by E b and E i , respectively. Due to the Coulomb repulsion between localized electrons, we have E L = E& + U where U is the Hubbard energy of the 2-fold occupied center (see sections 3.2, 3.4 and 3.5). We assume U > 0 so that EL > EL. The total energy of the two electrons at the 2-fold occupied center is 2 E i - U = 2(E& - $U) 2EE where E E means the ionization level of the 2-fold occupied donor. The ground state oneparticle wavefunctions (Da)' and of the donor center in the two occupancy states will also differ, but the difference will be neglected here. The details may be ascertained using first order perturbation theory with respect to the perturbation Vfi of the Hartree potential of equation (3.18). With this approximation, we again have only one type of one-particle wavefunctions

2 ) 1 2 = 1D)21 f $) of the 2-fold occupied center, thereby omitting all excited

I D a ) l % IDu) 2 = p a ) , (4.18)

as was the case before, and consequently also only one type of particle num- ber operators Nou, u = fi, Using the pertinent occupation numbers No, of localized states, the N-particle states of system may be characterized as I ND+r No-+, NJ, ... Nm), where N3, ..., N, denote the occupation numbers of ex&nded band states. We use these N-particle states as a basis for the calculation of the average particle number < N 1 > in localized state IDa) according to expression (4.8). We have

D2

(4.19)

The factors referring to extended states are the same in the numerator and denominator of expression (4.16), thus they cancel. In the remaining factors referring to localized states, the following four configurations are possible: No; = 0, N0-i = 0, N o + = 1, ND-l= 0, No+ = 0, ND-1= 1, No+ =

2

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464 Chapter 4. Electron system in thermodynamic equilibrium

1, N D - ~ = 1. Considering these configurations and their energies specified above, the evaluation of expression (4.16) results in

1 [,-P(%-P) + .-2P(E2d-d] , = f-. 2 (4.20)

The average number of particles occupying the center is the sum of < I$ 1 > and < iD-+ >, given by

D;i

Using this expression, we will now discuss how the average number of par- ticles localized at a center is influenced by configuration dependencies of one-particle energies due to Coulomb repulsion. In Figure 4.la this num- ber is shown as a function of chemical potential p for T = 0 K . As long as p < EA, we have < I? 1 + f i D - ~ > 0. If p exceeds EL, then

< N 1 + kD-& > suddenly increases to the value 1, which is maintained as long as E b < p < E g holds. When p becomes larger than E g , then < fi 1 + ND-+ > increases to the value 2, which is never exceeded even if p increases further. In regard to dependence on chemical potential one has, therefore, the values 0, 1 or 2 for the average particle number at the center. If there were no Coulomb repulsion effects, then E g = E& would hold and the 1-fold occupied state would cease to play a distinct role, as in the case of band states (see Figure 4.lb). For temperatures above zero Kelvin, the variation of < N 1 + kD-i > with p is qualitatively the same, except that the sudden changes which occur at zero temperature are smoothed out. The fact that the ionization level E$ rather than the oneparticle level E$ of the 2-fold occupied donor center determines the probability for two electrons to be at the center, is physically understandable: on a microscopic scale, ther- modynamic equilibrium is a dynamic state in which ionization (excitation) and de-excitation processes balance each other.

The above results for a double donor may be readily transferred to a double acceptor. If one replaces D by A , equation (4.21) gives the average number < $ 1 + > of electrons (rather than holes) at such a center, provided the energy levels EA and Eil in this expression are understood as electron rather than hole energies. Thus the average number of holes is (2- < NA+ 4 NA-+ >).

The method of calculating the average particle number at double donor and acceptor centers may be easily generalized to centers with more than

= D;i

D 5

D-i

D 2

AT

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4.2. Calculation of average particle numbers 465

1 21 21 . ElD E:' ED = ED E: ED+w

Figure 4.1: Average electron number < N D 3 + f i D - + > according to formula (4.21) of a double donor for T ---t 0 K as a function of chemical potential p. The energy levels E A and E$ are both negative but different from each other in case a), both negative and equal in case b), and only EA is negative in case c) while E S tends to +m. In the latter case one actually has a simply ionizable donor. In case b) there are effectively no configuration dependencies due to Coulomb repulsion which is actually only true for band states.

two ionization levels in the energy gap. For n-fold ionizable centers one has n + 1 charge states. In calculating the average number of particles at such centers one must include all of them, and take account of the fact that their oneparticle energies and ionization levels differ. We omit this calculation because it is rather cumbersome, and does not provide new physical insight. Instead we turn to simply ionizabIe centers which, from a practical point of view, are particularly important ones in semiconductors.

Simply ionizable donor and acceptor centers

Formally, the transition from a double donor to a simply ionizable donor can be represented by E E becoming infinitely large. In this case, E b plays the role of the binding energy E D of the simply ionizable donor, and all terms in equation (4.21) involving E g automatically vanish. Thus one obtains

which may be written in the form

(4.23)

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466 Chapter 4. Electron system in thermodynamic equilibrium

with (4.24)

as an effective distribution function. The latter can be interpreted as the probability that the simply ionizable donor is occupied by 1 electron. This carries the implication that f*(E) has the same meaning for localized states of simply ionizable donor centers as f(E) has for the doubly spin-degenerate band states.

One again calculates the average value < N 1 + kD-; > according to the general expression (4.8), but only takes into account the three states No; = 0 and N D - l = 0; N 1 = 1 and N D - ~ = 0; as well as N D f = 0 and No-+ = 1. This procedure, starting from equation (4.8), yields expression (4.22).

It is appropriate to clarify the consequences of Coulomb repulsion effects here. In the limiting case of strong degeneracy, the exponential term in the denominator of expression (4.24) for f*(Eo) can be neglected, and one arrives at ED) = 1 (see Figure 4 . 1 ~ ) . For band states, one obtains, under the same conditions, according to equation (4.19), the average value < Nyik + fiv-tk > = 2. Therefore, the energy level localized at the donor can host only half of the particles which can be accommodated by a band level. If statistical degeneracy exists, i.e. if the localized levels are almost completely occupied, then Coulomb repulsion limits the occupancy capacity of the localized states. In the non-degenerate limiting case, on the other hand, in which the ‘1’ in the denominators can be neglected and < N o + + No-4 > << 1 holds, expression (4.24) yields the Boltzmann distribution multiplied by 2. The same average particle number is obtained in this limiting case for a doubly spin-degenerate band state. This result is understandable, because in the non-degenerate case most of the localized levels are empty or simply occupied, and Coulomb repulsion does not play a significant role.

The results derived above for simply ionizable donor centers are imme diately applicable to simply ionizable acceptors. If one replaces D by A in equation (4.23) , then

Alternatively, equation (4.22) can be obtained as follows.

D %

Dg.

becomes the average number of electrons at the acceptor. The average hole number at the acceptor follows from this as

(4.26)

Here, the one-electron energy E A of the acceptor state represents the binding energy E B of the hole at this level.

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4.2. Calculation of average particle numbers 46 7

For a strongly degenerate hole distribution, i.e. for EA - p >> k T , the exponential term in the denominator of expression (4.26) may be neglected, and one gets 1 - ~ * ( E A ) M 1. This is half of the number of holes which under the same conditions, would be hosted by a valence band state. If the hole distribution is non-degenerate, expression (4.26) becomes the Boltzmann distribution function, multiplied by f. The latter factor reflects the fact that the population of the acceptor level by electrons is almost doubled due to the 2-fold spin degeneracy of the acceptor ground state.

Thus far, the average particle numbers at localization centers have been calculated under the assumption that, beside the spin degenerate ground state, no other energy levels exist with wavefunctions localized at the center. As we know from Chapter 3, there are, however, generally, several such states, indeed, infinitely many at hydrogen-like donors and acceptors. We refer to them as excited localized states. Below, we will determine the average particle numbers at centers having such excited localized states.

Centers with excited localized states

We restrict our considerations here to simply ionizable centers, although there are also excited states for multiply ionizable centers. The roles of these states are similar for both kinds of centers, and we will discuss the simpler case here. We examine a hydrogen-like donor, for which the localized states are characterized by the three orbital quantum numbers n, 1 , and m, and the spin-quantum number c. We denote the wavefunctions by IDnlmu), and the corresponding energy levels by ED,. The degree of degeneracy of level n is denoted by gDn. For the hydrogen atom, gDn = 2n2. w e seek the average number of electrons at the center, which is just the average particle number in the space of all states IDnlmc), < C n l m o N ~ d m o >. The averaging involved has to be carried out over states of the N-electron system in which either no particle is found at the center, i.e. with CdmU N I ) , ~ ~ ~ = 0, or there is exactly one, i.e. with Cnlmu N D ~ ~ ~ , , = 1. Then, according to the general formula (4.8) for the average value, we obtain

gDne-fl(EDn-p),

(4.27) This expression may be written in the form (4.24) if the ground state energy ED' is denoted by E D , and if the factor f multiplying the exponential term in the denominator in (4.24) is replaced by the temperature-dependent factor

I-' n < C fiDnl- > = 1 + C g D n e -P(ED,-P)

nlmu [ n

(4.28)

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468 Chapter 4 . Electron system in thermodynamic equilibrium

With this we have

(4.29)

The validity of this rewriting may be easily demonstrated by inserting y* of equation (4.28) into equation (4.29). Omitting the excited states in y*, considering therefore only the term with n = 1 and gD1 = 2, we obtain the value 4 for y*, and the average particle number at the donor center is given by equation (4.24). Considering the excited states, 7* becomes smaller than 4, provided the temperature T is larger than 0 K . At T = 0 K , y* retains the value 4 even in the presence of excited states. This value decreases only slightly with increasing temperature, as long as kT << [ED, - ED] holds for all excited states n 2 2. This inequality provides a simple criterion which helps decide whether or not the excited states are essential in the calculation of the average particle number at the center. The formal evaluation of the factor y* for hydrogen-like states with ED, = E, - Eg/n2 and g D n = 2n2 at arbitrary temperature T > 0 K leads to the result y* = 0, so that an apparently strong influence of the excited states occurs at all temperatures T > 0 K . In reality, however, the orbital radii U D ~ of the excited states are limited, if not through other factors such as finite sample size, then by the adjacent donor atoms - their distances from each other must be large compared to 2UDn, so that the n - t h electron shell may be formed unhindered by the adjacent donors. For a donor concentration of 1015 ern-' this condition yields 2 u ~ ~ << 1000 A. In silicon, 2UDn equals about 100 A xn , so that n must clearly be smaller than 10 and the overall influence of the excited states for T > 0 K is not so very great.

The factor y* significantly influences the average particle number at a donor center only for weak occupation of the center, which occurs when y* exp[P(ED - p ) ] >> 1 or E D - p >> kT holds, whence equation (4.29) yields

Since y* < 4, the average particle number at the center is enlarged as com- pared to the case without excited states where y* = 4. The excited states therefore lead to an increase of the average particle number at the cen- ter. This is physically understandable because more states are now avail- able for occupancy than without excited states. If, on the other hand, p - E D >> kT, i.e. the ground state is almost completely occupied, then one has y* exp[P(ED - p) ] << 1, and correspondingly

(4.31)

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4.3. Density of states 469

so that one gets the same average particle number at the donor center as without excited states. This is also physically clear immediately - because the donor can host only 1 electron, and, with high probability it will occupy the ground state, so excited states play practically no role.

As in the case without excited states, the results for donors can also be transferred directly to acceptors but we will not write down the correspond- ing expressions explicitly.

4.3 Density of states

Having determined the average particle numbers in single one-particle states and sets of such states, we can now proceed to explore the distribution of the electrons of a semiconductor over these states.

4.3.1 Total electron concentration

The total electron system is composed of the valence electrons of all atoms of the semiconductor. This number determines the average total particle number < fi > of the system,

We denote their number by Nbtal .

< N > = Nbtal. (4.32)

This relation may also be understood as a condition which determines the chemical potential p in the statistical operator j which is used in the aver- aging procedure. Considering equation (4.3) for N as a sum of one-particle number operators f i i for the various one-particle states li), one obtains from equation (4.32)

(4.33) i= I

The one-particle states l i ) comprise band states l v k f i) and localized states l y f i) where y abbreviates the orbital state quantum numbers and the posi- tion of the center. The band states are assumed to be 2-fold spin-degenerate, as before. With this, we presuppose that spin-orbit splitting is negligibly small which, as we know, is often not justified. The generalization of the results to the case in which spin-orbit splitting has to be taken into account is straight forward. The localized states (7 f i) are taken to be the ground states of simply ionizable donors (y = D ) or acceptors (y = A ) at various positions in the crystal. We assume that there are no excited localized states. The i-sum of equation (4.33) is decomposed in two partial sums, one over the band states, and one over the localized states. Correlations between the two sets of states which disturb this decomposition are neglected. With this, we obtain the expression

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470 Chapter 4 . Electron system in thermodynamic equilibrium

For the band part, equation (4.17) yields

(4.35)

with f(E) as the Fermi distribution function. The localized part in equation (4.34) can be written as a sum over the average values < k 1 + k7-; > for the various centers 7, because they are independent of each other, whence,

7 F

Using the effective distribution function f*(E) for localized states from equa- tions (4.24) and (4.25) we have

Altogether, we obtain

NtOta1= 2 C f ( E u ( k ) ) + Cf*(E,). (4.38)

The total electron number Ntotd is an extensive quantity. Generally, the corresponding intensive quantity, i.e. the total electron concentration related to the volume of the periodicity region, is of interest. It is defined by the expression

uk 7

(4.39) Ntotal 0 ' nbtal = -

In the further evaluation of the relation (4.38) we employ n t h l instead of Ntotal.

4.3.2 Density of states of ideal semiconductors

We first consider equations (4.38) and (4.39) in the absence of localized states, i.e. for an ideal semiconductor. Localized states will be taken into account in the next subsection dealing with real semiconductors. For the total concentration ni%$ of electrons occupying the extended Bloch states of an ideal semiconductor one obtains, using (4.38) and (4.39), the expression

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4.3. Density of states 471

(4.40)

Introducing a &function we rewrite the sum on k in the following way:

Substituting this expression in equation (4.40) for n i i l yields

where

2 pid"'(E) = - C 6 ( E - E , ( k ) ) .

uk

(4.41)

(4.42)

(4.43)

is the density of states (DOS) of the ideal semiconductor considered in Chap- ter 2 (see equation (2.210). The introduction of the density of states in ex- pression (4.42) for nh%i is significant in that it provides a physically mean- ingful division of the information needed for the calculation of the average total particle number and other statistical average values into two parts: first, information about the possible stationary quantum states of the sys- tem is contained in the density of states pidea'(E), and second, information about the statistical occupation of the states of the system is reflected by the distribution function f(E). One may also say that p*'(E) contains information relating to the dynamics of the system (classically, this would pertain to the solutions of Newton's equation of motion), and f(E) informa- tion about the statistics of the system (classically this would reflect on the statistical distribution of initial conditions or constants of motion). These two types of information are largely independent of each other and both are needed to calculate statistical average values in thermodynamic equilibrium.

Isotropic parabolic two band model

We consider a model of a semiconductor which has only one valence band and one conduction band. Both bands are assumed to be parabolic and isotropic for energies close to the band bottoms. The DOS pikd(E) of the conduction band is then given by equation (2.220), with v identified as the conduction band index c, and E d as the gap energy E,, so that

(4.44) ideal I 2m* 3/2

p c ( E ) = - 2x2 (L) h2 B(E - E , ) d G .

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472 Chapter 4. Electron system in thermodynamic equilibrium

Figure 4.2: Density of states of an ideal semiconductor with one conduction and one valence band. The full line corresponds to a parabolic dispersion law, the dashed line is closer to reality.

E

ENERGY 1 GAP

The DOS of the conduction band thus exhibits square root-like behavior at the band edge (see top part of Figure 4.2). Formally, it continues to follow a square root law also for energies deep in the band. However, for such energies, expression (4.44) does not apply. Qualitatively, the shape shown by the dashed curve in the top part of Figure 4.2 is more realistic. Expression (2.220) may not be used directly for the valence band, since its derivation presupposes that the effective mass m t is positive, while for the valence band it is in fact negative. The calculation above can, however, be modified easily to the case m: < 0. With v = v, E,o = 0, and mz -mt > 0, we have

(4.45)

The density of states of the valence band is thus seen to have square root-like dependence at the upper band edge and formally also continues to increase according to a square-root law for energies deeper in the band (see bottom part of Figure 4.2). The dashed curve again shows the more realistic shape. The entire density of states pideaz(E) of an ideal semiconductor having one valence and one conduction band is given by the expression

In the energy gap the density of states pideal(E) vanishes, as a direct con-

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4.3. Density of states 473

sequence of the fact that no stationary electron states exist there (see Fig- ure 4.2). Finally, we write down the total electron concentration n&%: of the ideal semiconductor having one conduction band and one valence band. From equation (4.42) it follows that

Bands with more complex structures

In the case of the elemental semiconductors Si and Ge, the conduction band edges consist of several valleys, and the iso-energy surfaces of the valleys are anisotropic. This has consequences for p,(E). For Si, for example, the conduction band density of states p p ( E ) takes the form

The valence band edges of diamond and zincblende type materials are de- generate. The degree of degeneracy depends on whether spin-orbit splitting is important or not. In the case of Si, the spin-orbit splitting energy A is small in absolute units (44 meV), but still large enough to compete with k T (25 meV for T = 300 K ) . Thus, for temperatures that are not extremely high, most of the holes are hosted by the I?$ valence band while the spin- split I?$ band remains practically devoid of holes. This means that there are two hole states per k-vector (apart from the 2-fold spin degeneracy), one for heavy holes, and one for light holes. Thus , the corresponding hole density of states p p of Si, calculated in the isotropic parabolic approximation of equation (4.45), is given by the expression

The same expression applies for Ge and GaAs. In cases where non-parabolicity effects are essential, the energy depen-

dence of the density of states no longer follows a square-root law. This is surely true for energies far removed from the critical point. In narrow gap semiconductors this occurs, however, even close to the critical point. For example, as shown in section 2.7, (see equation (2.404), the electrons and light holes in semiconductors having small energy gaps are, within the Kane model, described by the dispersion relation

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4 74 Chapter 4. Electron system in thermodynamic equilibrium

(4.50)

The conduction band density of states pP”(E) can be derived from equa- tion (4.50) as follows: First of all, the general relation

p p ( E ) = 4 k2S(E - E,(k)) (4.51) 27r

is still applicable. Here, we introduce E, as new integration variable, with the function k ( E c ) taken from the dispersion law (4.50). In this way, we obtain p p ( E ) for E-values above the conduction band edge at Ec = E, + 4 as follows:

Kane(E) - 1 (-) 3 3/2 (. - 5 - 3) /-. 27r2 2 P 2 3 2 Pc

(4.52) If E lies close to E,, then p p ( E ) given by (4.52) takes the form of ex- pression (4.44) for a parabolic band. To see this, the effective mass rn; has to be replaced by the expression from Table 2.13. That is to be expected, because the band structure is also almost parabolic for energies close to the edge within the Kane model.

4.3.3 Density of states of real semiconductors

We now consider doping the ideal semiconductor discussed above with N D donor atoms and N A acceptor atoms per cm3 to obtain a keal semiconduc- tor’. Each donor atom is assumed to give rise to a 1-fold ionizable donor level E D , and each acceptor atom gives rise to a 1-fold ionizable acceptor level EA. Excited donor or acceptor states are omitted from this considera- tion. The assignment of the donor and acceptor levels to impurity atoms is not essential in what follows. In principle, these levels can be generated by any simply ionizable point perturbation, including structural defects, with the same results. This also implies that the levels need not be shallow, they may be deep, although the shallow case is more typical than the deep case for the considerations below.

The electron system of a semiconductor with simply ionizable donors and acceptors may be thought to arise from that of the corresponding ideal semiconductor with one electron added per donor atom, and one electron is removed per acceptor atom. The electron concentration nL7; of the real semiconductor is thus given by the relation

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4.3. Density of states 475

The concentration (1/0) &=D,A < k 1 + kT-+ > of electrons occupying the donor and acceptor levels may be calculated by means of equations (4.23), (4.24), and (4.25), with the result

7 2

As in the context of the total electron concentration (4.42) of an ideal semi- conductor, we introduce here the density of states pTal(E) of the real semi- conductor. The contribution arising in the energy gap is denoted by pE$(E) and we set

For this relation to correspond to equation (4.54), pi$ must be defined by the expression

pgap Teal ( E ) = NDG(E - E D ) + N A ~ ( E - EA) . (4.56)

The densities of states of the valence and conduction bands of the real semi- conductor will be denoted by pLd(E) and p:"'(E), respectively. The band states of real semiconductor are, of course, no longer pure Bloch states, but Bloch states perturbed by scattering from the impurity atoms. Similar to pure Bloch states, they are, however, still spread out over the entire crys- tal. Correlation effects therefore play no important role in their occupancy. This means that the total electron concentration in the bands of the real semiconductor may be expressed by p;T'(E) in the same way as the total electron concentration of the ideal semiconductor was expressed by p Z d ( E ) in equation (4.47). Therefore,

0 ,Teal

total = 1, dE P:az(E) f (E) + /Or dE PF%) f ( - w Eg

00

+ I , d E PEW f*(E) . (4.57)

The band densities of states pLY'(E) of the real semiconductor differ from p ; Y ' ( E ) in accordance with the considerations of sections 3.4 and 3.5, where it was shown that localized states occur at the expense of the band states

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476 Chapter 4. Electron system in thermodynamic equilibrium

(Levinson's theorem). The number of conduction band states therefore de- creases by the number of donor states, and the number of valence band states decreases by the number of acceptor states, related to the volume unit in each case. Thus we have

0 dE pLal(E) = dE pikd(E) - N A ,

--M (4.58)

(4.59)

The entire density of states

of the real semiconductor satisfies the conservation law of the density of states, such that

m l, dEp'"l(E) = Jrn dEpiM(E) . (4.61) -m

Concerning the band state densities p,':'(E), we initially know, apart from the integral relations (4.58) and (4.59), nothing further than that it must reproduce p Z z ( E ) if N D and N A are zero. Below, we will see that it is possible, under certain conditions, to replace the real state densities p T f ( E ) approximately by the ideal state densities p : p z ( E ) even when N D # 0 and

In the following, we calculate the average electron concentrations in the energy bands and the donor and acceptor levels separately. Such separate calculations of the concentrations are necessary because the electrons in these different parts of the energy spectrum behave quite differently. The electrons localized at an impurity atom, for example, do not contribute to electric charge transport. The electrons of the conduction band do, just as do the electrons of the valence band, the latter, however, provide such a contribution in a different sense, not as electrons, but as holes. Therefore, facilitate to calculation of the conductivity of a semiconductor, one needs the average electron concentrations in the conduction band, in the donor and acceptor levels, and in the valence band, separately. The same remarks apply to the calculation of other measurable quantities.

N A f 0.

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4.4. fiee carrier concentrations 477

4.4 Free carrier concentrations

We start with general considerations which refer to both ideal as well as real semiconductors. The object is to derive a condition which fixes the total number of electrons of these systems.

4.4.1

Consider (4.53) for the total electron concentration nL%t of a real semicon- ductor. In this relation, the total concentration n&%! of the ideal semicon- ductor may be taken from equation (4.47), which we write down for T = 0 K . Since the Wrmi level of an ideal semiconductor lies in the energy gap ac- cording to the considerations of Chapters 1, one has f(E) = O(EF - E ) for T = 0 K . With this, relation (4.47) takes the form

nzdeal total - - S _ _ d E P"Vd"YE). (4.62)

This equation says that the total concentration nitg: of electrons equals the total concentration of valence band states of the ideal semiconductor. This statement is, of course, independent of temperature. The total electron concentration of the real semiconductor may therefore be written for any temperature as

Conservation of total electron number

0

0 n T f d total - - d E pid"'(E) + N o - N A . (4.63) 1,

Employing equation (4.58) this yields

(4.64)

On the other hand, we have relation (4.57) for nTzL, where we replace p z z ( E ) using (4.56) to obtain

From this equation we subtract equation (4.64), h d i n g

(4.65)

(4.66)

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478 Chapter 4. Electron system in thermodynamic equilibrium

Consider the abbreviations

(4.67)

These quantities have direct physical meaning: n is the concentration of electrons in the conduction band, p the concentration of holes in the valence band, Ngf is the concentration of non-occupied (ionized) and therefore simply positively charged donor atoms, and NX the concentration of occupied and thus simply negatively charged acceptors. The electrons in the conduction band and the holes in the valence band are freely mobile charge carriers, as pointed out in Chapter 1. In these terms, equation (4.66) becomes

n - p - ND+ + N; = 0. (4.71)

This expresses the conservation of total electron number for a real semicon- ductor. The electrons may be redistributed between the four energy regions, i.e. valence band, acceptor levels, donor levels and conduction band, but their total number cannot change thereby. One may also understand rela- tion (4.71) as the neutrality condition for a real semiconductor - the negative charge of the electrons of the conduction band and of the ionized acceptors must compensate for the positive charge of the holes of the valence band and the ionized donors.

4.4.2 Free carrier concentration dependence on Fermi en- ergy. Law of mass action.

We now derive an explicit relation between the concentrations n , p of elec- trons and holes in the bands, as they are given by equations (4.67) and (4.68), and the Fermi energy EF. The Fermi energy, in turn, is determined by the neutrality condition (4.71). In the non-degenerate case of an ex- ponential Boltzmann distribution for f(E), the position of the Fermi level has no influence on the relative distribution of the electron occupancy over the various available energy levels, only the total electron concentration de- pends on EF. However for a degenerate electron gas, i.e. in the regime of quantum statistics, the Fermi distribution function f(E) mandates that the relative distribution of electrons over the energy levels is also determined by

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4.4. nee carrier concentrations 479

the Fermi energy. In particular, this means that the relative distribution depends on the total particle concentration. By changing the total concen- tration, the ratios of the average particle numbers in the various energy levels can be altered. This is a clear manifestation of the statistical correlation be- tween the electrons of an ideal Fermi gas. It also shows that the chemical potential, or Fermi energy, is much more important in quantum statistics than in classical statistics.

In evaluating the integrals of (4.67) and (4.68), we note that the occu- pancy factors f (E) and [l- f ( E ) ] decrease monotonically with, respectively, increasing or decreasing energy and effectively cut off the integration range. The cutoff energies are separated by an interval of about kT from the respec- tive band edges. We assume that the densities of states of a real semicon- ductor, which are required in equations (4.67) and (4.68) for the effectively contributing energy intervals, may be approximately replaced by the state densities of the ideal semiconductor. This approximation is justified as long as the concentrations of donors and acceptors is small compared to the con- centration of significantly contributing band states. Furthermore, we assume that in the contributing energy intervals, the densities of states (4.44) and (4.45) of the ideal semiconductor, which are predicated on parabolic band structure, are approximately valid. Thus we obtain

Defining the Fermi integral as

the relations (4.72) and (4.73) may be written as

with

(4.72)

(4.73)

(4.74)

(4.75)

(4.76)

(4.77)

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480 Chapter 4. Electron system in thermodynamic equilibrium.

In the frequently occurring case that the Fermi level lies not only in the energy gap, but is separated by at least several k T from the two band edges, the arguments of the Fermi integrals in (4.75), (4.76) are negative and have absolute values large compared to 1. For such arguments the Fermi integral is approximately given by

so that

n = Nce(EF-Eg)/kT , ( E , - E F ) >> k T ,

(4.78)

(4.79)

p = NVevEFfkT , EF >> k T . (4.80)

The same result for n would be obtained if the Fermi distribution f(E) for the electrons of the conduction band were approximated by the Boltzmann distribution (4.13) from the very beginning, as is possible for ( E - E F ) >> k T . An analogous approximation for the electrons of the valence band is not available, since there E < EF holds. One may apply this approximation, however, to the holes of the valence band, i.e., to

1 - f(E) = e(E~-E)/rCT + 1'

For ( E F - E ) / k T >> 1, we have, approximately,

(4.81)

(4.82)

As in the case of conduction band electrons, the holes of the valence band are also described by a Boltzmann distribution if the Fermi energy is sufficiently remote from the band edges. One may say in this situation that the electron system of the conduction band and the hole system of the valence band are non-degenerate carrier gases. We emphasize that in the case of the valence band this statement refers to the holes and not to the electrons, which are highly degenerate. In further considerations we will always assume that the electrons of the conduction band and the holes of the valence band form non-degenerate carrier systems. The expressions (4.75) and (4.76) for n and p , respectively, provide a simple criterion for the validity of this assumption: The electrons are non-degenerate if n << Nc and the holes are non-degenerate if p << Nu. This may be seen as follows. If n << Nc holds, then the Fermi integral F L [ ( E F - E, ) /kT] in (4.75) must be much smaller than 1, which is possible only if - (EF - E,) /kT >> 1, meaning that degeneracy does not exist. The same argument holds for holes.

The quantities Nc and Nv in equation (4.77) have the dimensions of spa- tialdensities and are termed the effective densities ofs tates of the conduction

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4.4. f iee carrier concentrations 48 1

and valence bands, respectively. This terminology is clarified by examining relations (4.72) and (4.73) for n and p using Boltzmann distribution func- tions for both electrons and holes. In the corresponding expressions, the densities of states are summed over all energies, multiplied by weighting fac- tors which account for the occupancy of the various energy states, that is, for their effective contributions to average particle numbers. Equation (1.24) for Nc in Chapter 1 also demonstrates this explicitly (the symbol pc of (1.24) for a pure semiconductor corresponds to p:&’ here). We now estimate the order of magnitude of the effective densities of states. Generally, we have

Evaluating this with T M 300 K and m:,, FZ 0.5 m, then Nc,, M 10’’ follows. This order of magnitude may also be obtained in a strikingly simple way: The ordinary density of states of a band v is given by equation (4.43) as

(4.84)

which is of the order of magnitude of the number of primitive unit cells per cm3, i.e. M ~ m - ~ . With this number, the order of magnitude of the effective band densities of states is obtained by multiplying it by the ratio of k T to the whole band width. With a value of k T M 10 meV and a band width of 10 e V , N,,, FZ lo1’ follows.

The values of n and p determined by equations (4.75), (4.76) still depend on the Fermi energy E F , which is, as yet, unknown. The equation which determines EF is the neutrality condition (4.71). In this equation all quan- tities may be considered to be known except the Fermi energy, for which the equation is to be solved. Even without any calculation it is clear that the position of the Fermi level will depend on the donor and acceptor concentra- tions N D and NA. This dependence is transferred directly to n and p. There is, however, a simple function of the two concentrations, their product np, which in the case of non-degenerate carrier systems does not depend on E F , and hence not on N D and NA. This may be seen using equations (4.79) and (4.80), from which it follows that

np = ni, 2 (4.85)

ni = (NcNV)3e-iEg/ICT. (4.86) with

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482

n,

Chapter 4. Electron system in thermodynamic equilibrium

GaP GaAs Si Ge I d s InSb

2.7 x 10' 1.8 x lo6 1 .5 x 10" 2.4 x 1013 8.6 x lOI4 1.6 x 10l6

Table 4.1: Intrinsic carrier concentrations ni at T = 300 K (in ~ m - ~ ) .

This equation may be understood as the law of mass action for a 'chemical reaction' in which bound electrons in the valence band and at the donor atoms on the one hand, and bound holes in the conduction and at the ac- ceptor atoms on the other hand, generate free electrons in the conduction band and free holes in the valence band. The concentration of the bound electrons can approximately be set equal to the effective density of states Nu of the valence band, since in the non-degenerate case, the concentra- tions of holes and of occupied donors are small in comparison to N,,. A similar statement holds for the bound holes which, here, also include the non-occupied states of the conduction band. Their concentration roughly equals the effective density of states Nc of the conduction band. According to the mass action law, one then has, for an ideal gas, n p = K ( T ) N , N , , with K ( T ) = exp[-(gc + g,,)/,kT] as mass action constant. Here, gc and g,, are, respectively, the free enthalpies of an electron in the conduction band and of a hole in the valence band. With gc + gv = E,, equation (4.86) is indeed the mass action law, as stated above.

4.4.3 Intrinsic semiconductors

We consider the particular case of an ideal semiconductor, such that N D and N A both vanish. Then N& and N A are also zero, and the neutrality condition (4.71) becomes

n - p = 0. (4.87)

This equation reflects the fact that the electrons of the conduction band are created solely by thermal excitation of electrons from the valence band, leaving behind an equal number of holes. If we substitute p = n in equation (4.85), then we have

n = n i , p = ni. (4.88)

This relation was already stated in Chapter 1 on the basis of empirical facts (see equation 1.20). One refers to ni as intrinsic concentration, because it

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4.4. Free carrier concentrations 483

depends, aside from temperature, only on quantities which are determined by the ideal semiconductor itself. In this context, the latter is called an intrinsic semiconductor. For the intrinsic carrier concentration, with Nc = N , = lo1' ~ r n - ~ , E , = 1 eV,T = 300 K , we obtain the order of magnitude ni --" 10" ~ r n - ~ . Table 4.1 gives an overview of the ni-values for some important semiconductors at room temperature. In this, there is a strikingly large variation between a few carriers per crn3 for the wide gap semiconductor Gap, and 10l6 carriers per cm3 for the narrow gap semiconductor InSb. In Figure 4.3 the variation of ni with temperature is shown. Evaluating all numerics and fundamental constants, we find

The position of the Fermi level E F ~ for a non-degenerate intrinsic semicon- ductor follows from equations (4.79) and (4.86) by taking the logarithm. The result is

1 1 E F ~ = -E,+ - k T h

2 2

Replacing Nc and N , here by equation (4.77), it follows that

1 3 E F ~ = -Eg + -kT h

2 4

(4.90)

(4.91)

For T = 0 K , the Fermi level thus lies exactly in the middle of the gap between the valence and conduction bands. It remains there even for finite temperature if rn; = rn:. If rn; # m:, a temperature dependence emerges which is schematically shown in Figure 4.4. For rn: > rn; or N , > N,, EF( moves away from the valence band edge, and for rn; > rn; or Nc > N,, away from the conduction band edge. This may be understood as follows: If the effective density of states in the valence band is larger than that in the conduction band, one would have more holes in the valence band than electrons in the conduction band if the Fermi level were to stay in the middle of the gap. It must move away from the valence band edge to assure the equality of n and p . An analogous interpretation can be given in the case of Nc > N,. In short, one can say that the Fermi level is repelled by regions of high density of states, and is attracted by regions of low density of states.

We next determine n and p for the case of real semiconductors with donor and acceptor atoms present. In Chapter 1, the designation eztrznsic semiconductors was already introduced to describe them.

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484 Chapter 4 . Electron system in thermodynamic equilibrium

Figure 4.3: Intrinsic concen- trations of several semicon- ductors as functions of tem- perature (calculated values).

&T/OC

IO~/T/K-’ -. Figure 4.4 Variation of Fermi level with tempera- ture for an intrinsic semi- conductor.

4.4.4 Extrinsic semiconductors

n-type-semiconduct ors

Initially, we assume that there are only donors, and no acceptors are present, N D # 0 and N A = 0. The neutrality condition (4.71) is then transformed into an equation for the electron concentration n. In this regard, we need the concentration Ngf = N D ( ~ - ED)] of ionized donors. The donor population probability f *(ED) is given by equation (4.24) as

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4.4. fiee carrier concentrations 485

(4.92)

where EB denotes the donor binding energy. In general, the donor levels, unlike band energies, do not lie well above the Fermi level. They may be close to, or even below it. This means that the degeneracy of the Fermi distribution, which could be neglected for carriers in the bands, must in general be taken into account in the effective donor distribution function !*(ED). The expression (4.92) for f*(Eo) must therefore be used in its original form. The Fermi energy can be eliminated from it by introducing the electron concentration n given in equation (4.79), with the result

(4.93)

(4.94)

Apart from the factor 1/2, the quantity nl represents the electron concen- tration in the conduction band of a non-degenerate ideal semiconductor with a (fictitious) Fermi level at the position of the donor level. If we replace p in terms of nT/n using equation (4.85), then the neutrality condition takes the form

From this equation one may easily derive the magnitude relations

(4.95)

(4.96)

In a semiconductor containing donors but no acceptors, the electron concen- tration is always larger than the intrinsic concentration ni, and the latter is larger than the hole concentration p. It is therefore called an n-type semi- conductor. The electrons are majority carriers in this case, and the holes are minori ty carriers. To solve the neutrality condition (4.95) for n, we transform it into the third-order polynomial equation

2 2 n3 + nln2 - (ni + n l N D ) n - ninl = 0. (4.97)

The solution n of equation (4.97) can be obtained in closed analytical form. The minority carrier concentration p follows from n by means of the relation n p = nz. We examine here an approximate solution of equation (4.97) in two typical limiting cases: 1) the case of extremely small donor concentrations N o , in which N D << ni holds, and 2) the case of large donor concentrations with N o >> ni.

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486 Chapter 4. Electron system in thermodynamic equilibrium

1) N D <( ni :

Considering the inequalities (4.96), the term proportional to N D in equation (4.97) may be neglected in comparison with the second term. Thus we have, approximately,

n3 + n1n2 - nfn - ninl 2 = 0. (4.98)

The only positive real solution of this equation is

n = ni. (4.99)

For p , one gets p = p i . This means that in case l), the carrier concentrations n and p do not differ from those of an intrinsic semiconductor. If the values of N D are not extremely large, then the inequality N D <( ni may always be ful- filled, simply by making the temperature and therefore also ni large enough. For sufficiently high temperatures and not overly large N D values, one thus always has the intrinsic case. Only if the donor concentration becomes suffi- ciently large, and the temperature so low that the inequality N D << ni does not hold, then the donors can influence the carrier concentrations n and p consider ably. We turn now to the other extreme case

2 ) N D >> ni :

Because n > nil one then also has NDn >> n:. Under this condition, the last term in (4.97) may be neglected in comparison with the term nlNDn. With this, one has the approximate equation

n2 + n1n - n l N D = 0. (4.100)

Its positive solution is given by

(4.101)

If N D << nl, i.e. for relatively light doping, but sufficiently high tempera- tures, it follows, approximately, that

n = N o . (4.102)

All donor atoms are ionized and have ‘donated’ their additional electrons to the conduction band. Understandably, for this to occur one needs sufficiently high temperatures and, maintaining the validity of N D >> nil sufficiently small donor concentrations. The latter restriction is necessary in order to

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4.4. nee carrier concentrations

"I

7 Id6 1 - 5 ! I

tion of temperature for N D =

loi5 ~ r n - ~ . (After Smith, 1979.) - I - -

E : I t - *

487

I I I I

INTRINSIC RANGE SLOPE = Eg

I SATURATION RANGE

c $ E ; lo"=

10'3-

b

I I \

f \ n l

1 1 1 1 I I 1 I I 1 I I

3

insure that all donor electrons can be hosted by the conduction band. One terms this case the exhaustion of doping centers or the saturation of carrier concentration.

If, conversely, N o >> n1 holds, which means very high donor concentra- tions and low temperatures, equation (4.101) approximately yields

n = JnTNo. (4.103) Because of the assumption nl << N o , this concentration is substantially smaller than N o , i.e., only a small fraction of the donor atoms have been ionized and have transferred their electrons to the conduction band. One terms this a reserve of doping atoms or a freez-out of carriers at the im- purities. Even in this case n can be substantially larger than the intrinsic concentration ni, provided n1 is larger than or at least of the same order of magnitude as ni. This occurs for shallow levels with nl >> ni, and is also still valid for levels ED = ( 1 / 2 ) E g at the middle of the gap for which nl = ( 1 / 2 ) n i follows. The property of impurity atoms to be effective electron donors for the conduction band is not restricted, therefore, to the case where the impurity levels lie just beneath the conduction band edge. Even from the middle of the forbidden zone they can increase n far above the intrinsic concentration. This can change only for yet deeper levels with n1 << ni. The largest increase, however, is caused by shallow levels. This explains why impurity atoms forming shallow levels are particularly important for semiconductor doping. Figure 4.5 shows the temperature dependence of the free carrier concentration in such a case. The three qualitatively different

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488 Chapter 4. Electron system in thermodynamic equilibrium

0 EC

ED

Figure 4.6: Temperature de- pendence of the Fermi level for n-type Ge assuming typ-

and N D c ~ T - ~ ) . (After Bonch-Bruevich and Kalash- nikov, 198L)

1 ical values of Eg (10 m e v )

0 20 40 60 80 T/K -

regions - freez-out of carriers for low temperatures, saturation for medium, and intrinsic behavior for high temperatures - may be clearly recognized in this figure. The minority carrier concentration p in case 2) is always substantially smaller than pi of the intrinsic case.

We have yet to determine the position of the Fermi level in the case with considerable n-type doping where N D >> ni holds. Employing the relation

(4.104)

we obtain, for N D << n1, i.e. for high temperatures and low doping, the result

(4.105)

and, for N D >> nl, i.e. for low temperatures and high doping, we find that

(4.106)

The dependence of the Fermi level on temperature is shown in Figure 4.6 for n-type germanium with typical values of N D and EB. At T = 0 K the Fermi level lies exactly in the middle between the donor level and the conduction band edge, the donors being only partially ionized. With increasing temper- ature the degree of ionization increases. In order for this to be possible, the Fermi level must shift to lower energies. This may again be understood as a repulsion of the Fermi level by the group of states with higher density - the effective conduction band density of states is larger than the donor density of states, which equals the donor concentration.

p-type semiconductors

All the above considerations concerning carrier concentrations for n-type semiconductors can be transferred without difficulty to semiconductors which

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4.4. free carrier concentrations 489

contain acceptors instead of donors, i.e. for which N A # 0 and N D = 0. They are called p-type semiconductors. The holes are the majority carriers in this case, and the electrons are the minority carriers. We omit details of derivation here and only display the results. In place of equations (4.92), (4.93), and (4.94), we have the relations

(4.107)

(4.108)

where EB denotes the acceptor binding energy, and p i is given by

p l = 2 N , e - E /kT . (4.109)

With p instead of n as independent variable, the neutrality condition (4.71) takes the form

(4.110)

From this, we obtain the equation for p as

P 3 + p i p 2 - (ni 2 + p i N A ) p - n i p 1 2 = 0. (4.111)

For N A << ni the solution is

P = ni,

which results in n = ni. For N A >> ni we obtain

(4.112)

P = N A (4.113)

if NA << p l , and P = & K (4.114)

if N A >> p l . In both cases the minority carrier concentration n is much less than p . For the Fermi level position, we have results analogous to those obtained for an n-type semiconductor.

4.4.5

Above, we considered real semiconductors which contained only donor or only acceptor atoms, but not both kinds of atoms simultaneously. Now, we consider the presence of both types of impurities in comparably large concentrations. Such a state would hardly be created intentionally, but it may occur unintentionally in the fabrication of a semiconductor material,

Compensation of donors and acceptors

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490 Chapter 4 . Electron system in thermodynamic equilibrium

where unwanted chemical pollution cannot be avoided. Such pollution could involve, for example, some boron atoms in a silicon crystal which one would like to n-dope by means of phosphorus. Also, technological aspects of the fabrication of semiconductor devices can play a role, for instance, a p-doping of partial areas of a previously n-doped silicon wafer causes the simultaneous occurrence of donors and acceptors.

The question arises as to how carrier concentrations adjust under these circumstances in the semiconductor. Do the respective electron and hole concentrations simply add without interfering with each other, or do they strengthen or compensate among themselves completely or partially? A gen- eral answer to this question can be readily obtained from the mass action law (4.85). According to this law, the addition of acceptors to a mate- rial which previously contained only donors, cannot lead to an increase of the hole concentration without simultaneously lowering the concentration of electrons. The acceptors therefore compensate the effect of the donors to a certain degree. The physical reason for this is illustrated in Figure 4.7. The introduction of acceptor levels slightly above the valence band edge in a material which already contains donor levels slightly below the conduction band edge, changes the ground state of the crystal. Instead of transfer- ring into the conduction band accompanied by a small energy increase, the donor electrons will transfer into the empty acceptor levels with a substan- tial lowering of the total energy of the crystal, and the acceptors on their part preferentially trap these electrons rather than those from the valence band, since the latter must first be excited. Analogously, holes will move from acceptor levels to the donor levels instead of being excited into the valence band. The capture of the free carriers can be complete, whereupon one speaks of complete compensation If the concentration of acceptors is too small to host all donor electrons, one has a partially compensated n-type semiconductor (p < n < N D ) , and if the concentration of donors is too small to capture all acceptor holes, one has a partially compensated p-type semiconductor (n < p < N A ) .

In regard to occupancy statistics, the source of electrons (valence band or donor levels) which occupy the acceptor levels is completely irrelevant. What does matter is only that levels which can be occupied by electrons exist below the donor levels. Instead of shallow acceptor levels these can also be deep levels which do not at all act as acceptors because of their overly large energy separation from the valence band edge. If they can bind electrons, they can still be important for the compensation of donors - acting as trapping centers for electrons or simply as electron traps. Analogous statements hold for the compensation of acceptors by deep levels which can bind holes, one has hole traps. As we have seen in section 3.5, deep levels are caused both by structural defects as well as by chemical impurities. Since semiconductor materials always contain such perturbations - in larger numbers for lower

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4.4 . fiee carrier concentrations 49 1

Figure 4 . 7 Semiconductors with a) a donor level, b) an acceptor level, and c) with both kinds of levels. In parts a) and b) the upper drawings correspond to the ground state and the lower to the excited state. In part c) the lower drawing corresponds to the ground state, the acceptor level has captured the donor electron.

cost fabrication - the complete or partial compensation represents, so to say, the usual state of a semiconductor material. To overcome this requires strong efforts in material cleaning and in crystal growth. But, even then, certain semiconductor materials still resist intentional doping. Known examples occur among the wide-gap 11-VI compound semiconductors, such as ZnS and ZnSe, which can be n-doped with relative ease, but p-doped only with difficulty. This problem was solved only recently, when the application of modern epitaxial growth techniques made it possible to also fabricate p- doped 11-VI-semiconductors with wide energy gaps.

Quantitatively, the free carrier concentrations n , p in the case of both n- and p-type doping may be obtained from the same relations which were used in the case of pure n- or ptype doping. The Fermi level again follows from the neutrality condition (4.71), which here takes the form

(4.115) n+n1 N D + nf +pin

where n and p are to be taken from equations (4.79) and (4.80). The relation (4.115) can be transformed into a fourth-order equation for n or p. Since this is not solvable in compact form, we consider certain limiting cases which allow for simple solutions and provide some physical insight. Under the assumption n << n1, p << pl we find, approximately,

n - ni - n(No + N A ) = 0.

n: n1

n n - - - - N A = 0,

(4.116)

This equation corresponds to the limiting case of completely ionized donors and acceptors. Its solution is given by

2 2

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492 Chapter 4. Electron system in thermodynamic equilibrium

Figure 4.8: Calculated tempera- ture dependence of the electron con- centration of partially compensated n-type germanium. ( E B = 10 meV, mt = 0.25 m). An electron concen- tration N D - N A = 10l6 is assumed. The various curves corre- spond to different degrees of com- pensation N A : 1 - 0, 2 - 3 - loL5, 4 - loL6 (After Bonch- Bruevich and Kalashnikov, 1982.)

t ’OI6 T E < 1015

10’4

C

0 2 4 6 8 1 0 1 2 IOOT-’/K-L

For N D = N A , it follows that n = ni. The compensation is complete and we have the intrinsic case. For N D # N A and ( N o - N A ) ~ >> n?, the result is

(4.119) n?

p = ( N A - N o ) , n = if N A > ND. ( N A - N D ) ’

Thus a partially compensated n-type semiconductor occurs for N D > N A , and a partially compensated p type semiconductor occurs for N A > ND. In both cases the majority carrier concentration is JND - NAJ. For not completely ionized donors or acceptors, this simple relationship is no longer valid. In Figure 4.8, partial compensation as a function of temperature is illustrated using the example of n-doped germanium.

4.4.6 More complex cases

Several different single donors and acceptors

In the analysis above, we considered at most two kinds of simply ionizable doping centers simultaneously. But, in fact, more can occur, of course. It is not difficult to expand the theory to this more general case. Each additional type of a simply ionizable donor (described by an index j) introduces an

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4.4. f iee carrier concentrations 493

additional term N D j / ( n + nj ) into the neutrality condition, and each addi- tional type of such an acceptor (described by an index k ) gives rise to an additional term NAkn/(n: + n p k ) . The neutrality condition for this more general case therefore takes the form

n2 n - 2 - z ~ D j - n j + ) : N A ~ z p k n = 0.

n j n+nj ni + P k n (4.120)

As before, it determines the position of the Fermi level uniquely. For the doping centers themselves, excited bound states were excluded up to now, and we admitted only two charge states, the ground state and the simply ionized state. If we give up these assumptions, then the neutrality condition (4.71) changes. This means that the position of the Fermi level then depends on N D , NA, N,, Nu, and T differently than in the case of centers having excited states. The particulars of this dependence will be analyzed below. In this, we may use the fact that the relations (4.79) and (4.80) between the carrier concentrations n, p and the Fermi level remain the same as for the case of simple impurities.

Excited bound st at es

We consider a donor which, as before, is only simply ionizable, but has excited levels ED,, n = 2 , 3 , . . . in addition to the ground state level ED ED^. In section 4.2, the average number < Cdms N D ~ ~ ~ > of electrons at such an impurity was calculated. Using equation (4.29) we find that

with y* given by equation (4.28). Formally, the occurrence of excited bound states modifies condition (4.97) in that nl is replaced by

61 = y*2n1.

(4.121)

the neutrality

(4.122)

Since y* < f, and therefore 61 < nl, the concentration n of free electrons is decreased in general. This is readily understandable since the excited elect,ron states of the donors result in more electrons being bound. For acceptors with excited bound states one gets analogous results. The concentration p l is replaced by

$1 = y*-12-lp1. (4.23)

in the neutrality condition (4.111). Now, j1 > p l holds, which means that the number of free holes is increased. This is immediately clear because

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494 Chapter 4. Electron system in thermodynamic equilibrium

the acceptors bind fewer holes if there are excited electron states at the acceptors.

Multiply ionizable centers

n-type semiconductors

We consider double donors, such as substitutional sulfur atoms in silicon. The average electron number < fi 1 + ND-; > at such centers has been calculated in section 4.2. The result is shown in equation (4.21). To evaluate the neutrality condition

0,

n - p - N D ~ = 0, (4.124)

one needs the hole concentration Ngf localized at the donors. It is given by the relation

Substituting < fi I + fiD-l > as given by equation (4.21), replacing the chemical potential p by E p l Z E b by Eg - Eh, and E g by Eg - Ei l where E L and Ei l are, respectively, the ionization energies of one electron at the 1- or 2-fold occupied center, it follows that

D I

As before, in the case of simply ionizable donors and non-degenerate carrier gases, one can eliminate the Fermi energy from this expression by introducing the electron concentration n given by (4.79). Instead of the one concentration constant n1, two now occur, namely

Because E h > E i l , we always have n; < n;. In terms of these constants, equation (4.126) may be written in the form

(4.128)

With this result (and using n!/n for p ) , the neutrality condition (4.124) takes the form

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4.4. Free carrier concentrations 495

(4.129)

It may be transformed in a fourth-order equation for the determination of n. Because the difficulty of solving this in closed form, we again consider only limiting cases. Generally, the doping concentrations ND >> ni should be sufficiently large that the intrinsic term in (4.129) can be neglected. If one additionally assumes that

n << n;, and with this, also n << na,

which means that the doping should not be too heavy, nor the temperature too low, then (4.129) approximately yields

n = ~ N D . (4.130)

In this case, all donors are 2-fold ionized and one has the case of exhausted doping atoms. To fulfill the condition n << n7, it is necessary that N D << 7x7. The exhaustion of a doubly ionizable donor is thus characterized by an inequality similar to that which formerly was found for simply ionizable donors . In the limiting case

n >> n;,n$

which means sufficiently strong doping and not-too-high temperatures, equa- tion (4.129) approximately results in

(4.131)

Here, we have set n* = na2/n7 for brevity. The positive solution of this equation is given by

n = n* Jq- I] (4.132)

With the assumption N D << n*, it yields n = N o , i.e. each donor atom is simply ionized. One has exhaustion of the doping centers with respect to 1-fold ionization, and reserve with respect to 2-fold ionization. This result can be understood in the following term: In order to fulfill the conditions ND << n* and ND >> n7 simultaneously, n* >> ni and with this also nz >> nT must hold. This means that n* << n$ and ND << na, so that n? << N D << na

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496 Chapter 4. Electron system in thermodynamic equilibrium

follows. 2-fold occupied center, but not for the 1-fold occupied center.

from (4.132), approximately,

The condition for donor exhaustion is therefore fulfilled for the

A different situation occurs in the case ND >> n*, in which one finds

n = Jm. (4.133)

Because nT < n4, the condition ND >> n* also insures that ND >> na. This means that the reserve case also occurs with respect to the 2-fold occupied center. In comparison with a one-particle center having an ionization energy E B equal to the ionization energy EB of the simply occupied two-particle center, the carrier concentration due to the two-particle center is enlarged by the factor 2(n$/n;). The ‘2’ is a consequence of the doubling of the electron number at the non-ionized donors. The factor (n;/n;), which exceeds 1, reflects the fact that the ionization of the 2-fold occupied center has a larger probability than that of the simply occupied center.

p-type semiconductors

We consider double acceptors such as substitutional zinc atoms in silicon. The neutrality condition reads, in this case,

p - - - 2 N ~ n5

P P2 + 2PIP + P2

with

(4.134)

(4.135)

Here EL and Eil refer to electrons (rather than holes) and denote, respec- tively, the ionization energies of the acceptor with, respectively, 2 holes (no electron) or 1 hole (1 electron) at the center. Because Eil > EL, one always has pT > pz. The neutrality condition will again be solved in limiting cases. In these cases the acceptor concentration NA will always be assumed to be large compared to the intrinsic concentration p i = ni of holes, so that the intrinsic term in (4.134) can be neglected. We first assume that

p << p ; , and with this, also p << p ;

holds, which means not-too-strong doping and low temperatures. The neu- trality condition then approximately yields the relation

P 2NA, (4.136)

corresponding to all acceptors being 2-fold ionized, the case of exhaustion. In the opposite case

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4.4. Free carrier concentrations 497

Figure 4.9: Average electron num- ber q at an Au-atom in Ge as a func- tion of the position of the Fermi level (After Bonch-Bruevich and Kalash- nikov, 1982.)

we find from (4.134) that

(4.137)

P = $%lP;, (4.138)

follows, corresponding to the reserve case. For p << p ; , equation (4.137) approximately yields

p = NA. (4.139)

All acceptors are simply ionized, and there is exhaustion of the acceptors with respect to 1-fold ionization, and reserve with respect to 2-fold ionization.

Amphoteric doping centers

Finally, we discuss a special feature of multiply ionizable doping centers. As mentioned above in section 3.5, there are amphoteric deep centers simulta- neously acting as donor and acceptor. In silicon, for example, a neutral Au atom gives rise to a donor level Au(O/ f ) in the lower half of the energy gap, and an acceptor level Au(-/O) in the upper half. In germanium, there is a donor level Au(+/2+) of the 1-fold positively charged Au(f1) center which lies just above the valence band edge (see Figure 4.9). The acceptor levels Au(2 - /--) and Au(3 - / 2 - ) of, respectively, the 1- and 2-fold negatively charged Au(1-) and Au(2-) centers are found in the upper part of the gap. Whether Au is a donor or an acceptor thus depends on the charge state or, in other terms, on the occupancy of the Au center by electrons and, with

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498 Chapter 4 . Electron system in thermodynamic equilibrium

this, on the position of the Fermi level (see Figure 4.9). The latter is co- determined by the concentration of Au atoms, and it also depends, however, on the concentrations of other donors and acceptors if there are such. In p-type doped germanium, Au acts as donor because, in this case, the Fermi level lies just above the valence band edge so that the charge state Au(l+) occurs. In n-doped germanium, Au forms a double acceptor due to the Fermi level position lying slightly below the conduction band edge, which results in the charge states Au(1-) and Au(2-).

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499

Chapter 5

Non-equilibrium processes in semiconductors

In the preceding chapter we considered semiconductors which were spatially homogeneous in a macroscopic sense and were not subject to external per- turbations or fields. The thermodynamic equilibrium state of electrons in such semiconductors is characterized in terms of thermodynamic variables, in particular a temperature and a chemical potential, which do not depend on space and time coordinates. Semiconductors differ from other materi- als inasmuch as they can easily be driven out of equilibrium. The electron temperature and chemical potential may then differ relatively strongly from their equilibrium values, provided such state variables do exist at all. If this is the rase, they may depend on space and time coordinates. In addition, there may be relatively large electric fields and currents. ‘Relatively’ means with respect to other solid state materials. In metals, by way of comparison, macroscopic electric fields practically cannot exist, because they are almost completely screened out over distances of about 1 A by the huge number of mobile electrons available in these materials. The same may be said for spatial variations of the chemical potential in metals, which also practically cannot occur because of the large electron concentration.

Semiconductors have the unique property that external manipulations can easily bring them into states which deviate strongly from thermody- namic equilibrium. It is on this unique property that essentially all ap- plications of semiconductors in electronic and optoelectronic devices ulti- mately rest. The semiconductor photoresistor, for example, works because non-equilibrium electrons and holes are generated by irradiation with light, greatly changing the resistance of the semiconductor sample. In a field effect transistor, the resistivity of a particular semiconductor region is changed by means of an external voltage, and in a light emitting semiconductor diode, non-equilibrium minority carriers, created at a forward biased pn-junction,

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500 Chapter 5. Non-equilibrium processes in semiconductors

recombine with the equilibrium majority carriers accompanied by the emis- sion of light. While the latter two examples refer to semiconductor junctions, the following considerations will initially deal with bulk semiconductors, i.e. semiconductors with spatially constant doping and chemical composition.

5.1 Fundamentals of the statistical description of non-equilibrium processes

Thermodynamic equilibrium is the only macroscopic state of the electron system of a bulk semiconductor which does not change in time if one leaves the semiconductor alone. In all other macroscopic states, i.e., all non- equilibrium states, the system undergoes temporal changes. This means that time-dependent macroscopic processes occur if the semiconductor electron system is out of equilibrium. One terms them non-equilibrium processes, and it is these which we will discuss in the present chapter.

Within the framework of a semiclassical microscopic statistical theory, non-equilibrium states of an electron system are described by occupation probabilities f of one-electron states i which depend not only on the energies Ei of these states, as in thermodynamic equilibrium, but on the quantum numbers themselves. In general, they are also functions f ( i , x, t ) of space and time coordinates x, t . The transition from equilibrium to non-equilibrium states is thus described by the replacement

Since, ultimately, the temperature T and chemical potential p are defined as parameters in the equilibrium distribution function f(Ei), these quantities are not, in general, meaningful for macroscopic non-equilibrium states. Ac- cordingly, there is also no corresponding non-equilibrium distribution func- tion f(i, x, t ) which would depend on them. Moreover, there is no handy general expression for f(i , x, t ) at all which would depend on macroscopic parameters to be adjusted to the particular non-equilibrium state, in con- trast to the case of equilibrium. Instead of analytical expressions one may formulate equations from which the distribution function f ( i , x, t ) can be determined. Of such equations, the Boltzmann equation is the simplest for the electron system of a crystal.

Boltzmann equation

We will write this equation down explicitly for the extended one-electron excitations of the system, i.e. the Bloch states uk. The distribution function f (uk, x, t ) in this case is the probability that, at a position x and at a time t , the Bloch state having quasi-wavevector k and band-index u is occupied

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5.1. Fundamentals of the statistical description of non-equilibrium processes 501

by an electron. Since, in reality, the Bloch states are infinitely extended, the specification ‘at a position x’ must be understood in the sense of a region centered at x, whose spatial extension A is large compared to the electron wavelength X = 2n/JkJ. For the concept of a spacedependent distribution function f(vk, x, t ) of Bloch states to make sense, f(vk, x, t ) must change very slowly over a length of the order of magnitude A. A similar condition holds for the variation of the distribution function with time.

In the absence of external perturbations of the type which drive a current or energy transfer, the system of Bloch electrons is in thermodynamic equi- librium, such that it may be described by the Fermi distribution function (4.13) with E = E,(k). To drive the system out of equilibrium, appropriate external perturbations must be applied, such as particle density gradients or an external electric field E(x, t ) , as we will assume here. In addition to ex- ternal perturbations, internal perturbations also act on the Bloch electrons, such as scattering by phonons and impurities. The term collisions is used in this context which means that the scattering events take place in a very short time, practically instantaneous. Both types of perturbations, external ones and collisions, result in temporal changes of the distribution function, denoted as (df(vk, x, t ) / d t ) e z t and (df(vk, x, t ) / d t ) , d l , respectively. Accord- ing to a general theorem of statistical mechanics, the Liouville theorem, the total change o f f , taken at a position x(t), iik(t) of phase space which moves together with the particles, must vanish. Thus,

Performing the derivative, the first term of this equation may be written as

Here, (dx(t)/dt) may be replaced by (l/h)V&,(k) according to equation (2.193), and ( d k ( t ) / d t ) by -eE/h. according to equation (3.310). Combining equations (5.2) and (5.3) yields the Boltzmann equation

[i + iVkE,(k) . V, - eE(x, t ) . Cdl

The collision term is in general an integral over products of two distribu- tion functions, one of which, f(vk, x, t ) , involves the same Bloch state vk occurring on the left hand side, but the other, v’k’, involves any Bloch state which can participate in a collision. The characteristic difficulties of the

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502 Chapter 5. Non-equilibrium processes in semiconductors

Boltzmann equation are mainly due to its collision term, a statement which applies both to the derivation as well as to the solution of this equation. Let’s suppose that the Boltzmann equation has been solved. Then, as in the case of equilibrium, one may use the known distribution function f(vk, x, t ) to determine statistical average values < A > of any one-particle quantity A of the many-electron system. Such quantities represent sums over cor- responding individual electron quantities a. Like the distribution function itself, the average values are functions < A > (x, t ) of space and time coor- dinates, such that

< A > (x, t ) = C f ( v k , X , t)(vk)alvk). (5.5) uk

In addition to the physical quantities which are of interest in both equi- librium and non-equilibrium, such as the electron concentration and the energy density, non-equilibrium also involves other quantities to be deter- mined, in particular, currents related to equilibrium quantities such as, for example, particle or energy currents. The latter differ from zero‘only in non-equilibrium, and thus are of no interest in equilibrium. Moreover, those quantities whose average values do not vanish in equilibrium must also be re- considered, because under non-equilibrium conditions they exhibit temporal and spatial variations which are not present in equilibrium. As an example, one may take the total number of free electrons in a spatially homogeneous semiconductor sample. Under non-equilibrium conditions this number may have larger or smaller values than in equilibrium. If one withdraws external influences, equilibrium and its parameter values will be restored by non- equilibrium processes.

The two classes of phenomena, the appearance of currents, on the one hand, and, of time and space changes of quantities which are non-zero but constant in equilibrium, on the other hand, constitute the totality of non- equilibrium processes. For semiconductors, both types of processes are im- portant. Their theoretical description can be based on the Boltzmann equa- tion - one solves this equation and then employs the resulting distribution function f(vk, x, t ) to calculate the average values of the various quantities of interest. Alternatively it can be based on empirical equations for the av- erage currents and the temporal or spatial change rates. We refer to these as phenomenological equations. Ohm’s law for electrical conduction and the first and second Fick’s laws for diffusion are examples.

Phenomenological equations and their derivation from the Boltzmann equation

Insofar as information about the state of the system is concerned, the Boltz- mann equation provides much more detail than the phenomenological equa-

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5.1. Fundamentals of the statistical description of non-equilibrium processes 503

tions. The latter can be derived from the Boltzmann equation with addi- tional assumptions, but the reverse, i.e. the derivation of the Boltzmann equation from the phenomenological equations, is not possible. The transi- tion from the Boltzmann equation to phenomenological equations represents, essentially, a coarse graining of the length and time scales on which the non- equilibrium processes proceed. If processes must be analyzed on the space scale of the average mean free path length and the time scale of the mean free flight time, then Boltzmann’s equation is required. If a macroscopic average description suffices, one can use the phenomenological equations. In semiconductor physics, the latter case often applies. Accordingly, we will r e strict our considerations to the phenomenological equations in our treatment of non-equilibrium processes.

These equations are valid only within prescribed limits. One can con- sider these limits, like the equations themselves, to be empirically given. However, deeper insight can be gained by using Boltzmann’s equation. The question to be answered in this context is: Under what assumptions can the phenomenological equations be derived from the Boltzmann equation? The answer depends on the phenomenon under consideration. Equations as, for example, that for the temporal change rate of a given physical quantity, may be obtained by multiplying the Boltzmann equation by the value of this quantity in a Bloch state vk and subsequently summing over all states. The simplest equation of this type is that for the temporal change rate of the free carrier concentration of a single band, which later will explicitly be written down. To obtain this equation from Boltzmann’s equation in the manner indicated above, simplifying assumptions are necessary concerning the quan- tum mechanical transition probabilities between bands and localized states occurring in the collision term: These probabilities must be approximated by their values in thermodynamic equilibrium.

In the derivation of phenomenological equations for particle currents, to consider another example, one assumes that the non-equilibrium state deviates only slightly from a spatially and temporally local equilibrium state. In the latter state, the distribution function depends on the Bloch states only through their energies, and the form of the dependence is that of the Fermi distribution function f (E) , just like in global equilibrium. However, the temperature T and chemical potential p occurring in f(E), are understood as functions T(x, t ) and p(x, t) of space and time coordinates. Thus we have,

This particular non-equilibrium state may only be assumed if the tempera- ture and chemical potential are still well defined quantities, albeit in a local sense, not globally. For this assumption to be valid, adjacent parts of the

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504 Chapter 5. Non-equilibrium processes in semiconductors

system with measurably different values of temperature and chemical poten- tial are effectively decoupled from each other spatially and temporally. From the microscopic theory of non-equilibrium processes it is known that spatial decoupling (in the statistical sense of this term) occurs over a distance of the order of magnitude of the mean free path length l f , and temporal decou- pling occurs over a time interval of the order of magnitude of the mean free flight time t f . The characteristic lengths of the spatial inhomogeneities of T(x, t ) and p(x, t ) may be expressed by the ratio TIIVTI and p/IVpL(, and the characteristic time of the temporal inhomogeneities of these quantities is given by T/laT/atl and p/lOp/OtL(. In order for local equilibrium to be approximately realized, the conditions

If << TIIVTI, P I P P I ?

and tf << TllaTlatl, P l l a P l a t l (5.7)

must be satisfied. Typical values for the mean free path length of electrons and holes in semiconductors are of the order of magnitude 100 A. This means that local equilibrium is a reasonable zero order description if important spatial inhomogeneities occur on or above a length scale of 1000 A = 0.1 pm. The mean free flight time has typical values close to s. Temporal changes with frequencies of 10l2 s-’ can therefore still be accepted within the local equilibrium framework, although no light frequencies are allowed which lie in the region of 1014 s-’. The regime of validity of the local equilibrium assumption is nevertheless still relatively broad.

If the local equilibrium distribution were to describe the non-equilibrium state exactly, no currents would flow at all since the corresponding average values would be zero. In the presence of currents, the local equilibrium distribution function can, therefore, only be a zero-th approximation. It must be corrected by a perturbation term fl(vk, x, t ) , such that one has

For an electric current driven by an electric field E(x , t ) , Ohm’s law rep- resents the relevant phenomenological equation. To obtain this law, the correction term is written as a linear relation

with respect to the electric field E(x, t ) . The vector g(vk, x, t ) is understood to be field-independent. It may be calculated from Boltzmann’s equation in linear approximation with respect to E. For this approximation to be justi- fied, the field must be sufficiently weak. This represents a second condition for the validity of Ohm’s law.

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5.2. Systematics of non-equilibrium processes in semiconductors 505

The above derivation of this law yields not only a phenomenological equation, but also an expression for the electrical conductivity involved in it. It relates this coefficient with characteristic microscopic quantities of the system such as, for example, its band structure, as well as with external state parameters like temperature and chemical potential. This is to say that derivation from the Boltzmann equation yields the phenomenological equation, Ohms’s law, and, simultaneously, a microscopic theory for the conductivity, or, if other phenomenological equations are considered, for the other respective kinetic coefficients. If the phenomenological equations are taken as empiricaI laws, these coefficients are material parameters whose magnitudes and tempera- ture dependencies can only be determined from experimental investigations. However, by means of Boltzmann’s equation the kinetic coefficients can be predicted. Without the use of such an equation, this opportunity is lost. Be- side the somewhat lower precision of the phenomenological equations, due to simplifications and approximations, the principal loss lies in the inability to predict the kinetic coefficients, given only phenomenological equations in- stead of Boltzmann’s equation. For the purposes of this book, where we do not intend to develop a detailed understanding of the magnitudes and state dependencies of material parameters, we will accept this loss and use the phenomenological equations to gain insight into non-equilibrium dynamics quickly.

In the next section we present, first of all, a systematic survey of the non- equilibrium processes which are essential in semiconductors. More detailed treatment of each process is provided in following sections.

5.2 Systematics of non-equilibrium processes in semiconductors

The following survey of non-equilibrium processes is based on a gradual introduction of temporal and spatial inhomogeneities into the semiconductor. In the first step we admit only a temporal inhomogeneity, while spatial homogeneity will be maintained. The non-equilibrium processes which then occur are commonly called relaxation processes.

5.2.1

In this case, all state quantities, particularly the charge carrier concentra- tions n and p , depend only on time, and their values are different from those in thermodynamic equilibrium. The latter will be denoted by no and po henceforth, so that

Temporal inhomogeneity and spatial homogeneity

(5.10)

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506

with

Chapter 5. Non-equilibrium processes in semiconductors

(5.11)

Temporal changes of n and p occur because electrons in the conduction band and holes in the valence band are generated or annihilated. If the corresponding generation or annihilation rates are denoted by, respectively, (an/at)gmer and (an/at)annihal as well as (aP/ at)gmer and (aP/at)annihal, we have

(%) = -

(5.12)

With the right hand sides known, these equations determine the relaxation of non-equilibrium charge carrier concentrations.

5.2.2

Spatial inhomogeneity may be introduced in various ways. Firstly, we con- sider the case in which a homogeneous electric field exists while the charge carrier concentrations n and p do not depend on x:

Spatial inhomogeneity and temporal homogeneity

1) n = no, p = po, E = const . # 0.

The potential energy of an electron is eE . x in this case, i.e. the system is in fact spatially inhomogeneous despite its homogeneous charge carrier concentration. An electric current will flow in this case. This phenomenon is commonly referred to as drift, and the current itself is called drift current. The pertinent current density is spatially homogeneous and decomposes into a current density for electrons, j,, and another one for holes, j,. With nn and a, as conductivities for electrons and holes, respectively, Ohm’s law yields

j, = onE, j, = apE. (5.13)

Secondly, we also admit spatial inhomogeneity of the charge carrier concen- trations n ,p . Then the electric field also becomes inhomogeneous. Accord- ingly, we consider the case

2) n = n(x), p = p(x), E = E(x) # 0.

Under these conditions one has, in addition to drift, also a diffusion current of charge carriers which is superimposed on the drift current. Writing Dn and D, as diffusion coefficients for, respectively, electrons and holes, the

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5.2. Systematics of non-equilibrium processes in semiconductors 507

total electron and hole current densities j, and j, obey the phenomenological equations

jn(x) = o,E(x) + eD,Vn(x), (5.14)

j,(x) = oPE(x) - eD,Vp(x). (5.15)

Yet another new phenomenon arises in the inhomogeneous case considered here, namely the formation of space charge, reflected in the occurrence of a non-vanishing local charge density p(x). To describe this, we recall the neutrality condition n - - p - N & + N i = 0 derived in Chapter 4. It states that, in a homogeneous semiconductor, the negative charge density of the freely mobile electrons and of the spatially fixed ionized acceptors compensates the positive charge density of the holes and the ionized donors everywhere. In the spatially inhomogeneous semiconductor considered here the free electrons and holes are redistributed while the ionized doping atoms are kept fixed, so that local charge neutrality is perturbed. The local net charge density p(x) is given by the expression

P(X) = --e [.(XI - P(X) - Nof(x) + NJX)] , (5.16)

where we have allowed for the possibility that a spatially inhomogeneous distribution of ionized donors and acceptors also exists, either due to inho- mogeneous doping or to inhomogeneous ionization of the impurity atoms. Charge neutrality holds only for the semiconductor as whole,

s, d3xp(x) = 0. (5.17)

The electric field E(x) due to the charge distribution p(x) follows from the pertinent potential p(x) by means of the relation

E(x) = -Vp(x),

and the potential p(x) itself is governed by Poisson’s equation

1 =0

V2p(x) = --p(x)

(5.18)

(5.19)

with E as the static dielectric constant of the semiconductor material and €0

the vacuum dielectric constant. Finally, we consider in our survey of non-equilibrium processes the most

general case in which both space and time inhomogeneities exist.

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508 Chapter 5. Non-equilibrium processes in semiconductors

5.2.3 Space and time inhomogeneities

In this situation we have

n = n ( x , t ) , p = p ( x , t ) , E = E ( x , t ) # 0.

Here, the generation and annihilation of charge carriers is also possible be- cause of the fact that there are sources and sinks of the current densities j n and j , . This follows from the continuity equations for j n and j , , where the rates of change of the charge carrier concentrations due to relaxation enter as source terms. Thus. we have

The current densities continue to obey the phenomenological equations] even with temporal variation in the present case, so that

j p ( x , t ) = upE(x, t ) - ~ D , ~ P ( x , t )

The Poisson equation for the potential p ( x , t ) reads

(5.23)

Summing up, we have the following non-equilibrium processes in semicon- ductors:

- Generation and annihilation of free charge carriers (relaxation)

- Drift

- Diffusion - Formation of space charge and of the concomitant inhomogeneous elec-

tric field and potential.

Below we treat each of these non-equilibrium processes in greater detail.

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5.3. Generation and annihilation of free charge carriers 509

5.3 Generation and annihilation of free charge carriers

Since the total number of electrons in a semiconductor is conserved, free charge carriers can neither be produced without a well defined origin, nor can they vanish without a well defined destination. The generation of free charge carriers in a spatially homogeneous semiconductor must therefore be understood in the sense that electrons or holes, which are not freely mobile but bound to impurities and defects, are excited into the conduction and valence bands where they can move freely. The generation of free charge carriers thus takes place at the expense of localized carriers that are not free to move. The same is true for the annihilation of free carriers. They do not simply disappear, but are captured by states localized at impurities or defects in which they cannot move freely. According to Chapter 4 the free carriers of a semiconductor in equilibrium are mainly provided by ther- mally ionized shallow donor or acceptor centers. However, for generation and annihilation of non-equilibrium carriers these centers play only a mi- nor role if their ionization is almost complete under equilibrium conditions. In Chapter 4 we saw that this is in fact often the case, mainly because the characteristic thermal energy kT is comparable to the binding energies of the shallow centers. One may expect that deep centers should be more effective for the generation and annihilation of non-equilibrium free carriers than are the shallow ones. Under equilibrium conditions, deep centers are generally in their ground state, in which they are occupied by a number n of electrons (see section 3.5). By the application of an external perturbation, such as light, for instance, they can be transferred into excited states, in which they have exchanged electrons and holes with the valence and conduction bands. Such an exchange is, however, exactly the generation and annihilation of free carriers in which we are interested. Deep centers are therefore important for the generation and annihilation of non-equilibrium carriers.

Moreover, valence band states can play a role similar to that of deep levels for non-equilibrium electrons to be generated or annihilated, and con- duction band states can do so for non-equilibrium holes. In considering this, one must bear in mind the fact that electrons in the valence band and holes in the conduction band are also immobile charge carriers. The excitation of electrons from the valence band into the conduction band generates free carriers in the latter. However, it simultaneously creates holes, i.e., free charge carriers in the valence band, which stands in remarkable contrast to the generation of free electrons by exciting deep levels, in which case the r e maining hole is immobile. An analogous statement holds for the annihilation of free electrons. When an electron falls back from the conduction band into the valence band, then a free hole simultaneously vanishes together with the

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5 10 Chapter 5. Non-equilibrium processes in semiconductors

free electron, while the transition of a conduction band electron into a deep level only removes a free electron. A free hole, which could vanish, does not exist in the latter case. If free electrons and holes are simultaneously gen- erated or annihilated in equal numbers, we speak of bipolar generation and annihilation processes. If only one type of free charge carrier is generated or annihilated, we have unipolar processes. We first treat generation processes, including uni- and bipolar ones.

5.3.1 Generation processes

To excite free charge carriers from deep levels or from bands, an appropriate amount of energy must be provided. That can be done by exposing the semiconductor to light or electron beams, and in many other ways. The creation of free carriers by applying an external voltage to a semiconduc- tor junction, so-called injection, is excluded here because we are confining our attention to spatially homogeneous semiconductors. Injection is, by its nature, a unipolar process and it will be treated later. Of the remaining processes, excitation by means of light will be chosen as a representative example. We refer to it as optical excitation, and other excitation processes can be described in similar terms.

Unipolar optical excitation processes of electrons and holes are illustrated in Figure 5.1. An electron from a deep center, which in thermodynamic equi- librium is in its ground state, is raised into the conduction band by a photon of proper energy. As a result, a free electron appears in the conduction band. Alternatively, an electron from the valence band makes a transition into a deep level with the absorption of a photon. A free hole is generated in the valence band as a result of this process. We denote the number of photons absorbed per volume and time unit by, respectively, gn and g p . Then the rate of unipolar-generated carriers may be expressed as

(g) =gnI = g p . unip.gen. unip.gen.

(5.25)

In the case of bipolar generation by means of light (see Figure 5.1) one has gn = g p = g , so that

(5.26)

To estimate the magnitude of the optical generation rates, we consider the bipolar case. With I as light intensity and a as the absorption coefficient associated with the excitation of electrons from the valence band into the conduction band. we have

9 = (E). (5.27)

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5.3. Generation and annihilation of free charge carriers 511

a)

I .....

Figure 5.1: Unipolar generation of electrons (a), holes (b), and bipolar generation of electron-hole pairs (c).

For light intensity of 1 Wcrn-’, photon energy hw = 1 e V , and absorption coefficient lo3 ern-’, which is not too large for the absorption edge of a semiconductor, we obtain g = 6 x loz1 ~rn-~s- ’ . This indicates that op- tical excitation represents a very effective generation process. The size of g does not tell, however, how large the steady value of the free charge car- rier concentration will actually be. This value also depends on the speed of annihilation processes.

5.3.2 Unipolar annihilation of free charge carriers: capture at deep centers

As outlined above, free charge carriers are annihilated when they make a transition into deep levels. One refers to this process as the capture of free charge carriers by deep centers (see Figure 5.2). We first consider the capture of electrons, and assume that the deep center D can bind one and only one more electron than it has in its ground state. If the center is neutral in its ground state, this means that the D(-1/0) donor level lies in the gap, while the 0( -2 / - 1) level does not appear there. For brevity, the ground state of the deep center will be referred to as ‘unoccupied’, and the state with one electron more at the deep center as ‘occupied’. The ionization energy of the occupied center is denoted by Et.

Electron capture

After an electron of the conduction band is captured by a deep center, there is a finite probability that it may be re-emitted back into this band. The anni- hilation rate -(an/dt),,,ihii is therefore the net capture rate (dn/dt),pture, defined as the negative of the difference between the gross capture rate C, and the emission rate En, i.e.

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5 12 Chapter 5. Non-equilibrium processes in semiconductors

Figure 5.2: Capture (on the left) and emission (on the right) of electrons (above) and holes (below) at a deep level. .........

Y EP -- (5.28)

The capture rate C , is, on the one hand, a function of the concentration of free electrons available for capture, i.e. of n. On the other hand, it depends on the concentration of deep centers available for a capture. If we denote by Nt the total concentration of deep centers, and by f the probability that they are already occupied by an electron, then the concentration of available centers is Nt(1 - f ) . Thus,

c, = c,n(l - f)Nt. (5.29)

The proportionality factor c, is called capture coefficient. It has the di- mension cm3s-' and describes the capture capability of an unoccupied deep center.

The emission rate En depends on how many deep centers are already occupied by electrons, i.e. on Nt f . The concentration of existing free elec- trons has no influence on En, since the number of unoccupied states in the conduction band which must host the emitted electron is relatively large. Therefore,

En = en f Nt. (5.30)

The proportionality factor en is called emission coefficient. It measures the emission capability of an occupied center. For one occupied center the emission probability per second is just en. The net capture rate follows as

($) = - [cnn(l- f )Nt - e n f ~ t ~ . (5.31)

Consider first the special case of thermodynamic equilibrium. In this state all macroscopic quantities are independent of time, so that (dn/bt)capture = 0 holds. The concentrations n , p take their equilibrium values no and PO, and

capture

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5.3. Generation and annihilation of free charge carriers 513

the same holds for the occupation probability f of the deep level, which is given by

(5.32)

where -ytn denotes the ratio of the statistical weights of the unoccupied and occupied center. Equation (5.31), with vanishing left hand side, yields a relation between en and cn in thermodynamic equilibrium, namely

with (5.34)

The quantity nt has a structure similar to that of the concentration nl previously introduced in Chapter 4. It denotes, apart from the factor Ytn, the concentration of free electrons in the case when the Fermi energy lies exactly at the deep level. For an n-type semiconductor, EF is actually found to be much closer to the conduction band, i.e., nt is, in general, substantially smaller than the equilibrium concentration no.

The relation (5.33) between emission and capture coefficients is a con- sequence of the equilibrium between the deep level and conduction band. The term detailed balance is used in this context, since the two levels are in equilibrium already themselves.

If there are deviations from equilibrium, the net capture rate (5.31) is not fully determined inasmuch as the occupation probability f of the deep center has not yet been further specified. In the general case, this probability will also deviate from the equilibrium value fo. The exact value which it will have cannot be predicted a priori. One can only write down an equation for its temporal rate of change. With the assumption that the exchange of electrons occurs only between the deep level and the conduction band, this equation reads

(5.35)

If we also assume that beyond a particular point in time t o no further gener- ation of free charge carriers will take place, but only annihilation processes will occur through capture , we have

(g) = (g) f o r t > to. capture

(5.36)

The relations (5.35) and (5.36) together form a system of non-linear differen- tial equations, which can be solved exactly in closed form. However, we will

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514 Chapter 5. Non-equilibrium processes in semiconductors

not exhibit this solution here, restricting our discussion to a linear approxi- mation. In this, we assume that the functions n and f deviate only weakly from their equilibrium values. Formally, this means that n = no + An and f = f o + A f with lAnl << no and lAf I << 1, whence

(g) = - c n N [-(nt + no)A f + An(1- f o ) ] . (5.37) capture

Bilinear terms were neglected in this derivation, and equation (5.33) was used. We consider that the generation of non-equilibrium electrons before time t o arises from an initial equilibrium distribution with subsequent exci- tation of electrons from deep levels into the conduction band. The resulting total concentration of electrons in deep levels and in the conduction band jointly have the equilibrium value no + Ntfo at all points of time t before and after to. Therefore

An + NtAf = 0. (5.38)

With this, the linear expression

follows from (5.37), where for the sake of brevity we have defined

1 7, =

cn[nt + no + Nt(1 - f o ) ] ’

The time development of An is governed by the equation

An

(5.39)

(5.40)

(5.41)

The solution of this equation reads

where An0 = An(t = t o ) has been used. This result demonstrates that rn represents the decay time of a non-equilibrium charge carrier distribution. One also calls rn the lifetime of non-equilibrium electrons with respect to capture at deep centers. In the case of completely unoccupied deep centers, i.e. at fo = 0, and for very high concentrations Nt, more exactly for Nt >> nt + no, rn is given by

1 I rno = -. (5.43)

CnNt

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5.3. Generation and annihilation of free charge carriers 515

In accordance with its meaning, rn0 is the time interval in which the totality of Nt unoccupied deep level per cm3 capture one non-equilibrium electron from the conduction band. One therefore calls r,,o the capture time. The life- time r,, and the capture time rno differ because of the occupation dependence of the capture rate and the occurrence of reemission. If Nt(1- fo) >> nt+no, then one has r,, > rno. This result is immediately understandable because the lifetime r,, contains contributions only from unoccupied deep centers, whereas all deep centers, including occupied ones, enter in rno.

Hole capture

As in the case of electrons, deep centers can also capture and reemit holes. The capture of a hole means that a deep center, which in its ground state binds a certain number of electrons, transfers one of these electrons to the valence band and annihilates a hole there. The emission of a hole is to be understood in the sense that a deep level captures an electron from the valence band (see Figure 5.1). All considerations involving the capture and emission of electrons, and all relations derived for these processes, can be immediately transferred to holes. The capture and emission rates Cp and Ep are given by the expressions

In this, cp and e p denote the capture and emission coefficients for holes. They are related to each other by the equation

e p = PtCp 7 (5.45)

where pt is the hole concentration

(5.46)

The meaning of ytp is analogous to that of y h . The hole lifetime rp obeys the relation

1 rp =

Cpbt + P O + Ntfol '

and for the hole capture time ~~0 of a deep center we have

(5.47)

(5.48)

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516 Chapter 5. Non-equilibrium processes in semiconductors

Center

Au(l+)

Au(0) Au( 1-)

1 4 0 ) In(l-)

Bi(0) Bi(1-)

Bi(2-)

Fe(l+)

W O )

Zn(0) Zn(1-)

Table 5.1: Experimental capture cross sections for particular deep centers in Si and Ge at T = 300 K in units cm2.(Data compiled from various sources.)

on UP

35-63

1.7-5

10-110

0.1

170

0.025

0.6

0.6-1

16

3-7

10

0.0001-1 300

n Si

Cu(1-)

Cu(2-)

Ni(0) Ni(1-)

Ni(2-)

Fe(0)

Fe(2-)

Fe( 1-)

Zn(1-)

0.1-0.18 500

0.0036-0.3 1-3.6

1-8

3-6 1000

5 - 20 x 1000

1-10

30

100

< 0.001

Ga(0) < 0.01 Ga(0)

Experimental determination of capture coefficients

< 0.01

Experimentally, the emission coefficients e , of deep centers can be deter- mined by the same time-resolved current or capacitance measurements at pn-junctions that were discussed in section 3.5. The capture coefficients c, follow from en by means of equation (5.33). The experimental data for the capture efficiency of deep centers are generally expressed as capture cross- sections a, and up. These are related to the capture coefficients c, and cp by a simple relation. With vn and up as average thermal velocities of, respectively, electrons and holes, we have

(5.49)

Experimentally determined capture cross sections for some deep centers are listed in Table 5.1.

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5.3. Generation and annihilation of free charge carriers 517

Finally, we present a short discussion of the microscopic mechanisms respon- sible for the capture of free charge carriers at deep centers.

Capture mechanisms

We restrict our considerations to electrons, as the transfer of the discussion to holes is evident. To understand capture processes, the first question that has to be answered is how an electron, during its transition from the con- duction band into the deep level, can transfer away its surplus energy. There are essentially three possibilities. Firstly, the energy can be emitted as light, secondly, it can be transferred to the remaining electrons of the semiconduc- tor, and thirdly, it can be transferred to the lattice. The last case is the most important and most universal. The transfer of electron energy to the lat- tice involves the interaction of the electrons with phonons. This interaction was ignored in Chapter 2 by treating the interacting system of electrons and atomic cores within the adiabatic approximation. In order to account for the electron-phonon interaction, one must go beyond this approximation. More- over, it must be recognized that one single phonon is insufficient to make possible the transfer of energy during capture, and many phonons must be emitted simultaneously. This is termed a mult i -phonon process. Conse- quently, a perturbation theoretical treatment of the electron-phonon interac- tion is not adequate for this purpose, and one must employ non-perturbative calculations of the capture coefficients due to multi-phonon processes at deep centers (see Peuker, Enderlein, Schenk, and Gutsche, 1982).

5.3.3 Bipolar annihilation of carriers at deep centers

If an electron of the conduction band is annihilated by a transition into a hole of the valence band, we have a recombination of a n electron-hole pair. The physical mechanisms for the transfer-away of the surplus energy, which in this context is at least as big as the gap energy E,, are largely similar to those for the capture of carriers at deep centers. Firstly, an electron can make the transition from the conduction band into a hole in the valence band by emitting a photon. This process is called an optical band-band re- combination The term radiative recombination is also commonly used. It is known that spontaneous optical band-band transitions, i.e. transitions not stimulated by light, have only a relatively small probability. Often, they can be completely neglected. Optically stimulated processes, which play a cru- cial role in semiconductor lasers, are considerably more effective. Secondly, the surplus energy of a recombining electron-hole pair can be transferred to another electron in the conduction band, which is thereby lifted to higher energy in the band. In this process, referred to as Auger recombination, three free charge carriers are involved, two electrons in the conduction band and

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5 18 Chapter 5. Non-equilibrium processes in semiconductors

one hole in the valence band. Auger recombination is important mainly for high charge carrier concentrations, since its probability is determined by the product of the concentrations of the three carriers involved. Thirdly, the en- ergy released during recombination can be transferred again to the lattice. This direct band-band recombination with emission of as many phonons as needed to match the energy gap, proves to be extremely unlikely. The reasons for this are, on the one hand, that the number of phonons which would have to be emitted simultaneously is very large, and secondly that the electron-phonon interaction for electrons and holes in band states is rel- atively weak. Recombination processes in which deep levels are involved as intermediate states for an electron making the transition from the con- duction band to a hole in the valence band are much more effective. This mechanism is called Shockley-Read-Hall recombination Auger recombina- tion and Shockley-Read-Hall recombination together are jointly referred to as non-radiative recombination processes. Because of its dominating role in electronic devices, we treat Shockley-Read-Hall recombination below in more detail.

S hockley-Read-Hall recombinat ion

For a deep center X to be effective in mediating the recombination of an electron-hole pair, it must be able to capture both an electron and a hole. This means that both the X(-l/O)-donor level and the X(-l/O)-acceptor level of the center must lie in the gap. The capture of a hole by a deep level entails the emission of an electron into the valence band. The simultaneous capture of an electron and a hole (see Figure 5.3) corresponds, therefore, to the transition of an electron from the conduction band into a hole in the valence band, i.e. to the recombination of an electron-hole pair. The occupa- tion state of the deep center after recombination has taken place is the same as that before recombination - it is in the ground state, the captured elec- tron and the captured hole have compensated each other. Nevertheless the deep center plays a decisive role in the recombination process. It enhances its probability by many orders of magnitude. The deep center works, so to say, as a catalyst for recombination. The reason for this follows immediately from our discussion of the multi-phonon processes above.

A deep center not only captures electrons and holes, but it also emits them (see Figure 5.2). In order that a deep center actually function as a ten-

ter for recombination, a particular relation between the capture and emission rates C,, C, and En ,Ep for, respectively, electrons and holes, must hold. On the one hand, an electron must be captured from the conduction band more quickly than an electron from the valence band. If that were to be reversed, the electron from the conduction band could not be captured at all because the deep level would already be occupied. Then recombination

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5.3. Generation and annihilation of free charge carriers 519

Figure 5.3: Capture and emission of an electron and a hole at the same deep center.

Figure 5.4: Effects of deep levels for differing relations between the capture and emission rates for electrons and holes.

would be impossible. Since the capture of an electron from the valence band is equivalent to the emission of a hole to this band, the first condition to be satisfied may be expressed as Cn >> Ep. On the other hand, the electron captured from the conduction band must make the transition to the valence band more quickly, or, in other words, the deep center must capture a hole from the valence band more quickly, than the time it takes the electron cap- tured by the deep center can be re-emitted back to the conduction band. Thus Cp >> En must hold. If different order of magnitude relations exist between the four rates, the deep center can no longer function as a recom- bination center, but has other effects (see Figure 5.4). We summarize these in the following survey: The deep center works as

a) recombination center for Cn >> E p , Cp >> En

b) electron capture center for C, >> Ep, Cp << En

c) hole capture center for C p > > En, C,<< Ep

d) generation center for C p << En, Cn << Ep.

Verification of the three last cases, which were not explicitly discussed above,

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520 Cbapter 5. Non-equilibrium processes in semiconductors

is left to the reader. With the background of these considerations, the unipo- lar annihilation of charge carriers at deep centers appears in a new light. It is a special case of the more general situation considered here, in which the deep center can capture and emit both electrons and holes.

Thus far, recombination through deep centers has been considered only qualitatively. To exhibit a quantitative treatment, we present equations for the temporal rates of change of the electron concentration n, the hole concentration p, and the occupation probability f of the deep center. These equations are generalizations of the relat,ions which were already employed above in describing the capture of carriers by deep centers. Here, they read

(5.50)

(g)Tmb - ( Z ) recomb + N t (2) = 0. (5.52)

By means of relations (5.33) and (5.45) between, respectively, en, cn and ep, cp, and using notation rno and rPo, introduced above, we have

1 = - [(l - f ) P t - fPl .

(%)recomb TPo

(5.53)

(5.54)

Consider the case of stationary recombination. Then the occupation of the deep level does not change in time, so that

(g) = 0,

whence it follows from (5.52) that

(5.55)

(5.56)

i.e., the numbers of electrons and holes vanishing per volume and time unit, are the same. Their common value is the number of electron-hole pairs recombining per unit volume and time, so that the recombination rate R is

R = - ( $ ) =-(a) recomb recomb

Equating (5.53) and (5.54), we have

(5.57)

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5.3. Generation and annihilation of free charge carriers 52 1

from which R may be obtained as

n p - n t p t R = ~ p ~ ( n + n t ) + Tnoo) + P t )

The relation

CI

ntpt = nopo = n:,

permits R to be rewritten in the more compact form

(5.59)

(5.60)

with

as recombination coefficient. The expression (5.61) for the recombination rate may be interpreted as follows: The driving force of recombination is the deviation of the concentration product n p from its equilibrium value nope, since the recombination rate is proportional to this deviation. The propor- tionality factor T in general depends on the non-equilibrium concentrations n and p . Only for small deviations from equilibrium is T approximately constant. We will consider this case in more detail below.

Small deviations from equilibrium

Setting n=no+An, p = p o + A p (5.63)

with [An1 << no and lApl p o , we first assume Ap = 0, supposing that the non-equilibrium state has been created by unipolar electron excitation and that it decays by recombination:

An#O, A p = O .

It follows that

with

An T n

R = -

7, = ~ n o + n t T p O I P o + P t , n O PO PO

(5.64)

(5.65)

as the recombination lifetime of non-equilibrium electrons. In the case of a n-type semiconductor, i.e. if p o << p t << no holds, then T n >> ~ p o , ~ n o , which means that the recombination lifetime is substantially longer than the capture times for electrons and holes. This result is due to the fact that the holes necessary for recombination of the non-equilibrium electrons

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522 Chapter 5. Non-equilibrium processes in semiconductors

exist in the n-type semiconductor only in very low concentration, so that the probability of recombination is relatively small and the recombination lifetime is relatively large. In the case of a p-type semiconductor, i.e. for no << nt << po, then r, = rn0. This reflects the fact that the number of holes is now large enough for every non-equilibrium electron to recombine. The ‘bottleneck’ for recombination is electron capture at the deep center. For unipolar hole excitation, i.e. with

Ap#O, An=O,

one obtains analogous relations, and

with no + nt PO + pt

rp = ~ rpo + - TnO n0 n0

(5.66)

(5.67)

as the recombination lifetime of non-equilibrium holes. For an n-semiconduc- tor one has rp = rPo, and for a p-type semiconductor rp >> rno, rpo.

The above results for the case of unipolar excitation may be summarized by stating that the recombination lifetimes of minority carriers are substan- tially smaller than those of majority carriers - provided the deviation from equilibrium is weak.

Finally, we consider the case of bipolar excitation of non-equilibrium carriers. In this case,

An = Ap.

The recombination rate is now given by

with

and

(5.68)

(5.69)

r = rpo(no + nt) + T,O(PO + P t ) (5.70)

For an n-type semiconductor one has r = rpo, and for ap-type semiconductor r = r,~. The recombination lifetime therefore equals the capture rate for minority carriers. For an intrinsic semiconductor, we have no + p o = Pni, so that the recombination lifetime in this case is given by the expression

no +PO

(5.71)

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5.4. Drift current 523

Table 5.2: Recombination lifetimes q and intrinsic carrier concentrations ni for several semiconductors. c ~ u i u . = T / V , denotes the capture cross section corre- sponding to ri and ni according to formula (5.71). T = 300 K . (Data compiled from various sources.)

Material

Si

Ge

GaP

GaAs

InP

InSb

Gap indirect

indirect

indirect

direct

direct

direct

r, (s) ni ozuiv.(cm2)

3.9 x lo4 1.5 x i o 1 O i o P 2

0.4 2.4 x 1013 5 x

3.4 x 10l2 2.7 5 x 10-21

5.4 x lo2 1.8 x lo6 1 0 - l ~

5.8 x 10 8.2 x lo6 10-15

7.3 x 10-7 1.6 x 1016 10-17

In Table 5.2 typical values of the lifetimes q for some semiconductors are listed. It should be noted that there is a large variation of q, between 3.4 x 10l2 s for the indirect wide gap semiconductor GaP and 7.3 x s for the direct small gap semiconductor InSb. For extrinsic semiconductors, we have no + P O > 2n i , so that T < q. This means that the recombination lifetime through a particular deep center reaches its maximum value when the semiconductor is in its intrinsic state. Doping reduces the recombination lifetime of bipolarly excited non-equilibrium carriers.

5.4 Drift current

Consider a semiconductor in a spatially uniform electric field E. Apart from its potential, which depends linearly on x, all other characteristic quantities of the semiconductor, in particular, the concentrations n and p of free charge carriers, are assumed to be spatially uniform. If the electric field vanishes, the free charge carriers just perform thermal motions. The velocities of the latter have the same probability in all directions in space, thus the average momenta < mix, > of electrons and < m;xp > of holes are zero ( m i and m; denote the effective masses of, respectively, electrons and holes). In the following treatment of charge carrier drift, we will confine our considerations to electrons as free carriers. The results can be transferred to holes without any complications, involving only notational changes.

In the presence of the electric field, the electrons are accelerated and the average value of electron momentum < mixn > has a non-vanishing

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524 Chapter 5. Non-equilibrium processes in semiconductors

component in the direction of the field. Its rate of change [$ < m i i n > 1 field per second, according to equation (3.110), equals the force impressed by the field on a Bloch electron. whence

= -eE. 1 field [& < mkkn > (5.72)

In addition to the electric field there are frictional forces acting on the Bloch electron which arise from the semiconductor itself. The interactions of an electron with lattice vibrations and impurity atoms or defects result in scat- terings which act to brake its acceleration along E, producing a frictional effect similar to that in a viscous medium. Often, this frictional resistive force grows stronger in proportion to the average electron speed, so that one may put the corresponding rate [$ < mkxn >Iez of momentum change per second due to collisions with phonons and crystal imperfections, proportional to the average momentum, whence

(5.73)

with l / r i as proportionality factor. If there is a momentum excess at initial time t = 0, < m;xn(0) >, the solution of this equation reads

< mkxn(t) > = < rn:jcn(o) > e-t/T:. (5.74)

Thus, the momentum excess decays exponentially, relaxing to zero, as one may say. In this context, one terms r;E as the momentum relaxation time. This time should not be confused with the previously introduced electron lifetime 7, for capture or recombination. The two characteristic times de- scribe completely different physical phenomena. In general the momentum relaxation time is shorter than the capture and recombination lifetimes.

The total-momentum balance is governed by Newton’s equation of mo- tion,

whence

d 1 - < mkxn > = -eE- - < m*k n n > . (5.76)

For steady state dc current flow, the left hand side of this equation vanishes and the concomitant constant value of the average velocity is

d t r;

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5.4. Drift current 525

which is referred to as the drif i velocity. For a field strength of 1 Vcm-’ the drift velocity has absolute value

(5.77)

(5.78)

This quantity is a measure of the mobility of an electron per unit applied force, therefore pn is simply referred to as electron mobility. Using equation (5.78), the drift velocity of (5.77) may be written in the form

< X, > = pnE.

The average electron current density j,, defined as

j, = - e n < kn >, may be re-expressed in terms of p, as

(5.79)

(5.80)

j , = enp,E. (5.81)

For the electron conductivity D,, as it occurs in Ohm’s law, one t,hus obtains

on = enp,. (5.82)

This conductivity expression will be used frequently below. It can be trans- ferred to holes without difficulty, reading

‘ T p = e p p p ,

where the hole mobility p p is given by

er;

m P P p = - - y I

(5.83)

(5.84)

with r; signifying the momentum relaxation time of holes. Since electrons and holes in a semiconductor are always present simultane- ously, the total current density is the sum of the electron and hole current densities,

j = .in + j p , (5.85)

and the total conductivity D is the sum of the two partial conductivities,

CT = un + up. (5.86)

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526 Chapter 5. Non-equilibrium processes in semiconductors

Table 5.3: Electron and hole mobilities for pure semiconductor materials at T = 300 K . (After Landoldt-Bornstein, 1982.)

1000 Si

40

20

200

100 N= 10"

Temperature ("C) I O U -50 0 50 100 150 200

Temperature ("C 1

Figure 5.5: Temperature dependence of the electron and hole mobilities of Si for various doping concentrations.

For extrinsic semiconductors of n- or p-type, one of the two charge carrier concentrations exceeds the other by many orders of magnitude. To a good approximation, the total current density for an n-type semiconductor can be identified with j,, and for a p-type semiconductor with j,.

We will tabulate some values of the transport parameters above, and discuss aspects of their characteristic dependencies. Since the charge carrier concentration of a given semiconductor can vary over a wide range, it is more meaningful to discuss pn and p p rather than of u, and up. The mobility unit one commonly uses is cm2V-ls-'. If the mobility is unity in this unit, then a field of 1 Vcm-' causes a drift velocity of

1 cm2v-1s-l x 1 v c m - l = 1 cms-'. (5.87)

In Table 5.3 the mobilities of electrons and holes are given for some important semiconductors at room temperature. That the mobilities of electrons exceed those of holes, is to be attributed to the smaller effective masses of the electrons in these materials. The mobility depends on the interaction of the carriers with phonons through

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5.5. Diffusion and annihilation of free carriers 52 7

Figure 5.6: Dependence of the

doping concentration for Ge, Si and GaAs at T = 300 K . (After Sze, 1981.)

electron and hole mobilities on 5 lo474 T- 300 K

=- 103

,102

N

5 -

Doping concentration ( ~ r n - ~ )

the momentum relaxation time, and the same may be said for temperature dependence. In Figure 5.5 this dependence is shown for Si. The decrease of mobility with rising temperature is to be attributed to the growth of the amplitudes of the lattice oscillations or, in other terms, of the num- ber of phonons. In extrinsic semiconductors the mobility also depends on the doping concentration, because doping atoms constitute collision centers. Generally, the momentum relaxation times, and through them also the mo- bilities pn and p p , decrease with rising doping concentrations. Examples of such dependencies are shown in Figure 5.6.

5.5 Diffusion and annihilation of free carriers

In this section, we consider the concentrations n and p of electrons and holes to be spatially inhomogeneous. A direct consequence of this assumption is that the total charge density, which consists of the charge density of the free carriers and that of the ionized doping atoms, is no longer zero locally, although the total charge vanishes upon integration over the entire sample. The non-vanishing local charge distribution is accompanied by spatially in- homogeneous electric fields. Later, we will take account of these fields, but for the moment we omit them. To be consistent, we correspondingly ignore the charges carried by electrons and holes. Spatial inhomogeneities of the electron and hole concentrations n ( x ) and

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528 Chapter 5. Non-equilibrium processes in semiconductors

p(x) imply the existence of non-vanishing gradients On(x) and Vp(x). The latter cause spatially inhomogeneous diffusion currents in and ip of, respec- tively, electrons and holes. Considering first the case of electrons, the diffu- sion current is given by

in = -DnVn. (5.88)

Because of its inhomogeneity, this current leads to temporal changes of the local electron concentration, whereby the latter may take values which de- viate from local equilibrium. As a result, generation and annihilation pro- cesses will take place which tend to restore local equilibrium values. The overall balance of the electron concentration change is described by the con- tinuity equation (5.20). The right-hand side of this equation is specified such that generation processes are omitted while capture and emission pro- cesses at deep centers are taken into account. Then the annihilation term - ( d ~ ~ / d t ) ~ ~ ~ i h i l equals (dn/dt),,pture. For small deviations from equilib- rium, (an/dt)capture may be identified with expression (5.39), whence

n - no d n

at 7, - + V . i n - (5.89)

For the sake of simplicity, we suppose that the electron concentration n(x) changes only in x-direction. Then only the x-component of in is non-zero and we denote it by in. In the continuity equation, we eliminate V.in = (&,/ax) by means of the diffusion equation (5.88), with the result

d 2 n n-no at nax2 - rn . - - D d n

For steady state this yields

(5.90)

(5.91)

The determination of a unique solution of this equation requires specification of boundary conditions, which we choose as follows: at x = 0 a particular non-equilibrium value

n(x = 0) = no + An0 (5.92)

of the electron concentration is maintained, and n ( x ) will decay to the equi- librium value no at x = 00. In the interval 0 < x < 00, the solution is written in the form

(5.93)

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5.5. Diffusion and annihilation of free carriers 529

t Figure 5.7: Profile of the non- equilibrium electron concentra- tion as a function of position x in the presence of diffusion and charge carrier annihilation (schematically).

0 X-

Then it is readily seen that

n(2) = no + Ange-"ILn, (5.94)

where L,= &. (5.95)

The solution (5.94) admits the following simple physical interpretation (see also Figure 5.7). Diffusion tends to disassemble existing concentration gra- dients, transferring part of the distribution near z = 0 further to the right. However, since the lifetime of non-equilibrium charge carriers is limited, this process cannot be fully accomplished, as electrons are captured during the transfer process, leading to the exponentially decaying profile of the electron concentration given in equation (5.94). The characteristic decay length L , can be interpreted as the distance to which an electron diffuses, on the aver- age, before it is captured. One refers to L , as the dzfluszon length. L , is the larger for larger lifetime 7,. One can illustrate the interpretation above by means of a fluid current moving over porous ground. At a particular point x = 0 the fluid current level is maintained at a fked height. In the direction of the current flow, the fluid level drops because fluid is absorbed into the porous ground. Sufficiently far downstream, practically no fluid will arrive because most of the h i d was absorbed into the ground further upstream. Nevertheless, one has a continuous fluid current at x = 0, which provides exactly as much fluid as is absorbed along the entire pat,h between x = 0 and z = 00. The analogy to the diffusion of electrons with finite lifetime is obvious. Furthermore, the current flow at z = 0, in this case i,(z = 0), may be determined from (5.88) and (5.94) as

(5.96)

These considerations for electrons can be immediately transferred to holes. In analogy to equation (5.91) one obtains

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530 Chapter 5. Non-equilibrium processes in semiconductors

Table 5.4: Diffusion coefficients and diffusion lengths for several pure semiconduc- tors at T = 300 K . Diffusion coefficients are calculated by means of mobility data of Table 5.3 using Einstein's relation.

Si

Ge

GaAs

100.8

227.5 10.3 10-8 3.2

(5.97)

For boundary conditions analogous to those in equation (5.92), we have

with (5.99)

as the diffusion length of holes. In Table 5.4, diffusion lengths of electrons and holes are listed for several

semiconductor materials. The lifetimes r, and rp depend strongly on the degrees of purity and crystallographic perfection of the semiconductors. In Table 5.4 a value of s was assumed for T,,~. However, in very pure and perfect materials, considerably larger values are possible. The diffusion lengths are then much larger than those given in Table 5.4, and they can reach hundreds of microns.

5.6 Equilibrium of free carriers in inhomogeneous- ly doped semiconductors

In this section we consider semiconductors with spatially inhomogeneous doping, and explore the thermodynamic equilibrium state of such a spa- tially inhomogeneous system. The more general case, which includes semi- conductor materials having an inhomogeneous chemical composition, will he discussed in Chapter 6. As before, we can consider the semiconductor to be composed of a subsystem of atomic cores and another of the free charge carriers. Both subsystems are taken to he spatially inhomogeneous here.

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5.6. Equilibrium of free carriers in inhomogeneously doped semiconductors 531

Complete equilibrium of the total system can only exist then if the inho- mogeneities have equalized out in both partial systems. This means also that the doping of the semiconductor material must have become spatially homogeneous. The process which brings this about, the diffusion of dopant atoms, is so very slow, however, as compared to the carrier processes, that one can completely neglect it. This means that, the subsystem of atomic cores effectively remains in the non-equilibrium state in which it started; this state is frozen, so to say. The subsystem of the free charge carriers, however, will relatively quickly pass into a new stationary state. On the one hand, this state will differ from the equilibrium state of free charge carriers in a homogeneous semiconductor - the concentrations n and p of the elec- trons and holes will depend on position coordinate x, as a consequence of the inhomogeneity of the dopant concentrations. On the other hand, all current densities of the free charge carriers will also vanish in the inhomogeneous case, just as in the homogeneous case. A stationary final state conforming to this characterization may be understood as a thermodynamic equilibrium state of the free charge carriers of the spatially inhomogeneous semiconduc- tor. The nature of thermodynamic equilibrium in this sense will be discussed below.

We consider only spatial inhomogeneities in the x-direction. The com- ponents j, and j, of the electric current densities of electrons and holes in this direction are given by the phenomenological equations

dP dx

j, = a,E(x) - eDp-.

(5.100)

(5.101)

Here E(x) represents the electric field that occurs in consequence of the spatially inhomogeneous charge carrier distributions, initially omitted in the previous section. In equilibrium the current densities of electrons and holes must vanish. That this must hold for the electron and hole currents sepa- rately, and not just for the sum, follows from Boltzmann’s equation which states that in equilibrium the partial currents of the subsystems of electrons having a fixed energy must vanish separately (not just after summing over all energy values). The holes are, of course, also a subsystem of the total electron system with particular energy values, namely the energies of the unoccupied states of the valence band. Hence, in equilibrium we have

dn dx

0 = ~ , E ( x ) + eDn--, (5.102)

dP 0 = apE(x) - eDp-. dx

(5.103)

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532 Chapter 5. Non-equilibrium processes in semiconductors

The two electron current contributions in (5.102), the drift current vnE and the diffusion current eD,(dn/dx), need not vanish separately. The same holds for the hole current contributions apE and -eDp(dp/dx) in equation (5.103). Only the sum of these contributions must be zero. Thus we arrive at the conclusion that in thermodynamic equilibrium the drift and diffusion currents of each type of carrier must exactly compensate each other.

Our object now is to transform this observation into a general condition for thermodynamic equilibrium in systems with spatially inhomogeneous doping. To this end we express the field strength E of equations (5.102) and (5.103) in terms of the electrostatic potential cp, and replace cn and up, respectively, by the right hand sides of equations (5.78) and (5.83). Thus, we obtain

dYJ d dx o = - ~ n - + Dn- dx [In n ( x ) ] ,

dcp d dx dx

O = - p p - - Dp- [ h p ( ~ ) ]

(5.104)

(5.105)

In the case of local equilibrium, the charge carrier concentrations n(x) and p(x) depend on the chemical potential p(x) in just the same way that n and p in expression (4.59) depend on E F ,

Substituting these expressions in (5.104) and (5.105) we have

(5.106)

(5.107)

(5.108)

In order to satisfy these two relations simultaneously, it is necessary that

(5.109)

Equations (5.107) and (5.108) will now be applied to a particular spatially inhomogeneous system for which the space variation of the chemical potential is known. This consideration will result in the Einstein relation.

Einstein relation

Because the chemical potential for holes is the same as that for electrons, we can restrict ourselves to the latter. We assume that at a given position z = xo there is an infinitely high potential wall for these particles (see Figure 5.8). Electrons can reside on the right side of the wall, but not on the left, as

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5.6. Equilibrium of free carriers in inhomogeneously doped semiconductors 533

. . Figure 5.8: Illustration of the derivation of the Einstein rela- - tion. X -

4 aJ I 01

w

I I

x o X

they are reflected by the wall if they move toward the left. The situation is comparable with that of molecules of air above the surface of the earth. As in the latter case, in which the equilibrium vertical distribution of particles is given by the ‘law of atmospheres’, we have here

Comparing equations (5.110) and (5.106) it follows p(z) = ecp(z). Since equation (5.107) applies also in this special case, the factor multiplying ( d p l d z ) must necessarily be ( l / e ) , whence we conclude that

(5.111)

This equation is the Einstein relation It holds for holes as well as electrons,

kT D, = - p P . (5.112)

A generalization of Einstein’s relation for degenerate charge carrier gases is possible, but we will not discuss this here.

Condition for thermodynamic equilibrium of free carriers

Combining equations (5.111) and (5.107), we obtain the desired general con- dition for thermodynamic equilibrium of free charge carriers,

or

9- e - d v = 0 dx d x

p ( z ) - ecp(z) = const

(5.113)

(5.114)

The quantity on the left hand side of this equation is referred to as the electrochemical potential. In the case of vanishing electric potential, the electrochemical potential is just the same as the simple chemical potential. This also remains true if the electric potential has a non-zero but spatially

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534 Chapter 5. Non-equilibrium processes in semiconductors

constant value which, as we know, is irrelevant. The terms Fermi energy or Fermi level which previously, in the absence of spatially varying electric potential, were used for the simple chemical potential, will now be used for the electrochemical potential. With this terminology, relation (5.114) becomes

(5.115)

In thermodynamic equilibrium, the electrochemical potential of the free charge carriers is spatially constant, and the constant value is equal to the Fermi energy. The chemical potential and the electric potential themselves can change, however. It is appropriate to bear in mind here that the ini- tially assumed spatial inhomogeneity of the semiconductor is based on the inhomogeneity of its doping. The formal expression of this inhomogeneity is the spatial variation of the chemical potential. To properly adjust the ther- modynamic equilibrium state, the spatial variation of the electric potential must just compensate the spatial variation of the chemical potential.

The equilibrium state of equation (5.114) represents an equilibrium in both the local and in the global senses. That in the local sense was pre- supposed (approximately) at the outset with the use of phenomenological equations for the current densities of the charge carriers. Moreover, the equi- librium state described by (5.114) represents a relative equilibrium, because it was derived under the assumption that the initial spatial inhomogeneity of doping, and thus the inhomogeneity of the chemical potential, remains unchanged. Over very long periods of time, this is assumption ceases to be valid. However, this does not affect the equilibrium condition because it is only essential that the inhomogeneity of doping does not change during the relaxation of the charge carriers towards equilibrium, a condition which is always fulfilled.

As one may expect, the equilibrium condition (5.114) or (5.115) will play a central role in the treatment of semiconductor junctions in the next chapter.

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535

Chapter 6

Semiconductor junctions in thermodynamic equilibrium

The distinguished role of semiconductors in the modern world stems from devices which can be fabricated from these materials and, like transistors and semiconductor lasers, have opened the door to the present computer and communication technologies. The operation of semiconductor devices depends largely on effects which occur at semiconductor junctions, in ad- dition to the properties of homogeneous semiconductors which have been treated almost exclusively in this book thus far. A semiconductor junction is understood as a structure composed of two interconnected layers, of which at least one is a semiconductor material. Various semiconductor junctions are shown in Figure 6.1 schematically. In case (a) the two layers are made of the same semiconductor material, e.g. silicon, but one of the layers is p-doped and the other n-doped. This is called a pn-junct ion In case (b) two layers of different semiconductor materials are connected. The doping type may, but need not be different. This structure is called a semiconductor heterojunction. We encountered such semiconductor heterostructures above in section 3.7. If a semiconductor layer is combined with a metal layer, then one has a metal-semiconductor junction, and if it is connected with an insulator layer it is an insulator-semiconductor junct ion Formally, one can also include semiconductor surfaces here, which correspond to vacuum- semiconductor junctions.

Each of the semiconductor junctions identified above is associated with an important physical discovery. Several of these discoveries were even hon- ored with Nobel prizes. Moreover, each of these junctions plays a role in electronic devices, and in most cases a semiconductor junction is crucial for operation of the device. Electric rectification at a metal-semiconductor junction, the so-called Schottky contact, was historically the first semicon- ductor junction to be used technologically. In this particular case, funda-

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536 Chapter 6. Semiconductor junctions in thermodynamic equilibrium

a)

n -type

b)

cn Semic. Metal

Figure 6.1: The various semiconductor junctions.

mental general laws of semiconductor junctions were first recognized, par- ticularly by Schottky and Mott. I t is thus astonishing that, in this case of the metal-semiconductor junction, a complete microscopic understanding is elusive even now. The pn-junction, more precisely the combination of a pn- and a np-junction, underlies the bipolar transistor, for which the inventors Shockley, Bardeen and Brattain received the Nobel prize for physics in 1956. Also, the tunnel diode, for which the Japanese physicist Esaki earned the Nobel prize in 1973, rests on the pn-junction. This device employs electron tunneling between the valence and conduction bands of such a junction. The unipolar field effect transistor (MOSFET) contains, beside a pn-junction, also an insulator-semiconductor junction as an active structural element. In experimental studies of the 2-dimensional electron gas in specially tailored silicon MOSFET's, the quantized Hall effect was discovered by von Klitz- ing. This accomplishment was honored with the Nobel prize for physics in 1985. The 2-dimensional electron gas and the quantized Hall effect may also be observed in semiconductor heterojunctions. Combining heterojunctions with pn-junctions, efficient light emitting diodes (LEDs) and semiconductor injection lasers may be fabricated. Heterojunctions also form the key struc- tural elements of which the superlattices and quantum wells of section 3.7 are composed. The advent of these systems has opened a completely new and fruitful area in semiconductor physics and electronics, that of semiconductor microstructures.

In general terms, the operation of electronic devices based on semicon- ductors may be described as follows: The thermodynamic equilibrium of a semiconductor junction is disturbed by an external perturbation, for exam-

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6.1. pn-junction 537

ple, by applying a voltage or by exposing the sample to light. The result of the perturbation are large changes of some properties of the junction, for example, its electric resistivity. These changes are the effects employed to advantage in semiconductor devices. To understand the perturbation of an equilibrium state of a semiconductor junction, one must first deal with the equilibrium state itself. This will be done in the present chapter. Non- equilibrium processes will be treated in Chapter 7. To initiate discussion of the equilibrium state, we start with the pn-junction.

6.1 pn-junction

pn-junctions are fabricated such that either the surface region of a homoge- neously doped n (or p)-type sample, is p (or n)-doped by means of diffusion or ion implantation, or that a p (or n)-type layer is deposited on a n (or p)-type substrate by means of epitaxy. In the particular case of Si, neutron bombardment may also be used. Due to the nuclear reaction ;gSi + n ---f

:aSi + y -+ + p, a previously p-doped sample becomes n-doped in the surface region exposed to the neutrons.

In the following theoretical description of the pn-junction we assume homogeneously doped p- and n-regions left and right of the plane normal to the z-axis of a Cartesian coordinate system (see Figure 6.2a). The p - region extends to -00 and the n-region to +co. The transition from the p- to the n-region takes place abruptly at z = 0. We refer to this position as the ‘nominal transition’. In the transverse plane normal to the z-axis, the two semi-infinite semiconductor samples are taken to extend to infinity in all directions. Thus, there are no dependencies on y and z , provided the semiconductor materials are homogeneous in the transverse plane, which we take to be the case. With this, the problem is effectively one dimensional. The carrier concentration distribution along the x-axis before equilibrium is established is shown schematically in Figure 6.2a. Conceptually, this corre- sponds to the instant at which the junction is formed, so that the carrier distribution has not yet had enough time to adjust to form a new equilibrium state. Considering that the p-region also has electrons which are minority carriers in it, and that the n-region also has holes as minority carriers there, one must distinguish the region in which these concentrations are meant. We indicate this by affixing a subscript to n and p . The concentrations in the n-region will be denoted by n, and p,, and those in the p-region by np and p p . Thus, n,, and p p are majority carrier concentrations, while np and p, are minority carrier concentrations. The different values of the charge carrier concentrations in the n- and p-regions are related to different values EF, and E F ~ of the bulk Fermi energies in these regions, such that

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538 Chapter 6. Semiconductor junctions in thermodynamic equilibrium

D i f f h o n tc Drift

-ecp (x)

C)

Recombi - nation

I I I I

1------ I

1 I I I I

I I I

I X

I I

Figure 6.2: Non-equilibrium state of a pn-junction (a), from which the establish- ment of thermodynamic equilibrium develops (b,c).

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6.1. pn-junction 539

6.1.1 Establishment of thermodynamic equilibrium

The situation shown in Figure 6.2a certainly does not represent an equilib- rium state. The charge carrier concentrations are spatially varying functions n(x) and p ( z ) , and because np << n, and p, << pp, strong gradients of n(x) and p(x) exist in the vicinity of the nominal transition x = 0. They cause electrons to diffuse to the left into the p-region, and holes to the right into the n-region (Figure 6.2b). Consequently, the n-region becomes positively charged, and the p-region negatively. In the terminology of Chapter 5 , one can say that a space charge region is formed. It has an associated electric field E ( z ) and an electric potential cp(z). The dependence of these quantities on x is shown in Figure 6.2b (right column), illustrating results which will actually be obtained later by detailed analysis. For our immediate consid- erations, only their qualitative behavior is important. In Figure 6.2b the band edges are shown as position-dependent quantities. This representa- tion is surprising initially, since the band edges are energy eigenvalues of the Schrodinger equation, and as such they do not depend on space coor- dinates. Such a representation in terms of position-dependent band edges is nevertheless meaningful in the present case, based on the fact that cp(z) is a smoothly varying potential in the sense of section 3.3. Therefore, the effective mass equation can be applied. For the envelope function Xc(x) of the conduction band electrons this equation reads

and for the envelope function Xv(x) of the valence band holes it has the form

According to equation (6.3), the electrons at the conduction band edge ex- hibit a potential energy E, - ep(z), and the holes at the valence band edge exhibit a potential energy ecp(z). These energy functions are depicted in Figure 6.2b. In this, one has to recognize that the energy e p ( z ) of the holes counts negatively, so that on the positive electron energy scale one must therefore plot - ep (z). The electric field in the space charge region drives a current in the direction opposite to that of the diffusion current (Figure 6.2b). The concentrations

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540 Chapter 6 . Semiconductor junctions in thermodynamic equilibrium

of carriers which diffuse from the region where they are majority carriers into the regions where they are minority carriers, are much larger than the equilibrium minority concentrations there. Thus they will recombine with the abundantly available majority carriers in these regions (Figure 6.2~) . In this process, the concentrations of both types of free charge carriers decrease in the space charge region, as shown in Figure 5.2. The electric charge bal- ance as well as the electric field and potential distributions are, however, not influenced by such recombination since the annihilation of electrons and holes takes place simultaneously, and in the same measure. Overall, a pn- junction, in the time interval immediately following its formation, undergoes a relatively complicated interplay between various non-equilibrium processes - the diffusion and drift of majority carriers, the formation of a space charge region and of related electric field and potential distributions, as well as re- combination of non-equilibrium minority carriers with equilibrium majority carriers.

The processes under discussion act in such a way that thepn-junction ap- proaches thermodynamic equilibrium. The state to which the system relaxes is, however, not a static but a dynamic equilibrium state. Of course, it has no net current, but only because the drift and diffusion currents fully compen- sate each other. In regard to the equilibrium state which finally emerges, the details of the processes which establish equilibrium do not matter, what is important is only that there are such processes. Independently of the nature of these processes, specific local values of the chemical potential p ( z ) , electric potential p(z), and charge carrier concentrations n ( z ) , p(z) are established in equilibrium. The latter are given by the relations (5.106) discussed in Chapter 5, and we display them here once again:

Far from the nominal pn-transition at z = 0, recombination processes cause the concentrations n(z ) and p ( z ) to take their equilibrium values, whence

n(-m) = np, n(+m) = nn, (6.6)

~ ( - 0 0 ) = ~ p , P(+w) = Pn. (6.7)

P ( - m ) = EFp, P ( + m ) = EFn. (6.8)

Considering (6.2), we have

As discussed in Chapter 5, thermodynamic equilibrium is characterized by a spatially constant electrochemical potential ~ ( z ) - ep(z). The calculation of the electric potential p(z), is carried out using the Poisson equation (5.19). In this, one only needs to specify the charge density p(z ) on the right-hand side in accordance with the relationships which hold for the pn-junction.

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6.1. pn-junction 541

Before doing this and solving the Poisson equation, we will first determine a relatively easily accessible integral quantity, namely the potential difference U D , which exists between the left and the right end of the pn-junction in consequence of the charging of the two regions. This potential difference is called &;fusion voltage, because diffusion is the agent actually responsible for its existence. The diffusion voltage provides one of the boundary conditions needed for the solution of the Poisson equation.

6.1.2 Diffusion voltage

The diffusion voltage U D cP(m) - cP(-m)

can be determined by writing down the electrochemical potential p ( z ) - ecp(z) once at z = -m and again at z = +m, and requiring the two values to be equal, whence

p(-m) - ecp(-oo) = p ( + w ) - ecp(+m). (6.10)

Using (6.8) and (6.9), we have

1 U D - [ E F ~ - E ~ p l . (6.11)

The Fermi levels may be expressed in terms of the majority carrier con- centrations nn and p, employing equations (6.5) to (6.8). First of all, we have

nnPp - - N ~ N , -Eg/kTe ( E F ~ - E ~ p l / k T . (6.12)

Introducing ni from formula (4.86) and taking the logarithm, it follows that

(6.13)

To provide an estimate of the order of magnitude of the diffusion voltage, we proceed as follows: Let be T = 300 K , i.e. k T M 25 m e V . With n, L- p, = 10I6 cm-3 and ni = 10" ~ r n - ~ (roughly corresponding to the intrinsic concentration of Si), we obtain U D = 0.7 e V . A comparable value follows if one assumes that the Fermi level in the n-type semiconductor lies just below the conduction band edge, and in the p-semiconductor just above the valence band edge. Correspondingly, the difference of the two Fermi energies, which equals the digusion voltage, takes a value which is somewhat smaller than the energy gap. Thus, under realistic assumptions, we find that diffusion voltages have substantial magnitude.

Diffusion voltages cannot be measured or used, however, in any way involving interconnection of the two ends of the p- and n-regions by wires

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542

+ -

Chapter 6. Semiconductor junctions in thermodynamic equilibrium

bl - + + - P I: n ;: - +

I Metal I

I 1 + + +

- + - + - +

Figure 6.3: Measuring the diffusion voltage of a pn-junction.

(Figure 6.3a), or by bringing the two ends directly in contact with each other, bending the p - and n-regions (Figure 6.3b). No current will flow through closed circuits obtained in this way. The total voltage drop in such circuits is zero, since, beside the original pn-junction already present a second one is created by closing the circuit, acting in the opposite direction. The metallic connections in Figure 6.3a only play a passive role and do not affect the voltage balance of the circuit - the sum of the diffusion voltages of the two metal-semiconductor junctions exactly equals the negative of the diffusion voltage of the pn-junction. Nevertheless, the diffusion voltage can actually be measured, namely, in an electrostatic experiment. This can be done, for example, when the two ends of the p- and n-regions are bent to be close to each other, leaving only a small gap between them (Figure 6 .3~) . In the gap there is an electric field associated with the diffusion voltage, whose existence may be demonstrated by means of a small test charge.

6.1.3 Spatial variation of the electric and chemical potentials: Schottky approximation

The electric potential p(z) obeys the Poisson equation (5.19) with p ( z ) taken from (5.16). A particularly important feature of this equation is that its charge sources on the right-hand side themselves depend on the potential p(z). We consider this dependence initially for the free charge carrier part of the charge sources. One obtains this from equations (6.5) for n(z) and p ( z ) , expressing the chemical potential p ( z ) in terms of the electric potential p(z) using the relation

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6.1. pn-junction 543

whence

and with 6.13) we find 2

n ( x ) p ( z ) = ni

(6.16)

(6.17)

The mass action law, which was written down for spatially homogeneous semiconductors in equation (4.85), therefore also holds locally for inhomo- geneous semiconductors. Such a result is not surprising since this law is predicated on the existence of thermodynamic equilibrium, and local equi- librium was assumed in the inhomogeneous case considered here.

The concentrations of ionized donors NDf and acceptors N Z depend on the potential (p(z), at least in the general case. In a frequently occurring special case, namely that of complete ionization, this dependence no longer exists. We will assume that this case applies, so that

~ o f ( x ) = N&), N A ( X ) = N A ( Z ) . (6.18)

Since donor atoms are present only in the n-region, and acceptor atoms only in the p-region, and since they are homogeneously distributed there in accordance with our initial assumption, it follows that

0, x > 0, NA, x < 0.

N ~ ( X ) =

The Poisson equation for cp(x) thus reads

(6.19)

(6.20)

(6.21)

We seek a solution of this equation which accounts for the existence of the diffusion voltage (6.9). In order to get a unique solution one more condition is needed. This may be obtained from the observation that far from the nominal transition at x = 0, everything should behave as it would in an in- finite bulk sample. This means, in particular, that the electric field strength has to vanish at x = fco, whence

(6.22)

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544 Chapter 6. Semiconductor junctions in thermodynamic equilibrium

Figure 6.4: Illustration of the Schottky approximation.

The first part of this relation holds because of total charge neutrality of the pn-junction. It is not an independent condition, and does not provide additional information.

The scalar potential p(z) is determined only up to an additive constant. Without any loss of generality we may chose this constant such that

(6.23)

The explicit determination of the solution of the Poisson equation (6.21) is a difficult task, mainly because the function p(z) to be calculated enters the right-hand side of this equation via the charge carrier concentrations n ( z ) and p ( z ) as given by (6.15) in an extremely non-linear way, namely expo- nentially. It is therefore necessary to resort to approximations or numerical calculations.

Here, we will discuss the former. First of all, one must consider which approximations are suitable under existing conditions. The answer may be obtained from a qualitative estimate of the potential profile p(z) illustrated in Figure 6.4. The result of the estimate has already been used in Figure 6.2 to construct -ep(z). Far from the nominal pn-transition at z = 0 there is local charge neutrality, as for a homogeneous semiconductor sample. In consequence of the Poisson equation and the boundary condition (6.22), the potential will approximately retain the values p(-m) and p(+oo) which it has, respectively, at z = --oo and z = +m, in approaching the position x = 0 from --oo and +m. In the vicinity of the nominal transition at z = 0 a space charge region is formed, as outlined above, and it is here that the

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6.1. pn-junction 545

n = N, I Depletion region

- - - - I+++ ,+ I Bulk region Ispace charge region1 Bulk region

Figure 6.5: Space charge region, depletion region and bulk region of a pn-junction.

potential will undergo significant variation. We will denote by x , the first position in the n-region where, in the approach from +CQ, space charge starts to develop substantially. Correspondingly, the potential starts to undergo major change at x,, where it is approximately given by cp(x,) = cp(00). Since the total change of cp(x) over the space charge region amounts to UD, and since U D is, in general, substantially larger than k T , the point is quickly reached where cp(x) has fallen off to the value cp(x,) - k T / e . We denote this point by x n - A x , , so that

V ( x n - AX,) = cp(xn) - k T / e . (6.24)

The electron concentration n ( x ) of (6.15) at 2,- A x , has already decreased down to the e-th part of its equilibrium value nn (see Figure 6.4). Since it decreases further as x decreases, one can approximately set it to zero for x , - A x , , whence

0, x < ~ n - A x , ,

n,, x > x,. n ( x ) = (6.25)

An analogous conclusion can be drawn for the hole concentration p(x). Us- ing similar notations x p and A x p for holes as were used for electrons, we approximately have

(6.26)

If it turns out, as is actually the case, that A x , << x , and A x p <<I x p I hold, then we can make the further approximation of taking the two intervals A x , and Axp to be vanishingly small. We will do this, understanding this change to be made in equations (6.25), (6.26) without rewriting these formulas. The simplification described in the resulting equations (6.25) and (6.26) is called the Schottky approximat ion In the pn-junction, we can now identify four regions in which the total charge density has constant values, with jumps from region to region (see Figure 6.5). For x < xp and x > x , the

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546 Chapter 6 . Semiconductor junctions in thermodynamic equilibrium

charge density is zero, because the charge of the ionized impurity atoms is compensated by the charge of the free electrons and holes. Between x p and I , , free charge carriers are absent, so for x p < x < 0 we have the charge density - e N A of the ionized acceptors, and for 0 < x < x , the charge density e N D of the ionized donors. Because of the absence of free charge carriers between x p and 0 and between 0 and x,, these two regions are called depletion regions. Jointly, the two regions together form the space charge region, The adjacent regions on the left and right hand sides are called bulk regions. The Poisson equation associated with the charge distribution described above reads

d2P - N o , O < X < x n , --

d x 2 - { N A , x p < x < 0, (6.27)

I 0, x < xp .

The boundaries x , and x p of the space charge regions are undetermined so far. One relation between them follows from the total charge neutrality condition,

X , N D + X ~ N A = 0. (6.28)

It states that the total positive charge of the n-side of the space charge region equals the total negative charge of the pside of this region. A second relation for the determination of x , and x p follows from the boundary condition (6.22), which will be used later.

In solving the Poisson equation (6.27), attention must be given to the fact that, notwithstanding discontinuity of the right-hand side at x = 0, the only admissible potentials p(z) are those which are continuous everywhere together with their first derivatives. The solution of the boundary value problem (6.27), (6.9) and (6.22) which meets this requirement is given by

(6.29)

x < x p .

At the position x = 0, cp(x) is automatically continuous and vanishes, as it should according to equation (6.23). The continuity of the h t derivative at x = 0 follows from the charge neutrality condition (6.28). The diffusion voltage condition (6.9) yields

(6.30)

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6.1. pn-junction 547

Introducing the abbreviation w = I, - xp for the width of the space charge region, it follows from (6.28) that

N D w , x p = - N A 2, =

N D + N A + N A ~ . Applying equation (6.30) we get the result for w as

(6.31)

(6.32)

The potential values at x, and x p are given by

The calculated potential profile cp(x) corresponds in every detail to that which we assumed qualitatively in motivating the Schottky approximation. We also can now specify the validity limits of this approximation quanti- tatively. In fact, substituting (6.29) for cp(z) into the condition (6.24), we obtain

(6.34)

This length, which (apart from a factor a) is the Debye screening length for the potential U D in the n-region, must be small in comparison with the width x, of the n-part of the space charge region, in order for the Schottky approximation to be valid. A corresponding relation must be satisfied for the p-region. With the help of equations (6.31) and (6.32) these conditions may be written in the form

f i < d-m, n - region,

fi<< J-J~TT;;, p-region. (6.35)

In order for these relations to be fulfilled, the potential energy of an electron due to the diffusion voltage must be much larger than its average thermal energy. Considering equation (6.13) for UD this is the case if

1 < [ ln- n;y]”’ (6.36)

holds. At sufficiently high doping concentrations the Schottky approxima- tion is therefore always applicable provided N A and N D are of comparable size, so that the numerical factors d N A / ( N A + N O ) and ~ N D I ( N A -t N g )

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548 Chapter 6. Semiconductor junctions in thermodynamic equilibrium

of relations (6.35) are close to 1. With n, = p, = 1017 cm-' and ni = lo1' cm-3 the value 5.7 follows for the right hand side of (6.36), i.e. the Schottky approximation is valid under these conditions.

Some consequences of the relations derived above will now be reconsid- ered in more detail. In regard to the width w of the space charge region at room temperature, doping concentrations ND = N A = 1OI6 cm-', and values of E and UD corresponding to silicon, equation (6.32) approximately yields w = 0.4 p m . In order for the space charge region of a pn-junction to develop fully, the p- and n-regions must have lengths which clearly ex- ceed this value. In the case above, which is typical, they must lie in the micrometer range. This is one of the reasons why the length scale of doping structures in electronic devices cannot be made arbitrarily small.

Equation (6.32) for w also implies that the width of the space charge region is smaller for heavier doping of the two semiconductors. For N D = N A = 10'' cm-3 the space charge width w is ten times smaller than for N D = N A = 10l6 cm-'. This is understandable because larger space charge densities enable a narrower region to bridge the diffusion voltage. For the same reason, the n-region contributes less to the total width of the space charge region if it is more heavily doped. An analogous statement holds for the p-region (see Figure 6.4). Since p(0) = 0, the energy of a junction electron has the same value at x = 0 as an electron would have in the cor- responding infinite n- or p-type bulk semiconductor. The total potential difference UD is distributed between the p and n-regions according to for- mula (6.33), i.e. the contribution from the n-region is larger for larger N A , and that of the p-region is larger for larger N o . This is due to the larger width of the space charge region in the material with the lower doping.

After the electric potential p(x) is known, the chemical potential p(z ) can be calculated by means of relation (6.14). Also the uniform Fermi level EF of the pn-junction, which by definition equals the electrochemical potential p(x) - ep(x), can be determined easily. One only needs to calculate p(z ) - ecp(z) at one particular position x where p ( x ) is known. We take x = zn. Then it follows from equations (6.14), (6.29), (6.31) and (6.32) that

or

The uniform Fermi level of the junction is uniquely determined by the indi- vidual Fermi levels and doping concentrations of the n- and p-regions. The fact that p - and n-type materials exist at a semiconductor junction despite the common, uniform Fermi energy, is due to the bending of the conduction and valence band edges. This bending is such that, on the left-hand side

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6.2. Heterojunctions 549

Figure 6.6: Lineup of the band edges and the chemical potentials for a pn-junction with N D = 2N.4. The chemical potential p(z) and the Fermi level are also shown.

of Figure 6.6, the valence band edge is close to the Fermi level so that the material is p-type there, and on the right-hand side it is the conduction band edge that is close to the Fermi level so that material is n-type there.

e 6 . 2 Heterojunctions

The pn-junction is a semiconductor structure based on a single semiconduc- tor material, except the state of the semiconductor, specifically, the value of the chemical potential, is different on the two sides of the junction. In the case of semiconductor heterojunctions not only are the states different but also the materials differ - left of the nominal transition at z = 0 one has a particular material 1, and right of it there is a different material 2. The material inhomogeneity means that the valence and conduction band edges E,, E,, the energy gap E,, the effective band densities of states N,, N,, the Fermi level EF and other quantities have different values on the two sides

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550 Chapter 6. Semiconductor junctions in thermodynamic equilibrium

I * Material 1

X

Figure 6.7: Semiconductor heterojunction.

of the junction (see Figure 6.7). We indicate these differences by a material index, i.e. we write Evl, Ev2, Eel, Ecz etc. Heterostructures of this kind have already been encountered in section 3.7, where they were seen to play a key role in semiconductor microstructures. The discussion of the electronic structure of heterostructures, in particular, their valence and conduction band lineups as well as their band discontinu- ities, need not be repeated here. We will concentrate on the effects of free carriers at heterojunctions. Such carriers were absent in most of the consid- erations of section 3.7, and to the extent that they were discussed, results were merely quoted without derivation. Now, we will present a detailed anal- ysis of free carrier effects at heterostructures, similar to that above for the pn-junction. As in the latter case, we initiate this analysis with discussion of the thermodynamic equilibrium state of the free carrier system.

6.2.1 Equilibrium condition

As seen above, the thermodynamic equilibrium state of the free carrier sys- tem of an inhomogeneously doped semiconductor is characterized by the condition that its electrochemical potential is uniform in space. Here, we generalize this condition to heterostructures, i.e. to semiconductor systems having not only inhomogeneous doping but also inhomogeneous materials composition. Again, the characterization of thermal equilibrium is focused on the vanishing of the net electron and hole currents. However, the phe- nomenological equations for these currents differ in the case of systems hav- ing heterogeneous materials composition from those of systems involving a single material only. This can be seen as follows. Assume for the moment that the band edges Ec and Ev of the heterostructure change gradually as functions of position coordinate 2 at the interface, rather than abruptly.

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6.2. Heterojunctions 551

Then, within the framework of effective mass theory, the continuous func- tions Ec(z) and E,(z) describe position-dependent potential energies of elec- trons and holes, and one may expect that the gradients VE, and VE, play the role of driving forces, which result in additional contributions to the electron and hole currents. The same may be said for the gradients V N , ( z ) and V N v ( z ) of the effective densities of states in contributing to diffusion current. The exact forms of these current parts are not known a priori. They can be determined either by means of Boltzmann’s equation, or one can formulate them phenomenologically on the basis of experience as we did in the case of the electric drift and diffusion currents in equations (5.100) and (5.101). Using Einstein’s relations (5.111) and (5.112) for, respectively, D, and D,, as well as equations (5.82) and (5.83), the electron and hole diffusion currents in these equations can be written as u , ( k T / e n ) V n and - u , ( k T / e p ) V p , respectively. Since the negative gradients of N c and N , should play a role similar to that of the gradients of n and p , we write the generalized phenomenological equations for the electron and hole currents as

k T n

- ecp) + -Vn -

kT Nw 1 1 k T j, = up; [O(,T~ - ecp) - - 0 p + - V N , .

n

(6.39)

(6.40)

The concentrations n and p of electrons and holes can, as above, be expressed in terms of the chemical potential p ( z ) , where now, in contrast to equations (6.5), the band edges and the effective densities of states are position depen- dent. Thus,

Using these expressions for n ( z ) and p ( z ) , the two phenomenological equa- tions (6.39) and (6.40) may be written in the more compact form

Accordingly, the vanishing of the two current densities j, and j, is guaran- teed by the fact that the electrochemical potential p ( z ) - ecp(z) is spatially uniform. The constant value is again referred to as Fermi energy EF. With this, the spatial constancy of the Fermi energy EF is seen to be a prin- cipal feature of thermodynamic equilibrium for any carrier systems, those of single semiconductor materials with inhomogeneous doping, and those of

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5 52 Chapter 6. Semiconductor junctions in thermodynamic equilibrium

semiconductor junctions composed of different materials. In view of this, the chemical potential follows immediately if the electric potential is known in thermodynamic equilibrium. The latter must be determined independently. The problems involved in such calculations are discussed in the subsection below.

6.2.2 Electrostatic potential. GaAs/Gal-,Al,As heterojunc- tion as an example

Again, the determination of the electric potential is based on the Poisson equation (5.17). Generally, the dielectric constants €1 and € 2 of the two materials entering this equation are different in the present case. The dopings of the materials may either be of the same type or of different types. Even if they are the same type, the dopant concentrations may differ which, of course, may also happen if the doping types are different. As in the case of a pn-junction, one can divide the entire structure into a space charge region about the nominal transition point at x = 0 and two electrically neutral bulk regions on the right and left-hand side of it. The Schottky approximation, which in the case of the pn-junction greatly simplified the solution of the Poisson equation, is, however, generally not applicable for heterojunctions. The solution method must be adjusted to the particulars of the situation, as we will see below. In any case, boundary conditions must be applied. One such condition is obtained by replacing the potential difference between x = 00 and x = -00 by the difference of the values E F ~ E p(+00) and E F ~ 3 p(-00) of the chemical potential at x = +oo and x = -00, respectively. Since the electrochemical potential p(x) - ecp(x) is spatially uniform, we have

(6.44)

The right-hand side of this equation is, in fact, the diffusion voltage in- troduced above in the discussion of the pn-junction. The second boundary condition is again the vanishing of the electric field at x = +00 and x = -00,

as in the case of the pn-junction. Because one of the latter two conditions follows from the other by means of charge neutrality, it suffices to require

dio dx l2=--00 - - 0. (6.45)

The calculation of the electrostatic potential along the general lines described above will now be demonstrated in the particular case of a heterojunction between GaAs (material 1) and Gal-,Al,As with x < 0.4 (material 2), illustrated in Figure 6.8. Accordingly, we assume that the relations Ec2 > Ecl, Ncl = Nc2, and €1 = € 2 t are valid. Moreover, we suppose that both

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6.2. Hetero junctions 553

Figure 6.8: Lineup of the conduction band edges at a heterojunction between n-type GaAs (material 1) and n-type GaAs/Gal-,Al,As with x < 0.42 (material 2). Part (a) shows the situation before the establishment of equilibrium and (c) after equilibrium has been established. The poten- tial variation in space is shown in part (b) of the figure.

0 X

materials are n-type which means that N A ( z ) = p ( z ) = 0, approximately. With these simplifications, the Poisson equation for the heterojunction reads

(6.46)

We further assume that the donor concentrations in both materials are the same, so that the Fermi energies E F ~ and E F ~ of the two bulk semiconductors have the same energy separations from the respective conduction band edges E,1 and Ec2 of the two materials, whence

Considering that Ec2 > Eel, we also have EFZ > E F ~ , and consequently electrons originating in material 2 diffuse into material 1. There, they accu- mulate close to the interface, and one thus identifies an accumulation layer. Correspondingly, in material 2, a depletion layer is formed, and n < N o holds so that the Schottky approximation can be applied. However, in the accumulation layer, one has n >> N o , thus the Schottky approximation is not valid in material 1. The character of the Poisson equation here is fun- damentally different from that of the depletion layer - the source term on the right-hand side depends on the potential p(z) itself through the carrier concentration n(z) . In fact, from relations (6.41) and (6.44), it follows that

with no1 as equilibrium carrier concentration in the infinite bulk material 1. At large distances from the nominal transition at z = 0, the concentration

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554 Chapter 6 . Semiconductor junctions in thermodynamic equilibrium

n(x) becomes n01 , as one should expect. Using (6.48) the Poisson equation takes the form

(6.49)

which is a non-linear equation that must be solved self-consistently. This entails finding a solution p(z) which obeys the equation with the source term depending on cp(x) as shown on the right-hand side.

Some insight into the solution may be gained as follows: We note that the potential energy difference -e[cp(z) - p(--oo)] of an electron in the accu- mulation layer and at --oo cannot exceed the conduction band discontinuity AEc Ec2 - Eel, and also will not fall significantly short of it. In any case, it is large compared to RT. Thus, in the accumulation layer, the second term on the right side of (6.49) representing the concentration of free car- riers, is substantially larger than N o . In these circumstances it is possible to make a simplification which is, in a sense, complementary to that of the Schottky approximation, namely the neglect of the concentration of b e d charge carriers in comparison with that of the freely mobile ones. We retain a constant value for N D , nevertheless, in our analysis below because it does not interfere with the determination of an exact first integral of equation (6.49), as given by

This relation is in fact a non-linear first order differential equation. The boundary condition (6.45) is already accounted for in it. Equation (6.50) cannot be processed further exactly. It can be solved approximately by iteration. This means that one first substitutes an educated guess for an approximation to the solution p”(x) into the inhomogeneous right-hand side of equation (6.50) and then solves for (dp/dx) on the left and integrates it to determine an improved solution cpl(x). The result for the improved solution pl(x) is then substituted into the inhomogeneous right-hand side of (6.50) and again on solves for ( d c p l d z ) on the left and integrates it to obtain a further improved solution p2(x). This cycle is repeated as often as needed to achieve a desired degree of accuracy.

Our interest here lies solely in the qualitative shape of the potential cp(x) in the accumulation layer of material 1, so we will limit our considerations l o an estimate. The width WA of this layer, over which most of the whole potential change in material 1 occurs, can be easily estimated from equation (6.50). For this purpose we neglect N D and take account of the fact that the potential drop -e[p(x) - cp(-oo)] is smaller than, but of the same order of

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6.2. Heterojunctions 555

magnitude as, the conduction band discontinuity AE,. We approximately replace -e[cp(z) - cp(-oo)] by AE,/2 in the exponential term of equation (6.50). On the other hand, it is clear that the derivative e(dcp/dz) has the or- der of magnitude AEJwA. Therefore, it follows from (6.50), approximately, that

(6.51)

where the '-1' in the rectangular bracket of (6.50) has been neglected in comparison with exp(AE,/kT). The square root factor in equation (6.51) is the Debye length, which already played an important role in the case of the pn-junction. There the Debye length was small compared to the width of the space charge region, which was a depletion region for a pn-junction. For the accumulation region of a heterojunction, the result is just the opposite - the width of this region is reduced relative to the Debye length by a factor which, in general, is less than 1. With AE, = 0.25 eV and k T = 25 meV (T = 300 K ) it is 0.8. The Debye length itself has a value of about 400 A if no1 = 10l6 ~ m - ~ , T = 300 K and t = 12 are assumed. Thus WA amounts to about 320 A in the present case.

In material 2, one has a depletion region and the Schottky approximation is applicable. The potential profile has a parabolic shape, as in equation (6.29) for the n-doped depletion region of a pn-junction. The potential change extends over a width which is large compared to the Debye length, varying much more slowly than in material 1. In Figure 6.8b1 the potential is shown schematically throughout the entire heterostructure. Below it (Figure 6.8c), we plot the effective position dependent conduction band edge E,(z) - ecp(z). Considering equations (6.44) and (6.47), we have

El21 - eY4-w) Ec2 - ecp(+w), (6.52)

i.e., the band discontinuity is completely compensated by the electrostatic potential cp(x). Since this potential is a consequence of the redistribution of free electrons which adjust under the influence of the band discontinuity AE,, one may also say that AE, is dielectrically screened out by these electrons.

The spatial variation of the effective band edge E, - ecp(z) in Figure 6 . 8 ~ displays an interesting property, in that the GaAs accumulation layer is bounded on the right-hand side by the Gal-,Al,As potential barrier. Thereby, a triangular potential well is formed. The Fermi level lies above the bottom of this well for sufficiently heavy doping, as was assumed in Figure 6.8. The average width of the well, which was estimated above to be 360 A, is of the same order of magnitude as the width of the quantum wells considered

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5 56 Chapter 6. Semiconductor junctions in thermodynamic equilibrium

Magnetic Field Strength (Testa)-

Figure 6.9: GaAs/Gal-, A1,As-heterojunction at T = 8 m K . Probst, and Ploog, 1982).

Quantized Hall effect of the 2-dimensional electron system of a (After Ebert, van Klitzing,

in section 3.7. This means that in a single GaAs/Gal_,Al,As heterojunc- tions one can expect similar confinement effects as in Gal-,Al,As/GaAs/ Gal-,Al,As quantum well structures. In the well region, the conduction band splits into subbands. Since the Fermi level lies above the well bottom, these subbands are partially occupied, and the electrons of the accumula- tion layer form a 2-dimensional electron gas. As pointed out above, one can observe the quantized Hall effect in such a 2-dimensional gas (see Figure 6.9).

The formation of subbands has yet another implication. Namely, that equation (6.41) for the electron concentration n ( z ) , which is based on the assumption that the electrons are hosted by a 3-dimensional parabolic band, is no longer valid. Instead, the 2-dimensional subband structure of the triangular well has to be considered. The latter can only be determined, however, after the potential is calculated, which, in turn, cannot be accom- plished without knowing the subband structure. Thus, a new type of self- consistency problem emerges, beside those already considered: the Poisson equation must be solved self-consistently with the effective mass Schrodinger equation (6.3).

In the above discussion of the electric potential of a GaAs/Gal-,Al,As heterojunction, a particular assumption was made concerning the dopings of the two materials; they were both n-type, and their donor concentrations were identical. These assumptions are not essential, however, as one can easily see. Even if the dopant concentrations differ, a triangular quantum well is formed. A case of particular importance is that of modulation doping in which only the Gal_,Al,As region is n-doped while the GaAs region

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6.3. Metal-semiconductor junctions 557

is undoped. In this case the electrons in the GaAs well are not exposed to ionized donors because there are none. Therefore the mobility of these electrons is particularly high, it can reach several millions crn2V-ls-'. An electronic device which exploits this effect is the High Electron Mobility Transistor (HEMT), which was already mentioned several times in somewhat different contexts.

6.3 Metal-semiconductor junctions

Metal-semiconductor junctions are fabricated by evaporation of the metal and the subsequent deposition of its vapor on the colder semiconductor surface. These junctions play an important role as electrical contacts for semiconductor samples and devices. The dependence of their resistance on the polarity of an applied voltage can be used for electric current rectifi- cation purposes. In GaAs MESFET's (Metal-Semiconductor-Field-Effect- Transistors), particular metal-semiconductor junctions (so-called Schottky contacts) also have an active electronic function: The resistance of the GaAs layer below the metal can be varied by means of a voltage applied to the metallic gate-electrode.

6.3.1

The positions of the energy levels at a metal-semiconductor junction at the instant of its formation, i.e. before the junction had enough time to relax to thermodynamic equilibrium, are shown in Figure 6.10 schematically. All energies are referred to the vacuum-level Eo. In Figure 6.10a a situation is assumed in which the highest occupied energy level of the metal, which is its Fermi energy E F M , lies below the Fermi level EFS of the semiconductor, which is here assumed to be n-type. The relation EFM < EFS is typical of metals paired with n-type semiconductors, although the reverse case can also occur (see Figure 6.10b). For p-type semiconductors, one has essentially the same relations. The two possible cases, in which the Fermi-level of the p-type semiconductor either lies below that of the metal, or above it, are shown in Figures 6 .10~ and 6.10d) respectively.

The Fermi level of the metal is given by the negative of its work function 0, so that EFM = -0. The work function is defined as the smallest energy which must be transferred to the electron system of the metal to enable an electron to escape. It can be obtained from photoemission experiments in which the kinetic energies &in of electrons emitted from the metal by the absorption of photons of a particular energy hw are measured. Since an emitted electron has absorbed one photon, its kinetic energy is Ekin = hw + Einitbl where Einitbl is the energy which the electron had in the metal

Energy level diagram before establishing equilibrium

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558 Chapter 6 . Semiconductor junctions in thermodynamic equilibrium

I L

0 X

0 X

Figure 6.10: Positional dependence of the Fermi levels of a metal and a semicon- ductor, as well as of the band edges of the semiconductor in a metal-semiconductor junction at the instant of its inception, before equilibrium is established. Four dif- ferent cases are shown: n-type semiconductor (a, b), p-type semiconductor (c, d), EFS > EFM (a, d), and EFS < EFM (b, c).

prior to emission. There are as many different values of Einitfal as there are occupied energy levels of the metal. Thus the kinetic energies &in span the whole occupied part of the energy spectrum of the metal, shifted by hw. The work function represents the lower threshold energy of this distribution.

The energy separation between the conduction band of the semiconduc- tor and the vacuum level is referred to as the electron affinity X . It, too, is a quantity which can be determined experimentally. One measures the photothreshold a P h of the semiconductor (see section 3.7), and subtracts the energy gap Eg from it, obtaining the electron affinity X from the relation X = QPh - Eg.

If a voltage U is applied to a metal-semiconductor junction, a current will flow through it. In Chapter 7 we will show that the magnitude of this current essentially depends on the energy EA,(M + SC) necessary, at minimum, for

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6.3. Metal-semiconductor junctions 559

an electron to be transferred from the metal into the semiconductor. In the case of a p-type semiconductor, it will be the smallest energy E,+,(M -+ SC) necessary for the transfer of a hole from the metal into the semiconductor that determines the current. In both cases, one calls these energies Schottky barriers and denotes them by QB, so that

(6.53)

For a given semiconductor material, the sum of the two Schottky barriers, one referring to n-type doping and another referring to p-type doping, should be independent of the metal and equal the gap energy E , of the semicon- ductor. The Schottky barrier determines the current density j through the metal-n-type semiconductor junction by means of the relation

(6.54)

as will be shown in Chapter 7. Here B is a coefficient which depends only weakly on temperature. Using equation (6.54), and measuring the current- voltage characteristics of the metal-semiconductor junction, the Schottky barrier @B can be determined experimentally.

Moreover, if the level ordering at the metal-n-type semiconductor junc- tion is that shown in Figure 6.10a, then the equation

(6.55)

relates the Schottky barrier Q B of the junction, the work function Q of the metal and the electron affinity X of the semiconductor. In employing this relation we have, however, jumped ahead of the thermodynamic analysis of energy level ordering at the metal-semiconductor junction. In Figure 6.10, in fact, the energy levels are shown at the instant when the metal- semiconductor junction has just been formed. Since the chemical potentials of the metal and the semiconductor are different, the situation shown in Figure 6.10 surely does not represent thermodynamic equilibrium. Electrons or holes of the semiconductor will diffuse into the metal (Figure 6.10a,c) or from the metal into the semiconductor (Figure 6.10b,d). Thereby a space charge region will arise at the interface (see Figure 6.11) with its concomitant electrostatic potential cp(z) varying in space. This potential will now be considered in more detail.

6.3.2 Electrostatic potential

In equilibrium, the electrochemical potential EF = p ( z ) - ecp(z) is spatially uniform. In the metal, the spatial changes of p ( z ) and p(z) are limited to a

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560

M

b

Chapter 6. Semiconductor junctions in thermodynamic equilibrium

S I \ *

X

very thin boundary layer at the interface. The thickness of this layer is about one A. This value follows from relation (6.34) if one takes the electron con- centration of a metal to be of the order of magnitude crnV3, and replaces the average thermal energy k T by the Fermi energy of the metal, switch- ing over in this way from the Debye length to the Thomas-Fermi screening length. Over a distance of an A the potential ~ ( x ) is practically constant. Deeper in the metal, because of screening by electrons in the boundary layer, the potential is essentially constant anyway. It is therefore reasonable to ap- proximate the potential in the metal as spatially uniform. At the nominal transition from the metal to the semiconductor, the potential could be un- derstood ~ as in the case of semiconductor heterojunctions considered in the previous section - to exhibit a discontinuity [cp(+O) - cp(-O)] because of the formation of a dipole layer. At the outset, we wish to ignore this and explore the possibility that the potential cp(;c) may be continuous at 2 = 0, i.e. assume that

Cp(+O) - cp(-0) = 0. (6.56)

With this, it is clear that expression (6.55) for the Schottky barrier Cpg

will be valid even after thermodynamic equilibrium is established at the junction by the formation of a space charge layer and its associated potential. Both levels, the Fermi level of the metal and the conduction band edge, are lifted by the same energy -ecp(O), so that their difference remains unchanged (Figure 6.12).

Between the metal (z 5 0) and the bulk region of the semiconductor at x = 00, a potential difference

UD = Cp(m) - Cp(0) (6.57)

evolves which corresponds to the diffusion voltage of a pn-junction. The

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6.3. Metal-semiconductor junctions

M

56 1

E, -elp( x 1

I c 0 X

Figu

I b

0 X

2 6.12: Positional dependence of the energy levels of the metal-semiconducta junction of Figure 6.11 after equilibrium is established. In cases (a) and (c) one has non-ohmic contacts, and in cases (b) and (d) ohmic contacts.

spatial uniformity of the electrochemical potential implies that

If the Fermi energy of the semiconductor lies close to the conduction band bottom, the diffusion voltage approximately equals the Schottky barrier QB. The latter typically has values of several tenths e V , which are also typical values that one should expect for the diffusion voltage UD.

The details of the spatial dependence of the potential p(z) depend on the relative positions of the Fermi levels in the metal and semiconductor. For an n-type semiconductor, if EFS > EFM holds, then electrons transfer from the semiconductor into the metal and an electron depletion layer develops in the semiconductor close to the interface. The diffusion voltage UD is positive,

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562 Chapter 6. Semiconductor junctions in thermodynamic equilibrium

i.e. the two bands are bent up at the interface (Figure 6.12a). One speaks of a non-ohmic or Schottky contact in this case. If, for a n-type semiconductor, the reverse ordering EFS < EFM holds, then electrons from the metal trans- fer into the semiconductor, and an electron accumulation layer develops in the semiconductor at the interface (Figure 6.12b). The diffusion voltage UD is negative, i.e., the bands are bent down at the interface, as in the case of the heterojunction of Figure 6.8, expect that here there is no development of a potential well in which electrons are confined to two dimensions. This is called an ohmic contact.

Consider next p-type semiconductors. For EFS > EFM there is a hole accumulation layer, because electrons transfer from the semiconductor into the metal or, what is the same, holes transfer from the metal into the semi- conductor. The diffusion voltage UD is positive and the bands are bent up at the interface (ohmic contact, see Figure 6.12d). For EFS < EFM, there is a hole depletion layer, UD is negative, and the bands are bent down (non-ohmic contact, see Figure 6.12~). Below we first consider metal-n-type semiconductor contacts which are non-ohmic.

Non-ohmic metal-n-type semiconductor contacts.

The potential cp(x) can be calculated in the same way as for a pn-junction. In particular, the Schottky approximation can, again, be used, provided that the conditions (6.35) are satisfied. We denote the width of the depletion layer in the semiconductor by XB. The arbitrary constant of the potential will be chosen such that cp(x) vanishes deep in the semiconductor.

cp(x) Iz=w= 0. (6.59)

The two boundary conditions for p(x) are taken to be the same as in the case of the pn-junction. Firstly, the potential difference cp(00) - cp(0) must match the diffusion voltage UD in accordance with (6.58). Secondly, the field strength deep in the semiconductor approaches zero, i.e.

- dcp 12=w= 0. (6.60) dx

With these conditions the potential cp(x) and the width XB of the depletion layer can be calculated from the Poisson equation, with the result

(6.62)

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6.3. Metal-semiconductor junctions 563

With a diffusion voltage of several tenths e V , the width X B of the space charge region takes values which correspond to those of a pn-junction for comparable doping concentrations. For N o = 1OI6 ~ r n - ~ , X B is several tenths of ,urn or less.

here in detail. Qualitatively, the conditions in the semiconductor are similar to those in the accumulation layer of a heterojunction considered in section 6.2. The potential energy - e v ( x ) of an electron in the semiconductor in- creases very rapidly, i.e. over a distance which is small compared to the Debye length, from its negative value -eUD at the interface up to a value close to zero in the semiconductor bulk (see Figure 6.12b).

The potential profile of a metal-p-type semiconductor junction is analo- gous to that of a metal-n-type semiconductor junction, as shown in Figures 6.12c,d.

The case of ohmic metal-n-type semiconductor contacts will not be treated

6.3.3 Schottky barrier

The importance of the Schottky barrier @B for the current-voltage charac- teristics of a metal-semiconductor junction was pointed out at the beginning of this section. Under the assumption (6.56) that excludes discontinuities of the potential at the interface, equation (6.55) is valid for @B. We refer to it as the Mott relation since Mott was the first to propose it. In Figure 6.13, Schottky barriers are shown for a series of n-type-Si-metal junctions as a function of the metal work function a. The electron affinity of silicon is 4.05 eV. The straight line in Figure 6.13 corresponds to the prediction of Mott’s relation (6.55). One readily sees that the linear variation of @B as a function of @ with slope 1, predicted by this relation, is not at all ful- filled. Rather, an approximate independence of the Schottky barrier and the metal work function can be recognized from the figure. One may con- clude from this that the assumption of continuity of the potential at the metal-semiconductor interface is not valid, just as in the case of a semicon- ductor heterojunction. Admitting a potential discontinuity to consideration, equation (6.55) is supplanted by the the generalized relation

@B = @ - X - e[cp(+O) - p(-O)]. (6.63)

It is important to recognize that a discontinuous potential jump occurs only if a dipole layer exists. Such a dipole layer at a semiconductor-metal in- terface can arise as a result of interface states, i.e. electron states of the semiconductor which are localized at the interface and have energy levels in the semiconductor gap. The existence of such states follows from the same considerations which led us to predict the occurrence of surface states in section 3.6.

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564 Chapter 6. Semiconductor junctions in thermodynamic equilibrium

Figure 6.13: Schottky barriers for various metal-n-Si junctions versus the work function of the metal (After Bechstedt and En- derlein, 1988.)

Work Function @ (eV)-

Below, we will assume that interface states are also present at the metal- semiconductor junction. We denote their energy levels with respect to the vacuum level by EA. The concentration of the semiconductor interface state levels will be taken to be sufficiently large to host all electrons which, in their absence, would otherwise occupy states at higher energies. We will show that under these circumstances, the energy EL of the interface states is also the electrochemical potential, i.e. the energy to which the common Fermi level of the metal-semiconductor junction adjusts in equilibrium The proof rests on the general relation (4.57) between electron concentration, Fermi distribution and density of states, which will be applied to the case under consideration. For simplicity, we consider zero temperature T , but the conclusions to be derived also hold for T > 0. The semiconductor will be taken as n-type, and the energy levels EL of the interface states lie below the Fermi energy of the bulk semiconductor. In reality, the interface levels do not concentrate exactly at Ei , but are spread over a narrow interval about EL. The interface state density is therefore not an exact &-function, but a bell curve of finite width.

Conceptually, we regard the process of establishing thermodynamic equi- librium at the metal-semiconductor junction as taking place in two steps, firstly, the establishment of equilibrium within the semiconductor, i.e. be- tween the semiconductor bulk and the interface at the metal, and secondly, the establishment of equilibrium across the interface between the semicon- ductor and metal. In the first step, electrons transfer from the semiconductor bulk into the interface levels. Thereby, these levels fill starting from below up to a certain maximum energy. By definition, this maximum energy rep-

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6.3. Metal-semiconductor junctions 565

resents the Fermi energy of the semiconductor. Since it lies within the bell curve of the state density it practically equals Ei. An increase or reduction of electron concentration in the semiconductor bulk by changing its doping will increase or reduce the number of electrons filling the interface levels. Because of the assumed large density of interface states, however, the posi- tion of the Fermi level needs to change only very little to account for such an increase or reduction. Thus, the Fermi level is fixed at the interface state energy levels. This is called a pinning of the Fermi level.

In the second step, the establishment of equilibrium across the interface between the metal and the semiconductor, electrons are transferred across the interface, strictly speaking between the thin charged boundary layer in the metal and the interface states of the semiconductor. Thereby, a dipole layer is formed at the interface. This is accompanied by a potential jump [cp(+O) - p(-O)] between semiconductor and metal, which in equilibrium is sufficiently large that the Fermi level of the metal is raised or, respectively, lowered, to match that of the semiconductor, whence

EFM - ecp(-O) = EL - ecp(+O). (6.64)

Combining relations (6.63) and (6.64), and using EFM = -@, we obtain @B = -E i - X . Referring the interface levels EL to the top of the valence band rather than to the vacuum level as done so far, the redefined value of the interface levels is E , = EL - E,. Using - X - E, = E,, these relations yield

@B = E, - E,. (6.65)

The latter expression for the Schottky barrier was first derived by Bardeen. In it, the Schottky barrier, in contrast to Mott’s expression, is seen to be independent of the work function of the metal. Figure 6.13 shows that the experimental data for Si are closer to Bardeen’s relation than to that of Mott. This statement also holds for other group-IV semiconductors, such as Ge. For the 111-V semiconductors, the dependence of the Schottky barrier @ B on @ is noticeable, although smaller than that in Mott’s relation, and for II- VI semiconductors it corresponds approximately to Mott’s linear dependence with slope 1. These observations suggest a relation for @B which interpolates between Mott’s and Bardeen’s formulas. Accordingly, we write

@B = S ( @ + E F S ) + ‘Po, (6.66)

where S is a slope parameter which varies between 0 and 1, @o is a constant, and EFS is the Fermi energy of the semiconductor with respect to thevacuum level. For n-type semiconductors, EFS approximately equals - X . The Mott relation follows from (6.66) if one sets S = 1 and @o = 0, and the Bardeen relation emerges if one takes S = 0 and = Eg - E,.

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566 Chapter 6. Semiconductor junctions in thermodynamic equilibrium

Figure 6.14: Dependence of the slope parameter S on electroneg- ativity difference (After Kurtin, McGill, and Mead, 1969.)

A t c------- 0 I zns ZnC I I

I 0.6 )CdS

0 0.4 0.8 1.2 1

1.6 -

2. I

Electronegativity difference - In accordance with above remarks about Schottky barriers for the various semiconductor material classes, the value of S should increase monotonically starting from the covalent group-IV semiconductors Si and Ge through the moderately polar 111-V semiconductors up to the strongly polar 11-VI semi- conductors. To examine this conjecture, S is plotted in Figure 6.14 as a function of the electronegativity difference between the two types of atoms paired in a given semiconductor, which represents a measure of the polar- ity of the semiconductor material. The results verify the expected behavior of S. The different properties of the interfaces of various semiconductors with metals are, therefore, reflected by S in an overall way. Interface effects are evidently more pronounced in the case of completely or preferentially covalently bound semiconductors than in the case of more ionically bound materials.

Experimental investigations show that Schottky barriers also depend on the chemical and structural imperfections of a particular interface. This means that, depending on the preparation of the interfaces, different Schott- ky barriers may be obtained for the same metal-semiconductor junction. Thus, for a theoretical understanding of Schottky barriers, it is particu- larly important to study absolutely pure and structurally perfect metal- semiconductor junctions, fabricated under UHV-conditions. From experi- mental investigations of such junctions as, for example, with GaAs, con- clusions have been drawn about the nature of the interface states assumed in Bardeen's model. In the unified defect-model by Spicer for 111-V semi- conductors, it is assumed that these states are due to point defects of the semiconductor like vacancies or anti-site defects being induced by positioning the metal on top of the semiconductor. In all, the microscopic understanding of semiconductor-metal junctions still needs further investigation.

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6.4. Insulator-semiconductor junctions 567

6.4 Insulator-semiconductor junctions

Interest in insulator-semiconductor interfaces is, on the one hand, of purely scientific origin - one wants to understand the particular macroscopic and microscopic properties of these junctions. On the other hand, insulator- semiconductor interfaces are of great importance for electronic semiconduc- tor devices. In such devices, they are either used as layers for isolation or passivation purposes, or they are involved in active electronic functions, such as tuning the resistance of a lower-lying semiconductor layer, as occurs in the case of the Si-MOSFET (Metal-Oxide-Field-Effect-Transistor). Without the latter device, modern highly integrated microelectronics would not be con- ceivable. Therefore, the insulator-semiconductor junction, particularly that between SiOz and Si, represents one of the technological most important solid state systems of all. Here, we will discuss the properties of insulator- semiconductor junctions in thermodynamic equilibrium. For MOSFET op- eration, the changes of these properties due to the application of a voltage between the insulator and the semiconductor are important, and these will be treated in section 7.3.

6.4.1 Thermodynamic equilibrium

Just as insulators differ from semiconductors only by the size of the energy gap, i.e. quantitatively, the differences between semiconductor heterojunc- tions and insulator-semiconductor junctions are mainly of a quantitative nature. A certain qualitative difference exists in the occurrence of interface states which is more likely for insulator-semiconductor junctions than for semiconductor- heterojunctions. Initially, we will omit interface states from consideration. The energy diagram of an insulator-semiconductor junction at the instant of its inception, i.e. before establishing equilibrium, is shown in Figure 6.15. The depicted situation, in which the energy gap of the semi- conductor lies completely within the gap of the insulator, is not the only possible one, but it is typical. The SiOz/Si-junction is of this type. The relative positions of the valence bands of the insulator and semiconductor, and the relative positions of their conduction bands, may be characterized in the same way as was done in the case of heterojunctions. Again, the band discontinuities cannot be determined by simply comparing the bulk values of the band edges, since they are modified by the discontinuity of the electrostatic potential induced by the dipole layer at the interface.

Immediately after fabrication of the junction between an insulator and a semiconductor, the chemical potentials of the two materials have different positions. Equilibrium adjustments involve the exchange of electrons and holes across the interface. This leads to a space charge region and an elec- trostatic potential p(r) which equilibrates the inhomogeneity of the chemical

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568 Chapter 6. Semiconductor junctions in thermodynamic equilibrium

Figure 6.15: Lineup of the band edges and the Fermi levels at an insulator- semiconductor junction (a) before and (b) after equilibrium is established.

potential, such that the electrochemical potential EF = p(z) - ecp(z) is spa- tially uniform in the final equilibrium state.

We explore the positional dependence of the electrostatic potential p(z) below. To be specific, we assume a n-type semiconductor, and suppose that the free carriers in the insulator are predominantly holes. Then we have a pn-junction which differs, however, from that considered in section 5.1 be- cause it is composed of two different materials. Another particular feature of this pn-junction is that the acceptor concentration NAI of the insula- tor is smaller by several orders of magnitude than the donor concentration Nos of the semiconductor. It is natural to assume that the Fermi level EFS of the semiconductor lies above the Fermi level EFI of the insulator. Then, in order to establish equilibrium, electrons have to transfer from the semiconductor into the insulator. Consequently, the semiconductor becomes positively charged and the insulator is negatively charged.

The Poisson equation for the concomitant potential cp(z) has the general form (6.21). The material inhomogeneity of the junction enters this equation through the relation between the electron concentration TI(.) and the poten- tial cp(z) as well as through the static dielectric constant which has different values €1 and c in, respectively, the insulator and the semiconductor. If n(z ) does not appear explicitly, as occurs within the Schottky approximation, the material inhomogeneity simply results in the replacement of ( N A / E ) by ( N A / E I ) , and ( N o / € ) remains ( N o / € ) .

Below, we employ the Schottky approximation and later verify its justi- fication. The normalization and boundary conditions are again chosen ac- cording to relations (6.9) and, respectively, (6.22) and (6.23). The continuity of electric field strength - ( d p / d x ) at x = 0 is replaced by the continuity of

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6.4. Insulator-semiconductor junctions 569

the electric displacement - e ( z ) ( d ~ / d z ) at this position. The solution of the Poisson equation is then given by the previously derived expression (6.29). Relations (6.35) and (6.36), which determine the validity limits of this solu- tion also retain their applicability. In the case considered here, N M << N D S holds. Then, as equation (6.31) shows, the width x p ~ of the space charge region in the insulator is considerably larger than the width 2,s of the cor- responding region in the semiconductor. The space charge region in the insulator can become so large that it exceeds the sample size. If this oc- curs, the jump discontinuity of the chemical potential cannot be completely screened out by the electrical potential in this circumstance. Hence, the model of a junction of semi-infinite media, which underlies our calculations, is no longer valid. The general conclusions which we derive here are not affected by this fact. From formula (6.33) one finds that practically the en- tire diffusion voltage drops off within the insulator. Thus the electric field strength in the semiconductor is well approximated by zero. As far as the common Fermi level EF is concerned, equation (6.38) shows that it adjusts approximately to the Fermi level of the semiconductor, whence

EF = EFS. (6.67)

The application of the Schottky approximation still has to be justified. This may be done by means of relations (6.35) which yield the following results. For the p-region, i.e. the insulator, the Schottky approximation holds if f i << m, which we can assume. For the n-region, i.e. the semicon- ductor, the Schottky approximation can indeed become problematic because the factor J N A / ( N A + N o ) is small; however, it yields qualitatively correct results in this case too.

Summing up and generalizing the above findings on the equilibrium state of a insulator-semiconductor junction one can say that a relatively small number of free carriers, transferred from the semiconductor to the insula- tor, is sufficient to adjust the Fermi level of the insulator to that of the semiconductor. This number is so small, viewed from the point of view of the semiconductor, that the state of the semiconductor is almost completely unaffected by the insulator. Moreover, because of the small space charge density and the practically vanishing electric field in the semiconductor, the field strength must also be small in the insulator - recall that the dielec- tric displacement penetrates continuously into the insulator. Below, we will neglect the electric field in the insulator completely.

In the above treatment of the insulator-semiconductor junction, interface states were omitted from consideration. In reality, such states may exist in the semiconductor. Their effect on the electrostatic potential of the junction will be explored now.

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5 70 Chapter 6. Semiconductor junctions in thermodynamic equilibrium

6.4.2 Influence of interface states

As in the case of the metal-semiconductor junction above, the interface states cause the chemical potential of the semiconductor to take a different posi- tion at the insulator-semiconductor interface than it has in the bulk. We denote these positions by, respectively, Eg and Ek. In the case of an n-type semiconductor and acceptor-like interface states which can host electrons, a partially compensated semiconductor layer will arise at the interface. There, electrons are captured by the acceptor levels, so that in the semiconductor a negative interface charge density -ens (per unit area) is formed. This comes about at the expense of the semiconductor bulk, where a positively charged depletion region arises below the interface. The width xn of this depletion region is determined by the conservation of electron number, which may be expressed as

ns = NDSX,. (6 .68)

According to electrostatic laws, the interface charge density results in a jump discontinuity [F(+O) - F(-O)] of the electric field strength given by -(e/tto)n”. Since the field in the insulator vanishes, a non-zero field strength F S arises on the semiconductor side of the interface, as given by

(6.69)

One can also understand this interface field as being due to the positive space charge region in the semiconductor below the interface, which in Schottky approximation gives rise to a field strength F ( x ) that grows linearly towards the interface, and a potential p ( x ) which decays quadratically. The value of the potential at the interface,

d o ) = us, (6.70)

must have just the right magnitude so that the electrochemical potential is uniform, i.e. that the difference [Ek - Ek] between the chemical potentials at the interface and in the bulk of the semiconductor is equilibrated. Hence,

- eUS = [Ek - E$] (6.71)

must hold. Solving the Poisson equation in Schottky approximation subject to the boundary conditions that the potential and the field strength vanish deep in the semiconductor at x = m, and that the potential at the interface is given by (6.71), we obtain

(6.72)

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6.4. Insulat or-semiconduc tor junctions 571

with

x,= /s. eND (6.73)

Combining equations (6.68) and (6.73) yields a relation between the interface charge density ns and the interface potential U s , which reads

For the interface field F S , it follows that

(6.74)

(6.75)

We use the latter two relations to estimate the orders of magnitude of ns and Fs, with the resulting relations

ns = 1.05 x 103cm-2[(d/~/cm-3)(1 Us I /V)]1/2, (6.76)

(6.77)

An estimate of about lev for e I Us I follows from (6.71) if we assume the bulk Fermi level to be close the conduction band bottom, and the surface Fermi level to be deep in the energy gap. cm-3 and 6 = 10 we find ns = 3 x 10l1 cmA2 and F S = 6 x lo4 V/cm. The width zn of the depletion region is 0.3 pm. Thus, it is to be expected that the fields and interface charge densities are of considerable size at an insulator- semiconductor interface.

Our assumption that the Fermi level at the interface lies deep in the energy gap is justified only if the sheet concentration of interface states has at least the same order of magnitude as the calculated electron concentration ns, i.e. the order of magnitude lo1' cm-'. This may be valid in some cases, but in others it is definitely not. An example of the latter case is the interface between SiOa and Si, which is largely free of interface states. As we will see in Chapter 7, this is decisive for the operation of the MOSFETs, - indeed, it is downright necessary to make this device possible. A large interface charge density means, as we know from section 6.3, that the Fermi level is pinned at the energy of these states, while MOSFET operation depends upon having the capability to tune the Fermi level by means of an external voltage. The interface of GaAs with its own oxide or with foreign insulators, behaves otherwise: The interface state density is large, hence MOSFETs made of GaAs are difficult to achieve. In the case of GaAs one must turn to the MESFET to fabricate a field effect transistor.

1 F S I = 18.97 x 10-4(V/cm)[(~-1Ng/cm-3)(1 U s I /V)]1/2.

With ND =

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5 72 Chapter 6. Semiconductor junctions in thermodynamic equilibrium

Figure 6.16: Lineup of the band edges of an insulator-semiconductor junction in equilibrium in the pres- ence of interface states.

6.4.3 Semiconductor surfaces

Everything discussed above pertaining to insulator-semiconductor junctions also holds analogously for free semiconductor surfaces. In this regard, the insulator is replaced, so to speak, by vacuum, which may be thought of as a special insulator with an infinitely large energy gap and (relative) static dielectric constant 6 of unity. The positional dependence of the energy levels after equilibrium is established is essentially given by Figure 6.16, except that the potential curve on the insulator side is now exactly zero, since no free carriers exist in vacuum.

Instead of interface states, surface states can occur, as is known from section 3.6. These have the same effect that interface states have in the case of the insulator-semiconductor junction. They result in the formation of a surface charge density ns , which is accompanied by a surface field F S and a surface potential U s . For these quantities, equations (6.71), (6.74) and (6.75) hold without change. Therefore, at free semiconductor surfaces, sheet charges and electric fields of considerable size occur, just as in the case of an insulator-semiconductor junction. These are utilized in surface photoeffects, wherein optically excited free carriers screen the surface field, partially or completely, depending on the intensity of the absorbed light. Photoreflectance is a kind of electroreflectance (see section 3.8) in which this effect is used to modulate the surface electric field of a semiconductor by means of light.

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573

Chapter 7

Semiconductor junctions under non-equilibrium conditions

The treatment of semiconductor junctions above was limited to conditions of thermodynamic equilibrium. In this chapter, non-equilibrium states of these junctions will be explored. Such states occur when the junctions are subjected to various external perturbations, which can include the applica- tion of an electric voltage, irradiation with light, exposure to pressure, or the introduction of a temperature gradient. Here, we limit our considerations to the application of a voltage and irradiation with light, because they are by far the most import ant perturb at ions generating non-equili br ium conditions in semiconductor junctions. In sections 7.1 and 7.2 we treat the pn-junction subject to an external voltage. The interaction with light is considered in section 7.2. The influence of a voltage on a metal-semiconductor junction forms the subject of section 7.3, and in section 7.4 we examine voltage effects on an insulator-semiconductor junction.

In the introduction to Chapter 6, we pointed out that perturbation of the equilibrium state of a semiconductor junction provides the basic mechanism of operation for most semiconductor devices. Examination of the effects of an applied voltage and interaction with light for such junctions therefore con- stitutes an exploration of the physical principles of a number of such devices. In section 7.1, the devices involved are rectifying semiconductor diodes, the bipolar transistor and tunnel diode; in section 7.2 the photodiode, the so- lar cell and the semiconductor injection laser. Section 7.3 deals with the metal-semiconductor rectifier, and section 7.4 with the field effect transis- tor. In this preview of the content of the present chapter, it is worthwhile to note that we have omitted perturbation of the equilibrium of semicon- ductor heterojunctions. There are two reasons for this omission. The first

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5 74 Chapter 7. Semiconductor junctions under non-equilibrium conditions

concerns conventional semiconductor devices. Although heterojunctions are used in many of them, in none do they play a decisive role for the operation of the device. They mainly just improve its operation as, for example, in the case of an injection laser using a quantum well instead of a homogeneous material region as active layer. The second reason concerns devices whose principle of operation actually relies on heterojunctions. As an example, we recall the High Electron Mobility Transistor (HEMT) mentioned in Chap- ter 6 which utilizes the %dimensional electron gas of the quantum well at a GaAs/(Al,Ga)As interface. While the (A1,Ga)As alloy region is n-doped to provide the electrons for the 2-dimensional gas, the GaAs well region itself is undoped to avoid carrier scattering by ionized donors which would lower the mobility. The detailed treatment of such quantum devices which, with a few exceptions (the HEMT among them), are currently subject to research would exceed the framework of this book as an introduction to the principles of the semiconductor physics and devices. Therefore, we omit them here. Readers interested in quantum devices will find an overview in a recently published monograph (Kelly, 1995).

Restricting external perturbations to the application of a voltage and irradiation by light means that devices which are sensitive to physical influ- ences other than electricity and light, like pressure, temperature, chemical concentrations or magnetic fields, are ruled out. Such devices can convert non-electric signals into electrical signals. They are called semiconductor sensors. One of them, the photodetector, which involves the conversion of light, will be treated in section 7.2.

7.1 pn-junction in an external voltage

The application of an external voltage ( s e e Figure 7.1) causes the free charge carriers of the pn-junction, which originally had been in global thermody- namic equilibrium, to be driven into a global non-equilibrium state. To start, it is important to clarify how this state can be adequately described. It turns out that the understanding of non-equilibrium states used above, which rests on the concept of local equilibrium and a spatially varying chem- ical potential p(x) , must be modified in an important respect if applied to biased pn-junctions.

Local equilibrium means, in particular, that a local chemical potential p ( x ) exists for the entire system of free carriers, including both electrons and holes. That this can no longer hold when a voltage is applied to the pn- junction is easily understood. Indeed, on that side of the junction where the potential energy of electrons is raised by the applied voltage, the electron concentration will decrease compared to the unbiased case. On the same side, the potential energy of the holes is necessarily lowered and their con-

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7.1. pn-junction in an external voltage 575

Figure 7.1: pn-junction under an applied voltage.

centration is raised in the presence of the voltage. In view of the relations ( 6 . 5 ) between p ( 2 ) and the carrier concentrations n(z ) , p ( z ) , this means that, on the above considered side of the pn-junction, the chemical potential of the electrons must be lower, and that of the holes must be higher than for the unbiased junction. In any case, the chemical potential pn of elec- trons differs from the chemical potential pP of holes provided - and this is an essential restriction - these potentials exist at all.

In order to clarify whether they do exist or not, we recall that the chem- ical potential is defined as a parameter of an equilibrium distribution func- tion. By assuming different chemical potentials for electrons and holes, the energy distributions of these carriers within their respective bands are sup- posed to be in equilibrium. Only the distribution of carriers between the bands does not correspond to an equilibrium state if the chemical poten- tials are different. This assumption is justified if the processes which tend to establish equilibrium proceed much faster among electrons in the same band than among electrons in different bands. In the first case one has en- ergy relaxation of the carriers in their respective bands. The characteristic energy relaxation time is typically of the order of magnitude lo-'' s. In the second case one has capture or recombination with a characteristic time typically of the order of magnitude lo-' s, i.e. 100 times larger than the energy relaxation time. In such circumstances it is reasonable to assume that the two types of carriers are in equilibrium with respect to the energy distribution within their respective bands, even if no equilibrium exists with respect to their distribution between these bands. The assumption of sepa- rate chemical potentials for electrons and holes is also justified under these conditions.

At a pn-junction, both chemical potentials are functions p n ( 2 ) and pP(;c) of position. These functions must be determined separately in conjunction with the electric potential cp(z). We first solve for cp(z), and later for the two chemical potentials. The concentrations n(z) of electrons and p(z) of holes needed in these calculations are, by definition of the separate local chemical potentials, given by the relations

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5 76 Chapter 7. Semiconductor junctions under non-equilibrium conditions

7.1.1 Electrostatic potential profile

Formally, the presence of an external voltage U manifests itself in a change of the potential drop across the pn-junction, which in equilibrium was the diffusion voltage U g , to the new value

Here, the external voltage U , as usual, is counted as positive if it leads to a decrease of the potential energy of a positive charge carrier at z = +m as compared to x = -m. That the diffusion voltage does not contribute to the voltage balance in a closed circuit containing the pn-junction was already explained in section 6.1.

The calculation of the electrostatic potential p(z) at an unbiased p n - junction was based upon the Poisson equation (5.19) and the Schottky ap- proximation, which was used to eliminate the free charge carrier concen- trations n ( z ) and p ( z ) from this equation. If one also assumes complete ionization of the donors and acceptors in the space charge region, as done in section 6.1, then the chemical potential does not explicitly appear in the Poisson equation. It is only through the boundary conditions, strictly speaking through the diffusion voltage U g , that the electrostatic potential function depends on chemical potential in the p- and n-regions. Whether the Schottky approximation retains validity or not in the presence of an ex- ternal voltage must indeed be examined. If one repeats the reasoning used to justify this approximation in section 6.1 for the unbiased case, then it is readily seen that there are no changes, except that the diffusion voltage UD has to be replaced by U D - U . From this we may conclude that the Schottky approximation in the presence of an external voltage is applicable provided

holds and the doping concentrations N A and N D are of comparable size. For positive U equation (7.4) is not always satisfied. It becomes critical when U approaches the diffusion voltage, and is definitely invalid when U > UD. For negative external voltages, condition (7.4) may in fact be valid even if it does not hold in the absence of an external voltage. If it is valid in the unbiased case, then it is even better satisfied in the presence of a negative voltage. In the following we will assume that condition (7.4) is fulfilled and that the doping concentrations NA, N D are comparable. Then the Schottky

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7.1. pn-junction in an external voltage 577

junction on applied voltage U , measured in units of U D . Sym- metric doping with N A = N D =

10l6 ~ r n - ~ is assumed. The static dielectric constant is that of Si ( E = 12.1).

-$,,

1.0

95

-

-

approximation is applicable, as are the concepts of the depletion and bulk regions which were introduced in the context of an unbiased pn-junction. In conjunction with this, all the results of section 6.1 remain valid in regard to the potential function cp(z), except that UD must be replaced everywhere by UD - U . For the potential values at the boundaries of the space charge region, we have

and the width w of the space charge region is given by

In the presence of a negative voltage of 100 V , doping concentration N D = N A = 1OI6 ~ r n - ~ (using silicon values for E and U D ) the space charge width w is about 4 p a . For U = UD, w formally takes the value zero, i.e., no depletion region exists at all. This signifies that the Schottky approximation breaks down (see Figure 7.2).

The calculation of the two chemical potentials p"(z) and pp(x) for elec- trons and holes, respectively, is more complicated than that of the electric potential. It requires simplifications which are only understandable if one already has a qualitative understanding of the nature of current transport through the pn-junction. We will now address this point.

7.1.2 Mechanism of current transport through a pn-junction

That an applied voltage causes current flow through a pn-junction, is obvi- ous. Less evident, however, is the manner in which it happens. Far from the nominal transition at z = 0, one may assume that current transport does not

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5 78

0 0 0

0 0 0 0

0 0 0 0

0 0 0

Chapter 7. Semiconductor junctions under non-equilibrium conditions

0 . .

0 0 0 0

0 . 0 0

0 0 0

a1 u - 0

0 0 0 9 0 . 0 4 0 0 0-0

Cl u>o

.to 0 .c

0 0 0 -0 0 0-0 .to 0 o c

. . . . ( ~ + ~ o o o o ( . . . . + l o o o 0 / 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Figure 7.3: Mechanism of current flow through a pn-junction, a) pn-junction with- out voltage, b) Injection of minority carriers by applying a (positive) voltage, c) Recombination of the injected minority carriers with available majority carriers, d) Refilling of the emptied band states by majority carriers from the bulk region. There a majority charge carrier current flows.

differ from that in an infinite semiconductor sample, i.e. the current in the p-region will be carried mainly by holes, and that in the n-region mainly by electrons. In this matter, one has the peculiar situation that a hole current in the left part of the junction passes over into an electron current in the right part of the junction. The mechanisms which can realize such a tran- sition are the recombination and generation of electron-hole pairs, whence we may conclude that the latter processes should play an important role in current flow through the pn-junction.

In our further considerations we assume CJ > 0. In order for recombina- tion to occur, non-equilibrium carriers must be available. These appear in the p-region in consequence of the fact that electrons move over from the n- region to the p-region because their potential energy in the n-region is lifted under the applied voltage by the amount eU. Therefore they overcome the potential barrier between the n- and p-regions more easily. This transfer of carriers across a potential barrier is called injection In the p-region the electrons are minority charge carriers. Therefore we have an injection of mi- nority charge carriers into thep-region. An analogous process takes place on the n-side of the junction. There, holes from the p-region move over into the n-region, so that one has an injection of minority holes. These relationships are illustrated in Figure 7.3. The injected minority carriers (Figure 7.3b) recombine with the already present majority carriers (Figure 7.3~) . The

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7.1. pn-junction in an external voltage 5 79

states in the bands which become unoccupied in this way will be filled by majority carriers from the bulk regions of the junction (Figure 7.3d), where majority carrier currents therefore flow. The magnitudes of these currents are determined by the speeds of the injection and recombination processes. In the stationary state both speeds must be equal - all carriers which are injected must recombine, and all carriers which recombine must have been previously injected.

If one reverses the sign of the voltage, corresponding to U < 0, then one has extraction of minority charge carriers instead of injection, and generation instead of recombination. Since extraction and generation processes consume energy, in contrast to injection and recombination wherein energy is released, one must expect the current through the pn-junction to be much smaller for U < 0 than for U > 0. This is in fact the case, as we will formally prove below.

The above mechanism for current transport through a pn-junction will now be formulated quantitatively. The total current density j (x) consists, according to formula (5.85)) of the electron current density jn(x) and the hole current density jp(x). For the two current constituents the continuity equations (5.20) and (5.21) hold. In the present case, the generation term is zero, and the annihilation term is determined by recombination. In the stationary state one obtains

djp - = -eR(x). dx

where R ( z ) represents the recombination rate given by equation (5.61). Adding (7.7) and (7.8), it follows that the total current density j (x) is free of sources and must be spatially constant, so that

j (x) = j n ( 2 ) + jp(x) zz j = const. (7.9)

We consider a particular position X I in the p-region and another particular position x2 in the n-region. The fact that the total current density j is const ant yields

Integration of equation (7.7) provides the result

(7.10)

(7.11)

Combining (7.10) and (7.11)) we have

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580 Chapter 7. Semiconductor junctions under non-equilibrium conditions

Setting

(a) 21 = -oo,z2 = +m, we find

(7.12)

(7.13)

On the other hand, considering

(b) z1 = xp,z2 = zn, it follows that

X,

j = j n ( 4 +jp(zn) + e / d 4 z ) . (7.14)

We will later see that, under certain conditions, the two minority charge carrier contributions jn(-oo) and jp(+oo) in (7.13) can be approximately neglected. The same holds for the recombination contribution in (7.14). Consequently, equation (7.13) means that the total current, in its essence, represents a recombination current. Alternatively, in formula (7.14) the total current is seen to be the sum of the minority carrier currents at the two space charge boundaries. There, they are determined by the injection of minority carriers. The total current represents, therefore, an injection current. The two interpretations are equivalent, but they emphasize different aspects of the total current.

With these considerations concerning the mechanism of the current trans- port, we are sufficiently prepared to calculate the spatial profiles of the chem- ical potentials of the two types of carriers.

XP

7.1.3

We first calculate the charge carrier concentrations n ( z ) and p ( z ) . Once they are known, the chemical potentials follow immediately from relations (7.1) and (7.2). Since the current is due to recombination of injected non- equilibrium minority carriers in the bulk regions, we will restrict our con- siderations to these carrier concentrations in particular. The space charge region will therefore be omitted initially. Moreover, we will use the fact that the relative change of the majority carrier concentrations caused by injection is substantially smaller than the corresponding relative change of the minor- ity carrier concentrations. For the majority carrier concentrations in the bulk regions, the values without injection, i.e. without an applied voltage, can be used. This means that

Chemical potential profiles for electrons and holes

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7.1. pn-junction in an external voltage 581

n(x) = n, for 2 > x,, (7.15)

p ( z ) = p p for x < xp. (7.16)

The minority carrier concentrations in the bulk regions are calculated from the combined diffusion-recombination equations (5.81) and (5.87). To solve these equations uniquely, boundary conditions are required, in the case of electrons at -m and xp, and in the case of holes at xn and +oo. For n(-m) and p(00) the equilibrium values

apply. The concentrations at the boundaries of the space charge region may be taken from relations (6.15) for n ( z ) and p ( z ) . Of course, the latter expressions were written down for equilibrium conditions, but they provide approximately correct values for n(zp) and p(x,) even when a voltage is applied. The only change to be made is the replacement of [cp(zn) - (p(zp)]

by U D - U instead of by UD. It follows that

(7.18) -e(UD-u)/kT = eU/kT n(q,) = n,e npe ,

(7.19)

For U = 0, the values n ( x p ) and p(;cn) are the minority carrier concen- trations np and p , in equilibrium. If U > 0, the factor multiplying np in (7.18), and the factor multiplying p , in (7.19) is larger than 1, i.e., the minority carrier concentrations exceed the values they would have in the absence of an external voltage. This is the formal expression of the injec- tion of minority charge carriers. How effective injection is can be recognized through the following estimate. With U = 0.25 V and T = 300 K we obtain exp(eU/kT) M elo M 2 x lo4. This is to say that the small voltage of 0.25 V suffices to increase the minority carrier concentrations by more than 10000.

The steady state solutions of the diffusion-annihilation equations (5.90) for n(x) and (5.97) for p ( z ) , under the respective boundary conditions (7.15), (7.16) and (7.17), are given by

P(X) = P , + [~(z,) - P,I~-( x-x*) lLp , > ",, (7.21)

where L , and L, are the diffusion lengths of, respectively, electrons and holes. For positive external voltages one has n ( x p ) > n, and p(x,) > p,.

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582 Chapter 7. Semiconductor junctions under non-equilibrium conditions

Figure 7.4: Lineup of the quasi Fermi levels of a pn-junction under an applied voltage, (a) flow direction (U > 0) , (b) blocking direction (U < 0)). Dashed curves correspond to the unbiased pn-junction. In the space charge region interpolated values are used. The decay of the quasi Fermi levels in the two bulk regions is drawn greatly exaggerated.

This means that in the two bulk regions, both minority carrier concentrations are larger than their respective equilibrium values np and p,. The chemical potential p n ( z ) of the electrons in the pregion, is therefore shifted to higher energies with respect to its equilibrium value, and that of the holes in the n-region is shifted to lower values. The same holds for the pertinent non- equilibrium electrochemical potentials

E F ( ~ ) = p n ( x ) - ecp(x), E ; ( X ) = p P ( z ) - ecpG), (7.22)

which are also referred to as quasi Fermi levels. In Figure 7.4a the spatial variation of the two quasi Fermi levels is shown schematically, together with the valence and conduction band profiles.

For a negative voltage U < 0 one has n(zp ) < np and p ( z n ) < p,. The chemical potential of electrons in the p-region therefore lies below the equilibrium value, and that of the holes in the n-region lies above it. The same is true again for the quasi Fermi levels (see Figure 7.4b). The elevation or depression of these levels in the bulk regions is effective up to a distance from the depletion region which roughly equals the diffusion length of the pertinent minority carrier. In the depletion region between the two bulk regions, we cannot make such statements, or any others, because the above consideration excludes this region. Fortunately, the diffusion lengths are, as a rule, an order of magnitude larger than the width of the space charge

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7.1. pn-junction in an external voltage 583

region (see section 5.4), so that this lack of knowledge is not important. The space charge regions function solely as potential barriers over which non- equilibrium carriers are injected. The spatial expansion of the barriers can be neglected in a first approximation.

7.1.4

We can now carry out the calculation of the total current density through a pn-junction without difficulty, starting from the relations derived for the recombination current density (expression 7.13, method a) and for the injec- tion current density (expression 7.14, method b). Both methods must lead to the same result. Formally, it would therefore suffice to consider only one of them. We will study both to demonstrate explicitly that the recombination current (7.13) and the injection current (7.14) are in fact identical.

Dependence of current density on voltage

Method a

From the qualitative discussion of current flow through the pn-junction in subsection 7.1.2, we know that the currents at -co and +co do not differ from the currents in an infinite p or n-type semiconductor. This means, in particular, that the minority carrier currents at -00 and +co are neg- ligibly small. In regard to the remaining integral in (7.13), recombination processes in the bulk regions to the left and right of the depletion region contribute significantly only up to depths which roughly equal the diffusion lengths of the minority carriers - only there do the concentrations of these carriers differ substantially from their equilibrium values, so it is only there that recombination occurs. The diffusion lengths are, as has already been mentioned, generally much larger than the width of the depletion region. Taking advantage of this magnitude relation, we neglect the contribution of the depletion region to the integral in equation (7.13), whence we obtain, approximately,

(7.23)

The recombination rate R is given by relations derived above in section 5.2. According to formula (5.64), in the presence of a small excess concentration An(x) of electrons (here, of electrons in the p-region), we have

(7.24)

and according to formula (5.66), in the presence of a small excess concentra- tion Ap(z ) of holes (here, such in the n-region), the corresponding relation is

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584 Chapter 7. Semiconductor junctions under non-equilibrium conditions

(7.25)

with rn and rp given by (5.65) and (5.67) as minority carrier lifetimes of electrons and holes, respectively. For An(.) and A p ( x ) we employ relations (7.20) and (7.21) in the form

~ p ( x ) = b ( x n ) - pnle-(z-xJ'Lp, x n < 2 . (7.27)

Substituting these formulas in equations (7.24), (7.25) and (7.23) and using relations (7.18) for n ( x p ) and (7.19) for p(xn ) , we obtain

1) (7.28) eUlkT - j = j s ( e

with

(7.29)

The same result is obtained if one proceeds in accordance with method b, which will be verified below.

Method b

In equation (7.14) for j we again neglect the recombination integral over the depletion region. The minority carrier current densities jn(zp) and j p ( x n ) at the boundaries of the depletion region follow from the general phenomeno- logical equations (5.100) and (5.101), wherein the electric field strength has to be set zero since the electric potential is constant in the bulk regions. The charge carrier concentrations n ( x ) and p ( x ) are taken from expressions (7.20) and (7.21), yielding

(7.31)

Adding these equations and applying relations (7.18) and (7.19), the expres- sion (7.28) follows for j with

(7.32)

The two expressions (7.32) and (7.29) for j , are identical, since Ln = (see equation 5.95) and L, = fi (see equation 5.99).

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7.1. pn-junction in an external voltage 585

Figure 7.5: Current-voltage char- acteristic of a pn-diode made of Ge. Note the different voltage scales for forward and reverse biasing. For ex- tremely large negative voltages, the diode undergoes electrical break- down. (After Seeger, 1973.)

With this observation, the task of calculating the current through a pn- junction under an applied voltage is completely solved, within the framework of the conditions and approximations set forth above. We will now discuss the results. The current-voltage characteristic (7.28) is extremely non-linear. It exhibits the expected asymmetry with respect to a change of the sign of voltage U . For a positive U of a few tenths of a Volt, j is several orders of magnitude larger than j,, while for a negative U of same absolute value, j approaches - j , (see Figure 7.5). The current density j,, which cannot be surpassed at even larger negative voltages, is called the saturation current density. To estimate the size of this current density in the case of Si, we assume typical values for L,,p of 10 pm and for 7n,p of lo-' s. For the minority carrier concentrations, we obtain, from np = n?/NA and p, = n!/ND with ni = lo1' cm-3 and N A = N D = 1OI6 cmV3, the values np = p, = lo4 c ~ L - ~ . Using e = 1.6 x 10-lgA s, it follows that j, 10-l' A/cmz. The saturation current density is therefore extremely small. Thus, for U < 0 practically no current flows, the pn-junction blocks the current flow. One says that it is reverse biased or biased in blocking direction, The biasing U > 0 refers to the forward bias or Bow direction because the current density in this direction is orders of magnitude larger than j, as we have seen above. The pn-junction operates as an electrical rectifier. It is called a rectifyingpn- diode in this context. In Figure 7.5 a measured current-voltage characteristic of a pn-diode made of Ge is shown.

7.1.5 Bipolar transistor

The pn-junction has important application in the bipolar transistor. This device consists either of two n-regions, which are separated by a p-region, or of two p-regions separated by an n-region (see Figure 7.6). In the first case one speaks of an npn-transistor, and in the second case of a pnp-transistor.

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586 Chapter 7. Semiconductor junctions under non-equilibrium conditions

Figure 7.6: Bipolar npn- and pnp-transistor.

m Emitter Base Collector

Figure 7.7: npn-transistor in the common-emitter configuration (left-hand side). Illustration of the current flow (right-hand side).

In the following considerations we confine our attention to the npn-case. In Figure 7.7 the npn-transistor is shown in one of the possible switching modes, called common-emitter configuration (for reasons explained below). The left n-region is connected both with the p-region in the middle as well as with the n-region on the right. The voltage source of the np-circuit puts the left n-region at a potential P E , and the p-region at a potential p ~ . The voltage source in the npn-circuit puts the potential of the right n-region at (pc. We consider the case in which

PE PB (PCI

and, accordingly,

(7.33)

(7.34)

holds. In this case the left pn-junction is biased in the flow direction, and the right in blocking direction (see Figure 7.8). From the left n-region, electrons are injected into the p-region in the middle, while the right n-region extracts electrons from the p-region. One therefore calls the left n-region the emitter, and the right the collector. The p-region in the middle is called the base. Accordingly, the np-circuit will be referred to as the emitter-base circuit or, in short, the base circuit and the npn-circuit as the emitter-collector circuit,

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7.1. pn-junction in an external voltage 587

or in short, the collector circuit. Let the current in the base circuit be ig, and that in the collector circuit ic. Then the current i E through the emitter follows from Kirchhoff’s current branching theorem as the sum of the two currents,

i~ = iB + ic, (7.35)

One can also say that the emitter current splits in two partial currents, one flowing through the base, and one flowing through the collector (see Figure 7.7 on the right).

Our goal is the calculation of the three currents i ~ , iB, ic. In this matter, we can employ the results obtained above for the current flow through an individual pn-junction, with the valence and conduction band edges of the npn-transistor lined up as shown in Figure 7.8. The pn-junction on the emitter side is subject to the flow voltage U g , and the pn-junction on the collector side is subject to the blocking voltage U c - UB. The emitter current i~ is the current flowing through the emitter-side pn-junction. As such it is given by expression (7.12), multiplied by the emitter area A . Accordingly, it consists of the injection current j n ( z p ) A of the minority electrons into the

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588 Chapter 7. Semiconductor junctions under non-equilibrium conditions

base, the injection current jp(x,)A of the minority holes into the emitter, and the recombination current in the depletion region. Neglecting the latter contribution as before, and using relations (7.30) and (7.31), we have

Here the signs are opposite to those of relations (7.30) and (7.31) above, due to the fact that the emitter-base junction has thep-region on the right-hand side and the n-region on the left, whereas it was the opposite above. If both regions were expanded infinitely, as we always assumed above in the treatment of the pn-junction, then the minority charge carrier concentra- tions n(x) and p(x) in (7.37) could be replaced by the previously derived expressions (7.20) and (7.21). This procedure would result in expressions for j,(xp) and j p ( z n ) which are just the negatives of those in relations (7.30) and (7.31). In the transistor, however, only the emitter can still be consid- ered to be infinitely extended, while the width b of the base must be treated as finite because it is not large in comparison with the diffusion length L, of the minority charge carriers. Thus, only the injection current density jp(x,) of holes may be taken from the previously derived expression (7.31). Ad- justing this expression to the relationships at the emitter-base junction, we find along with (7.27),

(7.38)

The injection current density jn(zp) of electrons, however, must be calculated anew. This can be done on the basis of the following considerations. To start, it is clear that, because of the finite width of the base, only part of the injected minority electrons recombine in the p-region, while the remainder diffuse through this region and reach the depletion region at the collector side of the pn-junction. The negative electric field there pulls the electrons into the bulk region of the collector, from which they are sucked up by the applied positive voltage. The equilibrium value np of the electron concentration is therefore reached not only at x = 00, but it is already realized at x = b. This means that the boundary condition An(x = b) = 0 must now be imposed. The solution of the diffusion-recombination equation (5.91), which accounts for this new boundary condition, results in

(7.39)

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7.1. pn-junction in an external voltage 589

Substituting this into (7.37), we find

Dn Ln

j n ( zp ) = enp- (eeuBjkT - 1) coth[(b - zp ) /Ln] . (7.40)

The current ig in the base circuit would be identical with the emitter current iE, and both currents would equal the current through the emitter-base pn- junction, if the base were to be infinitely expanded. For finite base width, however, the emitter current differs from the base current because the latter only takes contributions from the portion of the electrons injected into the base which also recombine in the base. We denote the pertinent current density as jnT. The first term of formula (7.23) expresses it as

(7.41)

The upper boundary of the recombination region, which in (7.23) is located at infinity, is replaced here by the finite base width b. Furthermore, xn and x p are interchanged. The recombination rate R(x) in (7.41) can, as before, be calculated from the minority charge carrier concentration n ( z ) = np+An(z) in the p-region by means of relation (7.24). In this, expression (7.39) has to be used for An(x), yielding

The base current ig, like the emitter current in (7.36), also involves the injection current j p ( z n ) A of holes from the base into the emitter, besides the electron current j,A. Altogether, we therefore have

The collector current ic need not be calculated separately, because it is al- ready determined by Kirchhoff's law (7.35). The main source of this current are electrons which leave the emitter and do not recombine in the base but diffuse further to the collector where they are sucked up. This current contri- bution is entirely due to the coupling between the collector-base pn-junction and the emitter-base np-junction; it would not appear at all at a single, separate pn-junction. Another current which flows through the collector- base pn-junction is the true pn-current, which represents the current which would exist if this junction were not coupled to a second one. However, the collector-basepn-junction is biased in blocking direction, so that the true pn- current is an extraction or generation current, which because of its smallness

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590 Chapter 7. Semiconductor junctions under non-equilibrium conditions

Figure 7.9: Characteristics of a bipolar transistor made of Si in the common- emitter configuration. (After Volz, 1986.)

may be completely neglected. This approximation was used above tacitly, as we identified the current through the emitter-base pn-junction with the true pn-current of this junction, without adding the hole extraction current flowing in from the collector-base pn-junction on the right-hand side .

With this, the three currents iE , ig, ic we sought are fully determined. In Figure 7.9 some characteristic dependencies for i g and ic are shown in the case of a npn-transistor made of Si. We can now proceed to the question of what conditions are needed for the bipolar transistor to operate as an amplifier. As we will see below, significant amplification by a bipolar transistor results when the base current is only a small portion of the emitter current. In order for that to be the case, on the one hand, j,,A must be small. Considering (7.42)) this means that the base width b must be small in comparison with the diffusion length L , of the minority charge carriers. On the other hand, the hole injection current j p ( z , ) A is not allowed to be too large. This can be achieved by low doping of the base in comparison with the emitter, because jp(z,) is proportional to p, (as may be seen from (7.38)) and jn(zp) is proportional to np (see relation 7.39). If the doping n, of the emitter is substantially higher than the doping p, of the base, i.e. if nn >> p, holds, then it follows from the mass action law that p, << np. The hole injection current j p ( z n ) A from the base is therefore actually small compared to the electron emission current j n ( z p ) A from the emitter. For simplicity, we neglect it completely in what follows.

Suppose that the potential p~ of the base is changed by dpg, while the potentials p~ and pc of the emitter and collector are kept constant.

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7.1. pn-junction in an external voltage 591

Correspondingly, the voltage U B in the base circuit changes by d u B = d p B , while the voltage U D in the collector circuit remains constant. The currents, however, will change in both circuits in accordance with

(2)~~ = cosh[(b - z p ) / L n ] (2%) d u g uc , (7.45)

The current amplification ,13 is defined as the ratio of the current change in the collector circuit to the current change in the base circuit. From (7.44) and (7.45) it follows that

(7.46)

Here, the extent x p of the space charge region in the base has been neglected in comparison with the entire base width b. If b << L,, a current amplifi- cation occurs according to relation (7.46) in the sense that a small current change in the base circuit leads to a large current change in the collector circuit.

What is important, however, is not primarily amplification of current, but amplification of electric power. Initially, one might think that power amplification would be determined by the ratio of the power change in the collector circuit to the power change in the base circuit. This ratio would have, however, no value because in the switching scheme of Figure 7.7 all the power of the collector circuit is transferred to heat. For electric power to be useful, a working element must be included in the collector circuit, say a resistor RL. The voltage drop across it is ~ c R L . Taken together with the voltage between the emitter and the collector of the transistor, the total voltage in the collector circuit is U c + icRL. The transistor resistance is ap- proximately given by the resistance of the blocked pn-junction between base and collector. As such, it is practically independent of the base voltage. We consider, in particular, the case in which RL equals the collector resistance of the transistor. That means that the voltage drop across the working resistor R L is the same as the voltage U c between the collector and emitter. What we have to calculate are the power changes d P L in the working resistor RL and d P B in the base circuit, if the base voltage U B changes by d U B while keeping the collector-emitter voltage at a constant value Uc. We have

(7.47)

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592 Chapter 7. Semiconductor junctions under non-equilibrium conditions

Figure 7.10: Structure of a bipolar npn-transistor with a buried collec- tor.

Aluminum / I \ SiOD

.......... .......... .......... n-Silicon .......... .......... .......... :::::::::A ............ ........... .......................................... I/.._.J n*-Silicon

p3iG-I

(7.48)

The ratio of the first term in the square bracket of (7.48) to the second is kT to U g . Under normal conditions this is small, and neglecting it the ratio of the power changes dPL and dPB takes the simple form

( Z ) = P U B . UC (7.49)

Therefore, the power amplification turns out to be proportional to the cur- rent amplification. As far as the proportionality factor is concerned, we note that the voltage drop across the collector-base np-junction (biased in blocking direction), is larger than the voltage UB across the baseemitter np-junction (biased in flow direction). Because of this, the collector-emitter voltage U c is larger than U g , and the the power amplification is in fact larger than the current amplification. Small changes of electric power in the base circuit are therefore correlated with large power changes in the working resistor. Of course, this does not mean amplification of power in the sense that a large power is generated by a small power. Rather, it means that a large useful power is tuned by a small control power.

The base circuit is also called the input circuit, and the collector circuit the output circuit. Using this terminology one can also say that the bipolar transistor allows the control of a large output power by means of a small input power. The input resistance of the bipolar transistor, i.e. that of the base circuit, is relatively low, since the corresponding pn-junction is biased in the flow direction. The output resistance, i.e. that of the collector circuit, is relatively large compared to the former, because the second pn-junction operates in the blocking direction. This particular feature of the bipolar transistor essentially determines its application in electronic circuits. In the case of an unipolar transistor like the MOSFET, the relationships are

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7.1. pn-junction in an external voltage 593

P I n n

Figure 7.11: Lineup of the valence and conduction band edges at a pn-junction between moderately doped regions (a) and heavily doped regions (b).

reversed, as we will see below; there the input resistance is high and the output resistance low.

Due to the planar fabrication technology of microelectronics, the bipolar transistor, as it is actually used in integrated circuits, is structured differently than the one used in our theoretical analysis. A more realistic structure is shown in Figure 7.10.

7.1.6 Tunnel diode

The injection-recombination mechanism of current flow in a pn-junction is valid as long as the doping of the two regions is not extremely heavy ~

strictly speaking, as long as the bulk Fermi levels Epn and EF, do not lie in the respective bands (Figure 7.11a). When the latter occurs (Figure 7.11b), then there are energy levels at the conduction band bottom which are at the same position as some of the energy levels at the valence band top. The two energy bands partially overlap. In the overlap region each energy level belongs simultaneously to valence and conduction band states. However, the corresponding wavefunctions are localized in different regions of the pn- junction - those of conduction band states in the n-region, and those of valence band states in the pregion. An electron from a conduction band state with an energy level in the overlap region can transfer into a valence band state at the same energy. One then says that the electron tunnels from the conduction band into the valence band (also see section 3.8 on tunneling due to an external electric field). Analogously, electrons from the valence band can tunnel into the conduction band. The probability for a tunneling transition is the same in both directions.

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594 Chapter 7. Semiconductor junctions under non-equilibrium conditions

Figure 7.12: Current-voltage char- acteristics of a tunnel diode.

). 0 0 2 084

0 I

u/v-

I

In order for an electron of a given energy to actually be able to tunnel, it must find unoccupied states of the same energy in the other band. In equilibrium at temperature T = 0, the only possibility would be electrons having energies above the Fermi level, but there are none such. This means that in equilibrium at T = 0 no current can flow due to tunneling transitions between the valence and conduction bands. This equilibrium result also remains valid for T > 0. For simplicity, we proceed with the case T = 0 below. Applying a weak positive voltage to the pn-junction, the valence band in the p-region is lowered relative to the conduction band in the n-region, and the same happens with the Fermi levels in the two regions. Then there are electrons of the conduction band in the n-region that can find unoccupied states of the same energy in the valence band on the p-side of the junction. Therefore, an electron tunneling current flows from the n-region into the p-region of the junction. Further increasing the positive voltage, a point is reached at which the valence band top in the p-region drops below the conduction band bottom in the n-region. Then there are no more common energy levels on the two sides of the junction, and tunneling is no longer possible. The tunneling current through the junction vanishes. pn-junctions with the above described properties are called tunnel or Esaki diodes.

In Figure 7.12 the current-voltage characteristic of a tunnel diode is shown schematically. The dashed curve marks the tunneling current con- tribution to the total current which also contains, besides the tunneling current, a drift-diffusion current. Moreover, tunneling can resume with the aid of phonons, so that an energy difference between the tunneling states on the two sides of the junction is allowed when bridged by the absorption or emission of phonons. This process is referred to as phonon-assisted tunneling (in analogy to photon-assisted tunneling treated in section 3.8).

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7.2. pn-junction in interaction with light

7.2 pn-junction in interaction with light

595

7.2.1 Photoeffect at a pn-junction. Photodiode and photo- voltaic element

If a homogeneous semiconductor sample is irradiated with light, electron-hole pairs are generated. Only part of them is removed by recombination, so that in the stationary state there remain excess concentrations An of electrons and A p of holes. These may be calculated from the pair of equations (5.11)) (5.12) which state the equality of the generation rate g and recombination rate R in the present case. With R of (5.68) and g of (5.27), it follows that

(E) An = Ap = I-g = I- (7.50)

where I- is the recombination lifetime of the electron-hole pairs and Tzw is the photon energy of the radiation (the latter is taken to be monochromatic), Using equations (7.11) and (7.12), the corresponding quasi Fermi levels p n and pP of, respectively, electrons and holes may be calculated. Due to the optical excitation they have different values - that of the electrons is raised and that of the holes is lowered in comparison with the common equilibrium value. If a voltage is applied to the sample, the current flow is stronger when the sample is illuminated than without illumination. This phenomenon is called the internal photoeffect. Alternatively, if one keeps the current constant, then the photoeffect lowers the voltage drop across the sample. The illumination leads, so to say, to a negative pre-voltage of the sample. The latter is just the difference of the quasi Fermi levels of the electrons and holes.

In these considerations, a spatially homogeneous distribution of light in- tensity within the semiconductor sample has been assumed. However, since the light is absorbed in the excitation of the electron-hole pairs, this assump- tion is not justified. The intensity decays exponentially in the direction of light propagation. This leads to a corresponding spatial inhomogeneity of the electron and hole concentrations, which gives rise to diffusion. If the dif- fusion coefficients of electrons and holes are the same, then diffusion makes no contribution to electric charge transport since the electron and hole cur- rents exactly compensate each other. If the diffusion coefficients are different, however, as often actually happens, either a net current or voltage will arise in the light propagation direction, depending on whether the two ends of the sample are electrically connected or not. This phenomenon is called the Dember effect.

The homogeneous semiconductor photoeffects discussed above are rela- tively small in comparison with the photoeffects which occur at semicon-

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596 Chapter 7. Semiconductor junctions under non-equilibrium conditions

ductor junctions with reverse biased space charge regions. As examples of the latter, we have the depletion region at a pn-junction, also at a metal- semiconductor junction, an insulator-semiconductor junction or a free sur- face. In the case of a pn-junction, various configurations are possible: the light can propagate parallel to the boundary plane between the p- and n- regions or normal to it, and in the latter case, the depletion region can be smaller or larger than the penetration depth of light, etc.

Below, we will deal with a pn-junction and assume that the light prop- agates perpendicular to the boundary plane between the p- and n-regions, (parallel to the x-axis of our Cartesian coordinate system, see Figure 7.13). The corresponding pn-structure is, contrary to what has been assumed thus far, no longer infinitely extended in both directions but bounded in the pos- itive z-direction by a plane surface at 2 = 20. We take this surface to lie on the n-side of the junction, far enough from the depletion region so that the light is completely absorbed before reaching this region. This assump- tion simplifies our theoretical description, although it does not essentially influence its results. Under the assumed conditions, the optically excited electrons and holes diffuse away from the surface, towards the depletion re- gion. While the holes are attracted from the negatively charged p-side of this region, the electrons are repelled from there (see Figure 7.13). In the space charge field of the depleted p-region the holes drift deeper into the sample where they recombine. The electrons stay in the surface surface re- gion and recombine there. Consequently, a net hole current &hob flows from the surface into the bulk. Its magnitude is determined by the hole diffusion current at the boundary x = x,, of the depletion region. The maximum pos- sible value j p h t o of the hole current is given by the photon current density ( I l h w ) multiplied by -el i.e.

(7.51)

This upper limit is reached if all optically excited holes diffuse as far as the plane 3: = xn, which means that they do not recombine, neither at the surface nor in the bulk of the sample. We assume this condition to be fulfilled.

The flow of positive charge through the np-junction from its n- to its p- side, is just the opposite of what happened at the np-junction in establishing equilibrium without illumination by light. In the latter case, the transfer of charge results in the diffusion voltage. Therefore, exposure to light reduces the diffusion voltage. The voltage change U$’ooto is called the photovoltage. In the case of the open pn-circuit considered here, in which the outer planes of the p- and n-regions are not electrically connected, the total current density jgbl in a stationary state must vanish. The latter is composed not only of the photocurrent density, jphoto, but also of the current density j through the pn-junction under the photovoltage U$’ooto. This voltage takes a value which

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7.2. pn-junction in interaction with light 597

Figure 7.13: pn-junction subject to light illumination.

Figure 7.14: Electric circuit with an illuminated pn-junction.

guarantees the vanishing of the total current density jgtal. Using expression (7.28) for the current through the pn-junction, it follows that

(7.52)

If the outer planes of the p- and n-regions are electrically connected (see Figure 7.14), and if the resistance R in this circuit is 0 (short-circuit), then the voltage drop across the pn-junction is also zero, and the current

tota at .O = iphoto. (7.53)

flows. If the resistance in the pn-circuit is neither 0 nor infinity, but has a finite value R , then the photovoltage takes a smaller value UpRhoto than Up”hoto, and the total current density takes a smaller value jEtal than j:oM. The two quantities jEbl and Ugot0 are determined by the equations

(7.54)

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598 Chapter 7. Semiconductor junctions under non-equilibrium conditions

Here, A is the illuminated area of the pn-junction. Depending on the size of the working resistor R, the pn-junction can either be used to transform small light-signals into voltage signals, i.e. as a photodetector, or for the generation of electric power, i.e. as a photovoltaic element.

P ho t odet ect or

In the case of the photodetector, the working resistance R must be chosen as large as possible, so that the largest possible photovoltage occurs. For R = 00 it follows from equations (7.54) and (7.55) that

k T (7.56)

This relation shows that Urhto is more sensitive to the light intensity I for smaller saturation current density j, of the pn-junction. In subsection 7.1.4, j, was estimated to be 10-"A/cm2 for a pn-junction made of Si. From this, one can conclude that for hw = 1 eV, a light power of 10-l' W/cm2 will create a photovoltage of the order of magnitude 10 mV. The upper limit of photovoltage is given by the diffusion voltage UD. This cannot be surpassed because the space charge region would cease to exist if Upmbto were to exceed Uo. In the latter case, no separation of optically excited electron-hole pairs could take place, which is crucial for operation of the device. Photodetectors involving semiconductor junctions are referred to as photodi- odes. Beside the pn-photodiode discussed above one has also pin-photodiodes with particularly wide depletion regions (symbolized by the 'i' in pin). Other examples are met al-semiconductor or hetero j unction photodiodes.

Photovoltaic element

In the case of the photodetector, the usable electric power is very small and, in fact, for R = 00 it vanishes. If the illuminated pn-junction is to produce electric power, then R cannot be made infinitely large. On the other hand, R also cannot be taken arbitrarily small, because then there would be no voltage drop at the pn-junction at all and the power would vanish for this reason. For finite values of R , a non-vanishing electric power is generated, with light energy converted directly into electric energy. The useful electric power R(Ajztd)2 can be calculated by solving equations (7.54) and (7.55) with respect to jzhl and Upbto. The calculation shows that the efficiency of photovoltaic energy conversion can, theoretically, reach up to about 40 %. In practice, it is not much more than 10 % in most cases currently. Photo- voltaic elements have wide and important practical application as solar cells, particularly those made of silicon.

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7.2. pn-junction in interaction with light 599

7.2.2 Laser diode

If a pn-junction is biased in flow direction, the minority carriers diffuse into the depth of the two bulk regions, where they recombine. The manner in which recombination proceeds is unimportant if one is only concerned with current transport. In our previous calculations we assumed radiationless recombination through deep centers (Shockley-Read-Hall recombination). This was justified inasmuch as the recombination rate RmT of this process appreciably exceeds that of other recombination processes in many cases. Under appropriate conditions, however, radiative recombination processes can also play an important role. In such processes electron-hole pairs anni- hilate with the emission of photons (see section 5.2). If the emission proceeds spontaneously, then the recombination rate Rap, is given by the expression

(7.57)

where rSwT is the corresponding spontaneous radiative recombination life time and An = n - no is the excess concentration of electrons beyond the equilibrium value no. On the other hand, light emission can be induced by the radiation itself. Then one speaks of stimulated emission processes. In the following considerations we assume monochromatic light with a definite phase and propagation direction. Its photon energy hw is taken to match the band gap energy of the semiconductor. The light propagation is de- scribed by the photon current density S (flux of photons crossing unit area in unit time), which follows from the light intensity I by means of the rela- tion I = hwS. In terms of S, the radiative recombination rate RstimT due to stimulated emission may be written as

(7.58)

where g is the so-called gain coefficient. The latter is defined as the negative of the absorption coefficient, strictly speaking, that part of the absorption coefficient which is due to the excitation of electrons from the valence into the conduction band. The dimension of g is, therefore, that of a reciprocal length. In order for g to be greater than zero, the semiconductor must be put into an excited state. The energy of this excited state (relative to that of the ground state) is the energy emitted in stimulated emission when irradiating the semiconductor with light. If the emitted radiation is to prevail over that absorbed, the number of stimulated transitions from the excited state to the ground state must be larger than the number of transitions in the opposite direction. This case is realized when, in the excited state, the conduction band edge is occupied by more electrons than the valence band edge. This is called an occupation inversion In the case of equal occupation of the two

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600 Chapter 7. Semiconductor junctions under non-equilibrium conditions

band edges, absorption and stimulated emissions compensate each other, which means that the semiconductor is transparent (not counting sponta- neously emitted photons and the losses by absorption processes which are not related to transitions of electrons from the valence to the conduction band, like absorption by free carriers or phonons). The gain coefficient g is zero in the transparent state. Generally, in accordance with the remarks above, g is a function g(n) of the electron concentration n in the conduc- tion band. The particular concentration which corresponds to the case of transparency g ( n ) = 0, is called the transparency concentration nt,. In the vicinity of nt,, g ( n ) may be represented by a linear Taylor expansion with respect to (n - ntr), such that

9 = d ( n - ntr), (7.59)

where g’ is a constant. Although this expansion holds for both signs of (n - nt,), we assume (n - nt,) > 0 below. .This is to say that we consider only cases in which stimulated emission dominates over absorption due to optical transitions from the valence to the conduction band.

In order to achieve occupation inversion, heavily doped pn-junctions are required, for which the bulk Fermi levels of the p- and n-regions lie, respec- tively, in the valence or conduction bands (see Figure 7.15). By applying a sufficiently large flow voltage to such a junction, a non-equilibrium state can be created in which, close to the nominal transition region at x = 0, the two quasi Fermi levels differ by more than the energy gap. This indi- cates occupation inversion according to the definition above. The layer of the pn-junction in which inversion occurs is termed the active region There, the first of the two laser conditions, the occurrence of occupation inversion, is fulfilled. To also meet the second laser condition, namely the presence of radiation feedback, one has to provide that the active region operates as a Fabry-Perot resonator in one of the two directions perpendicular to the x-axis (here we take the y-direction, see Figure 7.16). This can easily be achieved by simply cleaving the semiconductor sample perpendicular to the y-axis. The cleavage planes function as mirror planes because of the high refractive index of semiconductor materials. A pn-junction prepared in this way is called a laser diode.

The useful laser output is the radiation which passes out through the resonator planes, as they are not ideal mirrors. This radiation is completely determined by the radiation within the resonator. Thus, the latter must be calculated to understand the laser diode quantitatively, and we will do this below on the basis of the qualitative considerations above.

As we have seen, occupation inversion at thepn-junction comes about by the injection of minority charge carriers under the influence of a flow voltage. In section 7.1, the corresponding injection current density j was determined

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7.2. pn-junction in interaction with light 60 1

Figure 7.15: Heavily doped pn-junction without voltage (a) and with voltage in flow direction (b). In case (b) occupation inversion occurs in the vicinity of the pn-boundary at x = 0.

as a function of applied voltage. Here, we need the relation between j and the photon current density S, taking the electric current density j be given. As we shall see, the functional dependence of S ( j ) can be calculated just from the continuity equations for the carrier currents j,, j , and the photon current density S. For simplicity, we assume a symmetric pn-junction, for which it

Figure 7.16: Geometry of a laser diode.

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602 Chapter 7. Semiconductor junctions under non-equilibrium conditions

suffices to consider only the electron current density j,. Also, the active region is symmetric with respect to x = 0 under this assumption, so that it extends between -(1/2)x,4 and ( 1 / 2 ) z ~ with ;CA being the total length of the region. The continuity equation for the electron current density j , was written down in Chapter 5 in general form. The derived relation (5.20) will be specialized here in the following way. First of all, we recall that the total current density j = j n ( z ) +jp(z) is spatially constant in the stationary state, while the two components j n ( x ) and &,(I) vary with I. On the right- hand side of the active region, j n ( z ) is almost identical with j , while &(z) practically vanishes. On the left-hand side of the active region, jp(x) has almost the same value as j , while jn(x) is close to zero. Consequently, in stationary state, we find for the active region the approximate relation

(7.60)

Relaxation processes which change the electron concentration in time include non-radiative recombination as well as spontaneous radiative and induced radiative recombinations. For the assumed symmetric pn-junction, the non- radiative recombination rate Rmr is determined by a common lifetime T for electrons and holes, as follows,

An R - p = -.

7- (7.61)

Applying expressions (7.57), (7.58) and (5.60), (5.61) to the continuity equa- tion (5.20) for the electron current, we have

(7.62) an j An An

g s . - - -= a t exA 78pDnT

To formulate the continuity equation for the photon current density S , an expression for the pertinent photon density is needed. According to electro- dynamics, this density is (S/c) where c denotes the group velocity of light propagation. The source density of the photon current is zero since S flows in y-direction and the pn-junction can be considered as homogeneous in this direction. The photon density does involve, external sources, however, due to spontaneous and stimulated radiative annihilation emission processes. Since each radiatively recombining electron-hole pair creates a photon, the corre- sponding source density is - apart from a modification which will be detailed immediately below - given by the sum of the radiative recombination rates, i.e. it is the negative of the last two terms on the right-hand side of equa- tion (7.62). The modification affects the second term (An/rSmT), which describes spontaneous emission. Only a small fraction of the spontaneously emitted photons have the same propagation direction and phase as those due to stimulated emission. Thus this term must be multiplied by a factor

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7.2. pn-junction in interaction with light 603

,B substantially less than unity if incorporated in the continuity equation for S. The source terms due to spontaneous and stimulated emission must still be complemented by an additional term, which accounts for light absorption processes which are not due to transitions of electrons from the valence to the conduction band (see above). We denote the corresponding absorption coefficient by a. The pertinent (negative) photon source density is -as . In a, one can also include losses due to the penetration of photons through the resonator planes which are unavoidable because otherwise the laser would be ineffective for the world outside. Finally, the continuity equation for photon current density takes the form

(7.63)

where the left-hand side is the change of the photon density per second. We are interested in the stationary state of the laser defined by

an d S at at - = 0. - -- (7.64)

We first consider g to be different from a. As we will see, in stationary state, this necessarily means that g is less than a:

g<cw The corresponding photon current density S< follows from equation (7.63) as

(7.65)

Since S<, An and rSpT are positive quantities, only the case g < a is meaningful, as indicated above. The photon current density S< given by equation (7.65) is relatively small because the spontaneous radiative lifetime T~~~~ has relatively large values and P is small compared to one. This is true on the proviso that g does not approach the value of a very closely, or, in other words, as long as the electron concentration n is distinctly smaller than the threshold value nth defined by the relation

0: = 9/(nttZ - nt7-1. (7.66)

The dependence of the photon current density S< on the carrier current density j follows from equations (7.62) and (7.63) as

(7.67)

Here, g has been replaced by g< to indicate that the value of g has to be taken as that which follows from g / ( n - ,ntT) for n < nth. If spontaneous emission

j e z ~ [ g < + ( l /P ) ( l+ rspa~/r)(a - g<)I

s< =

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604 Chapter 7. Semiconductor junctions under non-equilibrium conditions

is assumed to be extremely weak, i.e. if rSpar = 00, then it follows that S< = 0. This result indicates that under the assumption n < nth the photon current density S< is determined by spontaneous and not by stimulated emission. The pn-junction does not operate as a laser but as a spontaneous radiation emitter. Devices which rely on spontaneous emission, the so-called light emitting diodes (LEDs), are, however, constructed differently than laser diodes, in particular, they do not have a Fabry-Perot resonator. Also, their physical basis differs from that of the laser diode. Instead of focusing on the radiation propagating in one direction with the same phase and frequency, the total radiation is important. We will not discuss LEDs in greater detail here.

The photon current density S remains relatively small if the lifetime rsponr for spontaneous emission is identified with the value that actually occurs in semiconductors, instead of replacing it by 00. The contribution of radiative recombination to the carrier current density j in equation (7.62) can be neglected under these circumstances, and we have, approximately,

(7.68)

This equation relates the electric current density j and the electron con- centration n below the laser threshold. The threshold concentration nth corresponds to a threshold carrier current density j th as

(7.69)

Laser operation can occur only if the threshold current density j th is ex- ceeded. We will assume that this is the case. Then, by definition, we have g equal to a:

The photon current density can now take non-vanishing values if rVar = 00.

This does not contradict the photon current continuity equation (7.63) under the stationary conditions (7.64) because (g - a)S now vanishes since (g - a) does so. The actual value S> of S for a given current density j > j th may be obtained from the carrier continuity equation (7.62). In the approximation rspar = 00, it yields

(7.70)

We will show that S> is in fact considerably larger than Sc. To this end, we consider S> given by equation (7.70) for j = 2jth, and examine the ratio of s>(2jth) to s< of (7.67) for j = jlh, obtaining

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7.2. pn-junction in interaction with light

3

0 \ laser diode as a function of current den- sity j (schematically). v)

605

I / * J th j 1a.u.

Generally, the spontaneous radiative lifetime 7spaT is much larger than the non-radiative lifetime T. One has (7spaT//37) > 1 since ,B << 1. The ratio (g</m) is a positive number measurably smaller than 1. Therefore, the right-hand side of equation (7.71) is large compared to 1, meaning that the value of the photon current density S, above the threshold current density substantially exceeds the value S< below threshold. This is the manifestation of the laser effect. One can also say that the pn-junction functions as a laser diode when j > j t h holds. The functional dependence of the radiation intensity over the entire current density range is shown in Figure 7.17 schematically. The characteristic feature of the curve is its steep rise at the threshold density jth. This point marks the transition from spontaneous to stimulated emission, thus the onset of the laser regime.

To estimate the order of magnitude of the threshold current and of the radiation intensity of a laser diode, we choose parameters close to those of G ~ A S . We take ntr = c m ~ ~ , nth = 2 . 1 0 ~ ' cmv3, XA = 1 pm, r = lo-' s, and a = lo3 cm-'. With these values the threshold current density of (7.69) is j t h = 103A ern-'. The corresponding current strength amounts to 1 m A if a contact area of 10 x 10 ,urn2 is assumed. For current strength 2 m A the radiation intensity RwS within the resonator is about 10 mW. This example shows that the radiation intensities of semiconductor laser diodes are large enough to be employed for practical purposes, in particular in information processing and communication. Currently, semiconductor laser diodes are widely used in optical fiber communication, CD-players, scientific instruments and in many other ways.

The spontaneously emitting diodes (LEDs) have also attained wide prac- tical application, e.g., for signal generation and pattern display. Since the radiation emitted by laser diodes and LEDs lies in the photon energy range close to the band gap, one can theoretically cover the entire frequency spec- trum from the infrared to the near UV by a proper choice of semiconductor

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606 Chapter 7. Semiconductor junctions under non-equilibrium conditions

material. The practical difficulty of accomplishing this goal increases with frequency. There are various reasons for this, the most general being the separation of the operating non-equilibrium state from equilibrium: it be- comes larger, and therefore more difficult to achieve, if the gap increases. At this time, usable laser diodes and LEDs exist from the infrared up to the green spectral region. Blue and ultraviolet light emitting devices are still the subject of research.

7.3 Metal-semiconductor junction in an external voltage. Rectifiers

Only non-ohmic metal-semiconductor junctions, i.e. Schottky junctions are suitable for active electronic functions. Therefore, we restrict our consider- ations here to the latter. Moreover, we suppose that the semiconductor is n-type. For p-type semiconductors analogous results may be derived. The two most important electronic devices based on Schottky contacts are the metal-semiconductor rectifier and the MESFET. The physical principle of MESFET operation is similar to that of the MOSFET. It relies on the change of charge carrier concentration due to an external voltage in a thin layer of the semiconductor at the interface with, respectively, the metal or insulator. We will discuss this in section 7.4 in connection with the MOSFET. Here, we concentrate on the physical principle of operation of the metal-semiconductor rectifier, involving current transport through a biased Schottky junction.

As in the case of the pn-junction, the application of an external voltage U to a Schottky junction results in a reduction of the diffusion voltage UD of (6.58) by U , so that

(7.72)

We will assume that the Schottky approximation can be used in the semi- conductor, i.e., that the inequality (7.4) is fulfilled. Then the division of the semiconductor into depletion and bulk regions also retains meaning here. The width X B of the depletion region follows from (6.62) if UD there is re- placed by U D - U . The current in the bulk region of the semiconductor is carried by majority carriers, just as in the case of the pn-junction. For the assumed metal-n-type semiconductor junction, the majority carriers are electrons on both sides of the junction. At the interface between the two materials, electrons are thermionically emitted from the metal into the semi- conductor, and vice versa, electrons from the semiconductor are emitted into the metal. We denote the particle current density from the metal by i M S , and that from the semiconductor by i S M . The two partial current densities add up to the total current density i through the junction. All three current

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7.3. Metal-semiconductor junction in an external voltage. Rectifiers

a) Metol

607

0 - r 8 u-0 n- Semiconductor

bl

c

---I---: i " :

u7 0 ___.

L- I

x x x i ------ I I

I I L

XB X

c) +t+ UCO

- - -T ' \

x x A _ - I I I I

Figure 7.18: Illustration of current flow at a Schottky-contact, a) without voltage, b) with voltage in flow direction, and c) with voltage in blocking direction. The crosses in b) and c) indicate that the Fermi level is not defined in the space charge region.

densities are functions i~s(U),is~(U) and i ( U ) of the external voltage U , and

i ( U ) = iMS(U) - iSM(Cr). (7.73)

In the absence of any voltage (see Figure 7.18a) one has i(0) = 0, i.e., the two partial currents compensate each other, so that

i M S ( 0 ) = 1'SM(O). (7.74)

The application of a voltage U causes the space charge barrier of the semi- conductor to change, for U > 0 it is lowered (Figure 7.18b), and for U < 0 it

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608 Chapter 7. Semiconductor junctions under non-equilibrium conditions

is increased (Figure 7.18~). Accordingly, the current i s ~ ( U ) from the semi- conductor into the metal increases (U > 0) or decreases (U < 0). However, the current from the metal into the semiconductor remains constant since the potential barrier to be overcome by the electrons, the Schottky barrier Cpg, does not change with applied voltage. Thus,

i M S ( U ) = i M S ( 0 ) . (7.75)

The total current density i in the presence of the voltage U may then be written as

i ( U ) = i ~ s ( 0 ) - i s ~ ( U ) . (7.76)

We first calculate i ~ s ( 0 ) . In this, we consider a metal electron which, after passing into the semiconductor, has quasi-wavevector k and energy Ec(k). Since no momentum and energy change takes place in crossing the interface, the quasi-wavevector of this electron is k and its energy Ec(k) in the metal as well. The probability to find an electron in the metal with energy E = Ec(k) is given by the Fermi distribution f i ~ ( E ) of the metal,

(7.77)

The current density i ~ s ( 0 ) is the statistical average of the speed component h-'(Ll/LlkX)Ec(k) in the positive 2-direction, per unit volume. According to the general rule (5.3) for calculating statistical averages we have

The boundaries of the first B Z were put to infinity here, involving only a slight error. In fM(Ec(k)) we replace -EFM by the work function @ of the metal. This also indicates that, henceforth, the energy origin will no longer be taken at the top of the semiconductor valence band, but at the common vacuum level of the two materials. Since [Ec(k) +@] >> kT, the Fermi distri- bution (7.77) can be approximated by the Boltzmann distribution, whence

(7.79)

For Ec(k) we use an isotropic parabolic dispersion law with effective mass m:. Then relation (7.78) takes the form

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7.3. Metal-semiconductor junction in an external voltage. Rectifiers 609

The conduction band edge, referred to the vacuum level, lies at E, = -X so that @ + E, = CI, - X is equal to the Schottky barrier @B. After a short calculation we find

with

-OB/kT, i ~ ~ ( 0 ) = 4

(7.81)

(7.82)

as the average speed component in positive z-direction and n as the electron concentration in the semiconductor. Somewhat surprisingly, through mE and n quantities characteristic of the semiconductor occur in expression (7.81) for i ~ s ( O ) , although the current originates in the metal and one might have expected the appearance of characteristic metal quantities only. The reason is that for U = 0 (which is the case we are considering now) thermodynamic equilibrium exists, and the current from the metal into the semiconductor must equal the current from the semiconductor into the metal. That the latter depends on semiconductor quantities is obvious.

The interpretation of i ~ ~ ( 0 ) as a current i ~ ~ ( 0 ) from the semiconductor into the metal in thermodynamic equilibrium may also be used to calculate i s ~ ( U ) in the presence of a voltage. For U = 0, first of all, this interpretation yields the expression

If a voltage U is applied to the junction, the conduction band edge of the semiconductor at the interface with the metal shifts from its equilibrium position E, to the new position E, - eU. This change in (7.83) leads to the relation

i S M ( U ) ,eU/kT Z S M ( 0 ) . I (7.84)

Correspondingly, the total particle current density i ( U ) is

For the electric charge current density j = -ei it follows that

(7.85)

(7.86)

with I -@p,/kT j , = Tenvoe Lf

(7.87)

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6 10 Chapter 7. Semiconductor junctions under non-equihbrium conditions

For positive voltages U , the current density j grows exponentially with in- creasing U , while for negative U of increasing magnitudes, j approaches the saturation current density -js. The latter occurs when 1UI >> kT holds. The Schottky junction therefore operates as a rectifier, with positive voltages corresponding to the flow direction, and negative to the blocking direction. The following estimate shows that j , can assume quite large values. For T = 300 K , N D = 10l6 cm-3 and @B = 0.25 e V , we have j , 0.2 A / c m 2 . From this we may conclude that Schottky junctions are suitable for rec- tification of relatively strong currents. This distinguishes them from pn- junctions where the saturation current density is commonly much lower (see section 7.1). The reason for this difference is that the saturation current of a pn-junction is due to minority carriers, while the saturation current of a Schottky junction is caused by majority carriers.

The conditions of validity for the current-voltage characteristic of a metal- semiconductor junction derived above will now be examined. The applica- tion of formula (7.83) to the calculation of the current density iSM(0) reflects the assumption that thermionic emission of electrons from the semiconduc- tor into the metal proceeds unhindered. To appreciate the significance of this assumption one must first recognize that the emitted electrons originate in the bulk region of the semiconductor, for in the depletion region none are available. Unhindered emission can only occur, therefore, if the electrons, during their flight through the depletion region suffer no collisions. That is assured if the mean free path length Zf is larger than the space charge width "B,

lf > XB. (7.88)

For practical Schottky junctions, operating in blocking direction, this con- dition is often fulfilled, provided the blocking voltage is not too large. For flow voltages of sufficient magnitude it is always correct. Condition (7.88) excludes the possibility that the depletion region can even approximately be in a local equilibrium state. In particular, chemical and electrochemi- cal potentials cannot be meaningfully defined, not even in a local or 'quasi' sense. This implies that an essential requirement for the applicability of the phenomenological equations (5.14) and (5.15) for the current densities is no longer valid. If the inequality (7.88) is satisfied, drift and diffusion lose their meaning as transport mechanisms in the depletion region of a metal- semiconductor junction. The transport proceeds by electrons flying through the depletion zone unimpeded, which is termed ballistic transport. If, instead of l f > ZB, the condition

lf << X g (7.89)

holds, then equation (7.83) is no longer valid for use in calculating iSM(0) and i s ~ ( U ) . The magnitude relation (7.89) is just the upper part of the general condition (5.9) for the validity of the phenomenological equations,

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7.3. Metal-semiconductor junction in an external voltage. Rectifiers 611

I

Figure 7.19: Population and depopulation of interface states at an insulator n-type semiconductor junction.

applied to the Schottky junction. If the condition (7.89) is satisfied, these equations can be used to calculate i ~ ~ ( 0 ) . In doing so one obtains the so-called diffusion theory of current flow through the Schottky junction. A detailed outline of this theory will not be given here. Qualitatively it yields results similar to the above theory of i s ~ ( U ) , which in this context is called the thermionic emission theory. In fact, the two cases (7.88) and (7.89) must also be distinguished in the theory of current flow through a pn-junction. Starting from the phenomenological current equations in section 7.2, we also assumed the validity of a condition similar to (7.89) when we considered the case of a pn-junction. This means that the diffusion theory of current flow was developed, and the diode theory was omitted, in our earlier discussion of the pn-junction. For the pn-junction this is justified inasmuch as the diffusion theory is applicable in most cases, while the thermionic emission theory is not. If the depletion region of the metal-semiconductor junction becomes so nar-

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612 Chapter 7. Semiconductor junctions under non-equilibrium conditions

row that the electrons can tunnel through the potential barrier, then even the thermionic emission theory is no longer valid. This occurs in heavily doped semiconductors. In this case the Schottky contact conducts well in both directions, and it becomes an ohmic contact. This effect is exploited in manufacturing contacts in silicon devices. The frequently used contact- metal, aluminum, would in fact result in a non-ohmic Si-metal contact if the doping were not sufficiently high. The heavy doping of Si makes this contact ohmic, as is necessary for the proper operation of devices.

7.4 Insulator-semiconductor junction in an exter- nal voltage

In Chapter 6 we saw that the properties of an insulator-semiconductor junc- tion in thermodynamic equilibrium depend on whether interface states are present or not. If such states do exist, the bands at the interface are already bent without an applied voltage, and one has a depletion or accumulation layer there. These arise in consequence of the spatial redistribution of free charge carriers in the presence of interface states - some of the carriers are either captured or generated by these states (see Figure 7.19). The semi- conductor as a whole thereby remains electrically neutral, of course. This changes if a voltage is applied.

7.4.1 Field effect

Consider an insulator layer of finite thickness d on top of a semi-infinite semiconductor of either n- or p-type (see Figure 7.20). To applying voltage, a metal layer must be deposited on the surface of the insulator at x = -d. I f this layer is put at potential p ( - d ) , and the outer semiconductor boundary at x = 00 is at potential p(00), then the voltage U across the structure is given by the relation

U = p ( - d ) - p(00). (7.90)

The electric field strength deep in the semiconductor, i.e. at x = 00, contin- ues to be zero even in the presence of the external voltage, whence

(7.91)

From electrostatics it follows that under these circumstances, electric charge is induced in the semiconductor. The amount of this charge turns out to be the negative of the charge which must be put on the metal layer to establish the voltage U across the insulator-semiconductor junction. This applies, strictly speaking, to the case in which the static dielectric constants E I and E of the insulator and semiconductor, respectively, are equal. If E I differs

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7.4. Insulator-semiconductor junction in an external voltage

u<o

U’O

613

Figure 7.20: Formation of depletion and accumulation layers at an insulator- semiconductor junction subject to an applied voltage.

from t, then the charge on the metal contact has to be multiplied by the factor ( € I / € ) to get the induced charge. Since, in general, E I is smaller than t (ferroelectrics are an exception), the induced charge is generally a fraction of the charge placed on the metal contact. Inducing charge at an insulator- semiconductor junction by applying a voltage is also referred to as field effect. This effect has played an important role in the development of semiconductor physics. The thought of exploiting it to induce a change of semiconductor resistance by means of an external voltage, hence, to develop a kind of a transistor, is obvious and was followed by several scientists in the late twenties and thirties of this century. Initially, the efforts were not successful, however. Instead of the field effect transistor, the bipolar transistor was invented in 1949 - in a sense as a byproduct of the search for the former.

The main reason for the failure of the early work on the field effect transistor

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614 Chapter 7. Semiconductor junctions under non-equilibrium conditions

is the existence of interface states at the junction of the insulator and semi- conductor. If such states are present in sufficiently high density, which was in fact the case in the early work, all of the induced carriers are captured by them. Since carrier mobility in these states is zero, the resistance of the semiconductor does not change. Generally, the number of electrons cap- tured depends on how many unoccupied interface states are pushed below the Fermi level by applying the external voltage (see Figure 7.19c), or how many previously occupied states are lifted above the Fermi level (see Figure 7.19b). The amount of captured charge represents, in this way, a measure of the number of interface states in an energy interval at the Fermi level of the size of the applied voltage. By measuring this charge as a function of applied voltage U , which can be done, e.g., by measuring the capacitance change of the insulator-semiconductor junction with U , one may obtain experimental data about the density of interface states at different energies.

7.4.2 Inversion layers

For a field effect transistor to work well, the density of interface states must be as small as possible. In the following, we will neglect these states completely and will study the effect of an applied voltage on an insulator- semiconductor junction under these circumstances. Qualitatively, it can be ascertained immediately, that for not too large voltages U , an accumulation or depletion layer is formed depending on the sign of U and the type of semiconductor (see Figure 7.20). Below, we assume a p-type semiconductor. Starting from U = 0, an increase of U bends the bands of the semiconductor at the interface with the insulator downward. Increasing U further, sooner or later, a point is reached at which the conduction band energy Eg - ecp(0) at the interface dips down below the Fermi level EF (see Figure 7.21). This happens if the voltage drop across the semiconductor Us = cp(0) - cp(00) is larger than (Eg - E F ) / e 3 Cro, i.e. if

(7.92)

holds. Increasing the voltage U further, the conduction band edge E,-ecp(O) at x = 0 is pushed further down, and the region of the semiconductor in which the conduction band edge lies below the Fermi level extends to the right to a point xi > 0 given by the relation

E g - ecp(zi) = E F . (7.93)

(see Figure 7.21). In this region the free charge carriers are not predomi- nantly holes, as otherwise in a p-type semiconductor, but rather, they are electrons. The conduction carrier type is inverted in this region, which is therefore called an inversion layer. The electrons in the inversion layer of a

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7.4. Insulator-semiconductor junction in an external voltage 615

Figure 7.21: Inversion layer at an insulator-ptypesemiconductor junction.

psemiconductor form a degenerate electron gas. Their concentration is of the order of' magnitude of the effective density of states Nc of the conduction band. This is much larger than the concentration of holes in the p-type semi- conductor which hosts the inversion layer, provided the doping of the latter is not too strong which means as long N A << Nc holds. This requirement will be assumed to be satisfied here. In order that inversion occur at all, the voltage drop Us across the semiconductor must obey condition (7.92), which is also assumed to be valid.

We will now examine the spatial variations of the charge carrier con- centrations .(.) and p ( z ) in the semiconductor, which are determined by the Fermi distribution f ( E ) of equation (4.13) with the replacement of EF by the spatially varying chemical potential p ( z ) . The latter follows from p ( z ) - ecp(z) = E F as

p(.) = EF + e c p ( X ) . (7.94)

With this expression, the appropriate position-dependent Fermi distribution f ( E , x) is

(7.95)

If the Fermi energy EF lies below the conduction band edge, then the ma- jority of electrons in this band have energy values at and a few kT above the band edge E, - ecp(x). As long as [E, - ecp(z) - EF] is substantially larger

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616 Chapter 7. Semiconductor junctions under non-equilibrium conditions

than kT, f(E, x ) is approximately zero for the energy values mentioned. If, on the other hand, the Fermi level lies above the conduction band edge, i.e., if the expression [E, - ecp(x) - EF] is negative and its absolute value is simultaneously much larger than k T , then f(E, x ) is approximately 1. The energy interval in which f(E, x ) is neither 1 nor 0 decreases, as compared to the whole potential interval eU, spanned by the potential e p ( x ) in the semi- conductor, as eU, increases in comparison with k T . We therefore suppose that

k T << eU, (7.96)

holds. In a sense, this relation represents a generalization of condition (6.35) for the validity of the Schottky approximation. Then, considering (7.95) for energy values of the conduction band, we can approximately write

f(E, X ) = B [ E F - E + ecp(x)] E > E,. (7.97)

Substituting this expression in equation (4.67) for the electron concentration n ( x ) , we have

Furthermore, beyond the inversion boundary x i , the valence band edge -ecp(x) initially still lies far below the Fermi energy. Thus, for energy val- ues E < 0 within the valence band, f(E) = 1, provided x does not lie too far to the right of x i . The hole concentration p ( x ) therefore vanishes for not-too-large values of x . At very large x , however, the valence band edge -ecp(x) approaches the Fermi energy so closely that, approximately, the same conditions exist as in the bulk. There, p = NA holds if completely ionized acceptors are assumed, as we do. In the Schottky approximation, the transition between the two z-regions occurs abruptly, at a point of transition x p of the depletion region which has yet to be determined. Thus, the hole concentration p ( x ) , is approximately given by

(7.99)

Evaluating the energy integral for n ( x ) in (7.98) with the square-root ex- pression (4.44) for the density of states p:&&(E) x p;"'(E), a concentration n(z ) follows which is proportional to [ e p ( x ) + EF - Eg]3/2 . Substituting this into the Poisson equation (5.24), the solution leads to inverse elliptic func- tions which are difficult to handle analytically. We therefore replace the true density of states by a model state density Fc which is constant in the energy

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7.4. Insulator-semiconductor junction in an external voltage 617

interval between E, and EF + ecp(x), with the constant 7, of the model de- termined as follows: We form the average value of the true square-root-like density of states p?”’(E) in the occupied energy interval at x = 0, i.e. be- tween Eg and EF + eUs. If we were to use this average value for the density of states throughout the entire inversion layer without any correction, then the resulting average electron concentration in the inversion layer would be markedly larger than it really is. Therefore, a correction factor y having a value between 0 and 1 must be introduced in consideration of the decrease of the upper limit of the averaging interval with increasing I . Thus, we make the substitution

(7.100)

Using this constant density of states, the electron concentration n(x) in the inversion layer may be rewritten in the form

where

(7.102)

is the effective density of states of our model. The substitution (7.100) has the consequence that the true density of states, and with that also the true electron concentration, is underestimated in the left part of the inversion layer, and overestimated in the right part, provided that a reasonable choice of y has been made. Because of this, the potential profile p(x) of the in- version layer, whose calculation we will now discuss, has a curvature in our model which differs somewhat from that in reality. Qualitatively, however, no significant differences occur. For a quasi 2-dimensional electron system, the results of our model even apply exactly, since in this case the density of states is constant from the outset.

With the resulting concentrations of free charge carriers as derived above, the Poisson equation (5.24) in the semiconductor takes the form

-d _< I I: 0, I O1

(7.103)

The solution of this equation must satisfy the boundary conditions (7.90) and (7.91). The arbitrary constant of the potential is chosen such that ( ~ ( 0 0 ) = 0

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618 Chapter 7. Semiconductor junctions under non-equilibrium conditions

holds. From the Poisson equation it immediately follows that the potential and the field strength also vanish at x = xp, i.e. that

V b P ) = 0, (7.104)

(7.105)

The field strength in the insulator is spatially constant and we denote it by FJ. Then, in particular, we have

(7.106)

For the potential p(x) at x = -d, relation (7.90) and p(00) = 0 yield the value

p(-d) = U . (7.107)

The four conditions (7.104) to (7.107), as well as the continuity condition for p(x) and (dpldx) at x = 0 and x = xi, are sufficient to determine the potential profile p(x) uniquely, along with the three as-yet unspecified parameters xi,xp and FI. The solution of the Poisson equation under the above conditions is determined as

u - (X + d)FI, -d 5 x < 0,

Uo + (Us - Uo) cosh(x/L) - FsL sin(x/L), 0 5 x < xi,

with

xi i x < Xp,

(7.108)

(7.109)

as the characteristic screening length of the electron gas of the inversion

(7.1 10)

as the electric field strength in the semiconductor at the interface with the insulator, and

Ua = U - FId (7.1 11)

as the potential drop across the semiconductor. The inversion boundary xi and the field strength F obey the equation

tanh(xi/L) = ( U , - Uo)/F,L, (7.112)

which follows from (7.108) for x = zi immediately. Substituting equation (7.112) into the matching condition for the field strength at xi, we obtain

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7.4. Insulator-semiconductor junction in an external voltage 619

cosh(xi/L) [(Us - Uo) 2 - (LF,) 2 1 = =-(xi N A - xP)Fs(U, - Uo). (7.113) N ,

By means of (7.112) and (7.113), xi and Fs can be determined as functions of Us. The value of xi is finite for all Us. We consider here, however, only the limiting case N A << Fc, in which the right-hand side of (7.113) can be neglected, so that

Fs = (Us - U o ) / L . (7.114)

Substituting this expression in (7.112), we arrive at the conclusion that the width xi of the inversion layer should be infinitely large. This is, however, only an artifact due to the neglect of the depletion region to the right of xi. In reality, under the condition N A << N, we have only that

xi >> L. (7.115)

The number ns of electrons stored in the inversion layer per cm2 may be obtained from (7.101) and (7.108) as

- ns = N,L. (7.116)

and L taken from, respectively, (7.102) and (7.109), this expression With may be rewritten as

The sheet density ng of electrons in the inversion layer is zero if the voltage drop Us across the semiconductor equals the threshold voltage Uo marking the onset of inversion. Above this value, ng has a weak ‘overlinear’ depen- dence on Us. Thereby, U , depends on the applied voltage U through the relation

(7.118)

The ratio of insulator thickness d to the screening length L of the electron gas of the inversion layer is decisive for the size of the variation of ns as a function of U . For d >> L , ns is practically independent of U . In order that the accumulated charge density ns can be tuned by means of U as effectively as possible, we must have d << L. With Us - UO M 1 V and mz M m, the corresponding value of L is about 300 A. The insulator layer must therefore be extremely thin. In the limiting case d << L and Us >> Uo, we have

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620 Chapter 7. Semiconductor junctions under non-equilibrium conditions

(7.119)

With U M 5 V, this yields (dn,/dU) M 1013 crn-'V-l. A voltage increase of 1 V results in about lOI3 electrons being induced into the inversion layer per cm2.

Electrons in an inversion layer have been the subject of many physical investigations. Like carriers at a semiconductor heterojunction (see section 6.2), these electrons are confined in a potential well. Just above the bottom (ri 0.1 e V ) , the width of this well is still smaller than L. Thus, carriers of such energy are freely mobile only parallel to the well. One has a quasi 2-dimensional electron gas. In it, confinement effects, like the formation of subbands, can be observed. By means of an applied voltage, the density of the electron gas can be varied. In this context, the accumulation layer at the interface between SiOz and Si has been of particular interest. Historically, it was the 2D gas of this layer in which carrier confinement effects were studied first, and in which the quantized Hall effect was discovered.

7.4.3 MOSFET

Whether or not the charge density of a semiconductor inversion layer suffices to cause a resistance change of sufficient magnitude to function as a transis- tor, depends (among other things) mainly on the specific resistivity of the semiconductor material in the absence of an applied voltage. This must be as high as possible. The highest possible resistivity, or the smallest conduc- tivity, of a semiconductor material is observed when it is in its intrinsic state. This fact is used to advantage in the most important type of field effect tran- sistor, the so-called MISFETs (Metal Insulator Semiconductor Field Effect Transistor). Here, one exploits the fact that in a pn-junction a depletion region is formed wherein the charge carrier concentrations have intrinsic val- ues. The p-region is embedded between two n-regions, as shown in Figure 7.22. If one applies a positive voltage between the left n-region (source) and the right n-region (drain) (Figure 7.22), then the left pn-junction is biased in the flow direction, and the right in the blocking direction. Between the two n-regions, one therefore has a p-region which is almost completely depleted of holes. In the theory developed above, this may be taken into account formally by adding a positive prevoltage Uv to the voltage U applied to the insulator-semiconductor junction from outside, i.e. by the replacement

u - + u t u v . (7.120)

For U = 0, the semiconductor region between source and drain, the so-called channel, has high intrinsic resistivity. If a positive voltage U = UG is applied between the bulk of the p-semiconductor (substrate) and the metal layer on

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7.4. Insulator-semiconductor junction in an external voltage 62 1

a1

I Substrat P

Figure 7.22: Structure of a MISFET (a). In part (b) the switching scheme of the MISFET is shown.

E n ... v)

Y

- 0 4 8 12 16 20 24

Figure 7.23: Influence of the gate voltage UG on the I s 0 - versus - U s 0 current- voltage characteristics for a n-channel MOSFET of enhancement-type. (After Vzilz, 1986.)

top of the insulator (gate) , large enough to create an inversion layer, then the channel becomes a good conductor. The current in the source-drain circuit can be tuned by the voltage UG of the gate-substrate circuit since the electron density of the inversion layer depends on UG. The voltage UG is called gate voltage, and the minimum gate voltage UG necessary to a achieve inversion is the threshold voltage. If one also includes a working resistor in the source-drain circuit and compares the power changes in this resistor with those in the gate-substrate circuit, one finds that the former exceeds the latter appreciably. In this sense, the MISFET operates as an amplifier, just like the bipolar transistor. Because of the insulating layer below the gate electrode, the MISFET has a much larger input resistance than the bipolar

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622 Chapter 7. Semiconductor junctions under non-equilibrium conditions

transistor. The output resistance of the MISFET, i.e. that of the source drain circuit, is small because of the accumulation layer between source and drain.

If the MISFET is realized using Si as the semiconductor material and SiO2 as the insulator, one has the Metal Oxide Semiconductor Field Effect Transistor, abbreviated MOSFET, which is by far the most important MIS- FET. The basic requirement that the electrons of the inversion layer shall not be captured by interface states, are met by the MOSFET extremely well. Just as the base of a bipolar transistor can be made either of p or n-material, one also has two possibilities in the case of the unipolar MOS- FET - the charge carriers in the conducting channel can either be electrons, as has been assumed thus far, or holes. In the first case one speaks of a n-channel MOSFET, and in the second of p-channel MOSFET. The prevolt- age UV, which was introduced above only formally to simulate the intrinsic state, can really exist for various reasons, e.g., because of positively charged centers within the oxide arising during its formation. The prevoltage can take such large values that inversion exists even without an applied gate voltage. Then the transistor is already in its conducting state at zero gate voltage. To switch it into its blocking state, one must apply a negative gate voltage in the case of the n-channel MOSFET, and a positive gate voltage in the case of the p-channel MOSFET. One says that the transistor oper- ates in the depletion mode. If the transistor is blocked without an applied voltage and changes into the conducting state by applying a positive (for n-channel) or negative (for p-channel) gate voltage, one has the enhance- ment mode. The current-voltage characteristics of the source-drain circuit of a n-channel-enhancement-mode MOSFET are shown in Figure 7.23 for different gate voltages. One recognizes how the sourcedrain current at a h e d value of U s 0 increases with increasing gate voltage UG. Somewhat unexpectedly, for a ked gate voltage UG, the current saturates at higher source-drain voltages USD. This is a consequence of the fact that the effec- tive gate voltage in the vicinity of the drain electrode becomes smaller and smaller as the drain potential grows larger. This creates a ‘pinch off’ of the inversion layer, making the current stay constant.

The MOSFET represents the most important electronic component of digital circuits in microelectronics. Here it is used as an electrically control- lable switch, meaning that its control function is reduced to two states only, one with maximum output power, corresponding to a binary ‘l’, and another having minimum output power, corresponding to a binary ‘0’. Today, MOS- FETs can be made as small that millions can be integrated in one single Si chip, more than of any other electronic component. It is this ultra-large scale integration (ULSI) technique which has made modern computer and communication technologies possible.

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623

Appendix A

Group theory for applications in semiconductor physics

A. l Definitions and concepts

A . l . l Group definition

A group G is defined as a set of elements g which possesses the following properties:

1) There exists an assignment instruction which associates an ordered pair of elements g1 and 92 of the set uniquely with another element which also is a member of the set. One terms such an assignment instruction a 'mul- tiplication', and writes g1 . 92 for the associated element. The totality of the assignment instructions forms the multiplication table of the group. This table determines the individual nature of a group.

2) The multiplication obeys the associative law, i.e. elements 91, 92, 93 one has the identity

for three arbitrary

3) The set of elements contains the identity element. This is an element e defined by the property that its right or left product with any other element g of the group yields again g. Thus,

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624 Appendix A. Group theory for applications in semiconductor physics

4) For each element 4, the set also contains its im-erse element 9-l: such that

g . g - ' =; g -1 - g - e . (A.3)

Onc may show that it suffices to dcmand the existence of only the left-sided or only the right-sided inverse. This one-side inverse is thcn uniquely deter- mined and its existcnce automatically leads t o the existence and uniqueness of the other-siticd iiiverse. The other-sidd inverse thusly determined is iden- tical with the original oneside in?-erse. Analogous dstements hold €or the identity dement discussed above.

The inverse (91 . g2)-' of tl product gl . g:! is given by gz1 . gl1: as one may easily verify by explicit. mult.iplicat.ion of the two products.

The existence of the inverse ensures that t.he products g1 . g and g2 . g

of two elements g1 and g~ with an arbitrary group element g are different, if the elements g1 and 92 differ from each other. The same is true for the re-ordered products g .g1 and 9 92.

The number N of elements of a group is callrd its order. Depending on whether N is finite or infinite, one has finite or iu,~%itt grotqs. ' h e dements of infinite groups may be either discrehe or continuous. In the latter rase one has a coatinuow group.

The group multiplication which i s associative by definition, need not also be commutative. In general: g1 . $2 differs from g2 . gl. If the multiplication is also commutative, the group is call& A h e l i a ~ ~

We now introduce some goup-thcoretical concepts which are n d e d in this book.

A.1.2 Concepts

A suihject of a group which forms a group by itself with respect to the multiplication tahle defined by that of the full group, is termed a subgroup.

Subgroups

For finite groups G, the theorem of Lagrange holds, wherein the order 1L" of a subgroup G I is a divisor ol thp order N of the group. Thc proof of this theorem will be briefly sketched because it also provides some insight in other group properties. Let be ,g! = E,g$, . . , gh, the elements of a subgroup G" of G We denote an arbitrary element of G not contained in G' by gz. 'I'he products 92 gi, g 2 . gh,. . . ,g2 . gh, of g:! with dl elements of the subgroup G' form a 5et M,<, the elements of which are different from each other as well as from the dementb of G'. For the sake of uniformity of notation, we set C' M,<. If the two sets M1< and M; together do not cover trhe d i r e group C , WP choose an element 93 of G not contained in AT,< and M,<. We proceed in the same way with this element as we &d with

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A.1. Definitions and concepts 625

92. The thus emerging set M3< contains N' mutually different elements, just like M,< and M 2 , and none of these elements occurs in M1< and h42. If the entirety of the sets M:, and M Z still do not yet cover the full group G, we choose yet another element 94, and so on. After a finite number N f f of steps a complete division of the group G into N" distinct subsets with N' elements in each will be achieved. Therefore, one has N = N' ' N" which proves the theorem.

Lagrange's theorem allows us to draw several conclusions. For groups of prime order it means that only trivial subgroups can exist, namely the identity element, and the group itself. This statement may also be reversed - the order of a group which has no subgroups apart from the trivial ones, must be a prime number.

Cyclic subgroups

Cyclic subgroups consist of integer powers g', k = 0,1, . . . , K - 1, of an element g where K is a positive integer. For finite groups the value of K is finite and g K coincides with the identity element. If the cyclic subgroup with respect to a certain element is the entire group, then the same necessarily holds also for the cyclic subgroups with respect to all other elements. The group itself is then said to be cyclic. Cyclic groups are also Abelian. A group whose order is a prime number must necessarily be a cyclic group - if this were not so, the group would have subgroups, cyclic ones, which is not allowed in this case.

Normal divisor and factor group

The sets M;, i = 1 , 2 , . . . , N" defined above are called left cosets of G with respect to the subgroup G' for the elements gi. Analogously, one defines right cosets M,' of G with respect to the subgroup G' for the element gi, namely as sets of elements g: ' g i , g a . g i , . . . , g h , . g i . If all the right cosets with respect to a particular subgroup G' are identical with all the left cosets with respect to this subgroup, then the subgroup G' is called a normal divisor. One can readily see that such a subgroup consists only of classes of conjugate elements of G (for the definition of classes see below).

We can go a step further and understand the cosets with respect to the normal divisor G' as group elements and define the product of two such elements as the set of the products of all their elements. Since these products are again cosets, the multiplication thusly defined constitutes a group. It is called factor group. Normal divisors and factor groups are helpful concepts for the characterization of the internal structure of groups.

Direct product

We consider two groups G and G' of, respectively, orders N and N', and

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626 Appendix A . Group theory for applications in semiconductor physics

examine the products between all elements g k of G and all elements gh, of G'. The whole set of these N . N' products gg i , = g k . gLi forms a group G" of order N . N'. This statement may be easily verified by showing that the product of two of the elements of GI' is again an element of G". The group GI' is called the direct product of the two groups G and GI, for which one writes G" = G x G'.

Conjugate elements and classes

In regard to representation of groups by matrices, which we will consider later, the division into classes of conjugate elements is of great importance. An element g 1 is said to be conjugate to an element 92, if the group contains an element g such that g 2 = g . g1 . 9- l . From this definition it follows immediately that if g 1 is conjugate to 92, then 92 is conjugate to 91, and g l l is conjugate to g T 1 . Moreover, it holds that with g1 conjugate to g2 and 92 to 93, then g 1 is also conjugate to 93.

Now we consider a particular element g 1 and form all elements g . g 1 . g - l conjugate to it. This may be done by allowing g in the latter expression to run through the entire group. Thereby, a certain set of elements K1 emerges. Consider an arbitrary element of this set and again form all elements conju- gate to it. The set of elements thusly obtained coincides with the previously obtained set K1. One calls K1 a class of conjugate elements or simply a class. If one applies the same procedure to an element g2 of the group not contained in K1, one obtains a set K2 which has no common elements with K1. It forms another class of conjugate elements. If one continues to apply this procedure until no element remains which does not already lie in one of the prior classes, one achieves a complete division of the group into distinct classes of conjugate elements. One may say, therefore, that each group con- sists entirely of classes of conjugate elements. Each class is defined by one of its elements, the remaining elements follow by conjugating the defining ele- ment. The identity element forms a class by itself in all groups. In Abelian groups each element forms its class.

Homomorphism and isomorphism

We may associate the elements g of the group G uniquely with the elements f of another group F , as expressed by the mapping relation

f = f ( 9 ) . (A.4)

Such an association, or mapping, is termed homomorphic if it obeys the condition

f ( g 1 ' 92) = f (91) . f (92) 7 (A.5) where g1 and 92 are two elements of G. If the mapping is not only unique but is also bi-unique, i.e. if not only each element of G corresponds to exactly one

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A.2. Rigid displacements 62 7

element of F , but also, conversely, each element of F corresponds to exactly one element of G, then the mapping is isomorphic. Two groups which are related to each other by an isomorphic mapping, are called isomorphic. Since groups are completely characterized by their multiplication tables, and since the latter are necessarily the same for isomorphic groups, one may say that two isomorphic groups are essentially two different realizations of one and the same group. They differ only in the meaning of the elements.

For the description of microscopic properties of crystals, three kinds of groups are of particular interest ~ the translation groups, the point groups and the space groups. The elements g of these groups are in all three cases rigid displacements of certain point sets, which transform these sets into themselves. As we will see, the point sets for the translation groups are infinite lattices, those for the point groups are finite spatial bodies, and those for the space groups are crystals, i.e. several infinite lattices of the same Bravais type put into each other. The translation-, point- and space groups are summed up under the term displacement groups.

A.2 Rigid displacements

A.2.1 Definition

Formally one may define rigid displacements as transformations g of 3- dimensional space which shift its points x by vectors a(x) in such a way, that the distance I x1 - x2 I between two arbitrary points ~ 1 ~ x 2 does not change. Thus, we have

or ( A 4 2 2(x1 - x2) ’ Ia(x1) - a(xz)l+ I 4x1) - a(x2) I = 0.

The multiplication ‘dot’ in (A7) and (A.8) signifies the internal or scalar product. In order to satisfy equation (A.8), the displacement field a(x) cannot be of higher order in x than of the first, i.e. a(x) must have the general form

a(x) = px + a (A.9) where /3 is a linear operator in x-space and a a constant vector. From (A.6) it follows that

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628 Appendix A. Group theory for applications in semiconductor physics

gx = ax+ a, (A.lO)

with a = 1 + 0. The thusly defined operator cr is also linear. It is subject to certain constraints due to the rigidity condition (A.7). If one enters this equation with expression (A.lO) for gx, then one gets, first of all, (XI - xg). ( X I - xg) = a(x1 - xg) . a(x1 - x2). Using the adjoint operator a+ of a, the last relation may be put into the form

(XI - xg) ' (XI - x2) = a+ ' a(x1 - x2) . (XI - x2). ( A . l l )

The multiplication 'dot' between the two operators symbolizes the consecu- tive application of the operations. Since (XI - xp) represents an arbitrary vector, relation ( A . l l ) implies that

a+. a = 1. (A.12)

By this identity, a+ is seen to be the inverse operator a-l of a, whence

,+ = ,-I. (A.13)

Linear operators which act in a real vector space (as we assume here) and satisfy the relations (A.12) or (A.13), are called orthogonal. The transfor- mation they describe are orthogonal transformations.

A.2.2 Translations

For p = 0 or a = 1, g reduces to a rigid displacement of all points through the same vector a, i.e. to a translation. We use the symbol t, for it, so that

tax = x + a. (A.14)

With ta, one can rewrite equation (A.lO) for gx in the form

whence

which is stated as theorem below.

Theorem

(A.15)

(A.16)

Any rigid displacement g m a y be generated by a n orthogonal transformation a followed by a translation t,.

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A.2. Rigid displacements 629

This theorem remains valid if one reverses the sequence of the orthogonal transformation and the translation, but one must then use another transla- tion in order to obtain the same displacement g as before, specifically

t , ' a = a!. ta-la. (A.17)

A.2.3 Orthogonal transformations

Orthogonal transformations a have a number of important properties, which we will now explore in detail. For this purpose we introduce a Cartesian coordinate system with orthogonal unit vectors el, e 2 , e 3 . The effect of a on the basis vectors can be described by means of a matrix A ( a ) having real elements given by

a i k ( a ) = ei. a e k . (A. 18)

The transformation a itself may be expressed in terms of the dyadic struc- tures e i e k composed of two vectors e i and e k as follows:

a = C C a i k ( a ) e z e k . i k

(A. 19)

The application of a as given in equation (A.19) on a vector v is defined in terms of the formation of the scalar product of e k with v, i.e.

av = c C a i & ) e i e k ' v. (A.20) i k

It follows that

a e i = C a k i ( a ) e k = C a $ ( a ) e k . (A.21)

where a:(.) signify the elements of the matrix [A(a)IT, the transpose of A ( a ) . Using of A ( a ) , we may also describe the effect of the transformation a on a point

k k

x = C x i e i , 1

where x i = e i . x

are the Cartesian coordinates of x. We obtain

(A.22)

(A.23)

(A.24)

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630 Appendix A. Group theory for applications in semiconductor physics

By this relation one may understand the coordinates to be the transformed quantities instead of the basis vectors, as an alternative interpretation. The corresponding transformation equation reads

(A.25)

This is to say that the coordinates themselves transform with the matrix A ( a ) , in contrast to the basis vectors whose transformation is governed by the transposed matrix [A(a)IT.

The product 01 . a2 of two orthogonal transformations a1 and ag, un- derstood in the sense of sequential application, is represented by a matrix A ( a 1 . ag) which one can easily determine with the help of relation (A.19), obtaining

i.e. the matrix of the product transformation is represented by the product of the matrices of the two individual transformations.

Furthermore, other properties of the transformation operators a are car- ried over to the corresponding matrices A ( a ) . In particular, it follows from equation (A.13) that [ A ( a ) ] + = [ A ( a ) ] - l holds. Since the adjoint matrix [A(a)]+ of a real matrix A(&) equals the transposed matrix [ A ( a ) l T , one also has

[ A ( a ) l T = [ A ( a ) ] - l . (A.27)

Real matrices with this property are called orthogonaL Thus orthogonal transformations are represented by orthogonal matrices with respect to or- thonormalized basis sets.

Relations which follow immediately from (A.27) are A . AT = 1 and A T . A = 1, which may be written in terms of matrix elements as

This means that orthogonal matrices have the property that different rows are orthogonal to each other, and so are different columns.

The product of two orthogonal matrices is again an orthogonal matrix. Combining this result with equation (A.26), we arrive at the conclusion that the product of two orthogonal transformations is also an orthogonal transformation. The implicit assumption behind this result is that both transformations have a common point which remains unchanged (the point x = 0 here).

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A.2. Rigid displacements 63 1

Figure A.l: Definition of the Euler angles. t$

A.2.4 Geometrical interpretation

Orthogonal transformations may be interpreted geometrically. In order to develop such an interpretation one needs some results concerning the eigen- values and eigenvectors of orthogonal matrices. As is known from linear algebra, these matrices have complex eigenvalues with magnitude 1. The corresponding eigenvectors form a basis in 3-dimensional complex vector space. In this space the matrices can be transformed in such a way that they become diagonal (in the real 3-dimensional vector space this is not pos- sible). For each eigenvalue X the corresponding complex conjugate A* is also an eigenvalue, so there must be at least one real eigenvalue of magnitude 1. To this real eigenvalue corresponds a real eigenvector e. The real eigenvalue of unit magnitude can only be either +1 or -1. The determinant Det[A] of A is +1 or -1, respectively. In the case of Det[A] = +1 one has

ae = e. (A.29)

Since only the direction, but not the length, of e is determined by this relation, equation (A.29) means that all points given by xo = <e with < as arbitrary real number which lie on a straight line in the direction of e through the origin, are transformed into themselves. One says that they are fixed points of the transformation a. The orthogonal transformation (Y with Det[A] = +1 therefore corresponds to a rotation about the axis defined by xo. A special rotation is the identity transformation a = E. In the case Det[A] = -1 one has

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632 Appendix A. Group theory for applications in semiconductor physics

ae = -e. (A.30)

The transformation a therefore describes a rotation about the axis xo = e followed by a reflection in the plane normal to the rotation axis through the origin. It is called an improper rotation or a rotation-reflection . Among the improper rotations are also pure reflections in the plane normal to the axis, and inversion with respect to the origin. In the first case the rotation angle is 27r, and in the second case T. Since an improper rotation is constituted of the consecutive application of a rotation and a reflection, all orthogonal transformations may be traced back to pure rotations and pure reflections.

The orthonormality relations of (A.28) mean altogether six independent equations. The nine elements of an orthogonal matrix may thus be thought of as functions of three independent variables. The choice of these vari- ables is not unique. The most common is the one by Euler, who uses three angles $,8 and p. Two of these angles, $ and 8, determine the direction (sin 8 sin $, - s in0 cos $, cos 0) in which the z-axis is turned, and cp is the rota- tion angle about the transformed z-axis (see Figure A.l). The corresponding transformation matrix reads

sin$ cos‘p + ws6 cos $ sin9 -sin $ sin ‘p + cos6 cos $cos ‘p - sin 6 cos $

cos $ cos‘p - cos 6 sin$sin‘p - cos$ sin ‘p - cos 6 sin $ cos ‘p sin6 sin$

A = ( sin 6 sin ‘p sin 6 cos ‘p cos 6

Using this matrix, a reflection in a plane normal to the direction (sin8 sin$, - sin 8 cos $, cos 0) may be traced back to a reflection in a plane normal to the z-axis, which is easily described. The matrix A‘ of the reflection in the plane normal to the above cited direction thus reads

(A.32)

cos 28 sin $ + cos $ sin 2$ sin 0 - sin 28 sin $

cos 28 cos $ + sin $ sin 28 cos $

sin 28 cos $ - cos 28 - sin 28 sin $

A.2.5

The geometrical interpretation of the orthogonal transformations makes it possible to provide concrete illustrations of the theorem on the decomposi- tion of rigid displacements into translations and orthogonal transformations. One can show, specifically, that a rotation about a particular axis followed

Screw rotations and glide reflections

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A.2. Rigid displacements 633

Figure A.2: a) Rotation followed by a translation normal to the rotation axis, b) Reflection followed by a translation normal to the mirror plane.

I

I b)

Figure A.3: a) Rotation followed by a translation parallel to the axis (screw rotation), b) Reflection followed by a translation parallel to the mirror plane (glide reflection).

by a translation normal to the rotation axis, can be represented as a pure rotation around an axis parallel to the former, but shifted with respect to it in the normal plane. A reflection with respect to a particular plane followed by a translation normal to the plane can be represented as a reflection in a plane parallel to the former, but translated with respect to it in the normal direction. Using Figures A.2 and A.3 one can readily convince oneself of the validity of these assertions. A formal proof may be given as follows.

We consider, first of all, rotations followed by translations normal to the rotation axis. In this case, the transformation g is of the general form (A.10), where the hed-points xo of cy are solutions of the equation

ax0 = XO, (A.33)

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634 Appendix A . Group theory for applications in semiconductor physics

with xo‘a= 0. (A.34)

The identity element Q = E must be excluded here because no rotation axis is defined for it. What has to be proved is that, under the conditions of equation (A.33), the transformation g itself has hed-points xb which form a straight line, i.e. that the equation

cvxb+a=xb (A.35)

has a 1-dimensional solution manifold. To verify this, we express xb in the form

x’o = (2x0 + ra + sa-la (A.36)

with q , r, s as coefficients which still have to be determined. Since the three vectors xo, a and a-la, are linearly independent, such a representation is always possible expect for the special case aa = -a in which the rotation angle p of Q about the axis xo is equal to T. The latter case has to be considered separately. Employing equation (A.36) for xo jointly with equa- tion (A.35), we obtain a vectorial relation for q, r and s which is satisfied by arbitrary values of q. Consequently, the result is an inhomogeneous vector equation in the plane perpendicular to xo for r and s only. Considering the components of this equation with respect to a and cra, we find a system of two inhomogeneous equations, whose determinant equals laI2 2 sin2 cp. It never vanishes since the two cases cp = 0 and cp = T have been excluded, and thus there is a unique solution for r and s. In the excluded case cp = n one may take s = 0, and then the above procedure yields a unique solution for r. With this, our assertion is proved for a rotation with consecutive translation normal to the rotation axis. In the case of a reflection accompanied by a translation normal to the mirror plane, the proof is analogous.

There remain rotations and reflections followed by translations parallel to, respectively, the rotation axis and the reflection plane. That these cannot be replaced by pure rotations or reflections is obvious. The rotation axis and the reflection plane are unaffected by the translation. Therefore a pure rotation would have to be one about the same axis, and a pure reflection one with respect to the same plane. But these have to be followed by translations in order to get the complete transformations. Therefore pure rotations and reflections do not exist in this case.

In summary, this means that rigid displacements can be traced back to rotations and reflections only if translations which are directed parallel to the rotation axis or mirror plane are not involved (see Figures A.2 and A.3). Rotations followed by translations parallel to the axis are called screw rotations, and reflections followed by translations parallel to the mirror plane

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A.3. 'Ikanslation, point and space groups 635

are glide reflections. These results allow one to express the above stated theorem in the following, more specific form:

Theorem 2

A n y rigid displacement is either a screw rotation ore a glide reflection, o r a product of the two.

A.3 Translation, point and space groups

A.3.1 Lattice translation groups

Translations which transform crystal lattices into themselves only may have displacement vectors a equal to lattice vectors R. We will show that the set of all lattice translations t R forms a group. The product tR1 . tR, of two translations tR1 and tRz is defined as the translation which results from the consecutive application of the two individual translations. One may express this by writing

which yields

tR1 ' tRz = t R ~ + R ~ (A.38)

The latter relation means that the product of two lattice translations defined in equation (A.37) is again a lattice translation. This set therefore forms a group if points 2), 3) and 4) of the general group definition are also ful€illed. That this is so may be demonstrated easily. Using relation (A.4) one can readily show that the product (A.38) obeys the associative law (point Z), that an identity element exists, namely the translation through the lattice vector 0 (point (3)), and that the inverse of each element t R belongs to the group, namely the translation t-R through the lattice vector -R (point 4).

Each lattice translation can be expressed as a product of translations through three primitive lattice vectors al, az, ag, so that

(A.39)

Since the sequence in which the primitive translations are executed plays no role, all other lattice translations also commute with each other, i.e. the translation group is Abelian.

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636 Appendix A . Group theory for applications in semiconductor physics

A.3.2 Point groups

Definition

Point groups are sets P of orthogonal transformations a, namely of rotations, reflections, rotation-reflections , rotation-inversion s and the inversion itself, which transform geometric bodies having planar boundaries, i.e. prisms, tetrahedrons, cubes or other polyhedrons, into themselves. One may thus define point groups as the symmetry groups of these bodies. In this context their elements are called point s ymmet ry elements or point s ymmet ry oper- ations. That the set of all point symmetry operations of a geometric body actually forms a group, may be seen as follows. Firstly, as has already been demonstrated above, the product a1 . a2 of two orthogonal transformations a1 and a2, understood as their consecutive application, is itself again an or- thogonal transformation. The condition under which this statement holds, namely the existence of common fixed-points, can be satisfied in the case at hand. One has to keep in mind, however, that the various symmetry rotation axes and mirror planes of the geometric body have to be chosen to have at least one point in common. Secondly it is also clear that the transformation a1 . a2 maps a body into itself if this is true for a1 and a2 separately. Thus the product also belongs to the set P of all point symmetry operations as well. The validity of the associative law (a1 .a2).ag = al.(aq.ag) is evident, likewise the existence of a identity element and of the inverses. This means that, with respect to the above defined product, P does in fact form a group. Since rotations about two different axes and reflections at different planes do not, in general, commute the point groups are in general non-Abelian.

The number n of the different rotations about a particular axis is finite in all point groups. From this it follows that the rotation angles may only differ by multiples of (an/.). Moreover, only a finite number of different rotation axes and mirror planes occur. Point groups are therefore finite groups. This follows from the restriction to bodies with planar boundaries. If we would had also allowed for the sphere as geometrical body, then the symmetry operations would have included rotations about arbitrary axes and through arbitrary angles, likewise for reflections at arbitrary planes. Then one would no longer have, however, a point group, but the ful l orthogonal group. If one removes from the latter group all elements which involve a reflection or inversion and retains only pure rotations the remainder is the ful l rotation group. Both groups, the full orthogonal group and the rotation group, are infinite. The point groups may be understood as finite subgroups of the full orthogonal group.

Beside geometric bodies having planar boundaries, there are yet other point sets which are transformed into themselves by point groups. In the

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A.3. Tkanslation, point and space groups 637

main text we have shown that such sets include infinite point lattices. In this case, however, only 7 special point groups are admissible as symmetry groups. As far as ideal crystals are concerned, these are neither geometric bodies nor infinite lattices. Therefore, the symmetry groups of crystals, the so-called space groups, are not necessarily constituted of point groups. This is not so for the symmetry groups of equivalent crystal directions, which were also considered in the main text. These are point groups, but again only special point groups are possible, altogether 32. One calls them point groups of equivalent crystal directions or, in short point groups of directions.

Description of point groups

Generating symmetry elements

The lattice symmetry translations can be set up as products of translations through the three primitive lattice vectors. It turns out that likewise also all point symmetry groups can be generated from a small number of elements by forming products of them. The set of generating elements in the case of point groups is, however, larger than for translation groups. In neither of the two cases is the choice of generating elements unique. Here we con- sider the transformations which are commonly used for the construction of point groups. Their actions are illustrated in Figure A.4. A list of these transformations follows:

a) Rotation C, by an integer fraction 2 n l n of 2 n about a vertical axis. As ‘vertically’ one refers to the axes of highest symmetry. Thus it depends on the point group which axis is named vertical. On each axis one defines a direction to which the axis points, so that rotations by positive and negative angles can be distinguished. Positive angles correspond to a right handed screw sense of the rotation. The whole set of rotations by 2 ~ / n about the axis is called an n-fold rotation axis or a rotation axis of multiplicity TL The symbol for such an axis in the Schonflies notation is C,, just as for the single rotation through 2nln as indicated above. Commonly, one tolerates this ambiguity of notation and adds, if necessary, whether the axis is meant or the rotation. The international notation for an n-fold axis is n.

b) The rotation U:! by n about an axis which forms a right angle with a vertical axis, i.e. about a horizontal axis. In the international notation scheme a horizontal axis is abbreviated by a ‘2’ which follows the symbol n for the n-fold vertical axis to which it belongs, as in n2.

c) Reflections nV or Od in a plane which contains a vertical axis. This plane is called a vertical plane or a diagonal plane. In the international scheme one uses an ‘m’ for such a plane, following the n for the vertical axis which it contains, as in nm.

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638 Appendix A. Group theory for applications in semiconductor physics

cn .t m -__ _---

l

"2 I

I b)

el

Figure A.4: Generating symmetry elements of the 32 point groups of crystals.

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A.3. ZhnsJation, point and space groups 639

S ni I ?:' f )

I

Figure A.4: Generating symmetry elements of the 32 point groups of crystals (continuation).

d) A reflection u h in a plane orthogonal to a vertical axis, i.e. in a horizontal plane. The international notation is an 'm' under the 'n' for the vertical axis, as in

e) A rotation by 2 ~ / n about a vertical axis followed by a reflection at the horizontal plane ffh containing the axis, i.e. The Schonflies symbol for this element is Sn. By definition we have

a rotation-reflection .

s n = U h ' cn=cn ' O h ' . (A.40)

The two factors in this product commute because the reflection Uh does not change the directionof the rotation axis and the absolute value of the rotation angle, if it is applied before instead of after the rotation. The direction to which the rotation axis points is reversed, of course, however, with this, the sign of the rotation angle is also changed, so that in both cases we have the same rotation.

The symbol Sn is also used to denote an n-fold rotation-reflection axzs, which means the whole set of powers of 27r/n.

f ) A rotation by 2 ~ / n about a vertical axis followed by an inversion I at

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640 Appendix A. Group theory for applications in semiconductor physics

Figure A.5: Principle of the stereographic projection.

Figure A.6: Graphical sym- bols for symmetry elements of point groups. Shown in the left column are 2-, 3-, 4- and 6- fold rotation axes, in the mid-

A A

* @

dle column 2-, 3-, 4 and 6-fold rotation-reflection axes, and in the right column vertical and di- agonal mirror planes.

the fked-point (see Figure A.4g), or an rotation-inversion The Schonflies symbol for it is S,i. By definition we have

S,i = I . c, = c,. I. (A.41)

The two factors commute for the same reason as in the case of rotation- reflections considered above. Rotation-inversion axes, which are understood as the set of all powers of S,i, are used in the international system. The notation for an n-fold rotation-inversion axes is E.

Stereograms

Before investigating the various point groups, we will provide a procedure for the 2-dimensional illustration of these groups, namely the method of stereograms. This method relies on the fact that the elements of point groups are orthogonal transformations which transform a point on the surface of a

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A.3. Banslation, point and space groups 64 1

sphere around the common fixed-point into another point on the surface of the sphere. The elements of a point group are thus put into one-to- one correspondence with points on the surface of a sphere. At best, one imagines the sphere as a globe. To achieve a planar representation) one utilizes, as in cartography, the stereographic projection. This is illustrated in Figure A.5 with the south pole as the reference point. In this case the northern hemisphere is projected onto the region within the equator, and the southern hemisphere onto the region outside the equator. In order that only interior points need to be considered) one takes the north pole as the reference point for projection of the southern hemisphere. Points of the northern hemisphere are shown as full circles in the projection, and points of the southern are blank circles (see Figure A.5). It can be shown that the stereographic projection preserves circles and angles, i.e. circles (including straight lines as circles of infinite radius) are projected into circles, and the angles between intersecting tangents of circular segments also remain the same in the projection. As for the elements of the point group, one can also represent rotation axes and mirror planes by means of stereographic projection. Figure A.6 displays the corresponding graphic symbols. A n- fold rotation axis, for example, is depicted as a small regular n-angle placed at the projection of the intersection of the axis with the northern hemisphere. A mirror plane is illustrated by the projection of the arc on which the plane intersects the northern hemisphere. That the equatorial plane is a mirror plane may be recognized by the fact that for each full circle one has a blank circle at the same place. The procedure described here allows a 2-dimensional illustration of the point groups. The result of the projection is called a stereogram.

The stereograms may also be used to obtain the product of two group elements. As an exercise, the reader should try to verify, in this way, the following helpful relations between generating elements:

u2 . Uv = U v . u2 = U h ,

u2. U h = U h . u2 = UV ,

where U2 is the horizontal axis given by the intersection of the two planes uv and Uh. Analogous relations hold with U d replaced by uv. Furthermore, one has

U h . c 2 = c 2 ' U h = I , (A.43)

I . U h O h ' I = c 2 , c2 ' I = I . c2 = Uh,

I . uv = UV' I = u2 , u2. I = I . u2 uv.

Here U2 is understood to be orthogonal to uv.

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6 42 Appendix A. Group theory for applications in semiconductor physics

Figure A.7: Stereogram of the point group C3 (3).

General rules for the division of point groups in classes

E

The division of point groups in classes of conjugate elements is facilitated by some general rules which follow immediately from the definition of the conjugation given in section 1. These rules read as follows:

1) Rotations through the same angle with respect to symmetrically equiva- lent rotation axes are mutually conjugate.

2) If the group contains a symmetry operation which reverses the direction of a rotation axis, as with a 2-fold axis normal to the rotation axis, or a symmetry operation which changes the sign of the rotation angle, as with a mirror plane parallel to the rotation axis, then the axis is called two-sided Each rotation around a two-sided axis is conjugate to its inverse.

3) Reflections at symmetrically equivalent planes are conjugate with respect to each other.

Depending on whether a point group contains only rotations or, in addition, reflections, rotation-reflections , rotation-inversion s or inversion, one speaks of point groups of the f irst kand or point groups of the second kind. The point groups of the first kind may also be understood as finite subgroups of the full rotation group. Below the full set of these groups will be described.

Point groups of the first kind

cn (4 The simplest point group of the first kind consists of the n powers CA, C$- . . . , Cg of the rotation C, around a vertical axis. These are rotations through, respectively, ( 2 ~ / n ) , (2n/n) . 2,. . . , (21r/n) . n = 2 ~ . The rotation Cc is the unit element E. Using the terminology introduced above one may

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A.3. Translation, point and space groups 643

Table A.1: Multiplication table of the point group C3.

U U

briefly say that the group consists of an n-fold rotation axis. Thus the nota- tion for the group is C, in the Schonflies system, and n in the international system. The multiplication of two rotations Ck and C i leads to

c,” . c i = cy. (A.44)

If k + j is larger than n, one may replace C;+j by C,k+j-.. The product thus belongs to the set of elements CA, C:, . . .. Table A . l shows the complete multiplication table of the group CJ. The product in its kth line and Z t h column is the result of the multiplication of the lcth element as left factor and the l th as right. In the group C, the multiplication order plays no role, because this group is cyclic and thus also Abelian. Each element forms a class of conjugate elements by itself. The stereogram of the point group C3 is shown in Figure A.7.

Starting from the point group C , one may set up other point groups of the first kind by complementing the n-fold rotation axis with new generating elements, along with the products of these elements among themselves and with the elements of the original group.

D, (n2, n22)

The group C, is first of all complemented by a horizontal rotation U2. This may also be expressed by saying that a horizontal 2-fold rotation axis is added, for the identity element is already contained in C,. However, in this way no group is yet obtained. The multiplication of the powers C$-’, k = 1,2, . . . , n, with U2 produces n - 1 new elements, namely U2k = Ck-’ . U2 (here Uzl is merely another notation for U2). The set of these elements is identical to the set of elements U2 . Ck-’ arising by multiplication with U2 from the left, but the numbering is different. The n elements U2k correspond to n different 2-fold rotation axes. Each pair of adjacent axes forms an an angle x/n. For odd n all n rotation axes U2k are symmetrically equivalent to each other, while for even n they fall into two distinct classes of inequivalent axes, namely U21, U23, .., U2,-1 for odd 1, and U2, U22, .., U2n-2 for even

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644 Appendix A. Group theory for applications in semiconductor physics

a ) b)

Figure A.8: Stereograms of the point groups D2(222) (a) and D3(32) (b).

Table A.2: Multiplication table of the point group D3

k . One must add all n rotations u 2 k to the group C, in order to get a new group. This group then contains 2n elements. Their designation in the international system is n2 if n is odd, and n22, if n is even. The ‘2’ following the odd ‘n’ stands for the n symmetrically equivalent horizontal rotation axes, and the ‘22’ following the even ‘n’ stands for the two symmetrically inequivalent classes of axes. In the Schonflies system the designation of the group is D, in both cases.

One may interpret the group D, as the symmetry group of all rotations of a straight prisms through a regular n-angle (the full symmetry group of this body still contains reflections and is therefore of the second kind). Figure A.8b shows the stereogram of the group D3, and Table A.2 shows the multiplication table of D3. The latter may easily be checked using the stereogram in Figure A.8b. The group D2 with three mutually orthogonal 2-fold axes, called also V , is shown in Figure A.8a.

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A.3. ?f-anslation, point and space groups 645

Figure A.9: Rotation symmetry of a tetrahedron (left) and stereogram of the point group T (23)(right).

We still have to specify the division into classes of conjugate elements for the point group D,. Applying the first rule for class division derived above one finds that the n 2-fold rotations form one class for odd n, and two for even n. The second rule implies that a rotation Ck and its inverse CLk belong to the same class. These form (n - 1)/2 classes if n is odd, and n/2 if n is even. In addition one has the identity element E which forms a class by itself. Altogether, this yields (n - 1)/2 + 2 classes for odd n, and n/2 + 3 classes for even n. The group Dz has the 4 classes E , C2, U21 and U22, i.e. each element forms a class by itself. The group D3 has 3 classes, namely E , the three 2-fold horizontal rotations (U21, U22 , U231, and the %fold vertical rotations (C3, C:} (here and below elements of the same class are shown in curly brackets).

T (23) Another point group of the first kind is the group of all rotations which transforms a tetrahedron into itself (the full symmetry group of a tetrahedron also contains rotation-reflections , and is therefore of the second kind). The rotation axes of a tetrahedron are the four %fold vertical axes C3 through a corner and the center of the opposite face, and the three vertical axes C2 through the centers of opposite edges (see left-hand part of Figure A.9). We will not name the axes differently. The international symbol of the group reads '23', and the Schonflies symbol T . The four %fold axes yield a total of 8 different rotations, 4 of the form C3, and 4 of the form C:. The three 2-fold axes yield a total of 3 elements of the form Cz. Along with the identity

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646 Appendix A. Group theory for applications in semiconductor physics

Figure A. 10: Stereogram of the point group 0 (432).

element, the group T therefore consists of 12 elements. That the set of these elements does in fact form a group is most easily verified by means of the stereogram in Figure A.9. Each of the following sets of elements forms a class: the 4 rotations C3, the 4 rotations Ci, the 3 rotations C2, and E. There are, altogether, 4 classes.

0 (432)

The group of all rotations which transform a cube into itself is also a point group of the first kind (the full symmetry group of a cube is of the second kind). The group consists of the three 4-fold rotation axes C4 through the centers of two opposite cube faces, the four %fold vertical axes C3 formed by the diagonals of the cube, and the six 2-fold axes C2 through the centers of opposite cube edges. Altogether, this comprises 24 symmetry operations. They are illustrated in the stereogram of the group in Figure A.lO. The international symbol of the group is 432, and the Schonflies symbol is 0. The classes of 0 are the following: all 6 rotations C4 and C: (for the axes C4 are two-sided), all 3 rotations Cz, all 8 rotations C3 and Ci (the C3-axes are likewise two-sided), and all 6 rotations C2 and E. One therefore has 5 classes. We abbreviate them in the form E , 6C4, 3C2, 8C3, and 6C2, the symbol for each class being a representative element with the number of elements in the class positioned in front.

Beside those already mentioned, there is only one other point group of the first kind, namely the rotation symmetry group Y of a pentagondodec- ahedron. This refers to a body with twelve faces, each of which forms a regular pentagon. The group Y , which is also known as the icosahedral group, contains &fold rotation axes which cannot occur in the symmetry groups of crystals. Therefore we will not discuss the group Y here further.

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A.3. fianslation, point and space groups 647

Point groups of t h e second kind

sn ( z , ii), n even

The simplest point group of the second kind consists of one n-fold rotation- reflection axis s, of even order n. In the Schonflies system this group is denoted by S,. It contains rotation-reflections S," with k = 1,2, . . . , n. Since for even n the relation S," = (Uh . C,)" = E holds, there are no other elements. For odd n one has S," = Oh, which means that the set of the n powers Sk, with k = 1,2,. . . , n does not form a group. Thus, we exclude odd n.

The group S, is evidently cyclic. Each of its elements forms it own class, for a total of n classes. The even powers Sk describe rotations through angles ( 2 ~ k / n ) = (27rg/?). Thus the n-fold rotation-reflection axis also automatically generates a (n/2)-fold ordinary rotation axis. This also means that the group C,/2 is a subgroup of Sn. For even n not divisible by 4, the group S , contains the inversion I , namely the element S,"12 = (Uh.Cn)n'2 = a h . C2 = I . The group S 2 consists only of I and the identity element E. One also denotes S 2 as Ci. Each group S, with even n, not divisible by 4, can be understood as the direct product C,l2 x Ci of the two groups C,/z and Ci.

In the international notation system one employs, instead of rotation- reflections s,, the rotation-inversion s S,i as generating elements. In the case of even n, the powers Sii with k 1, 2, . . . , n already form a group, because then Szi = E holds. For odd n one needs the 2n powers Ski, k = 1, 2, . . . , 2n, to get a group. In both cases the international symbol reads 6. If n is odd, the group ii is equal to S2n For even n one must distinguish between those n which are divisible by 4 and those which are not. In the former case, we have ii = S,, i.e 4 = S4, 8 = SB etc. In the latter case the group f i can be generated by a (n/2)-fold rotation axis and a vertical mirror plane q. The group 2 consists of only this mirror plane, which implies that it contains only the two elements E and f f h . It is denoted by C,. Generally, the group +i can be written as the direct product C , / 2 x C, of C,p and C, if n is even and not divisible by 4.

Applying the above results to our original groups S, with even n, we recognize that they can all be traced back to 6' groups. One has S2 = T, S 4 = 1, s6 = 3, s8 = 8, Slo = 5, S12 = Tz etc. Some of these point groups are illustrated by their stereograms in Figure A.ll.

=

The group C, gives rise to a new group if one adds the reflection O h at the plane normal to the rotation axis. It contains besides the n elements C,"

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648 Appendix A . Group theory for applications in semiconductor physics

@ Figure A.11: Stereograms of the point groups Sz(T) = Ci, S4 (4), s6 (3), and 2 = c,.

with k = 1 , 2 , . . . ,n, also the n elements ffh . ck = ck . ffh. Its order is thus 2n. The international symbol reads E, and the Schonflies symbol Cnh. The groups Cn and C, are subgroups of Cnh. One has c n h = Cn x C,. For even n, the group c n h contains the inversion and can be written as the

Figure A.12: Stereogram of the point group CB~;).

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A.3. Tkansfation, point and space groups 649

a ) b)

Figure A.13: Stereograms of the point groups C3,(3m) (a) and C4v(4mrn) (b)

direct product C, x Ci of the groups C, and Ci. For odd n, Cnh is equal to z, as shown in the previous subsection. In any case, the group Cnh is Abelian. Each element forms a class by itself, thus there are 2n classes. The stereogram of the group C3h is shown in Figure A.12 as an example.

The group C, may also be complemented by a normal mirror plane uv as opposed to a horizontal one. Multiplying the powers Ck-', k = 1,2, . . ., n, with u,, n different elements U& = Ck-'.uv arise, where awl represents u,. The set of these elements coincides with the set of elements uv. C2-l obtained by multiplying by C6-l from the right, but the numbering is different. The elements u,k correspond to n different normal mirror planes. Each adjacent pair of them forms an angle r /n . For odd n all n mirror planes Uvk are symmetrically equivalent, for even n they fall into two distinct classes of symmetrically equivalent planes, the uwl, uv3, .., uvn-l with odd k , and the uvr uv2, .., u,,-~ with even k . One must add all n mirror planes to the group C, in order to get a new group. The latter then contains, along with the n rotations C," a total of 2n elements. The international symbol of the group is nm, if n is odd, and nmm, if n is even. The 'm' following the odd 'n' describes the system of n symmetrically equivalent normal mirror planes, and the 'mm' following n describes for the two inequivalent classes of n/2 such planes. The Schonflies symbol of the group is, uniformly, Cnv. The group may be understood as the symmetry group of a straight pyramid upon a regular n-angle. The stereograms of the two point groups C z and C4, are shown in Figure A.13 as examples. If one replaces the n mirror planes by n 2-fold rotation axes, then the group C,, becomes the gropu D , already considered. We conclude that the two groups C,, and D , are isomorphic.

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650 Appendix A. Group theory for applications in semiconductor physics

0 0 i-_)@ 0 0 0

a) b)

Figure A. 14: Stereograms of the point groups Dzh($$$) (a), and Dsh(6rn2) (b).

The division into classes of D, corresponds to that of C,. This means that C,,, like D,, has 1 + (n + 1)/2 classes if n is odd, and 3 + n/2 classes if n is even.

Dnh (---, n 2 2 - 2nm2) m m m

In this case the group Cnh is complemented by the n 2-fold horizontal axes U2 of D,. Multiplying the 2-fold rotations with ‘Th, one automatically obtains n normal mirror planes, each of which cuts the horizontal plane f f h of Cnh

along one of the n 2-fold rotation axes. This yields 4n elements altogether. The group can be interpreted as the symmetry group of a straight prism upon a regular wangle. The Schonflies symbol reads Dnh. For odd n the identity = 5 holds. This gives rise to the international symbol z 2 m in this case. For even n, no uniform relation exists between the subgroup n - and an rotation-inversion axis. Instead, in this case each mirror plane m is normal to one of the 2-fold axes. This leads to the international symbol $26. Among the subgroups of Dnh are, in each case, the two groups D, and C,. Their direct product D, x C, is the full group Dnh. For even n, D,h contains the inversion. Then the group may also be written as the direct product Dn x Ci of the two groups D , and Ci. Figure A.14 shows the stereograms of D2h and D3h.

The division of D,h into classes may easily be obtained from that of D,. Each class of D, is also a class of Dnh. In addition one has the classes which emerge from those of D , by multiplying them with f f h or, in the case of even n, also with I . This yields n + 3 classes if n is odd, and n + 6 if n is even.

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A.3. Tkanslation, point and space groups 65 1

a) b )

Figure A.15: Stereograms of the point groups D&2m) (a), and D3d(3$) (b).

Consider the division of a prism having the symmetry group D , normally to the n-fold rotation axis in two half-prisms, and then rotate them towards each other by an angle xln. In doing so a body emerges whose symmetry group forms one more point group of the second kind. It includes a 2n-fold vertical rotation-reflection axis S2,, n 2-fold rotation axes U2 perpendicular to it, and n mirror planes a d which contain the rotation-reflection axis and intersect the angle between each pair of adjacent 2-fold axes half way. This yields, altogether, 4n elements. The Schonflies symbol of the group is Dnd.

We already know that for even n the relation S2n = 2n holds. The inter- national symbol is therefore z 2 m . If one adds the inversion to this group, then the rotation-inversion axis 5 becomes an ordinary rotation axis Cz,, and the n diagonal mirror planes f f d change over into n diagonal 2-fold axes U2 by multiplying them with I . A diagonal mirror plane f f d of Dnd, if multi- plied with the 2-fold axes U2 it contains, yields the horizontal mirror plane ah. The resulting group is thus nothing but &,h. This means that for even n we have D d x Ci = D2,h.

For odd n, S2,, = n from which the international symbol f i 6 follows. For both even and odd n the group D , is a subgroup of Dnd. In the case of odd n the group contains the inversion (while D n h did so for even n ) , and may be written as the direct product D, x Ci. The stereograms of the two particular groups D2d and D3d are shown in Figure A.15.

of D d by n horizontal 2-fold rota- tion axes Uz in these planes, then the rotation-reflection axis S2, becomes the rotation axis C2n, and t,he group Dnd becomes the group D2n. These groups

-

If one replaces the n mirror planes

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652 Appendix A . Group theory for applications in semiconductor physics

Figure A. 16: Stereogram of the point group ~ h ( 6 3 ) .

7 are therefore isomorphic. Knowing this, the classes of Dnd may be obtained directly from those of D2n. According to what we found earlier, the number of classes of D2,, amounts to 2n/2 + 3 = n + 3, whether n is even or odd. The corresponding n + 3 classes of D d are E , the n classes {Szn, S::,"-' }, { sin, s::-'}>, . . . , { SF;', ST$+' 1, { ~ 2 ~ ) of powers of rotation-reflections around the two-sided rotation-reflection axis Szn, the class of the n 2-fold rotations U2, and the class of the n reflections uv.

The group Th may be obtained from the group T of all symmetry rotations of a tetrahedron by adding the inversion I . The group is therefore the direct product T x Ci, and contains 24 elements. Besides the inversion these are the following: In forming the product of the three 2-fold axes C2 with I one obtains three mirror planes uh perpendicular to these axes. Each of the four %fold rotation axes gives rise to a 6-fold rotation-reflection axis sf3 = 3, which means eight new group elements. In Figure A.16 we show the stereogram of the group. Its international symbol is $3. The classes of Th follow from those of T : each of the 4 classes of T is also a class of Th, and in addition there are 4 further classes obtained by multiplying the former by I . The group Th then has 8 classes altogether.

The full symmetry group T d of a tetrahedron follows from T on replacing each of the three 2-fold rotation axes C2 by a 4-fold rotation-reflection axis S4 and adding the six vertical mirror planes uo which contain these axes. In this way 12 new elements occur, yielding 24 altogether. That the set of these elements does form a group can be seen from the stereogram in Figure A.17. The international symbol of the group reads 43m, which reflects its

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A.3. Banslation, point and space groups 653

Figure A. 17: Stereogram of the point group ~ d ( 1 3 m ) .

Figure A. 18: Stereogram of the point group oh($$$).

construction and is self-explanatory. The group T d is isomorphic to the group 0, where the rotation-reflection axes S4 of T , are assigned to the ordinary 4-fold axes C4 of 0, the uv to the C2, and the C3 again to the C3.

The classes of T d are the 6 elements S4 and Si , the 3 elements ,942 = C:, the 8 rotations C3 and Cz, as well as the 6 reflections uV. Along with E , there are 5 classes. We abbreviate them by the previously introduced notation E , 6S4, 3C2, 8C3 and 6uv.

The full symmetry group Oh of a cube may be obtained from the symmetry group 0 of its rotations by adding the inversion I . Therefore Oh = 0 x Ci. The 24 new elements include, beside the inversion, the reflections at the six planes Ud normal to the six 2-fold rotation axes of 0, and the three mirror planes Uh normal to the three 4-fold rotation axes of 0. The or&-

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6 54 Appendix A. Group theory for applications in semiconductor physics

nary rotation axes become rotation-inversion axes, i.e. one has four %fold rotation-inversion axes 3 in the direction of the %fold ordinary axes of 0, and three 4-fold rotation-inversion axes in the direction of the 4-fold ordi- nary rotation axes of 0. The four %fold rotation-inversion axes lead to 8 new elements, and the three 4-fold ones to 6 . The 48 elements of Oh are depicted in the stereograms of the group in Figure A.18. The international symbol of o h is $3$ (abbreviated as m3m).

The six planes Uh of O h may be interpreted as 6 planes uv which contain the three 4-fold axes C4. If one observes that each rotation-inversion axis 4 is nothing but a 4fold rotation-reflection axis S4 , then it becomes evident how those elements of T d arise in o h , which are not already contained in 0 (these are the four %fold rotation axes and the even-numbered powers S t of 5'4). From this one may conclude that, with 0, T d is also a subgroup of o h . Since T d does not contain the inversion, one may understand O h as the direct product T d x I of T d with I .

The classes of o h may be derived either from those of 0 or those of T d .

The 5 classes E , 6C4,3C;, 8C3 and 6C2 of 0 are complemented by the classes I , 6 1 . C 4 , 3 I . C:, 8 1 . C3 and 6 I . (32 = Oh. If one refers to T d , one has first of all the 5 classes E , 6 S 4 = 6 I . C4,3C,2,8C3 and 6u,, which, by multiplication with I , then yield the 5 missing classes I , 6 1 . S4 = 6 C 4 , 3 I . C:, 8 1 , C3 and 6 1 . uv = 6C2.

A further point group is the full symmetry group Yh of a pentagondodec- ahedron. It follows from the icosahedral group Y by adding the inversion. For the description of the symmetries of crystals, Yh is of as little importance as Y .

It may be proven that the set of the possible point groups is exhausted by the above list.

A.3.3 Space groups

Space groups are sets G of rigid displacements g = t a . (Y of crystals which transform crystals into themselves. That these sets form groups is proved as follows. The product g1 . 92 of two displacements g1 = t,, . a 1 and 92 = taz . a2 is again defined by their successive application. Thus, the equation

91 ' 9 2 = tal+alaz . 02, (A.45)

holds which means that the product is again a rigid displacement. The crystal will transform into itself under the action of the product g1 . 92, if it does under the factors g1 and 92 separately. Thus g1 . 92 belongs to the set G . The other group properties can also be similarly verified using the product (A.45). The group G represents the spatial symmetry group of a crystal. If one imagines the crystal as the composite of several superimposed

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A.4. Representations of groups 655

lattices of the same Bravais type, then it becomes evident that the lattice translation group must be a subgroup of the space group.

Theorem 2 of subsection 2.1, wherein each rigid displacement is either a screw rotation, a glide reflection or a product thereof, is important for the construction of the space groups. It means that the generating elements of these groups are screw rotations and glide reflections. Lattice translations and orthogonal transformations belong to them as special cases. How space groups may be set up from these elements is explained in the main text.

A.4 Representations of groups

A.4.1 Introduction

The concept of representations

We consider a group G of rigid displacements, say a translation group, point group or space group. Although representations will be investigated only for these particular groups, most of our considerations are independent of particular group properties and hold for groups in general.

A representation of a group is defined in a certain space. Here, we use the Hilbert space R of quantum mechanical state vectors p of an electron. Spin will be omitted initially (it will be included later). The components of a vector 'p are written as functions p(x) of position vector x, where x varies over the entire (infinite) coordinate space. Since for each x, the value g-lx is also in coordinate space, p(g-lx) belongs to the Hilbert space if 'p(x) does. If g runs through the entire group G, then a particular function po(x) gives rise to as many different functions 'po(g-lx) as there are elements in the group. Among these functions some number d are linearly independent. They span a d-dimensional subspace Rd of the Hilbert space R. Let p(x) be an arbitrary function of R d . Through the relation

9 CpW = (P(g-lx) (A.46)

a h e a r operator j in R d is uniquely assigned to each element g of G . we will show that the operator 91.92 which according to equation (A.46) cor- responds to the product g1 . g2 of two group elements g1 and 92, is equal to the product of the two operators gl and 92, i.e. that the relation

91 ' 92 = Sl . 9 2

is valid. To this end we form ='p(x). By definition we have

(A.47)

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656 Appendix A. Group theory for applications in semiconductor physics

Applying the operator g1 to 92 p(x) = Cp(gF1x), the relation

(8.48)

(A.49)

follows. Comparison of equations (A.48) and (A.49) immediately yields equation (A.47). This equation means that the linear operators g defined by equation (A.46) form a group G.

The unique assignment of the original group G to a group G of linear operators in a certain space Rd, is called a representation of the group G in Rd. Applying the concept of ‘homomorphy’ introduced in section 1, one may also say that a representation of a group G is a homomorphic assignment of G to a group of h e a r operators in a space Ed. As a general symbol for a representation we choose D. The space R d is called the representation space. The dimension d of the space is referred to as the dimension of the represen- t a t i o n Not every subspace of the Hilbert space R can be a representation space, although the multiplication rule (A.47) holds everywhere in R. In order that a subspace Rd have this property, it must contain no vectors p ( x ) such that ~p(9-l~) ( g being any group element) does not also belong to Rd. The statement that a particular subspace Rd forms a representation space of a group G is therefore not trivial.

A formal difficulty for the representations of the displacement groups under consideration, more strictly speaking, for the translation and space groups, lies in their being infinite groups, i.e. their having infinitely many different elements. The representations of infinite groups are more difficult to treat than those of finite groups. Because of this we will focus our consid- erations on finite translation and space groups. This may be done by means of the the periodicity region concept introduced in Chapter 2 (the physical meaning of this concept was also explained there). Instead of the Hilbert space of arbitrary one-electron quantum states p(x), one uses that of states being periodic with respect to the periodicity region. Then

(A.50)

holds for arbitrary integers n1, n2, ng. With respect to these states the translations t R and tnlGlal+nzG2a2+n3G3a3 are represented by the same op- erator ifi. Thus, as long as one deals with representations in the Hilbert space of functions obeying the periodicity condition (A.50), one can render the translation and space groups finite by considering only translations t R through lattice vectors R belonging to the same periodicity region. The thusly reduced translation group T contains only G3 different elements, as many as there are different primitive unit cells in the periodicity region.

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A.4. Representations of groups 657

The applications of group theory in quantum mechanics rest almost exclu- sively on the theory of representations. The reason for this is that in quantum mechanical investigations, as a rule, it is not the groups by themselves which are of interest, but the operations of their elements on wavefunctions. For this reason the following theorem, already cited in Chapter 3 is of particular importance.

Theorem 1

T h e subspace of eigenfunctions of a Hamil tonian having the s a m e energy eigenvalue f o r m s a representation space of the fu l l spatial s ymmet ry group of the Hamil tonian

Below we will demonstrate that so-called irreducible representations are of particular importance in this context.

Representations depend not only on the groups which are represented but also on the space which underlies the representation. Here, we have used the Hilbert space of wavefunctions cp(x) of a single electron without spin. This gives rise to a particular type of representations (referred to as vector representations). Later we will consider the space of spin-dependent wavefunctions cp(x, s ) which introduces representations that do not occur in normal Hilbert space (so-called spinor representations).

We will now show that the particular representation of the group G of rigid displacements g defined by equation (A.46) exhibits an important prop- erty, which other conceivable representations do not have. We are referring to its unitarity property. An operator g is called unitary if its Hermitian adjoint operator g+ equals the inverse operator, such that

(A.51)

holds. That these relations are in fact valid for the operators ij defined by equation (A.46), may be shown as follows. We form the scalar product

(A.52)

of two arbitrary vectors cpl(x) and cp2(x) of Hilbert space R obeying the periodicity condition (A.50), where the integral is taken over a periodicity region. The integration variable x in (A.52) is then replaced by g-lx. Since a rigid displacement does not change the volume element, we find

(A.53)

According to the definition of the hermitian adjoint operator gf one has (jp2 I g(01) = (g f . j c p 2 I cpl). Substituting this relation in (A.53), it follows

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658 Appendix A. Group theory for applications in semiconductor physics

(cp2 I cpl ) = (3+. 3 9 2 I cpl). (A.54)

Because 9 1 and cp2 are two arbitrary vectors of Hilbert space R, the relation (A.51) must hold in R.

Representation matrices

It is convenient to describe the operators g of a representation as matrices with respect to a particular orthonormalized basis set cp', p2 , . . . cpd in the representation space R d . Since the vectors @+$, gcp2,. . . gpd are likewise vec- tors in E d , they may be written as linear combinations of basis functions cpl, p2, . . . cpd, whence

(A.55)

with Dpl(g) as expansion coefficients. The Diq(g) are elements of a quadratic matrix D(g) with d rows and d columns. The operators 9 are thus described by d-dimensional square matrices D(g). Since the 3 form a representation of the group G, the same also holds for the matrices D(g) representing them. The 'product operation' between the elements of the matrix group is ordinary matrix multiplication. This may be seen by calculating the matrix D ( m ) of a product element 91.92. By definition, we have

(91 .92) cpZ(X) = cDZIl(g1 . g2)cpZI(x). (A.56) I'

On the other hand. it is also true that

Comparison of equations (A.56) and (A.57) yields

(A.57)

(A.58)

D(g1 ' 92) = D(g1) ' D(92). (A.59)

The properties of the representation operators g are carried over to the representation matrices D ( g ) . Thus, the unitarity of the operators 3 results in unitarity of the matrices D(g) such that

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A.4. Representations of groups 659

D+(g) . D(g) = 1 or D+(g) = o-'(g). (A.60)

The Hermitian adjoint matrix D'(g) is defined as the complex conjugate transpose matrix [oT(g)]*.

Special representations. Equivalence

Each group allows for several more or less obvious representations. One of them is the so-called identity representation.

Identity representation

Every group element is associated with the identity operator or the identity matrix, in this representation space. The 1-dimensional identity representa- tion can be constructed as follows: One considers an arbitrary vector p(x) of Hilbert space, and forms the new vector cgl p(g'-'x), where the summation runs over all elements of the group G . This vector is then a 1-dimensional representation space of G, and the representation in it is the identity repre sentation. To prove this statement we apply an arbitrary element g of the group to Cgl p(g'-'x). In this, the sum reproduces itself because g'-lg-l runs through all group elements if g varies over the entire group. Thus we may write

(A.61)

According to this relation, all g are identity operators in the 1-dimensional space cgl p(g'-lx), and the representation is indeed the 1-dimensional iden- tity representation, as claimed. It is also now clear how one can construct identity representations of higher dimensions. One selects linearly indepen- dent vectors cpl(x), cp2(x) etc. in such a way that the sums Cgl cpI(g'-'x), Cgl (02(g'-lx) etc. are linearly independent of each other. Then the vectors Cgl pl(g'-'x), Cgl 'p2(g'-'x) etc. span a space in which all vectors trans- forms into themselves under the action of all elements of G , in other words it forms a multi-dimensional identity representation.

From a known representation V, additional representations of the same dimension can be derived. Below, we give the most important ones.

Complex conjugate representation V*

This representation is obtained when the representation space Rd of V is replaced by the complex conjugate space Rfi. The operators g* representing an element g in the representation V* are defined by the equation

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660 Appendix A. Group theory for applications in semiconductor physics

g*cp*(x) = q"(g-1x). (A.62)

The matrices D*(g) of the complex conjugate representation D* are the complex conjugate matrices [D(g)]* of the original matrices D(g), whence

D*(!?) = [D(g)l*. (A.63)

A representations is called real if its matrices are real, i.e. if

(A.64)

- Conjugate representation D

The conjugate representation 5 of a representation D is defined by assigning matrices b(g) to the elements g given by the equation

= [D(g-l)lT. (A.65)

One can easily demonstrate that the matrices b(g) form actually a repre- sentation if the matrices D(g) do so. In the case of a unitary representation, the conjugate representation equals the complex conjugate representation because

(A.66)

Equivalent representations

Transforming the representation space Rd by means of an arbitrary non- singular operator M likewise results in new representations of the same dimension. A representation DM is said to be equivalent to another rep- resentation V if it is associated with the group GM of operators

j M - - M . j. M-1. (A.67)

The group property of GM follows directly from that of the original operator group G. The associated matrices DM(g) of the equivalent representation read

DM(g) = M . D(g) . M - l , (A.68)

where M denotes the representation matrix of the transformation opera- tor A?. Since A? can be any non-singular operator in the representation

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A.4. Representations of groups 661

space R d , one can produce infinitely many equivalent representations from a given representation 2). It turns out that it is often unimportant to make a distinction between equivalent representations, and that only inequivalent representations are to be considered as essentially different. Therefore, one often refers to equivalent representations as representations which are equal ‘apart of equivalence’. Occasionally, one omits the qualification ‘apart from equivalence’ and just calls equivalent represent ations ‘equal’.

A.4.2 Irreducible representations

Reducibility and irreducibility

In the above definition of a group representation we did not exclude the pos- sibility that the representation space R d contain several genuine subspaces which are also representation spaces. If this is the case, the representation space is called reducible and the representation of the group in it is called a reducible representation. If the representation space does not contain such genuine subspaces, which are also representation spaces, one calls it irre- ducible. The representation of a group in an irreducible space is called an irreducible representation. The following important theorem can be proved:

Theorem 2

Each reducible unitary representation can be decomposed in to irreducible uni- tary representations.

The proof of this theorem will be briefly sketched below. By definition, in every reducible representation space Rd there is at least one subspace which is itself a representation space. Let R1 be such a subspace, either reducible or irreducible. Using the unitarity of the representation one can easily show that the complementary subspace %of R1 in R d is orthogonal to R1. Under the operations of a unitary representation the complementary subspace is then transformed into a subspace of Rd which is again orthogonal to R1.

Thus is mapped into itself, and it forms a representation space contained in Rd as well. If a smaller representation space R2 also exists in K, then its complement in K also forms a representation space contained in R d . One can continue this procedure until one reaches a remainder space which no longer contains a genuine subspace which is also a representation space. This space is then, by definition, irreducible. The same procedure can be applied to the representation spaces R1, R2 etc., which one has chosen in decompos- ing R d into subspaces. If these paces are reducible at the outset, then one can also decompose them into irreducible representation spaces by means of the above procedure. In this way a complete decomposition of the originally

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662 Appendix A. Group theory for applications in semiconductor physics

considered reducible representation space Rd into irreducible representation spaces R,,, v = 1,2,. . . has been accomplished. The corresponding repre- sentations D,, are irreducible. The original reducible representation D is decomposed, therefore, into a number of irreducible representations D,, and the theorem is proved.

The decomposition may be visualized by means of the representation matrices D(g) of a reducible representation D. We consider a basis in the representation space R d , which is composed of basis sets of the irreducible representation subspaces of R d . In this basis the d-dimensional matrices D ( g ) of the reducible representation Ig decompose into the matrices D,(g), D,t(g), D,tt(g), . . ., of the irreducible representations D,, D,,, D,tt, . . . con- tained in D. The matrices D,(g), Dy , (g) , D,,t(g), . . . are situated along the diagonal of D ( g ) as shown below:

..... 1 \ One says that the matrices D ( 9 ) are block-diagonalized. The block-diagonal form is thereby the same for all g , i.e. the matrices D,(g), Dvt(g), D,,i(g), . . ., for all group elements are, respectively, situated at the same positions on the main diagonal, and have, respectively, the same dimensions.

An irreducible representation V, exists for a subspace R, of Hilbert space such that, firstly, every g maps R, into R, (which means that R, indeed forms a representation space) and secondly that there is no yet smaller nontrivial subspace of R , with this property. We already know a special 1- dimensional irreducible representation, namely the 1-dimensional identity representation. From this we conclude that each group has at least one irreducible representation. In general there may be several.

That the degenerate eigenfunctions of a particular one-electron Hamilto- nian having the same energy eigenvalue form a representation space of the full symmetry group of the Hamiltonian, was already stated in theorem 1. If the energy eigenvalues are degenerate only because of spatial symmetry, i.e. if no accidental degeneracy exists due to the particular values of the potential or to time reversal symmetry, then it is possible to make an even stronger statement:

Theorem 1'

The subspace of eigenfunctiom of a given Hamiltonian having the same en- ergy eigenvalue is a representation space for an irreducible representation of the f i l l spatial symmetry group of the Hamiltonian

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A.4. Representations of groups 663

We will now discuss important relations which hold for matrices of irreducible representations.

Orthogonality relations

The elements D$(g) of the representation matrices may be understood as complex scalar functions defined on the group. In analogy to the ordinary scalar product between vectors we define the scalar product of two such arbitrary functions y ( g ) and 6 ( g ) through the relation

(A.70)

The factor 1/N with N being the number of elements of G is added for reasons of normalization. Orthogonality in the sense of this scalar product has the usual meaning (6 1 y) = 0. With this definition, we have the following theorem:

Theorem 3

The matrix elements Drk(g) and Dt",(g) of two inequivalent unitary irre- ducible representations are orthogonal functions over the group, i.e. they obey the relation

(A.71)

If the two representations Dp and V" are equivalent, then the matrix ele- ments D&(g) and D z k , ( g ) are orthonormal functions, such that

(A.72) 1

- d , (DCk, I D U ) - - 6ij~6kkf,

with d, = d, as the common dimension of the two representations. We know that the 1-dimensional identity representation D1, which as-

signs each group element to the number 1, forms an irreducible representa- tion for any group. If one identifies the representation D, with V1 in relation (A.72), then the equation

(A.73)

follows. Important instruments for investigation of the properties of representa-

tions are their characters. These will be treated now.

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664 Appendix A. Group theory for applications in semiconductor physics

Characters of representations

A d,-dimensional irreducible representation D, is composed of N matrices D'(g) of dimension d,. The sum of diagonal elements of a matrix D , or 'trace'

(A.74)

has the property that it does not change under a unitary transformation of the basis with a given matrix U . This is clear if we recall that the matrix D in the new basis is given by U-IDU and that under the trace symbol 'Tr' the factors of a product of two matrices can be interchanged. We have, therefore, TrIUDU-l] = Tr(U- lUD] = T r [ D ] . This property gives rise to the definition of the character of a representation in the following way. By means of the relation

(A.75)

each element g is assigned a particular complex number X,(g). This number is called the character of the element in the representation D,. The totality of the character values X,(g) for the various elements g of a group constitute the character X , of the representation One can thus say that the character is a complex scalar function defined on the group. The character of the d- dimensional identity representation has the constant value d over the whole group.

Two different elements g' and g", which are conjugate to each other in the sense of section A. 1, i.e. such that there is an element g of the group for which gN = gg'g-' holds, have the same character X,(g') = X,(g"). This follows, again, because of the commutativity of the operators under the trace symbol. One can also say that the character function X,(g) has a uniform value for each class of conjugate elements. The values change only in passing between different classes.

In subsection A.4.1, starting from a given representation, a series of other representations was constructed, among them the complex conjugate repre- sentation, the conjugate representation and the equivalent ones. The char- acters of these representations bear well defined relationships to those of the original representation. For the character Xv* ( 9 ) of the complex conjugate representation D: of a given representation D, one obtains

(A.76)

Real representations have real characters. From the reality of the character one cannot, however, deduce the reality of the representation, since complex representations can also have real characters.

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A.4. Representations of groups 665

For the character X c ( g ) of the conjugate representation 6u of a given rep- resentation Vur it follows that

(A.77)

For unitary representations this relation can also be written as

Thus the character X , ( g ) of an element g must be real in such a represen- tation if g is in the same class as the inverse element g-'.

The characters X,M (9) of all equivalent representations D,M of a given representation D, are equal and coincides with the character of V,. Thus,

If the complex conjugate representation V* is equivalent to the representa- tion V itself then the character of D must be real.

Relation (A.79) is, again, a consequence of the commutativity of matri- ces under the trace symbol. That equivalent representations do not differ by their characters underlines the above-made statement that there are no essential differences between such representations.

Theorems on irreducible representations

There are a number of important theorems relating to the irreducible rep- resentations of a group. Here, we will discuss the ones which are of direct relevance for semiconductor physics and which are used in the main part of the book.

Theorem 4

A f inite group has exactly as m a n y inequivalent irreducible representations as it has classes of conjugate elements.

It follows that finite groups can have only a finite number of inequivalent irreducible representations. In all cases, one of them is the 1-dimensional identity representation.

Theorem 5

T h e s u m of the squares d: of the dimensions of all non-equivalent irreducible representations Dv yields exactly the number N of the elements of the group, so that

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666 Appendix A. Group theory for applications in semiconductor physics

Ed: = N . U

(A.80)

This statement is referred to as Burnside's theorem. An immediate conse- quence is that the dimension d, of any irreducible representation of a finite group must also be finite. There is an even stronger statement relating to dv :

Theorem 6

The dimension d , of any of the irreducible representations of a finite group i s a divisor of the group order N .

Using theorems 4, 5 and 6, and considering the fact that among the irre- ducible representations there is always a 1-dimensional representation (namely the identity representation), one often can determine the dimensions of the irreducible representations of a group without knowing the representation matrices themselves. The group &d, for example, has 8 elements and 5 classes. There is only one way to obtain the group order 8 by adding 5 squares, namely by taking four squares of 1 (12 + l2 + l2 + l2 = 4), and one of 2 (22 = 4). This means that the group D w has four 1-dimensional and one 2-dimensional irreducible representations. For an Abelian group Burn- side's theorem immediately implies that only 1-dimensional representations are allowed for in such a groups each element forms its own class, so that the number of classes equals the number N of group elements. Because of (A.80) then all d , must then be 1.

Theorem 7

The characters of the irreducible representations of a group are orthonormal with respect to each other, whence

Since one of the irreducible representations is the 1-dimensional identity representation D1, whose character has the value 1 for all group elements, we obtain from equation (A.81)

(A.82)

By means of this theorem the characters of the irreducible representations

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A.4. Representations of groups 667

can be often already determined without knowing the representation matri- ces.

Each unitary reducible representation V can be decomposed into irre- ducible representations, as was shown at the outset. A particular irreducible representation may occur repeatedly. If m, is the multiplicity of the irre- ducible representation V, in the reducible representation V, then the char- acter X(g) of ’D may be written in the form

(A.83)

where the sum is extended over all distinct irreducible representations V, of the group. Using the orthonormality relation (A.81), one can determine the multiplicities m,r immediately. Upon scalar multiplication of X with Xu(, it follows that

(xV/ I X ) = x(g) = Cm,(x,/ I x,) = m y / . (A.84)

This relation provides a simple procedure for determining if and how often a particular irreducible representation V, occurs in a supposedly reducible representation 2). The reduction of V into VU1s may be written in the symbolic form

Y

= Cm,D,. V

(A.85)

The summation in (A.85) is to be understood in terms of placing mu copies of the dv-dimensional matrices on the main diagonal of the d-dimensional matrix D. One has, of course, d = Cu mu . d,.

A.4.3 Products of representations

Definitions and theorems

We now introduce the concept of the direct produet of two representations. Let R, be a representation space for the representation V, of G, and Rv be a representation space for V, of G. We will show that the product space R , x Ru of the two spaces is also a representation space. Let cp,(x) be an arbitrary vector of R,, and cp’(x’) an arbitrary vector of R,. Then cpf’(x) . cpV(x’) forms a general vector of the product space. Since, with v p ( x ) . cp’(x’) we also have yP(g-lx) . q ~ ” ( g - ~ x ’ ) in the product space, the equation

g cp,(x). cp”(x’) = yP(g-1x) . (p”(g-1x’) (A.86)

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668 Appendix A. Group theory for applications in semiconductor physics

assigns each element g of the group an operator g. This assignment is a representation of the group in the product space, called the product repre- sentation D, x V,. Along with (A.86) we also have

g p ” ( x ) . p,(x’) = p”(g-1x) ’ p y g - l x ’ ) (A.87)

The direct product representation is therefore independent of the ordering of the factors, i.e. we have

D , x V , = V , x ~ , . (A.88)

The matrices DPXu(g) of the direct product representation V, x D, can be easily calculated from the matrices Dp(g) and D”(g) of the two fac- tors. To this end we introduce a basis in the product space. Let the vectors p y ( x ) , p g ( x ) , . . ., p$@(x) be a basis in R,, and the vectors pi(x’), p?(x’ ) , . . ., pzp(x ‘ ) are a basis in R,. As a basis of R, x R,, we therefore have the prod- ucts pf(x) . p;E(x’) with 1 = 1,2,. . . , d , and k = 1 , 2 , . . . , d,. By definition,

D:tlyk = D:l . DLfk. (A.91)

For the character X P x , of the direct product representation it follows from (A.91) that

or

X,xv(g) = & ( g ) . X d g )

This is summarized in the following theorem:

Theorem 8

(A.93)

The character of the direct product of two representations is the product of the characters of the factors.

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A.4. Representations of groups 669

The direct product representation is in general reducible even if the factors were irreducible. To find all irreducible parts one may use relation (A.84). It is of particular interest to know whether the identity representation is contained in the direct product or not. The answer to this question is given by the following theorem:

Theorem 9

T h e direct product Vz x V, of two unitary representations V*, and V, con- tains the identity representation exactly i f and only if the representations Vp and V, are equivalent t o each other. T h e identity representation then occurs exactly once.

We will now consider the direct product of a representation with itself.

Direct product of a representation with itself

In this case one has

The product space Rux , can always be decomposed in this case into two representation subspaces R t x u and R;,,, where RExu is spanned by the symmetrical combinations (l/a)[Cp;(x) . $(XI) + $(x) . (p;(x’)] of the products of basis functions, and RExu by the antisymmetric combinations (l /&)($(x). cpi(x’) - (pi(x) . &(x’)]. The dimension of REx, is given by d(d f 1)/2, and that of RExu by d(d - 1)/2. The representation [V, x V,I8 of the group G in RExu is called symmetric product and the representation [V,x V,], in RExu is the antisymmetric product. For the matrices DL:Jc(g) of [Vv x V,], we find from (A.89) and (A.90),

and for those of [V, x V,],, we have

The pertinent characters X i x u and XExu are

(A.95)

(A.96)

(A.97)

(A.98)

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670 Appendix A. Group theory for applications in semiconductor physics

Selection rules for matrix elements

We consider a particular group G of rigid displacements. In practical cases, G is the symmetry group of the Hamiltonian. The following considera- tions are not limited, however, to this meaning of G. Given two irreducible representations 'D, and 'D, of G , we take cp';(x), ( O ~ ( X ) , . . . , cpzv(x) as a set of basis functions in the representation space R, belonging to V,, and cpy(x), &(x) , . . . , cpZp(x) as a basis set in the representation space R, be- longing to V,. In the case considered above, the v y ( ~ ) , &'(x), . . . , cp$Jx) and cpy(x), &(x) , . . . , & J x ) form complete sets of eigenfunctions of the Hamiltonian for particular energy eigenvalues E, and E,, respectively. We will examine the matrix elements of a linear operator A in Hilbert space with respect to these basis functions. The operator A may be a tensor of arbitrary rank. We denote its components with respect to a particular basis el, e2, . . . of the tensor space by A1, A2 , . . .. The basis vectors e k are tensors of the same rank as A. To be able to use the theory of representations, we must assume that the components A k transform according to a particular representation DA of G. Thus, we should have

(GAg- ' )k = D $ k A k l . k f

For a scalar operator A, relation (A.99) takes the form

(A.99)

A. (A. 100)

If G is the symmetry group of H , then H itself can be taken as scalar operator. For the components ( A 1 , 6 2 , A g ) of a vector operator A with respect to a Cartesian coordinate system e l , e2, e3, one obtains from (A.99) the transformation

(A.lO1) k'

with Dgfk(g) as matrices of the representation VV of G which governs the transformation of 3-dimensional vectors. The momentum operator p serves as an example of a vector operator.

we now form the matrix elements of the operator components ( c p t I A k 1 (py) of A between the basis functions (pr(x), 1 = 1,2, . . . , d,, and &(x), m = 1,2,. . . , d,. Because of the unitarity of the operators g, we have

(A.102)

or, considering equation (A.99),

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A.4. Representations of groups 671

This means that the matrix elements (&&l(Pr) transform according to the direct product representation 23; x V A x D,. Summing over all elements g , we obtain from (A.103) the relation

(A.104) If the identity representation does not occur among the irreducible parts of the direct product representation V; x DA x D,, then the sum over g vanishes according to relation (A.73). In this case, one correspondingly has

i.e. all matrix elements ($7kl&(pfl) vanish for symmetry reasons. If the identity representation is contained in the direct product representation 21: x DA x Dp, the sum over g in (A.104) does not vanish and the matrix elements ($&1&1$7r) can be non-zero. Nevertheless, they may vanish for reasons unrelated to spatial symmetry.

If some of the elements ( & [ A k l p r ) do not vanish by reason of symmetry, questions remain open as to which of the elements are non-zero and what symmetry-induced relations exist between them, in other words, how many independent non-zero matrix element exist. We now proceed to discuss this question. To this end we start with equation (A.103) and rewrite it using the representation matrices

(A.106)

(A.107)

Because of the reducibility of the direct product representation D; x V A x D,, there is a unitary transformation U of the basis [cpf(x)]* . e k . cpL(x‘) of the product space such that the matrix D$j$Akl(g) in the transformed basis takes the block-diagonalized form (A.69). We denote the transformed basis by [&(x)]*. e K . cp&(x’), and the matrix elements with respect to this basis

p ’ x A x u ($7h I A k I $7;) = Dm’k’l’mkl(g)($7&’ I Ak’IpF). l‘m‘k’

~ ‘ X A X U by D M ! K i L i M K L ( g ) . Then We have

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672 Appendix A. Group theory for applications in semiconductor physics

where the DLtKtL!MKL(g) denote the block-diagonal matrices in the product space. The one non-vanishing block is the representation matrix d i ) ( g ) of an irreducible representation V A ( i ) of G . For different i, X ( i ) can have the same value, i.e. a particular irreducible representation matrix D x ( g ) can appear as a block in several matrices D L I K , L I M K L ( g ) . The number of these matrices is the multiplicity mA of equation (A.84) with which Vx is contained in VoT/ x V A x V p .

To facilitate the use of the decomposition (A.108) in relation (A.107), one must transform relation (A.107) into the new basis. In this, the summation over all elements g of the group G yields

All their blocks are zero except for one.

(A.109) Because of the orthogonality relation (A.73), the g - s u m results in a non-zero value only for i such that Vx(i) is the identity representation. We assume that this occurs in the matrices DLlKtLiMKL ( 9 ) for i = 1,2, . . . ml. For these values of i, the DL,,,,,,,,(g) are diagonal matrices with only one non-vanishing element. The one non-zero element has the value 1. We denote the position M'K'L' = M K L of this element by MiKiLi. Then equation (A.109) may be written as

Since 1 5 i 5 ml, these relations mean that exactly ml matrix elements ( c p L l A ~ I p ; ) can be different from zero. The matrices (&IAklpy) with re- spect to the original basis follow from (pLlA~lp;) by means of the transfor- mation U + inverse to U . In the back-transformed matrix, which is the orig- inal matrix (pklAklpr), generally, all elements (rpklAk1pf) differ from zero, but they all depend linearly on the mi non-vanishing elements ( p & M ( A ~ ( p z ) of the transformed matrix. Altogether, there are mi independent non- vanishing matrix elements (&I A k l p r ) . By means of the unitary transforma- tion U , the matrix (pk,lAkJ(~f) can be calculated explicitly from the known matrix (pLl2~lpL). The results above are summarized in the following theorem:

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A.5. Representations of the full rotation group 673

Theorem 10

T h e matrix elements (p$lAklp[) of the operator vanish f o r symmet ry reasons i f and only if the direct product representation VE x V A x V, does n o t contain the ident i ty representation. If the ident i ty representation i s in VE x VA x V,, n o t all elements (p&]Aklp[) vanish by reason of symmetry . There are as m a n y independent elements among t h e m as the number of t imes the identity representation i s contained in Vi x V A x D,.

From theorem 9, and the commutivity of the factors of a direct prod- uct representation, we conclude that the identity representation appears in VE x V A x V v as often as the representation product V; x V, contains the irreducible representation VA.

A.5 Representations of the full rotation group

Although the full rotation group has no direct meaning for crystals, indi- rectly, however, it plays an important role in their study. Its representations define the space of electron states in which the operators of quantum me- chanics act and, consequently, in which the representations of the symmetry groups of crystals need to be considered.

This distin- guishes it from the groups considered heretofore, which were finite and dis- crete. Many concepts and theorems formulated for the latter can be utilized directly for infinite groups. Among them is the group definition itself and the concepts of conjugate elements and classes of such elements. In the full rotation group, all rotations through the same angle, but with respect to different axes, are conjugate to each other. The corresponding classes, therefore, contain an infinite number of elements. The number of different classes is also infinite because the number of the different axes is infinite. The concept of representations may be used for the infinite rotation group as well. The dimension of the representation matrices is, however, no longer necessarily finite, as in case of finite groups. Matrices of any size are allowed, among them, of course, also finite ones. The reducibility and irreducibility of representations are defined in analogy with the corresponding definitions for finite groups.

A number of theorems which were formulated for finite groups lose their validity, however, or must be modified when applied to infinite groups. That this happens is then evident, if the group order or the number of classes enters the theorem directly. For example, one can only speculate whether the number of inequivalent irreducible representations of the rotation group is actually infinite, as it follows formally from theorem 4 and the fact that the number of classes is finite, for it is not clear whether this theorem still

The full rotation group is an infinite continuous group.

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6 74 Appendix A. Group theory for applications in semiconductor physics

holds in the case of infinite groups at all. We shall see below that there is in fact an infinite number of inequivalent irreducible representations of the rotation group, so that the formal application of theorem 4 is justified. Generally, one ought to use care in transferring results for finite groups to infinite ones. Luckily, it turns out that the construction of the irreducible representations of the rotation group (being the aim of this section) does not require the theorems derived above.

O n e of the irreducible representations of the full rotation group we know already, namely that in the 3-dimensional vector space, which is called the vector representation The corresponding representation matrices follow from the matrix A of equation (A.31), if the Eulerian angles are allowed to take all real values. The matrix A represents rotations in a special ba- sis, namely that of the Cartesian coordinate system before the rotation. As we will see, there exists in each representation space, also including that of dimension 3, exactly one (inequivalent) irreducible representation. Each 3- dimensional irreducible representation may, therefore, be obtained from the matrix A of equation (A.31) by a unitary transformation. The reason that we will once again deal with the 3-dimensional vector representation is that this representation is suitable for introducing a concept which is of impor- tance for irreducible representations of any dimension, namely the concept of generators of infinitesimal rotations.

A.5.1 Vector representation of the rotation group and gen- erators of infinitesimal rotations

The following construction of the vector representation of the rotation group G R relies on the fact that each rotation can be described by a directed axis and a rotation angle cp with respect to this axis. We denote the unit vector in the axis-direction by n. Let the Cartesian unit vectors be e,, ey, e,. We maintain that the coordinates x’, y’, z’ of the rotated vector x’ may be obtained from the coordinates 2, y, z of the vector x before rotation by means of the relation

(A. 11 1)

(A. 112)

is the rotation matrix. The components Iz, Iyl I, of I are called generators of the infinitesimal rotation. They are given by

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A.5. Representations of the full rotation group 675

I, = 0 0 -i , I p =

O i 0

0 0 i

0 0 0

-i 0 0

0 -i 0

0 0 0 113)

The exponential function in (A.112) is to be understood in the sense of a power series expansion with respect to -i(I. n)p. In the following we sketch the proof of assertions (A.ll l) and (A.112).

Initially, we consider a rotation about the z-axis through the infinitesimal angle dp. The coordinates after rotation, obtained using the matrix A of (A.31), are given by 2’ = 2, y’ = y - zdq,, z’ = z + ydp,. In matrix form these equations read

(A.114)

with D(ex, dp,) = E - iIxdvz, (A. 11 5)

and with E the 3-dimensional identity matrix. Analogous relations hold for infinitesimal rotations with respect to the y- and z-axes. If the three in- finitesimal rotations are executed one after another, then this corresponds, apart from higher order corrections, to a rotation about an axis in the direc- tion of the vector d$ = e,dp,+eydpy+e,dp, . With n as unit vector in the direction of d$, d p as the absolute value of d$, and I = Izex + Igey + Izez, the result of the rotation can be written as

with D(n, dp) = E - i ( I . n)dp.

(A. 116)

(A. 117)

Now consider a rotation through the finite angle cp with respect to the same rotation axis n. The pertinent rotation matrix is D(n,p). After this ro- tation we apply the infinitesimal rotation (A.117). The total rotation is described by the matrix D(n, p + dp). Because of the group property of the representation matrices, we have

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6 76 Appendix A . Group theory for applications in semiconductor physics

On the other hand, expanding D(n, cp+dcp) with respect to the infinitesimal angle dcp, we obtain

(A.119)

Comparing with (A.117) it follows that

(A. 120) -D(n, cp) = - i ( I . n)D(n, p), dcp

This differential equation is to be solved with the initial condition D(n, 0) = E. The result is just expression (A.112) for D(n, cp).

d

A.5.2

Equation (A.112) relates the rotation matrix with the three generator ma- trices I z , Iy, Iz. Through the latter D(n, cp) is uniquely determined. If, in turn, the rotation matrix D(n, cp) is known, the three generator matrices can be determined from it as follows

Representations for dimensions other than three

d d d I , = i--D(ez, cp)lrP=0, Iy = i-D(eY, cp)lrPP=o, I , = i -D(e, , cp)lrP=o.

dcp dP dcp (A.121)

These relations may be understood as the defining equations of the gen- erators matrices. As such, they may also be applied to representations of dimensions other than 3. With this in view we shall consider D(n, cp), and I%, Iy, Iz as matrices of any dimension henceforth. With a knowledge of the generator matrices for dimension d, one can then determine the rotation matrices of a d-dimensional irreducible representation by means of equa- tion (A. 112). In this way, the construction of the irreducible representations of the rotation group is traced back to the determination of the generator matrices. To accomplish the latter task, the commutation relations among the matrices are of great value. In the 3-dimensional case the explicit form (A.113) of the matrices I,, Iy, I, may be employed to show that

[I,, Iy] = iI, , [Iyr Iz] = iI, , [Iz, I,] = iIv. (A.122)

In quantum mechanics, the same commutation relations hold for the com- ponents of the angular momentum operator. This reflects a specific relation between the two sets of quantities, which we will discuss further below. The definition (A.121) can be used to demonstrate that the commutation rela- tions (A.122) apply to generator matrices of any dimension. One may thus use them for the construction of these matrices. Before we can do this, we must address two questions which, thus far, have been ignored the questions

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A.5. Representations of the full rotation group 677

which dimensions are allowed for the irreducible representations, and how many inequivalent representations exist for a given dimension.

Consider an irreducible representation 2) of the rotation group GR in a d-dimensional space Ed. Being a representation space of GR, Rd will also simultaneously be a representation space of the subgroup G, of all rotations about the z-axis. Due to the Abelian type of G,, its representations are 1-dimensional. Thus, D considered as a representation of G,, falls into a series of 1-dimensional representations. Let the corresponding basis vectors be bl , b2, . . . , bd. we will relate the rotation matrices D of the representation D to them in the following. For a rotation through an angle cp with respect to the z-axis. we have

D(e, , cp) b, = e-amuv b,, Y = 1,2,. . . ,d, (A. 123)

with m, an arbitrary integer (including zero). Differentiating relation (A. 123) with respect to cp and applying equation (A.121), it follows that

I,b, = m,b, , Y = 1,2, . . . , d. (A.124)

Instead of the generator matrices I , and I y , it is expedient to use the two linear combinations

I+ = I , + i I y , I - = I , - i Iy . (A.125)

By j we denote the largest of the integers ml, m2,. . . ,mu. Using of the commutation relations (A.122), it may be shown that the vectors

ej-1 = I-bj , ej-2 = I-ej-1 , ej-3 = I-ej-2 , ...

are all eigenvectors of I,. They satisfy the equations

(A.126)

I re , = m e , , m = j , j - 1,j -2 , . . . , (A. 127)

where for the sake of uniformity we have set e j bj. The ej-k are mutually orthogonal. Also they all lie in the representation space R d , so that not more than d of them can be different from zero. For a particular value k = T , we must therefore have

ej-T = I-.bj-T+l = 0. (A.128)

Similarly, by using the commutation relations, one can show that the oper- ator I' = I," i- I: + I," satisfies the eigenvalue equation

(A.129)

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6 78 Appendix A . Group theory for applications in semiconductor physics

with E as the identity matrix in Rd. To prove this relation, one first verifies the equation

I + e j = 0 (A.130)

arguing indirectly: If the vector I+ej were to be non-zero, then it would be an eigenvector of I , having eigenvalue j + 1. We have already assumed, however, that j is the largest eigenvalue of Iz in R d . On the basis of this contradiction, the assumption that relation (A. 130) does not hold must be rejected.

Using this result, one first proves equation (A.129) for m = j , by express- ing Ix and I, in terms of I+ and I-. To do the proof for arbitrary values of m, equation (A.126) and the commutativity of I," + I; + I," with I- are sufficient.

Equation (A.129) makes it possible to calculate the effect of I+ on the basis vectors em with m # j . To this end, one replaces em in terms of I-e,+l. For I+I-, the relation I+I- = I 2 + I, - I," may be employed, with the result

I+em = [ j ( j +l) - m(m +l)]em+l , m = j - 1 , j - 2 , . . . , j - r + l . (A.131)

Owing to above derived relations it is now clear that the matrices I+, I-, Iz transform the space spanned by e j , e j -1 , . . . , ej-,.+l into itself. The same holds, of course, for the generator matrices Ix, Iy, I, and the matrices D of the irreducible representation 2, of the rotation group defined by them. According to the initial assumption, the dimension of this representation is d , whence we have r = d.

There is an intimate relation between d and j , which may be seen as follows. The diagonal matrix elements (em I 1," + I; + I," 1 em), may be evaluated using equation (A.129), with the result

(em I 12 I em) = j ( j + 1) = m2 + (em I I: +I; I ern). (A.132)

Since here (em I I: + 1; 1 em) is positive, then m2 < j ( j + 1). The largest negative integer which satisfies this condition is m = - j . Thus m takes the (2j + 1) values - j , - j + 1, . . . , j - 1, j, and consequently d = 2 j + 1, or,

d - 1 j = - 2 - (A. 133)

Thus j is a half integer if d is even, and an integer if d is odd. Relations (A.126) to (A.131) define the elements of the matrices I+, I-, 1,

and, therefore, also those of the generator matrices Iz, Iy, I, with respect to the basis e j , ej-1, ..., e- j . The disadvantage of this basis is that it is not yet normalized, so that the generator matrices, and the corresponding

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A.5. Representations of the full rotation group 679

representation matrices are not yet unitary. Therefore, we now normalize the em and denote the normalized basis vectors by 2,. The latter obey the following relations in place of (A. 126):

with

i m = Nm I-6,+1 (A.134)

1 Nm = (A.135)

J ( j + m + l)(j - m )

as normalization factor. The elements of the matrices I+, I - and I, in the normalized basis become

( E m I Iz 1 i m l ) = m6,,1. (A.137)

The generator matrices of a given dimension result in an irreducible repre- sentation of the rotation group of the same dimension. This is evident from equation (A.112), which determines this representation uniquely. One can show that this is also the only irreducible representation of this dimension. An irreducible representation of the full rotation group is thus uniquely de- termined by its dimension d or, in view of equation (A.133), by the parameter j. One usually employs j and denotes the various irreducible representations by Dj. The corresponding representation matrices are denoted by Dj . Their elements (im I D j 1 Cm,) may be calculated by using equation (A.112), as has already been discussed, but other, methods based directly on the differential equation (A.120) are more practical. We will not discuss this matter here in detail, but only give the final results. Using Eulerian angles for the rotations a, one finds

3 - m)! ( j + m')! X

with p = cos 0. The following relations may be derived:

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680 Appendix A. Group theory for applications in semiconductor physics

( im 1 Dj l i m t ) * = (- 1)m-ml (kLm 1 Dj l L m t ) . (A.140)

To calculate the character X j ( g ) of a representation Dj it suffices to consider a rotation about the z-axis, i.e. to set 6 and $J to zero. Rotations g with all other values of 6 and space $, but the same value of cp, belong to the same class and thus have the same character X j ( g ) . Therefore

(A.141)

Sometimes the direct product of two irreducible representations Dj and Dji is needed. It contains each of the representations Dl with values of 1 between I j - j ' I and j + j ' exactly once. Thus the decomposition of the product reads as

j + j t

I = ( j- j l I vj x vjt = c n. (A.142)

It is also worth mentioning that the generator matrices give rise to an irre- ducible representation of the rotation group in yet another sense than that discussed before: they are themselves the basis vectors of a representation. The dimension of this representation is 3, i.e. the generator matrices trans- form according to the vector representation of the rotation group. If one adds inversion to this group, the result is the pseudovector representation. The reason for this is that the sign of the generator matrices cannot change under an inversion because of the commutation relations (A. 121). The transforma- tion of the generator matrices as pseudovectors is employed in the invariant method for construction of the k. p-Hamiltonian (see section 2.7). For that purpose, the explicit form of these matrices is needed. By means of formu- lae (A.136) and (A.137), one obtains, in the 2-dimensional case, the Pauli spin-matrices multiplied by 1/2. For d = 3, it follows that

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A.5. Representations of the full rotation group

and for d = 4 the result is

( 8 , 11, I &d) =

( 0 0 -1

0 id3 0

Id3 2 0 1

0 0 $d3 0 1 0

0 -$& 0

id3 0 -i

0 0 id3 0 i o

% o 0

0 ; 0

0 0 -4 0 0 0 - 3 2

68 1

(A.143)

(A.144)

Again, transformation of the pseudovector of three generator matrices re- flects the close relation between these matrices and the angular momentum operator J. This relation may be expressed as follows: The generator matri- ces of dimension d are the matrices which represent the angular momentum operator in the basis of the irreducible representation of the full orthogonal group of this dimension. The basis itself may be characterized in this con- text as the set of simultaneous eigenvectors of the square 5' of the angular momentum operator J for eigenvalue 7i2j(j + 1) with j = (d - 1)/2, and of the z-component J z of J.

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682 Appendix A. Group theory for applications in semiconductor physics

A.6 Spinor representations

A .6.1 S pace-dependent spinors

Having completed the discussion of irreducible representations of the full rotation group in the previous section, which, as noted above, have only in- direct significance for us, we now return to the representations of the finite groups of rigid displacements which are of immediate interest here. The smallest possible dimension of an irreducible representation of the full rota- tion group is 2, corresponding to the smallest possible value ; of j. For the pertinent representation matrix D i ( a ) , it follows from (A.138) that

2

(A'. 145)

The basis vectors -2 1, - 2 - ~ which transform according to this representation, are referred to as two-component spinors. We abbreviate them by Ii), I - i). In quantum mechanics, the states cp of electrons are described by just such spinors with the two components (slcp) cp(s), s = i, -$, corresponding to the two possible spin states. The two spinor components are themselves vec- tors in ordinary Hilbert space R having components (xlcp(s)) z p(x, s) with respect to the position vector x. For the quantum mechanical description of electrons, the representations of the group of rigid displacements G in the Hilbert space R of the two-component position dependent spinor functions cp(x) are of central importance (the notation R should not be mixed with that for a lattice vector). By definition, these representations imply that the elements g of the displacement group G are uniquely assigned to a group of linear operators g in the spinor Hilbert space R. Therefore, we must have

5 2

g1 . g2 = g 1 . g 2 . (A.146)

We now proceed to construct this assignment explicitly. To this end, we set

where S(g) denotes an operator in the 2-dimensional spin space. Since trans- lations do not act on the spin coordinate s, for elements g = t , . a we have

S(g) S(t , . a ) = D I (a) . (A. 148)

For pure translation groups the only representation in spin space is the identity representation. The group of rigid displacements can therefore be

2

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A.6. Spinor representations 683

restricted to point and space groups. Using the basis I s) in spin space, relation (A.147) can be written in the form

(A.149) S’

Since, according to the definition (A.46) of g, the function (o(g-lx,s’) in (A.149) equals 3 cp(x, s’), and since the operator 9 can be factored out of the sum, we have

The representations of the group G of rigid displacement in the spinor Hilbert space R are thus the direct products of representations in the Hilbert space R of scalar functions with the representation Vl of the full rotation group. In the next subsection we therefore explore the ;epresentation V1 in greater detail.

5

A.6.2 Representation D+

The two Euler angles cp and $ both vary between 0 and 2n. The ro- tations a[cp + 2a, 8,$] and a[cp, 8, $.J + 2n] are identical with the rotation a[cp,O,$]. Strangely enough this does not also hold for the pertinent ma- trices D g ( a [ p + 2n, 8, $]) and D;(a[cp, 8, $ + 2n]) from relation (A.145). Both equal -Dr(a[cp, 8, $1). As long as one considers only single rotations a[p, 8, $1 this does not cause any difficulty because one can always restrict cp and $ to the interval between 0 and 2n, and assign + D i ( a ) to the rotation a. Difficulties arise, however, if two rotations a1 and a 2 are multiplied. Then it may happen that one of the two angles (o12 or $12 of the product a1 . a 2

takes a value larger than 2n. In this case, the representation (A.145) assigns the matrix - D ~ ( a 1 . a 2 ) to a 1 . a 2 . As it is itself an element as well, a1 . a 2

has, however, already been assigned the matrix + D l ( a l . 4, which cannot be changed because the assignment in a representation must be unique. The only resolution of this difficulty is to allow the multiplication rule for the two representation matrices D l ( a l ) , D i ( a 2 ) to differ from that of the two group elements a 1 and a 2 by a minus sign. More precisely, the sign in this rule must be allowed to depend on the particular combination of the two elements a1 and a2. It must be a function w(a1, a 2 ) of a 1 and a 2 , with two possible function values +1 and - 1. The corresponding multiplication rule reads

2

3

2

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684 Appendix A. Group theory for applications in semiconductor physics

It is obvious that the matrices Di(cu) obeying this multiplication rule no longer form a representation of the group of rotations in the sense of section A.4 ~ according to the understanding there the factor w ( q , az) had to be +1 for all cu1, a 2 which is not the case here. In fact, assignments of matrices to group elements obeying the multiplication rule (A. 151) are representations in a generalized sense, that of so called spinor representations. They form a special case of projective representations or representations with factor systems which will be considered below in greater detail.

For the complete determination of the representations of point and space groups in spinor space we still need the matrix D i ( I ) for the inversion I . In determining it one can take advantage of the fact? that I commutes with all elements g of the space group. Therefore D i ( I ) must also commute with all D i ( g ) . It follows from a theorem of group theory (known as Schur’s l emma) that D i ( I ) must then be a multiple of the identity matrix 1. Because of [ D I J I ) ] ~ = 1, only the identity matrix itself or its negative are allowed. The two assignments

2

I

D.(I) = +1 or D l ( 1 ) = -1 I

(A.152)

give rise to two inequivalent irreducible representations of point and space groups containing the inversion.

A.6.3 Irreducible spinor representations

In the spinor Hilbert space R, the two spinor components p(s) are func- tions p(x, s) of the position coordinate X. It has already been demonstrated in subsection 1 that the representations of the displacement group G in R are given by direct products of the representation V L of G and a particular representation of G in the Hilbert space R of scalar functions. While Vi signifies a spinor representation, the representations of G in the ordinary Hilbert space R mean representations in the ordinary sense, often referred to as vector representations. The direct product of a spinor representation with a vector representation is again a spinor representation for the corre- sponding multiplication rule is of the general form (A.151). From this we may conclude that any representation of G in the spinor Hilbert space R will not be an ordinary, but a spinor representation. On the other hand, the direct product of two spinor representations yields a vector representation because the multiplication rules for the product representation are of the general form (A.59).

Of particular interest among the spinor representations of G are, of course, the irreducible ones. We now set out to show how to determine them explicitly.

2

2

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A.6. Spinor representations 685

A.6.4 Double group method

The irreducible spinor representations of G may be obtained by means of the irreducible vector representations of a fictional group, the so-called double group G‘ which has twice as many elements as G. The double group G’ is defined as follows. To begin with, we introduce a group ??’ of linear operators in spinor space, which consists of all representation operators 9 . 0 1 ( a ) where j is an element of the representation operator group G of G, and D 1 ( a ) is given by equation (A.145). The first half of the elements of g, which will be denoted by &, are taken to be those with the Euler angles ‘p and $ of a [ p , 0, $1 in D.(a) restricted to the interval between 0 and 27r, and the second half, which will be denoted by g>, are those elements for which one of the angles cp and $ lies in the interval between 27r and 47r, with the other one in the interval between 0 and 27~ as before. The operator group ??’ thus contains twice as many elements as G. The second half of the elements can be obtained by multiplying all elements of the first half by a rotation through a[27~, 0, 01 or a[O, 0,27~]. Let Q be the operator for a rotation through a[27r,O,O] . It then follows from (A.145) that D + ( Q ) is the negative of the identity matrix, -1. Therefore, D i ( Q ) . D L ( Q ) = 1. According to equation (A.145), the rotation a[O, 0,27r] is represented by the same operator D i ( Q ) . Since D$Q) acts only in spin space, one has D l ( Q ) x = x. Moreover, it holds that D L ( Q ) . j = g . D 1 p ( Q ) , i.e. the element D 1 ( Q ) commutes with all remaining elements of G’. Finally, we reverse the assignment of group elements and operators and assign the operator fj< to the displacement g,

and the operator g> to the displacement Q . g = g . Q. The group of displacements g and Q ‘ g is the double group G’ we need for

the construction of the spinor representations of the simple group G. The elements of G’ will be denoted by 9’. The set G of elements g , generally, does not form a subgroup of G’ because the multiplication of two elements g: and g b of G may lead to an angle cp or $ between 27r and 4%. Analoguously, the set Q . G = G . Q of elements Q . g or g . Q is not a subgroup of G‘. Whether the product g$ . gb belongs to G or Q . G can easily be determined. The square 1’ of the inversion I , for example, is the identity element E , and the square cr2 of a reflection equals Q. This follows from the relations Oh = I . C2

for Oh or uz, = I . U2 for uv (see equation (A.42)). Representing the double group G‘ in spinor Hilbert space R means

uniquely assigning to each element g’ a linear operator g’ in this space. This assignment is determined by that for the simple group in equation (A.153), according to

T

T

2

2

2

F

(A.153)

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686 Appendix A. Group theory for applications in semiconductor physics

It may easily be demonstrated that the assignment (A.153) forms an or- dinary or vector representation of G'. It also holds that each irreducible representation space R, of the double group G' simultaneously forms a rep- resentation space of the group G because the elements of G cannot map out of R, if none of the elements of G' does so. Moreover, as a representation space of G, R, cannot be reducible. Indeed, if R, were reducible, then there would be at least two irreducible subspaces of R,, one denoted by R,l, and another denoted by R,2 which forms the complement of R,l in R,. The two subspaces Rvl and Rv2 would be transformed only into themselves by all elements g of G. However, R,l and R,2 would also be mapped into themselves by all elements Q . g , because the application of Q results solely in multiplication by -1. Thus R, would also be reducible as a representa- tion space of GI, which contradicts the initial assumption. Each irreducible vector representation of G', therefore, is also an irreducible representation of G.

The latter may be either a vector representation or a spinor representa- tion, both cases must be taken into account. Since one can also prove the converse, i.e. that each of the irreducible vector or spinor representations of G gives rise to a vector representation of the double group GI, it follows that the set of irreducible vector and spinor representations of the simple displacement group G is identical to the set of irreducible vector represen- tations of the associated double group GI. Here we are only interested in the spinor representations of the original displacement group G. These can easily be selected from the whole set of irreducible vector representations of G' - they are all those which are not also vector representations of G. These irreducible vector representations of the double group are called eztra representations. Using this term one may state the following theorem:

Theorem 11

The irreducible spinor representations of a displacement group G (point or space group) are the extra representations of i ts double group GI.

The decomposition of double groups into classes is of importance for the explicit construction of the irreducible representations of these groups. Like the identity element E , Q also forms its own class. The reason for this is the commutativity of Q with all elements g and Q . g . From this, it follows that elements g which are conjugate in G are also conjugate in G'. The same holds for the pertinent elements Q . g , also they are in the same class of the double group G' if the elements g are in the same same class of the simple group G. If each element Q . g fell into a class different horn that of g , then the double group G' would always have twice as many classes as the simple group G. However, this is not the case. Indeed, if a rotation axis of the

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A. 7. Projective representations 687

order n is twesided (see section A.2), the elements Ck and CE-k = Q . Cqk belong to the same class, and so do the elements Q . Ck and CGk, however, the two classes are different. If an n-fold rotation axis is not two-sided, then C;, C i k , Q . Ck and Q . CLk form four different classes. For n = 2 the two elements C2 and C,' of the simple group G are identical, but in the double group Gt they differ for it holds C; = Q which results in C2 = Q . C,' and C,' = Q . C2. The two elements C2 and C,' 3 Q . C2 form one class if and only if the 2-fold axis is two-sided. An analogous statement holds for reflections uh. In this case uh and Q . uh belong to one class if the 2-fold rotation axis C2, associated with ah by the relation uh = I ' C2, is two-sided. Similarly, u,, and Q . cr, are in one class if the rotation axis U2, associated with u, by the relation u,, = I . U2, is two-sided.

We illustrate the above results using the tetrahedral group Td as an exam- ple. In section A.3 we found 5 classes of the simple group Td: E , 8C3,6S4,3C; and 60,. From them the following classes of the double group Td are obtained by means of the rules mentioned above: ( E } , {Q}, {4C3,4Q . Cj}, (4Q . C3,4C$}, {3C4,3Q.C,3},{3Q.C4,3c,3}, {3c,2,3Q.C,2}, { 6 u v 7 6 Q . ~ v } . The 48 elements of the double group Ti belong therefore to 8 different classes. Since the number of classes equals the number of different irreducible rep- resentations, and since the simple group T d has 5 classes, there must be 3 irreducible spinor representations of the group Td. The sum of squares of the dimension of these representations must be 24. There is only one way to obtain the number 24 by summing three squares, namely 2' + 2' + 4' = 24. Thus two of the three spinor representations of Td have the dimension 2, and one has the dimension 4.

The spinor representations of point and space groups were characterized above as projective representations with a particular factor system. Later, we will see that projective representations will also occur in another context, namely the ordinary vector representations of space groups. These represen- tations may be traced back to the projective representations of point groups of equivalent directions with a particular factor system determined by the crystal structure. This demonstrates that projective representations play an important role in the applications of group theory in solid state physics. In the following subsection, we discuss properties of these representations which will be of use later.

A.7 Projective represent at ions

A.7.1 Factor systems

Representations of groups are unique assignments of their elements to ele- ments of groups of linear operators in a particular space. In regard to the

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representations defined in section A.4, the multiplication rules of the op- erator group could be derived from that of the original group because the assignment was given a priori. According to these rules, the product of two operators assigned to two group elements was equal to the operator assigned to the product of the two elements. In the projective representations to be considered below one takes the opposite approach: Not that the operators ij assigned to the group element g are specified a priori, but rather the mul- tiplication rules of the assigned operator group G are defined at the outset. The assigned operators are later derived from these rules. The multiplica- tion rules of the operator group are defined as follows. Let gl and 32 be the two operators assigned, respectively, to the two group elements g1,gz. Then complex numbers w(g1,gz) depending on g1,g2 are introduced and the product 31 . 92 is defined as

91.92 = w(g1,92) 31 ’ 92 . (A.154)

The set of factors w(g1, 92) is called the factor system of the representation. The assignment of G to the operator group G defined by the multiplication rules (A.154) is referred to as a projective representation or representation with factor system.

The factors w(gI,g2) are restricted by the group properties of the oper- ator group G. The associative property results in the condition

It may be shown that this condition is also sufficient to assure that G is a group, i.e. to make G a projective representation of G. If the absolute value of the factor system w(g1,gz) is 1, i.e. if

holds, then the corresponding projective representation of the displacement group is unitary, as in the case of ordinary representations.

The ordinary representations are projective representations with the par- ticular factor systemwt(gl,g2) = 1 for an elements g1,g2. We callw!(g1,g2) = 1 the identity system. If a particular factor system w(gl,g2) is given, an infi- nite number of other factor systems w’(gl,g2) can be constructed by means of an arbitrary complex function u(g), namely by setting

(A. 1 5 7)

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A. 7. Projective representations 689

This can easily be verified by demonstrating that equation (A.155) holds for w’(g1, g 2 ) if it holds for w(g1,92). Two projective representations D’ and D whose factor systems w’(gl,g2) and w(gl ,g2) are related by the equation (A.157), are called pequiwalent. The representation operators $ ( g ) and $ ( g ) assigned to the group element g in the two p-equivalent representations obey the equation

(A.158)

The entirety of all p-equivalent factor systems is referred to as a class of factor systems. If the factor system has the particular form

(A.159)

then it is p-equivalent to the identity system w t ( g 1 , g a ) = 1, and the cor- responding class is called identity class. In this case the projective repre- sentations are p-equivalent to representations in the ordinary sense. The number of different classes of factor systems of a finite group is also finite. The numbers and particular forms of these classes are characteristic for each group. The identity class occurs for all groups. For cyclic groups, it is the only one possible. Generally, the number of different factor system classes is a power of 2 , therefore it is either 2O, or 2 l , or 22 etc. Most of the 32 point groups of crystal structures have several classes of factor systems, for example, D4h has 8, Td has 2, and Oh has 4. If there are only 2 factor sys- tem classes, then the second class beside the unit class necessarily includes the factor system of spinor representations because the spinor representa- tions also always exist. In special cases (in particular, for the point groups c1, c 2 , c 3 , c4, c6, c,, s4, C3h, D3, c3,,), the spinor representations coincide with the vector representations. Then only the identity class of factor sys- tems exists and the factor system of the spinor representations is p-equivalent to the identity system.

A.7.2 Definitions and theorems

Unitary equivalence

Let us consider two projective representations 2, and VM with the same factor system, and let R, and R,M be the corresponding representation spaces. The representations two are said to be equivalent to each other if, as in the case of ordinary representations, a unitary operator M exists which transforms the two representation spaces R, and R,M into each other. To

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avoid confusion with the term p-equivalence, one speaks of unitary equiv- alence rather than just of ‘equivalence’, as we did in the case of ordinary representations.

Reducibility a n d irreducibility

The concepts of reducibility and irreducibility can be defined for projective representations as well. This can be done in the same way as for ordinary representations, and we need not repeat the definition here. It also holds that any unitary projective representation with a particular factor system may be decomposed into irreducible projective representations having the same factor system as the original projective representation.

Characters

The operators g representing a group element g in a particular projective representation may be written as matrices D ( g ) with respect to a particular basis of the representation space R,. As for ordinary representations, the character X ( g ) of an element g is defined as the sum of the diagonal elements of its representation matrix D(g) . Also, the characters of unitarily equivalent projective representations are identical. However, unlike the case of ordinary representations, for projective representations the character of an element is no longer only a function of its class. As a consequence of this, the number of unitarily inequivalent irreducible projective representations of a group G no longer equals the number of classes of conjugate elements G, but is smaller than the latter one, unless the factor system class of the projective representation is that of the identity system. In the case of the irreducible spinor representations of the point group T d considered above, we already noted that there are 5 classes of conjugate elements in T d , but only three spinor representations.

If one knows the character X ( g ) of a projective representation, then the character X’(g) of a p-equivalent representation defined by equation (A.157) is given by the relation

X’(g) = 4 7 ) ’ x ( g ) . (A ,160)

If the matrices D ( 9 ) form an &dimensional irreducible projective represen- tation with a particular factor system, then it follows from (A.160) that the matrices D’(g) = u ( g ) D ( g ) form an irreducible projective representation of the same dimension d with the pequivalent factor system given by equation (A. 15 7).

Regarding the dimensions of irreducible projective representations of groups G possessing more factor system classes than just the unit class, one

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A. 7. Projective representations 69 1

may show that the representations for factor system classes different from the identity class must be 2- or higher dimensional. That this rule works may be seen from the irreducible representations of the group T d considered in the section on spinor representations. This group has two classes of factor systems and, therefore, one set of non-ordinary representations. These are the spinor representations of T d , and none of them is 1-dimensional.

Orthogonality

The orthogonality relations (A.71), (A.72) and (A.81) for the matrix el- ements and characters of ordinary irreducible representations also remain valid for projective irreducible representations with the same factor system. Matrix elements and characters of irreducible representations with differ- ent factor systems are always orthogonal to each other. Owing to these properties of the characters, the same procedure for the decomposition of a projective representation into irreducible parts can be applied as that used for ordinary representations, namely, in accordance with equation (A.83), the projection of the character of the representation onto the characters of the irreducible representations. Being a consequence of the orthogonality relations, Burnside’s theorem holds also for projective representations:

Burnside theorem

The sum of the squares of the dimensions of all unitarily inequivalent irre- ducible projective representations of a group having the same factor system equals the order of the group.

From this theorem and the statement on the dimensions of p-equivalent representations in connection with equation (A.160) it follows that for each projective irreducible representation with a particular factor system, there is exactly one projective irreducible representation with a p-equivalent factor system having the same dimension. If one, therefore, knows the projec- tive irreducible representations for one factor system of a class, one also knows these representations for all the other factor systems in the same class. The projective irreducible representations with factor systems from different classes will, however, differ in general from each other regarding their dimensions and numbers.

Direct product

The direct product of two projective representations is defined as it was for ordinary representations, i.e. as the representation induced in the product of the representation spaces. The factor system of the product representation equals the product of the factor systems of the two projective representations

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692 Appendix A. Group theory for applications in semiconductor physics

which are multiplied. A particular consequence of this rule is that the factor system of the product of two spinor representations for which the factors are either 1 or -1, equals the identity system. This confirms our former conclusion that the product of two spinor representations forms an ordinary or vector representation.

The consequences of spatial symmetry for the matrix elements of opera- tors as expressed in theorem 10 are also valid if the wavefunctions transform according to projective and, in particular, to spinor representations.

Finally, we sketch how irreducible projective representations may be con- st ruct ed.

A.7.3 Construction of projective representations

The irreducible projective representations may be constructed in a manner similar to the construction of the spinor representations in section A.6. Us- ing the particular factor system for spinors defined by equations (A.145) and (A.151) a double group G was introduced in addition to the original group G. The irreducible vector representations of the double group resulted in two sets of irreducible representations of the group G , namely the vector and spinor representations. It has already been pointed out that the lat- ter are projective representations of G with the particular factor system of spinors. The above procedure may be generalized to obtain the irreducible projective representations for all possible classes of factor systems. In fact, this procedure was originally derived in the general case of projective repre- sentations by Schur, and later Bethe applied it to representations in spinor space. Based on the various classes of factor systems of G , Schur constructed a super-group G’ of as many times more elements as there are different factor system classes. He then demonstrated that the ordinary irreducible represen- tations of this super-group G’ yield all irreducible projective representations of all classes of factor systems of the original group G . The explicit construc- tion of the super-group which delivers all the possible factor system classes exceeds the limitations of this introduction to group theory. The reader can find it in the book by Bir and Pikus (1974). This book gives also the irre- ducible projective representations of the 32 point groups of crystals for all of their factor system classes.

A.8 Time reversal symmetry

The state of an electron generally depends on time t , i.e. its components cp(x) in ordinary Hilbert space or cp(x, s ) in spinor Hilbert space are, respectively, functions cp(x, t ) or cp(x, s, t ) of t. As well as transformations in coordinate and spin spaces, one can also consider transformations on the time axis. Examples are translations of time by a constant to , i.e. t becomes t + t o , and

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A.8. Time reversal symmetry 693

time inversion i.e. the transition from t to -t. The latter transformation is commonly referred to as t i m e reversa l , and is denoted by the symbol K .

A.8.1 Time reversal operator

As in classical mechanics, the basic dynamical laws of quantum mechanics are invariant with respect to time reversal. In short, we say that they have t i m e reversa l s y m m e t r y . This symmetry holds, in particular, for the time- dependent Schrodinger equation governing the time development of electron states lp(x, t ) without spin, as well as for the Pauli equation which applies to electron states lp(x, s, t ) with spin. From this observation one can deduce the transformation of the wavefunctions under time reversal. If the time t is reversed in the Schrodinger equation, then the vector lp must be replaced by the complex conjugate vector lp* in order that the equation remain valid. The time reversal transformation K therefore causes the transformation of lp to lp*. In other words, in the Hilbert space of scalar functions the time reversal operation K is given by the operator K defined by the relation

Klp = lp*. (A.161)

If there is a magnetic field, one also has to reverse its direction in order that lp* satisfy the Schrodinger equation.

In order that the Pauli equation remain valid under time reversal, the operation K must be represented by an operator K in spinor Hilbert space defined by

being the y-component of Pauli’s spin matrices. One can easily show that for spinors the relation (klp I lp) = 0 holds, i.e. Klp is orthogonal to lp. If lp is an eigenstate of the Pauli Hamiltonian for a given energy, and if there is no magnetic field, then Klp forms a second linearly independent eigen- state for this energy. That means that, because of time reversal symmetry, all eigenvalues of the Pauli equation are at least doubly degenerate in the absence of a magnetic field.

Without spin it is evident that K 2 = E , and K 2 = -E with spin. This implies that without spin the unit element E and the time reversal operation K form a group, while with spin the elements E , K , K 2 and K 3 do so. We denote both groups by 2. Since K commutes with all elements of the group G, the direct product G x 2 of G and Z is itself also a group. We call it the space- t ime g r o u p and denote it by 6 . We are interested in the

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694 Appendix A. Group theory for applications in semiconductor physics

joint effect of spatial and temporal symmetry of a system. At first glance one could think of exploring these by using the irreducible representations of the space-time group Q defined above. This is, however, not possible, because of the unusual properties of the time reversal operator K . This operator is not linear, since one has %[clcpl + c2p2] = [crkpl + czkpn], i.e. one gets the complex conjugate coefficients c?, cf while the coefficients themselves would be needed for f? to be linear. The operator K is also not unitary, since one has (Ifpl 1 Ifpz) = (pl I p2)*, while (91 I p 2 ) should follow for I? to be unitary. In the group representations considered so far, the linearity of the group operations played a fundamental role. In particular important it was for theorem 1, which states that a subspace of eigenfunctions of the one-electron Hamiltonian for a particular energy eigenvalue is also a representation space for the full symmetry group of the Hamiltonian. This theorem does no longer hold if time reversal is included, i.e. if the space-time group 4 is taken to be the symmetry group instead of the space group G alone. For these reasons, the powerful tools of group representations, which apply in the case of space symmetry, are no longer available if the joint consequences of space and time symmetry are to be studied.

Mainly two questions need to be addressed, namely, whether time re- versal symmetry of the Hamiltonian causes additional degeneracy of energy eigenvalues, and secondly, what additional relations emerge among the ma- trix elements of operators if time reversal symmetry is taken into account. These two questions will now be directly answered.

A.8.2 Additional degeneracies of energy eigenvalues

We start by solving the first problem. To this end we consider an irre- ducible representation 2) of G in the Hilbert space of scalar functions p(x) or of spinors p(x, s) in spinor Hilbert space - both possibilities are admit- ted below. Let 2) be a vector representation or a spinor representation, as needed. We denote the representation space in both cases by RD, and a basis therein by 91, p 2 , . . . , v d . The vectors Kpl,Kpa,. . . , K p d span a space R p which is likewise an irreducible representation space of G for K commutes with all elements of G. The representation of G in RD* is the complex conjugate representation D* of D. This follows from the relation (kpi I g I KPk) = (cpi I g 1 pk)* for the matrices of the two representations, which holds both with and without spin. If the two spaces RD and RD* are joined, a representation space RD of G emerges which is generally reducible. The dimension of any irreducible representation subspace in RD cannot be smaller than d , or else D and D* are reducible. The dimension 6 of RD can be at most 2d. The possibility that 6 lies between d and 2d has to be ex-

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A.8. Time reversal symmetry 695

cluded, however, because then an irreducible representation space of smaller dimension than d would exist, namely the complementary subspace of RD in RD. Under G , this subspace would transform into itself as RD and RD do. It would, therefore, form an irreducible representation space of dimension d’ - d < d in RD, which is impossible. It follows that the dimension of 720 will be either d or 2d. An equivalent formulation of this result is that the basis functions Kpl, Kp2,. . . , KqJd of RD* are either all linearly dependent on the basis functions p1,p2,. . . , p d of RD, or all linearly independent of them.

If G is understood as the spatial symmetry group of the crystal Hamilto- nian H , and if in addition there is time reversal symmetry, the two basis sets p1, p 2 , . . . , pd and Kpl, Kpz,. . . , K p d together can be identified with the totality of eigenfunctions of H for a particular energy eigenvalue ED. The degeneracy of E D is d if the two basis sets are linearly dependent on each other, and 2d, if they are linearly independent. The former case is referred to as case a). In the latter case a further distinction is necessary. The rep- resentations V and V* which belong, respectively, to the two subspaces RD and R p of 7 2 ~ , are either inequivalent, which will be case b), or equivalent, which will be case c). In case a) the degeneracy of an energy eigenvalue does not change because of time reversal symmetry. Using the fact that the two equivalent representations V and V* are representations in the same space RD, one can show that a real representation exists in RD which is equivalent to V and V*. In cases b) and c), the degeneracies double b e cause of time reversal symmetry, i.e. they are twice the degeneracies due to spatial symmetry alone. The doubling occurs in both cases because two dif- ferent irreducible representation spaces of the space group are joined to one eigenspace of the Hamiltonian because of time reversal. In case b), the two joined representations are inequivalent, and in case c) they are equivalent. We can also say that in case b) the characters of the joined representations are both complex and one is the complex conjugate of the other. In case c) the two joined representations have the same characters, and the characters are real. This does not mean, however, that the representations are them- selves real. They are complex, in fact, unlike case a) where they are real. The relationships described above are summarized in Table A.3. They are commonly referred to as the Herring criterion.

The question of which of the three cases a), b) or c) applies to a particular representation V, is answered by a theorem which establishes a connection between the above three cases and the sum C X ( g 2 ) of the characters of all squared elements g2 of G.

Theorem of Frobenius and Schur

This theorem is expressed by the relation

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696 Appendix A . Group theory for applications in semiconductor physics

Table A.3: Properties of the time-reversed irreducible representations of space groups and consequences of time reversal symmetry for additional degeneracies of energy eigenvalues.

Herring case

Relationship Relationship of V Degeneracy of RD and Rb and D*

R D = R b l? equivalent to V* Unchanged and both equivalent to real representation Xb = X D

R D # R b 2) inequivalent to V* Doubled X b f X D

R D # R b 2) equivalent to I?* Doubled but both not equivalent to real representation Xb = X D

K~ in case a )

0 in case b)

- K ~ in case c ) ,

1 - c X ( g 2 ) =

g E G (A.163)

where K 2 = 1 for vector representations, and K 2 = -1 for spinor repre- sentations. We note that cases a) and c) are interchanged if one switches over from an ordinary to a spinor representation. The value ‘+l’ on the right hand side of equation (A.163) means no additional degeneracy for vec- tor representations, but an additional degeneracy for spinor representations, while the value ‘-1’ means additional degeneracy for vector representations, but no additional degeneracy for spinor representations. The definitions of cases a) and c) were originally made using the values ‘+l’ and ‘-1’ rather than K 2 and - K 2 as we do here (for good reasons). One has to keep this in mind to avoid confusion.

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A.8. Time reversal symmetry 697

A.8.3 Additional selection rules for matrix elements

Time reversal symmetry has particularly simple implications for matrix el- ements of operators A(t) in Hilbert space which are either even or odd functions of time, such that

KA(t)K-’= nA(t) with n = 1 or n = -1. (A.164)

As in section A.4, we consider matrix elements (p& 1 A k ( t ) 1 cpr) of tensor components A k ( t ) of operators A(t) between the basis vectors {py, cpz, . . ., ~p :~ } and {cpy, p$, . . . , of particular irreducible representations V, and Dp of the space group G. For the matrix elements of &(t) between the time- reversed basis vectors {Kpy, Kp!j’, . . . ,Kcpzp} and { K c p y , Kpt, . . ., K p z U } it follows

(G4 I A&) I Kcp3 = 4 P k I A k ( t ) I cpY)*. (A.165)

This relation implies that the matrix elements with respect to timereversed basis vectors are completely determined by the original elements involving the non time-reversed basis vectors. If either or both of the two represen- tations D, and Vp belongs to case a) defined above , then { K c p y , Kp;, . . ., KpzY} are linearly dependent on {cpy, pz, p$“} or {Kpy, Kcp!,. . . , K p S p } are linearly dependent on py, pg, . . . , p k or there is such a linear depen- dence for both. Therefore equation (A.165) leads to relations between the elements (cpc 1 A k ( t ) I cpr) themselves and the number of independent matrix elements (qk I A k ( t ) 1 pr) is reduced by time reversal symmetry. Below, we will outline a procedure which allows one to calculate the reduced number o i independent elements in the special case p = v.

If neither of the representations Vv and Dp belongs to case a), then equation (A.165) does not impose an additional condition on the matrix elements (cp& I A k ( t ) I pr), rather it connects these elements with matrix elements between basis vectors of other representation spaces.

Consider now the mixed matrix elements (Kcp2 I A k ( t ) I pr) and (cpk I A k ( t ) I I?&) between one original basis vector and a time-reversed one. For these, one can easily derive the relations

(& I I KPK) = .K2(cpF I Am I &4J . (A.167)

These equations mean that the number of independent matrix elements (gp& ( A k ( t ) 1 pr)? as well as the number of independent matrix elements (pg 1 A&) 1 Kply)) is reduced because of time reversal symmetry. The factors n and K 2 in (A.166) and (A.167) can take, independently of each

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698 Appendix A. Group theory for applications in semiconductor physics

other, the values +1 or -1, so that the product K K ~ will be either +1 or -1. The matrices (I?& I A k ( t ) I pr) and (pg I A k ( t ) I Kpr) are thus either symmetric (nX2 = +1) or antisymmetric (&K2 = -1) with respect to the exchange of the pairs of indices p m and vl. How the number of independent matrix elements follows from this observation in the general case is shown in the book by Bir and Pikus (1974).

In the special case p = v, one can apply the procedure developed in section A.4.3 for the determination of independent matrix elements without taking account of time reversal symmetry, provided one makes a distinction between symmetric products [D, x D,], and antisymmetric products [D, x D,Ia, as defined in section A.4.3. The matrix elements (Rp; 1 A k ( t ) I p?) transform themselves according to the representation [a, x v,]~ x Dd, if nK2 = +1, and according to the representation [D, x V,Ia x VA, if K K ~ = -1. Following section A.4.3 the matrix (zpk I A( t ) I (or) then has as many independent elements as the number of times the identity representation occurs in the product representation [V, x D,], x V A for K K ~ = +l, or in the product representation [D, x V,la x DA for r;K2 = -1. Analogous statements hold for the matrix (p; I A&) I kp?), where V, is to be replaced by VE.

A.9 Irreducible representat ions of space groups

Below, we discuss the irreducible representations of the space groups in the Hilbert space of one-electron states p(x). Initially, electron spin will be omitted, i.e. we start with the ordinary or vector representations. As out- lined in section A.4.1 the wavefunctions p(x) are taken to be periodic with respect to a periodicity region of the crystal, so that the originally infinite space groups become finite. The aim of the present section is to develop a procedure to derive the general form of the irreducible representations of space groups. Using this procedure, the representations of two particular space groups, those of diamond and zincblende crystal structures, will be obtained in section A.lO.

A.9.1 Representations of translation groups

Each irreducible representation of a space group G is also necessarily a rep- resentation of the translation group T contained in it. As such, it is in general reducible. Since T is an Abelian group, the irreducible represen- tations of T are 1-dimensional. This means that the representation space of an irreducible representation of a space group G can be set up from 1- dimensional irreducible representation spaces of the translation subgroup T of G. We mark these spaces by a vector k whose meaning will immediately become clear. The spaces themselves, which are 1-dimensional as we know,

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A.9. Irreducible representations of space groups 699

are denoted by (pk . The irreducible 1 x 1 representation matrices of the G3 elements t~ of T with respect to (pk are complex numbers c k ( R ) , given by the relation

-ik.R c k ( R ) = e (A. 168)

as was demonstrated in Chapter 2 of the main text. In this, k i s a vector in a vector space reciprocal to the space of position vectors x. We have

k = kibi + k2b2 + k3b3 , (A.169)

with bl, b2, b3 being primitive lattice vectors of the reciprocal lattice and k l , k2, k3 being real numbers. The periodicity condition with respect to the periodicity region causes the components of k to be of the form k l = (2n/G)ll, k2 = (27r/G)l2, k3 = (2n/G)13 with 11,12,/3 as integers. If K is an arbitrary vector of the reciprocal lattice, then the irreducible representation v k of the translation group T belonging to k is identical with the irreducible representation V k f K belonging to k+K, since e x p ( 4 K . R ) = 1. One there- fore gets all different irreducible representation of the translation group if k varies within a primitive unit cell of the reciprocal lattice. Altogether, there are G3 such vectors and thus also G3 different irreducible representations of the translation group. This number also follows from the Burnside theorem, - since the irreducible representations are 1-dimensional, there are as many inequivalent irreducible representations as there are group elements, i.e. G3. By definition, the effect of the translation operator t R on the basis function vk(x) of the irreducible representations is given by

(A. 1 70)

The basis functions (pk(x) may be thought of as Bloch functions, i.e. as plane waves modulated by a lattice periodic factor u ~ ( x ) ,

(A.171)

Bloch functions (pk(x) of the same wavevector k, but different Bloch factors U k ( x ) , give rise to the same representation of the translation group. As long as one considers only the translation group, no relations exist between the Bloch factors of the basis functions p k for different values of k - any Bloch function can be used for a given k. Below, starting from an arbitrarily chosen (but later h e d ) Bloch function ( p k ( x ) of wavevector k, we will construct basis functions for an irreducible representation of the space group G. It turns out that basis functions cpk,(x) will emerge in this process which transform themselves according to irreducible representations of the translation group with wavevectors k' # k. The Bloch factors up(x) of the basis functions

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700 Appendix A . Group theory for applications in semiconductor physics

~pk~(x) of these representations cannot, however, be chosen arbitrarily, but are determined by the Bloch factor U k of ' p k . The determination of the wavevectors k' and the pertinent Bloch factors Uk' is the essential task which has to be solved in constructing the irreducible representations of a space group G from those of its translation subgroup T. The primitive unit cell in which the wavevector k can vary will be chosen below to be the Wigner-Seitz cell of the reciprocal lattice, i.e. the first B Z . The reason for this is that the latter cell, generally, is the only primitive unit cell of the reciprocal lattice which exhibits the full symmetry of the point group of equivalent directions.

A.9.2 Star of wavevectors

According to section 1, each element g of the space group G may be written as product g = t ~ . tT(a) ' a of an orthogonal transformation a, a translation tT(.) by a fractional lattice vector ?(a) associated with a, and a lattice translation t R belonging to the translation subgroup T of G. The orthogonal transformations a form the point group of equivalent directions P of the crystal. The effect of an element g of the space group on 'pk(x) is described by

3 rPk(X) = Pk(g-lx) = ,-iak.R cPlc(a-l[x - 5(a)1). (A. 172)

Here, ak denotes the vector to which k transforms if the orthogonal trans- formation a which originally acted on the lattice vector R, is transferred to the vectors k of reciprocal space, as was explained in Chapter 2 of the main text. Rewriting equation (A.172) in the form

it becomes clear that [tT(a) . a] 'pk(x) is a function which transforms accord- ing to an irreducible representation of the translation group with wavevector ak. The set of vectors ak generated as a runs through all elements of the point group of equivalent directions P , is called the star {k} of k.

Henceforth, we must distinguish between vectors k with and without symmetry. In the case of non-symmetric k, we have ak # k + K for all p elements a of the point group P , where K is a vector of the reciprocal lattice. For symmetric k we have ak = k+K for at least one element (Y different from the identity element of P. Under the action of such an element, the function [tT(a) . a] (pk(x) transforms according to the same irreducible representation of the translation group as does the function 'pk(x) .

We consider, initially, non-symmetric k. The star {k} has p different points crlk 3 k = k1,agk = k2,. . . , kp = q,k, where a j , j = 1,2, . . . , p , denotes the elements of the point group P of equivalent directions. With

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A.9. Irreducible representations of space groups 70 1

each star point kj , there is an associated basis function ( P g . We define the latter by the relation

(A. 174)

For the Bloch factor U k i ( x ) of ' p k j ( x ) , one then has

One can easily show that the space R { k } spanned by the p functions ' p k j ( x ) of equation (A.174), forms a representation space for the space group G. To prove this, we write the elements t R . tT(aj t ) . ay of the space group G in the form

z= kT(aj,) . aj ' l ' tajrlR. k T ( o I j ) ' a j l -1 I (A.176)

where j', at fixed j , takes all values between 1 and p . That (A.176) is indeed a possible way of writing of the space group elements g can be seen as follows: first one commutes the translation t R in t R . tT(a. , ) . ajt with [tT(a.,) . ajt],

simultaneously replacing t R by t -1 in accordance with equation (A.17).

Multiplying [tT(ajt) . aj,] . t -1 with the particular space group element

[t,(,,, . aj]-', one obtains again an element of the space group. If j' runs through all values between 1 and p , and R through all lattice points, then all elements will be different, in accordance with the general group properties derived in section 1. The set of these elements coincides, therefore, with the space group G. This proves that the space group elements can indeed be written in the form (A.176).

The application of g in the form (A.176) to one of the basis functions f , D k j ( x ) of R { k } of (A.174) yields, in view of (A.173), the expression

ujt R

ail R

(A. 177)

The function f,Dk,, (x) and therefore also g'pk2 ( x ) lie in R { k } . This means that R { k } forms a representation space. R { k } is also irreducible, we omit formal proof here. For non-symmetric k, the irreducible representations of the space group can therefore be characterized solely by the k-vector itself or, more strictly, by the star {k} of this vector. Thus, in this context we write Z){k] for the irreducible representations. The character x { k ) ( g ) of an element g in the representation Z ) { k } reads

P x { k } (9) = e- ikj-R- (A. 178)

j=1

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702 Appendix A . Group theory for applications in semiconductor physics

A.9.3 Small point groups and their projective representa- tions

Now let k be a symmetric point. The elements Q of P which transform k into itself or into an equivalent vector k+K, form a subgroup Pk of the point group P . It is called the small point group ofk. The corresponding elements [tR.t,(,) .a] of the space group form a subgroup G k of the space group G. It is called the small space group of k. The latter subgroup Gk of G plays a role similar to that of the translation subgroup T in the case of non-symmetric k-vectors. Unlike T , Gk is not an Abelian group; thus the irreducible repre- sentations of Gk, unlike those of T , are not necessarily 1-dimensional. This also implies that there may be more than one irreducible representation Dk for a given k (later we will distinguish these representations by an additional index v, here we keep the general notation Dk).

w e assume that a particular representation Dk of Gk is known. The pertinent representation space will be denoted by Rk, and a basis therein by pkl , p k 2 , . . . , p k d . According to the assumption, for an element g of Gk, the function

g p k n ( X ) [ t R . t7(cr) . a ] V k n ( X ) = (oh(a - l [X - .‘(a) - R]) (A.179)

also lies in Rk. Therefore, it may be written as

3 (Pln(x) = ,-ik.[?(u)+R] C ~ m n ( Q ) V h ( X ) (A.180)

where Dmn(a) denote the expansion coefficients. This relation means that the assumed representation of the small space group assigns to each element Q of Pk a specific matrix Dmn(a). Below, we will demonstrate that the matrices Dmn(a) form a group with multiplication rules of the general form (A.154), and thus, a projective representation of the small point group 9. To this end we consider two elements g 1 = [tR1 . t7(,l) . ‘311 and 9 2 = [tRz ’ tT(a2) ’ az] of Gk. For brevity we set T ( Q ~ ) = 7-1, and 7 - ( ~ 2 ) = 7-2. Equation (A.180) associates the elements g1 and 9 2 , respectively, with the matrices Dmn(Qi1) and Dmn(~12). For the effect of the product element [tR1 . t , . al l . [ tRz . t , . a21 on pkn, one obtains from (A.179) the relation

m

mmf (A. 18 1)

On the other hand, using (A.45), we also have

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A.9. Irreducible representations of space groups 703

(A. 182)

- - ,-wRl+fi+al(R2+ii)I CDm,n(al . a2)cPkm,(y+ (A.183)

Comparing equations (A.181) and (A.183), one finds that the matrices D(a1), D(a2) and D(a1 . a 2 ) are related by the equation

m'

where

Hence, the assertion is proven. The matrices D ( a ) do indeed form a pro- jective representation of the small point group Pk of k. The factor system is given by equation (A.185). The projective representation of the small point group Pk arises from the representation of the small space group Gk assumed at the outset. If the assumed representation Z)k of Gk is irreducible, the same is true of the pertinent projective representation of Pk. Vice versa, an irreducible projective representation of PI, gives rise to an irreducible representation Z)k of Gk.

The factors w ( q , a2) of (A.185) will be referred to as the crystallographic factor system. We will rewrite these factors in a more convenient form. The scalar product k . al(R2 + ?2) in the exponent may be expressed as a1 k.(R2+7'2). The exponent in (A.185) then becomes i(k-a11k).(R2+?2). Here, a1 is an element of the small point group of k, i.e. q k may differ from k by a reciprocal lattice vector K only. Because exp(-zK . Rz) = 1, we may omit R2 from the exponent, whence

-1

(A. 186)

For symmorphic space groups, no fractional translations 7' occur, i.e. the vectors ?(a) are zero for all elements a of the small point group of a sym- metric k-vector. Thus, the crystallographic factor system equals the identity system for all symmetric k-vectors, and the projective representations of the small point groups become representations in the ordinary sense. For non- symmorphic space groups, a factor system different from the identity system may occur, but only for vectors k on the surface of the first B Z . In fact, if k is an internal point of the first B Z , then ak also lies in the interior of

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704 Appendix A. Group theory for applications in semiconductor physics

the first B Z and cannot differ from k by a non-zero reciprocal lattice vector K. For surface points, however, it is possible that ak deviates from k by a vector K # 0. If that is the case, then factors differing from 1 may occur in the crystallographic factor system (A.185). The basis ( p k l , ( p k 2 , . . . , (pkd of the irreducible representation v k of the small space group G k then gives rise to a projective irreducible representation of the small point group P k of k.

A.9.4

For any irreducible representation of the full space group G one needs a basis set which contains basis functions for every point of the star of k. These will be defined now. We denote the number of star points by s (k) . With p ( k ) as the order of the small point group of k, we have s (k) = p / p ( k ) . From the full point group P of equivalent directions we select s(k) elements a1 = E , 122, . . . , a S ( k ) each generating a different star point kl , k 2 , ..., k s ( k ) .

Thus, we can write alk The choice of elements aj, j = 1,2, . . . , s ( k ) is not unique because, together with aj , each element generated by multiplication with an element ai of the small point group P k j of k j forms another element which transforms kl into k j . The choice of these elements made at the outset will be maintained throughout.

The basis functions of an irreducible representation of the small space group Gkj will be denoted by ( p k j 1 , ( p k j 2 , . . . , ( p k j d . They can be generated from the basis functions p k n ( x ) , n = 1,2, . . . , d, of the irreducible repre sentation D k of the small space group G k in the same way in which, for non-symmetric k , the basis functions C p k j ( x ) were obtained from the ( p k ( X )

using relation (A. 174) above. Accordingly, we set

Representations of the full space group

k = k l , a 2 k = k 2 , . . . , a , ( k ) k = k , ( k ) .

One can easily show that the functions ( p k 3 n ( X ) , n = 1,2, . . . , d , defined this way, actually form a basis of an irreducible representation of the small space group G k 3 if the ( p k n ( X ) , n = 1,2, . . . , d do so for the small space group G k

as we assume. We will prove now that the d x s(k)-dimensional space R k d spanned

by the basis functions ( p k j n ( x ) , ? 2 = 1,2, . . . , d ,g = 1,2, . . . ,s(k), of equa- tion (A.187) forms an irreducible representation space of the full space group G . To this end, we decompose the full point group P into the small point groups Pk3 of the various star points k 3 . we denote the el- ements of P k l by a l l , ~ ~ 1 2 , . . . , a l p ( k ) . Using the latter elements, the el- ements of Pk3 may be written in the form a 3 1 = ff3 . a 1 1 . a3-' ,a32 =

-1 -1 a3 . a 1 2 . a3 , . . . , a j p ( k ) = a3 . ' Y l p ( k ) . a3 . The products a3 . alz . a;' cover the whole point group of equivalent directions P if g and i take all

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A.9. Irreducible representations of space groups 705

allowed values, i.e. j = 1,2 , . . . , s(k), and i = 1,2 , . . . , p(k). If, in the prod- ucts aj . ali . a;' the left factor aj is replaced by a y , and j ' and i are varied within their respective definition regions at a fixed value of j , then one likewise obtains all elements of P. Each element a of P may, therefore, be written in the form a = ajl . a l i . c y j ', where j can be arbitrarily chosen and j ' and i are determined by a and j uniquely. An analogous statement holds for the elements of the space group. They allow for a representation of the form

-

(A.188)

with a fixed value of j which can be arbitrarily chosen and values of j ' and i depending on g. The lattice translation ta;iR in the second angular bracket

of expression (A.188) guarantees that the lattice translation described by the whole product (A.188) becomes tR , as it has to be. If g in the form (A.188) is applied to one of the basis functions qkjn(x), then one obtains by means of (A.180) and (A.187), the relation

This means that the resulting functions are again in the space R k d spanned by the basis functions (pk,n(x), n = 1 , 2 , . . . , d , j = 1 , 2 , . . . , s(k). Moreover, the general definition (A.46) of the operators 9 ensures that the group mul- tiplication relations 91.92 = g1 . 92 for any two elements g1,g2 of the space group G hold also in RM. This means that the space R k d forms a repre- sentation space. It is irreducible, if the partial basis set for the particular star points k used at the outset, gives rise to an irreducible projective rep- resentation of the small space groups Gk, as we have supposed. Thus, our assertion above is proved.

The question of whether one obtains all irreducible representations of a space group for a symmetrical wavevector if one proceeds in the way de- scribed above, is answered in the affirmative: By taking all stars {k} of the first B Z and all irreducible projective representations Dkv of the small point groups of the respective k-points with the crystallographic factor system, all irreducible representations D{k}v of the space group follow. For the character X { k } y ( g ) of an element g = tR++) . a in the space group representation D{klv, equation (A.189) yields

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706 Appendix A. Group theory for applications in semiconductor physics

Here ali is that element of the small point group of k = kl associated with the element a through the relation ali = a;' . a . aj. Furthermore, j is defined by the equation ak = kj, and the operators aj are the special star- generating elements introduced above for the point group P of equivalent directions, and s(k) is the number of different points of the star {k}. With

(A.191)

as the character of the projective representation of the small point group of k one obtains from (A.190)

Because of the commutativity of operators under the trace operation, X k v ( a l i )

may be replaced by X k v ( a ) . In section A.10, we will write down explicitly all irreducible projective

representations of the small point group, with the crystallographic factor system, for the space groups of the diamond and zincblende structures.

A.9.5

The representations of space groups derived above are ordinary or vector representations in the Hilbert space of scalar functions. They are character- ized by a star {k} of wavevectors and associated with a particular projective representation of the small point group of k with the crystallographic factor system. If one considers space group representations in spinor space instead of ordinary function space, the only aspect which changes is the meaning of the projective representations of the small point groups - in the presence of spin their factor system is given by the product of the crystallographic factor system and the factor system of the spinor representation discussed in sec- tion A.6. Projective representations with such a factor system can again be constructed by means of the double group method. Here one needs projective rather than ordinary representations of the small double point group with the crystallographic factor system. The extra projective representations of the small double point group are the spinor projective representations of the original small point group. Instead of the double point group method one may also directly use the general method of section A.7 for constructing projective representations. This method covers any factor system, includ- ing also the product factor system of the crystallographic and spinor factor systems of interest here. As the factor systems and pertinent projective rep- resentations of point groups are well-known (see Bir and Pikus, 1974), one only needs to inspect to the corresponding tables. In section A.10 we give

Spinor representations of space groups

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A.9. Irreducible representations of space groups 707

the projective representations of the small point groups for crystals with diamond and zincblende structure both with and without spin.

A.9.6

The implications of time reversal symmetry for the irreducible representa- tions of space groups have already been explored in general terms in section A.8. The procedure discussed there, i.e. the Herring criterion in conjunction with the Frobenius-Schur theorem is, however, very cumbersome - one must sum over all elements of the space group including the G3 translations in order to utilize it. Below, we will apply the concrete form (A.189) of the representations in order to express the general criterion of section A.8 in a more transparent way to facilitate its use.

Implications of time reversal symmetry

Stars {k} and {-k}

We consider an irreducible representation v { k } u of the space group G in a particular space R{k},. From section A.8 we know that the time-reversed space KR{k}u gives rise to the complex conjugate irreducible representation VTklu. Owing to the construction of the space group representations, V*

CkIv belongs to the star {-k} of the wavevector -k. There are two possible cases, case 1 with {k’} = {-k} and case 2 with {k’} # {-k}.

In case 2 the two irreducible representation spaces R{k}, and KR{k}u are necessarily linearly independent of each other, i.e. case b) of the general Herring criterion of section A.8 applies, and an additional degeneracy exists because of time reversal symmetry. The two representation spaces R{k}y and KRtklu are joined to the same eigenspace of the Hamiltonian by the time reversal operation.

In case 1, one cannot immediately tell whether the two representation spaces R{k},, and KR{h}, are linearly independent or not. In order to deter- mine this, the Frobenius-Schur theorem must be applied.

Theorem of F’robenius and Schur for space groups

For the application of this theorem, the characters of the elements g 2 of the space group G have to be calculated for a particular irreducible repre sentation z){k}, . According to relation (A.45), the square of a space group element g = t ~ + ( ~ ) . a is given by the relation

(A.193)

We apply this result to expression (A.192) for the characters X { q u ( g 2 ) of representations D{k}p It follows that

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708 Appendix A. Group theory for applications in semiconductor physics

Summing over the entire space group yields

(A.195)

The R-sum extends over all lattice points of the periodicity region and the a-sum over all elements of the point group P of equivalent directions. The R-sum differs from zero only for those a and j for which

a-lk, = - k, + K, (A.196)

holds with K, being a reciprocal lattice vector. The condition (A.196) may be interpreted in a simple way. For iixed 3 and varying a , the vector a-lk, runs through all points of the star of k, and, since k, belongs to the same star as kl k, through all points of the star of k. The condition (A.196) can, therefore, be fulfilled if among the various points k of the star {k} there are such which are equivalent to -k. This means that -k belongs to the star of k, or that the stars of k and -k are identical. This is case 1) defined above. If the condition (A.196) cannot be fulfilled, then there is no point in the star of k equivalent to -k, i.e. the stars of k and -k are different. The latter is case 2) from above. In this case the two representation spaces R { k } , and K R { k } y are linearly independent, and are joined by time reversal symmetry to an eigenspace of the Hamiltonian with the same energy eigenvalue. The degree of degeneracy is doubled, and the representation in KR{k}, forms the complex conjugate D;,}, of the representation D{+ in R{+. While the star of D:,+ is {-k}, the projective representation of the small point group of -k belonging to D:,}, is Z)kv* = D;,. If D,& is real, then the two energy eigenvalues at k and -k, which are degenerate because of time reversal symmetry, belong to the same projective representation of the small point group of k, and if z)ky is complex, these eigenvalues belong to different projective representations complex conjugate to each other.

In case 1) the evaluation of the R-sum yields the value G3 for elements a which obey condition (A.196). For all other a this sum is zero. One thus may restrict the a-sum for a particular value of J to those elements a which obey relation (A.196). We denote the set of these elements a by "5. With this, equation (A.195) becomes

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A.9. Irreducible representations of space groups 709

Since the ali run through the set Mkl if the a vary over a set Mki, the expres- sion under the j- sum in (A.197) is independent of j . Moreover xkv(afi) may be replaced by x k , ( ( r 2 ) as has already been mentioned above. One therefore gets

where Mk is the set of all (r from P such that

(A.199) - a ' k = - k + K ,

K being a reciprocal lattice vector. In case 2) the R sum yields zero, whence

cx(I,>v(s2) = 0. (A.200)

Using the last two relations, the left hand side of the Frobenius-Schur theo- rem (A.163) for representations of space groups becomes

9

Ip(k)]-' CaEMk e-iK'7'(a)Xkp(Lu2) {k} {-k}

{k} # {-k) 1 0 (A.201)

Here, we have used the relation N = G3 . s(k) .p(k) for the total number N of space group elements. For {k} # {-k} one obviously has the Herring case b) , as anticipated at the outset. For {k} = {-k}, all three Herring cases are possible, depending on which of the relations

1 - cx{I,}u(g2) = N L l

is satisfied. As above, one has K 2 = 1 for vector representations, and K 2 = -1 for spinor representations.

The set Mk does not form a group in general, for a2 is generally not contained in MI, if a does (since a-2k = k - K + a-lK). The whole set MI,

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710 Appendix A. Group theory for applications in semiconductor physics

Table A.4: Characterization of the three Herring cases for space groups.

Herring case

Characteristics

a)

{k} f {-k} no further condition

{k} = {-k} [p(k)]-l xaEMk e-iK'P(a) X k , ( a 2 ) = - K 2

{k} = {-k} Ip(k)]-l e-iK'P(at X k u ( a 2 ) = K 2

can, however, be generated from a single element a 0 of Mk by multiplying it with the elements of the small point group Pk of k; thus MI, = ao. Pk. If k and -k are equivalent, then a0 may be taken as the identity element, and one has Mk = 9. For inequivalent k and -k, one can take a0 in form of the inversion I , provided it belongs to P . Then one has

Mk = I ' Pk. (A.203)

Using the above results we will now discuss the implications of time reversal symmetry for the energy bands E,(k) in a more explicit form.

Additional degeneracy of energy bands

The characterization of the three Herring cases for space group representa- tions is summarized in Table A.4. Unlike the Frobenius-Schur theorem in its general form (A.136), the sums in Table A.4 are carried out only over the subset Mk rather than over the whole space group. Let us first consider a star with {k} = {-k}.

{k} = {-k}.

In cases b) and c) time reversal symmetry joins the irreducible represen- tations R{k}, and K R { k } , = R{k}fi into one eigenspace of the Hamiltonian. One therefore has E,(k) = E,(k), meaning that each energy band is at least 2-fold degenerate because of time reversal symmetry. In case b), the two bands E,(k) and E p ( k ) belong to different irreducible representations

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A.9. Irreducible representations of space groups 71 1

of the space group, and in case c) to the same one. For case a), p formally becomes v, and the relation E,(k) = E,(k) is a tautology, devoid of new information. For stars with {k} # {-k},

time reversal symmetry joins the irreducible representation spaces R { k ) , and KRjk},, = R { - k } , into one eigenspace of the Hamiltonian. w e have E,(k) = E,(-k), which means that the two bands Ey(k) and E,(k) have equal energies in different parts of the first B Z . The two bands, therefore, overlap, but need not be doubly degenerate as in the case above.

Example

We demonstrate the application of the Herring criterion using the example of a non-symmetric point k from the interior of the first BZ. The small point group of k contains, therefore, only the identity element. Two cases have to be distinguished: i) the complete point group P of equivalent directions contains the inversion I , and ii) P does not contain I .

In case i) the star {-k} of -k is identical to the star {k} of k. In order to use the Herring criterion, the sum over the set Mk needs to be calculated. In the present case Mk consists only of the inversion. The reciprocal lattice vector K in (A.199) is zero, and the only representation of the small point group is the identity representation with Xku(0") = 1. The sum upon MI, therefore yields 1. Without spin this corresponds to the Herring case a), i.e. there is no additional degeneracy. With spin the Herring case c) is obtained in which an additional degeneracy exists. It occurs between the two Bloch spinors pk and K p k = pi both of which belong to the only spinor representation of the point group Pk = E , but they do this in orthogonal spaces. The two bands E(k) and E'(k) belonging to pk and K p k z q$ , respectively, are therefore identical. There is a 2-fold degeneracy of all bands E(k) in the presence of spin, if spatial inversion symmetry and time reversal symmetry exist. If spin-orbit interaction is omitted, the degenerate states p k and pk may be interpreted as 'spin up' and 'spin down' states.

In case ii), i.e. without inversion, one has {-k} # {k}. The Herring case b) applies, and the two Bloch functions or spinors ' p k and Kpk = ' p I k belong to the same energy: we have E(k) = E'(-k) where again E(k) and E'(k) are, respectively, the energies of the states p k and pk. This means that both with and without spin, time reversal symmetry causes an additional band overlap.

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712 Appendix A . Group theory for applications in semiconductor physics

A.9.7 Compatibility

From the main text we know that the Bloch functions p k u and the pertinent energy eigenvalues E,(k) are continuous functions of wavevector k within the first B Z . Therefore the irreducible representations z ) (k} , , of the space group in the eigenspace of the Hamiltonian for a given energy eigenvalue E,(k) must likewise be continuous functions of k. From this observation one obtains compatibility relations between the small representations v k p

for a symmetry line k and the small representations v k o u for a symmetry center ko, which lies on this line. Here, we will study these relations in greater detail. To this end we as- sume that there is an energy eigenvalue E,(ko) at the symmetry center ko. The corresponding eigenfunctions should belong to the irreducible repre- sentation Vh,, of the small point group P k o of ko. The small point group P k of a symmetry line on which the symmetry center ko is located forms a subgroup of Pb. A particular irreducible representation Vb,, of P b is therefore simultaneously also a representation of P k , in general, however, not an irreducible but a reducible one. Let Z ) b / k p , D b / k p ! , . . . be the ir- reducible parts into which v k o u decomposes as representation of P k . Be- cause of continuity of the energy eigenvalues and the representations with respect to k, E,,(ko) must split along the symmetry line k into bands E,(k), E,!(k), . . ., and their corresponding eigenfunctions belong to the irre- ducible representations Vb/k,, D b / k p , . . .. One says that the representation D b v at the symmetry center ko must be compatible with the representa- tions V b / k , , v b / k , ! , . . . along the symmetry line k through ko. In partic- ular cases, the set v k , , / k p , v b / k p j , . . . may reduce to the one representation v b / k p alone. Then no band splitting occurs, and V b / k , is compatible with

Similar results hold for compatibility between representations at a sym-

In section A.10 the compatibility relations are given for symmetry centers

P k O U .

metry line and a symmetry plane which is bounded by the symmetry line.

and lines of crystals with diamond and zincblende type structure.

A.10 Irreducible representations of small point groups

A.lO.l Character tables

We will exhibit the characters of the projective irreducible representations of the small point groups for crystals having zincblende and diamond structure at symmetry points I?, A, X , A, L , C and K of the first B Z . The point groups

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A.10. Irreducible representations of small point groups 713

k r

to be considered are listed in Table A.5. We start with the structure of lower symmetry, that of zincblende type crystals.

A X A L K C

Zincblende structure

The symmetry group of crystals with zincblende structure is given by the symmorphic space group F13m (T j ) . The irreducible representations of the space group are therefore associated with vector or spinor representations of the small point groups of wavevectors k. The point k = 0 or

has the highest symmetry, namely that of the full point group T d of equiva- lent directions. The star of I' consists only of I' itself.

The 24 elements of Td belong to 5 classes according to section 2: E , 6S4, 6S2, 8C3, 60~. As before, each class is characterized by a representative element, the number in front indicating the number of different elements in the class. Since there are as many inequivalent irreducible representations as there are classes, the number of these representations equals 6 in the present case. To satisfy the Burnside theorem two of them must be 1-dimensional, one 2-dimensional and two 3-dimensional. The two 1-dimensional represen- tations are denoted by I'l(A1) and I'2(A2), the 2-dimensional one by I ' lz(E), and the two 3-dimensional ones by I'Is(T2) and r25(Tl). The notations in parenthesis are the ones which commonly are used to describe the symmetry of localized one-electron states of of molecules and point perturbations in

Table A.5: Small point groups P and number s of star points for symmetric points k of the first BZ in zincblende and diamond type crystals. Properties of the time reversed k-point are also indicated.

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714 Appendix A. Group theory for applications in semiconductor physics

Table A.6: Irreducible representations of the small point group T,j of !? for zincblende type crystals. Here and below, the vector representations are given in the upper part of the table, and the spinor representations in the lower part. The right column gives basis functions transforming according to the irreducible representa- tions of the same line. The symbols for the representations given in parenthesis are the ones commonly used in the context of localized oneelectron states of molecules and point perturbations in crystals.

crystals. The characters of the 5 representations are shown in Table A.6. Since the character is a class function it suffices to give it for whole classes of elements. In Table A.6 the following notations are used:

(A.205) 3T 1 3 3 i I - - ) = -[Iz - i y t ) + 212 .!)I, I--) = -Ix - iy l), 2 2 dz 2 2 f i

11 1 i i i I--) = ---[I. + iy 1) + Iz t)], I - - ) = -[-I. - iy t ) + Iz l)]. (A.206) 2 2 J?; 2 2 J3

For each representation, Table A.6 also gives a basis of linear combinations of products of the three Cartesian components x, y, z of the position vector x, or of the components Jz, Jg, Jz of the angular momentum pseudovector J. According to their interpretation as basis vectors, these components transform under the action of an orthogonal operator a-' according to the

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A.lO. Irreducible representations of small point groups 715

transposed matrix A-' of equation (A.31), therefore, according to the matrix A itself. For inversion, the transformation is given by the negative of the identity matrix if one deals with a vector, and by the identity matrix if one has a pseudovector. If the coordinate axes align with the cubic crystal directions, then the elements of T d transform the components of the position vector as shown in Table A.7. Using this table it may easily be verified that the basis functions of Table A.6 do, indeed, transform according to the corresponding representations. We demonstrate this using the representation I'15 as an example. The identity element E transforms zyz into zyz. This corresponds to the 3-dimensional identity matrix. The trace of this matrix is 3, in agreement with the character of E in the representation r 1 5 in Table A.6. In each representation the character of the identity element equals the dimension of the representation. As a representative of the class 6S4 we take the element S42.. According to Table A.7, it transforms xyz in Zzy. This corresponds to a transformation matrix whose zz-element is -1 and whose yy- and zz-elements vanish. The trace of the matrix is therefore -1, as expected. The representative Czz of the class 3cz generates zg2, the trace of the corresponding matrix being -1. The element C3(111) of 8C3 yields zxy. The matrix of this transformation has trace 0. Finally, the reflection uZy of the class 60, gives Gxz. The trace of the corresponding matrix is 1. In all cases the traces agree with the character values of r15 in Table A.6.

In the double group of T d , the 48 elements are distributed among 8 classes, as was shown in section A.8, where Q forms its own class, and the classes {C3, C ] } and {C4, C2) of the simple group each give rise to two classes. The double group therefore has 8 different irreducible represen- tations. Among them are the 5 ordinary representations, which we have already come to know in the case of the simple group. Consequently, there are 3 extra representations. Because of the Burnside theorem, the sum of the squares of the dimensions of these representations must be 24. From this one may conclude that there must be two 2-dimensional spinor representa- tions and one 4-dimensional one. The first two are denoted by r6 and r7, and the last by r8. The pertinent characters are given in Table A.6. If the irreducible spinor representation D I of the full orthogonal group is taken as a representation of T d , it remains irreducible and yields r7. By multiplying 2 ) ~ with one of the 5 ordinary irreducible representations, one obtains:

I

2

rl D~ = rr , r2 x D~ = r6, r12 x D~ = r8, r15 x D$ x r6 + rs , r25 x D~ = r7 + rs . (A.207)

As far as time reversal is concerned, the star {-k} equals the star {k} in the case of I", and all 8 representations of the space group belong to the

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716 Appendix A. Group theory for applications in semiconductor physics

Table A.7: Transformation of the position vector components x, y, z with respect to the cubic axes under the symmetry operations of point groups T d and Oh.

Herring case a). This means that time reversal symmetry does not result in an additional degeneracy of energy bands at I?.

A

The points (0,0,2~</a) with 0 < C < 1 and all points on symmetrically equivalent directions are denoted by A. The star of A has 6 points. The small point group is C2,. This group is Abelian, i.e. each of the 4 elements E , C2, C T ~ ~ , c,,2 forms its own class. Thus, there are 4 different 1-dimensional irreducible representations A,, A2, A3, A4 as shown in Table A.8. In the

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A.10. Irreducible representations of small point groups 717

Table A.8: Irreducible representations of the small point group Caw of A for zincblende type crystals. In the lower right corner those group elements are given which transform the basis functions (shown in the last column) according to the respective irreducible representation. The components z, y, z refer to the cubic axes.

double group one has, in addition, the class Q. The double group therefore has 5 different irreducible representations (see Table A.8). The one extra representation As must be 2-dimensional according to Burnside's theorem. It coincides with the spinor representation V1 of the full orthogonal group taken as a representation of CzV. Moreover, we have

1

A1 x Vl= A2 x Vr. = A3 x 2 ) ~ = A4 x DL = As. (A.208)

The time reversed star {-k} differs from the star {k} in the case of A (see Table A.5). Thus, all representations of the space group at A correspond to the Herring case b). Since all 5 representations of the small point group are real, time reversal symmetry joins a certain representation at k with the same representations at -k. At A one therefore has

Eai(k) = EAi(-k), i = 1 ,2 ,3 ,4 ,5 . (A.209)

X

The boundary point (0, 0,27r/a) of the first BZ and all symmetrically equiva- lent points are denoted by X . The star of X has 3 points, because (0, 0,27r/a) and (0, 0, -27r/a) are equivalent (in the sense of reciprocal lattice transla- tions). It contains the 8 elements E , C2, S4, ,943, odl, ad2, U21, U22 which form the 5 classes { E } , {Cz}, (54, S,"}, (U21, U22}, {adl, a d z } . Consequently, there are 5 irreducible representations among which 4 are 1-dimensional, namely XI, X2, X3, X4, and one, X5, is 2- dimensional (see Table A.9). In the double group one has the class Q in addi- tion, and, instead of (S4, Sj}, the two classes (S4, QS;} and (QS4, S;}. The

The small point group of X is D M .

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718 Appendix A. Group theory for applications in semiconductor physics

Table A.9: Irreducible representations of the small point group &d of X for the zincblende structure

16 elements of the double group are distributed, therefore, among 7 classes. The two extra representations of the double group must be 2-dimensional according to Burnside's theorem. These are denoted by x6 and X7. The full orthogonal group representation V 1 , taken as a representation of the point group D u , coincides with X7. In addition, the following relations hold:

5

The time reversed star {-k} coincides with the star {k} of X . All represen- tations at X belong to the Herring case a), so that no additional degeneracy occurs due to time reversal symmetry.

A

The point (./a)(C, C, 6) with 0 < C < 1 and all the points on equivalent di- rections are denoted by A. The star of A has 4 points. The small point group is C3v. The 6 elements E , C3, Ci, uvl, av2, av3 are distributed among the 3 classes { E } , {C3, C:}, {uvl, r V 2 , u v 3 } . There are therefore two 1-dimensional representations, A1 and A2, and one 2-dimensional representation, A3. In the double group, each of the three classes of the simple group again forms a class, and each gives rise to an additional class by multiplying it with &. Thus there are three spinor representations, two 1-dimensional ones, A4 and As, and one 2-dimensional one, As. Table A.10 shows all six representations. The spinor representations are p-equivalent to the vector representations. D 1 z taken as a representation of C3v coincides with As. One has the relations

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A. 10. Irreducible representations of small point groups 719

Table A.lO: Irreducible representations of the small point group C3v of A and L for the zincblende structure

In the case of A, the time reversed star of -k differs from that of k. Thus, all representations of the space group at A correspond to the Herring case b). The real representations A1, A2, A3 and at the point k are joined by time reversal symmetry with the same representations in the point -k, and the two complex representations A4, A5 at k with the complex conjugate representation at -k. From this it follows that at A

L

The point L is the point of A with = 1. Therefore it also has 4 star points, and the small point group remains also the same, i.e. C3,. Everything said about the representations at A applies, therefore, to L as well (see Table A.lO). The only difference between L and A is that for L the star {-k} does not differ from, but coincides, with the star {k}. The representations 151, L2, LJ and Lg correspond to the Herring case a), i.e. no additional de- generacy occurs between the corresponding bands. In the case of L4 and L5 one has the Herring case b), so that at L the additional degeneracy

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720 Appendix A . Group theory for applications in semiconductor physics

Table A.ll : Irreducible representations of the small point group C, of C and K for zincblende type crystals.

(A .2 13)

occurs because of time reversal symmetry.

The point ( 3 ~ / 2 a ) ( < , < , O ) , 0 < < < 1 and points on equivalent directions are denoted by C. The point with C = 1 is denoted by K. The number of star points amounts to 12 in both cases, and the small point group is C,. Each of the two elements E and ah of C, forms it’s own class. Thus, there are two 1-dimensional irreducible representations, C1 and Cz or K1 and Kz (see Table A.ll) . In the double group of C, each class of C, gives rise to yet another class through multiplication by &. This means that there are also two 1-dimensional spinor representations, C3 and C4 or K3 and K4 . These are p-equivalent to the two vector representations. If taken on the point group C,, Vi becomes reducible and decomposes into the 1-dimensional spinor representations C3, C4 or K3, K4, respectively.

The star {-k} differs from the star {k} in the case of C and K , so that the Herring case b) applies. Thus the following relations hold at C:

5

The same relations also hold at K .

Diamond structure

The diamond structure follows from the zincblende structure if the two prim- itive f.c.c sublattices are occupied with identical, as opposed to different,

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A.lO. Irreducible representations of small point groups 721

atoms. The operation which transforms the two sublattices into each other thus forms an additional symmetry element of the diamond structure. One can take it in the form of a glide reflection or the inversion joined with a frac- tional translation. We choose the second possibility. The inversion is carried out at the center of the line connecting the two atoms of a primitive unit cell, and is accompanied by a translation along the connecting line by the amount a / 4 . The point group of equivalent directions for diamond structure follows, therefore, from the corresponding point group for the zincblende structure, i.e. T d , by forming the direct product T d x ci = oh,

As for the zincblende structure, we consider the irreducible representa- tions of the space group at the symmetric points I?, A, X , A, L , C and K . The corresponding small point groups are shown in Table A.5. Adding inversion, the number of star points either doubles in comparison with the point group of the zincblende structure, or remains the same while doubling the number of elements of the small point group. For all symmetry points mentioned above the stars {k} and {-k} are the same in the diamond structure (see Table A.5).

Due to the occurrence of fractional translations in the space group of the diamond structure, not only vector and spinor representations of the small point groups are required, as in the zincblende case, but also projective representations with other factor systems. Actually, this applies only to k-points at the boundary of the first B Z , i.e. to X , L , and K , for only these k-points can have a crystallographic factor system different from the identity system. As we will see below, only that of X differs essentially from the identity system, while that of L is p-equivalent to the latter, and that of K is equal to it. Those of I', A, A and C are equal to the the identity system anyway.

The small point groups C3w and CzW occur in both the diamond and zinc- blende structures. Since the crystallographic factor systems of the respective symmetric points A, K , and C are the identity systems in the diamond structure as well, the same irreducible representations of these groups apply as in the zincblende case.

On the other hand, there are small point groups in the diamond struc- ture which can be obtained from small point groups in the zincblende struc- ture by multiplying with Ci. In fact only one of the remaining small point groups of the diamond structure cannot be generated in this way, namely the point group C4w. In all other cases this construction is possible, one has oh = 0 x Ci, D4h = D u x Ci and D u = D3 x Ci. If, in these cases, the crystallographic factor system is the identity system also in the diamond structure, then the irreducible representations found for zincblende case can also be used to obtain the representations for diamond case. Let us consider r: The small point group oh of I? in the diamond structure follows from the corresponding small point group 0 of I' in the zincblende structure by

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722 Appendix A . Group theory for applications in semiconductor physics

multiplying with the inversion I , and the crystallographic factor system is the identity systems also for oh. The irreducible representations of Oh may be obtained from the irreducible representations of 0 by multiplying with the two irreducible representations r+ or r- of Ci. In this, r+ denotes the identity representation of Ci, and emerges from r+ by assigning the in- version to -1 instead of +l. In this way, each irreducible representation of a small point group Pk in the zincblende structure gives rise to two irreducible representations of the corresponding point group Pk x Ci in the diamond structure. The irreducible representations formed in this way are all there are, for they satisfy the Burnside theorem. There is only one small point group, namely the symmetry group D4h of X , whose irreducible represen- tations cannot be obtained in this way, since in this case the factor system differs substantially from the identity system.

The five irreducible vector representations of T d generate 10 vector repre- sentations of oh, and the three spinor representations of T d give rise to 6 spinor representations of oh. The 16 representations are listed in Table A. 12. Vector representations for which the inversion is associated with +1 are de- noted by the same symbol as for T d . Vector representations with -1 for the inversion are marked by a prime on the symbol for the corresponding repre sentation of T d . The two 3-dimensional representations are an exception, in their case the symbols of the representations are exchanged in the transition from T d to o h : r15 with +I for I becomes the representation rL5 of oh, and r15 with -1 for I becomes the representation r 1 5 of oh, while r 2 5 with $1 for I becomes ri5 and r25 with -1 for I becomes r25 (see Table A.13). These somewhat strange relations result from the fact that the notations I'15

and r25 were originally introduced for the 3-dimensional representations of the cubic group 0 rather than the tetrahedral group T d . In the presence of spin, the representations of oh with fl for I are denoted by an upper index ' + I , and those with -1 for I by an upper index ' - I on the symbol for the corresponding representations of T d . Simultaneously, the designations r6 and r7 are exchanged in the transition from T d to oh.

The representation V1 of the full orthogonal group becomes I': on oh. In addition, we have the following relations:

T

r12 x D~ = r$ , r:, x v1 x r, ,

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A.lO. Irreducible representations of small point groups

a" 0 - 1 - (3 0 3 13 I 3 3 13 I 3 3

0 - 3 1 3 0 0 3 3 1 3 0 c

Q 3 I 3 0 I 3 3 3 H 0 k3 -

+ b 3 0 0 0 0 0 01;;

3 3 I 3 I 3 13 - I $ $

0 0 0 0 0 0

N N V N N C"(h

Table A.12: Irreducible representations of the small point group oh of I? for dia- mond type crystals.

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724 Appendix A. Group theory for applications in semiconductor physics

Td

r15

rz5

r,

r7

O h Basis functions

rh5 for I = 1 ~ y , y ~ , ~ l :

r15 for I = i c , ~ , z

ri5 for I = 1 Jx, J y , Jz

rz5 for I = I X(VU - 12). ~ ( Z Z - ax), Z(XO - W )

r$ for I = 1

r; for 1 = i t 1 4 $) , I& J)> for I = 1 {I T),I 1))

r; for I = i

r15 x vr = r; +r, , r;, x ~1 = rg+r,+ , 5

r25 x vr = rF+r;, r;, x v+ = r;+r,+ . (A .2 15)

As far as time reversal symmetry is concerned, all representations at belong to the Herring case a), so that no additional degeneracy of energy bands occurs.

A

The star of A has 6 points, and its small point group is C4,,. The 8 elements are assigned to classes as follows: { E } , {C4,C43}, {Ci}, {cl,c3}, {ff2,c4}.

Thus there are five ordinary irreducible representations, among them four 1-dimensional ones, Al, A2, A3, A4, and a 2-dimensional one, As (see Table A.14).

In the double group we find the new class {Q}, and the class {C4, C,"} of the simple group splits into the two classes {C4, Q . Cl} and {Q . C4, Cz}. There are thus two spinor representations, A6 and A?, of dimension 2. The representation Vi coincides with A6 on C4,,. We have the relations

A1 X V L =a: X Dl = A s , (A.216)

(A .2 17)

The time-reversed star {-k} equals the original star {k} in the case of A. All representations of the space group in A correspond to the Herring case a), i.e. no additional degeneracy occurs because of time reversal symmetry.

A2 x Vr = A h x Vr = A ? , A5 x V + = & + A ? .

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725 A . 10. Irreducible representations of smd point groups

Table A.14: Irreducible representations of the small point group CdV of A for diamond type crystals.

X

The star of X has 3 points. Its small point group is D4h. According to section A.3, we have D4h = D2d x ci. For elements of the subgroup D2d of D4h the crystallographic factor system w ( q , a 2 ) has the constant value 1 because D2d forms a subgroup of T d and none of the elements of T d is joined with fractional translations in the space group of the diamond structure. For the remaining elements, the factors w(cr l ,cq) generally differ from 1 and must be calculated separately. For a(1, I), for example, it follows that exp[i(4~/a).(a/4)] = -1. It turns out that the crystallographic factor system of D a is not pequivalent to the identity system in the case of X and the diamond structure. It belongs to a class different from that of the identity system.

In constructing the vector representations of D4hr one may take advan- tage of the identity D4h = D4 x Ci proved in section A.3. Then the 16 elements of D4h can be associated with the 5 classes of D4 as well as 5 fur- ther classes, which are generated by the previously mentioned ones through multiplication by I . Altogether one has, therefore, 10 classes and thus also 10 vector representations of D4h. These are shown in the upper part of Table A.15 and are denoted by Xp, XH, ..., Xg and Xp', X;', ..., Xg'. For the space group representations at X without spin the vector representations of D4h

are not helpful, one needs the projective representations of this point group with the crystallographic factor system discussed above. These are shown in the middle part of Table A.15. The characters are given for each element individually because the character is no longer a class function for the pro- jective representation under consideration. Remarkably enough, none of the

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726 Appendix A. Group theory for applications in semiconductor physics

X E

xt" xz" x; x; xs" x;' xz"' x;' xi' XgVI

x1 x2

x3

x4

x5

Table A.15: Projective irreducible representations of the small point group D4h of X with three different factor systems corresponding to diamond type crystals. The vector representations (those with unit factor system) are given in the upper part, the vector representations with crystallographic factor system in the middle part, and the spinor representations with crystallograhic factor system in the lower part.

1

1

1

1

2

1

1

1

1

2

2

2

2

2

4

E

1

2

1

1

2c4

1

1

i

i 0

1

1

i i 0

I 1

0 0

1 1

I i

1

0

0 0 0

0 0 0

0 0 0

0 0 0

C4I c;

0

0 0

0

i 1 i 2 0 0

0 0 0 0

2 0 1 0

0 0 0 0

2 0 2 0

0 0 0 0 0 0 0

cir U21I U23' U22I u24r

- - I

1

1

1

1

2

i

I

1

-

i

1

0

0

0

0

0

I

-

- - - -

four representations Xi, X 2 , X3, X 4 is 1-dimensional. This corresponds to the general rule for irreducible projective representations with a factor sys- tem which are not p-equivalent to the identity system, mentioned in section A.7, whereupon 1-dimensional representations are not possible. The Burn- side theorem allows for yet two possibilities in this case, either there are four 2-dimensional irreducible representations or a 4-dimensional one. Table A.15 shows that in the spin-free case considered here the first possibility applies. For the projective spinor representations of D4h with the crystallo- graphic factor system, the second possibility is valid - one has exactly one 4-dimensional representation X5, which is shown at the bottom of Table A.15. The product of two projective irreducible representations of D a with the same factor system decomposes into irreducible vector representations Of D4h.

The time reversed star {-k} equals the original star {k} in the case of X . Time reversal symmetry does not result in additional degeneracy at X,

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A.lO. Irreducible representations of small point groups 727

Table A.16: Irreducible representations of the small point group D 3 d for diamond type crystals.

L E 2 c 3 3U" 1 z c 3 1

L1 1 1 1 1 1

L2 1 1 I 1 1

L3 2 i 0 2 i

L; 1 1 1 i i

L; 1 1 i i i

Lb 2 I 0 2 1

3avI

1

I

0

1

1

0

0 0

1

1 0

Basis functions

*I; zx f uu

J Z

{ J z - Jz, Ja, - J r )

x i l l i =

(. - U)(# - 2 x 2 - x)

t= - 2 , u - 2 )

Lq-basis from Tab.A.6

with (xu.) - ( J z J Y J z )

L g - h i s from Tab.A.6

with ( z v z ) - ( J z J Y J z )

II t), Ill} Lq-basis from Tab.A.6

Lg-basis from Tab.A.6

since all representations of the space group belong to the Herring case a).

A

The star of A has 8 points. The small point group is C B ~ , and the crys- tallographic factor system (A.186) equals the unit system, just as in the case of the zincblende structure. Thus the same irreducible representations apply as those shown in Table A.9 for the zincblende structure. Unlike the zincblende structure, however, the time reversed star {-k} of A equals the original star {k} of k in the case of the diamond structure. The rep- resentations A l , A2, h3, correspond to the Herring case a), and the two representations A4, A5 to the Herring case b). The energy levels A4 and As are therefore degenerate because of time reversal symmetry. One has

EA,(k) = EA,(k) at A. (A.218)

L

The star of L has 4 points and its small point group is DM = D3 x Ci.

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728 Appendix A . Group theory for applications in semiconductor physics

Therein elements occur which, like the inversion, are joined with fractional translations in the space group of diamond type crystals. Some of these elements, among them again the inversion, transform L into an equivalent, but not identical point, whereby the crystallographic factor system differs from the identity system. It can, however, be shown that this system is p- equivalent to the identity system. The projective irreducible representations of D3d to be considered here are, therefore, pequivalent to the ordinary irreducible representations of this point group. Due to the relation D3d = D3 x Ci the latter can be generated from the irreducible representations of D3 by multiplication with the two representations of Ci. The results are shown in Table A. 16.

In the double group, each of these classes gives rise to a further class. There fore, two 1-dimensional representations L1, L2, and a 2-dimensional one, L3, exist without spin, and also two 1-dimensional representations, Lq, L5 , and a 2-dimensional one, L6, with spin. In the group D u = D3 x Ci, one has the vector representations L1, L2, L3, and the spinor representations Lsf, L,f , Lg corresponding to +1 for the inversion, as well as the vector representations L i , Lh, L3‘, and the spinor representations L 4 , L g , L; , corresponding to -1 for the inversion (see Table A.16). The repTesentation D1/2 of the full orthog- onal group coincides with L$ on D3d. In addition, the following relations hold:

The 6 elements of D3 belong to the 3 classes { E } , {C3, C,”}, {Uz l , U22, U23}.

L3 x V+ = L z + L.$+ L.$ , L$ x D I = L 4 + L , + L;. (A.219)

The time reversed star {-k} equals the original star {k} in the case of L. The representations L1, La, L.3, L i , Lh, L i as well as L$ and L; belong to the Herring case a), and the spinor representations L z , L,: and L 4 , L c to the Herring case b). Thus, time reversal symmetry joins L4f, L z as well as L.4, L , to representations in the same energy eigenspace. We conclude that at L,

z

(A.220)

The small point groups of C and K are both CzV, and the stars both have 12 points. The crystallographic factor systems (A.186) are identical and in both cases equal the identity system. Therefore, only ordinary and spinor representations have to be considered. We formulate the results for C. The point group CzV has already been treated as the small point group of A

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A.lO. Irreducible representations of small point groups 729

Table A.17: Irreducible representations of the small point groups CzV of C and K for diamond type crystals.

av2 Basis functions u

for the zinrblende structure. According to results obtained earlier, there are four 1-dimensional irreducible vector representations, C1, Cp, C3, C4, and one 2-dimensional spinor representation, C5 (see Table A.17). The latter corresponds to the representation DL of the full rotation group considered as a representation of the subgroup Cpv.

The timereversed star {-k} coincides with the original star {k}, and the Herring case a) holds for all representations at C. Thus, no additional degeneracy of energy bands occurs at C due to time reversal symmetry. Results identical to those for C follow for K .

A. 10.2 Multiplication tables

Zincblende structure: Point group Td

Table A. 18: I' without spin.

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730 Appendix A. Group theory for applications in semiconductor physics

Table A. 19: r with spin.

Table A.20: A without spin.

Table A.22: X without spin.

Table A.21: A with spin.

Table A.24: A and L without spin.

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A. 10. Irreducible representations of small point groups 73 1

Table A.27: C and K with and without spin.

%

A. 10.2 Multiplication tables

Diamond structure: Point group Oh

Table A.28: F without spin.

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732 Appendix A . Groilp theory for applications in semiconductor physics

Table A.29: I' with spin.

Table A.30: A without spin Table A.31: A with spin.

Table A.32: X without spin.

x: + x; + xs" x3" + x; + xgu XX' + x;' + xs"' x3"' + xquI + xs"' x; + x; + xgv xi' + Xl' 4- x;t xy + xz"' + XguI

x,.+x;+xs" x;+x;+xs" xl" + x; + xs"

Table A.33: X with spin.

1-1 xl" + xz" + x; + x; + 2x;+ x1"' + x;' + xg + xi' + 2xs"'

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A. 10. Irreducible representations of small point groups 733

Table A.34: A without spin.

Table A.36: L without spin.

Table A.38: C and K without spin.

Table A.35: A with spin.

11 I Ad A5 A6 n

Table A . 3 7 L with spin.

Table A.39: C and K with spin.

-1

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734 Appendix A . Group theory for applications in semiconductor physics

A. 10.3 Compatibility relations

Zincblende structure

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A. 10. Irreducible representations of small point groups 735

Diamond structure

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736 Appendix A . Group theory for applications in semiconductor physics

r; - A; r; ~ h2 r; - c2 L 1 ~

L2 - r; ~ A; r; - ra ~ c3 L 3 -

L; ~

c3 Lg - A3 r;, < c2 L a _ _ _ r;,< A; r;,-

Ah

Page 751: Fundamentals of Semiconductor Physics and Devices

737

Appendix B

Corrections to the adiabatic approximat ion

In this Appendix we estimate the corrections terms (2.24) and (2.25) of section 2.2 to the adiabatic approximation. The difficulty of this estimation is that the eigenfunctions of the total crystal Hamiltonian $(x, X)$(X) are not explicitly given; one knows only certain general properties. First of all, the functions $(x, X), being eigenfunctions of a Hermitian Hamiltonian, may be assumed to be mutually orthogonal, i.e., for two different functions $(x, X) and $'(x, X) one has the relation

($'(X)l$(X)) / d3x$'*(x, X)$(x, X) = 0, $' # $. (B.1)

In section 2.2 it is verified that the factors @(X) at @(x, X)d(X) in the total wavefunctions $(x, X)$(X) obey a Schrodinger equation with a Hermitian Hamiltonian as well. Thus they also may be assumed to be orthogonal,

(4'14) = 0, 4 # 4. (B.2)

A second property of the total eigenfunction $(x, X)$(X) follows from the invariance of the crystal potential Vee,ec(~, X) + Vcc(X) under a common translation of the spatial coordinates of all electrons,

and all atomic cores,

x --+ X+ t = (XI + t,Xz + t , . . . ,XJ+ t) , 03.4)

thus a displacement of the whole crystal, through the vector t. Then the eigenfunctions $(x, X)$(X) of the crystal Hamiltonian can simultaneously

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738 Appendix B. Corrections to the adiabatic approximation

be chosen as eigenfunctions of the total momentum operator xi pi + cj Pj. We will do so and also assume that the crystal as a whole is at rest, so that

The third property we will use involves strong localization of the wavefunc- tions +(X) at atomic cores. This follows from the large masses of cores - in order for the kinetic energy operator Tc = ‘&(l/Mj)P! in the Schrodinger equation (2.25) to compete with the potential energy term, large second derivatives of the core wavefunction +(X), i.e., strong spatial changes, are necessary. These can be achieved only through strong localization of these functions, because the normalization condition requires that d(X) shall de- cay as X approaches infinity. In the case of a crystal, the core wavefunc- tions +(X) can have non-zero values only in direct proximity of the lattice points. In classical terms this corresponds to cores executing small oscilla- tions around their equilibrium positions.

Fourthly and finally, we use that the electron wavefunction $(x,X), taken as a function of X, is smooth compared with +(X) which was found to vary rapidly with X. This may be understood as follows: First, we note that, because of the small mass of electrons, the argument for strong local- ization of the core wavefunctions +(X) with respect to core positions X is not applicable to the variation of the electron wavefunctions +(x, X) with respect to electron coordinates x. The electron wavefunctions @(x, X), taken as functions of x, are rather spread out more or less uniformly over the entire crystal. Second, we form the internal product of relation (B.6) with +(X). Using the relation

(4IPjld) = 0 (B.6)

which holds because of the oscillatory character of the core motion, we arrive at

The total number of terms on the left hand side of (B.7) is of the same order of magnitude as the total number of terms on the right hand side, because the number of (valence) electrons equals, apart of a factor of the order of magnitude 1, the number of cores. Moreover, the order of magnitude of the terms with different i is the same throughout, as is the order of magnitude of the terms with different j . Therefore, the order of magnitude relations

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Appendix B. Corrections to the adiabatic approximation 739

follow for all i, j, indicating that $(x, X) is as smooth with respect to X as it is with respect to x. From relation (B.8) it follows also that

P?$(X, X) M Pj2$(X, XI. (B.9) The four properties of the wavefunctions $(X) and $(x, X) discussed above will be used to rewrite the two correction terms ($‘4’I4Tc$) and E j ( l /Mj )x ($’#’lPj$Pj4). The first term has the explicit form

(B.lO)

In rewriting this expression we use the third property concerning the wave function 4(x) and the fourth concerning the wavefunction $(x, X). Accord- ingly, the matrix element with respect to the electron states in B.10 depends only weakly on X, and the factor @’*(X)#(X) differs substantially from zero only in small environments around the equilibrium positions of cores. Thus, the matrix element can be approximately evaluated at the equilibrium values of the core coordinates X and be factored out of the integral over X. In this way one gets

(@’+’l4Tc$) - (4’ 14) ($’ I Tc I$ ) . (B.ll)

In a similar way one obtains for the second term

In relation (B.ll) we replace the kinetic energy operator Tc of cores by the kinetic energy operator Te of electrons using equation (B.9). This results in the order of magnitude relation

In a similar way the second term may be rewritten as

Here the product of non-diagonal matrix elements (~,’ITe1/21$)(#’IT~/214) has the same order of magnitude as the product ($]T~/21$)(+IT~’214) of the corresponding diagonal elements. According to section 2.2, the kinetic energy (+IT&) of cores, on statistical average, is smaller than the average

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740 Appendix B. Corrections to the adiabatic approximation

kinetic energy ($lTel+) of electrons so that (+ITe)+) sets an upper limit for (iITe1/21i)(41Tc11/214). Using (B.13) we arrive at

(B.15)

The two order of magnitude relations (B.14) and (B.15) are the equations (2.23) and (2.24) used in section 2.2.

Page 755: Fundamentals of Semiconductor Physics and Devices

741

Appendix C

Occupation number representation

Within the oneparticle approximation, Slater determinants @vly...vN(x~, x2, . . . , XN) of an N-electron system describe stationary many-particle states composed of oneparticle states vi. In a Slater determinant, a particular oneparticle state is not ascribed to an individual electron, but all electrons of the system may be associated with this state with the same probability. This reflects the principle of indistinguishability of elementary particles in quantum mechanics. Owing to this principle, a many-electron state cannot be characterized by statements like ‘electron 1 is in one-particle state v1, ‘electron 2 is in oneparticle state v2’, etc. One can only describe the oc- cupation of a particular oneparticle state by electrons, without identifying any individuality of the occupying particles.

Slater determinants in occupation number representation

For electrons, a particular one-particle state vi (of definite spin) can either be occupied with one particle, or not be occupied at all. This observation is most naturally described by means of an occupation number Nv - if the state v is occupied, then one has Nu = 1, and if the state is empty, then Nu = 0 holds. For simplicity, we identify the quantum numbers v of one-particle state with integers i, where i runs from 1 to 00, as there are idnitely many one-particle states.

The selection {v} = v1,v2,. . . , UN of one-particle states entering a partic- ular Slater determinant @iV}, determines the entirety of occupation numbers for all one-particle states, i.e. the sequence N1, N2,. . . , N,, as follows: the Ni-value of a state whose quantum number i coincides with one of the quan- tum numbers vl, v ~ , . . . , VN, is 1, and the Ni-value of a state whose quantum number i differs from all quantum numbers v1, v2,. . . , v ~ , is 0. Conversely,

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742 Appendix C. Occupation number representation

a particular sequence N1, N2 , . . . , N , of occupation numbers Ni of which N have the value ‘1’ and all other the value ‘O’, defines a particular Slater deter- minant of the N-electron system. This means that the many-particle states of N-electron systems may be represented by vectors I N l , N 2 , . . . , N,) whose components are occupation numbers of one-particle states with N of them having the value ‘1’ and all other the value ‘O’ , One refers to this description as the occupation number representation of many-particle states. Slater determinants @ju} (x l , x2, . . . , X N ) describe the same states in coordi- nate space representation. Each Slater determinant @{ul (x l , x 2 , . . . , X N ) is uniquely assigned to an occupation number vector I N1, N2, . . . , N - ) , and, vice versa, each occupation number vector I N1, N2 , . . . , N,) corresponds to a Slater determinant t,b{u}(xl, x 2 , . . . , X N ) :

@ { V } ( X l , x 2 r . . ., XN) @I N1, N2, ’ . ., N,) . (C-1) As the set of all Slater determinants @{, } (x l , x2, . . . , x ~ ) forms a basis in the Hilbert space of the N-electron system, the set of all occupation number vectors I N1, N2, . . . , N,) does the same. The space spanned by the occu- pation number vectors is sometimes referred to as occupation number space or Fock space.

Creation and annihilation operators

To facilitate use of the occupation number representation, one needs opera- tors whose effects on the basis functions I N1, N2 , . . . , N,) of Fock space are known. To this end one introduces so-called annihilation operators ai and creation operators a’. They are defined by the relations

(N1, N 2 , . . . , 0, . . . , N,), Ni 1 1,

0, Ni = 0. ai I N1, N2 , . . . , N i , . . . , N,) =

0, Ni = 1,

JN1, N2 , . . . , 1 . . . , N,) , N.j = 0. a t I N1, N2, . . . , N i , . . . , N,) =

According to this definition, ai yields a non-zero state only when Ni = 1, and the resulting state is one with Ni = 0, while a’ results only then in a non-zero state if Ni = 0 , and the resulting state is one with Ni = 1. The operator ai, therefore, annihilates a particle in state i, and the operator a: creates a particle in state i. By means of creation operators, one can generate any stationary many particle state of the N-electron system from

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Appendix C. Occupation number representation 743

the so-called vacuum state I O,O, . . . , O ) =I 0) in which all Ni = 0 are zero. In the sense of the correspondence (C.l), one may therefore write

N

$v(xl, x2,. . * 7 XN) * rl[ a& I 0). (C.4) j=l

Initially, the '+'-symbol in a: means nothing more than an upper index. We will now show that a: is in fact the Hermitian adjoint of the operator ai , as the notation suggests. To do so, we consider the operator product a t a i . From the definitions of ai and a t one obtains the relation

a:ai 1 N1, N 2 , . . . , Ni, . . . , N,) = Ni 1 N1, N 2 , . . . , N i , . . . , N,) . (C.5)

The states 1 N1, Nq, . . . , Ni, . . . , N,) are therefore eigenvectors of a:ai, and the particle number Ni of state i is the pertinent eigenvalue. One calls a r a i the particle number operator of the state i. We abbreviate it by I?i, i.e. we set

Now we multiply equation (C.4) from the left by ( N i , NB,. . . , N L 1. follows that

It

N

Using the definition of the Hermitian adjoint operator (a+)h.a. of a:, we may write

For relation (C.6) to be valid, = ai must hold. From this it follows that a t =

The ai and aT do not commute with each other. Simple commutation relations can be derived in terms of the anticommutator. For two arbitrary operators A , B the anticommutator [A , B]+ is defined as

i.e. a: is in fact the Hermitian adjoint operator of ai.

[A , B]+ = A B + BA.

With this definition we have

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744 Appendix C. Occupation number representation

[a:, 4 + = b i j , (C.10)

[ U i , U j ] + = [a:, a;]+ = 0. ((2.11)

These relations can be easily verified by means of the defining equations ((2.2) and (C.3) for, respectively, ai and a'. We leave this to the reader.

Operators in occupation number representation

In order to obtain the matrix elements of a quantum mechanical operator Q in occupation number representation it suffices to express Q in terms of cre- ation and annihilation operators because the elements of the latter operators are known by virtue of the defining equations (C.2) and (C.3). We demon- strate such an expression in terms of creation and annihilation operators for an operator Q(x1, x 2 , . . . , XN) which, in coordinate space representation, is additivily composed of one-particle operators q ( x i ) , so that

N

Q ( x 1 , ~ 2 , . . . ,XN) = c q ( x i ) . (C.12) i= 1

Some operators of important physical quantities of many-electron systems are of this form, e.g., the total particle number operator, the total momentum operator, and the total current density operator. The total energy operator may be one of them if the particles do not interact mutually (which is not true for electrons, of course). We will demonstrate that operators Q having the form (C.8) in coordinate space representation, may be written as operators

in occupation number representation. To prove this assertion it suffices to demonstrate that the matrix elements

of &(XI, x 2 , . . . , XN) between Slater determinants $ $ v l ( ~ l , x 2 , . . . , XN) and

$~y~(x~, x 2 , . . ., XN), are the same as the matrix elements of Q between the pertinent occupation number vectors 1 Ni, N;, . . . , N k ) and 1 N 1 , N 2 , . . . , Nm). We have to prove, therefore, that the identity

is valid. As far as the left hand side of this equation is concerned, one easily verifies that the only non-vanishing matrix elements are those between Slater determinants which differ solely in one column. This means that in

Page 759: Fundamentals of Semiconductor Physics and Devices

Appendix C. Occupation number representation 745

(v;, vi, . . . , v h ) only one component, say vi, can have a value different from v1 in (v l , v2, . . . , v ~ ) . The non-vanishing matrix element reads

(@{vO I Q I @(v}) (v: I q 1 ~ 1 ) . (C.15)

For the two occupation number vectors I N i , N i , . . . , N L ) and I N1, N2, . . . , N,) , the above described form of the Slater determinants means Nu, = 1, N,,; = 0, and NLl = 0, N ’ , = 1, while for all other vi the occupation numbers N , and N ’ , must be equal. With these specified values of the NUi and NL!, the right hand side of equation (C.14) may be expressed as

v1

vi

I

( N i , N L , . . . , N L I a$uu I N i , N 2 , . . . , N m ) . (C.16)

The matrix element of u$uU in (C.16) differs from zero only when v = v1

and v’ = vi hold. Thus, it follows that

This is the same result as in equation (C.15). If the two vectors 1 N i , N i , . . . , N L ) and I N1, N 2 , . . . , N,) differ in more

than one occupation number, then the matrix element on the right hand side of equation (C.14) vanishes, as does the matrix element on the left hand side between Slater determinants differing in more than one column. With this equation (C.14) is proven for all states. The operator Q ( x l , x 2 , . . . , X N ) in coordinate space representation is in fact given by the operator Q of (C.12) in particle number representation.

One can similarly proceed in the case of operators of the N-electron system which are additively composed of two-particle operators q(xi, x j ) ,

i.e. which may be written as

The corresponding occupation number representation is

Page 760: Fundamentals of Semiconductor Physics and Devices
Page 761: Fundamentals of Semiconductor Physics and Devices

74 7

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Page 771: Fundamentals of Semiconductor Physics and Devices

757

Index

Abelian group 86, 624, 626, 666 ab initio methods 133 absorption 41, 433 acceptor level 474 acceptor transition 304, 321 acceptor 271, 276, 277, 279, 490,

accumulation layer 553, 555, 613 actinides 231 activation enthalpy 239, 240 active region 600 additional degeneracy 710 adiabatic approximation 57, 59, 61,

adiabatic potential 60 adiabatic 61, 62, 64 Airy-function 436 all electron problem 134 alloy 27, 403 a-Sn 29, 210 amorphous semiconductor 2, 27 amphoteric center 305, 497 amplification 592 angular momentum basis 192 angular momentum matrix 187 angular momentum operator 681 Angular Resolved UPS (ARUPS)

anisotropic effective mass 273 annealing 238 annihilation operator 742 annihilation 506, 527, 529 anthracene 28 anti-bonding energy level 291, 292,

507

355, 737

374, 379

3 14

anti-bonding orbital 166, 172, 177 anti-resonance state 301 anticommutator 743 antisite defect 228 APW method 139 arsenic 44 artificial semiconductor microstruc-

As-antisite defect 331 As-rich surface 388 associates 232 asymmetric dimer model 384 atomic core 51, 53, 54, 55, 60, 63,

atomic force microscopy (AFM) 373 atomic layer 339 atomic orbital 135, 140, 141, 285 atomic structure 54, 64, 82, 177,

234, 336, 371 Au-center 497 Auger recombination 517, 518 augmented plane waves (APW) 139 average value 460

ture 402

70

ballistic transport 610 banddegeneracy 112,116, 117,184 band discontinuity 555, 567 band edge 539, 549, 550, 551 band index 108 band model 38, 41 band structure 105, 108, 109, 112,

113, 133, 211 band-band recombination 517 Bardeen’s relation 565 barrier 422

Page 772: Fundamentals of Semiconductor Physics and Devices

758 INDEX

base 586, 588, 589 basis of crystal 17, 23 BenDaniel-Duke boundary condi-

binary compound 3 binding energy 273, 275, 276, 280,

282 bipolar generation 511 bipolar transistor 4, 50, 536, 573,

585, 590 Bloch electron 501, 524 Bloch factor 92,101,129,135,179,

255, 256, 415, 425, 699 Bloch function 87, 92, 109, 132,

254, 255, 357, 699 Bloch integral 256 Bloch oscillation 438, 439 Bloch state 120, 437 Bloch sum 135, 364 Bloch theorem 85, 87, 89, 92, 130,

356 blocking direction 585, 591 bluegreen laser diode 407 body-centered 10 Bohr radius 267, 270 Boltzmann distribution 42, 461, 480 Boltzmann equation 500, 501 bond orbital approximation 171,173 bonding energy level 291, 292, 314 bonding orbital 166, 172, 177 Born-Oppenheimer approximation

boron 45, 46, 47 Bose distribution function 462 bound interface state 390, 391 bound state 268 bound surface band 361, 374 bound surface state 361, 386, 387 Bragg reflection line 358 Bragg reflection plane 101, 102, 103,

104, 105, 358, 410 Brattain 536 Braun 4

tion 417

57

Bravais indices 336 Bravais lattice 9, 10, 11, 13, 15, 17,

Bravais type 10 Brillouin zone 106, 108 buckling model 375, 377 buckliig 383 bulk region 545, 546, 577 bulk state surface energy band 359 bulk state 358, 389 Burger’s vector 240 buried collector 592 Burnside theorem 666, 691

83, 94, 110, 336, 342

capture at deep centers 511 capture center 519 capture coefficient 512, 516 capture cross section 516 capture mechanisms 517 capture rate 512, 515, 523 capture time 515 capture 284, 511 carrier concentration 34, 43, 537 CD-player 605 cds 222 CdTe 221 cell methods 134, 138 central cell corrections 243, 244,

274, 276, 281 cesium chloride structure 178 chalcopyrite structure 33 channel 620 character of representations 663, 664,

character table 712 charge carriers 50, 506, 508, 509 charge state 251 chemical bonding 51, 53, 138, 165,

174, 250 chemical potential 458, 461, 479,

499, 542, 575 chemical shift 276, 280 class of conjugate elements 626,665

668, 680, 690

Page 773: Fundamentals of Semiconductor Physics and Devices

INDEX 759

class 642, 686 clean surfaces 334, 336 cleavage plane 387 cleavage 335 cluster method 294 cluster 294 cohesion energy 175 collective many-particle excitation

78 collector 586, 587, 589 collision term 502 collisions 501, 524 common-emitter configuration 586,

compatibility relation 118 compatibility 712, 734 compensation 284, 489, 491, 492 complete compensation 490 compositional microstructure 402,

compositional disorder 226 compound semiconductor 29 conduction band discontinuity 394,

conduction band edge 265 conduction band minimum 180 conduction band 39, 41, 42, 134,

138, 267, 271, 472, 473 conductivity 1, 36, 505, 525 configuration dependence 78, 301 configuration interaction 78, 301,

configuration 67, 70,457,460, 462,

conjugate elements 626 continuity equation 508, 528, 579,

continuous group 624, 673 copper oxide 4 core electron 52, 53, 133, 134, 137 core state 135 corecore interaction 56, 60 core 52, 242, 265

590

406

403

305, 307

465

601, 602, 603

correlation 78, 301, 475 Coulomb interaction 56, 72, 165 Coulomb potential 70, 119 Coulomb repulsion 462, 463, 464,

465, 466 covalent bonding 174, 175 creation operator 742 critical point 119, 126, 473 critical thickness 404, 406 crystal class 13, 14, 345 crystal field splitting 222, 324 crystal lattice 6 crystal structure 14, 15, 21, 30 crystal system 9, 13, 14, 345 crystal without basis 12 crystal with basis 12 crystal 6 cubic crystal system 10 cubic lattice 10 current-voltage characteristic 559,

585, 594, 610 cyclic subgroup 625 cyclotron motion 455 cyclotron resonance 211, 443, 455,

456

d-electron 53, 134, 321 d-orbital 138 d-shell230, 232, 307 dangling bond 382 dangling hybrid 286, 377 Debye screening length 547, 555 deep center 281,282,284,299,307,

Deep Level Transient Spectroscopy

defect molecule model 285, 287, 312,

defect molecule 321 defect structure 32 defects 227 degenerate electron gas 615 Dember effect 595

318, 393, 462, 509, 519

(DLTS) 309

323, 365, 367

Page 774: Fundamentals of Semiconductor Physics and Devices

760 INDEX

density functional theory 79, 136 density of states (DOS) 42, 120,

123, 124, 126, 127, 128, 268, 426, 469, 472, 472

density of states mass 126 depletion mode 622 depletion region 545, 546, 577 detailed balance 513 device 490, 499, 535, 548, 567, 573 diamagnetic 449 diamond structure 15, 17, 18, 19,

diamond type semiconductors 148,

diamond 3, 16, 29 diffusion coefficient 239, 506, 530 diffusion current 506, 528, 539 diffusion length 529, 530, 581 diffusion theory 611 diffusion voltage 541, 542, 561, 576 diffusion 239, 502, 508, 527, 528,

Dimer-Adatom-St acking-Fault (DAS)

dimerization 382 Diophantin equation 337 direct gap semiconductor 224, 215 direct product 625, 626, 667, 691 displacement groups 627 distribution function 472, 500 divacancy 233 DLTS 310 donor level 270, 474 donor transition 303, 321 donor-acceptor pair 232 donor 270, 271, 276, 486, 490, 507 doping microstructure 402, 408 doping 35, 36, 398, 486, 488 DOS 432,454, 455 double acceptor 464 double donor 304, 463, 465, 494 double group 130, 685, 686, 715 double heterostructure 397, 400, 401

116, 118, 720

198

529, 537

model 381, 383

drain 620 drift current 506, 523 drift velocity 525 drift 506, 508 DX-center 233, 235, 333 Dyson equation 79

effective density of states 43, 480,

effective g-factor 449, 450 effective mass anisotropy 274 effective mass equation 252, 253,

258, 259, 266, 272, 418, 420, 434, 444, 449, 452, 539

48 1

effective mass tensor 121, 262 effective mass theory 412, 414 effective mass 119, 121, 126, 182,

183, 211, 216, 219, 440, 449, 472

effusion cell 398 Einstein relation 532, 533 EL2 center 231, 333, 234 electric field 45, 433, 440, 499, 501,

electric potential 539, 542 electrical conductivity 34, 35, 41,

electro-optic frequency 436 electrochemical potential 533, 552 electroluminescence display 31 electron affinity 558 electron concentration 34, 42, 469,

electron diffraction 371 Electron Energy Loss Spectroscopy

electron gas 478 electron system 70, 72, 457 electron trap 490 electron-core interaction 56, 242 electron-electron interaction 70, 72,

506

49

470, 484, 492

(EELS) 375

178, 245

Page 775: Fundamentals of Semiconductor Physics and Devices

INDEX 76 1

electron-hole pair 78, 511 Electron-Nuclear-Double Resonance

Electron-Paramagnetic Resonance

electron-phonon interaction 62, 517 elect ronegat ivit y 566 electronic elementary excitation 65 electronic structure of surfaces 363 electronic structure 51, 354, 371,

409, 420 electroreflectance 211, 443, 444, 572 electrostatic potential 552 elemental semiconductor of group

elemental semiconductor 4, 28, 29,

elementary excitation 65, 72 emission coefficient 512 emission rate 512 emission 511 emitter 586, 587, 588, 589 empirical TB (ETB) method 159,

empty lattice band structure 214 empty lattice 110, 113 energy band 2,39, 81,98, 108, 116,

energy gap 39, 41, 104, 108, 134,

enhancement mode 622 enthalpy of formation 236 envelope function equation 258 envelope function 256, 259, 271,

278, 414, 435, 443 epitaxial growth 336, 396 EPR 310 equilibrium distribution function 500 equivalent represent ation 660 equivalent directions 12, 14 equilibrium position 54, 63, 64 Esaki diode 536, 593 Euclidean algorithm 338

(ENDOR) 308

(EPR) 308

IV 289

31

163

125, 149

219, 255, 403

Euler angle 130, 631, 632, 679, 683 exchange energy 249 exchange integral 249 exchange potential 77, 133, 247,

26 5 exchangecorrelation energy 80 exchangecorrelation potential 8 1,

241, 245, 428 excitation energy 73 excitation level 304 excited state 274, 467, 468, 469,

493 excitonic effects 433 exciton 78 extended state 246, 457 extended zone scheme 96, 97 Extended-State X-ray Absorption

Fine Structure (EXSAFS) 308

extra representation 686 extraction 579 extrinsic semiconductor 35, 43,47,

483, 484

f-shell230, 232 , 307 Fabry-Perot resonator 600 factor group 625 factor system 687, 690, 703 Faraday 3 fcc lattice 11, 13, 113, 118, 131 Fermi distribution 39, 42, 461, 462 Fermi energy 40, 58, 461, 478 Fermi gas 479 Fermi integral 479 Fermi level 41, 250, 305, 477, 483,

Fermi statistics 461 Feynman theorem 64 Fick’s law 239, 502 field effect transistor 4, 386, 613,

620 fieId effect 612, 613 fine structure splitting 307

484,488,489, 534, 548

Page 776: Fundamentals of Semiconductor Physics and Devices

762 INDEX

finite group 624 first Brillouin zone 98, 106, 107,

110, 114, 116, 700 first SL BZ 410, 411 flow direction 585 Fock space 742 folding 107, 108, 411 forbidden zone 39, 108 forward bias 585 Franz-Keldysh effect 442 Franz-Keldysh oscillations 442 free carriers 527, 533 freez-out of carriers 487 Frenkel defect 233 frozen-core approximation 53, 133 full orthogonal group 636 full rotation group 636, 673

Ga-rich surface 388 GaAs 4, 30, 31, 220, 395, 384, 387,

398, 411, 429, 431 gain coefficient 599 GaN 30, 31 gap discontinuity 394, 403 GaP 5, 220 gate voltage 621 gate 621 generating elements 637, 638, 639 generation center 519 generation of free carriers 509 generation 506 generators of infinitesimal rotations

Ge 2, 29, 36, 218, 444, 456, 488,

Gibbs free energy 235 glide-reflection 14,15,632,633,635 grain boundaries 240 grand canonical ensemble 458 Green’s function method 295, 368 Green’s function 79, 295, 298, 299,

6 74

492, 497

369, 370

ground state 70, 73, 274, 279, 468, 469, 491

group IV semiconductors 28, 31, 313, 566

groups 7, 82, 87, 623 group theory 623 GW approximation 134 gyromagnetic ratio 448

Hall constant 49 Hall effect 43, 44, 46 Hartree approximation 68, 66, 76 Hartree energy 248 Hartree potential 69, 70, 77, 241,

245, 247, 265, 428 Hartree-Fock approximation 68, 76,

134 Hartree-Fock equation 77, 81 heavy hole 189, 197, 204, 210 Hellman-Feynman forces 63, 64 HEMT 407, 557, 574 hermiticity 417 Herring cases 710 Herring criterion 695, 707, 711 heterojunction 388, 390, 394, 396,

535, 536, 549, 550, 552, 574

heterojunction bipolar transistor (HBPT) 406

hexagonal lattice 22, 23 HgTe 209, 221 High Electron Mobility Transistor

Hittorf 3 hole capture 515 hole subbands 429 hole trap 490 hole 46, 47, 48, 428, 429, 489 holohedral point group 8, 14 homomorphism 626 Hubbard energy 249,280,305,309,

hybrid orbital 168, 285, 286

(HEMT) 402

310, 328, 463

Page 777: Fundamentals of Semiconductor Physics and Devices

INDEX

hydrogen atom 119, 141, 267 hydrogen model 266, 273

ideal surface 340, 341, 343, 354,

ideal crystal 5, 50, 225, 226 ideal semiconductor 470, 472, 482 ideal wurtzite structure 25 identity element 623 identity representation 659, 669 impact ionization 440 imperfect semiconductor crystal 225 improper rotation 632 impurity atom 35, 50, 226, 230,

237, 238, 250, 251, 312, 474, 501

indirect gap semiconductor 215,221 indirect gap 394 infinite group 624 infinitesimal rotation 674 infrared detector 31 injection current 580 injection 510, 571 inner shell electron 51 inorganic semiconductor 28 input power 592 input resistance 592 insulat or-semiconductor junction 53 5,

6 12 insulator 1, 34, 41 integrated circuit 622 inter-valley coupling 273, 274 interband transition 438 interband tunneling 440 interface charge density 571 interface field 571 interface potential 571 interface resonance 390 interface state 564, 565, 567, 570,

interface 388 internal photoeffect 595 internal transition 303

367

614

international notation 9, 15, 637 interstitial impurity 228, 239 intrinsic concentration 482, 484 intrinsic semiconductor 35, 47, 49,

inverse element 624 inversion layer 614, 615, 617, 620 inversion 7, 200, 204 inverted band edges 408 ion bombardment 335 ion implantation 238, 537 ionic bonding 177, 178 ionic conductor 2 ionic crystal 179 ionic model 324 ionization energy 283, 320, 322, 326 ionization level 304, 464 iron group 231 irreducibility 66 1 irreducible crystal slab 348, 354 irreducible part of first BZ 111, 117 irreducible representation 86, 95,

117, 118, 278, 362, 657, 661, 665, 679, 698, 690, 698, 704, 712

482, 483, 484

iso-energy surface 125, 217 isocoric impurity atom 242, 244 isomorphic 627 isomorphism 626 isovalent impurity atom 243, 252,

313 iteration 63

Jahn-Teller distortion 310, 315 Jahn-Teller effect 234

Kane model 199, 200, 209, 264, 473, 474

kinetic coefficient 505 Kirchhoff’s theorem 587 KKR method 139 Kohn-Sham equation 81 Koopman theorem 73,81,247,302

Page 778: Fundamentals of Semiconductor Physics and Devices

764 INDEX

Koster-Slater equation 296, 317 Koster-Slater method 295 Kronig-Penney problem 422

Lagrange’s theorem 625 Landau level 453 Landausubband454 laser diode 30, 50, 599, 600 lattice constant 10, 16, 18, 25, 64,

lattice matched heterostructure 404 lattice misfit 404 lattice mismatched heterostructure

404 lattice mismatch 403 lattice oscillations 64, 65 lattice plane 336, 338 lattice relaxation 234 lattice translation 7, 82, 87, 635 lattice 6, 7, 9, 336 LCAO 135 lead sulfide 4 LEED (Low Energy Electron Diffrac-

tion) 372, 373, 386 Levinson theorem 268, 270, 301,

361, 476 lifetime of non-equilibrium electrons

5 14 ligand field theory 324 light emitting diode (LED) 30, 31,

light hole 189, 197, 208 line defect 240 line perturbation 227 linear approximation 504 Liouville theorem 501 liquid semiconductor 2 LMTO method 139 local density approximation (LDA)

local equilibrium 503, 574 localized state 246, 460, 457, 462 long-range order 5

403

604, 605, 606

80, 134

long-range potential 243, 244, 252,

low index surfaces 344 luminescence diode 50 Luttinger parameters 196, 419 Luttinger-Kohn functions 135, 180,

260 Lut tinger-Kohn Hamiltonian 417,

452 Luttinger-Kohn model 189,196,198,

428 Lowdin orbital 144 Lowdin theorem 144

281

Madelung constant 178, 179 Madelung energy 178, 179 magnetic field 45, 444, 451, 452,

magnetic susceptibility 308 magnetoresistance 211 main groups 230, 238, 311 majority carriers 48, 485, 489, 520,

522, 537 many-body effects 309, 315 many-electron effects 284 many-electron system 65, 460 many-particle excitations 78 many-valley semiconductor 215 mass action constant 482 mass action law 478, 482, 543 mass operator 79 mass spectroscopy 308 matching conditions 415 material parameter 505 mean free flight time 504 mean free path length 504, 610 MESFET 557, 571, 606 Metal Insulator Semiconductor FET

454

(MISFET) 386, 620, 621, 622

Metal Organic Vapor Deposition (MOCVD) 397

metal sulfide 4

Page 779: Fundamentals of Semiconductor Physics and Devices

INDEX 765

metal-semiconductor junction 535,

metal-semiconductor rectifier 573 metal 1, 2, 29, 34, 41 metastable interstitial state 332 method of invariants 187, 451 microelectronics 4, 29, 593, 622 microstructures 396, 402, 406, 408,

409, 414, 420, 431 migration 238, 239 Miller indices 336 miniband structure 400 miniband 410, 423 minigap 410, 423, 424 minority carriers 48, 485, 499, 522,

537 mirror plane 9, 20 mismatch strain 404 mixed crystal 29, 30 mobility 34, 35, 525, 526, 527 MOCVD 398 modulation doping 556 modulation spectroscopy 211 Molecular Beam Epitaxy (MBE)

388, 397, 336, 409 MOMBE (Metal Organic MBE) 399 momentum relaxation time 524 MOSFET 536, 567, 571, 592, 606,

Mott relation 563, 565 MTO method 139 muffin-tin method 134, 138 multi-phonon process 517 multiband effective mass equation

259 multiple donor 304 multiple heterost ructure 397 multiple quantum well 424 multiplet structure 325 multiplication table 623, 643, 644,

multiply ionizable acceptor 271,280 multiply ionizable center 494

557, 606

620, 621, 622

729

multiply ionizable donor 270, 280,

multi-vacancy 233 462

n-channel MOSFET 622 N-electron system 66, 70, 73 n-type semiconductor 47, 50, 484,

nanostructures 402 naphthaline 28 narrow gap semiconductor 224,209,

483 nearest neighbor 21 nearly-free-electron approximat ion

98 negative differential drift velocity

43 1 negative effective mass 431 negative-U center 310 net capture rate 512 neutrality condition 478, 491 Newton’s law 440 nipi-structures 408 noble metal 328 non-adiabatic 62 non-equilibrium carriers 284, 509,

non-equilibrium electrons 514 non-equilibrium holes 522 non-equilibrium processes 50, 499,

500, 508 non-equilibrium state 500, 574 non-ohmic contact 561, 562 non-polar surface 387 non-radiative recombination 284, 518 non-symmetrical k-vector 95 non-symmorphic 15, 81, 119 normal divisor 625 npn-transistor 585, 586, 587, 590,

592 nuclear reaction 53 nucleus 52

489, 494

522

Page 780: Fundamentals of Semiconductor Physics and Devices

766 INDEX

occupation inversion 599, 600 occupation number representation

occupation number space 742 occupation number 458, 741 Ohm’s law 502, 504, 505, 525 ohmic contact 561, 562 ohmic metal-semiconductor contact

one-electron approximation 355 oneelectron potential 133, 134 one-electron Schrodinger equation

oneelectron state 74 oneparticle approximation 57, 66,

68, 71, 73, 77, 741 one-particle energy 72, 73, 309 one-particle excitation energy 73 oneparticle excitation 73, 78, 81 oneparticle Schrodinger equation

one-particle state 67, 70, 71, 76,

oneparticle wavefunction 66, 67,

optical absorption spectrum 38 optical excitation 510 optical fiber communication 31, 407,

605 optical transition 432 optoelectronics 29 OPW functions 135, 136 orbital motion 75 organic semiconductors 28 orthogonal transformation 7, 12, 130,

628, 629, 630 output power 592 output resistance 592 oxidation state 250, 251, 324, 331

p-channel MOSFET 622 p-equivalent 6 8 9, 6 90 p-orbital 138, 143, 150

458, 741

6 12

241, 355

66, 70, 71, 74, 76, 77

457, 460, 462

71

pshell230, 231 p-type semiconductor 47, 50, 489,

496 palladium group 231 partial compensation 490 Pauli principle 39, 42, 67, 76 PbTe 223, 224 Peierls instability 377 Peierls transition 377 periodic continuation 113 periodic potential 94, 100 periodic table 4, 232, 285 periodicity condition 93, 656 periodicity region 54, 55, 57, 66,

persistent photoconductivity 332 perturbation theory 98, 99 phenomenological equation 502, 503 phonon assisted tunneling 594 phonons 65, 501 phosphorus 45, 274 photo-conduction 49 photocell 4 photocopying 31 photocurrent 596 photodetector 50, 407, 574, 598 photodiode 598 photoeffect 595 photoemission spectroscopy (PES)

photoluminescene 433 photon assisted interband tunnel-

photoreflectance 572 photothreshold 558 photovoltage 596, 597 pho t ovolt aic element 598 pinning of Fermi level 565 planar defect 240 plane crystal system 342, 343 plane Bravais lattice 342, 343, 358 plane perturbation 227 plane space groups 345, 346, 347

85, 656

373, 375

ing 442

Page 781: Fundamentals of Semiconductor Physics and Devices

INDEX 767

plasma oscillations 78 plasmon 78 platinium group 231 pn-diode 585 pn-junction 535, 536, 537, 538, 540,

pnp-transistor 585 point contact rectifier 4 point defect 228 point group of equivalent directions

14, 345, 637 point group symmetry 343 point group of directions 637 point group 8, 9, 17, 82, 86, 95,

354, 362, 627, 636, 637, 6412, 647

542, 546, 555, 574, 595

point perturbation complex 232 point perturbation 227, 239, 241,

242, 474 point symmetry operation 8, 636 point symmetry 7, 9, 83 polyacetylene 28 Poisson equation 507, 508, 546, 554,

362, 570 polar surface 387 precipitate 234 primitive crystal slab 339 primitive lattice vector 6, 8, 17,

338, 699 primitive lattice 6, 7, 9, primitive unit cell 6, 30 projected bulk band structure 360 projective representation 118, 684,

pseudo-wavefunction 137 pseudopotential method 136, 139 pseudopotential 137, 138

687, 688, 690, 702, 703

quantized Hall effect 536, 556 quantum device 574 quantum dot 402 quantum statistics 479

quantum well (QW) laser diode 400, 402, 433, 536, 556

quantum wire 402 quasi Fermi level 582, 595 quasi-crystal 8 quasi-particle method 134 quasi-particle 65 quasi-wavevector 92, 94, 95, 104,

500 QW laser diode 406

radiative recombination 284, 517 rare earth atom 329 rare earths 231, 238 real semiconductor crystal 226, 474,

475 reciprocal basis 90, 91 reciprocal lattice vector 89, 90, 96,

reciprocal lattice 89, 93, 94, 699 reciprocal SL 410 reciprocal vector space 90 recombination center 519 recombination coefficient 521 recombination current 580, 583 recombination lifetime 522, 523 recombination rate 520 recombination 50, 517, 575, 578,

579, 583 reconstructed surface 348 rectification 535, 557 rectifier 4, 31, 610 reduced zone scheme 96, 97 reducibility 661 reducible representation 661 Reflection High Energy Electron Diffrac

reflection 14, 632, 633 reflectivity 380 reflectance spectroscopy 211 relaxation 505 relaxed surface 348 remote bands 201. 205

101

tion (RHEED) 373

Page 782: Fundamentals of Semiconductor Physics and Devices

768 INDEX

representation matrices 658 representations of groups 656 representations with factor system

repulsive forces 56 resonance state 301 reverse bias 585 RHEED 398 rigid displacement 7,627,635,654,

682 rocksalt structure 16, 17, 18, 21,

22, 178 rotation group 679 rotation-inversion 7, 14, 640 rotation-reflection 7, 14, 632, 639 rotation 14, 631, 633 Rutherford Backscattering (RBS)

308 Rydberg energy 267

684, 688

s-atom 146 s-orbital 138, 143, 150 s-shell 230, 231 saturation current density 585,610 saturation 487 scanning tunneling microscopy (STM)

373, 382 Schottky approximation 542, 544,

545, 547, 576 Schottky barrier 559, 563, 564 566,

608, 609 Schottky contact 535, 557, 562, 563,

607, 612 Schrodinger equation 57, 59, 62,

67, 74, 82, 98 Schonflies notation 637 Schonflies symbol 640 screw-dislocation 28, 240, 241 screw-rotation 14, 632, 633, 634 selection rules 670 selenium structure 27, 28 Se 2, 4, 31, 224 self-consistent 74, 133, 134, 428

self-energy operator 79 self-interstitial 228 self-organized growth 402 semiconductor device 4, 50, 241 semiconductor heterojunctions 388 semiconductor heterostructures 443,

semiconductor microstructures 402 semiconductor optoelectronics 5 semiconductor surfaces 572 semiconductor 1, 34, 41 sensor 574 shallow acceptor 276 shallow center 462 shallow donor 276 shallow level 265, 268, 269, 487 Shockley-Read-Hall recombination

518 Shockley 536 short-range order 5 short-range potential 244,245, 258,

28 1 Shubnikov-de-Haas effect 2 11 S i c 31 silicon wafer 490 silver sulfide 3 simply ionizable acceptor 465, 466 simply ionizable donor 465 single donor 304 single heterostructure 388, 397 SiOz/Si-interface 571 Si02/Si-junction 567 Si 2, 4, 29, 36, 37, 38, 188, 189,

212, 213, 375, 490 slab method 364 Slater determinant 76, 78, 301, 305,

457, 458, 741 small point group 116, 118, 131,

362, 702, 703, 712 smooth potential 252, 258 solar cell 50 solid state shift 52 solid state 5

535

Page 783: Fundamentals of Semiconductor Physics and Devices

INDEX 769

solubility 237, 238, 311, 328 sp-bonding elements 230 space charge region 539 space charge 507, 508 space group symmetry 343 space group 14, 15, 17, 81, 82, 85,

86, 95, 119, 131, 354, 362, 627, 654, 698, 704, 706

spin degeneracy 473 spin-orbit interaction 74, 75, 129,

189, 216, 218, 264, 277, 448

spin-orbit splitting 193, 194, 219, 279, 473

spin-orbit-split band 210 spinor function 75 spinor representation 130, 131, 682,

684, 690, 691, 706, 709, 714

spinor 77, 128, 129, 130, 131, 682 spin 74, 128, 129, 131, 132 spontaneous emission 602 stacking fault 28, 240 stacking vector 338 star degeneracy 117 star of wavevector 116, 700, 707,

710, 713 statistical average value 502 statistical correlation 479 statistical degeneracy 461 statistical operator 458 step dislocation 28, 240, 241 stereogram 20, 640, 641, 642 stereographic projection 640 sticking coefficient 398 stimulated emission 599, 603 strained layer 404 structural defect 227 structural perturbation 226 subband 410, 423, 424 subgroup 624, 625 sublevel 424 substitutional impurity 227, 238,

239, 242, 250, 265, 269, 283, 289, 313, 319

substitutional RE impurity 330 substrate 397, 620 supercell method 294, 367 supercell 294, 367 superlattice 397, 400, 421, 424, 536 surface antiresonance 361 surface band structure 365 surface Brillouin zone 35, 358, 359 surface energy band 357, 363, 358 surface field 572 surface photo-efFects 572 surface potential 572 surface reconstruction 350, 352, 353,

354 surface relaxation 349, 350, 386 surface resonance 361 surface state 361 symmetrical k-vector 95, 116 symmetry operation 14, 15, 83 symmetry point 111, 119, 120 symmorphic 15

Te 2, 27, 31, 224 ternary compound 3 tetrahedral semiconductor 238, 315,

320 thermal annealing 335 thermal donor 233 thermionic emission theory 611 thermodynamic equilibrium 49,457,

458, 499, 501, 503, 530, 533

thermoelectric properties 32 thermoreflectance 211 threshold current density 605 tight binding (TB) method 135, 139 tight binding approximation 140,

tight binding matrix elements 153, 148, 170

159

Page 784: Fundamentals of Semiconductor Physics and Devices

770 INDEX

tight binding parameters 159, 160, 169 392, 403

time reversal symmetry 95, 132, 204, 692, 693, 707

time reversal 693 total energy 70, 72, 79, 82, 165,

173, 174, 175, 303, 355, 490 194, 269, 472, 473

valence band discontinuity 328,391,

valence band edge 266 valence band maximum 180 valence band offset 392 valence band structure 140, 209 valence band 39, 42, 46, 184, 188,

total particle number 458, 469 valence electron 30, 51, 52, 133, transfer matrix method 370 transistor 31 valence shell orbital 142 transition groups 230 valence shell 51 transition metal (TM) 231,232,237, Valley 224, 271

251, 320, 393 van Hove singularity 125, 126, 442 translation group 7, 81, 86, 95, 627, vector potential 444

635, 698 vector representation 674, 684, 686, translation symmetry 6, 7, 348 709, 714 translation 6, 7, 85, 628, 635 Vegard’s rule 30, 403, 407 transmission electron microscopy (TEM) von Klitzing 536

134, 138, 250

373, 382 transparency concentration 600 tunnel diode 50, 573, 593 two-particle excitation 78 type I heterojunction 396, 397 type I1 heterojunction 396, 397 type I1 misaligned 396 type I1 staggered 396 type I11 heterojunction 396, 397

unified defect-model 566 unipolar annihilation 511 unipolar generation 511 unipolar transistor 50 unit cell 6, 11 universal tight binding parameters

Ultraviolet Photoemission Spectroscopy 159, 160

(UPS) 374, 374

vacancy 228, 236, 239, 285, 286,

vacuum-semiconductor junctions 53 5 309, 310

Wannier functions 296 warping of energy bands 198, 188 Weare-Thorpe Hamiltonian 169 wide gap semiconductor 483, 523 Wigner-Seitz cell 7, 97, 98, 105,

work function 557 wurtzite structure 16, 17, 18, 23,

114, 138, 700

24, 31, 178

X-ray diffraction 371, 355 zero gap semiconductor 210 zincblende structure 16, 17, 18, 19,

31, 178, 713 zincblende type semiconductor 148,

163, 198 ZnSe 491 ZnS 31

11-VI semiconductors 31, 221, 566 111-V semiconductors 30, 219, 384,

N-VI semiconductors 32, 224 387, 566

Page 785: Fundamentals of Semiconductor Physics and Devices

INDEX 771

2 x 1 reconstruction 376 7 x 7 reconstruction 376, 381, 382 IL x m reconstruction 352 sp2-bonding 333, 334 sp3-bonding impurity atom 314 sp3-hybrid orbital 166, 290, 330 sp3-hybrid 165, 334 &doping structures 409, 430 A-line 111, 112 r-point 112, 118 A-line 112 n-bonded chain model 381,378, 377,

r-bonded chain 377 k . p-interaction 182 k . p-perturbation 185 k . p-method 179 k . p-perturbation theory 259, 262 k-space 93 k-vector 93, 95, 105 (100) surface 379, 387, 388 (110) surface 384, 387, 388 (111) surface 376 2-dimensional electron gas 536, 552,

2D lattice 337 2D quasi-wavevector 356 3d-TM atom 320 4d-atom 320 4f-level 331 4f-orbital329 5d-level 331 5d-TM atom 320

379, 380, 381

620

Page 786: Fundamentals of Semiconductor Physics and Devices