Mahdi˜Pourfath The Non-Equilibrium Green's Function …

Computational MicroelectronicsSeries Editor: Siegfried Selberherr

Mahdi Pourfath

The Non-Equilibrium Green's Function Method for Nanoscale Device Simulation

Computational Microelectronics

EditorSiegfried SelberherrTechnical University ViennaVienna, Austria

For further volumes:http://www.springer.com/series/1263

http://www.springer.com/series/1263

Mahdi Pourfath

The Non-EquilibriumGreen’s Function Methodfor Nanoscale DeviceSimulation

123

Mahdi PourfathSchool of Electrical and Computer EngineeringUniversity of TehranTehran, Iran

ISSN 0179-0307ISBN 978-3-7091-1799-6 ISBN 978-3-7091-1800-9 (eBook)DOI 10.1007/978-3-7091-1800-9Springer Wien Heidelberg New York Dordrecht London

Library of Congress Control Number: 2014943949

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To my parents, my wife, and my daughters

Preface

This book bridges the gap between elementary quantum transport books and morerigorous graduate-level material on the quantum field theory of many-body systems.The book presents a simple, intuitive understanding of Green’s function theory andits application for the analysis of nanoelectronic devices. It attempts to explain theunderlying physics with a consistent theoretical footing. This book targets graduate-level students and researchers in electronics and physics. One of the stimulatingfactors for the writing of this book was the many requests I received from scientistsand students who wanted to receive a copy of my dissertation, where I addresseda similar topic. This book, however, includes more materials on the underlyingprinciples, numerical techniques, and applications. It is my hope that the inclusionof these elements will help young scientists to contribute something new to thefrontiers of nanoelectronics.

In this book after a short introduction in Chap. 1, the postulates of quantummechanics are briefly presented in Chap. 2. As electrons in solids experience variousscattering mechanisms, an accurate study of electron transport in solid state devicesrequires the knowledge and techniques of many-body theory. Chapters 3 and 4,respectively, review the basic principles of many-body systems and band theoryof electrons in solids. With the aid of statical mechanics, which is discussed inChap. 5, we relate microscopic and macroscopic quantities in many-body systemsand study systems both under equilibrium and non-equilibrium condition. Next, theGreen’s function formalism is presented in Chap. 6. As the exact solution of theGreen’s function for a realistic system cannot be obtained, approximation methodsare needed. Such approximations and the related methods are discussed in the restof this chapter. After building a solid theoretical foundation, numerical methodsfor calculating Green’s functions are presented in Chap. 7. All the elements ofthe kinetic equations, which are the device Hamiltonian, contact self-energies, andscattering self-energies are carefully studied and efficient methods for evaluationare explained. Finally, these methods are applied to the study of electron, spin,and phonon transport in nanoribbons in Chap. 8. Additionally, device characteristicsof tunneling transistors and photo-detectors are investigated using the outlinedmethodologies.

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I am deeply grateful to my family for their understanding and loving acceptanceof my engagement in writing this book. Some contribution to this text, however,has come from my students: Nayereh Ghobadi, Hossein Karami-Taheri, ShoeibBabaee Touski, Zahra chagazardi, Nima Djavid, Kaveh Khaliji, Sahar Pakdel, andMohammad Tabatabaee. I would specially like to thank Prof. Hans Kosina for hissupport during the preparation of this work. Finally, I owe thanks to Prof. SiegfriedSelberherr for his encouragement and long lasting patience.

Tehran, Iran Mahdi PourfathApril 2014

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Review of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1 Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Postulates of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Quantum States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.2 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.3 Measurements and Expectation Values . . . . . . . . . . . . . . . . . . . . . . . . 202.2.4 Schrödinger Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3.1 Spinors and Pauli Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3.2 Spin-Orbit Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Many-Body Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.1 First Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1.1 Indistinguishability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.1.2 Slater Determinants and Permanents . . . . . . . . . . . . . . . . . . . . . . . . . . 313.1.3 Operators in the First Quantization Representation . . . . . . . . . . . 34

3.2 Second Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2.1 Creation and Annihilation Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2.2 Operators in the Second Quantization Representation . . . . . . . . 403.2.3 Basis Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2.4 Field Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2.5 Quasi-particles and Collective Excitations . . . . . . . . . . . . . . . . . . . . 463.2.6 Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.2.7 Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.2.8 Interaction with Photons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

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4 Band Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.1 Crystal Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.2 Electrons in Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2.1 Bloch States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.2.2 Tight-Binding Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.2.3 The Hubbard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.3 Phonons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.3.1 Phonon Interaction Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.3.2 Scattering of Bloch States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5 Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.1 Historical Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.2 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.2.1 Macro and Microstates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.2.2 Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.2.3 Classical and Quantum Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.3 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.3.1 The Laws of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.3.2 Closed Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.3.3 Systems in Contact with a Heat Bath . . . . . . . . . . . . . . . . . . . . . . . . . . 825.3.4 Systems in Contact with a Heat and Particle Reservoir . . . . . . . 835.3.5 Thermodynamic Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.3.6 Thermodynamic Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.3.7 Connection to Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.4 Statistical Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.4.1 Micro-canonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.4.2 Canonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.4.3 Grand-Canonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.5 Quantum Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.5.1 Density Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.5.2 Fermi-Dirac Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.5.3 Bose-Einstein Statistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.5.4 Maxwell-Boltzmann Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.6 Non-equilibrium Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.6.1 Boltzmann Transport Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.6.2 Validity of the Boltzmann Transport Equation . . . . . . . . . . . . . . . . 975.6.3 Density Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985.6.4 Wigner Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.6.5 Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6 Green’s Function Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056.1 Historical Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056.2 Quantum Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.2.1 Schrödinger Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066.2.2 Heisenberg Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

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6.2.3 Interaction Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.2.4 The Evolution Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086.2.5 Imaginary Time Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.3 Equilibrium Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136.3.1 Zero Temperature Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136.3.2 Finite Temperature Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . 1156.3.3 Matsubara Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.4 Non-equilibrium Green’s Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1186.4.1 Non-equilibrium Ensemble Average . . . . . . . . . . . . . . . . . . . . . . . . . . . 1186.4.2 Contour-Ordered Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.4.3 Keldysh Contour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216.4.4 Real-Time Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216.4.5 Langreth Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1236.4.6 Non-interacting Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1246.4.7 Non-interacting Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.5 Perturbation Expansion of the Green’s Function . . . . . . . . . . . . . . . . . . . . . . 1286.5.1 Wick’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1296.5.2 Feynman Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1316.5.3 First-Order Perturbation Expansion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1316.5.4 Dyson Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1346.5.5 Electron-Electron Self-Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1366.5.6 Electron-Phonon Self-Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.6 Quantum Kinetic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1386.6.1 The Kadanoff-Baym Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1396.6.2 Keldysh Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1396.6.3 Steady-State Kinetic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.7 Variational Derivation of Self-Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1426.7.1 Electron-Electron Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1436.7.2 Screened Interaction, Polarization, and Vertex Function . . . . . 1446.7.3 Electron-Phonon Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1476.7.4 The Phonon Green’s Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1506.7.5 The Phonon Self-Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1516.7.6 Approximation of the Self-Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

6.8 Relation to Observables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1526.8.1 Electron and Hole Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1526.8.2 Spectral Function and Local Density of States . . . . . . . . . . . . . . . . 1536.8.3 Current Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

7 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1577.1 Basis Functions and Matrix Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

7.1.1 Free Transverse-Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1597.1.2 Real-Space Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1607.1.3 Coupled Mode-Space Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1627.1.4 Decoupled Mode-Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

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7.2 Contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1657.2.1 Matrix Truncation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1667.2.2 Surface Green’s Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1687.2.3 Sancho-Rubio Iterative Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1697.2.4 Contact Self-Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1727.2.5 Wide-Band Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

7.3 Scattering Self-Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1757.3.1 Electron-Phonon Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1757.3.2 Acoustic Phonon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1777.3.3 Optical Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1777.3.4 Polar Optical Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

7.4 Recursive Method for Calculating Green’s Functions. . . . . . . . . . . . . . . . . 1807.4.1 Retarded and Advanced Green’s Functions . . . . . . . . . . . . . . . . . . . 1817.4.2 Lesser and Greater Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . 182

7.5 Evaluation of Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1847.5.1 Carrier Concentration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1847.5.2 Current Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1857.5.3 Transmission Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

7.6 Selection of the Energy Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1877.6.1 Confined States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1877.6.2 Non-adaptive Energy Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1887.6.3 Adaptive Energy Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

7.7 Self-Consistent Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1927.7.1 Self-Consistent Iteration Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1937.7.2 Convergence of the Self-Consistent Simulations . . . . . . . . . . . . . . 194

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

8 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2018.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2018.2 Electronic Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

8.2.1 Transport Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2058.2.2 Line-Edge Roughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2068.2.3 Substrate Corrugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

8.3 Spin Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2168.3.1 Multi-orbital Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2168.3.2 Transport Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2188.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

8.4 Phonon Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2208.4.1 Phonon Bandstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2218.4.2 Phonon Green’s Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2228.4.3 Phonon Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2238.4.4 Ballistic Phonon Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

8.5 Graphene-Based Tunneling Transistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2308.5.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2318.5.2 Self-Consistent Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2358.5.3 Device Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

Contents xiii

8.6 CNT and GNR-Based Photodetectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2418.6.1 Electron-Photon Self-Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2428.6.2 Quantum Efficiency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

Notation

Symbols

x Scalarx� Complex conjugate of xOO OperatorOO� Hermitian Conjugate of the operator OOOI Unity operatorx Vectorex Unity vector in direction xx � y Scalar inner product˝ Convolution@t .�/ Partial derivative with respect to trx Gradient of xA MatrixAij Elements of the matrix AA� Conjugate transpose of the matrix AG Green’s function for electronsG0 Non-interacting Green’s functionGr;a Retarded and advanced Green’s functionG? Greater and lesser Green’s functiong Surface or incomplete Green’s functionD Green’s function for phonons˙ Self-energy˙C Contact self-energy˙S Scattering self-energyOVe-� Electron-photon interaction potentialOVe-ph Electron-phonon interaction potentialH Hamiltonian in the first quantizationH0 Single-particle or non-interacting HamiltonianOH Hamiltonian in the second quantization

xv

xvi Notation

O Field operatorOb Annihilation operator for bosonsOc Annihilation operator for fermionsOa General annihilation operatorOb� Creation operator for bosonsOc� Creation operator for fermionsOa� General creation operatorh�i Ensemble statistical averager Position operatorOp Momentum operatorOL Angular momentum operatorOS Spin operatorV VolumeA Surface˝ Volume of unit-celltij Hopping parameter� Spin-orbit coupling constantu Lattice vibration vectorR Position of atomsR0 Equilibrium position of atoms"q� Phonon and photon polarization vector˚ Force constant� Acoustic deformation potential constantD0 Optical deformation potential constantqD Inverse Debye screening lengthE EnergyEF Fermi energyEc Conduction band-edge energyEv Valence band-edge energyEG Band gap energyI CurrentJ Current densityk Wave vector of electronkt Transverse wave vector of electronq Wave vector of phononqt Transverse wave vector of phononm Mass of electronsM Mass of atomsnB Bose-Einstein distribution functionf Fermi-Dirac distribution function� DensityH Hilbert spacet TimeT Temperature

Notation xvii

U Potential energyE Electric fieldD Electric displacement fieldB Magnetic field� Dielectric permittivity

Abbreviations

ACF Auto-correlation functionAGNR Armchair GNRAP Acoustic phononsBTE Boltzmann transport equationCMOS Complementary MOSCMS Coupled mode-spaceCNT Carbon nanotubeDFT Density functional theoryDOS Density of statesDP Deformation potentialFCM Force constant methodGNR Graphene nanoribbonDOS Density of statesFET Field-effect transistorITRS International technology road-map for semiconductorsIR Infra-redLDOS Local DOSLER Line-edge roughnessMOS Metal-oxide-semiconductorNEGF Non-equilibrium Green’s functionOP Optical phononsPOP Polar optical phononsRS Real-spaceSCBA Self-consistent Born approximationTB Tight-bindingTFET Tunneling FETVFF Valence force fieldVTGFET Vertical graphene TFETVTGNRFET Vertical GNR TFETZB Zone boundary phononsZGNR Zigzag GNR

Chapter 1Introduction

The increasing demand for higher computing power, smaller dimensions, andlower power consumption of integrated circuits leads to a pressing need to down-scale semiconductor components. Moore’s law, which has continued unabated for40 years, is the empirical observation that component density and performance ofintegrated circuits doubles every 2 years. The microelectronics industry has driventransistor feature size scaling from 10m to �20 nm during the past 40 years.However, growing technological challenges and costs are limiting the scaling ofSi-CMOS technology [84]. To meet the requirements of the ITRS roadmap [88]novel structures, channel materials, and even state variables, such as spin [79], areexpected to be used. On the one hand, in nano-scaled devices quantum mechanicaleffects play an important role in the operation of such devices. On the other hand,novel devices, such as tunneling transistors [86], which operate based on quantummechanical effects, have been proposed.

Technology computer-aided design (TCAD) is a branch of electronic designautomation that models semiconductor device fabrication and semiconductor deviceoperation. TCAD is one of the few enabling methodologies that can reducedevelopment cycle times and costs. The development of new modeling capabilitygenerally requires long term research, and increasingly involves interdisciplinaryactivities. The degree of TCAD success depends on the accuracy and efficiency ofthe employed model. Common TCAD tools for semiconductor device analysis arebased on the semi-classical drift-diffusion approach [87]. However, the appearanceof short channel effects in sub-micron devices led to the development of thehydrodynamic model [5]. The Boltzmann transport equation (BTE) is the basisof all semi-classical transport models. In BTE electrons are assumed to obeyNewton’s laws between their collisions, and quantum mechanics only describesscattering process which is assumed to be instantaneous [66]. However, an exactsolution of the BTE for a realistic system can be computationally expensive. Variousapproximations and approaches, such as the method of moments [21], sphericalHarmonic expansion [20], and Monte-Carlo [35] have been developed for solvingBTE with reasonable computation time and accuracy.

M. Pourfath, The Non-Equilibrium Green’s Function Method for Nanoscale DeviceSimulation, Computational Microelectronics, DOI 10.1007/978-3-7091-1800-9__1,© Springer-Verlag Wien 2014

1

2 1 Introduction

Modeling nanoelectronic devices, however, requires quantum mechanical mod-els [12] and/or consideration of other degrees of freedom, such as spin [108].Due to the presence of various interaction mechanisms in solids, an accuratemodeling of quantum transport in nanoelectronic devices requires knowledge ofmany-body theory, which is mainly based on the second quantization language andthe application of Green’s function [56, 59]. The non-equilibrium Green’s function(NEGF) has been developed to study the many-particle quantum system underequilibrium or non-equilibrium condition. The applications of the NEGF formalismhave been extensive including quantum optics [27], quantum corrections to theBoltzmann transport equation [36, 67], high field transport in bulk systems [4], andelectron transport through nano-scaled systems. Over the last decade, the NEGFformalism has become widely used for modeling high-bias, quantum electron andhole transport in a wide variety of materials and devices: III–V resonant tunneldiodes [3, 6, 8, 10, 22, 41, 42, 44, 45, 53–55, 57–59, 113, 115], electron waveguides[69], quantum cascade lasers [52, 61, 109, 110], Si tunnel diodes [82, 83], ultra-scaled Si-MOSFETs [38,60,92,94,98], Si nano-pillars [47,80,81], carbon nanotubes[24,70,72,74,76,78,95,96,101,104], graphene nanoribbons [73,75,111,112,114],Si-nanowires [64, 65], metal wires [7, 62], organic molecules [11, 15, 16, 18, 19,33, 89, 97, 102, 105, 106], spintronic devices [26, 63, 85, 108], thermal and thermo-electric devices [39, 40, 43]. Physics that have been included are full-bandstructure[42, 59, 64, 65, 83], the self-consistent Hartree potential [46, 58, 115], exchange-correlation potentials within a density functional approach [7,11,14,58,96,106,107],acoustic, optical, intra-valley, inter-valley, and inter-band phonon scattering, alloydisorder and interface roughness scattering [22, 48, 49, 55, 59, 61, 93, 111], photonabsorption and emission [1, 17, 25, 29, 61, 73, 77, 91], single-electron charging andnon-equilibrium Kondo systems [9, 23, 31, 32, 71, 99, 103], topological insulatingphases [34,68], shot noise [10,30,113], A.C. [2,8,13,37,50,90,100], and transientresponse [28, 51, 100].

Due the complexity of this formalism, however, one should have a deepunderstanding of the underlying principles and employ smart approximations andnumerical methods for solving the Green’s functions with the desired accuracy at areasonable computational time. We continue with a brief review of the postulates ofquantum mechanics. Next, the basis of many-body theory, which is formulated inthe language of second quantization, is presented. Thereafter, electrons in a solidas an example of a many-body system is discussed. Then we continue with aquick review of statistical mechanics. With the aid of this theory one can expressmacroscopic states of the system in terms of its microscopic states. All thesetheories will be employed to introduce the Green’s function formalism. However,as stated before, an exact solution of the Green’s function for realistic systemsis nearly impossible to achieve. Therefore, approximation methods for evaluatingGreen’s functions are introduced. To apply Green’s function for the analysis ofelectronic devices, numerical techniques need to be employed. After discussingthese techniques, the outlined model and techniques are applied to devices withdifferent operating principles.

References 3

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6 1 Introduction

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73. Nematian, H., Moradinasab, M., Pourfath, M., Fathipour, M., Kosina, H.: Optical propertiesof armchair graphene nanoribbons embedded in hexagonal boron nitride lattices. J. Appl.Phys. 111, 093512 (2012)

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Chapter 2Review of Quantum Mechanics

Quantum mechanics is the most successful physical theory. No phenomenonhas yet been found, which contradicts the predictions of quantum mechanics.Quantum mechanics is essential for understanding and modeling carrier transportin nanoelectronic devices. In this chapter basic elements of quantum mechanics arebriefly reviewed. For a more comprehensive review, interested readers are referredto standard textbooks, such as Refs. [7, 8, 12, 13].

2.1 Historical Background

At the end of the nineteenth century, classical physics offered a rather completeview of most processes in the natural world based on deterministic Newtoniandynamics, Maxwell’s equations of electromagnetism, and thermodynamics. In1900, Thompson (Lord Kelvin) gave a lecture titled Nineteenth-Century Cloudsover the Dynamical Theory of Heat and Light. He claimed that most problemsrelated to physics had already been solved, except the failure of Michelson-Morley’sexperiment to measure the velocity of light with respect to an absolute ether, and theproblem of black body radiation. The attempt to solve these problems gave birth toEinstein’s theories of special and general relativity, and quantum mechanics, thelatter of which provides a conceptual framework for understanding the physicalprocesses taking place at the atomic scale.

Classical electrodynamics predicts that objects emit radiation, but this idea failsto predict the observed spectrum of light at higher frequencies, which is oftenreferred to as ultraviolet catastrophe. In 1900, Planck supposed that light is notemitted continuously in a constant amount from all matter. He suggested that lightis always emitted and absorbed in discrete units referred to as quanta such thateach of these energy quanta � is proportional to the frequency with which eachindividually radiates energy [11]:

" D h ; (2.1)


9

10 2 Review of Quantum Mechanics

where h is Planck’s constant. Based on this assumption, Planck obtained a math-ematical equation which described the entire spectrum of black body radiation. In1905, Einstein explained the photoelectric effect by postulating that light can bedivided into a finite number of energy quanta [6], later came to be called photons.It explained why the energy of photo excited electrons was dependent only on thefrequency of the incident light and not on its intensity. In 1913, Bohr explainedthe spectral lines of the hydrogen atom, again by using quantization [3]. The ideathat each photon had to consist of energy in terms of quanta was a remarkableachievement, but the concept was strongly resisted at first because it contradictedthe wave theory of light that followed naturally from Maxwell’s equations. However,Einstein’s postulate was confirmed experimentally by Millikan and Compton overthe next two decades. Thus it became apparent that light has both wave-likeand particle-like properties. In 1924, de Broglie put forward his theory of matterwaves by stating that particles can exhibit wave characteristics and vice versa. Hesuggested that all particles, like electrons, must be transported by a wave into whichthey are incorporated [4]. With every particle of matter, a wave must be associated:

� D h

p; (2.2)

where � is the wavelength and p is the momentum. Building on de Broglie’sapproach, modern quantum mechanics was born in 1925, when Heisenberg devel-oped matrix mechanics [9] and Erwin Schrödinger invented wave mechanics andthe non-relativistic Schrödinger equation as an approximation to the generalizedcase of de Broglie’s theory [14–17]. Schrödinger subsequently showed that thetwo approaches were equivalent [18]. Starting around 1927, Paul Dirac started tounify quantum mechanics with special relativity by proposing the Dirac equationfor the electron. It predicts electron spin and led Dirac to predict the existence ofthe positron. He also pioneered the use of operator theory, including the bra-ketnotation. During the same period, von Neumann formulated a rigorous mathematicalbasis for quantum mechanics as the theory of linear operators on Hilbert spaces.

Quantum mechanics was successful at describing non-relativistic systems withfixed numbers of particles, but a new framework was needed to describe systems inwhich particles can be created or destroyed, for example, the electromagnetic field,considered as a collection of photons. Beginning in 1927, researchers, includingDirac, Pauli, Weisskopf, and Jordan, made attempts to apply quantum mechanicsto fields instead of particles, resulting in quantum field theories [5]. Quantizing theclassical theory of a single-particle gave rise to a wave function and quantizinga field appeared to be similar to quantizing a theory that was already quantized,leading to the term second quantization in the early literature, which is still used todescribe field quantization. Quantum field theory provides a theoretical frameworkfor constructing quantum mechanical models of fields and many-body systems.Interested readers can find further details on the history of quantum mechanics inRef. [1].

2.2 Postulates of Quantum Mechanics 11

2.2 Postulates of Quantum Mechanics

Theories are based on postulates. Postulates are rules of nature, which cannot beproven or derived. Their justification is from the fact that they are consistent withexperiment. Classical mechanics is based on two postulates: the state postulate andthe time evolution postulate. In classical mechanics, the change in the state of thesystem is characterized by a set of dynamical variables. Therefore, initial state ofthe system specifies the initial conditions of these dynamical variables, and the waythe state of the system changes, based on an equation of motion, is described byhow the dynamical variables change with time. In classical mechanics the stateof a system is described by the positions and velocities of the particles that formthe system. Given that the state of a point particle is known at some time t0, onecan predict its state at any other time t , by using Newton’s second law. Quantummechanics differs from classical mechanics by the employment of operators ratherthan dynamical variables. In addition, quantum mechanics involves a new postulate– the measurement postulate – that does not have a classical analogue. A moredetailed discussion of those postulates follows.

2.2.1 Quantum States

The state of a system at any instant of time may be represented by a wave function which is continuous and differentiable. For example the state of an isolated systemat some time t with a point particle is given by the wave function .r; t /. Theinterpretation of a wave describing a particle was the subject of discussion in earlyyears of quantum mechanics. Born paved the way for the statistical interpretation ofthe wave function describing a particle. He suggested the term guiding field whichdetermines at every point the probability of finding the particle there. The square ofthe amplitude of the wave function gives the probability density j j2 D �,where � is the complex conjugate of . The probability of finding a particle ina certain volume element dr at some time t is therefore given by j .r; t /j2dr. Theamplitude of is conventionally normalized:

ZV

dr .r; t / �.r; t / D 1 ; (2.3)

which implies that the particle must be somewhere in space V . The wave functioncan only be normalized if it is square integrable. Bound states, which describe closedsystems, are square integrable, whereas for free states this condition is not satisfied.Free states, however, can be normalized by imposing either a periodic or closedboundary condition to the system [7].

The state of a classical particle can be completely described by its position andmomentum, which can be mathematically represented by a point in a phase space. Ina classical system with N particles, the phase-space is in general 6N -dimensional.


In quantum mechanical systems, however, any state can be mathematically repre-sented as a vector in a Hilbert space. A Hilbert space H is a vector space over C(H ! C � C ) on which an inner product hji is defined that satisfies the followingconditions for all f , g, and h on H and ˛ and ˛, ˇ in C :

• Symmetry: hf jgi D hgjf i�.• Linearity: h f̨ C ˇgjhi D ˛hf; hi C ˇhgjhi.• Positivity: f ¤ 0, hf jf i > 0.

In addition H should be complete with respect to the norm defined by

jjf jj Dp

hf jf i : (2.4)

For describing states in Hilbert space, bra-ket notation is considered as a standardnotation. This notation was introduced by Dirac and is also known as Dirac notation.A state is represented by the ket j i. This state is an abstract vector in a Hilbertspace. A dual or adjoint-space state is represented by the bra h j. There is a one-to-one correspondence between the elements of the ket space and those of the relatedbra space. For every element of the ket space, there is a corresponding element inthe bra space. These elements are related by the adjoint operation

h j � j i� : (2.5)

The expressionR

dr�� is considered as an inner product of wave functions and� and is interpreted as the probability amplitude for the state to collapse into thestate �. Usually a shorthand notation is used

h�j i DZ

dr �� : (2.6)

In the Dirac Notation, the normalization condition reads

h j i D 1 : (2.7)

While, by orthogonality

h j�i D 0 : (2.8)

For any two kets from a countable orthonormal set indexed by integers, we maywrite

hmjni D ım;n ; (2.9)

where ıi;j is the Kronecker delta function. Using a complete orthonormal set of ketsjni as a basis in Hilbert space, one can expand any state in terms of these basisvectors:


j i DXn

cnjni; cn D hnj i : (2.10)

By writing hnj i D n, one can represent j i as a column vector with respect tothe set of basis states:

j i D

264 1 2:::

375 : (2.11)

To obtain a vector representation of a bra, one can form the inner product of twoarbitrary states as

h�j i DXn

h�jnihnj i : (2.12)

Using the notation �n D hnj�i and knowing that h�jni D hnj�i� one obtainsh�jni D ��

n and hence

h�j i DXn

��n n ; (2.13)

which can be written as

h�j i D ��1 �

�2 : : :

�264 1 2:::

375 : (2.14)

Here one can make an identification of the bra vector h�j as a row vector, which isthe complex conjugate of its corresponding ket vector:

h�j D ��1 �

�2 : : :

�: (2.15)

In Eq. (2.10) the summation is for discrete states. In many systems which wouldinclude, for example, a free particle, one has a continuous set of states. In suchsystems the discrete index n and summation in Eq. (2.10) are replaced by acontinuous variable and integral, respectively. For a continuous set of complete ketsindexed by the continuous variables x and x0, the orthogonality relation is written as

hxjx0i D ı�x � x0� : (2.16)

Here ı.x � x0/ is the Dirac delta function, which is an analogue to the Kroneckerdelta function. The Hilbert space of a point particle without spin degree of freedomcan be spanned by the spatial coordinate jri, where r extends over the set of allpoints in space. Since there are infinitely many vectors in the basis, this is an


infinite-dimensional Hilbert space. Using this basis set, an arbitrary state j i canbe represented by

j i DZ

dr c.r/jri : (2.17)

The coefficients c.r/ in Eq. (2.17) are simply the value of the wave function at eachpoint r, c.r/ D .r/. In Dirac notation, the wave function in the coordinate spaceis represented by

.r/ � hrj i ; (2.18)

that is, its projection on the r basis.In principle the wave function can be expressed in terms of any complete set of

eigenstates. By choosing different sets of basis functions, one can arrive at a numberof different representations of the same state. The eigenstates of any Hermitianoperator form a complete basis in Hilbert space, see Sect. 2.2.2. The eigenstatesof the momentum operator (see Eq. (2.49)), which are represented by the ket jpi,are plane waves in coordinate-space [7]

hrjpi D c.p/eip�r=„ : (2.19)

The normalization constant c.p/ can be obtained by using the orthonormality of themomentum eigenstates:

hpjp0i D ı.p � p0/ DZ

dr hpjrihrjp0i ;

DZ

dr c�.p/e�ip�r=„c.p0/eip0�r=„ ;

D c�.p/c.p0/Z

dr e�i.p�p0/�r=„ ;

D jc.p/j2 .2�„/3 ı.p � p0/ :

(2.20)

Thus jc.p/j2 D 1=.2�„/3. The eigenstates of the momentum operator in coordinaterepresentation can be written as

hrjpi D 1

.2�„/3=2 eip�r=„ : (2.21)

If one picks the eigenstates of the momentum operator as a set of basis functions, theresulting wave function .p/ is said to be the wave function in momentum space:

.p/ � hpj i : (2.22)


The interpretation of the wave function in momentum space is that j .p/j2dp isthe probability to find the momentum in the range Œp;p C dp . The momentumrepresentation of a wave function is very closely related to the Fourier transformand the concept of frequency domain. Since a quantum mechanical particle has afrequency proportional to the momentum (Eq. (2.2)), describing the particle as a sumof its momentum components is equivalent to describing it as a sum of frequencycomponents, which is a Fourier transform:

.r/ D hrj i DZ

dp hrjpihpj i DZ

dp hrjpi .p/ ;

D 1

.2�„/3=2Z

dp eip�r=„ :(2.23)

Here hrjpi represents the momentum eigenstate in coordinate space, which is aplane wave, see Eq. (2.21). Similarly one can write

.p/ D hpj i DZ

dr hpjrihrj i ;

D 1

.2�„/3=2Z

dr e�ip�r=„ .r/ :(2.24)

It is often practical to use wave vectors defined as k D p=„. Conventional momentaare denoted by p;p0;P and wave vectors are represented by k;K;q. The wavefunction for the state with the wave vector k becomes

hrjki D 1

.2�/3=2eik�r ; (2.25)

with the normalization condition

hk0jki D ı.k0 � k/ : (2.26)

Equation (2.24) can be expressed in k-space as

.k/ D 1

.2�/3=2

Zdr e�ik�r .r/ : (2.27)

For some applications it is more convenient to assume periodic boundary conditionsfor k enclosed in a huge box taken to be a cube with sides L and volume V D L3,yielding the wave function

hrjki D 1pV

eik�r : (2.28)


The normalization then reads

hk0jki DZV

dr hk0jrihrjki ;

D 1

V

ZV

dr ei.k�k0/�r D ık;k0 :

(2.29)

Boundary conditions imposed at the edges of the box, allow only discrete values foreach component of the wave vector:

kx D 2�

Lnx; ky D 2�

Lny; kz D 2�

Lnz : (2.30)

In macroscopic systems, the distance between two adjacent wave vectors is verysmall. If one has to sum a function f .k/ over allowed wave vectors, one cansubstitute the sum with integral, yielding the practical rule

Xk

f .k/ ! L3

.2�/3

Zdkf .k/ D V

.2�/3

Zdkf .k/ : (2.31)

2.2.2 Operators

An operator, OO , is a mathematical entity which transforms one state into another

OOj i D j�i : (2.32)

By expanding the state j i in terms of some basis set jni, one can write

j�i D OOj i D OOXn

jnihnj i DXn

OOjnihnj i : (2.33)

By multiplying both sides by the bra hnj one obtains

hmj�i DXn

hmj OOjnihnj i ; (2.34)

which can be written as

�m DXn

Omn n ; (2.35)

with

Omn D hmj OOjni: (2.36)


Equation (2.35) can be written as a matrix equation

26664

�1�2�3:::

37775 D

26664

O11 O12 O13 : : :

O21 O22 O23 : : :

O31 O32 O33 : : ::::

::::::: : :

37775

26664

1 2 3:::

37775 ; (2.37)

where the operator OO is represented by a matrix

OO D

26664

O11 O12 O13 : : :

O21 O22 O23 : : :

O31 O32 O33 : : ::::

::::::: : :

37775 : (2.38)

The quantitiesOmn are known as the matrix elements of the operator OO with respectto the basis set jni. If a different set of basis states is employed, the state vectors andoperators remain the same, but the column or row vector, or matrix representing thestate vector or operator, respectively, will change. Thus to give any meaning to a rowvector, a column vector, or a matrix, it is essential that the basis states be known. Animportant part of quantum mechanics is the mathematical formalism that deals withtransforming between different sets of basis states. For this purpose the projectionoperator plays an important role. This operator can be formed by the outer productof a ket and a bra. The outer product of a ket and a bra generates an operator:

OP D j�ihj : (2.39)

If applied to a vector, it projects the vector onto the state ji and generates a newvector in parallel to j�i with a magnitude equal to the projection:

OP j i D hj ij�i : (2.40)

Given a complete set of orthonormal basis states jni any state j i can be writtenas (see Eq. (2.10))

j i DXn

jnihnj i ; (2.41)

which implies that

OI DXn

jnihnj ; (2.42)


Table 2.1 Physical observables and their corresponding quantum mechanical operators. ex is theunit vector along the x-direction

Observable name Observable symbol Operator symbol Operator operation

Position r Or Multiplication by r

Momentum p Op �i„�

ex@

@xC ey

@

@yC ez

@

@z

�

Kinetic energy T OT � „22m

�@2

@x2C @2

@y2C @2

@z2

�

Potential energy V.r/ OV .r/ Multiplication by V.r/Total energy E OH OT C V.r/

Orbital angular momentum Lx OLx �i„�y@

@z� z

@

@y

�

Ly OLy �i„�

z@

@x� x

@

@z

�

Lz OLz �i„�x@

@y� y

@

@x

�

where OI represents an identity operator. By applying the identity operator from theleft and right side

OI OO OI DXm;n

jmihmj OOjnihnj ;

DXm;n

Omnjmihnj : (2.43)

Classical dynamical variables, such as position and momentum, are representedin quantum mechanics by linear Hermitian operators, which act on the wavefunction. An operator OO W H ! H in Hilbert space is called Hermitian or self-adjoint if OO D OO�, which is equivalent to (see Sect. 2.2.3)

h�j OO i D h OO�j i ; (2.44)

where j�i and j i are arbitrary states. Hermitian operators have the followingproperties:

• The eigenvalues are always real.• The eigenstates can always be chosen so that they are normalized and mutually

orthogonal; in other words, an orthonormal set.• Their eigenstates form a complete set. This implies that any state can be written

as some linear combination of the eigenstates.

In quantum mechanics the operators are also called observable operators, or justobservables. Table 2.1 shows common dynamical variables and their correspondingquantum mechanical operators. The wave function of a particle, .r/, representsthe probability density of finding the particle at some position r. Therefore, the


expectation value (see Sect. 2.2.3) of a measurement of the position of the particleis hri D R

dr rj j2. Accordingly, the quantum mechanical operator correspondingto position is Or, where

Or .r/ D r .r/ : (2.45)

It can be shown that the eigenstates of the position operator, represented in positionbasis, are Dirac delta functions [7]:

hrjr0i D ı.r � r0/ : (2.46)

The momentum operator can be derived from infinitesimal translations. The trans-lation operator OT .�/, where � represents the length of the translation, satisfies

OT ."/j i DZ

dr OT ."/jrihrj i ;

DZ

dr jr C �ihrj i DZ

dr jrihr � �j i ;

DZ

dr jri .r � �/ :

(2.47)

Assuming the wave function is differentiable, for infinitesimal values of �, one has .r � �/ D .r/ � �d =dr. Therefore, the translation operator can be written as

OT ."/ D 1 � � d

drD 1 � i

„��

�i„ d

dr

�: (2.48)

As the momentum is the generator of translation, the relation between translationand momentum operators is T .�/ D 1 � i� Op=„, therefore, the momentum operatorcan be written as

Op D �i„ d

dr: (2.49)

In classical mechanics, angular momentum is defined as L D r � p. This canbe carried over to quantum mechanics by reinterpreting r as the position operatorand p as the momentum operator. OL is then an operator, called the orbital angularmomentum operator. OL is a vector operator OL D . OLx; OLy; OLz/ with

OLx D Oy Opz � Oz Opy; OLy D Oz Opx � Ox Opz; OLz D Ox Opy � Oy Opx : (2.50)

However, there is another type of angular momentum, called spin angular momen-tum (more often shortened to spin), represented by the spin operator OS, see Sect. 2.3.Spin is an intrinsic property of a particle, unrelated to any sort of motion in space.Thus total angular momentum becomes OJ D OL C OS. Conservation of angular


momentum states that OJ for a closed system is conserved. However, OL and OS arenot generally conserved. For example, the spin-orbit interaction allows angularmomentum to transfer back and forth between OL and OS, with the total OJ remainingconstant.

It should be noted that the product of two operators in general does commuteOA OB � OB OA ¤ 0. Two operators commute if, and only if OA OB � OB OA D 0. This

expression is usually called commutator and is written as

Œ OA; OB � D OA OB � OB OA : (2.51)

In analogy an anti-commutator relation is defined as

Œ OA; OB C D OA OB C OB OA : (2.52)

If two operators commute with each other, they have simultaneous eigenstatesimplying that they can be measured together, see Sect. 2.2.3. For instance,

Œ Ox; Opx �j i D x�i„@j i@x

� �i„@xj i@x

D i„j i : (2.53)

This indicates that position and momentum operators along the same direction donot have common eigenstates, whereas position and momentum along differentdirections commute with each other:

Œ Ox; Opx � D Œ Oy; Opy � D ŒOz; Opz � D i„ ;Œ Ox; Opy � D Œ Ox; Opz � D Œ Oy; Opx � D Œ Oy; Opz � D ŒOz; Opx � D ŒOz; Opy � D 0 :

(2.54)

Using the commutation relations of position and momentum, one can show that thecomponents of the orbital angular momentum satisfy the following relations

Œ OLx; OLy � D i„ OLz ; Œ OLy; OLz � D i„Lx ; Œ OLz; OLx � D i„ OLy : (2.55)

Like any vector, a magnitude can be defined for the orbital angular momentumoperator OL2 � OL2x C OL2y C OL2z . OL2 is a quantum operator that commutes with the

components of OL:

Œ OL2; OLx � D Œ OL2; OLy � D Œ OL2; OLz � D 0 : (2.56)

2.2.3 Measurements and Expectation Values

The measurement postulate provides the bridge between the wave function, whichis an abstract object and cannot be probed directly, and actual measurements.


In classical mechanics, it is implicitly assumed that the accuracy of a measurement isonly limited by the accuracy of the measurement device, which, at least in principle,can be improved indefinitely. In addition, it is assumed that, at least ideally,measurements can be made such that they do not significantly affect the system.In quantum mechanics, the outcome of a measurement is, however, probabilistic. Ifthe wave function of a quantum mechanical system is known, one can only predictthe probability of a measurement, rather than the outcome itself. It is a fundamentallimitation of quantum mechanical systems and has nothing to do with experimentallimitations or the accuracy of measurement devices. Furthermore, there is no way toavoid the effect of the measurement on the system. Assuming un and juni representthe eigenvalues and eigenstates of OO , respectively, such that OOjuni D unjuni, then

• The outcome of the measurement is always one of the eigenvalues of OO ,• The probability for measuring the eigenvalue un is given by pn D jhunj ij2,• If the measurement gives the value un, after the measurement the state of the

system will collapse to the corresponding eigenstate, juni.In a quantum mechanical system, the only possible measurement outcomes are equalto the eigenvalues of the operator representing the observable, and the probability ofmeasuring this value is given by the absolute value squared of the inner product ofthe quantum state with the operator’s corresponding eigenstate. If two Hermitianoperators commute, there is a complete set of eigenstates that is common toboth. Under this condition it is possible to measure both quantities simultaneouslywith certainty. If they do not commute, one of the measurements alters the othermeasurement outcome.

Quantum mechanics shows an inherent statistical behavior. The measured out-come of an experiment will generally not be the same if the experiment is repeatedseveral times. Quantum mechanics does not, in fact, predict the result of individualmeasurements, but only their statistical mean. This predicted mean value is calledthe expectation value. If a state of a system is described by the wave function .r; t /the average of any physical observable OO is given by

hOi D h j OOj i DZ

dr �.r; t / OO .r; t / ; (2.57)

where hOi represent the expectation value. This integral can be interpreted as theaverage value that one would expect to obtain over a large number of runs of theexperiment. As stated before, any quantum mechanical operator OO associated witha measurable property must be linear and Hermitian. Physically, the Hermitianproperty is necessary in order for the eigenvalues or measurement values to beconstrained to real numbers.

As discussed earlier, if two physical quantities correspond to commuting Her-mitian operators, they have a common set of eigenstates. In these eigenstatesboth quantities have precise values at the same time and they can be measuredsimultaneously. However, if two operators do not commute, in general one cannot


specify both values precisely. In this case the uncertainty principle requires that ifone measures one of them more accurately, one increasingly loses track of the other.For example, the energy and momentum of a free particle can both be specifiedexactly, whereas position and momentum along one direction cannot be specifiedsimultaneously. To quantify how accurately physical quantities can be measuredtogether one can use the mean square deviation. The deviation from the mean valueof a quantum mechanical operator OO is defined as � OO D OO � hOi. The meansquare deviation is, therefore, expressed as

�O2 D h j�� OO

2 j i DZ

dr �.r; t /�� OO

2 .r; t / : (2.58)

One may consider two physical quantities described by Hermitian operators OA andOB . It can be shown that the mean values of the square of deviation are related as [7]

�A2�B2 ��1

2ihŒ OA; OB �i

�2: (2.59)

This is Heisenberg’s uncertainty principle [10] in its most general form. It indicatesthat two physical quantities cannot be simultaneously measured without uncertaintyif their corresponding operators do not commute. The commutator of the positionand momentum operators along the same direction is Œ Ox; Opx � D i„. As a result, theuncertainty relation for these quantities is obtained as

�x�px � „2: (2.60)

A similar relation holds between energy and time:

�E�t � „2: (2.61)

In the special theory of relativity, a coordinate in space-time is specified by a 4-vector consisting of position and time. Knowing that E D i„@=@t one obtains thecommutator Œ OE; t D i„. Therefore, by employing Eq. (2.59) one can achieve theuncertainty relation for energy and time [7]. In non-relativistic theory, however,time is considered as in independent variable of which dynamical variables arefunctions. In this context, �t is not the standard deviation of a collection of timemeasurements. The time-energy uncertainty principle is a statement about howstatistical uncertainty in the energy controls the time scale for a change in thesystem [8]. If the initial state of a system is an energy eigenstate, then the systemremains stationary and �E D 0, which forces �t ! 1, implying that the physicalattributes of the state never change.


2.2.4 Schrödinger Equation

The Schrödinger equation replaces Newton’s second law as the fundamentalequation of motion. Given that the state of the system at some initial time t0,this equation predicts the state of the system at another time t . Time-dependentSchrödinger equation is written as

i„ @@t� D OH ; (2.62)

The Hamiltonian operator OH replaces the classical Hamiltonian, which givesthe total energy in terms of the particle position and momentum. Using thecorresponding operators from Table 2.1, one obtains

OH D OT C OV D � „22m

r2 C V.r/ ; (2.63)

where r2 is the Laplacian. Mathematically the time-dependent Schrödinger equa-tion is a linear, second order, partial differential equation. Any linear differentialequation allows for the superposition of its solutions. This implies that, if 1 and 2are solutions of the Schrödinger equation, then any linear combination of �1 and �2are solutions as well:

�.r; t / D C1�1.r; t /C C2�1.r; t / ; (2.64)

where, C1 and C2 are some constants. This property is called superposition.If the potential energy does not explicitly depend on time, one can solve the

Schrödinger equation by separating the variables. Under this condition the wavefunction is assumed to be the product of a function of time and a function of position�.r; t / D .r/�.t/. Substituting this relation in Eq. (2.62) and dividing both sideby .r; t / the Schrödinger equation can be reformulated as

i„ 1

�.t/

@�.t/

@tD

OH.r/.r/

D E : (2.65)

The term on the left hand side of Eq. (2.65) is a function of time only, whereas thesecond term depends on the positions only. As time and position are independentvariables, this relation holds only if both sides are constant. This constant is denotedby E and has a unit of energy. Therefore, the Schrödinger equation can be writtenas two decoupled equations. The solution of the time-dependent equation is simplygiven by �.t/ D exp .�iEt=„/ and the other equation, which is called the time-independent Schrödinger, reads as

OH .r/ D E .r/ : (2.66)


Finally, the total wave function of the system is given by

�.r; t / D .r/ exp .�iEt=„/ : (2.67)

2.3 Spin

Spin is one of two types of angular momentum in quantum mechanics, the otherbeing orbital angular momentum, which is the quantum-mechanical counterpart tothe classical notion of angular momentum. Spin is an intrinsic form of angularmomentum carried by elementary particles. All elementary particles of a givenkind have the same magnitude of spin angular momentum, which is indicated byassigning the particle a spin quantum number. Pauli was the first to propose theconcept of spin. In 1925, Kronig, Uhlenbeck, and Goudsmit suggested a physicalinterpretation of particles spinning around their own axis. When Paul Dirac derivedhis relativistic quantum mechanics in 1928, electron spin was an essential partof it. There is a theorem in relativistic quantum field theory called spin-statistics,proven by Heisenberg. It says that any particle with integer spin, such as a photon,should obey Bose-Einstein statistics, while any particle with half-odd spin, such asan electron, should obey Fermi-Dirac statistics. The half-spin property of electronsresults in the Pauli exclusion principle, which in turn underlies the periodic table ofchemical elements.

2.3.1 Spinors and Pauli Equation

The existence of spin angular momentum is inferred from experiments, such asthe Stern-Gerlach experiment, in which particles are observed to possess angularmomentum that cannot be accounted for by orbital angular momentum alone. Spinis like a vector quantity that has a definite magnitude and also has a direction.Experiments suggest the existence of a spin vector OS D . OSx; OSy; OSz/ that should bean angular momentum vector operator. Therefore, the components of OS should obeythe same commutation relation as the components of the orbital angular momentum:

Œ OSx; OSy � D i„ OSz ; Œ OSy; OSz � D i„Sx ; Œ OSz; OSx � D i„ OSy : (2.68)

Furthermore, the components of the spin operator should be Hermitian ( OSi D OS�i )to guarantee that their expectation values are real. However, spins are not strictlyvectors, and they are instead described as a related quantity: a spinor. For therepresentation of the operators, it is common to use Pauli matrices �i [7]:

OSx D 12„O�x ; OSy D 1

2„O�y ; OSz D 1

2„O�z : (2.69)

2.3 Spin 25

Thus the commutation relations Eq. (2.68) take the following form

Œ O�x; O�y � D 2i O�z ; Œ O�y; O�z � D 2i O�x ; Œ O�z; O�x � D 2i O�y : (2.70)

Based on possible orientations, spin components have only two eigenvalues ˙„=2that are generally referred to as spin-up and spin-down. Therefore, the spin matricesshould be 2� 2 matrices. It is common to take the z-direction as the direction of thequantization. Then the z axis is the axis which the orientation of the spin is relatedto. Matrix O�z is diagonal in its eigenstates representation and has the eigenvalues˙1 as diagonal elements and O�2z D 1, where 1 represents 2 � 2 identity matrix.The matrices O�x and O�y take similar form in their eigenstate representation. As anidentity matrix remains unchanged under the change of representation, the identityO�2x D O�2y D O�2z D 1 holds in general. Using this identity relation and Eq. (2.70), thematrices O�x and O�y in the eigenstate representation of O�z are obtained as [7]

O�x D

0 11 0

�; O�y D

0 �ii 0

�; O�z D

1 00 �1

�: (2.71)

The unit matrix together with the Pauli matrices span the space of two-dimensionalmatrices. By taking spin into account, a further degree of freedom can be assigned toa particle. For describing this degree of freedom, one should additionally introducethe component of the spin in the z-direction Sz as an argument of the wave function.As Sz takes only two values, the wave function with spin can be mathematicallydescribed by a vector-like object known as spinor. The two components of the spinorare ".r/ D .r;C 1

2„/ and #.r/ D .r;� 1

2„/, while the total wave function is

written as [13]

� D ".r/ #.r/

�D ".r/

1

0

�C #.r/

0

1

�D ".r/�" C #.r/�# ; (2.72)

where the spin wave functions indicate only the state of spin, spin-up or spin-down,and � are unit spinors:

�" D1

0

�; �# D

0

1

�: (2.73)

The spinors in Eq. (2.73) are the eigenstates of O�z. One can write the total wavefunction as .r/�� , where � D f";#g is the spin index. In Dirac notation, spinstates can be represented by the kets j "i and j #i.

Particles with spin can have a magnetic dipole moment. The magnetic moment� of a spin-half particle with charge e, mass m, and spin angular momentum OS, is

O� D �gse

2mOS D ��B O� ; (2.74)


where �B D e„=.2m/ and the dimensionless quantity gs is called the spin g-factorand is approximately equal to 2 for electrons. Since spin interacts with magneticfields, the electron gains additional potential energy in the presence of a magneticfield �E D � O� � B. Therefore, the Hamiltonian of an electron with spin takes theform

OH� Dh OH0 C �B O� � B

i� ; (2.75)

where H0 is the spin-independent part of the Hamiltonian, � is the spinor wavefunction, and the second term on the right hand side is called the Zeeman term.Equation (2.75) is referred to as Pauli equation.

2.3.2 Spin-Orbit Coupling

Spin-orbit coupling is the interaction of a particle’s spin with its motion. Oneof the known example of this effect is that spin-orbit interaction causes shifts inan electron’s atomic energy level due to electromagnetic interaction between theelectron’s spin and the magnetic field generated by the electron’s orbit around thenucleus. Spin-orbit coupling can be viewed as the Zeeman term due to an effectivemagnetic field. In the framework of the electron that moves with velocity v relativeto an electric field E, it sees a magnetic field due to relativistic effects. In otherwords, the moving electron experiences a magnetic field in its rest frame that arisesfrom the Lorentz transformation of the static electric field; this field will affect theelectron spin and is given by

B D � v � E

c2p1 � v2=c2 � �v � E

c2D E � p

mc2; (2.76)

where terms of order .v=c/2 and higher order terms are neglected. The energy of theelectron in this field, due to its magnetic moment �, is

�ESOC D �� B D � e

mOS � B D � e

m2c2OS � .E � Op/ : (2.77)

The spin-orbit interaction potential consists of two parts. The Larmor part isconnected to the interaction of the magnetic moment of the electron with themagnetic field of the nucleus in the co-moving frame of the electron. The secondcontribution is related to Thomas precession. Equation (2.77) takes care of theformer contribution. The electron’s curved trajectory is taken into account inthe Thomas precession correction. In 1926, Llewellyn Thomas relativisticallyrecomputed the doublet separation in the fine structure of the atom [19]. The neteffect of Thomas precession is the reduction of the Larmor interaction energy byfactor 1=2, which came to be known as the Thomas half. In case of centrally

References 27

symmetric electrical fields – for example, the orbital motion of an electron in theelectric field of an atomic nucleus – one has

E D �1e

rr

dU

dr: (2.78)

Using Eq. (2.78), the definition of the angular momentum of a particle OL D Or � Op,and reducing the interaction potential due to Thomas precession, one can write thespin-orbit coupling Hamiltonian as

OHSOC D � e

2m2c2OS ��

�1e

rr

dU

dr� Op

�D 1

2m2c21

r

dU

drOL � OS D � OL � OS ; (2.79)

where the spin-orbit coupling constant � varies in the range 1.25–250 meV forvarious materials and orbitals. As spin-orbit coupling provides a way to manipulatethe spin of electrons with electric field, it plays an important role in spintronics [2].

References

1. Baggott, J.: The Quantum Story: A History in 40 Moments. Oxford University Press, Oxford(2011)

2. Bandyopadhyay, S., Cahay, M.: Introduction to Spintronics. CRC, Boca Raton (2008)3. Bohr, N.: The spectra of helium and hydrogen. Nature (London) 92, 231–232 (1913)4. De Broglie, L.: Recherches sur la Théorie des Quanta. Ann. Phys. 3(10), 22–128 (1925)5. Dirac, P.A.M.: The quantum theory of the emission and absorption of radiation. Proc. Phys.

Soc. A 114(769), 710–728 (1927)6. Einstein, A.: Über einen die Erzeugung und Verwandlung des Lichtes Betreffenden Heuristis-

chen Gesichtspunkt. Ann. Phys. 322(6), 132–148 (1905)7. Greiner, W.: Quantum Mechanics: An Introduction, 4th edn. Springer, Berlin/New York (2001)8. Griffiths, D.J.: Introduction to Quantum Mechanics. Prentice Hall, Englewood Cliffs (1995)9. Heisenberg, W.: Über quantentheoretische umdeutung kinematischer und mechanischer

beziehungen. Zs. f. Phys. 33(1), 879–893 (1925)10. Heisenberg, W.: Über den Anschaulichen Inhalt der Quantentheoretischen Kinematik und

Mechanik. Zs. f. Phys. 43(3–4), 172–198 (1927)11. Planck, M.: Über das Gesetz der Energieverteilung im Normalspectrum. Ann. Phys. 309(3),

553–563 (1901)12. Sakurai, J.J., Napolitano, J.: Modern Quantum Mechanics, 2nd edn. Addison-Wesley, Boston

(2011)13. Schiff, L.I.: Quantum Mechanics, 3rd edn. McGraw-Hill, New York (1968)14. Schrödinger, E.: Quantisierung als Eigenwertproblem. Ann. Phys. 384(4), 361–376 (1926)15. Schrödinger, E.: Quantisierung als Eigenwertproblem. Ann. Phys. 384(6), 489–527 (1926)16. Schrödinger, E.: Quantisierung als Eigenwertproblem. Ann. Phys. 385(13), 437–490 (1926)17. Schrödinger, E.: Quantisierung als Eigenwertproblem. Ann. Phys. 386(18), 109–139 (1926)18. Schrödinger, E.: Über das Verhaeltnis der Heisenberg-Born-Jordanschen Quantenmechanik zu

der meinen. Ann. Phys. 384(8), 734–756 (1926)19. Thomas, L.H.: The motion of the spinning electron. Nature (London) 117, 514 (1926)

Chapter 3Many-Body Systems

Quantum many-body theory is concerned with the study of the properties ofmicroscopic systems made of a large number of interacting particles. An accuratedescription of such systems requires the inclusion of the inter-particle potentialsin the many-body Schrödinger equation. The many-body wave function containsall possible information, but a direct solution for this problem is impractical. It istherefore necessary to resort to other techniques, and one shall rely on the secondquantization representation. These techniques are reviewed in this chapter.

3.1 First Quantization

In a quantum mechanical system of many particles, one is not able to relate a wavefunction to a certain particle. One can only determine the state of the totality of allparticles. The wave function of a system with N particles reads

�.x1; x2; : : : ; xN / D hx1; x2; : : : ; xN j� i ; (3.1)

where xi denotes the coordinates of the i th particle, including the spatial coordinateri and any discrete variables, such as spin coordinate �i . Assuming that one- orfew-body operators defined for one-body states remain unchanged when acting onN -body states, the Hamiltonian takes the form

OH DNXiD1

(Op2i2m

C U.xi /

)C 1

2

NXi¤jD1

V .xi ; xj / D OH0 C OHint ; (3.2)

where Opi represents the momentum operator, U is the background potential, and Vis the interaction potential between particles. The summation of kinetic and back-ground potential by itself is just as simple to solve as each particle alone. The term


29

30 3 Many-Body Systems

which makes the Hamiltonian hard to solve is the interaction potential. This term ismultiplied by one-half since the double summation counts each pair twice.

3.1.1 Indistinguishability

A fundamental difference between classical and quantum mechanics concernsthe concept of indistinguishability of identical particles. In quantum mechanics,identical particles are characterized by physical properties, such as mass, charge,and spin, and behave in the same manner under equal physical conditions. There aretwo ways to distinguish between particles. The first method relies on differences inthe particles’ intrinsic physical properties. If differences exist, one can distinguishbetween the particles by measuring the relevant properties. However, subatomicparticles of the same species have completely equivalent physical properties. Forinstance, every electron in the universe has exactly the same electric charge.

Even if the particles have equivalent physical properties, there remains a secondmethod for distinguishing between particles, which is to track the trajectory ofeach particle. As long as one can measure the position of each particle withinfinite precision, there would be no ambiguity of which particle is which. Thisapproach, however, contradicts the uncertainty principle. The spreading of thewave packets that describe the particles leads to an overlapping of the probabilitydensities in time [8]. It becomes therefore impossible to determine, in a subsequentmeasurement, which of the particle positions correspond to those measured earlier.Because of possible interactions, dynamical properties can also not be used todistinguish between them. The particles are then said to be indistinguishable. Asa quantum mechanical system consists of identical particles, the physical stateremains the same if particles j and k are interchanged. This operation is carriedout by the operator OPjk:

OPjk�.: : : ; xj ; : : : ; xk; : : :/ D ��.: : : ; xk; : : : ; xj ; : : :/ ; (3.3)

where � is an arbitrary constant factor. A second exchange of the two particlesresults in the original state

OP 2jk D �2 D ; (3.4)

yielding � D ˙1. Therefore, the wave function of a many-body system withrespect to the interchange of any two particles must be either anti-symmetric,with � D �1, or symmetric, with � D C1. In quantum mechanics, the spin-statistics theorem relates the spin of a particle to the particle statistics it obeys. Allparticles have either integer spin or half-integer spin in units of the reduced Planckconstant „. The theorem states that particles with half integer spin are describedby an anti-symmetric wave function under the interchange of two particles and

3.1 First Quantization 31

are called Fermions. On the other hand, particles with integer spin are describedby a symmetric wave function and are called Bosons. Examples of Fermions areelectrons and protons, while photons and phonons are Bosons. Fermions, unlikeBosons, do not share quantum states. The Pauli exclusion principle states that twoidentical Fermions can not occupy the same quantum state simultaneously. A morerigorous statement is that the total wave function for two identical Fermions is anti-symmetric with respect to exchange of the particles. This principle was formulatedby Wolfgang Pauli in 1925 [10].

3.1.2 Slater Determinants and Permanents

The basis states for a many-body system can be built from any complete orthonor-mal one-body basis states [3]. One can use a general notation for the one-body wavefunction k .xi / with i denoting a complete set of one-body quantum numbers [7].For example, for particles moving in a crystal lattice, an appropriate choice is thecomplete set of Bloch wave functions.

TheN -body Hilbert space can be written as a tensor product of one-body Hilbertspaces:

H N D H ˝ H : : :˝ H : (3.5)

If the set ji i forms an orthonormal basis in H , the states of H N are spanned bythe tensor products:

j1; 2; : : : ; N i D j1i ˝ j2i : : :˝ jN i ; (3.6)

which in the coordinate basis would correspond to the wave function product.The product states that

QNiD1 i .xi / D QN

iD1hxi ji i is a basis for the N -bodyHilbert space, however, it is not an appropriate basis since the coordinates shouldappear either in a symmetric or anti-symmetric way [7]. This requirement canbe satisfied by expanding the N -body state function as a linear superposition ofproduct states containing N factors of one-body basis states. This is accomplishedfor employing an anti-symmetrization operator or symmetrization operator forFermions or Bosons, respectively.

Fermions

The anti-symmetrization operator for Fermions is defined as

OS� D 1

N Š

Xp

.�1/p OP ; (3.7)


where the sum runs over all NŠ permutations, OP is the permutation operator for Nparticles, and the sign indicates whether the corresponding permutation is even orodd. Normalized anti-symmetric states are then given by

j1; 2; : : : ; N i D pNŠ OS� j1ij2i : : : jN i : (3.8)

For any antisymmetric N -body state there are NŠ physically equivalent statesobtained by a permutation of the one-body quantum numbers. Only one physicalstate corresponds to these NŠ states. By ordering the one-body quantum numbers,one can write the completeness relation for N particles as [5]

orderedX1;2;:::;N

j1; 2; : : : ; N ih1; 2; : : : ; N j D 1 ; (3.9)

while without ordering, the completeness relation can be written as

1

N Š

X1;2;:::;N

j1; 2; : : : ; N ih1; 2; : : : ; N j D 1 : (3.10)

Normalization for ordered states has the form

h01;

02; : : : ;

0N j1; 2; : : : ; N i D h0

1j1ih02j2i : : : h0

N jN i ;

D ı0

1;1ı0

2;2: : : ı0

N ;N:

(3.11)

If the states are not ordered, the result is in the form of a determinant

h01;

02; : : : ;

0N j1; 2; : : : ; N i D

2666666664

h01j1i h0

2j1i � � � h0N j1i

h01j2i h0

2j2i � � � h0N j2i

::::::

: : ::::

h01jN i h0

2jN i � � � h0N jN i

3777777775: (3.12)

The normalized N -body wave function of an anti-symmetric state in coordinaterepresentation is expressed as


1;2;:::;N .x1; x2; : : : ; xN / D hx1jhx2j : : : hxN j1; 2; : : : ; N i ;

D 1pNŠ

266666664

hx1j1i hx2j1i � � � hxN j1ihx1j2i hx2j2i � � � hxN j2i:::

:::: : :

:::

hx1jN i hxN jN i � � � hxN jN i

377777775;

D 1pNŠ

266666664

1.x1/ 1.x2/ � � � 1.xN / 2.x1/ 2.x2/ � � � 2.xN /:::

:::: : :

:::

N .x1/ N .x2/ � � � N .xN /

377777775:

(3.13)

Such a wave function is commonly called a Slater determinant [11]. The useof Slater determinants ensures an anti-symmetric wave function at the outset;symmetric functions are automatically rejected. Swapping two electrons is the sameas swapping two columns. It is known from linear algebra that a determinant thenchanges sign. Because of the properties of a determinant, the wave function fulfillsthe Pauli exclusion principle. When two coordinates are equal, for example xi D xj ,the two rows i and j become equal and one obtains � D 0. More generally, if ji iforms a linear dependent set, the Slater determinant vanishes.

Bosons

For Bosons, the symmetrization operator OSC is defined as

OSC D 1

N Š

Xp

OP : (3.14)

Symmetric states for bosons are given by

j1; 2; : : : ; N i Ds

NŠ

n1 Šn2 Š : : : nN ŠOSCj1ij2i : : : jN i ; (3.15)

withP

i ni D N . For Bosons, there is no restriction on the occupation of one-bodystates. All particles can occupy the same state. For a given symmetricN -body state,there areNŠ physically equivalent states, obtained by a permutation of the one-bodyquantum numbers. In addition, one can have multiple occupation of a one-body


state. Appropriate weighting of these states is obtained by including factors ni Š inthe completeness relation as [5]

X1;2;:::;N

n1 Šn2 Š : : : nN Š

N Šj1; 2; : : : ; N ih1; 2; : : : ; N j D 1 : (3.16)

By ordering of the states, this factor is not needed, as in the case for fermions

orderedX1;2;:::;N

j1; 2; : : : ; N ih1; 2; : : : ; N j D 1 : (3.17)

Normalization for ordered states has the form

h01;

02; : : : ;

0N j1; 2; : : : ; N i D h0

1j1ih02j2i : : : h0

N jN i ;

D ı0

1;1ı0

2;2: : : ı0

N ;N;

(3.18)

whereas for not ordered states one has

h01;

02; : : : ;

0N j1; 2; : : : ; N i D

1pn1 Šn2 Š : : : nN Š

Xp

h01j1ih0

2j2i : : : h0N jN i : (3.19)

The sum on the right-hand side is a sign-less determinant, so called permanent. Thenormalized N -body wave function of a symmetric state becomes

1;2;:::;N .x1; x2; : : : ; xN / D hx1jhx2j : : : hxN j1; 2; : : : ; N i : (3.20)

3.1.3 Operators in the First Quantization Representation

One-Body Operators

Any one-body operator OO acting in a one-body Hilbert space can be written in termsof basis sets ji and j�i.

OO DX�

j�ih�j OOjihj ; (3.21)

and is completely determined by all its matrix elementsO� D h�j OOji in a chosenone-body basis. In the N -body Hilbert space, the corresponding extension of thisone-body operator generally takes the form


OON DNXiD1

OO.i/ ; (3.22)

where OO.i/ is an ordinary one-body operator acting on a one-body state. Typicalexamples are the background potential and the kinetic energy operator, see Eq. (3.2).Using Eq. (3.21) the action of OO.i/ on a product state is given by

OO.i/j1i � � � jN i D j1i � � � ji�1i X

�i

j�i ih�i j OOji i!

jiC1i � � � jN i ;

DX�i

h�i j OO.i/ji ij1i � � � ji�1ij�i ijiC1i � � � jN i :(3.23)

The matrix elements of OO do not depend on which particle is considered as long asthe same quantum numbers are involved. Therefore, the matrix element h�i j OOji iin Eq. (3.23) are the same for any particle [5]. For the operator OON one can write

OON j1i � � � jN i D OO.1/j1i � � � jN i C � � � C j1i � � � OO.N/jN i ;

DX�1

h�1j OOj1ij�1i � � � jN i C � � � CX�N

h�N j OOjN ij1i � � � j�N i ;

DNXiD1

X�i

h�i j OOji ij1i � � � ji�1ij�i ijiC1i � � � jN i ;

DNXiD1

X�i

O�i i j1i � � � ji�1ij�i ijiC1i � � � jN i :

(3.24)

OON is symmetric, therefore, it commutes with the anti-symmetrization OS� or thesymmetrization operator OSC. As a result, the action of OON on an antisymmetricEq. (3.8) or symmetric Eq. (3.15) many-body state is given by

OON j1; : : : ; N i DNXiD1

X�i

O�i i j1; : : : ; i�1; �i ; iC1; : : : ; N i : (3.25)

Two-Body Operators

The two-body potential interaction in Eq. (3.2) is an example of an operatorinvolving the coordinates of two particles. A two-body operator OO acting on statesin the two-body space of product states can be written in terms of basis sets jij0iand j�ij�0i as


OO DX��0

X0

O��00 j�ij�0ih0jhj ; (3.26)

where the matrix elements are given by O��00 D h�jh�0j OOjij0i. In the N -bodyHilbert space the extension of this operator is given by

OON D

8̂ˆ̂̂̂ˆ̂̂̂ˆ̂̂̂<̂ˆ̂̂̂ˆ̂̂̂ˆ̂̂̂ˆ̂:

OO.1; 2/ C OO.1; 3/ C OO.1; 4/ C � � � C OO.1;N /

C OO.2; 3/ C OO.2; 4/ C � � � C OO.2;N /

C OO.3; 4/ C � � � C OO.3;N /: : :

:::

C OO.N � 1;N /

DNX

i<jD1OO.i; j / D 1

2

NXi¤jD1

OO.i; j / :

(3.27)

In a similar way to the case of a one-body operator, the application of OO.i; j / on aproduct state of N particles gives [5]

OO.i; j /j1i � � � jN i DX�i�j

O�i�j i j

�j1i � � � ji�1ij�i ijiC1i � � � jj�1ij�j ijjC1ijN i :(3.28)

The matrix elements O�i�j i j do not depend on the selected pair of particles as

long as the same quantum numbers are involved. Therefore, the application of OONyields

OON j1i � � � jN i D 1

2

NXi¤jD1

X�i�j

O�i�j i j j1i � � � j�i i � � � j�j i � � � jN i : (3.29)

As OON is symmetric it commutes with the anti-symmetrization and symmetrizationoperators. Therefore, the application of OON on a N -body state yields

OON j1; � � � ; N i D 1

2

NXi¤jD1

X�i�j

O�i�j i j j1; � � � ; �i ; � � � ; �j ; � � � ; N i :

(3.30)

3.2 Second Quantization 37

3.2 Second Quantization

The wave function formalism discussed in the previous section is often referredto as first quantization. This formalism denotes the transition from classical toquantum mechanics in which dynamical variables and the equation of motion fora system of particles are replaced by operators and the time-dependent Schrödingerequation, respectively. The wave function describes the state of a system with afixed number of particles that evolves in time. It neglects, however, the effects ofspecial relativity and ignores the quantum nature of force fields, especially thatof the electromagnetic field. Moreover, processes with creation and annihilation ofparticles are not described. Each of these points of views has led to the extensionof the frame of quantum mechanics, which is named quantum field theory. In thisapproach, classical fields are converted into operators acting on quantum states ofthe field theory. Quantizing a field appeared to be similar to quantizing a theorythat was already quantized, leading to the term second quantization in the earlyliterature, which is still used to describe field quantization.

Second quantization is a powerful procedure used in quantum field for describingmany-body systems by quantizing the fields using a basis that describes the numberof particles occupying each state. This differs from the first quantization, whichuses one-body states as basis. The second quantization is also known by themore descriptive name occupation number representation. Second quantizationoperators incorporate the statistics of particles, which contrasts with the morecumbersome approach of using symmetrized or anti-symmetrized products of one-body wave functions. The starting point of second quantization is the notion ofindistinguishability of particles combined with the observation that the determinantor permanent of one-body states forms a basis for the N -body Hilbert space.Quantum theory can be formulated in terms of occupation numbers of these one-body states. Using any ordered and complete one-body basis ji i, the basis statesfor an N -body system in the occupation number representation can be obtained bylisting the occupation numbers of each basis state jn1 ; n2 ; : : :i [3]. The notationmeans that there are ni particles in the state i with

Pi ni D N . In the previous

section we considered systems with a fixed number of particles, but in manyprocesses the particle number does change. Examples are electron hole annihilationsin metals or semiconductors, electron-phonon processes, and photon absorption oremission. Also, in order to formulate statistical mechanics in terms of a grand-canonical ensemble (see Sect. 5.4.3), one must be able to treat states with a differentnumber of particles. It is therefore useful to consider the Fock space F defined asthe direct sum of all N -body Hilbert spaces:

F D H 0 ˚ H 1 ˚ H 2 : : : (3.31)

As the Hamiltonian conserves the number of particles, the states containing adifferent number of particles are orthogonal.


3.2.1 Creation and Annihilation Operators

The Fock space as defined in Eq. (3.31) allows for a representation of states withdifferent particle numbers. To connect first and second quantization, annihilationand creation operators Ob and Ob� for Bosons and Oc and Oc� for Fermions are introduced.As the name implies, these operators generate or destroy one particular singleparticle. The annihilation operators, Obi and Oci , decrease the occupation numberof the state i by 1, while the creation operators, Ob�i and Oc�i , increase the occupationnumber of the state i by 1. That means, they connect subspaces of Fock space. Itshould be also noted that the definitions of the creation and annihilation operatorsare such that these operators are each other’s adjoint in the Fock space:

Ob�j D . Obj /� ;Oc�j D . Ocj /� :

(3.32)

Bosons

Since the wave functions of Bosons are symmetric with respect to the exchange ofparticles, one can demand that Ob�i and Ob�j should commute. Using Eq. (3.32), one

can show that Obi and Obj must commute. Creation and annihilation operators forBosons satisfy the commutation relation:

Œ Obi ; Obj � D 0 ;

Œ Obi ; Ob�j � D ıi ;j ;

Œ Ob�i ; Ob�j � D 0 :

(3.33)

Considering normalization condition, Bosonic creation and annihilation operatorscan be defined as

Ob�i jn1 ; : : : ; ni ; : : :i D pni C 1jn1 ; : : : ; ni C 1; : : :i ;

Obi jn1 ; : : : ; ni ; : : :i D pni jn1 ; : : : ; ni � 1; : : :i :

(3.34)

Obi and Ob�i are not Hermitian, whereas the product Obi Ob�i is Hermitian. UsingEq. (3.33), one can show that Obi Ob�i is the occupation number operator. Theeigenvalues of this operator are the number ni of particles occupying the state ji i

Oni D Obi Ob�i ;Oni jni i D ni jni i :

(3.35)


The occupation number of Bosons includes any non-negative integer. It is worthnoting that one of the states in Fock space is the vacuum. The wave function for thequantum system when it contains no particle is called vacuum and is represented byj0i, where h0j0i D 1. It should be noted that j0i is different from Ob j0i D 0.

Any many-body states with correct symmetry properties can be constructed byacting creation operators on the vacuum state:

jn1 ; n2 ; : : : ; nN i D 1pn1 Šn2 Š : : : nN Š

. Ob�1/n1 . Ob�2/n2 : : : . Ob�N /nN j0i : (3.36)

Fermions

As a result of the anti-symmetry property of a Fermionic wave function, the creationand annihilation operators of Fermions follow the anti-commutation rule:

Œ Oci ; Ocj C D 0 ;

Œ Oci ; Oc�j C D ıi ;j ;

Œ Oc�i ; Oc�j C D 0 :

(3.37)

The anti-commutation relation for two operators OA and OB is defined by Œ OA; OB C Df OA; OBg D OA OB C OB OA. All properties of these operators can be derived from theseanti-commutation rules. The creation and annihilation operators for Fermions aredefined as

Oc�i jn1 ; : : : ; ni ; : : :i D .�1/si jn1 ; : : : ; ni C 1; : : :i ;Oci jn1 ; : : : ; ni ; : : :i D .�1/si jn1 ; : : : ; ni � 1; : : :i ;

(3.38)

where si D Pi�1jD1 nj and the occupation numbers ni have to be considered

modulo 2 due to the Pauli exclusion principle for Fermions. It should be noticed thatphase factor .�1/si is due to sign change needed for the ordering of states after theaction of the annihilation or the creation operator on the many-body state. Therefore,the following relations hold for Fermionic creation and annihilation operators:

Oc j0i D 0 ; Oc� j0i D j1i ;Oc j1i D j0i ; Oc� j1i D 0 :

(3.39)

An immediate consequence of these relations is

. Oc�i /2 D 0 ;

. Oci /2 D 0 :(3.40)


In a similar way to the case of Bosons, the occupation number operator for Fermionsis defined as

Oni D Oci Oc�i ;Oni jni i D ni jni i :

(3.41)

An anti-symmetric N -body state can be generated by repeated application ofcreation operators to the vacuum state

j1; 2; : : : ; N i D Oc�1 Oc�2 : : : Oc�N j0i : (3.42)

3.2.2 Operators in the Second Quantization Representation

By applying creation and annihilation operators to the vacuum state, one can gener-ate the Fock space in general. It is possible to represent any operator in the secondquantization representation. Operators in the second quantization representationare composed of linear combinations of the products of creation and annihilationoperators weighted by the appropriate matrix elements. One of the advantages ofthe second quantization representation is that the permutation symmetry propertiesare taken care of by the creation and annihilation operators.

One-Body Operators

If Oa� and Oa denote either Boson operators Ob� and Ob or Fermion operators Oc� and Oc,one can write the general form for a one-body operator OO in the second quantizationrepresentation as

OO DX�

O� Oa�� Oa : (3.43)

The physical interpretation of this formula is quite simple. The operator Oa destroysa particle in state ji. Then, Oa�� creates a particle in state j�i. The net result is thatthere is still one particle in the system, but it has changed its quantum state goingfrom state ji to state j�i, see Fig. 3.1. The amplitude of such transition is given bythe matrix elements of the operator OO between the states O� D h�j OOji. To proveEq. (3.43) we consider first the following commutator for Fermions:


Oμν

|ν

|μ

Oμμ ν ν

|ν |ν

|μ |μ

Fig. 3.1 A graphical representation of the one-body (left) and two-body (right) operators inthe second quantization. The incoming and outgoing arrows represent initial and final states,respectively. The dashed and wiggled lines represent the transition matrix elements from the initialto the final state for the one-body and two-body processes, respectively. In the second quantizationrepresentation, the initial state is first annihilated and then the final state is created

Œ OO; Oc�i DX�

O�Œ Oc�� Oc; Oc�i ;

DX�

O�

�Oc�� Oc Oc�i � Oc�i Oc�� Oc

;

DX�

O� Oc�� Oc Oc�i C Oc�i Oc

�;

DX�

O� Oc��ı;i ;

DX�

O�i Oc�� ;

(3.44)

where the anti-commutation relation for Fermions Eq. (3.37) has been used for thetransition from the third to the forth line. Equation (3.44) can be employed formanipulating the following relation [5]:

OOj1; 2; : : : ; N i D OO Oc�1 Oc�2 : : : Oc�N j0i ;

D Œ OO; Oc�1 Oc�2 : : : Oc�N j0i C Oc�1 OO Oc�2 : : : Oc�N j0i ;

D Œ OO; Oc�1 Oc�2 : : : Oc�N j0i C � � � C Oc�1 Oc�2 : : : Œ OO; Oc�N j0i ;

DNXiD1

X�i

O�i i Oc�1 : : : Oc�i�1 Oc��i Oc�iC1: : : Oc�N j0i ;

(3.45)

which proves the equivalence of Eqs. (3.25) and (3.45) for an N -body Fermionicsystem. As Eq. (3.45) is valid for any N -body problem, one concludes thatEq. (3.43) has the required form of a one-body operator in Fock space. A similarprocedure can be applied for bosons to establish Eq. (3.43) [2].


Two-Body Operators

The second quantization representation of a two-body operator generally takes theform

OO D 1

2

X0

X��0

O��00 Oa�� Oa��0 Oa0 Oa : (3.46)

To prove this relation, one should employ the following commutator [2]:

Œ OO; Oa�i DX��0

X0

Oa�� Oa��V��0i 0 Oa0 ; (3.47)

which is valid for both Fermions and Bosons. Equation (3.47) is obtained in a similarway as Eq. (3.44) by making use of the symmetry OO.i; j / D OO.j; i/, which implies

V��00 D V�0�0 : (3.48)

Using these results and by applying Eq. (3.46) to Eq. (3.42) for Fermionic systems,one obtains [5]

OOj1; 2; : : : ; N i D 1

2

NXi¤jD1

X�i�j

O�i�j i j Oc�1 : : : Oc��i : : : Oc��j : : : Oc�N j0i : (3.49)

Equation (3.49) is equivalent to Eq. (3.30) and holds for any N . Therefore,Eq. (3.46) represents the extension of the two-body operator in Fock space. A similarapproach can be applied for Bosonic systems.

3.2.3 Basis Transformation

Depending on the studied system, quantum mechanical operators are naturallyexpressed in various representations. Assuming that ji and j�i are two differentcomplete and ordered one-body basis sets and using the completeness condition(Eq. (2.42)), one can write the basis transformation for a one-body operator as

j�i DX

jihj�i ;

DX

h�ji�ji :(3.50)

For one-body states one can define the creation and annihilation operators corre-sponding to the two basis sets and rewrite Eq. (3.50) as


Oa��j0i D j�i DX

h�ji� Oa� j0i ; (3.51)

which leads to the following transformation rules for creation and annihilationoperators:

Oa�� DX

h�ji� Oa� ;

Oa� DX

h�ji Oa :(3.52)

The general validity of Eq. (3.52) can be proved by applying the first quanti-zation one-body result Eq. (3.50) to the N -body first quantized basis states [3].Using Eq. (3.52) one can show that basis transformation preserves the Bosonic orFermionic particle statistics:

Œ Oa�i ; Oa��j ˙ DXk ;l

h�i jkih�j jli�Œ Oak ; Oa�l ˙

DXk ;l

h�i jkihl j�j iık;l ;

DXk

h�i jkihkj�j i D ı�i ;�j :

(3.53)

In addition, the total number of particles remains unchanged under the transforma-tion in Eq. (3.52):

X�

Oa�� Oa� DX�

Xi ;j

hi j�ih�jj i Oa�i Oaj ;

DXi

Oa�i Oai :(3.54)

3.2.4 Field Operators

It is often convenient to employ real space representation of the second quantization,leading to the definition of quantum field operators. Using the transformed basis setj�i in Eq. (3.52) as the continuous set of spatial coordinate kets jri and includingthe spin state in ji, one obtains

O .r/ DX

hrji Oa DX

.r/ Oa ;

O �.r/ DX

hrji� Oa� DX

� .r/ Oa� :

(3.55)


The field operator O .r/ annihilates and O �.r/ creates a particle at place r. O �.r/can be considered as the sum of all possible ways to add a particle to the systemat some position r through any of the basis states .r/ [3]. Since O �.r/ and O .r/are second quantization operators defined in a spatial coordinate, they are calledquantum field operators. By applying Eq. (3.55) to Eq. (3.43), one can express anyone-body operator in terms of field operators

OO DX�;

�Zdrh�jrihrj OOjrihrji

�Oa�� Oa ;

DZ

dr

X�

hrj�i� Oa��!O.r/

X

hrji Oa!;

DZ

dr O �.r/O.r/ O .r/ :

(3.56)

In a similar way, a two-body operator can be rewritten in terms of field operators as

OO DZ

drZ

dr0 O �.r/ O �.r0/O.r; r0/ O .r0/ O .r/ : (3.57)

For homogeneous systems, momentum representation is usually preferred.Assuming a separable spin wave function, substituting momentum and spin basisji D jkij�i in Eq. (3.55), and using Eq. (2.21) yields

O �.r/ DXk�

hrjki�� Oak� DXk�

eik�r�� Oak� ;

O ��.r/ D

Xk�

hrj�i�� Oa�k� DXk�

e�ik�r�� Oa�k� ;(3.58)

where �� is the spin wave function, see Eq. (2.73). Using Eq. (3.53), it is straightfor-ward to show that field operators satisfy simple commutation or anti-commutationrelations:

Œ O �.r/; O � 0.r0/ ˙ D 0;

Œ O �.r/; O �

� 0.r0/ ˙ D ı�;� 0ır;r0 ;

Œ O ��.r/; O �

� 0.r0/ ˙ D 0 ;

(3.59)

where the plus and minus sign refer to Fermion and Boson operators, respectively.As shown in Eq. (3.2), the Hamiltonian of a many-body system can be written in

terms of a non-interacting and an interacting part:

OH D OH0 C OHint : (3.60)


The non-interacting term can be rewritten in the second quantization representa-tion as

OH0 D OT C OU DX�

Zdr O �

�.r/

"Op22m

C U.r/

#O �.r/ : (3.61)

Assuming a two-body interaction term without spin-flipping and using Eq. (3.57),the interacting part of the Hamiltonian is given by

OHe-e DX�� 0

1

2

ZdrZ

dr0 O ��.r/ O �

� 0.r0/V .r; r0/ O � 0.r0/ O �.r/ : (3.62)

Using Eq. (3.58), each term of the Hamiltonian can be written in momentum space

OT DXkk0

X�;� 0

Zdr e�ik0�r��

� 0 Oc�k0� 0

�� „22m

r2

�eik�r�� Ock� ;

DXk;k0

X�;� 0


� 0 Oc�k0;� 0

�„2k22m

�eik�r�� Ock� ;

DXk;k0

X�;� 0

„2k22m

�Zdr ei.k�k0/�r

�„ ƒ‚ …

ık;k0

��

� 0��„ƒ‚…ı�;�0

Oc�k0� 0 Ock� ;

DXk�

„2k22m

Oc�k� Ock� DXk�

�.k/ Onk;� ;

(3.63)

OU DXk;k0

X�;� 0


� 0 Oc�k0� 0U.r/eik�r�� Ock� ;

DXkk0

X�� 0

�Zdr U.r/e�i.k0�k/�r

�„ ƒ‚ …

Uk0�k

��

� 0��„ƒ‚…ı�;�0

Oc�k0� 0 Ock� ;

DXkk0�

Uk0�k Oc�k0�Ock� D

Xkq�

Uq Oc�.kCq/� Ock� ;

(3.64)

where Uq is the Fourier transform of U.r/. Equation (3.64) describes a scatteringprocess in which an electron is scattered from momentum k to k C q. In case of aconstant potential independent of position U.r/ D U0, the Fourier transform of thepotential becomes a delta function in momentum space and the potential operatorbecomes diagonal in the momentum representation. Assuming a Coulomb electron-electron interaction

V.r; r0/ D V.r � r0/ D e2

4��

1

jr � r0j ; (3.65)


Vq

|k σ |k σ

|k+q σ |k −q σFig. 3.2 A graphicalrepresentation of Coulombscattering. The scattering isdescribed by the annihilationof the incoming electronswith momentum k and k0 andthe creation of the outgoingelectrons with momentumk C q and k0 � q

the interacting part of the Hamiltonian in momentum representation is given by [1]

OHe-e DX

k1k2k3k4

X�1�2�3�4

ZdrZ

dr0V.r � r0/ Oc�k1�1 Oc�k2�2 Ock3�3 Ock4�4 ;

��

e�i.k1�k4/�r ��1��4„ƒ‚…ı�1;�4

��e�i.k2�k3/�r0

��2��3„ƒ‚…ı�2;�3

�;

DX

k1k2k3k4

X�� 0

Oc�k1� Oc�k2� 0 Ock3� 0 Ock4� ;

��Z

dRV.R/e�i.k1�k4/�R„ ƒ‚ …

Vk1�k4

��Zdr0e�i.k2�k3Ck1�k4/�r0

„ ƒ‚ …ık1;k4Ck3�k2

�;

DX

k2k3k4

X�;� 0

Vk3�k2 Oc�.k4Ck3�k2/�Oc�k2� 0 Ock3� 0 Ock4� ;

DX

q

Vq

Xk;k0

X�;� 0

Oc�.kCq/� Oc�.k0�q/� 0 Ock0� 0 Ock� :

(3.66)

In the second line, the variables R � r�r0, � � �1, and � 0 � �4 are used. To obtainthe last line in Eq. (3.66), the following variables are introduced k � k4, k0 � k3,k0 � q � k2, which give k4 C k3 � k2 D k C q and k3 � k2 D q. The scatteringis described by the annihilation of the incoming electron with momentum k and k0,and the creation of the outgoing electrons with momentum k C q and k0 � q. Adiagrammatic representation of the scattering process is shown in Fig. 3.2.

3.2.5 Quasi-particles and Collective Excitations

As discussed in Sect. 3.1, in the case of a non-interacting many-body system,the Hamiltonian can be written as the sum of N independent single-particleHamiltonians. In the presence of interactions the problem is very difficult to solve.However, many-body systems usually behave as if the bodies of which they are


composed of hardly interact at all. The reason for this is that the bodies involved arenot real, but factious. In fact, a system composed of strongly interacting real bodiesacts as if it were composed of weakly interacting or non-interacting factious bodies.A simple example is a system composed of two masses held together by a strongspring. In other words, the system consists of two strongly coupled real bodies.The complicated motion can be decomposed into two independent simple motions:motion of the center of mass and motion about the center of mass [9]. These twofactious bodies move exactly as if they were independent bodies with differentmasses. This concept can be applied to many-body systems. When a particle movesthrough a system, it pushes or pulls on its neighbors and thus becomes surroundedby a cloud of agitated particles. The real particle plus its cloud is referred to as quasi-particle. These fictitious particles are typically called quasi-particles if they arerelated to Fermions and called collective excitations if they are related to Bosons [9].A quasi-particle is usually thought of as being like a dressed particle: It is builtaround a real particle at its core, but the behavior of the particle is affected bythe environment. For example, as an electron travels through a semiconductor, itsmotion is disturbed by interactions with other electrons and nuclei. However, itapproximately behaves like an electron with a different mass traveling unperturbedthrough free space. This electron with a different mass is called an electron quasi-particle. On the other hand, a collective excitation is usually imagined to be areflection of the aggregate behavior of the system, with no single real particle atits core. A standard example is the phonon, which characterizes lattice vibrations.In condensed matter physics, quasi-particles and collective excitations help simplifythe quantum mechanical many-body problem.

The concept of elementary excitations is an equivalent way of viewing thesefactious bodies which has the advantages of giving a unified picture of many-bodysystems. Light waves can be considered as quantized radiation oscillators or asconsisting of particles (‘quanta’) called photons where each of them has energy„!. A similar approach can be applied to lattice vibrations. Instead of consideringthe lattice vibration of wavenumber q as one harmonic oscillator having quantizedenergy Eq D „!q.nq C 1

2/ (see Eq. (3.71)), one can alternatively regard it as a set

of nq quanta each having energy „!, together with a ground state of energy 12„!.

These quanta of lattice vibrations are called phonons. It should be noted that fora given nq, there is only one quantized lattice vibration of wavenumber q, whichis just the fictitious body of energy Eq, but there are nq phonons of wavenumberq. Therefore, it is more appropriate to call phonon a quantum or particle of latticevibrations [9]. This section proceeds with the quantization of a harmonic oscillatorand electromagnetic fields. Lattice vibration quantization will be studied in Sect. 4.3.

3.2.6 Harmonic Oscillator

Electromagnetic fields and lattice vibrations can be quantized with the aid ofBosonic field operators. A Bosonic quantum field can be fundamentally described


by a set of linearly coupled oscillators [4]. In linear systems, the modes of oscillationcan be decomposed into a sum of independent normal modes. Each normalmode can be modeled by a simple harmonic oscillator, which provides the basicbuilding block for Bosonic field theories. To quantize collective, Bosonic fields, theHamiltonian is reduced to its normal modes in the first step. For translationallyinvariant systems, this is just a matter of Fourier transforming the field, and itsconjugate momenta. In the next step, one can quantize the normal mode Hamiltonianas a sum of independent Harmonic oscillators.

The one-dimensional harmonic oscillator in first quantization is characterizedby two conjugate variables appearing in the Hamiltonian: the position Ox and themomentum Op operators:

OH D 1

2mOp2 C 1

2m!2x2 ; (3.67)

with Œ Ox; Op � D i„. Equation (3.67) rewritten in the second quantization representa-tion by introducing two operators b and Ob� as

b �r

m!

2„�

Ox C i

m!Op�; Ob� �

rm!

2„�

Ox � i

m!Op�: (3.68)

It is rather straightforward to show that the introduced operators obey the Bosoniccommutation relations Eq. (3.35). The position and momentum operators can beexpressed in terms of b and Ob� operators:

Ox �r

„2m!

. Ob C Ob�/ ; Op � i

rm!„2. Ob� � Ob/ : (3.69)

This leads to the useful representation of the Hamiltonian

OH D „!�

Ob� Ob C 1

2

�D „!

�OnC 1

2

�; (3.70)

where On is the number operator. The eigenstates and eigenvalues are thereforegiven by

OH jni D Enjni ; En D „!�nC 1

2

�; jni D

� Ob�n

pnŠ

j0i : (3.71)

The excitation of a Harmonic oscillator can thus be interpreted as filling theoscillator with Bosonic quanta, where each of these quanta carries energy „!, andn counts the number of vibrational quanta added to the ground state, see Fig. 3.3.The ground state energy is E0 D „!. The creation operator can be used to generateeigenstates of the oscillator starting from the ground state (see Eq. (3.36)).


Fig. 3.3 Harmonic oscillatorpotential energy and energystates

3.2.7 Photons

In order to explain the photoelectric effect, Einstein assumed heuristically in 1905that an electromagnetic field consists of packets of energy. In 1927 Dirac applied thesecond quantization technique to electromagnetic waves to introduce the concept ofphoton to quantum mechanics [6]. The quantum nature of the radiation field, andthe associated concept of photons play a crucial role in the theory of interactionsbetween matter and light. The quantization of the electromagnetic field is based onthe observation that the eigenmodes of the classical field can be thought of as acollection of harmonic oscillators, which are then quantized.

Maxwell’s equations in the vacuum can be written as

r � E D �@B@t; r � B D 0 ; r � H D @D

@t; r � D D 0 ; (3.72)

where B D �0H, D D �0E, and �0�0 D c�2. In the Coulomb gauge, E and B aredetermined by the vector potential B D r �A and E D �@A=@t . With the Coulombgauge condition r � A D 0 one obtains

r2A D 1

c2@2A@t2

; (3.73)

which has plane wave solutions eiq�r at angular frequency !q and wave vector q thathave a linear dispersion relation !q D cjqj. Assuming periodic boundary condition,one can Fourier expand the vector potential enclosed in a finite cubic box of volumeV as

A.r; t / D 1pV

Xq

Aq.t/eiq�r ; (3.74)

where Aq.t/ are the expansion coefficients. The Coulomb gauge assumption isequivalent to q �Aq D 0, which makes the waves transverse. For any q, there are twotransverse directions, and hence two independent polarization directions, denoted bytwo orthogonal unit vectors �� with � D ˙1:


Aq DX�

��Aq� : (3.75)

Based on Eq. (3.73), both polarizations have the same oscillation frequency:

Aq.t/ D Aqe�i!qt : (3.76)

The coefficients Aq are generally complex. To obtain a vector potential with a realamplitude, one can combine positive and negative wave vectors together:

A.r; t / D 1

2pV

Xq

�Aq.t/e

iq�r C A�q.t/e�iq�r� : (3.77)

As the sum over q includes wave vectors in all directions, Eq. (3.77) is divided by 2to avoid double counting of q and �q. In order for the vector potential to be real, thesecond term on the right hand of Eq. (3.77) should be complex conjugate of the firstterm A�q D A�

q. Therefore, the vector potential in Fourier space can be written as

A.r; t / D 1

2pV

Xq

�Aq.t/e

iq�r C A�q.t/e

�iq�r� : (3.78)

In the next step one turns to the Hamiltonian of the system, which is simply the fieldenergy known from electromagnetism. One can write the Hamiltonian of photons interms of the radiation field

H� D 1

2

Zdr��0jE.r; t /j2 C 1

�0jB.r; t /j2

�;

D 1

2�0

Zdr�!2qjA.r; t /j2 C c2jqj2jA.r; t /j2� ;

D �0!2q

ZdrjA.r; t /j2 :

(3.79)

Using Parceval’s theorem in Fourier space, one obtains

H� DX

q

2�0!2q1

4jAq.t/j2 D

Xq�

2�0!2q1

4jAq�.t/j2 ;

D 1

2

Xq�

�0!2q

��<efAq�.t/g�2 C �=mfAq�.t/g

�2;

D 1

2

Xq�

�P 2

q� C !2qQ2q�

�:

(3.80)


In the last line of Eq. (3.80) normalized variables, Pq� and Qq� are introduced

Qq� � p�0 <efAq�g ; Pq� � !q

p�0 =mfAq�g : (3.81)

Similar to position and momentum, Pq� and Qq� are conjugate variables:

@Qq�

@tD Pq� ;

@H

@Qq�D �@Pq�

@t;

@Pq�

@tD �!2qQq� ;

@H

@Pq�D Qq� :

(3.82)

Therefore, radiation field A can be thought of as a collection of harmonic oscillatoreigenmodes, where each mode is characterized by the conjugate variables Qq� andPq�. Quantization of the radiation field can be achieved by imposing the commutatorof the variables and introducing the second quantized Bosonic operators for eachquantized oscillator [3]:

ŒPq�;Qq� � D „i: (3.83)

Qq� Ds

„2!q

� Ob�q� C Obq�

; Pq� D

r„!q

2i� Ob�q� � Obq�

; (3.84)

with Œ Obq�; Ob�q� � D 1. Therefore, Eq. (3.80) can be written as

OH� DXq�

„!q

�Ob�q� Obq� C 1

2

�: (3.85)

To represent the vector potential in the second quantization, one can express Aq� interms of Pq� and Qq�, which in turn are expressed in terms of Ob�q� and Obq�

OAq� D <ef OAq�g C i=mf OAq�g DOQq�p�0

C iOPq�

!qp�0

D 2

s„

2�0!q

Obq� ;

OA�q� D <ef OAq�g � i=mf OAq�g DOQq�p�0

� iOPq�

!qp�0

D 2

s„

2�0!q

Ob�q� :(3.86)

By substituting Eq. (3.86) in Eq. (3.78), the second quantization representation ofthe vector potential is obtained as


OA.r; t / D 1pV

Xq�

s„

2�0!q

� Obq�e�i!qt C Ob��q�ei!qt

eiq�r�� ;

D 1pV

Xq�

s„

2�0!q

� Obq�.t/C Ob��q�.t/

eiq�r�� ;

(3.87)

where in the sum of the second terms, q is substituted with �q, and the sum runsover the same values.

3.2.8 Interaction with Photons

Electromagnetic fields are always coupled to charged particles. The usual way forincluding the effect of electromagnetic fields in the Hamiltonian is by modifyingthe momentum of a particle with charge e, according to the effects of the field itexperiences Op ! Op�e OA. Therefore, the effective Hamiltonian for a charged particlein the electromagnetic field is given by

OH D 1

2m

�Op � e OA

2 C U.r/C OH� ;

D Op2m

C U.r/„ ƒ‚ …

OHel

� e

2m

� OA � Op C Op � OA

C e2 OA2

2mC OH� ;

(3.88)

where OH� is the Hamiltonian of free photons, which is described by Eq. (3.85).Assuming Coulomb gauge r � A D 0, one can show that Op � OA D OA � Op. Asa result, the interaction includes only two terms, one that is linear in OA and onethat is quadratic in OA. The linear term involves individual photon creation andannihilation, corresponding to single-photon processes, whereas the quadratic termcorresponds to two-photon processes which are usually negligible in comparisonwith single-photon processes. The electron-photon interaction potential is thereforesimplified as

OVe-� D � e

mOA � Op C e2 OA2

2m� � e

mOA � Op : (3.89)

By substituting Eq. (3.87) in Eq. (3.89), the electron-photon interaction potential canbe expressed as

OVe-� D 1pV

Xq�

� e

m

s„

2�0!q�� Op

� Obq�.t/C Ob��q�.t/

eiq�r: (3.90)

References 53

The electron-photon interaction Hamiltonian can be written as

OHe-� DZ

dr O �.r/Ve-� O .r/ D 1pV

Xq�

X;�

Mq�;�

� Obq�.t/C Ob�q�.t/

Oc�.t/ Oc�.t/;

(3.91)

where the matrix elements are given by

Mq�;� D � e

m

s„

2�0!q�� hjeiq�r Opj�i : (3.92)

For atomic wave functions that are localized to dimensions much shorter than anoptical wavelength, one can neglect the factor exp.iq � r/ for the evaluation of tran-sition matrix elements. This approximation is referred to as dipole approximation.

References

1. Altland, A., Simons, B.D.: Condensed Matter Field Theory. Cambridge University Press,Cambridge (2010)

2. Blaizot, J.P., Ripka, G.: Quantum Theory of Finite Systems. MIT, Cambridge (1986)3. Bruus, H., Flensberg, K.: Many-Body Quantum Theory in Condensed Matter Physics: An

Introduction. Oxford University Press, Oxford (2004)4. Coleman, P.: Introduction to Many Body Physics. Cambridge University Press, Cambridge

(2014)5. Dickhoff, W.H., VanNeck, D.: Many-Body Theory Exposed!: Propagator Description of

Quantum Mechanics in Many-Body Systems, 2nd edn. World Scientific, Hackensack (2008)6. Dirac, P.A.M.: The quantum theory of the emission and absorption of radiation. Proc. Phys.

Soc. A 114(769), 710–728 (1927)7. Fetter, A.L., Walecka, J.D.: Quantum Theory of Many-Particle Systems. McGraw-Hill,

San Francisco (1971)8. Greiner, W.: Quantum Mechanics: An Introduction, 4th edn. Springer, Berlin/New York (2001)9. Mattuck, R.D.: A Guide to Feynman Diagrams in the Many-Body Problem, 2nd edn. Dover,

New York (1992)10. Pauli, W.: Über den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der

Komplexstruktur der Spektren. Zs. f. Phys. 31, 765–783 (1925)11. Slater, J.C.: The theory of complex spectra. Phys. Rev. 34, 1293–1322 (1929)

Chapter 4Band Theory

Electronic transport in semiconductors deals with the response of electrons, whichare able to move, to external forces. Therefore, the determination of the statesavailable to the electrons in the crystals and their energies is of fundamentalimportance. However, due to its many-body nature, the problem is not easy.Electrons can interact with each other and the electrostatic potential of ions in thecrystal. The static part of the ionic lattice potential results in electronic bands whilethe dynamic part, which results from the vibration of ions, results in electron-phononinteraction. In this chapter the basic concepts of band theory and electron-phononinteraction in crystals are briefly discussed. A more comprehensive overview can befound in standard textbooks [1, 6–8, 10].

4.1 Crystal Lattices

A solid is said to be a crystal if atoms are arranged in such a way that their positionsare exactly periodic. A perfect crystal is formed by adding identical building blocks,which are called unit cells, that consist of atoms or groups of atoms. A unit cellis the smallest component of the crystal that, when stacked together with puretranslational repetition, reproduces the whole crystal. All crystals have translationalsymmetry, but some have other symmetry elements as well. The set of pointsdefined by translation vectors is called a lattice. The group of atoms or moleculesthat forms a crystal by infinite repetition is called basis. The basis is positionedin a set of mathematical/abstract points that form the lattice, which is also calledBravais lattice. Thus a crystal is a combination of a basis and a lattice. The simplestportion of a lattice that can be repeated by translation to cover the entire spaceis called unit cell. The primitive cell is a unit cell that only contains one latticepoint. A lattice system is a class of lattices with the same point group. In three-dimensions there are seven lattice systems: triclinic, monoclinic, orthorhombic,


55

56 4 Band Theory

tetragonal, rhombohedral, hexagonal, and cubic. By coupling one of the 7 latticesystems with one of the lattice centerings, 14 Bravais lattices are obtained.

4.2 Electrons in Crystals

Solids, such as metals, semiconductors, and insulators can be modeled as a systemof the interacting electron gas moving in a charge compensating for the backgroundof positively charged ions. Any atom consists of three parts: the positively chargedheavy nucleus at the center, the light cloud of the many negatively charged coreelectrons tightly bound to the nucleus, and finally, the outermost few valenceelectrons. The nucleus with its core electrons is denoted as an ion. The innerdegrees of freedom of the ions do not play a significant role, leaving the centerof mass coordinates and total spin of the ions as the only dynamical variables. Incontrast to the core electrons, the valence electrons are often free to move awayfrom their respective host atoms forming a gas of electrons around the ions. TheHamiltonian of the system is written as a sum of the kinetic and potential energy ofthe ionic system and the electronic system treated independently, and the Coulombinteraction between the two systems [2]:

OH D� OTion C Vion-ion

C� OTel C Vel-el

C Vel-ion : (4.1)

Ions at equilibrium positions form a periodic lattice. This lattice has an energyElatt and an associated electronic potential energy Vel-latt, both originating from acombination of OTion, Vion-ion, and Vel-ion in the Hamiltonian equation (4.1). At finitetemperature the ions can vibrate around their equilibrium positions with the totalelectric field acting as the restoring force. As will be demonstrated in Sect. 4.3,these vibrations can be described in terms of quantized harmonic oscillators givingrise to the concept of phonons. The non-interacting part of the phonon field isdescribed by a Hamiltonian OHph. Electrons are described by their kinetic energyTel, their mutual interaction Vel�el, and their interaction with both the static part ofthe lattice (Vel-latt) and the vibrating part (Vel�ph), which models the influence ofthe electronic potential caused by vibrating ions. Therefore, the Hamiltonian for thephenomenological lattice model of Eq. (4.1) can be rewritten as

OH D�Elatt C OHph

C� OTel C Vel-el

C �

Vel-latt C Vel-ph�: (4.2)

The Hamiltonian for phonons is presented in Sect. 4.3, however, to investigateelectronic transport one can simply drop this part of the Hamiltonian. The kineticenergy of electrons and the interaction of electrons with the static part of the latticeform a single-particle Hamiltonian, which is addressed in the section. Electron-phonon interaction is discussed in Sect. 4.3.1 and electron-electron interaction isbriefly discussed in Sect. 4.2.3. The one-body or single-particle Hamiltonian used

4.2 Electrons in Crystals 57

in Bloch’s theorem of free electrons moving in a static periodic ion lattice can bewritten as

OHBloch D OTel C Vel-latt.r/ ; (4.3)

where the lattice potential is periodic with the lattice vector R

Vel-latt.r C R/ D Vel-latt.r/ : (4.4)

4.2.1 Bloch States

A crystal in real space can be described in terms of the primitive lattice vectors a1,a2, a3 and the positions of atoms inside a primitive unit cell. The lattice vectors Rare formed by combinations of primitive lattice vectors, multiplied by integers

R D n1a1 C n2a2 C n3a3 ; (4.5)

where n1, n2, n3 are integers. The primite is defined as the volume enclosed by thethree primitive lattice vectors: The lattice vectors connect all equivalent points inspace; this set of points is referred to as the Bravais lattice [5]. When working withperiodic lattices it is often convenient to Fourier transform from the direct space tok-space, also known as the reciprocal space. Reciprocal space is defined as the setof equivalent points connected by reciprocal lattice vector

G D m1b1 Cm2b2 Cm3b3 ; (4.6)

where exp i.G � r/ D 1 and the reciprocal basis vectors fb1;b2;b3g are defined as

b1 D 2�a2 � a3

a1 � .a2 � a3/; b2 D 2�

a3 � a1a2 � .a3 � a1/

; b3 D 2�a1 � a2

a3 � .a1 � a2/;

(4.7)

with the obvious consequence aibj D 2�ıi;j . An important concept is the firstBrillouin zone, defined as all points in reciprocal space lying closer to G D 0 thanto any other reciprocal lattice vector G ¤ 0 [2]. Using wave vectors k in the firstBrillouin zone, as shown in Fig. 4.1a, any wave vector k0 in the reciprocal space canbe decomposed k0 D k C G.

The Fourier transform of any function that has the periodicity of the Bravaislattice f .r/ D f .r C R/ can be written as f .r/ D P

G f .G/eiG�R [5]. Because

of the periodicity of the lattice, any such function need only be studied within theprimitive unit cell. This statement applied to single-particle wave functions is knownas Bloch’s theorem. Therefore, the lattice potential is given by

58 4 Band Theory

a b c

Fig. 4.1 (a) The gray area shows the first Brillouin zone for a two-dimensional square lattice. (b)Bloch’s theorem for a one-dimensional lattice with lattice constant a. The dashed line indicates theparabolic energy band for free electrons. The Bloch bands viewed as a break-up of the parabolicfree electron band in Brillouin zones. (c) As all wave vectors are equivalent to those in the firstBrillouin zone, one can displace all the energy branches into the first Brillouin zone

V.r/ DX

G

V.G/eiG�R : (4.8)

Because of periodic boundary condition, the lattice wave function can be written asa sum of plane waves:

.r/ D 1pV

Xk

c.k/eik�r : (4.9)

By substituting Eqs. (4.9) and (4.8) into the Schrödinger, equation one obtains

Xk

".k/c.k/eik�r CXk;G

V.G/c.k/ei.kCG/�r D EX

k

c.k/eik�r : (4.10)

By using the orthogonality of plane waves and rewriting the second term on the lefthand side of Eq. (4.10) as

Pk;G V.G/c.k � G/eik�r, one obtains

c.k/".k/X

G

V.G/c.k � G/ D Ec.k/ : (4.11)

Schrödinger equation of the form Eq. (4.11) for c.k/ couples to an infinite numberof similar equations for c.k � G/. Each such infinite family of equations has exactlyone representative wave vector in the first Brillouin zone, while any wave vectoroutside the first Brillouin zone does not give rise to a new set of equations. Theinfinite family of equations generated by a given k in the first Brillouin zone givesrise to a discrete spectrum of eigenenergies "n.k/, where n is a natural number and isreferred to as band index, see Fig. 4.1b, c. The corresponding lattice wave functions nk.r/ take the form


nk.r/ DX

G

cn.k CG/ei.kCG/�r D 1

V

XG

cn.k C G/eiG�r!

eik�r � unk.r/eik�r :

(4.12)

As exp .G � R/ D 1 the function unk.r/ is periodic in the lattice unk.rCR/ D unk.r/and thus Bloch’s theorem is stated as [2]

OHBloch nk D "n.k/ nk ; nk.r/ D unk.r/eik�r : (4.13)

Through the rest of this book we restrict the wave vector values to the first Brillouinzone. For many applications it is reasonable to approximate the Bloch states byplane waves if at the same time the electronic mass m is replaced by a materialdependent effective mass m�, known as effective mass approximation.

For a system with a Hamiltonian that is invariant under the time-reversal, one canshows that for any state ".k/ D ".�k/, this is known as Kramer’s theorem. A moredetailed analysis, which takes into account spin states, explicitly reveals that, forspin-half particles, Kramers’ theorem becomes "".k/ D "#.�k/. For systems withan equal number of up and down spins, Kramer’s theorem amounts to inversionsymmetry in reciprocal space.

4.2.2 Tight-Binding Approximation

The tight-binding (TB) approximation is a versatile tool for obtaining the electronicbandstructure of periodic systems with adequate precision and considerable sim-plicity. The basic assumption in the TB approximation is that one can use orbitalsthat are very similar to atomic states as a basis for expanding lattice wave functions.As the wave function of atomic orbitals are tightly bound to the atoms, the termtight-binding is used for this approximation. The states that can be used as the basisfor the expansion of lattice wave functions should obey Bloch’s theorem

˚kmj.r/ D 1pN

XR0

eik�.R0Ctj /'m.r � tj � R0/ ; (4.14)

where the summation runs over all the N unit cells in the lattice, R0 representsthe vector associated with each primitive unit cell, 'm.r � tj / represents a set ofatomic wave functions, tj is the position of the j th atom in the cell, and 'm.r/is one of the atomic states associated with this atom. The index m can take theusual quantum numbers for an atom, for example, the angular momentum characters, p, d , : : :. The state 'm.r � tj / is assumed to be centered at the position ofthe atom with index j . Minimal basis can be as many orbitals as the number ofvalence states in the atom [5]. In this section, the spin-index in eliminated, but willbe addressed in Sect. 4.2.3. It can be easily shown that Eq. (4.14) is a Bloch state

60 4 Band Theory

˚kmj.r C R/ D ˚kmj.r/. Therefore, the lattice wave function can be expanded interms of the following basis function:

k.r/ DX

mj

cmj.k/˚kmj.r/ ; (4.15)

where cmj.k/ can be obtained by substituting Eq. (4.15) in OHBloch k D ".k/ k

Xmj

hh˚klij OHBlochj˚kmji � ".k/h˚klij˚kmji

icmj.k/ D 0 : (4.16)

As h kj k0i � ı.k � k0/, only matrix elements of states with the same wave vectorare considered. Equation (4.16) is called secular equation, its eigenvalues are theelectronic bands, and to solve it the following integrals need to be evaluated:

h˚klij˚kmji D 1

N

XR0;R00

eik�.tj�tiCR00�R0/h'l.r � ti � R0/j'm.r � tj � R00/i ;

D e�ik�ti0@ 1

N

XR;R0

eik�Rh'l.r � ti /j'm.r � tj � R/i1A eik�tj ;

D e�ik�ti

0B@X

R

eik�R h'l.r � ti /j'm.r � tj � R/i„ ƒ‚ …Slm;ij.R/

1CA eik�tj ;

D e�ik�ti Slmij.k/eik�tj ;(4.17)

where R � R00 � R0, and one of the sums over the lattice vectors is eliminated withthe factor 1=N , since in the last line of Eq. (4.17) there is no explicit dependenceon R0. The brackets in the last expression are the overlap matrix elements betweenatomic states and are represented by the matrix S , where Slm;ij.k/ and Slmij.R/ arelattice Fourier transforms of each other. The last line of Eq. (4.17) can be viewedas the matrix Slmij.k/ transformed by a unitary transformation. However, a unitarytransformation of a Hermitian matrix does not affect its eigenvalues, but merelytransforms the eigenvectors. In a similar way, one obtains the following:

h˚klij OHBlochj˚kmji D e�ik�ti

0B@X

R

eik�R h'l.r � ti /j OHBlochj'm.r � tj � R/i„ ƒ‚ …Hlmij.R/

1CA eik�tj ;

D e�ik�ti Hlmij.k/eik�tj ;(4.18)

The brackets on the right-hand side of Eq. (4.18) are Hamiltonian matrix elementsbetween atomic states. There are several important approximations in the TB


method. It is common to conjecture that the overlap matrix elements in Eq. (4.17)are assumed to be non-zero only for the same orbitals on the same atom Slmij.R/ Dıi;j ıl;mı.R/. This is referred to as an orthogonal basis, since any overlap betweendifferent orbitals on the same atom or orbitals on different atoms is taken to be zero.Similarly, one can assume the Hamiltonian matrix elements in Eq. (4.18) to be non-zero only if the orbitals are on the same atom or if the orbitals are on different atoms,but situated at nearest neighbor sites, denoted in general as dnn [5]:

Hlmij D8<:"lıi;j ıl;mı.R/ ;

�tlmijı..tj � ti � R/ � dnn/ ;(4.19)

where " and t are referred to as the on-site energies and hopping matrix elements,respectively. The hopping matrix element between neighboring states will generallybe given by an overlap integral of the wave functions with the negative crystallinepotential, and for this reason, it is taken to be negative. In Eq. (4.19), R is zero if thenearest neighbors are in the same unit cell; when the nearest neighbors are acrossunit cells, R can be one of the primitive lattice vectors. The real space representationof the TB Hamiltonian in second quantization language provides a rather intuitiveinterpretation:

OH DX

li

"l Oc�li Ocli �Xlmij

tlmij Oc�li Ocmj : (4.20)

As Oc�mn annihilates an electron on atomic orbital with label m at site j , and c�licreates one electron at atomic orbital l and site i , then the combination of these twooperators results in hoping of the electron between the respective atomic orbitals andsites. The real-space formulation of the Hamiltonian allows also for the introductionof disorder and non-periodicity, which can be implemented by spatially dependenton-site energies and hopping matrix elements.

4.2.3 The Hubbard Model

Neglecting the interaction of electrons, in Sect. 4.2.2, the Hamiltonian based on theTB approximation is presented. For electrons in a solid, the Hubbard model canbe considered as an improvement on the TB model. Most general band theories donot consider interactions between electrons explicitly. They consider the interactionof a single electron with the potential of nuclei and other electrons in an averageway only. By formulating conduction in terms of the hopping integral, however,the Hubbard model includes the so-called on-site repulsion, which stems from theCoulomb repulsion between electrons at the same atomic orbitals. This sets up acompetition between the hopping integral, which is a function of the distance andangles between neighboring atoms, and the on-site Coulomb repulsion, which is not

62 4 Band Theory

considered in the usual band theories. To keep our discussion simple, we focus ona single orbital model and drop the reference to the orbital index. Based on the TBapproximation, Eq. (3.62) can be re-expressed as

OHe-e DX�� 0

Xii0jj0

Uii0jj0 Oc�i� Oc�i 0� 0 Ocj 0� 0 Ocj� ; (4.21)

where the matrix elements of the interaction between electrons at different sites isgiven by

Uii0jj0 D 1

2

ZdrZ

dr0 '�.r � ti /'�.r0 � ti 0/V .r; r0/'.r0 � tj 0/'.r � tj / : (4.22)

If the atoms are well separated, and the overlap between neighboring orbitals isweak, the matrix elements are exponentially small in the inter-atomic separation. Inthis limit, the on-site Coulomb or Hubbard interaction, Uiiii � U=2, generates thedominant interaction mechanism:

OHe-e DX�� 0

Xi

Uiiii Oc�i� Oc�i� 0 Oci� 0 Oci� D

Xi

U Oni" Oni# : (4.23)

Taking only the nearest neighbor contribution to the hopping matrix elements, theeffective Hamiltonian takes a simplified form known as the Hubbard model:

OH DXi�

" Oc�i� Oci� �X

hi;j i�tij Oc�i� Ocj� C

Xi

U Oni" Oni# ; (4.24)

where hij i denotes neighboring lattice sites. This model can be phenomenologicallyinterpreted: electrons tunnel between atomic orbitals that are localized on individuallattice sites while double occupancy of a lattice site incurs an energetic penaltyassociated with the mutual Coulomb interaction. The Hubbard model has playeda central part in the theory of magnetism, metal-insulator transitions, and mostrecently, in the description of electron motion in high temperature superconduc-tors [3]. With the exception of one-dimensional physics, a complete understandingof the physics of this model has not been achieved.

4.3 Phonons

Lattice atoms vibrate about their equilibrium positions as a result of their thermalmotion. The forces which oppose this motion are those of the chemical bond. Bysupplying thermal energy to a lattice atom, it will rapidly be distributed throughoutthe entire lattice by the mutual interaction between the atoms. Local excitationswill therefore lead to collective vibrations of the whole ion system. It is therefore

4.3 Phonons 63

appropriate to use collective coordinates (normal coordinates) for the mathematicaldescription. By setting up equations of motion for the lattice atoms within theframework of classic mechanics, one can derive the energy and frequency of thenormal modes of lattice oscillations. In describing the dispersion relations of thesenormal modes, the concepts of the reciprocal lattice and the Brillouin zone areneeded. The lattice vibrations can be quantized and the associated quanta areelementary excitations called phonons [7].

It is a common to approximate the potential energy of a lattice by expandingin powers of the atomic displacements from their equilibrium position. Since allforces on all atoms are zero, the Taylor expansion does not have linear terms. Theharmonic approximation consists of neglecting all powers of displacements largeror equal to 3. Based on this approximation, the Hamiltonian can be resolved into asum of independent terms with the form of Hamiltonians of harmonic oscillators.This is the basis of the quantization and with it the description of lattice vibrationsas a non-interacting phonon gas. The inclusion of higher anharmonic terms in theexpansion means an interaction between phonons [7]. Being at a local minimum,the matrix of second derivatives must be positive definite, and thus will have onlypositive eigenvalues. The potential energy thus becomes

U.R1;R2; : : : ;RN / D U 0 C 1

2

XR0i R0j˛ˇ

u˛.R0i /

@2U

@u˛.R0i /@uˇ.R0

j /

ˇ̌ˇ̌uD0

uˇ.R0j / ;

(4.25)

where U 0 D U.R01;R

02; : : : ;R

0N / is the potential energy at equilibrium position, ˛

and ˇ run over the Cartesian coordinates, and u D P˛ e˛u˛ is the displacement

from the equilibrium position Ri D R0i C P

˛ e˛u˛.R0i / with e˛ representing unit

vectors of the ˛th Cartesian coordinate. The second derivative of the potentialenergy evaluated at the equilibrium position is called force constants matrix:

˚˛ˇ.R0i ;R

0j /D @2U

@u˛.R0i /@uˇ.R0

j /

ˇ̌ˇ̌uD0

D � @F˛.R0i /

@uˇ.R0j /

D �@Fˇ.R0j /

@u˛.R0i /

D ˚ˇ˛.R0j ;R

0i /

(4.26)

Given the expression for potential energy as a function of displacements, it is aneasy task to derive the Newtonian equations of motion:

Mi

d 2u˛.R0i /

dt2D F˛.R0

i / � �Xˇ;R0j

˚˛ˇ.R0i ;R

0j /uˇ.R

0j / ; (4.27)

where Mi is the mass of the atom located at R0i . Equation (4.27) is linear in the

atomic displacements, and can be interpreted as the particles being connected bysprings. The resulting set of linear equations defines the vibrational modes of the

64 4 Band Theory

system and has a harmonic solution in the form of u / cos.!t/. To solve thisequation system, however, a change of variable is needed. Because of the latticeperiodicity, one can use Bloch’s theorem and represent the atomic displacements as

u.R0i ; t / D 1

2pNMi

X˛q

�u˛qei.q�R0i �!t/ C u�˛qe�.iq�R0i �!t/

e˛ ;

D 1

2pNMi

X˛q

�u˛q.t/C u�˛�q.t/

�eiq�R0i e˛ ;

(4.28)

where the summation over the wave vector q runs over the first Brillouin zoneand the factor 1=

pMi is used to make the system of equations in Eq. (4.27)

symmetric. Due to the lattice periodicity, the force constant matrix depends onlyon the difference between any two atom positions ˚˛ˇ.R0

i ;R0j / D ˚˛ˇ.R0

i � R0j /.

By substituting Eq. (4.28) in Eq. (4.25), the expansion of the potential energy takesthe form

U � U 0 D 1

8N

X˛ˇ

XR0i R0j

u˛.R0i /˚˛ˇ.R

0i � R0

j /uˇ.R0j / ;

D 1

8N

Xq;q0

X˛ˇ

XR0i R0j

.u˛quˇq0 C u˛qu�ˇ�q0 C u�˛�quˇq0 C u�˛�qu�

ˇ�q0/

�˚˛ˇ.R0i � R0

j /pMiMj

eiq�R0i eiq0�R0j ;

D 1

8N

Xq;q0

X˛ˇ

.u˛quˇq0 C u˛qu�ˇ�q0 C u�˛�quˇq0 C u�˛�qu�

ˇ�q0/

�XR0i R0j

˚˛ˇ.R0i � R0

j /pMiMj

eiq�.R0i �R0j /ei.qCq0/�R0j ;

D 1

8

Xq

X˛ˇ

.u˛quˇ�q C u˛qu�ˇq C u�˛�quˇ�q C u�˛�qu�ˇq/D˛ˇ.q/ :

(4.29)

In the last line, the relationP

R0jexp.i.q C q0/ � R0

j / D Nıq0;�q has been used.In Eq. (4.29) the definition of the dynamical matrix D is used. It should be notedthat the definition of the dynamical matrix is very similar to that of the Hamiltonianmatrix in the TB formalism. The matrix elements of the TB Hamiltonian arehopping parameters between neighboring atoms, whereas the matrix elements ofthe dynamical matrix are the force constants, which also extend to the neighboringatoms only. Because of the symmetry properties of the force constant matrix, D.k/is a real and symmetric matrix. Using the Bloch theory, the force constants matrixis transformed into the dynamical matrix, which is block-diagonal, and atomicdisplacements are decoupled in the q-space, implying that a displacement with wave

4.3 Phonons 65

vector q does not couple to any other displacement associated with the vector q0.Thus Eq. (4.27) can be rewritten as

!2uq D D.q/uq : (4.30)

Similar to the TB Hamiltonian matrix,D.�q/ D D.q/� implies that the eigenvaluesare even functions of q. For any value of q, one can find a set of eigenvectors "q�,the so-called polarization vectors, that diagonalize dynamical matrix

D.q/"q� D !2q�"q� : (4.31)

Similar to the case of a lattice, the eigenvectors of the dynamical matrix are completeand orthonormal "�q� � "�0;q D ı�;�0 . One can finally write the potential energy in thenew basis as

U � U 0 D 1

8

Xq�

.uq�u�q� C uq�u�q� C u��q�u�q� C u��q�u�q�/!2q� : (4.32)

The kinetic energy of lattice vibrations can be evaluated in a similar way:

T D 1

2

X�;R0i

Mi .Pu�.R0i //

2:

D 1

8N

Xqq0;�

.Puq� C Pu��q�/.Puq0� C Pu��q0;�/XR0i

ei.qCq0/R0i :

D 1

8

Xq�

�!2q�.uq� � u��q�/.u�q� � u�q�/ :

D 1

8

Xq�

!2q�.�uq�u�q� C uq�u�q� C u��q�u�q� � u��q�u�q�/ :

(4.33)

Using Eqs. (4.32) and (4.33), the Hamiltonian is given by

H D T C U D 1

4

Xq�

!2q�.uq�u�q� C u��q�u�q�/ ;

D 1

4

Xq�

!2q�.uq�u�q� C u�q�uq�/ :(4.34)

The Hamiltonian in Eq. (4.34) describes a classic harmonic oscillator, which can bequantized by writing lattice vibrations in terms of the second quantization operators

Ouq� D 2

s„

2M!q�

Obq� ; Ou�q� D 2

s„

2M!q�

Ob�q� : (4.35)

66 4 Band Theory

a b

c

d

e

Fig. 4.2 (a) A generic phonon dispersion for a lattice with a diatomic unit cell. The six modesdivide into three acoustical and three optical modes. Various polarization in phonon modes(b) longitudinal acoustic, (c) transverse acoustic, (d) longitudinal optical, and (e) transverse optical

Lattice vibrations are therefore given by

u.R0; t/ D 1pN

Xq�

s„

2M!q�. Obq�.t/C Ob��q�.t//e

iq�R0i "q� : (4.36)

The quantum mechanical Hamiltonian is also obtained as

H D 1

2

Xq�

„!q�. Obq�Ob�q� C Ob�q� Obq�/ D 1

2

Xq�

„!q�.2 Ob�q� Obq� C 1/ ;

DXq�

„!q�


2

�:

(4.37)

The quantized oscillations are called phonons. The phonon dispersion relations of alattice with a diatomic unit cell are shown in Fig. 4.2a. Phonons can be categorizedinto acoustic and optical modes. Acoustic phonons are due to coherent movementsof atoms of the lattice out of their equilibrium positions. As shown in Fig. 4.2b, ifthe displacement is in the direction of propagation, then in some areas the atomswill be closer, in others farther apart, which is similar to the propagation of soundin air. Displacements perpendicular to the propagation direction are comparableto wave propagation in water, see Fig. 4.2c. Longitudinal and transverse acousticphonons are often abbreviated as LA and TA phonons, respectively. Acousticphonons exhibit a linear relationship between frequency and phonon wave vectorfor long wavelengths. The frequencies of acoustic phonons tend to zero with longerwavelength. Acoustic phonons with infinite wave length correspond to a simpledisplacement of the whole crystal.

Optical phonons are the out-of-phase movement of atoms in the lattice; adjacentatoms move in opposite directions (Fig. 4.2d, e). They are called optical becausein ionic crystals, they are excited by infrared radiation. The electric field of lightwill move every positive ion in the direction of the field, and every negative ion inthe other direction. Optical phonons are often abbreviated as LO and TO phonons,for the longitudinal and transverse, respectively. Optical phonons have a non-zero

4.3 Phonons 67

frequency at the Brillouin zone center and show no dispersion near that longwavelength limit. This is because they correspond to a mode of vibration whereadjacent atoms just vibrate against each other. In polar semiconductors, the oppositemovement of positive and negative atoms creates a time-varying electrical dipolemoment. Away from q D 0, the vibrations of an acoustic branch are not always inphase and those of an optical branch are not always out of phase. Complicated mixedmodes of the two limiting cases can be formed. For example, all modes are standingwaves at the zone boundary, @!=@q D 0, which is a necessary consequence ofthe lattice periodicity. In a diatomic lattice, the frequency-gap between the acousticand optical branches depends on the mass difference. In the higher energy, modeonly light atoms move, whereas in the lower energy mode, only heavier atomsmove.

In general, for a lattice with p atoms inside the unit cell, there exists 3p phononmodes with 3 acoustic (1 longitudinal and 2 transverse) and 3.p�1/ optical branches(p � 1 longitudinal and 2.p � 1/ transverse). The acoustical modes appear becauseit is always possible to construct modes where all ions have been given nearly thesame displacement, resulting in an arbitrarily low energy cost associated with sucha deformation of the lattice [2].

4.3.1 Phonon Interaction Potential

In the band model description, electrons in a solid are quasi-particles which occupyone-electron states and are described by Bloch functions with wave vector k [7].As discussed in Sect. 4.3, phonons are collective excitations of the lattice. Thevibrational state of the lattice is characterized by the number of phonons in theindividual oscillator states defined by the wave vector q and the branch �. Blochstates are eigenstate of a perfect crystal. However, electrons are scattered by latticevibrations propagating the crystal because the periodicity of the crystal potentialmay be disturbed for various reasons. The basic electron-phonon interaction processis the annihilation (absorption) or creation (emission) of a phonon with simultaneouschange of the electron state from k to k C q. The interaction mechanism can bedistinguished by the deformation of the lattice or electrostatic.

For acoustic phonon modes, neighboring atoms displace in the same direction,and hence the changes in lattice spacing are produced by the strain or differen-tial displacement. For optical phonons, neighboring atoms displace in oppositedirections; therefore, the displacement directly produces the change in latticespacing. Since the acoustic and optical phonon scattering can be expressed bythe deformation potential, which relates lattice vibrations with the changes in theband energies, they are referred to as deformation potential scattering. In covalentcrystals, the distortion of the lattice due to thermal vibrations is not accompaniedby polarization fields and the interaction of the vibrations with the electrons issimply due to the deformation of the lattice. In compound semiconductors, however,

68 4 Band Theory

in addition to acoustic and optical deformation potential scattering, a very stronginteraction due to the polar nature of bonds can take place. Displacement of thelattice caused by phonons perturbs the dipole moment between atoms, which resultsin an electric field that scatters carriers. Polar scatterings, which can be due toeither acoustic or optical phonons, are referred to as the piezoelectric and polaroptical scatterings, respectively. From the point of view of energy exchange, oneshall consider optical-phonon scattering as inelastic, while acoustic phonons carrya very small amount of energy, and the scattering can be considered elastic at roomtemperature.

Deformation Potential

In the long wavelength acoustic branch limit, one can employ the continuumapproximation by a displacement field:

Ou.r; t / DXq�

s„

2�V !q�. Obq�.t/C Ob��q�.t//e

iq�r"q� : (4.38)

It should be noticed that NM in Eq. (4.36) is replaced by �! in Eq. (4.38), where �is the density of the material and V is the volume of the crystal. For long wavelengthacoustic phonons, the following dispersion relation is satisfied:

!q D qvs ; (4.39)

where vs D pcL=� is the velocity of longitudinal elastic waves and cL is the

elastic constant of the material. Longitudinal acoustic waves can be consideredas compression waves in the continuum. A long wavelength acoustic phonondisplacement does not affect the energy bands because the neighboring atoms allmove by the same amount; only differential displacement, namely the strain, isof importance. A relative volume change is associated with a compression waveıV=V D r �u. A change in volume means a change in lattice constant, and hence inthe band model parameters, which depend on the lattice constant. A periodic changeof the lattice constant due to a compression wave will cause a periodic change in theconduction and valence band-edges:

ıEc D @Ec

@VıV D V

@Ec

@V

ıV

VD �r � u : (4.40)

� in Eq. (4.40) is called deformation potential constant. The interaction potentialdue to acoustic deformation potential (ADP) takes the form

OVADP D �r � u.r; t / ; (4.41)

4.3 Phonons 69

where the proportionality constant � is called the deformation potential. UsingEq. (4.38), one obtains

OVADP D 1pV

Xq�

iq � "q��

s„

2�!q�. Obq�.t/C Ob��q�.t//e

iq�r ;

D 1pV

Xq�

Mq�. Obq�.t/C Ob��q�.t//eiq�r :

(4.42)

As q � "q� D 0 for " ? q, only the longitudinal acoustic phonons are coupled to theelectrons. Therefore, the coupling constant for LA modes is given by

MADP.q/ D i

s„�2

2�!qq D i

s„�2q

2�vs: (4.43)

The scattering of electrons due to optical deformation potential (ODP) can bealso treated like the ADP. However, unlike the case of acoustic phonons, neighboringatoms oscillate in opposite direction. Therefore, long wavelength optical phononsmay affect the electronic energy directly. The interaction potential is assumed totake the following form:

OVODP D D0 � u ; (4.44)

where D0 is the optical deformation potential constant. As a result, the interactionpotential has a similar form to that of the ADP (see Eq. (4.42)), but with a differentcoupling constant:

MODP Ds

„D20

2�!q�: (4.45)

Polar Interaction

Vibrations of oppositely charged atoms in polar materials give rise to long-rangemacroscopic electric fields in addition to deformation potentials, and the interactionof the electron with these fields produces additional components of scattering.Polar scattering is indeed the dominant scattering mechanism in III-V and II-VI compounds. Polar scattering can be due to either acoustic or optical phonons(POP) [8]. Polar acoustic scattering, termed piezoelectric phonon scattering, isimportant at very low temperatures in pure semiconductors, but marginal at roomtemperature [4]. The major scattering mechanism in polar materials at roomtemperature is that associated with longitudinal optical modes [9]. For longitudinaloptical phonons, the relative displacement read as u D uC � u�, where uC and

70 4 Band Theory

u� are the displacements of positive and negative ions, respectively. The relativedisplacement can be expressed as (see Eq. (4.36))

u.r; t / D 1pN

Xq

s„

2M!op. Obq�.t/C Ob��q�.t//e

iq�r"q ; (4.46)

where N is the number of pair ions and M is the reduced mass of the positive andnegative ions 1=M D 1=MC C 1=M�. The polarization P caused by the motion ofions contributes to the dielectric displacement D, such that [9]

D D �0E C Pion C P„ ƒ‚ …Ptot

: (4.47)

In Eq. (4.47) �0 is the permittivity of vaccum, E is the electric field, Pion is due tothe polarization of ions, and P is the dipole moment due to motion of ions:

P D e�

˝u : (4.48)

The dipole moment in Eq. (4.48) depends on the effective charge e�. Fröhlichderived a relation for the effective charge in terms of the static and high-frequencypermittivity of the crystal. The difference between the permittivities at low and highfrequencies in polar materials is related to the effective charge on the ions. At highfrequencies, the contribution to the polarization made by ionic motions vanishes,whereas in the static case both contributions are present. The total polarization inthe static and high-frequency case are generally given by

Ptot.0/ D D � �0E D .1 � �0=�/D ;

Ptot.1/ D .1 � �0=�1/D ;

(4.49)

where �1 is the high-frequency permittivity. Therefore, the polarization caused byionic motions is

P D Ptot.0/ � Ptot.1/ D �0

�1

�1� 1

�

�D : (4.50)

As anticipated above, this polarization of the ions can be obtained also from theirdynamics. If !op is the frequency of the optical phonons, we may assume that theyhave a restoring force that tends to push them toward the equilibrium position givenby �M!2opu. Thus in the presence of the microscopic electric field E D D=�0, thedynamics of the ions are [8]

4.3 Phonons 71

M

�@2u@t2

C !2opu�

D e�D�0

: (4.51)

Using Eq. (4.48), one can rewrite this as an equation of motion for the polarizationassociated with ionic displacement:

M˝

e�

�@2P@t2

C !2opP�

D e�D�0

: (4.52)

For the static case one obtains

P D e�D

M˝!2op�0: (4.53)

By relating Eqs. (4.50) and (4.48) the expression for the effective charge is obtained:

e�2 D M˝!2op�20

�1

�1� 1

�

�: (4.54)

As the electric displacement associated with the polarization is zero, the inducedelectric field by the polarization is given by

Eind D � P�0: (4.55)

By substituting Eqs. (4.46) and (4.54) in Eq. (4.48), the induced potential can bewritten as

U.r/ D �Z

Eind � r D �i

s„!op

2N˝�p

q � uq2

; (4.56)

where �p is defined as

1

�pD 1

�1� 1

�0: (4.57)

The resulting interaction potential is

OVPOP D �eU.r/ D 1pV

Xq

ie

q

s„!op

2�p. Obq�.t/C Ob��q�.t//e

iq�r ; (4.58)

where the relation V D N˝ is employed. Thus the coupling constant for POPscattering reads as

72 4 Band Theory

MPOP D i

se2„!op

2�pq2: (4.59)

In the derivation presented here, the screening of electron charge is neglected. In thepresence of screening caused by the motion of charges, which is rapid enough torespond to lattice vibrations, one can assume a Thomas Fermi potential [8]:

V.r/ D e

4�jr0 � rje�qdjr0�rj ; (4.60)

where qD is the inverse Debye screening length. In the Debye formulation, for non-degenerate statistics [4]

qD Ds

e2n

�kBT; (4.61)

where n is the electron density. In this case, the coupling constant for POP scatteringis modified as

MPOP.Scr/ D i

se2„!op

2�p

q

q2 C q2D: (4.62)

For piezoelectric phonon scattering, the polarization is proportional to the acousticstrain P D ePZr � u where ePZ is the piezoelectric tensor. Piezoelectric constant hascomplicated directional dependence. For a simple treatment, one can use a suitablyaveraged piezoelectric constant denoted by ePZ. Following the same arguments asfor the POP scattering and assuming a linear dispersion !.q/ D qvs, one finds thecoupling constant for this mechanism as [8]

MPZ D i

se2„e2PZ

2��2vsq: (4.63)

4.3.2 Scattering of Bloch States

Considering a generic interaction potential as Eq. (4.42) and using Bloch states inEq. (4.13), the electron-phonon interaction Hamiltonian takes the form

4.3 Phonons 73

OHe-ph DZV

dr O �.r/ OVe-ph O �.r/ ;

DZV

dr O �.r/ O .r/ 1pV

Xq�

Mq�. Obq�.t/C Ob��q�.t//eiq�r ;

DZV

drXk;k0

k0.r/ k.r/ Oc�k0.t/ Ock.t/1pV

Xq�

Mq�. Obq�.t/C Ob��q�.t//eiq�r ;

DXk;k0

ZV

drei.k�k0Cq/�ruk0.r/uk.r/„ ƒ‚ …

I.k;k0/

� 1pV

Xq�

Mq�. Obq�.t/C Ob��q�.t// Oc�k0.t/ Ock.t/ :

(4.64)

In Eq. (4.64) only intra-band transitions are considered n0 D n and band index isomitted from the equations. To evaluate the integral I.k;k0/, one can split it as asum of integrals over the crystal cells labeled by the direct-lattice vectors Ri . Usingr D R C r0, one obtains

I.k;k0/ DXi

ei.k�k0Cq/�RiZ˝

dr0ei.k�k0Cq/�r0

uk0.r0/uk.r0/ ;

D ık�k0Cq;GN

Z˝

dr0eiG�r0

uk0.r0/uk.r0/ D N

Z˝

dr0eiG�r0

ukCq�G.r0/uk.r0/ ;

(4.65)

where N is the number of unit cells and ˝ D V =N is the volume of each unit cell.The factor N appearing in Eq. (4.65) is due to the normalization of the Bloch wavefunctions to unity in the volume of the crystal [4]. The summation on the right handside of Eq. (4.65) is only non-zero for k � k0 C q D G. The scattering processes,in which G in Eq. (4.65) is non-zero, are called umklapp. If G D 0 the process iscalled normal and k and k C q lie in the first Brillouin zone. In this book, we focuson normal scattering processes. The integral in Eq. (4.65) provides a selection ruledepending on the symmetries of the interaction and Bloch functions of the initial andfinal states. For normal-processes, I is equal to unity for nearly parabolic bands. Asa result, the electron-phonon interaction Hamiltonian takes the form

OHel�ph D 1pV

Xk

Xq�

I.k;k C q/Mq�. Obq�.t/C Ob��q�.t// Oc�kCq.t/ Ock.t/: (4.66)

74 4 Band Theory

References

1. Ashcroft, N., Mermin, N.: Solid State Physics. Holt, Rinehart and Winston, New York (1976)2. Bruus, H., Flensberg, K.: Many-Body Quantum Theory in Condensed Matter Physics: An

Introduction. Oxford University Press, Oxford (2004)3. Coleman, P.: Introduction to Many Body Physics. Cambridge University Press, Cambridge

(2014)4. Jacoboni, C.: Theory of Electron Transport: A Pathway from Elementary Physics to Nonequi-

librium Green Functions in Semiconductors. Springer, Berlin (2010)5. Kaxsiras, E.: Atomic and Electronic Structure of Solids. Cambridge University Press,

Cambridge (2003)6. Kittel, C.: Introduction to Solid State Physics, 8th edn. Wiley, Oxford (2005)7. Madelung, O.: Introduction to Solid-State Theory. Springer, Berlin (1978)8. Ridley, B.K.: Quantum Processes in Semiconductors. Oxford University Press, Oxford (1993)9. Tomizawa, K.: Numerical Simulation of Submicron Semiconductor Devices. Artech House,

Boston (1993)10. Yu, P.T., Cardona, M.: Fundamentals of Semiconductors: Physics and Materials Properties.

Springer, Berlin (2001)

Chapter 5Statistical Mechanics

Statistical mechanics is the branch of physics that deals with systems only partiallyknown. The lack of knowledge of the initial condition of the system is often due tothe large number of degrees of freedom. On the other hand, describing the motionof each particle in a many-body system is indeed impossible. The fundamentaltheoretical tool of statistical physics is the statistical ensemble, a collection of anarbitrary large number of systems, all prepared in the same way as the actual systemof interest. In classical statistical mechanics, the ensemble is usually represented asa distribution in a phase space with canonical coordinates. In quantum statisticalmechanics, the ensemble is a probability distribution represented by density matrix.In this chapter the basis of statistical mechanics is briefly reviewed and followed bythe connection to thermodynamics. The density matrix and the related equation ofmotion are introduced and various transport models are compared.

5.1 Historical Review

Thermodynamics grew essentially out of an experimental study of the macroscopicbehavior of physical systems. Through the work of Carnot, Joule, Clausius andKelvin, thermodynamics became an stable discipline of physics by 1850. Thetheoretical conclusions following from the first two laws of thermodynamicsappeared to be in agreement with experimental results [49]. At the same time, thekinetic theory of gases emerged. This theory aimed at explaining the macroscopicbehavior of gaseous systems in terms of the motion of molecules. In 1856, Krönigcreated a simple gas-kinetic model, which only considered the translational motionof particles [34]. In 1857, Clausius developed a similar, but more sophisticatedversion of the theory, which included translational, rotational, and vibrationalmolecular motions. In addition, he introduced the concept of a mean free path ofa particle [10]. In 1859, Maxwell formulated the Maxwell distribution of molecular


75

76 5 Statistical Mechanics

velocities, which gave the proportion of molecules having a certain velocity in aspecific range, and appeared as the first-ever statistical law in physics [42].

The kinetic theory of gases was successful at the beginning; however, a con-nection with thermodynamics could not be made until 1871, when Boltzmannestablished a direct connection between entropy, on the one hand, and moleculardynamics, on the other hand [4–6]. He generalized Maxwell’s achievement andformulated the Maxwell-Boltzmann distribution. He was also the first to state thelogarithmic connection between entropy and probability. Almost simultaneously,the kinetic theory began giving way to its successor: the ensemble theory. In thisapproach, the dynamical state of a given system, as characterized by the generalizedcoordinates qi and the generalized momenta pi , is represented by a point in aphase space of appropriate dimensionality. The time evolution of the dynamicalstate is depicted by the trajectory of the point in the phase space, the geometryof the trajectory being governed by the equations of motion of the system andby the nature of the imposed physical constraints. The most important quantity inthe ensemble theory is the density function of phase space points �.qi ;pi ; t /. Astationary distribution (@�=@t D 0) characterizes a stationary ensemble, which inturn represents a system in equilibrium. The appearance of Gibbs book in 1902marks a milestone in this direction [23]. He developed schemes which enabledone to compute a complete set of thermodynamic quantities of a given systemfrom the mechanical properties of its microscopic constituents. The techniques thatfinally emerged rendered thermodynamics as a natural result of the statistics andthe mechanics of the molecules constituting a given physical system. The resultingformalism was given the name statistical mechanics [43].

Bose was the first to take the position that the Maxwell-Boltzmann distribution isnot be true for microscopic particles, whereas the fluctuations due to Heisenberg’suncertainty principle is significant. Thus he focused on the probability of findingparticles in the phase space, and discarding the distinct position and momentumof the particles. His derivation of Planck’s law, in which he implicitly assumedthat photons are mutually indistinguishable, was published in 1924 [7]. Einsteinargued that what Bose had implied for photons holds for particles as well. Einsteinapplied Bose’s method to study an ideal gas [14] and thereby developed what isnow called Bose-Einstein statistics. Subsequently, he showed that the fundamentaldifference between the new statistics and classical Maxwell-Boltzmann statistics isdue to the indistinguishability of the molecules [15]. Following the work of Paulion exclusion principle [44], Fermi showed that certain physical systems would obeya different kind of statistics, namely the Fermi-Dirac statistics, in which not morethan one particle could occupy the same energy state [16]. Landau [36] and vonNeumann [54] introduced the so-called density matrix, which was the quantum-mechanical analogue of the density function of the classical phase space andsimilar to the classical ensemble theory in that they considered both micro-canonicaland canonical ensembles. In 1927, Pauli introduced grand-canonical ensembles inquantum statistics [45]. Later, Belinfante [1, 2] and Pauli [46] discovered the vitalconnection between spin and statistics. They showed that those particles whose spin

5.2 Basic Concepts 77

is an integral multiple of „, obey Bose-Einstein statistics, while those whose spin isa half-odd integral multiple of „, obey Fermi-Dirac statistics.

5.2 Basic Concepts

To study either a classical or a quantum mechanical system, one needs twocomponents. Firstly, the complete state of the system at a given time, mathematicallyrepresented as a phase point in classical mechanics or a pure quantum state inquantum mechanics, is needed. Secondly, an equation of motion, which predictsthe dynamics of the system, that is Hamilton’s equations in classical mechanicsor the time-dependent Schrödinger equation in quantum mechanics, is also needed.Knowing the initial condition of a physical system, one can use these two ingredientsto investigate the state of the system at any other time. If the system is only partiallyknown, however, one can employ statistical mechanics to predict the dynamics ofthe system. Generally, in statistical mechanics, one abandons following the precisechanges in the state of a particular system, and studies instead the behavior of acollection or ensemble of systems of similar structure to the system of actual interest,distributed over a range of different precise states. The fundamental theoreticaltool of statistical physics is thus the statistical ensemble, a mental collection ofan arbitrary large number of systems, all prepared in the same way as the actualsystem of interest. More precisely, taking at random a system in the ensemble, theprobability of finding it in a given state is equal to the probability of finding theactual system in that given state. This statement may be considered the definition ofthe statistical ensemble.

From knowledge of the average behavior of the systems in a representativeensemble, one can predict what may be expected on average for the particularsystem [52]. As is usual for probabilities, the ensemble can be interpreted in twodifferent ways: an ensemble can be taken to represent the various possible statesthat a single system could be in, or the members of the ensemble can be understoodas the states of the systems in experiments repeated on independent systems, whichhave been prepared in a similar but imperfectly controlled manner, in the limit of aninfinite number of trials. These two meanings are equivalent for many purposes. Thefundamental postulate of classical statistical physics is the equal a priori probability.According to this hypothesis, in the statistical ensemble of an isolated system,all points in the phase space are represented with equal probabilities. In quantumstatistical mechanics, this postulate is complemented with the hypothesis that allstates in the ensemble are present with equal probabilities [29].

Each state in the ensemble evolves over time according to the equation of motion.Thus the ensemble itself also evolves. The ensemble’s evolution is given by theLiouville equation in classical mechanics or the von Neumann equation in quantummechanics. One special class of ensemble is those ensembles that do not evolve overtime. These ensembles are known as equilibrium ensembles and their condition isknown as statistical equilibrium. Non-equilibrium statistical mechanics addresses


the more general case of ensembles that change over time, and/or ensembles ofopen systems [43]. In the following sections, the main principles and results ofthe statistical mechanics will be briefly reviewed. Interested readers are referredto standard textbooks, such as Refs. [43, 52].

5.2.1 Macro and Microstates

A macrostate of a thermodynamic system is described by a few thermodynamicvariables, such as E, V , N , where they represent the energy, the volume, and thenumber of particles, respectively. These quantities are the measure of collectivebehaviors of constituting particles. The specification of the actual values of theparameters N , V , and E then defines the macrostate of the given system. Amicrostate, however, specifies the system with the physical quantities of theconstituent particles. At the molecular level, a large number of possibilities stillexist because at that level there will in general be a large number of different waysin which the macrostates E, V , and N of the given system can be realized. One cannot assign P , V , and T for a single particle, but its physical state of motion can bespecified.

In the case of a non-interacting system, there will be a large number of differentways in which the total energy E of the system can be distributed among the Nparticles constituting it. Each of these (different) ways specifies a microstate of thegiven system. For a quantum mechanical system, the state of a system is describedby a wave function, which is usually specified by a set of quantum numbers.Various microstates can be regarded as independent wave functions .r1; : : : ; rN /,corresponding to the eigenvalue E of the relevant Hamiltonian. The total number ofmicrostates is a function of E, V , and N and is denoted by W.N; V;E/. From themagnitude of the number W , and from its dependence on the parameters E, V , andN , complete thermodynamical properties of a given system can be derived [43].

5.2.2 Ergodicity

Statistical mechanics bridge between the microstates and macrostates of a system.For this purpose, one can imagine many copies of the system, all characterized bythe same set of quantities, such as total energy, pressure, etc. But each copy maybe in a different possible state. The copies of the system are called ensemble. Byfocusing on the selected system, one could watch it for a long time. If one observesthe system for a long enough time T , one should observe it in all its possible states.The average value of a property O of the system can be obtained as

hOi D limT!1

1

T

Z T

0

dts O.ts/ ; (5.1)

5.2 Basic Concepts 79

where O.ts/ is the value of O at time ts when the system is in state s. By countingthe number of times, N.s/, that one observes the system in state s, during time T ,one could also write this time average as

hOi DP

s N.s/O.s/Ps O.s/

; (5.2)

where O.s/ is the value of O in state s observed at time t . Alternatively, one canfocus on N mental copies of the selected system and count the number of copiesin state s, which is in fact an ensemble average over all copies. In the second pointof view, the average of O in Eq. (5.2) is interpreted as a sum over all the statesappearing among the N mental copies. Based on this point of view, Eq. (5.2) can berewritten as

hOi DXs

N.s/

NA.s/ D

Xs

�.s/A.s/ ; (5.3)

where �.s/ is the fraction of copies in the ensemble observed in state s. The twopoints of view may not be equivalent. They are equivalent only when the systemcan visit all the possible states, many times, during a long period of time. This isthe ergodicity hypothesis. Not all systems are ergodic. For a non-ergodic system,statistical physics only apply to its ensemble. For an ergodic system, statisticalphysics also apply to the time average of the system. Therefore, for an ergodicsystem, provided that T is long enough and the number of copies N is largeenough, Eq. (5.1) is equal to Eq. (5.2).

5.2.3 Classical and Quantum Statistics

If the constituting particles of a system are relatively large or if the temperatureof the system is high enough, the system can be accurately described by classicalmechanics. In this case, the de Broglie wavelength is much smaller than the inter-particle spacing so that each particle can be clearly distinguished from another. Withthis distinguishability, one can trace the path of a single particle in the system andidentify it at each point. In this case, one can distinguish between states in which twoparticles have interchanged positions. The distinguishability of particles in classicalsystems leads to the Maxwell-Boltzmann statistics. However, in systems wherethe particle de Broglie wavelength is comparable or larger than the inter-particlespacing, quantum mechanics’ principles apply. In this case, one can not trace thepath of a particle in the system and thus cannot distinguish between particles. Thisimplies that in enumerating the states of a system, one can not distinguish betweenstates in which two particles are interchanged, thus one shall count them as thesame state. As discussed in Sect. 3.1.1, quantum mechanical particles have eithersymmetric (Bosons) or anti-symmetric (Fermions) wave functions with respect to


the exchange of particles. However, the Pauli exclusion principle states that one canonly place one Fermion in each single particle state. For the Boson case there is nosuch restriction and we may assign any number of particles to a single particle state.The restriction on Fermions leads to Fermi-Dirac statistics, whereas Bosons obeyBose-Einstein statistics. The three statistics hold, however, for interacting particlesas well. For classical particles, they remain distinguishable when they interact. Ina quantum mechanical system, the states are not independent single particle states,but the system wave function must again be either symmetric or anti-symmetric withrespect to the particle exchange.

5.3 Thermodynamics

A macroscopic system has many degrees of freedom, only a few of which aremeasurable. Thermodynamics is concerned with the relation between a smallnumber of variables, which are sufficient to describe the bulk behavior of the system.For many cases, appropriate variables are the pressure P , the volume V , and thetemperature T . If the thermodynamic variables are independent of time, the systemis said to be in a steady state. If, moreover, there are no macroscopic currents inthe system, such as a flow of heat or particles through the material, the systemis in equilibrium. A state variable is a property of a system that depends only onthe current state of the system, not on the way in which the system acquired thatstate. A state variable describes the equilibrium state of a system. For example,internal energy, enthalpy, and entropy are state quantities because they describequantitatively an equilibrium state of a thermodynamic system, irrespective of howthe system arrived in that state. In contrast, mechanical work and heat are processquantities because their values depend on the specific transition (or path) betweentwo equilibrium states. State variables are mostly classifiable as intensive andextensive variables. An intensive variable is one that is proportional to the size ofthe system while extensive variables are independent of the system size. Examplesof extensive variables are the internal energy E, the entropy S , and the mass ofthe different constituents or their number, while the pressure P , the temperature T ,and the chemical potentials � are intensive. The postulate that quantities, like theinternal energy and entropy, are extensive and independent of shape is based on theexistence of the thermodynamic limit. In the process of taking the thermodynamiclimit, one allows the size of the system to become infinitely large, with densitiesremaining constant [47].

5.3.1 The Laws of Thermodynamics

The four laws of thermodynamics define fundamental physical quantities (E, S , andT ) that characterize thermodynamic systems. The Zeroth law of thermodynamics

5.3 Thermodynamics 81

states that if two systems are in thermal equilibrium with a third system, they mustbe in thermal equilibrium with each other. The zeroth law can be thought of as thestatement that for a matter in equilibrium, one can assign values for the temperature,pressure and chemical potentials, which in principle can be measured. Since inequilibrium the forces are balanced, intensive variables are constant throughoutthe system. For example, a constant temperature leads to thermal equilibrium, aconstant pressure to mechanical equilibrium, and a constant chemical potential tochemical equilibrium. The first law of thermodynamics states that heat is a form ofenergy. Because energy is conserved, the internal energy of a system changes as heatflows in or out of it. This law restates the law of conservation of energy. However,it also partitions the change in energy of a system into two pieces: heat and work. Itstates that if heat dQ is added then this heat increases the internal energy dE of thesystem and goes into work done by the system

dQ D dE C dW ; (5.4)

where dW is the amount of work done by the system during an infinitesimalprocess [47]. The most common idea is that the body does work by expandingagainst the external pressure:

dW D PdV : (5.5)

The second law of thermodynamics states that the entropy of any isolated system,which is not in thermal equilibrium, almost always increases. This law introducesthe entropy S as an extensive state variable and states that for an infinitesimalreversible process at temperature T , the heat given to the system is

dQrev D T dS : (5.6)

Since the amount of heat dQirr exchanged in an irreversible process is alwayssmaller than that exchanged in a reversible process, it holds that

dQirr < dQrev D T dS : (5.7)

Combining the first and second laws one obtains

dE C dW T dS : (5.8)

This relation indicates that if one performs a cyclic process on a system thenT dS � 0. Thus the entropy S will either increase or will reach an upper limitas one continues these cyclic processes. In practice, it is observed that the systemeventually reaches a state from which no further changes occur, that is, it returnsafter subsequent cycles to the same internal state. Based on Eq. (5.8), the entropy ofthis state must be a maximum, which defines the equilibrium state of the system.


For isolated systems, one has dQrev D 0. Therefore, for such systems, the entropyis constant in thermodynamic equilibrium and it has an extremum because dS D 0.

All irreversible processes in isolated systems which lead into equilibriumincrease the entropy, until the entropy reaches its maximum, when equilibrium isreached. For isolated systems in equilibrium, it holds that

dS D 0 ; S D Smax ; (5.9)

and for irreversible processes it holds that

dS > 0 : (5.10)

In irreversible processes the entropy of the system grows until it reaches a maximumin equilibrium. Since the internal energy as well as the amount of heat are extensivequantities, the entropy is an extensive quantity as well. Therefore, when heat isexchanged at temperature T , the entropy is a quantity analogous to the volume,when compression work is performed against a pressure P [27].

5.3.2 Closed Systems

For an isolated system, E and V are constant, assuming the pressure in the systemcannot expand the system. Thus for a change in the system one has

dE C P dV D 0 T dS : (5.11)

Thus entropy increases until in the equilibrium state S gets its maximum values.From the second thermodynamics law, one observes that S is a function ofE and V ,S D S.E; V /. E and V are independent variables, which depend on the boundaryconditions of the isolated system, and are constant for such systems. Therefore, theentropy is ideally suited for describing an isolated system.

5.3.3 Systems in Contact with a Heat Bath

An isolated system is an idealization and practically all systems have at leastsome thermal contact with their surroundings. The exchange of heat then takesplace between the system and its surroundings and this brings the two to thesame temperature eventually. It is therefore more realistic to consider a system inequilibrium at constant temperature than to consider one in equilibrium at constantenergy. Thus one can consider a system in contact with a large heat reservoir atconstant temperature. The reservoir serves to supply or extract heat from the systemand is large enough so that its temperature remains constant during these transfers


of heat. To identify the equilibrium state of this system, one can extract heat dQfrom the heat reservoir and transfer it to the system

dQ D dE C P dV T dS : (5.12)

As the volume is assumed to be unchanged dV D 0, one has dE � T dS 0. Asthe heat transfer takes place at constant temperature, one obtains

d.E � T dS/ 0 : (5.13)

The Helmholtz free energy is introduced as

F D E � TS D F.T; V / ; (5.14)

and for changes at constant T and V , dF 0. This implies that for a given initialstate, the system changes in a way that F always decreases until the system reachesequilibrium where dF D 0. Thus under equilibrium condition, F is a minimumfor a system at constant T and V . F is in fact a natural thermodynamic function todescribe a system at constant temperature and volume. Once these values are set,then the equilibrium value of F is also defined. Therefore, F is a function of T andV as independent variables F D F.T; V /.

5.3.4 Systems in Contact with a Heat and Particle Reservoir

Both the isolated system and the system in contact with a heat reservoir that we haveconsidered so far are closed systems where the number of particles in the systemremains constant. One can extend the above arguments for systems with variablenumber of particles. Therefore, one shall consider the entropy as a function of N asan independent variable as well as E and V , S D S.E; V;N /, and Helmholtz freeenergy as a function of N as well as T and V , F D F.T; V;N /.

One can consider a system in contact with a heat and particle reservoir whereboth heat and particles can be exchanged between the system and the reservoir. Thissystem is referred to as an open system. To describe an open system thermodynami-cally, one needs to introduce the chemical potential �. If we add dN particles to thesystem, the external work done on the system for adding the particles is �dN . Thework done by the system is

dW D ��dN : (5.15)

In many practical cases, � is constant. In other words, the driving force for addingor removing particles remains constant and the same as the value in the heatreservoir. By generalizing the formulation of the first and second laws to allow for


the exchange of particles between the system under consideration and the reservoirsurroundings

dQ D dE C dW D dE C P dV � �dN T dS : (5.16)

For an exchange of heat and particles at constant T , V , and �, the system undergoesa change with

dE � T dS � �dN D d.F � �N/ D d˚G 0 ; (5.17)

where ˚G � F ��N is called the grand thermodynamic potential. The equilibriumis the state for which the thermodynamic potential is a minimum. The grandthermodynamic potential is a natural thermodynamic function to describe a systemat constant temperature and chemical potential. With these independent variablesheld constant, ˚G must be constant. As F is a function of N , in going from F to �,one has to transform to a function ˚G, which depends on � rather than N :

˚G.T; V; �/ D F.T; V;N / � �N : (5.18)

This type of transformation is called a Legendre transformation. The transformationis useful since in open systems � rather than N remains constant. With T , V and �constant, ˚G.T; V; �/ will be constant in equilibrium.

AlthoughE,F , and˚G represent equivalent ways of describing the same system,their natural independent variables differ in one important way. In particular, the set.S; V;N / consists entirely of extensive variables, proportional to the actual amountof matter present. The transformation to F and then to ˚G may be interpretedas reducing the number of extensive variables in favor of intensive ones that areindependent of the total amount of matter.

5.3.5 Thermodynamic Potentials

By extending the first and second laws of thermodynamics to systems with a variablenumber of particles, one obtains

dE D T dS � P dV C �dN : (5.19)

This relation indicates that changes in the entropy S , the volume V , or the numberof particles N , change the internal energy E. From a mathematical point of view,this equation shows that E can be expressed as a function of S , V , and N :

dE D�@E

@S

�V;N

dS C�@E

@V

�S;N

dV C�@E

@N

�S;V

dN : (5.20)


Hence, it must be true that

E D TS � PV C �N : (5.21)

Equation (5.20) provides a definition of the temperature, pressure, and chemicalpotential:

T D�@E

@S

�V;N

; P D ��@E

@V

�S;N

; � D�@E

@N

�S;V

: (5.22)

The evaluation of T , P , and � from Eq. (5.20) indeed requires that the energy E beexpressed as a function of the quantities N , V , and S . Alternatively, if S is knownas a function of N , V , and E, one can write

1

TD�@S

@E

�V;N

;P

TD�@S

@V

�E;N

;�

TD �

�@S

@N

�E;V

:

(5.23)

The function E.S; V;N / is the first instance of a thermodynamic potential, afunction that generates other thermodynamic variables by partial differentiation. Asdiscussed in Sect. 5.3.4, other instances are obtained by considering other choicesof independent variables. To relate the chemical potential to other thermodynamicfunctions, one can employ Eq. (5.14) to obtain

dF D dE � SdT � T dS : (5.24)

By substituting Eq. (5.19) in Eq. (5.24), one can write

dF D �P dV � SdT C �dN : (5.25)

Therefore, the chemical potential can be defined as

� D�@F

@N

�T;V

: (5.26)

5.3.6 Thermodynamic Equilibrium

The condition of equilibrium can easily be obtained from the principle of maximumentropy. One can consider two physical systems, A1 and A2, which are separatelyin equilibrium, see Fig. 5.1. The macrostate of A1 is represented by the parametersE1, V1, and N1 and the macrostate of A2 is represented by the parameters E2, V2,and N2 [43]. By bringing the two systems into contact with each other so that theycan exchange energy, particle, or even volume, one obtains


Fig. 5.1 Two systems thatexchange energy, volume, andparticles are brought intocontact

E D E1 CE2 D const. ; dE1 D �dE2 ;

V D V1 C V2 D const. ; dV1 D �dV2 ;

N D N1 CN2 D const. ; dV1 D �dV2 :

(5.27)

The total entropy is given by S.E; V;N / D S1.E1; V1;N1/ C S2.E2; V2;N2/.Therefore, the differential entropy reads as

dS D dS1 C dS2 : (5.28)

At equilibrium the entropy is maximal

dS D 0 ; S D Smax : (5.29)

Thus

dS D�@S1

@E1

�V1;N1

dE1 C�@S1

@V1

�E1;N1

dV1 ��@S1

@N1

�E1;V1

dN1

C�@S2

@E2

�V2;N2

dE2 C�@S2

@V2

�E2;N2

dV2 ��@S2

@N2

�E2;V2

dN2 D 0 ;

D"�

@S1

@E1

�V1;N1

��@S2

@E2

�V2;N2

#dE1 C

"�@S1

@V1

�E1;N1

��@S2

@V2

�E2;N2

#dV1

�"�

@S1

@N1

�E1;V1

��@S2

@N2

�E2;V2

#dN1 D 0 ;

(5.30)

where the relations in Eq. (5.27) are used. Using Eq. (5.23), the conditions ofequilibrium then become

T1 D T2 ; P1 D P2 ; �1 D �2 : (5.31)

5.3.7 Connection to Statistics

To connect thermodynamic variables to statistical mechanics, one can assume thatthe macrostate of A1 has ˝1.E1;N1; V1/ possible microstates, and the macrostateof A2 has ˝2.E2; V2;N2/ possible microstates. By bringing the two systems into

5.4 Statistical Ensembles 87

contact with each other, the total number of all macrostates of the total system,which is represented by ˝.E; V;N /, is given by the product of these numbers forthe subsystems

˝.E; V;N / D ˝1.E1; V1;N1/˝2.E2; V2;N2/ : (5.32)

In thermodynamic equilibrium, the most probable macroscopic state is the onewhich corresponds to the largest number of consistent microstates, i.e., ˝ D ˝max

and d˝ D 0. Taking the total differential of Eq. (5.32) one gets d˝ D ˝2d˝1 C˝2d˝1, and by dividing by Eq. (5.32), one gets

d ln˝ D d ln˝1 C d ln˝2 : (5.33)

The equilibrium condition reads

d ln˝ D 0 ; ln˝ D ln˝max : (5.34)

In comparing Eq. (5.33) with Eq. (5.28), and Eq. (5.34) with Eq. (5.29), one con-cludes that an equally intimate relationship exists between the thermodynamicquantity S and the statistical quantity ˝. Thus for any physical system, one canwrite

S D kB ln˝ ; (5.35)

where kB is called Boltzmann’s constant, which is a bridge between macroscopicand microscopic physics [27]. The second law of thermodynamics states that theincrease of entropy is related to the fact that the energy content of the universe, inits natural course, is becoming less and less available for conversion into work.Accordingly, the entropy of a given system may be regarded as a measure ofthe so-called disorder or chaos prevailing in the system. Equation (5.35) showshow disorder arises microscopically. Apparently, disorder is a manifestation of thelargeness of the number of microstates the system can have. The larger the choiceof microstates, the lesser the degree of predictability or the level of order in thesystem. The zero of entropy then corresponds to the special state for which only onemicrostate is accessible (˝ D 1). Such systems are, for instance, ideal crystals attemperature T D 0. The statement that such systems at T D 0 have the entropyS D 0 is referred to as the third law of thermodynamics [43].

5.4 Statistical Ensembles

A statistical ensemble consists of many identical systems that are all in the samemacrostate but, in general, in different microstates. A thermodynamic ensemble is aspecific variety of statistical ensemble that, among other properties, is in statistical


equilibrium and is used to derive the properties of thermodynamic systems fromthe laws of classical or quantum mechanics [31]. Gibbs noted that different macro-scopic constraints lead to different types of ensembles, with particular statisticalcharacteristics. He defined three important thermodynamic ensembles:

• The micro-canonical ensemble describes an isolated system where the totalenergy of the system and the number of particles remain constant.

• The canonical ensemble describes a system in contact with a heat bath wherethe temperature and the number of particles are fixed, however, the energy is notfixed.

• The grand-canonical ensemble describes a system in contact with a heat andparticle bath where the temperature and chemical potential are specified whileneither the energy nor particle number are fixed.

The calculations that can be made over each of these ensembles are explored furtherin this section.

5.4.1 Micro-canonical Ensemble

A micro-canonical ensemble is the statistical ensemble that is used to represent thepossible states of a system which has an exactly specified total energy. The systemis said to be isolated in the sense that the system cannot exchange energy or particleswith its environment. The system’s energy, volume, and the number of particles arethe same in all possible states of the system. In Sect. 5.3.3, the thermodynamics ofan isolated system were discussed. In such system E, V and N remain constant.As entropy S D S.E; V;N / is constant for an isolated system, it takes a singleand constant value at equilibrium and is therefore an appropriate function to use. Inpractice, the micro-canonical ensemble does not correspond to an experimentallyrealistic situation. Systems in thermal equilibrium with their environment haveuncertainty in energy, and are instead described by the canonical ensemble or thegrand-canonical ensemble.

5.4.2 Canonical Ensemble

A canonical ensemble is an assembly of mental copies of a system in contactwith a heat bath at constant temperature. Since energy can be transferred betweenthe system and the heat bath, the energy of the systems in the assembly is notconstant. In this case T , V and N of the system are constant and the appropriatethermodynamic function to describe the system is the Helmholtz free energy F DF.T; V;N /. Since at equilibrium the T , V , N are constant, F takes a single valueat equilibrium. If the system s has ˝s.Es/ number of states at energy Es and thethermal bath has ˝b.Eb/ number of states at energy Eb , then the number of states

5.4 Statistical Ensembles 89

˝.Es;Eb/ of the combined system for the partition Es C Eb D E0 with a totalenergy E0 is given by

ln˝.Es;Eb/ D ln˝s.Es/˝b.Eb/ D ln˝s.Es/C ln˝b.Eb/ : (5.36)

The definition of thermal bath requires that E0 D Es CEb Es , therefore

ln˝b.Eb/ D ln˝b.E0/C @

@Eln˝b.E0/.Eb �E0/ D ln˝b.E0/C ˇ.Eb �E0/ ;

(5.37)

where ˇ for an isolated system is defined as

ˇ � 1

kBTD @

@Eln˝.E/ D 1

kB

@S

@E: (5.38)

Substituting Eq. (5.37) in Eq. (5.36), one obtains

ln˝.Es;Eb/ D ln˝s.Es/C ln˝b.E0/ � ˇEs ; (5.39)

which can be rewritten as

˝.Es;Eb/ D ˝s.Es/˝b.E0/e�ˇEs : (5.40)

As a result, the probability of observing the system in states with energy Es in thisensemble is proportional to exp .�ˇEs/. On normalization, it becomes P.s/ [29]:

P.s/ D 1

Ze�ˇEs ; (5.41)

where Z is the canonical partition function

Z DXs

e�ˇEs : (5.42)

All the macroscopic equilibrium thermodynamic properties of a canonical ensemblecan be calculated from the partition function using the fundamental relation

F D �kBT lnZ : (5.43)

5.4.3 Grand-Canonical Ensemble

The grand-canonical ensemble is specific for open systems. The system is open,meaning that particles as well as heat can be exchanged at constant temperature and


chemical potential. In the combined system, however, the total number of particles isconstant, as is the energy. For such systems the appropriate thermodynamic functionis the grand thermodynamic potential ˚G D ˚.T; V; �/, which takes a singleminimum value at equilibrium. For such a system, one shall obtain the probabilitythat the system has energy Es and Ns particles. In analogy with the canonicalcase, one can assume that the energy Eb of the thermal bath and its number ofparticles Nb are much larger than the same quantities of the system of interestEt D Eb CEs Es and Nt D Nb CNs Ns . As a result,

ln˝b.Eb;Nb/ D ln˝b.Et ;Nt /C @

@Eln˝b.Et ;Nt /.Eb �Et/

C @

@nln˝b.Et ;Nt /.Nb �Nt/ ;

(5.44)

which can be simplified as

ln˝b.Eb;Nb/ D ln˝b.Et ;Nt /C ˇ.Eb �Et/ � ˇ�.Nb �Nt/ : (5.45)

By replacing Eq. (5.45) in Eq. (5.36), the result is

ln˝.Es;Ns/ D ln˝s.Es;Ns/C ln˝b.Et ;Nt / � ˇEs C ˇ�Ns ; (5.46)

which can be alternatively written as

˝.Es;Ns/ D ˝s.Es;Ns/˝b.Et ; Nt /e�ˇ.Es��Ns/ : (5.47)

In a similar way to the case of canonical ensemble, it is straightforward to show thatthe probability that the system of interest is found in a state s with energy Es andNs particles can be written as [29]

P.s/ D 1

Ze�ˇ.Es��Ns/ ; (5.48)

where the grand-canonical partition function is given by

Z DXs

e�ˇ.Es��Ns/ : (5.49)

All the macroscopic equilibrium thermodynamic properties of a grand-canonicalensemble can be calculated from the partition function using the relation

˝G D �kBT lnZ : (5.50)

5.5 Quantum Statistics 91

5.5 Quantum Statistics

In statistical classical mechanics it is assumed that a system can have manymicrostates for given macroscopic thermodynamic state quantities. In the frame-work of ensemble theory, one is able to derive the probability density of findingthe system in a certain microstate. This concept can be transferred to quantummechanical systems. In classical statistical mechanics, a microstate corresponds to acertain point in phase space, whereas in a quantum mechanical system, the classicalphase-space trajectory is replaced by the time evolution of the wave function ofthe system. A quantum mechanical system is said to be in a pure state if it is in acertain microstate, denoted by a state vector ji. On the other hand, a mixed state is astatistical ensemble of pure states. A mixed state can be in any of several microstatesji i with probabilities pi . In pure and mixed state, one deals with two forms ofprobability. One is related to the definition of a state. A state can be a superpositionof eigenstates of some given Hermitian operator and the squares of the coefficientsare then the probabilities that the system will be measured to be in one eigenstate oranother. The other form is related to the system, rather than a state, and correspondsto the degree of ignorance about the system. A mixed state cannot be described asa ket vector. Instead, it is described by the matrix elements of its associated densityoperator. Density matrices can describe both mixed and pure states, treating themon the same footing. All quantum mechanical and statistical averages of arbitraryobservables can be calculated if the density matrix is known [27].

5.5.1 Density Matrix

The density matrix is the quantum-mechanical analogue to a phase-space probabilitymeasure in classical statistical mechanics. The density operator for the ensemble ormixture of states j i i with probabilities pi is given by

O� DXi

pi j i ih i j : (5.51)

By choosing an arbitrary basis set ji i, one can resolve the density operator into thedensity matrix, whose elements are

�mn DXi

pi hmj i ih i jni ; (5.52)

where the diagonal elements �nn are the probabilities that the system is in the statejni. For this reason, diagonal elements are referred to as populations. On the otherhand, off-diagonal matrix elements �mn give the probability for a transition from astate jni to another state jmi. The density matrix is Hermitian �mn D �

�nm, as seen

from Eq. (5.52), it is therefore possible to choose a basis in which the density matrix


is diagonal. In such a basis, the off-diagonal elements are all zero. And as long asthe states are normalized, the density matrix has a trace of unity, since

TrŒ O� DXi

piXn

hnj i ih i jni DXi

piXn

h i jnihnj i i ;

DXi

pi h i j i i DXi

pi D 1 :(5.53)

The rules for measurement in quantum mechanics can be stated in terms of densitymatrices. For example, the ensemble average of a measurement corresponding to anobservable OO is given by

h OOi DXi

pi h i j OOj i i DXi

piXmn

h i jnihnj OOjmihmj i i ;

DXmn

�mnOnm D TrŒ O� OA :(5.54)

In other words, the expectation value for the mixed state is the sum of theexpectation values for each of the pure states weighted by the probabilities pi .

In working with statistical mixtures, one usually deals with systems at thermalequilibrium. The density matrix for a canonical ensemble at temperature T ischaracterized by thermally distributed populations in the quantum states

�nn D pn D e�ˇEnZ ; (5.55)

where Z is the canonical partition function. This follows naturally from the generaldefinition of the equilibrium density operator

O� D e�ˇ OH

Z; (5.56)

where the partition function is given by

Z D TrŒe�ˇ OH : (5.57)

In a similar way, the density operator of a grand-canonical ensemble at chemicalpotential � and temperature T is defined as

O� D e�ˇ. OH�� ON/

Z; (5.58)

where the grand partition function Z reads as

Z D TrŒe�ˇ. OH�� ON/ : (5.59)

5.5 Quantum Statistics 93

For any operator OO , the grand-canonical ensemble average is therefore obtained as

h OOi D TrŒ� OO D TrŒe�ˇ. OH�� ON/ OO TrŒe�ˇ. OH�� ON/

: (5.60)

5.5.2 Fermi-Dirac Statistics

Using the density matrix and partition functions, one can evaluate the equilibriumdistribution function of Fermionic or Bosonic systems. In a grand-canonical system,the number of particles fluctuates. Based on Eq. (5.18), the average number ofparticles in the system is obtained as

hN i D ��@˚G

@�

�T;V

: (5.61)

Using Eq. (5.50), Eq. (5.61) can be written in terms of the partition function:

hN i D kBT

�@ lnZ

@�

�T;V

: (5.62)

By writing Eq. (5.59) in detail with the complete set of states in the abstractoccupation number space, one gets

Z D TrŒe�ˇ. OH�� ON/ DXn1;n2;:::

hn1; n2; : : : je�ˇ. OH�� ON/jn1; n2; : : :i : (5.63)

Since these states are eigenstates of the non-interacting Hamiltonian OH0 and thenumber operator ON , both operators can be replaced by their eigenvalues:

Z DXn1;n2;:::

hn1; n2 : : : jexp

"�ˇ

Xi

Eini � �Xi

ni

!#jn1; n2; : : :i : (5.64)

The exponential operator is now a number and is equivalent to a product ofexponentials. Therefore, the sum over expectation values factor into a product oftraces

Z DXn1

hn1je�ˇ.E1n1��n1/jn1iXn2

hn2je�ˇ.E2n2��n2/jn2i : : : DYi

Xni

e�ˇ.Ei��/ni :

(5.65)


For Fermions the occupation numbers are either 0 or 1, and the sum in Eq. (5.65) isrestricted to these values:

ZF DYi

1XniD0

�e�ˇ.Ei��/�ni D

Yi

�1C e�ˇ.Ei��/� : (5.66)

Taking the logarithm of both sides, one gets

lnZF DXi

ln�1C e�ˇ.Ei��/� : (5.67)

The mean number of Fermions is given by the Fermi-Dirac distribution function:

nF � hN i DXi

1

eˇ.Ei��/ C 1: (5.68)

where the i th term in the summation gives the mean occupation number in the i thstate. In textbooks on semiconductors it is more common to denote the Fermi-Diracdistribution function with f .E/ and to also use Fermi energy EF instead of thechemical potential. In the next chapters, we also follow these notations.

5.5.3 Bose-Einstein Statistics

For Bosons, the occupation number is not restricted, so one must sum ni over allintegers in Eq. (5.65):

ZB DYi

1XniD0

�e�ˇ.Ei��/�ni D

Yi

�1 � e�ˇ.Ei��/��1 : (5.69)

The logarithm of Eq. (5.69) yields

lnZB D ln

Yi

�1 � e�ˇ.Ei��/��1

!DXi

ln�1 � e�ˇ.Ei��/� : (5.70)

As a result, the mean number of Bosons is given by the Bose-Einstein distributionfunction:

nB � hN i DXi

1

eˇ.Ei��/ � 1 : (5.71)

5.6 Non-equilibrium Statistics 95

5.5.4 Maxwell-Boltzmann Statistics

The classical regime, where the Maxwell-Boltzmann statistics can be used as anapproximation to Fermi-Dirac statistics, is obtained by considering the situationthat is far from the limit imposed by the Heisenberg uncertainty principle for aparticle’s position and momentum. If the concentration of particles is low enoughthat their wave functions have a negligible overlap, indistinguishability does notplay an important role. Mathematically, for energies much higher than the chemicalpotential .Ei � �/ kBT , the exponential term in the denominator of boththe Fermi-Dirac and Bose-Einstein distributions becomes dominant with respectto unity. Further, these two distribution function tend to the classical Maxwell-Boltzmann distribution

hN i � e�ˇ.Ei��/ D eˇ�e�ˇEi : (5.72)

5.6 Non-equilibrium Statistics

Established techniques used to address non-equilibrium quantum statistics can beclassified according to the state functions they are based upon: the density matrix,the Wigner function, and the non-equilibrium Green’s function (NEGF). All threeapproaches are based on fundamental equations of motion. The resulting systemof integral-differential equations for the density matrix �.r1; r2I t /, the Wignerfunction f .r;p; t /, or the Green’s function G.r1; t1I r2; t2/ would in many casesbe too complex to allow for a direct numerical solution. For example, the lesserGreen’s function G<.r1; t1I r2; t2/ in the coordinate representation depends on twoposition arguments r1; r2 and two time arguments t1; t2. For a numerical solution,each argument of the Green’s function needs to be discretized. In the case of athree-dimensional system, the total number of unknowns to be evaluated wouldbe Ntot D .Nx �Ny �Nz �Nt/2. Assuming 100 grid points for each argument, thisresults in the astronomical number Nt D 1016. Even in the two-dimensional case,the number of unknowns is still very large, Nt D 1010, resulting in a prohibitivelylarge memory requirement. Approximations and simplifications must necessarilybe incorporated in order to make the problem numerically tractable. It is mainlythese simplifying assumptions that make the difference between the approaches.The assumptions are usually physically motivated and may be different in thedifferent formalisms. For instance, the approximations to simplify the equations forthe Green’s functions in real-space may not be suitable to the Wigner equation,and vice versa. The hierarchy of the transport models is shown in Fig. 5.2. In whatfollows, we briefly outline strong points and shortcomings of techniques based onthe density matrix, the Wigner function, and the Green’s function.


Fig. 5.2 The hierarchy of quantum and semi-classical transport models (Reproduced withpermission from Ref. [22]. Copyright (2004), Springer)

5.6.1 Boltzmann Transport Equation

The transport phenomena result from an average over the behavior of a large numberof electrons. The model of semi-classical transport is thus based on the Boltzmanndistribution function f .r;k; t /, which corresponds to the probability density to finda particle at positions r and k of the phase space at some time t . The dynamicequation of this distribution function without collision is readily obtained from theprobability conservation

df

dtD @f

@tC rrf � dr

dtC rkf � dk

dtD 0 ; (5.73)

which leads to the Boltzmann transport equation (BTE)

@f

@tC v.k/ � rrf C F

„ � rkf D @f

@t

ˇ̌ˇ̌Coll

; (5.74)

where the term .@f=@t/Coll is added to include the effects of collision or scattering onthe distribution function. The collision term is commonly evaluated using the Fermigolden rule. Collisions are considered as instantaneous scattering events, whichmodify the wave vector. In this approach, electrons are assumed to have ballisticfree flights between two consecutive scattering events. The duration of free flight isdetermined from the scattering rate S.k;k0/ for an electron initially in the state k

to the state 0k:

S.k;k0/ D 2�

„ jh k0 j OH intj kij2ı.E.k0/ �E.k// ; (5.75)


where OH int represents the interaction potential. The net rate of increase f due tocollisions is a result of the competition between in-scattering and out-scattering:

@f

@t

ˇ̌ˇ̌Coll

DX

k0

f .k0/Œ1 � f .k/ S.k0;k/ �X

k0

f .k/Œ1 � f .k0/ S.k;k0/ D OCf ;

(5.76)

where OC is the collision operator. The BTE has been the foundation of mostdevice simulations. It has shown tremendous success in explaining and modelingmost phenomena seen in electron devices [29]. BTE can be solved by employing adirect approach or, more often, a statistical Monte Carlo approach. Approximationsof the BTE, such as hydrodynamic or drift-diffusion equations, are the basemodel for various commercial and academic device simulators. The most widelyused technique for solving the Boltzmann equation has been the Monte Carlomethod [28]. Transport models based on the BTE can be derived using the method ofmoments [3,39,50], which yields the drift-diffusion model [48], the energy-transportand hydrodynamic models [26], or higher-order transport models [25]. Furthermore,an approximate solution can be obtained by expressing the distribution function asa series expansion, which leads to the spherical harmonics approach [24, 37].

5.6.2 Validity of the Boltzmann Transport Equation

The BTE is an approximation because it is a single-particle description of a many-body system and correlations between particles are not considered. As electrons insemiconductor devices interact with each other through their electric fields, theirmovements are correlated. To model such a system, a many-particle distributionfunction is needed [38]. At low electron concentrations, electron-electron corre-lations are weak and the many-particle distribution function can be contractedto the single-particle distribution function. In the case of electron transport insemiconductors, the influence of other electrons is, however, considered throughthe self-consistent electric field.

In BTE, the wave packet of an electron is treated as a particle and only collisionsare described by quantum mechanics. Therefore, f .r;p; t / is a classical concept,which specifies both position and momentum at the same time, violating theHeisenberg uncertainty principle �p�r � „. Assuming the spread in particleenergy to be about kBT and using E D p2=.2m�/, then �p ' p

2m�kBT . Asa result,

�r � „p2m�kBT

D �B

2�; (5.77)


where �, which is typically 10–20 nm at room temperature, is the associated deBroglie length of an electron with thermal energy

�B D hp2m�kBT

: (5.78)

Therefore, to treat electrons as particles, the condition�r �B should be satisfied;that is, the potential should vary slowly with respect to �B. Another limitation of theBTE is that collisions are assumed to be binary and instantaneous in time and localin space. However, the uncertainty principle for energy states that an electron shouldstay a long time in a state to have a well-defined energy �E�t � „. Taking �t tobe the time between collisions and assuming that �E ' kBT , then

�t � „kBT

: (5.79)

Supposing further that the electron has the velocity v D p=m� � p2kBT=m�,

corresponding to the thermal energy, the distance between two collisions is

�L D v�t �r2kBT

2m�„

2�kBTD �B

�; (5.80)

which states that the mean free path must be longer than the De Broglie wavelengthif the BTE is to be valid [39].

5.6.3 Density Matrix

The most natural approach for rigorous modeling of a quantum mechanical systemis the density matrix formalism, see Eq. (5.51). To study transport of carriers in asystem, a dynamical equation for the density matrix is required. By taking a timederivative of the density operator, one obtains the Liouville equation:

@�

@tDXi

pi

@j i i@t

h i j C j i i@h i j@t

�D 1

i„Xi

pi

h OH j i ih i j � j i ih i j OHi

D 1

i„ ŒOH; � (5.81)

Assuming a single-particle Hamiltonian, Eq. (5.81) becomes

i„@�.r1; r2/@t

D � „22m

�@2

@r21� @

@r22

��.r1; r2/C .V .r1/ � V.r2// �.r1; r2/ :

(5.82)


Irreversible or energy-dissipating processes always involve transitions betweenquantum states. Such processes can be described, at the simplest level, by a masteror rate equation which takes into account only the diagonal elements of the densitymatrix. The time evolution of such systems is determined by the rates of transitionbetween states Wk0;k. These rates are usually estimated using the Fermi goldenrule. Assuming Markovian transitions, which occur independently within any smalltime interval, the transition between states k0 and k will produce changes in thecorresponding occupation factors d�k D �d�k0 D Wk0;k�k0dt . The occupation ofthe state k increases and that of the state k0 decreases as a result of this particularprocess. Further, the amount of change depends only on the occupation of the initialstate. The Pauli master equation [33] is a frequently used model of irreversibleprocesses in simple quantum systems. It can be derived from elementary quantummechanics along with a Markov assumption.

In order to evaluate transport, the device under consideration, however, must becoupled to external reservoirs. Coupling introduces carrier exchange between deviceand reservoirs, which are assumed to be in thermal equilibrium. The differencebetween the electrochemical potentials of the reservoirs causes current through thedevice. Therefore, it is essential to properly include the coupling to the reservoirs inthe master equation. In the Pauli master equation approach developed by Fischetti,this coupling is introduced in a phenomenological manner [17, 18]. Applicationof the Pauli master equation is restricted to stationary systems, since in the non-stationary case the current continuity would be violated [19]. Another issue is thatthe Pauli master equation can only be justified for devices where the quantumregion is shorter than the phase coherence length [18]. A solution free from theabove mentioned shortcoming of phenomenological coupling of the device to thereservoirs was suggested by Gebrauer and Car [20, 21]. They impose periodicboundary conditions upon the unperturbed system. This approach can be also usedto describe transients.

5.6.4 Wigner Representation

Another approach capable of handling both quantum coherent propagation anddissipative scattering effects is based on the Wigner distribution function. TheWigner quasi-probability distribution was introduced by Wigner in 1932 [55]to study quantum corrections to classical statistical mechanics. The goal wasto replace the wave-function that appears in the Schrödinger equation with aprobability distribution in phase space. A classical particle has a definite positionand momentum and hence is represented by a point in phase space. For a collection(ensemble) of particles, the probability of finding a particle at a certain position inphase space is given by a probability distribution. This does not hold in quantummechanics due to the uncertainty principle. Instead, the Wigner quasi-probabilitydistribution plays an analogous role; it is defined as the density matrix in a mixedcoordinate/momentum representation [32, 55]. But the Wigner quasi-probability


distribution does not satisfy all the properties of a probability distribution. On theother hand, it satisfies boundedness properties unavailable to classical distributions.For instance, the Wigner distribution can and normally does go negative for stateswhich have no classical model and a convenient indicator of quantum-mechanicalinterference.

To derive the Wigner function from the density matrix �.r1; r2; t /, one rewritesthe arguments .r1; r2/ as r D .r1Cr2/=2 and s D r1�r2 and then Fourier transformss into a momentum variable k:

fw.r;k; t / DZ

ds�.r C s2; r � s

2; t/e�ik�s : (5.83)

Applying the Wigner-Weyl transformation to the Liouville equation gives the kineticequation for the Wigner function:

@fw.r;k; t /@t

Cv �r rfw C F„ �rkfw �

Zdk0Vw.r;k�k0/fw.r;k0; t /C @fw

@t

ˇ̌ˇ̌Coll

D 0 ;

(5.84)

where the kernel of the potential operator is given by

Vw.r;k/ D 1

i„.2�/3Z

ds�V.r C s

2/ � V.r � s

2/C s � F

e�ik�s : (5.85)

The kinetic equation for the Wigner function is similar to the semi-classical Boltz-mann equation, except for a non-local potential term. In the case of a slowly varyingpotential, this non-local term reduces to the local classical force term, and the semi-classical description given by the BTE is obtained from the Wigner equation. Apractically used approximation to incorporate realistic scattering processes intothe Wigner equation is to utilize the Boltzmann scattering operator [32, 41], orby an even simpler scheme, such as the relaxation time approximation [30]. Toaccount for scattering more rigorously, spectral information has to be included inthe Wigner function, resulting in an energy-dependence in addition to momentumdependence [40].

5.6.5 Green’s Function

Green’s functions in general have two position and two time arguments:G.r1; t1I r2; t2/. As shown in Sect. 6.6.3, under steady-state condition, one canFourier transform the time difference � D t1� t2 to energyE to obtainG.r1; r2IE/,which can further be transformed via the Wigner-Weyl procedure into G.r;kIE/.The presence of energy dependence (or two time arguments) distinguishes theGreen’s function approach from the Wigner function. Because the Wigner functionmeasures the state of the device at a particular time and its evolution is described

References 101

by a first-order differential equation, it can comprehend only external interactionswhich occur instantaneously in time. Such behavior is termed Markovian. Theenergy dependence of the Green’s function permits a description of processes notlocal in time, or non-Markovian processes because the energy argument provides away to include convolution integrals over the past history of the system. An exampleof a process which is non-Markovian is the resonant absorption or emission of aphonon. For the energy of the phonon to be well defined, the interaction must occurover a time longer than the oscillation period of the phonon. A non-MarkovianGreen’s function approach can accurately describe such processes, whereas theMarkovian Wigner function approach can not.

The NEGF method addresses the problem of dissipative quantum transport in aconsistent and complete way. Therefore, the method is computationally expensiveand applied to systems under steady-state condition [12]. The quantum deviceregion is coupled to the reservoirs by contact self-energies, while dissipation isintroduced via the scattering self-energies. When scattering via a self-energy isintroduced, the determination of the Green’s function requires inversion of a matrixof huge rank. To reduce the computational cost, the local scattering approximationis frequently used [11, 35]. In this approximation, the scattering self-energy termsare diagonal in coordinate representation. It allows one to employ the recursivealgorithm for computing the Green’s functions [51]. The local approximation is welljustified for electron-phonon scattering caused by deformation potential interaction.In order to reduce the computational cost even further, systems with simplifiedgrid requirements are considered. The mode-space approach [53] takes only arelatively small number of transverse modes, Nmode, into consideration. For theremaining one-dimensional transport problems, the number of unknowns reducesto Ntot D Nmode �N2

x �NE , where NE is the number of energy grid points.It can be shown that quantum ballistic formalism can be fully recovered from

the NEGF formalism as a special case, where no dissipative scattering occurs inthe system [13]. Another important point is that the NEGF formalism looks verydifferent from Landauer-Buttiker formalism [8,9]. The NEGF formalism focuses onthe internal state of the conductor. By contrast, in the Landauer approach, the centralquantity is the transmission function from one contact to another. The internal stateof the conductor usually never appears in this formalism. However, the transmissionfunction can be expressed in terms of internal quantities. One can precisely obtainthis result from the NEGF formalism as well when non-dissipative transport isassumed.

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resonant-tunneling diodes. Phys. Rev. B 45(12), 6670–6685 (1992)36. Landau, L.D.: Das Dämpfungsproblem in der Wellenmechanik. Zs. f. Phys. 45, 430–441

(1925)37. Liang, W., Goldsman, N., Mayergoyz, I., Oldiges, P.J.: 2-D MOSFET modeling including

surface effects and impact ionization by self-consistent solution of the Boltzmann, Poisson,and Hole-Continuity equations. IEEE Trans. Electron Devices 44(2), 257–267 (1997)

38. Liboff, R.L.: Introductory Quantum Mechanics, 4th edn. Addison-Wesley, San Francisco(2003)

39. Lundstrom, M.: Fundamentals of Carrier Transport, 2nd edn. Cambridge University Press,Cambridge (2000)

40. Mahan, G.D.: Physics of solids and liquids. In: Many-Particle Physics, 2nd edn. Plenum Press,New York (1990)

41. Mains, R.K., Haddad, G.I.: Wigner function modeling of resonant tunneling diodes with highpeak-to-valley ratios. J. Appl. Phys. 64(10), 5041–5044 (1988)

42. Maxwell, J.C.: Illustrations of the dynamical theory of gases.-part I. On the motions andcollisions of perfectly elastic spheres. Philos. Mag. 19(124), 19–32 (1860)

43. Pathria, R.K.: Statistical Mechanics, 3rd edn. Elsevier, Amsterdam (2011)44. Pauli, W.: Über den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der

Komplexstruktur der Spektren. Zs. f. Phys. 31, 765–783 (1925)45. Pauli, W.: Über Gasentartung und Paramagnetismus. Zs. f. Phys. 41, 81–102 (1927)46. Pauli, W.: The connection between spin and statistics. Phys. Rev. 58, 716–722 (1940)47. Plischke, M., Bergersen, B.: Equilibrium Statistical Physics, 3rd edn. World Scientific,

Singapore, Hackensack (2006)48. Selberherr, S.: Analysis and Simulation of Semiconductor Devices. Springer, Vienna (1984)49. Simon, F.: Fünfundzwanzig Jahre Nernstscher Wärmesatz. In: Ergebnisse der Exakten

Naturwissenschaften, vol. 9, pp. 222–274. Springer, Berlin (1930)50. Stratton, R.: Diffusion of hot and cold electrons in semiconductor barriers. Phys. Rev. 126(6),

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quantum mechanical modeling of nanotransistors. J. Appl. Phys. 91(4), 2343–2354 (2002)52. Tolman, R.C.: The Principles of Statistical Mechanics. The Clarendon Press, Oxford (1938)53. Venugopal, R., Ren, Z., Datta, S., Lundstrom, M., Jovanovic, D.: Simulating quantum transport

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Chapter 6Green’s Function Formalism

The Schrödinger equation is the basis of all quantum transport models, but solvingthe Schrödinger equation for a many particle system is very difficult. To addressthis problem, different techniques and methods have been introduced. Among themthe Green’s function formalism appears to be appropriate for analyzing nano-scaledevices. However, the Green’s function formalism merely reformulates the problemand an exact solution for a realistic device can not be obtained; thus approximationmethods need to be used. This chapter outlines the theory of Green’s functiontechniques for modeling transport phenomena in semiconductor devices.

6.1 Historical Review

In the early 1960s, the non-equilibrium Green’s function (NEGF) formalism wasdeveloped by Martin and Schwinger [36,43], Kadanoff and Baym [24], and Keldysh[25]. The general formalism for NEGF calculations of current in devices wasfirst described in a series of papers in the early 1970s [6–8, 10]. In 1976, theformalism was applied to a multi-band model (two-bands) to investigate tunneling[2] and diagonal disorder [3], and in 1980 it was extended to model time-dependentpotentials [9]. The accelerated use of the approach was motivated by experimentalinvestigations of mesoscopic physics made possible by high quality semiconductorhetero-structures grown by molecular beam epitaxy. In 1988, Kim and Arnoldapplied the NEGF formalism to such a system, specifically, a resonant tunnelingdiode [26]. As experimental methods progressed allowing for the finer manipulationof matter and probing into the nano-scale regime, the importance of quantum effectsand tunneling continuously increased. The theory was adapted to address the currentsystems of interest ranging from mesoscopics to single-electronics, nano-scaledfield effect transistors (FETs), and molecular electronics [1, 27]. References [11,31, 32, 34, 40, 41] have been mostly used for the preparation of this chapter.


105

106 6 Green’s Function Formalism

6.2 Quantum Dynamics

A discussion of different pictures in quantum mechanics is essential for understand-ing and application of time-dependent perturbation theory. The time evolution ofoperators and state vectors in quantum mechanics can be expressed in differentrepresentations. The Schrödinger, the interaction or Dirac, and the Heisenbergrepresentations are discussed in this section.

6.2.1 Schrödinger Picture

In the Schrödinger picture, the operators remain constant

OOS.t/ D OOS ; (6.1)

whereas quantum states evolve in time and their time dependence is obtained fromthe Schrödinger equation i„@t j�S.t/i D OH j�S.t/i, which has the formal solution

j�S.t/i D e�i OHt=„j�S.0/i: (6.2)

6.2.2 Heisenberg Picture

In the Heisenberg picture, states do not change with time. This is accomplished byadding a term to the Schrödinger states to eliminate time-dependence:

j�Hi D ei OHt=„j�S.t/i ;

D j�S.0/i : (6.3)

In the Heisenberg picture, operators, however, are time-dependent. The operators inthe Heisenberg picture can be defined via expectation values. The expectation valuefor an operator OO in the Schrödinger picture can be written as

hOi D h�S.t/j OOSj�S.t/i ;

D h�S.0/jei OHt=„ OOSe�i OHt=„j�S.0/i ;

D h�Hjei OHt=„ OOSe�i OHt=„j�Hi : (6.4)

Therefore, a general operator in the Heisenberg picture is given by

OOH.t/ D ei OHt=„ OOSe�i OHt=„ : (6.5)

6.2 Quantum Dynamics 107

6.2.3 Interaction Picture

An interaction picture is an intermediate representation between the Schrödingerpicture and the Heisenberg picture. Whereas in the other two pictures either the stateor the operators carry time dependence, in the interaction picture both the states andthe operators are time-dependent. In order for the interaction picture to be usefulin simplifying the analysis of a problem, the Hamiltonian is divided into two parts(see Eq. (3.2)):

OH D OH0 C OH int ; (6.6)

where OH0 is the non-interacting part, which is assumed to be exactly solvable. OH int

can be time-dependent or time-independent and contains all the interactions, suchas carrier-carrier, carrier-phonon, impurity scattering, and so forth. In addition, atime-dependent perturbation Hamiltonian describing, for instance, the interactionwith an external time-dependent field, can be added to the total Hamiltonian.The development of the time-dependent perturbation theory was initiated by PaulDirac’s early work on the semi-classical description of atoms interacting withelectromagnetic fields [15]. Dirac, Wheeler, Heisenberg, Feynman and Dysondeveloped it into a powerful set of techniques for studying interactions and timeevolution in quantum mechanical systems, which cannot be solved exactly [14].

The quantum state in the interaction representation is given by

j�I.t/i D ei OH0t=„j�S.t/i : (6.7)

The equation of motion of this state is found by taking the time derivative

i„@t j�I.t/i D � OH0ei OH0t=„j�S.t/i C ei OH0t=„i„@t j�S.t/i ;

D ei OH0t=„Œ� OH0 C OH0 C OH int e�i OH0t=„j�I.t/i ;

D ei OH0t=„ OH inte�i OH0t=„j�I.t/i : (6.8)

Therefore, one obtains the following set of equations in the interaction picture

i„@t j�I.t/i D OH int.t/j�I.t/i ;

OH int.t/ � ei OH0t=„ OH inte�i OH0t=„ : (6.9)

The expectation value of an arbitrary operator OO in the Schrödinger picture can bewritten as

h�S.t/j OOSj�S.t/i D h�I.t/jei OH0t=„ OOSe�i OH0t=„j�I.t/i ; (6.10)


which suggests the following definition of an operator in the interaction picture:

OOI.t/ D ei OH0t=„ OOSe�i OH0t=„ : (6.11)

6.2.4 The Evolution Operator

To solve the equations of motion for the quantum state in the interaction picture,a unitary operator OS.t; t0/ that determines the state vector at time t in terms of thestate vector at time t0 is introduced:

j�I.t/i D OS.t; t0/j�I.t0/i ; (6.12)

where OS satisfies the initial condition OS.t0; t0/ D 1. For finite times, OS.t; t0/ can beconstructed explicitly by employing Eq. (6.7):

j�I.t/i D ei OH0t=„j�S.t/i ;D ei OH0t=„e�i OH.t�t0/=„j�S.t0/i ;D ei OH0t=„e�i OH.t�t0/=„e�i OH0t0=„j�I.t0/i ;

(6.13)

which therefore identifies

OS.t; t0/ D ei OH0t=„e�i OH.t�t0/=„e�i OH0t0=„ : (6.14)

Since OH and OH0 do not commute with each other, the order of the operators mustbe carefully maintained.

Equation (6.5) can be rewritten in terms of the interaction picture operator and OS

OOH.t/ D ei OHt=„e�i OH0t=„ OOI.t/ei OH0t=„e�i OHt=„ D OS.0; t/ OOI.t/ OS.t; 0/ : (6.15)

Several properties of OS can be deduced from Eq. (6.14) [16]:

OS�.t; t0/ OS.t; t0/ D OS.t; t0/ OS�.t; t0/ D 1 ; (6.16)

implying that OS is unitary OS�.t; t0/ D OS�1.t; t0/,

OS.t1; t2/ OS.t2; t3/ D OS.t1; t3/ ; (6.17)

which shows that OS has the group property,

OS.t; t0/ OS.t0; t/ D 1 ; (6.18)


indicating that OS.t0; t/ D OS�.t; t0/. Although Eq. (6.14) is the formal solution tothe problem posed by Eq. (6.12), it is not very useful for computational purposes.Instead, one can construct an integral equation for OS , which can be solved byiteration. It follows from Eqs. (6.9) and (6.12) that OS satisfies the differentialequation

i„@t OS.t; t0/ D OH intI .t/

OS.t; t0/ : (6.19)

Integrating both sides of Eq. (6.19) with respect to time yields,

OS.t; t0/ D OS.t0; t0/ � i

„Z t

t0

dt1 OH intI .t1/

OS.t1; t0/ ;

D 1 � i

„Z t

t0

dt1 OH intI .t1/

OS.t1; t0/ : (6.20)

By iterating this equation repeatedly, one gets

OS.t; t0/ D 1 � i

„Z t

t0

dt1 OH intI .t1/C

��i

„�2 Z t

t0

dt1

Z t1

t0

dt2 OH intI .t1/

OH intI .t2/C

: : :C��i

„�n Z t

t0

dt1

Z t1

t0

dt2 : : :Z tn�1

t0

dtn OH intI .t1/

OH intI .t2/ : : :

OH intI .tn/

D1XnD0

��i

„�n Z t

t0

dt1

Z t1

t0

dt2 : : :Z tn�1

t0

dtn OH intI .t1/

OH intI .t2/ : : :

OH intI .tn/ :

(6.21)

A graphical sketch of the contents of S.t; t0/ is shown in Fig. 6.1. Equation (6.21)has the characteristic feature that the operator containing the latest time standsfarthest to the left. At this point it is convenient to introduce the time-orderingoperator, denoted by the symbol Tt:

Ttf OA.t1/ OB.t2/g D �.t1 � t2/ OA.t1/ OB.t2/ C �.t2 � t1/ OB.t2/ OA.t1/ ; (6.22)

where �.t/ is the step function

�.t/ D

8̂<̂ˆ̂:

1 t > 0

1=2 t D 0

0 t < 0

: (6.23)


Fig. 6.1 The time evolution operator OS.t; t0/ can be viewed as the sum of phase factors due toOH int. The sum contains contributions from processes with 0, 1, 2, 3, : : : scattering events, which

happen during the evolution from time t0 to time t [5]

Using Tt, the second integral in Eq. (6.21) can be rearranged:

1

2Š

Z t

t0

dt1

Z t

t0

dt2Ttf OH intI .t1/

OH intI .t2/g D

1

2Š

Z t

t0

dt1

Z t1

t0

dt2 OH intI .t1/

OH intI .t2/C 1

2Š

Z t

t0

dt2

Z t1

t0

dt1 OH intI .t2/

OH intI .t1/ :

(6.24)

The second term on the right hand-side is equal to the first, which is easy to see byjust redefining the integration variables t1 ! t2, t2 ! t1. Thus one gets

1

2Š

Z t

t0

dt1

Z t

t0

dt2 Ttf OH intI .t1/

OH intI .t2/g D

Z t

t0

dt1

Z t1

t0

dt2 OH intI .t1/

OH intI .t2/ :

(6.25)

Time ordering can be generalized to higher order terms. In n-th order term, wheren factors H int.tj / appear, all nŠ permutations p 2 Sn of the n times tj are involved,and one can define [5]

Ttf OH int.t1/ OH int.t2/ : : : OH int.tn/g �Xp2Sn

OH int.t1/ OH int.t2/ : : : OH int.tn/�

�.tp.1/ � tp.2//�.tp.2/ � tp.3// : : : �.tp.n�1/ � tp.n// : (6.26)


Thus for the expansion of the OS.t; t0/, one obtains

OS.t; t0/ D1XnD0

1

nŠ

��i

„�n Z t

t0

dt1

Z t

t0

dt2 : : :Z t

t0

dtnTtf OH intI .t1/

OH intI .t2/ : : :

OH intI .tn/g

D Ttfexp

�� i

„Z t

t0

dt 0 OH intI .t

0/�

g : (6.27)

This expression is a symbolic representation of Eq. (6.21), in which it is understoodthat the time-ordering operator acts to order the operators in each term of theexpansion of the exponential. This expression for OS.t; t0/ is the starting point forinfinite order perturbation theory. In many applications, the perturbation OH int isweak compared to H0. It can therefore be justified to approximate OS.t; t0/ by thefirst order approximation

OS.t; t0/ � 1 � i

„Z t

t0

dt1 OH intI .t1/ : (6.28)

This simple time evolution operator forms the basis for the Kubo formula in linearresponse theory, which is applicable to a wide range of physical problems [16, 35].

6.2.5 Imaginary Time Propagation

At finite temperature under thermodynamic equilibrium, the state of a system isdescribed by the equilibrium density operator O�. For a given O�, the ensemble averageof any operator OO is calculated as (see Sect. 5.5.1)

h OOi D TrŒ O� OO ; (6.29)

where O� is called density operator

O� D 1

Ze�ˇ OH ; Z D TrŒe�ˇ OH : (6.30)

Using the time evolution operator, Eq. (6.29) can be rewritten as

h OOi D 1

ZTrŒe�ˇ OHT�f OS.0; t/ OOI.t/ OS.t; 0/g : (6.31)

Each operator OS in Eq. (6.31) can be expanded as a time-ordered exponential andcan be collected into a single time-ordered exponential. However, a problem appearsfor the density operator, which should also be expanded in powers of the interaction


Hamiltonian. The solution to this problem is to use imaginary times instead of realtimes. This has no real physical meaning, and is only a clever mathematical trick.Both the density operator O� and the time evolution operator OS are both exponentialfunctions of the Hamiltonian. In order to treat both OS and O� simultaneously, one canreplace the time argument by an imaginary quantity t D �i� , where � is real andhas the dimension of time. Under this condition, both OS and O� can be treated in justone expansion in powers of the interaction Hamiltonian. As for real time, one candefine an imaginary time Heisenberg picture by substituting it by � :

OOH.�/ � e OH�=„ OOS e� OH�=„ : (6.32)

Throughout this book, the imaginary time definition is used when the time argumentis a Greek letter and the usual definition when the times are written with romanletters. The interaction picture for imaginary times is introduced

OOI.�/ � e OH0�=„ OOSe� OH0�=„ : (6.33)

The Heisenberg and interaction pictures for imaginary time are related by

OOH.�/ D e OH�=„e� OH0�=„ OOI.�/eOH0�=„e�i OH�=„ ;

D OS.0; �/ OOI.�/ OS.�; 0/ ; (6.34)

where the imaginary time evolution operator OS.�; �0/ is given by (compare withEq. (6.14))

OS.�; �0/ D e OH0�=„ e� OH.��0/=„e� OH0�0=„ : (6.35)

It should be noted that OS.�; �0/ is not unitary, but still satisfies the group propertyand the boundary condition

OS.�1; �2/ OS.�2; �3/ D OS.�1; �3/ ; (6.36)

OS.�0; �0/ D 1 : (6.37)

In addition, the equation of motion of OS is calculated as

„@� OS.�; �0/ D e OH0�=„. OH0 � OH/e� OH.��0/=„e� OH0�0=„ ; (6.38)

D e OH0�=„. OH0 � OH/e� OH0�=„ OS.�; �0/ ; (6.39)

D OH int.�/ OS.�; �0/ ; (6.40)

6.3 Equilibrium Green’s Function 113

where OH int.�/ � e OH0�=„ OH inte� OH0�=„. It follows that the operator OS.�; �0/ obeys thesame differential equation as the unitary operator introduced in Eq. (6.12) and onecan write down the solution as (compare with Eq. (6.27))

OS.�; �0/ D1XnD0

1

nŠ

��1„�n Z �

�0

d�1

Z �

�0

d�2 : : :Z �

�0

d�nT� f OH intI .�1/ OH int

I .�2/ : : : OH intI .�n/g ;

D T� fexp

�� 1„

Z �

�0

d� 0 OH intI .� 0/

�g :

If � is set equal to ˇ„, Eq. (6.35) can be rewritten as

e�ˇ OH D e�ˇ OH0 OS.ˇ„; 0/ ; (6.41)

which relates the many-body density operator to the single-particle density operatorby means of an imaginary time evolution operator. Utilizing Eq. (6.41), Eq. (6.31)can be expanded as

h OOi D 1

ZTrŒe�ˇbH0 OS.ˇ„; 0/T�f OS.0; �/ OOI.�/ OS.�; 0/g ;

D 1

ZTrŒe�ˇbH0T�f OS.ˇ„; 0/ OOI.�/g ;

D hT�f OS.ˇ„; 0/ OOIgi0h OS.ˇ„; 0/i0

; (6.42)

where the averages h: : :i0 depend on e�ˇH0 andZD TrŒe�ˇ OH D TrŒe�ˇ OH0 OS.ˇ„; 0/ .The results indicate that the imaginary time indeed allows for a systematic expan-sion of expressions, including the density operator [5].

6.3 Equilibrium Green’s Function

In this section, the Green’s function for many-body systems under equilibriumcondition is discussed. To treat such a system, it is common to use the interactionrepresentation OH D OH0 C OH int, where H0 is the non-interacting Hamiltonianand H int is the perturbation, which contains all the interactions, that is OH int DOHe-e C OHe-ph.

6.3.1 Zero Temperature Green’s Function

The time-ordered single-particle Green’s function at zero temperature is definedas [16]


G.r; t I r0; t 0/ D � i

„h�0jTtf O H.r; t / O �

H.r0; t 0/gj�0i

h�0j�0i ; (6.43)

where j�0i is the ground-state of the interacting system in the Heisenberg pictureand Tt is the time-ordering operator defined in Eq. (6.22). The field operator O H.r; t /in the Heisenberg picture is given by

O H.r; t / D ei OHt=„ O .r/e�i OHt=„ : (6.44)

Inserting Eq. (6.44) into Eq. (6.43), the physical interpretation of the Green’sfunction becomes obvious:

G.r; t I r0; t 0/ D � i

„h�0jTtfei OHt=„ O .r/e�i OH.t�t 0/=„ O �.r0/e�i OHt 0=„gj�0i

h�0j�0i :

(6.45)

If t > t 0, the Green’s function G.r; t I r0; t 0/ is the probability amplitude that aparticle created at time t 0 at place r0 moves to time t and place r. This followsfrom the definition of G.r; t I r0; t 0/. At zero time the system is at the ground-state�0. The system then evolves to time t 0 with operator e�i OHt 0=„. At this time, O �.r0; t 0/creates a particle at place r0. Then, the system continues its evolution from t 0 to twith the operator e�i OH.t�t 0/=„, after which O .r; t / annihilates the particle at placer. The system returns to the initial ground-state with the operator ei OHt=„. In asimilar way, if t 0 > t , the field operator creates a hole at time t , and the systemthen propagates according to the Hamiltonian OH . These holes can be interpretedas particles traveling backward in time [17]. The probability amplitude that a holecreated at time t at place r moves to time t 0 and place r0 is again just the Green’sfunction for t < t 0.

To calculate G.r; t I r0; t 0/, a perturbation expansion is very useful. However, thedefinition of the Green’s function in Eq. (6.43) does not allow a direct solution, sinceit involves the exact ground-states of the interacting Hamiltonian OH , which is oneof the things to be calculated. As the ground state of the non-interacting part, OH0,can be calculated easily, one can express the ground state of the interacting systemj�0i in terms of the ground state of the non-interacting one j�0i. For that purpose,in Eq. (6.19) one adds to the operator OH int

I .t/ a factor e�j"jt , which switches theinteraction off at t ! ˙1 [16]. The non-interacting ground state j�0i is assignedto the system at t ! �1 and the connection to j�0i is formed by the Gell-Mann andLow theorem [20]:

j�.0/i D OS.0;�1/j�0i ; (6.46)

where the OS operator in defined in Sect. 6.2.4. The traditional argument is that onestarts from t ! �1 with a wave function �0 which does not contain the effectsof the interaction OH int. The operator OS.0;�1/ brings this wave function up to the


present, t D 0 [16]. Thus one has the wave function which contains the effects ofthe interaction OH int, so that it is an eigenstate of OH . As t ! 1, one gets j�.1/i DOS.1;�1/j�0i. One possible assumption is that j�.1/i must be related to �0.

The system returns to its ground state for t ! 1 except for a phase factor [35]j�.1/i D eiLj�0i which implies that h�0j OS.1;�1/j�0i D eiL. An alternativeto this assumption is discussed in Sect. 6.4.2. Finally, by using Eq. (6.46) for theground-state, Eq. (6.43) becomes

G.r; t I r0; t 0/

D � i

„h�0j OS.�1; 0/Ttf O H.r; t / O �

H.r0; t 0/g OS.0;�1/j�0i

h�0j OS.�1; 0/ OS.0;�1/j�0i;

D � i

„h OS.�1; 0/Ttf OS.0; t/ O I.r; t / OS.t; 0/ OS.0; t 0/ O �

I .r0; t 0/ OS.t 0; 0/g OS.0;�1/i0

h OS.�1; 0/ OS.0;�1/i0;

D � i

„h OS.1; 0/Ttf OS.0; t/ O I.r; t / OS.t; t 0/ O �

I .r0; t 0/ OS.t 0; 0/g OS.0;�1/i0

h OS.1;�1/i0;

D � i

„hTtf OS.1;�1/ O I.r; t / O �

I .r0; t 0/gi0

h OS.1;�1/i0;

(6.47)

where the short-hand notation h: : :i0 D h�0j : : : j�0i is introduced to representthe expectation value over the ground-state of the non-interacting system at zerotemperature. The transition from the first to the second line is achieved byusing Eq. (6.15) for converting the Heisenberg representation of operators into theinteraction representation. The second step is obtained by taking account of theproperties of the OS operators described in Sect. 6.2.4 and the return of the systemto its ground-state as t ! 1. In the forth line the operator OS.1;�1/ containsseveral time intervals: .1; t /, .t; t 0/, and .t 0;�1/. The Tt operator automaticallysorts these intervals so that they act in their proper sequences. Replacing operator OSwith its formal definition from Eq. (6.27), one gets

G.r; t I r0; t 0/ D � i

„hTtfexp

�� i

„Z 1

�1dt1 OH int

I .t1/

�O I.r; t / O �

I .r0; t 0/gi0

hTtfexp

�� i

„Z 1

�1dt1 OH int

I .t1/

�gi0

:

(6.48)

6.3.2 Finite Temperature Green’s Function

In Sect. 6.3 the Green’s function for a system under equilibrium at zero temperaturehas been introduced. Furthermore, the Green’s function was readily expressed


as perturbation expansion in the interaction picture. In this section, the Green’sfunction for a system under equilibrium at finite temperature is presented. TheGreen’s function at finite temperature has a simple perturbation expansion – similarto that for the Green’s function at zero temperature – and also enables us to evaluatethe properties of the system. At finite temperatures, one assumes that the particle,either electron or phonon, is interacting with a bath of other particles. The exactstate of all these other particles is not known, since they are fluctuating betweendifferent configurations. At finite temperature under thermodynamic equilibrium thestate of a system is described by the equilibrium density operator O� (see Sect. 5.5.1).In treating such systems, it will be most convenient to use the grand-canonicalensemble, which allows for a variable number of particles. Therefore, the system isconsidered to be in contact with a heat bath of temperature T and a particle reservoircharacterized by the Fermi energy EF. With the definition OK D OH � EF ON , whereON is the particle number operator, the statistical operator can be written as

O� D e�ˇ OK

TrŒe�ˇ OK ; (6.49)

where the short-hand notation ˇ D 1=kBT is used. The operator OK may beinterpreted as a grand-canonical Hamiltonian. Given the density operator, theensemble average of any operator OO can be calculated as

h OOi D TrŒ O� OO ;

D TrŒe�ˇ OK OO TrŒe�ˇ OK

:(6.50)

Therefore, the single-particle Green’s function at finite temperature is defined as

G.r; t I r0; t 0/ D � i

„hTtf O H.r; t / O �H.r

0; t 0/gi ;

D � i

„TrŒe�ˇ OKTtf O H.r; t / O �

H.r0; t 0/g

TrŒe�ˇ OK :

(6.51)

At this stage, this form of the Green’s function does not admit the Wick decomposi-tion because the Wick theorem described in Sect. 6.5.1 requires a dependence on thenon-interacting Hamiltonian OH0 for both the field operators and the thermal average.A way around this problem is the Matsubara technique [37], where one introduces acomplex time � D i t and a new physical quantity, the Matsubara (imaginary time)Green’s function G .r; � I r0; � 0/. The representation of operators with imaginary timearguments is given in Sect. 6.2.5.


6.3.3 Matsubara Green’s Function

The single particle Matsubara Green’s function is defined as

G .r; � I r0; � 0/ D �1„hT�f O K.r; �/ O �K.r

0; � 0/gi ;

D �1„TrŒe�ˇ OKT�f O K.r; �/ O �

K.r0; � 0/g

TrŒe�ˇ OK :

(6.52)

The Green’s function now may be rewritten in the interaction picture

G .r; � I r0; � 0/

D � 1„TrŒe�ˇ OKT� f O K.r; �/ O �

K.r0; � 0/g

TrŒe�ˇ OK ;

D � 1„TrŒe�ˇ OK0 OS .ˇ„; 0/T� f O K.r; �/ O �

K.r0; � 0/g

TrŒe�ˇ OK0 OS .ˇ„; 0/ ;

D � 1„TrŒe�ˇ OK0 OS .ˇ„; 0/T� f OS .0; �/ O I.r; �/ OS .�; 0/ OS .0; � 0/ O �

I .r0; � 0/ OS .� 0; 0/g


D � 1„TrŒe�ˇ OK0T� f OS .ˇ„; �/ O I.r; �/ OS .�; � 0/ O �

I .r0; � 0/ OS .� 0; 0/g


D � 1„TrŒe�ˇ OK0T� f OS .ˇ„; 0/ O I.r; �/ O �

I .r0; � 0/g


(6.53)

where Eq. (6.30) is employed for the transition from the first to the second lineand Eq. (6.34) for the transition from the second to third line. Equation (6.53) hasprecisely the structure analyzed in Eq. (6.47). Using Eq. (6.41), the operator OS canbe expanded as

G .r; � I r0; � 0/ D

� 1„

Trhe�ˇ OK0

P1

nD01nŠ .

�1„/n R ˇ„

0 d�1:::R ˇ„

0 d�nT� f OK int.�1/::: OK int.�n/ O I.r;�/ O �I .r0;� 0/gi

Trhe�ˇ OK0

P1

nD01nŠ .

�1„/n R ˇ„

0 d�1:::R ˇ„

0 d�nT� f OK int.�1/::: OK int.�n/gi ;

(6.54)

where the denominator is just the perturbation expansion of the grand partitionfunction. However, it serves to eliminate all disconnected diagrams, exactly asin the zero-temperature formalism. It is apparent that the perturbation expansionof the Matsubara Green’s function Eq. (6.54) is very similar to that of the zerotemperature Green’s function Eq. (6.94). Matsubara [37] has proved that there existsa generalized Wick theorem (see Sect. 6.5.1) that deals only with the ensembleaverage of operators and relies on the detailed form of the statistical operator eˇ OK0 .


6.4 Non-equilibrium Green’s Functions

In Sect. 6.3.2, the Green’s function for a system under equilibrium at finitetemperature has been introduced. In this section, a more general formalism forsystems under non-equilibrium conditions at finite temperature is presented. First,the ensemble average of an operator under non-equilibrium is defined. Then thecontour-ordered non-equilibrium Green’s function (NEGF) formalism is introducedand the equation of motion for the Green’s function is presented. Finally, it willbe shown that a perturbation expansion similar to the equilibrium theory can beachieved.

6.4.1 Non-equilibrium Ensemble Average

We employ the standard device for obtaining a non-equilibrium state. At time t0,prior to which the system is assumed to be in thermodynamic equilibrium witha reservoir, the system is exposed to a disturbance represented by the contributionOH ext to the Hamiltonian. The external perturbation can for instance be a time varying

electric field, a light excitation pulse, and so forth. The total Hamiltonian is thus

OH .t/ D OH0 C OH int C OH ext D OH C OH ext; (6.55)

where OH ext D 0 for t < t0. One is not restricted to using the statistical equilibriumstate at times prior to t0 as the initial condition. A non-equilibrium situation can bemaintained through contact with a reservoir [28, 42]. Non-equilibrium statisticalmechanics is concerned with calculating average values h OOH .t/i of physicalobservables for times t > t0. Given the density operator O�, the average of anyoperator OO is then defined as

h OOH .t/i � TrŒ O� OOH .t/ ; (6.56)

where OOH .t/ is an operator in the Heisenberg picture. The NEGF is defined as

G.r; t; r0; t 0/ D � i

„hTtf O H .r; t / O �

H .r0; t 0/gi ; (6.57)

where O H is the field operator in the Heisenberg picture evolving with theHamiltonian OH defined in Eq. (6.55), and the bracket h: : :i is the statistical averagewith the density operator defined in Eq. (6.56).

One can evaluate Green’s functions by using Wick’s theorem, which enables usto decompose many-particle Green’s functions into sums and products of single-particle Green’s functions (see Sect. 6.5.1). The restriction of the Wick theoremnecessitates that the field operators and the density operator have to be representedin the interaction picture, or equivalently, their time evolution is governed by the

6.4 Non-equilibrium Green’s Functions 119

non-interacting Hamiltonian OH0. The contour-ordered Green’s function, which isintroduced next, provides a suitable framework for this purpose.

6.4.2 Contour-Ordered Green’s Function

To express the field operators in the interaction representation, an operator OS isdefined in Sect. 6.2.4 and applied for calculating the Green’s functions in Sect. 6.3.1.In Eq. (6.47), the time integration limit is .�1;1/. The state at t ! �1 is welldefined as the ground-state of the non-interacting system j�0i. The interactionsare turned on slowly. At t D 0, the fully interacting ground state is j�.0/i DOS.0;�1/j�0i. The state at t ! 1 must be defined carefully. If the interactions

remain on, then this state is not well described by the non-interacting groundstate. Alternatively, one could require that the interactions are turned off at largetimes, which returns the system to the ground-state j�0i. Schwinger [43] suggestedanother method of handling the asymptotic limit t ! �1. He proposed thatthe time integral in the OS operator has two parts: one goes from .�1; t / whilethe second goes from .t;�1/. The integration path is a contour, which startsand ends at �1. The advantage of this method is that one starts and ends theS operator expansion with a known state j�.�1/i D j�0i. Instead of the time-ordering operator Eq. (6.22), a contour-ordering operator can be employed. Thecontour-ordering operator TC orders the time labels according to their order on thecontour C . Under equilibrium condition, the contour-ordered method gives resultsthat are identical to the time-ordered method. The main advantage of the contour-ordered method is in describing non-equilibrium phenomena using Green’s func-tions. Non-equilibrium theory is entirely based upon this formalism, or equivalentmethods.

Any operator OOH in the Heisenberg picture can be transformed into theinteraction picture, see Eq. (6.15):

OOH D OS.t0; t/ OOI OS.t; t0/ : (6.58)

Analogous to the derivation of Eq. (6.27), the OS operator is given by

OS.t; t0/ D Ttfexp

�� i

„Z t

t0

dt 0 OH extI .t 0/

�exp

�� i

„Z t

t0

dt 0 OH intI .t

0/�

g ; (6.59)

where the operators are in the interaction representation. The ordinary time-orderingcan also be written as ordering along contour branches C1 and C2 as depictedin Fig. 6.2:

OS.t; t0/ D TC1fexp

�� i

„ZC1

dt OH extI .t/

�exp

�� i

„ZC1

dt OH intI .t/

�g ;

OS.t0; t/ D TC2fexp

�� i

„ZC2

dt OH extI .t/

�exp

�� i

„ZC2

d� OH intI .t/

�g :

(6.60)


Fig. 6.2 The contourC D C1 [C2 runs on the realaxis, but for clarity its twobranches C1 and C2 areshown slightly away from thereal axis. The contour Ci runsfrom t0 to t0 � iˇ

By combining two contour branches, C D C1 [ C2, Eq. (6.58) can be rewritten as

OOH .t/ D TC f OS extC

OOIg D TC f OS extC

OS intC

OOIg D TC f OSC OOIg ; (6.61)

where,

OSC D exp

�� i

„ZC

dt OH extI .t/

�exp

�� i

„ZC

dt OH intI .t/

�D OS ext

COS intC : (6.62)

In Eq. (6.56), O� describes the equilibrium state of the system before the externalperturbation OH ext is turned on. Interactions OH int, which are switched on adiabati-cally at �1, are present in O�. However, to apply Wick’s theorem in Sect. 6.5.1, onehas to work with non-interacting operators. A methodology similar to the Matsubaratheory can be applied to express the many-particle density operator O� in terms of thesingle-particle density operator O�0, see Sect. 6.2.5. If the contour Ci D Œt0; t0 � iˇ is chosen (Fig. 6.2), then Eq. (6.30) takes the form

e�ˇ OK D e�ˇ OK0 OSCi : (6.63)

Therefore, Eq. (6.56) can be rewritten as

h OOH .t/i D TrŒe�ˇ OK0TCi OSCi OOH .t/

TrŒe�ˇ OK0TCi OSCi : (6.64)

Using Eqs. (6.61) and (6.64), the Green’s function in Eq. (6.57) becomes [40]

G.r; t; r0; t 0/ D � i

„TrŒe�ˇ OK0TCi OSCi TC OSC O I.r; t / O �

I .r0; t 0/

TrŒe�ˇ OK0TCi OSCi TC OSC : (6.65)

The twofold expansion of the density operator and the field operators may con-veniently be combined to a single expansion. The two contours Ci and C canbe combined together, C � D C [ Ci (Fig. 6.3), and a contour-ordering operatorTC� D TCi TC , which orders along C �, can be introduced. Hence, a point onC is always earlier than a point on Ci . Furthermore, we define an interactionrepresentation with respect to OH0 on C and with respect to OK0 on Ci . Therefore,the Green’s function in Eq. (6.57) is given by


Fig. 6.3 The contourC� D Ci [ C , runs from t0to t0 and from t0 to t0 � iˇ

G.r; t; r0; t 0/ D � i

„TrŒe�ˇ OK0TC�f OSC�

O I.r; t / O �I .r

0; t 0/g TrŒe�ˇ OK0TC�

OSC� ;

D � i

„hTC�f OSC�O I.r; t / O �

I .r0; t 0/gi0 ;

(6.66)

where h: : :i0 represents the statistical average with respect to O�0. From here weassume that all statistical averages are with respect to O�0 and drop the 0 from thebrackets h: : :i0.

6.4.3 Keldysh Contour

If one does not consider initial correlations, one can let t0 ! �1. Since we assumethat the Green’s function falls off sufficiently rapidly as a function of the separationof its time arguments, one can neglect the part of the contour Ci extending fromt0 to t0 � iˇ [39]. It has been shown that by explicitly taking the initial correlationsinto account [18,19,22,29], the neglect of this part of the contour corresponds to theneglect of initial correlations. The initial condition, that the system is assumed to bein equilibrium before the external perturbation is turned on, can then be imposeddirectly on the Dyson equation in integral form. This provides an independentdemonstration that, for cases where initial correlations can be neglected, one candiscard the contribution of the contour from t0 to t0 � iˇ. The contours C � and Cbecome identical, as they both start and end at �1. They can be extended beyondthe largest time by considering that the time-evolution operator is a unitary, and onethen obtains the contour CK introduced by Keldysh [25].

6.4.4 Real-Time Formalism

The contour representation presented in Sect. 6.4.2 is rather impractical in calcula-tions, and one prefers to work with real time integrals. The procedure of convertingthe contour into real-time integrals is called analytic continuation [23]. We followedhere the formulation by Langreth [33]. In this section we are only concerned withtemporal variables, therefore, spatial variables have been suppressed. The contourCK depicted in Fig. 6.4 consists of two branches: C1 and C2. Each of the time


Fig. 6.4 Keldysh contourbranches C1 D .�1;1/

and C2 D .1;�1/

arguments of the Green’s function can reside either on the first or second part of thecontour. Therefore, contour-ordered Green’s function thus contains four differentGreen’s functions:

G.t; t 0/ D

8̂ˆ̂̂̂<ˆ̂̂̂:̂

G>.t; t 0/ t 2 C2; t0 2 C1

G<.t; t 0/ t 2 C1; t0 2 C2

Gt.t; t0/ t; t 0 2 C1

GQt.t; t 0/ t; t 0 2 C2

The greater (G>), lesser (G<), time-ordered (Gt), and anti-time-ordered (GQt)Green’s functions can be defined as

G>.t; t 0/ D �i„�1h O H.t/ O �H.t

0/i ;G<.t; t 0/ D Ci„�1h O �

H.t0/ O H.t/i ;

Gt.t; t0/ D �i„�1hTtf O H.t/ O �

H.t0/gi ;

D ��.t � t 0/i„�1h O H.t/ O �H.t

0/i C �.t 0 � t /i„�1h O �H.t

0/ O H.t/i ;D C�.t � t 0/G>.t; t 0/ C �.t 0 � t /G<.t; t 0/ ;

GQt.t; t 0/ D �i„�1hTQtf O H.t/ O �H.t

0/gi ;D ��.t 0 � t /i„�1h O H.t/ O �

H.t0/i C �.t � t 0/i„�1h O �

H.t0/ O H.t/i ;

D C�.t 0 � t /G>.t; t 0/ C �.t � t 0/G<.t; t 0/ ;

(6.67)

where the time-ordering operator Tt is defined in Eq. (6.22). The anti-time-orderingoperator TQt can be defined in a similar manner. SinceGt CGQt D G>CG<, there areonly three linearly independent functions. The freedom of choice reflects itself in theliterature, where a number of different conventions can be found. For our purpose,the most suitable functions are the G?, and the retarded (Gr) and advanced (Ga)Green’s functions defined as

Gr.t; t 0/ D C�.t � t 0/ŒG>.t; t 0/ �G<.t; t 0/ ;

Ga.t; t 0/ D C�.t 0 � t / ŒG<.t; t 0/ �G>.t; t 0/ :(6.68)

It is straightforward to show that Gr �Ga D G> �G<.


Fig. 6.5 Deformation of contour C into contours C1 and C2

6.4.5 Langreth Theorem

As discussed in Sect. 6.5.4, within the Dyson equation one encounters the contourintegrals

D.t; t 0/ DZC

d� A.t; �/B.�; t 0/ ; (6.69)

and their generalizations involving products of three or more terms. The nextstep is replacing contour by real time integrals in the Dyson equation. To evalu-ate Eq. (6.69), one can assume that t is on the first half of the contour and t 0 ison the latter half. In view of the discussion of Eq. (6.67), we are thus analyzing alesser function. The next step is to deform the contour as indicated in Fig. 6.5. ThusEq. (6.69) becomes

D<.t; t 0/ DZC1

d� A.t; �/B<.�; t 0/ CZC2

d� A<.t; �/B<.�; t 0/ : (6.70)

Here, in appending the label < to the function B in the first term, we made use ofthe fact that as long as the integration variable � is confined on the contour C1, it isless than t 0 (in the contour sense). A similar argument applies to the second term.Considering the first term in Eq. (6.70), the integration can be split into two parts:

ZC1

d� A.t; �/B<.�; t 0/ DZ t

�1dt1 A

>.t; t1/B<.t1; t

0/

CZ �1

t

dt1 A<.t; t1/B

<.t1; t0/ ;

�Z 1

�1dt1 A

r.t; t1/B<.t1; t

0/ ;

(6.71)

where the definition of the retarded function Eq. (6.68) has been used. A similaranalysis can be applied to the second term involving contourC2, where the advancedfunction is generated. Putting the two terms together, one gets the first of Langreth’sresults [23]:


D<.t; t 0/ DZ 1

�1dt1 ŒA

r.t; t1/B<.t1; t

0/ C A<.t; t1/Ba.t1; t

0/ : (6.72)

The same result applies for the greater function just by replacing the < labels by the> labels. It is easy to generalize the result Eq. (6.72) to a product of three functions.The retarded and analogously the advanced component of a product of functionsdefined on the contour can be derived by repeated use of the definitions Eqs. (6.67)and (6.68), and the result of Eq. (6.72):

Dr.t; t 0/ D �.t � t 0/ŒD>.t; t 0/ �D<.t; t 0/ ;

D �.t � t 0/Z 1

�1dt1 ŒA

r.B> � B</C .A> � A</Ba ;

D �.t � t 0/Z t

�1dt1.A

> � A</.B> � B</

CZ t 0

�1dt1.A

> � A</.B< � B>/

#;

DZ t

t 0dt1 A

r.t; t1/Br.t1; t

0/ :

(6.73)

As shown in Sect. 6.5.1, in the self-energies another structure occurs:

D.�; � 0/ D A.�; � 0/B.�; � 0/ ; (6.74)

where � and � 0 are contour variables. The derivation of the required formula issimilar to the analysis presented above [23]:

D?.t; t 0/ D A?.t; t 0/B?.t; t 0/ ;

Dr.t; t 0/ D A<.t; t 0/B r.t; t 0/ C Ar.t; t 0/B<.t; t 0/ C Ar.t; t 0/B r.t; t 0/ :(6.75)

The rules provided by the Langreth theorem are summarized in Table 6.1.

6.4.6 Non-interacting Fermions

The non-interacting or free Green’s function is used in the perturbation expansionsdescribed in Sect. 6.5. The Hamiltonian for non-interacting electrons (Fermions) inmomentum representation is

H0 DX

k

Ek Oc�k Ock ; (6.76)


Table 6.1 Rules for analytic continuation derived from the Langreth theorem

Contour Real axis

D DZC

AB D? DZt

ŒArB? C A?Ba

Dr DZt

ArB r

D DZC

ABC D? DZt

ŒArB rC? C ArB?C a C A?BaC a

Dr DZt

ArB rC r

D.�; � 0/ D A.�; � 0/B.�; � 0/ D?.t; t 0/ D A?.t; t 0/B?.t; t 0/

Dr.t; t 0/ D A<.t; t 0/B r.t; t 0/C Ar.t; t 0/B<.t; t 0/

CAr.t; t 0/B r.t; t 0/

where Ek is the single-particle energy measured with respect to the Fermi energy,and Ock and Oc�k are the Fermion annihilation and creation operators, respectively. Thetime-evolution of the annihilation operator in the Heisenberg picture is

Ock.t/ D eiH0t=„ Ocke�iH0t=„ ; (6.77)

so the operator obeys the equation

i„@t Ock.t/ D Œ Ock.t/;H0 D Ek Ock.t/ ; (6.78)

which has the solution

Ock.t/ D e�iEkt=„ Ock : (6.79)

The creation operator for Fermions is the just the Hermitian conjugate of Ock, i.e.,

Oc�k.t/ D eiEkt=„ Oc�k : (6.80)

The non-interacting real-time Green’s functions (Sect. 6.4.4) for Fermions inmomentum representation are now given by

G<0 .k; t I k0; t 0/ � Ci„�1h Oc�k0.t

0/ Ock.t/i0 ;

D Ci„�1e�iEk.t�t 0/=„nkık;k0 ;

G>0 .k; t I k0; t 0/ � �i„�1h Ock.t/ Oc�k0.t

0/i0 ;

D �i„�1e�iEk.t�t 0/=„Œ1 � nk ık;k0 ; (6.81)


Gr0.k; t I k0; t 0/ � �i„�1�.t � t 0/h Ock.t/ Oc�k0.t

0/C Oc�k0.t0/ Ock.t/i0 ;

D �i„�1�.t � t 0/e�iEk.t�t 0/=„ık;k0 ;

Ga0.k; t I k0; t 0/ � Ci„�1�.t 0 � t /h Ock.t/ Oc�k0.t

0/C Oc�k0.t0/ Ock.t/i0 ;

D Ci„�1�.t 0 � t /e�iEk.t0�t/=„ık;k0 ;

where nk D hOc�k Ocki is the average occupation number of the state k. Assuming thatthe particles are in thermal equilibrium, one obtains nk D f .Ek/, where f .E/ is theFermi-Dirac distribution function (Sect. 5.5.2). The Green’s functions depend onlyon time differences. One usually Fourier transforms the time difference coordinate,t � t 0, to energy:

G<0 .k; E/ D C2� if .Ek/ı.E �Ek/ ;

G>0 .k; E/ D C2� iŒ1 � f .Ek/ ı.E �Ek/ ;

Gr0.k; E/ D 1

E �Ek C i�;

Ga0.k; E/ D 1

E �Ek � i�;

(6.82)

where � D 0C is a small positive number. The result Eq. (6.82) shows that G<

and G> provide information about the statistics, such as occupation f .Ek/ orun-occupation 1�f .Ek/ of the states, andGr andGa provide information about thestates regardless of their occupation. The spectral function A0.k; E/ for Fermionsis therefore defined as

A0.k; E/ D CiŒGr0.k; E/ �Ga

0.k; E/ D �2=mŒGr0.k; E/ D C2�ı.E �Ek/ ;

(6.83)

where the following relation is used:

1

x ˙ i�D P

�1

x

�� i�ı.x/ ; (6.84)

where P indicates the principal value. Under equilibrium, the lesser and greaterGreen’s functions can be rewritten as

G<0 .k; E/ D if .E/A0.k; E/ ;

G>0 .k; E/ D iŒ1 � f .E/ A0.k; E/ :

(6.85)


6.4.7 Non-interacting Bosons

The Hamiltonian for non-interacting phonons (Bosons) in momentum representa-tion is

H0 DXq�

„!q�


2

�; (6.86)

where „!q� is the energy of mode q with the polarization �, Obq�, and Ob�q� are theBosons annihilation and creation operators. In a similar way to the case of Fermions,the annihilation and creation operators are given by

Obq�.t/ D e�i!q�t Obq� : (6.87)

Ob�q�.t/ D eCi!q�t Ob�q�: (6.88)

The non-interacting real-time Green’s functions for Bosons in momentum represen-tation are now given by

D<0�.q; t I q0; t 0/ � �i„�1h OA�q0�

.t 0/ OAq�.t/i0 ;

D �i„�1h Ob�q0�.t 0/ Obq�.t/ C Ob�q0�.t

0/ Ob��q�.t/i0 ;

D �i„�1�e�i.!q�t�!q0�t0/h Ob�q0�

Obq�i0C e�i.!q0�t

0�!q�t/h Ob�q0�Ob��q�i0

�ıq;q0 ;

D �i„�1 he�i!q�.t�t 0/nq� C eCi!q�.t�t 0/.nq� C 1/iıq;q0 ;

� D<0�.qI t; t 0/ ;

D>0�.qI t; t 0/ D D<

0�.qI t 0; t / ;

D �i„�1 heCi!q�.t�t 0/nq� C e�i!q�.t�t 0/.nq� C 1/i;

Dr0�.qI t; t 0/ � �i„�1�.t � t 0/h OA�q�.t 0/ OAq�.t/ C OAq�.t/ OA�q�.t 0/i0 ;

D �i„�1�.t � t 0/he�i!q�.t�t 0/ � eCi!q�.t�t 0/

i;

Da0�.qI t; t 0/ D �i„�1�.t 0 � t /

heCi!q�.t�t 0/ � e�i!q�.t�t 0/

i;

(6.89)


where OAq�.t/ D Obq�.t/ C Ob��q�.t/, OA�q�.t/ D OA��q�.t/, !�q� D !q�, and

nq� D h Ob�q� Obq�i are the occupation number of the state .q�/, where under thermalequilibrium one obtains nq� D nB.„!q�/, with nB denoting the Bose-Einsteindistribution function (Sect. 5.5.3). The Green’s functions depend only on timedifferences. One usually Fourier transforms the time difference coordinate, t � t 0, toenergy

D<0�.q; E/ D �2� i

�nq�ı.E � „!q�/C .nq� C 1/ı.E C „!q�/

�;

D>0�.q; E/ D �2� i

�nq�ı.E C „!q�/C .nq� C 1/ı.E � „!q�/

�;

Dr0�.q; E/ D 1

E � „!q� C i�� 1

E C „!q� C i�;

Da0�.q; E/ D 1

E � „!q� � i�� 1

E C „!q� � i�:

(6.90)

6.5 Perturbation Expansion of the Green’s Function

In previous sections, Green’s functions at zero and finite temperatures have beendefined. It was shown that the Green’s functions can be written in terms of the OSoperator:

G.r; t; r0; t 0/ D � i

„hTtf OS O I.r; t / O �I .r

0; t 0/gi ; (6.91)

where OS includes the effects of interactions and external perturbations:

OS D exp

�� i

„Z

dt OH intI .t/

�: (6.92)

It is not possible to give an analytical solution forG.r; t I r0; t 0/, unless the interactionperturbation OH int is set equal to zero. As described in Sect. 6.4.6, this gives the non-interacting Green’s function

G0.r; t I r0; t 0/ D � i

„hTtf O I.r; t / O �I .r

0; t 0/gi ; (6.93)

which is central for any perturbation expansion. This section proceeds with thecalculation of the Green’s function by expanding the OS operator as a series ofproducts of OH int

I in the numerator and the denominator. By expanding the OSoperator, one obtains (see Eq. (6.21))

6.5 Perturbation Expansion of the Green’s Function 129

G.r; t I r0; t 0/

D � i

„

h1XnD0

1

nŠ

��i

„�n Z

dt1 : : :Z

dtn Ttf OH intI .t1/ : : :

OH intI .tn/

O I.r; t / O �I .r

0; t 0/gi

h1XnD0

1

nŠ

��i

„�n Z

dt1 : : :Z

dtn Ttf OH intI .t1/ : : :

OH intI .tn/gi

:

(6.94)

The expansion of the numerator of the Green’s function in Eq. (6.94) can bewritten as

GN D hTt f O I.r; t / O �I .r

0; t 0/gi„ ƒ‚ …G0N

C hTtf� i

„Z

dt1 OH intI .t1/

O I.r; t / O �I .r

0; t 0/gi„ ƒ‚ …

G1N

C : : : ;

(6.95)

where the superscript denotes the order of perturbation. The zero-order perturbationleads to a non-interacting Green’s function G0

N D i„G0. Wick’s theorem allowsus to write each of these brackets in terms of non-interacting Green’s function andthe interaction potential. The same procedure can be applied to the denominator.The terms in the expansion of the denominator, hS.1;�1/i, are called vacuumpolarization terms [17].

6.5.1 Wick’s Theorem

The Wick decomposition allows a perturbation expansion of Green’s functions. Italways holds for zero-temperature Green’s functions and only under the conditionthat field operators must be given in the interaction picture (Sect. 6.2.3). Their timeevolution is governed by the non-interacting Hamiltonian OH0, and OH int is treatedas a perturbation. If these conditions are fulfilled, Wick’s theorem states that theexpectation values of products of field operators is equal to the sum of expectationvalues of all possible pairs of operators and that each of these pairs will be a non-interacting single-particle Green’s function:

hTtf OO1 OO2 : : : OOngi DXPd

hTtf OO1 OO2gi0hTtf OO3 OO4gi0 : : : hTtf OOn�1 OOngi : (6.96)

The sum runs over all Pd distinct permutations of the n indices. It should be notedthat brackets such as Eq. (6.96) vanish if the number of creation and annihilationoperators is not the same. If the number of annihilated particles is not the same asthe number of created particles, then the system will not come back to its ground-state. As a result, the expectation value over the ground-state vanishes. With the


same reasoning, one concludes that if both operators appearing in a bracket areannihilation or creation operators, the expectation value disappears, otherwise oneobtains an expression proportional to the non-interacting Green’s function G0.

The most general proof of this theorem is due to [11], where it is shownrigorously that the theorem holds exactly if the operators to be averaged arenon-interacting, and the density operator, which appears in the finite temperatureformalism (see Eqs. (6.54) and (6.65)), is a single-particle operator. Therefore,one can use the Wick theorem to get a perturbation expansion for the Green’sfunction. The only formal difference from the equilibrium theory is the appearanceof integration over a contour instead of integration over the inverse temperatureinterval for the case of finite temperature or the real axis for the case of zerotemperature. Consequently, the contour-ordered Green’s function is mapped ontoits Feynman diagrams as in equilibrium theory.

Based on the Wick theorem, the following time-ordered product of electron fieldoperators gives only two non-vanishing pairs:

hTtf O I.r1; t1/ O �I .r2; t2/ O I.r3; t3/ O �

I .r4; t4/gi0D Œ ChTtf O I.r1; t1/ O �

I .r2; t2/gi0 hTtf O �I .r3; t3/ O I.r4; t4/gi0

�hTtf O I.r3; t3/ O �I .r2; t2/gi0 hTtf O �

I .r1; t1/ O I.r4; t4/gi0 ;D Œ Ci„G0.r1; t1I r2; t2/ i„G0.r3; t3I r4; t4/

�i„G0.r3; t3I r2; t2/ i„G0.r1; t1I r4; t4/ :

(6.97)

A few simple rules should be noted when making these pairings. The first is that asign change occurs each time the positions of two neighboring Fermi on operatorsare interchanged. An odd number of interchanges is the origin of the minus sign inthe second term of the example above.

The second rule concerns the time-ordering of combinations of operators repre-senting different excitations. For example, a bracket with a mixture of electron andphonon operators can be separated into electron and phonon parts, since electronoperators commute with phonon operators:

hTtf O I.r1; t1/ O �I .r2; t2/ OA.q; t / O I.r3; t3/ O �

I .r4; t4/ OA�.q0; t 0/gi0D hTtf O I.r1; t1/ O �

I .r2; t2/ O I.r3; t3/ O �I .r4; t4/gi0 hTtf OA.q; t / OA�.q0; t 0/gi0 ;

(6.98)

where OA.q; t / D Obq.t/C Ob��q.t/ represents a phonon operator in momentum space.Wick’s theorem can be also applied to brackets of phonon operators. Since phononsare Bosons, the sign does not change when exchanging positions of operators.

The third rule is a method of treating the time-ordering of two operators whichare applied at the same time. The time-ordered product is undefined at equal times.To remove this ambiguity, the following interpretation can be applied:

hTtf O I.r; t / O �I .r

0; t /g D limt 0!tC

hTtf O I.r; t / O �I .r

0; t 0/gi0 : (6.99)


Wick’s theorem has been applied to calculate first-order perturbation expansionsin Sect. 6.5.3. It can be also applied for higher-order perturbations.

6.5.2 Feynman Diagrams

The Wick theorem allows us to evaluate the exact Green’s functions as a perturbationexpansion involving expressions of free Green’s functions G0 and the perturbationpotential V (see Sect. 6.5.1). This expression can be analyzed directly in coordinateor momentum space, in time or energy domain. Different expansion terms achievedfrom the Wick theorem can be translated into Feynman diagrams. Feynmanintroduced the idea of representing different contributions obtained from the Wickdecomposition by drawings. These drawings, called diagrams, are very usefulfor providing insight into the physical process, which these terms represent. TheFeynman diagrams provide an illustrative way to solve the many-particle problemsand the perturbation expansion of the Green’s functions.

A diagram dictionary for electrons, which are Fermions, and phonons, which areBosons, is shown in Table 6.2. Diagrams for electrons are in coordinate-time space,while phonon diagrams are in momentum-energy space. As described in Sect. 6.3.1,the Green’s function can be interpreted as the creation of a particle at .r0; t 0/ inspace-time, and the propagation of the corresponding perturbation to the point .r; t /in space-time, where the particle is annihilated. Hence, the full Green’s function isrepresented by a double line joining these two points. The free Green’s function ischaracterized by a single line.

The Coulomb potential is represented by a wavy line with two inputs and outputswhich can be coupled together to describe a self-interaction. The Coulomb interac-tion is assumed to be instantaneous. It is convenient to consider the inter-particlepotential as a static instantaneous potential proportional to a delta function ıt1;t2 .

Intermediate variables describe events taking place between the two space-timearguments of the Green’s function, but without any constraints for exact time orplace. The overall amplitude involves an integration over these variables. Each timea Fermi on loop appears, the perturbation expression corresponding to this Feynmandiagram must be multiplied by a factor �1.

Electrons can also interact with phonons. For phonons it is more convenientto work in the momentum-energy rather than in the space-time domain. Diagramsconcerning a free phonon Green’s functions and the interaction between electronsand phonons are also shown in Table 6.2. The factor Mq refers to the electron-phonon interaction matrix elements.

6.5.3 First-Order Perturbation Expansion

The electron-electron interaction up to the first-order perturbation is studied here. Inthe interaction representation, the corresponding operator is given by


Table 6.2 Feynman diagrams for electrons (Fermions) and phonons (Bosons)

Expression Description Diagram

i„G.r; t I r0; t 0/ Full Green’s function

i„G0.r; t I r0; t 0/ Free Green’s function

�iV.r1; r2/=„ Coulomb interaction

Zr1

Zt1

Intermediate variable(s)

Factor �1 Any Fermion loop

i„D0.qI!/ Free phonons

�iMq=„ Electron-phonon interaction

OH e-eI .t1/ D 1

2

Zdr1

Zdr2 O �

I .r1; t1/ O �I .r2; t1/V .r1 � r2/ O I.r2; t1/ O I.r1; t1/ ;

(6.100)

where the Coulomb interaction potential is assumed to be an instantaneous potentialproportional to a delta function ıt1;t2 . The first-order term of the perturbationexpansion is given by

G1N D 1

2

��i

„�Z

dt1

Zdr1

Zdr2V .r1 � r2/

� hTtf O �I .r1; t1/ O �

I .r2; t1/ O I.r2; t1/ O I.r1; t1/ O I.r; t / O �I .r

0; t 0/gi„ ƒ‚ …F 1

N

; (6.101)

F 1N D ŒChTtf O I.r1; t1/ O �

I .r1; t1/gihTtf O I.r2; t1/ O �I .r2; t1/gihTtf O I.r; t / O �

I .r0; t 0/gi

�hTtf O I.r1; t1/ O �I .r1; t1/gihTtf O I.r; t / O �

I .r2; t1/gihTtf O I.r2; t1/ O �I .r

0; t 0/giChTtf O I.r2; t1/ O �

I .r1; t1/gihTtf O I.r; t / O �I .r2; t1/gihTtf O I.r1; t1/ O �

I .r0; t 0/gi

�hTtf O I.r2; t1/ O �I .r1; t1/gihTtf O I.r1; t1/ O �

I .r2; t1/gihTtf O I.r; t / O �I .r

0; t 0/giChTtf O I.r; t / O �

I .r1; t1/gihTtf O I.r1; t1/ O �I .r2; t1/gihTtf O I.r2; t1/ O �

I .r0; t 0/gi

�hTtf O I.r; t / O �I .r1; t1/gihTtf O I.r2; t1/ O �

I .r2; t1/gihTtf O I.r1; t1/ O �I .r

0; t 0/gi :(6.102)


a b c

d e f

Fig. 6.6 Feynman diagrams of the first-order perturbation terms G1N

By replacing the brackets by Green’s functions, one gets

G1N D 1

2

��i

„�Z

dt1

Zdr1

Zdr2V .r1 � r2/

� Œ Ci„ G0.r1; t1I r1; t1/ i„ G0.r2; t1I r2; t1/ i„ G0.r; t I r0; t 0/„ ƒ‚ ….a/

�i„ G0.r1; t1I r1; t1/ i„ G0.r; t I r2; t1/ i„ G0.r2; t1I r0; t 0/„ ƒ‚ ….b/

Ci„ G0.r2; t1I r1; t1/ i„ G0.r; t I r2; t1/ i„ G0.r1; t1I r0; t 0/„ ƒ‚ ….c/

�i„ G0.r2; t1I r1; t1/ i„ G0.r1; t1I r2; t1/ i„ G0.r; t I r0; t 0/„ ƒ‚ ….d/

Ci„ G0.r; t I r1; t1/ i„ G0.r1; t1I r2; t1/ i„ G0.r2; t1I r0; t 0/„ ƒ‚ ….e/

�i„ G0.r; t I r1; t1/ i„ G0.r2; t1I r2; t1/ i„ G0.r1; t1I r0; t 0/„ ƒ‚ ….f /

:

(6.103)

Feynman diagrams for the corresponding terms are shown in Fig. 6.6. In thefirst-order example, the connected diagrams .b/ and .f / are equal, as are thediagrams .c/ and .e/; they differ only in that the integration variables r1 and r2 areinterchanged, whereas the Coulomb potential is symmetric under this substitution.It is therefore sufficient to retain just one diagram of each type, simultaneouslyomitting the factor 1=2 in front of Eq. (6.103). For the nth-order perturbation, thereare n! possible interchanges of integration variables. Therefore, the repetition of thesame diagrams cancels the factor 1=nŠ in Eq. (6.94). Diagrams .a/ and .d/ containsub-units that are not connected by any lines to the rest of the diagram. Feynman


diagrams in which all parts are not connected are called disconnected diagrams.Equation (6.103) shows that such diagrams typically have Green’s function andinteractions whose arguments close on themselves. As a result, the contribution ofthis sub-unit can be factored out of the expression for GN. The same procedurecan be applied for the denominator. In this case, the second term of the expansionincludes only two non-vanishing terms, which are only disconnected diagramsof Eq. (6.103), namely, (a) and (d), which cancel the disconnected diagrams ofthe numerator. Thus the resulting Green’s function consists of only connecteddiagrams. It can be shown that in general the vacuum polarization terms cancel thedisconnected diagrams in the expansion of the Green’s function [21]. In other words,the numerator of the Green’s function can be written as Gc.r; t I r0; t 0/hS.1;�1/i,where Gc is the summation of connected diagrams. Finally, the Green’s function isjust the summation of all topologically different connected diagrams [23]

G.r; t I r0; t 0/ D

� i

„h1XnD0

1

nŠ

��i

„�n Z

dt1 : : :Z

dtn T f OH intI .t1/ : : :

OH intI .tn/

O I.r; t / O �I .r

0; t 0/giconn :

(6.104)

6.5.4 Dyson Equation

The Dyson equation can be achieved by classifying the various contributions in arbi-trary Feynman diagrams. Dyson’s equation summarizes the Feynman-Dyson per-turbation theory in a particularly compact form. The exact Green’s function can bewritten as the non-interacting Green’s function plus all connected terms with a non-interacting Green’s function at each end, see Eq. (6.104). This structure is shownin Fig. 6.7, where the double line denotes G and the single line G0. By introducingthe concept of self-energy˙ , the structure in Fig. 6.7 takes the form shown Fig. 6.8.The corresponding analytic expression is given by

G.r; t I r0; t 0/ D G0.r; t I r0; t 0/ CZ

d1Z

d2 G0.r; t I 1/ ˙.12/ G0.2I r0; t 0/ ;(6.105)

where the abbreviation 1 � .r1; t1/ andR

d1 � Rdr1

Rdt1 is used. The self-energy

˙ describes the renormalization of single-particle states due to the interactionwith the surrounding many-particle system and the Dyson equation determines therenormalized Green’s function.

Another important concept is the proper self-energy insertion, which is a self-energy insertion that can not be separated into two pieces by cutting a single-particleline. By definition, the proper self-energy is the sum of all proper self-energyinsertions, and will be denoted by ˙�. Using the perturbation expansion, one can


Fig. 6.7 The Green’s function expanded in terms of connected diagrams

Fig. 6.8 Feynman diagrams showing the general structure of G

ΣH = ΣF =

Fig. 6.9 Feynman diagrams of the first-order proper self-energies

define the proper self-energy ˙� as an irreducible part of the Green’s function.Based on this definition, first-order proper self-energies, which result from the first-order expansion of the Green’s function (see Sect. 6.5.3), are shown in Fig. 6.9.These diagrams are irreducible parts of Fig. 6.6b, c and are referred to as theHartree (˙H) and the Fock (˙F) self-energies. The self-energy can also in principlebe introduced variationally [11]. A variational derivation of the self-energies forthe electron-electron and electron-phonon interactions is presented in Sects. 6.7.1and 6.7.3, respectively. It follows from these definitions that the self-energy consistsof a sum of all possible repetitions of the proper self-energy

˙.r; t I r0; t 0/ D ˙�.r; t I r0; t 0/

CZ

d1Z

d2 ˙�.r; t I 1/ G0.12/ ˙�.2I r0; t 0/ C : : : :

(6.106)


Fig. 6.10 Feynman diagrams representing Dyson’s equation

Correspondingly, the Green’s function in Eq. (6.105) can be rewritten as

G.r; t I r0; t 0/ D G0.r; t I r0; t 0/

CZ

d1Z

d2 G0.r; t I 1/ ˙�.12/ G0.2I r0; t 0/ C : : : ; (6.107)

which can be summed formally to yield the Dyson equation, see Fig. 6.10:


d1Z

d2 G0.r; t I 1/ ˙�.12/ G.2I r0; t 0/ :

(6.108)

The validity of Eq. (6.108) can be verified by iterating the right-hand-side, whichreproduces Eq. (6.107) term by term. The Dyson equation can also be written as


d1Z

d2 G.r; t I 1/ ˙�.12/ G0.2I r0; t 0/ :(6.109)

6.5.5 Electron-Electron Self-Energy

An exact evaluation of the self-energy is possible only for some rather pathologicalcases. For real systems one has to rely on approximation schemes. Hence, a naturalapproach is to retain the single-particle picture and assume that each particle movesin a single-particle potential that comes from its average interaction with all ofthe other particles. Thus as a first-order approximation, one can keep just thefirst-order contribution to the proper self-energy ˙� � ˙1, see Fig. 6.9. Thisapproximation corresponds to summing an infinite class of diagrams containingarbitrary iterations of ˙1. Therefore, as shown in Fig. 6.7, any approximation for˙� generates an infinite-order series for the Green’s function. However, usingnon-interacting Green’s function in self-energies, which is referred to as Bornapproximation, is not fully consistent, since the background particles contributingto the self-energy are treated as non-interacting. In reality, of course, these particlesalso move in an average potential coming from the presence of all the otherparticles. Thus instead of non-interacting Green’s functions (single lines), one hasto use the exact Green’s function (double line) in the proper self-energy, as shown


ΣHSCBA = ΣF

SCBA =

Fig. 6.11 Feynman diagramsof the self-consistentfirst-order properself-energies

in Fig. 6.11. Since the exact Green’s function G both determines and is determinedby the proper self-energy ˙�, this approximation is known as the self-consistentBorn approximation, see Sect. 6.7.6. As discussed in Sect. 6.8.3, the self-consistentapproach preserves conservation laws, for example, the continuity equation holdsvalid. The self-consistent Hartree self-energy due to electron-electron interaction isgiven by [41]

˙e-e.r1; t1/ D �i„Z

dt3

Zdr3 ıt1;t3 V .r1 � r3/G.r3; t3I r3; t3/ ;

D �i„Z

dr3 V .r1 � r3/G.r3; t1I r3; t1/ ;

DZ

dr3 V .r1 � r3/n.r3; t1/ ;

DZ

dr3e2

4��jr1 � r3j%.r3; t1/

�eD �e�.r1/ ;

(6.110)

where %.r; t /=.�e/ D n.r; t / D �i„G.r; t; r; t /, see Sect. 6.8.1. The potential �resulting from the Hartree self-energy is in fact the solution of the Poisson equationwith the charge density %. The Hartree self-energy is instantaneous.

6.5.6 Electron-Phonon Self-Energy

Using Eq. (4.64), the electron-phonon interaction Hamiltonian reads as

OH e-phI D

Zdr O �

I .r/

0@ 1p

V

Xq�

Mq� OAq�eiq�r1A O I.r/ ; (6.111)

where OAq�.t/ D . Obq�.t/ C Ob��q�.t// and Mq� are the electron-phonon interactionmatrix elements. The zero-order perturbation gives the non-interacting Green’sfunction. The first-order term of the perturbation expansion must vanish becausethe factors h Obq�i and h Ob�q�i are zero [35]. Similarly, all the odd terms vanish becausetheir time-ordered bracket for phonons contains an odd number of creation andannihilation operators. Applying the Wick theorem, only the even terms contributeto the perturbation expansion for the electron-phonon interaction:


G1N D hTtf1

2

��i

„�2 Z

dt1

Zdt2 OH e-ph

I .t1/ OH e-phI .t2/ O I.r; t / O �

I .r0; t 0/gi ;

D 1

2

��i

„�2 Z

dt1

Zdt2

Zdr1

Zdr2

� hTtf O �I .r1; t1/ O �

I .r2; t2/ O I.r2; t2/ O I.r1; t1/ O I.r; t / O �I .r

0; t 0/gi„ ƒ‚ …F 1

N

� 1

V

Xq1;q2;�

eiq1�r1eiq2�r2Mq1;�Mq2;�h OAq1;�.t1/OAq2;�.t2/i

„ ƒ‚ …K1

N

;

(6.112)

whereF 1N has been calculated before, see Eq. (6.102). Additionally, h OAq1;�

OAq2;�i D 0

unless q2 D �q1, therefore

K1N D 1

V

Xq1;�

eiq1�.r1�r2/jMq1;�j2 i„D0�.q1; t1; t2/ ; (6.113)

whereD0�.q1; t1; t2/ is the non-interacting phonon Green’s function, see Sect. 6.4.7.Feynman diagrams for this expansion are similar to Fig. 6.6 but one shouldonly replace the Coulomb interactions with non-interacting phonon Green’s func-tions [35]. However, the contributions of the diagrams (a), (b), and (f) are zero. Theyare non-zero only if the phonon wave vector q is zero, but such a phonon is eithera translation of the crystal or a permanent strain, and neither of these are meant tobe in the Hamiltonian. The analytical expression regarding the contribution of theself-consistent Fock self-energy is given by [30]

˙e-ph.r1; t1I r2; t2/ D i„Xq�

eiq�.r1�r2/jMq�j2G.r1; t1I r2; t2/D�.qI t1; t2/ :

(6.114)

6.6 Quantum Kinetic Equations

In this section the equations of motion (in real time) for the NEGF are introduced.There are two different but equivalent formulations: the Kadanoff-Baym and theKeldysh formulation. These are treated in the following subsections. Finally, kineticequations under steady-state condition are presented.

6.6 Quantum Kinetic Equations 139

6.6.1 The Kadanoff-Baym Formulation

The starting point of the derivation is the differential form of the Dyson equation.By assuming that Œi„@t1 � OH0.1/ G0.12/ D ı1;2, Eqs. (6.108) and (6.109) can berewritten as [11]

hCi„@t1 � OH0.1/

iG.12/ D ı1;2 � i„

ZC

d3 ˙.13/G.32/ ;

h�i„@t2 � OH0.2/

iG.12/ D ı1;2 � i„

ZC

d3 G.13/ ˙.32/ :(6.115)

The singular part of the self-energy on the contour, which corresponds to the Hartreeself-energy, does not appear explicitly in the kinetic equations, but is included inthe potential energy of the single-particle Hamiltonian OH0, see Eq. (6.135). UsingTable 6.1 and fixing the time arguments of the Green’s functions in Eq. (6.115) atopposite sides of the contour, the Kadanoff-Baym equations read [11, 24]

hCi„@t1 � OH0.1/

iG?.12/ D

Zd3 ˙ r.13/G?.32/C

Zd3˙?.13/Ga.32/ ;

h�i„@t2 � OH0.2/

iG?.12/ D

Zd3 Gr.13/˙?.32/C

Zd3 G?.13/˙ a.32/ :

(6.116)

One should note that the delta-function term in Eq. (6.115) vanishes identicallybecause the time-labels required in the construction of G< and G> are on differentbranches of the contour. The Kadanoff-Baym equations determine the time evolu-tion of the Green’s functions but they do not determine the consistent initial values.This information is contained in the original Dyson equations (6.108) and lost in thederivation. To have a closed set of equations, the Kadanoff-Baym equations mustbe supplemented with Dyson equations for Gr and Ga. By subtracting the equationsin Eq. (6.116) from each other, the equations satisfied by Gr are obtained [11]

hCi„@t1 � OH0.1/

iGr.12/ �

Zd3 ˙ r.13/Gr.32/ D ı1;2 ;

h�i„@t2 � OH0.2/

iGr.12/ �

Zd3 ˙ r.13/Gr.32/ D ı1;2 :

(6.117)

Similar relations hold for the advanced Green’s functions.

6.6.2 Keldysh Formulation

For certain applications in classical transport theory, it is advantageous to writethe Boltzmann equation as an integral equation, rather than an integro-differential


equation. An analogous situation holds in quantum kinetics. Instead of workingwith the Kadanoff-Baym relations in Eq. (6.116), it may be useful to considertheir integral forms. Historically, Keldysh [25] derived his alternative form almostsimultaneously and independently of Kadanoff and Baym. However, the Keldyshand Kadanoff-Baym formalisms are equivalent.

By applying Langreth’s rules to the Dyson equation, see Eq. (6.108), one obtains

G< D G<0 C Gr

0 ˙r G< C Gr

0 ˙< Ga C G<

0 ˙a Ga : (6.118)

For convenience, a notation where a product of two terms is interpreted as a matrixproduct in the internal variables (space, time, etc.) has been used. One can proceedby iteration with respect to G<. By iterating once and regrouping the terms, oneobtains

G< D �1 C Gr

0 ˙r�G<0 .1 C ˙ a Ga/

C �Gr0 C Gr

0 ˙r Gr

0

�˙< Ga

C Gr0 ˙

r Gr0 ˙

r G< :

(6.119)

The form of Eq. (6.119) suggests that infinite order iterations result in [23]

G< D .1 C Gr ˙ r/G<0 .1 C ˙ a Ga/ C Gr ˙< Ga : (6.120)

Equation (6.120) is equivalent to Keldysh’s results. In the original work, however, itwas written for another function, GK � G< CG>. This difference is only of minorsignificance [23].

The first term on the right hand-side of Eq. (6.120) accounts for the initialconditions. One can show that this term vanishes for steady-state systems, ifthe system is in a non-interacting state in the infinite past [23]. Thus in manyapplications it is sufficient to only keep the second term. Similar steps can befollowed to obtain the kinetic equation for G >. In integral form, these equationscan be written as

G?.12/ DZ

d3Z

d4 Gr.13/ ˙?.34/ Ga.42/ : (6.121)

The relation between the Keldysh equation and the Kadanoff-Baym equation isanalogous to the relation between an ordinary differential equation plus a boundarycondition and the corresponding integral equation.

6.6.3 Steady-State Kinetic Equations

Under steady-state condition, the Green’s functions depend on time differences. Oneusually Fourier transforms the time difference coordinate, � D t � t 0, to energy

6.6 Quantum Kinetic Equations 141

G.r1; r2IE/ DZ

d�

„ eiE�=„G.r1; r2I �/ : (6.122)

Under steady-state condition, Eqs. (6.117) and (6.121) can be written as [12]

hE � OH0.r1/

iGr.r1; r2IE/ �

Zdr3 ˙ r.r1; r3IE/ Gr.r3; r2IE/ D ır1;r2 ;

(6.123)

G7.r1; r2IE/ DZ

dr3

Zdr4Gr.r1; r3IE/˙7.r3; r4IE/Ga.r4; r2IE/ ;

(6.124)

where ˙ is the total self-energy. A similar transformation can be applied to self-energies. However, to obtain self-energies one has to first apply Langreth’s rules andthen Fourier transform the time difference coordinate to energy. The evaluation ofthe Hartree self-energy due to electron-electron interaction is straightforward, sinceit only includes the electron Green’s function. However, the lowest-order self-energydue to electron-phonon interaction contains the products of the electron and phononGreen’s functions. Using Langreth’s rules (Table 6.1) and then Fourier transformingthe self-energies due to electron-phonon interaction, Eq. (6.114) takes the form

˙?e-ph.r1; r2IE/ D i

V

Xq�

ZdE 0

2�eiq�.r1�r2/ jMq�j2G?.r1; r2IE �E 0/D?

� .q; E0/ :

(6.125)

To calculate the retarded self-energy, however, it is more straightforward to Fouriertransform the relation ˙ r.�/ D �.�/Œ˙>.�/ �˙<.�/ , see Eq. (6.68). By definingthe broadening function � ,

� .r1; r2IE/ D iŒ˙>.r1; r2IE/ � ˙<.r1; r2IE/ D 2=mŒ˙<.r1; r2IE/ ;(6.126)

the retarded self-energy is given by the convolution of �i� .E/ and the Fouriertransform of the step function [13]

˙ r.E/ D �i� .E/˝�ı.E/

2C i

2�E

�; (6.127)

where ˝ denotes the convolution. The retarded self-energy is given by [31]

˙ r.r1; r2IE/ D � i

2� .r1; r2IE/C P

ZdE 0

2�

� .r1; r2IE 0/E �E 0 ; (6.128)

where P stands for principal part.


6.7 Variational Derivation of Self-Energies

In addition to the Feynman diagrams and the Wick’s decomposition, one can obtainequations of motion for the NEGF by taking the time evolution of the Green’sfunction under the action of the time-independent Hamiltonian OH D OH0 C OH int

and the time-dependent external perturbation OH ext. The latter is included throughthe evolution operator OS ext

C ,

G.12/ D � i

„hTC f OS extC

O H.1/ O �H.2/gi0 ; (6.129)

where the abbreviation 1 � .r1; t1/ is used. To obtain the equation of motion, onecan take the derivative of the Green’s function with respect to time:

i„@tG.12/ D ıt1;t2h O H.1/ O �H.2/C O �

H.2/O H.1/i0

ChTC f OS extC Œ

O H.1/; OH � O �H.2/gi0

ChTC f OS extC U.1/

O H.1/ O �H.2/gi0 ;

(6.130)

The first contribution results from @t�.t1; t2/. Due to the anti-commutation relationof the field operators, it can be reduced to ı1;2 D ır1;r2ıt1;t2 . The equation of motionfor the field operator, i„@t1 O H.1/ D Œ O H.1/; OH �, has been employed in the secondterm, and the third contribution results from @t OS ext

C . Inserting the commutator withthe Hamiltonian, one obtains

hi„@t1 � OH0.1/

iG.12/ D ı1;2 � i„

ZC

d3 V.1 � 3/G.1323/; (6.131)

where the two-particle Green’s function G.1234/ is defined by

G.1234/ D�

� i

„�2

hTC f OS extC

O H.1/ O H.2/ O �H.4/

O �H.3/gi0 : (6.132)

To evaluate the two-particle Green’s functions, one can either write a new equationof motion, which will be coupled with a three-particle Green’s function, and inturn coupled with a four-particle Green’s function, leading to infinite hierarchy, oralternatively, one can express it as products of single-particle Green’s functions,yielding an infinite perturbation expansion [4,11,34,41]. This can be accomplishedby utilizing the Green’s functions as generating functional. The two-particle Green’sfunction can be expressed by means of functional derivatives of the single-particleGreen’s functions with respect to the external potential. Based on the variationalmethod, the electron-electron and electron-phonon self-energies are derived next.

6.7 Variational Derivation of Self-Energies 143

6.7.1 Electron-Electron Interaction

By taking the functional derivative of Eq. (6.132) with respect to U , one obtains

ıG.12/

ıU.3/D � i

„1

hTC OS extC i0

hTC f ıOS extC

ıU.3/O H.1/ O �

H.2/gi0

C i

„hTC f OS ext

CO H.1/ O �

H.2/gi0hTC OS ext

C i02hTC ı

OS extC

ıU.3/i0 ;

D ��

� i

„�2 hTC f OS ext

CO .1/ O H.3/ O �

H.3/O �

H.2/gi0hTC OS ext

C i0

CG.12/�

� i

„� hTC f OS ext

CO H.3/ O �

H.3/gi0hTC OS ext

C i0;

D �G.12102/CG.12/G.33/ :

(6.133)

Equation (6.133) relates the two-particle Green’s function to the functional deriva-tive of the single-particle Green’s function, which allows one to write the equationof motion Eq. (6.131) as

�i„@t1 C „2

2mr 21 � Ueff.1/

�G.12/ D ı1;2 C i„

ZC

d3 V.1 � 3/ ıG.12/ıU.3/

;

(6.134)

where H0.1/ D � „22m

r 21 C U.1/ and the effective potential is given by

Ueff.1/ D U.1/ � i„ZC

d2 V.1 � 2/G.22/ : (6.135)

Since �i„G.22/ is nothing but the electron density, the second term in Eq. (6.135)can be easily identified as the Hartree potential. Exchange and correlation effectsare described by the functional derivative contribution, which still requires thecalculation of a two-particle Green’s function. In order to decouple the hierarchyformally, one can introduce the single-particle self-energy. This is accomplished bythe identity

G.12/ DZC

d4ZC

d5 G.14/G�1.45/G.52/ : (6.136)

Differentiating with respect to U , one obtains

ıG.12/

ıU.3/DZC

d4ZC

d5G.14/ıG�1.45/ıU.3/

G.52/ ; (6.137)


where the following relation is used:

ı

ıUeff.4/

ZC

d3G.13/G�1.32/ D ı1;2

U.4/D 0 ;

DZC

d3G.13/

U.4/G�1.32/C

ZC

d3 G.13/G.32/

U.4/:

(6.138)

This allows us to express the functional derivative of G by means of the functionalderivative ofG�1. Therefore, the equations of motion can be cast into a closed form:

�i„@t1 C „2

2mr 21 � Ueff.1/

�G.12/ �

ZC

d3 G.13/ ˙.32/ D ı1;2 ; (6.139)

where the self-energy is defined as

˙.12/ D �i„ZC

d3ZC

d4 V.1 � 3/ G.14/ ıG�1.42/ıU.3/

: (6.140)

6.7.2 Screened Interaction, Polarization, and Vertex Function

Equation (6.140) can be employed as a starting point for a diagrammatic expansion.One possible way is to iterate G.12/ in the functional derivative with respect toU.3/, starting from the non-interacting Green’s function G0. This procedure isdescribed, e.g., in [24], and specifically for the Keldysh formalism in [11]. Thisexpansion scheme is based on the non-interacting Green’s function. In order to avoidthe appearance of non-interacting Green’s functions in the diagrammatic expansionwithout simultaneously complicating the rules for constructing the diagrams, onehas to extend the equations for G.12/. Technically, this extension is based on therepeated change of variables and the consequent application of the chain-rule inthe evaluation of the functional derivatives. One usually generates the followingadditional functions:

• The self-energy ˙.12/, which contains information on both the renormalizationof the single-particle energies and the scattering rates.

• The longitudinal polarization function ˘.21/, which describes the possiblesingle-particle transitions as a result of a longitudinal electric field (which caneither be an external field or the result of charge density fluctuations in thesystem).

• The screened Coulomb potential W.12/, which differs from the bare Coulombpotential because of the possibility of single-particle transitions as described


by ˘ , brought about by charge density fluctuations, and because of the relatedpossibility of collective excitations.

• The vertex function � .123/, which serves to formally complete the set ofequations.

Although the expanded set of functions still does not lead to a closed set of equations(an additional function, ı˙=ıG, occurs), it allows for a perturbative solution bymeans of iterating ˙ in the derivative ı˙=ıG. The formal structure of theseequations will turn out to be essentially

˙ D WG� ;

˘ D G� G ;

W D V C V ˘W ;

� D 1C ı˙

ıGG� G :

(6.141)

By applying the chain rule for functional derivatives, one can introduce thederivative with respect to the effective potential. This allows one to write the self-energy Eq. (6.140) as [11]

˙.12/ D �i„ZC

d3ZC

d4 V.1 � 3/G.14/ıG�1.42/ıU.3/

;

D �i„ZC

d3ZC

d4ZC

d5 V.1 � 3/G.14/ıG�1.42/

ıUeff.5/

Ueff.5/

U.3/;

D �i„ZC

d3ZC

d4 W.51/G.14/ � .425/ ;

(6.142)

where the screened interaction is defined as

W.12/ DZC

d3 V.2 � 3/ıUeff.1/

ıU.3/; (6.143)

and the vertex function as

� .123/ D ıG�1.12/ıUeff.3/

: (6.144)

It should be noted that screening is defined by the inverse dielectric function. Anexternal potential induces a charge density in the system. This induced chargedensity gives rise to a change in the potential via the Coulomb interaction, which inturn yields an induced charge density and so forth. The result of this infinite seriesof charge redistribution process is the screening of the external potential.


��1.12/ D ıUeff.1/

ıU.2/; (6.145)

can be written in terms of the polarization function,

˘.12/ D �i„ ıG.11/ıUeff.2/

; (6.146)

in the following way:

ıUeff.1/

ıU.2/D ıU.1/

ıU.2/� i„

ZC

d3ZC

d4 V.1 � 3/ ıG.33/ıUeff.4/

ıUeff.4/

ıU.2/;

D ı1;2 CZC

d3ZC

d4 V.1 � 3/˘.34/ ıUeff.4/

ıU.2/:

(6.147)

As such, one obtains

��1.12/ D ı1;2 CZC

d3ZC

d4 V.1 � 3/˘.34/��1.42/ ; (6.148)

and from Eq. (6.143)

W.12/ D V.2 � 1/CZC

d3ZC

d4 V.1 � 3/˘.34/W.42/ : (6.149)

By using the relation Eq. (6.138), one can express the polarization in terms of thevertex function

˘.12/ D �i„ ıG.11/ıUeff.2/

;

D �i„ZC

d3ZC

d4 G.13/ıG�1.34/ıUeff.2/

G.41/ ;

D �i„ZC

d3ZC

d4 G.13/� .342/G.41/ :

(6.150)

The system of equations defining the self-energy is closed by the equation forthe vertex functions. For that purpose one needs an explicit expression for G�1in terms of G. One can multiply and integrate both sides of the equation ofmotion Eq. (6.139) by G�1

0 .32/ and G�1.32/, where G�10 .12/ D .i„@t1 C „2

2mr 21 �

Ueff.1//ı1;2. Finally, one obtains G�1.12/ D G�10 .12/ �˙.12/, which can be used

to rewrite the vertex function Eq. (6.144) as


� .123/ D ıG�1.12/ıUeff.3/

D ıG�10 .12/

ıUeff.3/� ı˙.12/

ıUeff.3/;

D �ı1;2ı1;3 �ZC

d4ZC

d5ı˙.12/

ıG.45/

ıG.45/

ıUeff.3/;

D �ı1;2ı1;3 CZC

d4ZC

d5ı˙.12/

ıG.45/

ZC

d6ZC

d7 G.46/ıG�1.67/ıUeff.3/

G.75/ ;

D �ı1;2ı1;3 CZC

d4ZC

d5ZC

d6ZC

d7ı˙.12/

ıG.45/G.46/� .673/G.75/ ;

(6.151)

where the relation Eq. (6.138) is employed. Contributions proportional to ı˙=ıGare referred to as vertex corrections and describe interaction processes at the two-particle level.

6.7.3 Electron-Phonon Interaction

The coupling of electrons and nuclei in the lattice to external sources is given by

OH ext DZdr U.r; t / .�el.r/C �n.r// C J.r; t /�n.r/ ; (6.152)

where h�el.r/i D ie„G<.r; t I r; t / is the density of electrons. The density of thenuclei in the lattice h�n.r/i is represented as a sum of local charge densities �

�n.r; t / DXR0i

�.r � Ri / ; (6.153)

where the actual positions of the nuclei, Ri D R0i C u.R0

i /, are defined in termsof the equilibrium lattice vector R0

i and the lattice displacement u.R0i /. In case of

bare nuclei, � would approximately be ı functions. However, it is more convenientto consider rigid ion cores instead of bare nuclei. In this case, � denotes the chargedensity of the ion cores.

For a simple derivation of the electron-phonon interaction, one has to add anadditional external source J.r; t / in Eq. (6.152) [11], which couples to the chargedensity of the nuclei and is merely a mathematical trick, see Eqs. (6.160)–(6.163).With similar steps for deriving Eq. (6.135), one can show that under the Hamiltonianin Eq. (6.152), the effective potential can be written as


Ueff.1/ D U.1/ � i„ZC

d2 V.1 � 2/ G.22/CZC

d2 V.1 � 2/h�n.2/i ;

D U.1/CZC

d2 V.1 � 2/ .h�el.2/i C h�n.2/i/ :(6.154)

The aim is the calculation of the total linear response of the system, including thecontribution from the nuclei, i.e., the variation of the total electrostatic potential withthe external potential [41]:

ıUeff.1/

ıU.2/D ıU.1/

ıU.2/� i„

ZC

d3 V.1 � 3/�ıh�el.3/iıU.2/

C ıh�n.3/iıU.2/

�;

D ı1;2 CZC

d3ZC

d4 V.1 � 3/ ıh�el.3/iıUeff.4/

ıUeff.4/

ıU.2/CZC

d3 V.1 � 3/ıh�n.3/iıU.2/

:

(6.155)

Solving with respect to ıUeff=ıU , one obtains

ıUeff.1/

ıU.2/D ��1.12/C

ZC

d3 W.13/ıh�n.3/iıU.2/

; (6.156)

where the dielectric function is

�.12/ D ı1;2 CZC

d3 V.1 � 3/ ıh�el.3/iıUeff.2/

; (6.157)

and W is the screened interaction. The derivative ıh�eli=ıUeff differs from thepurely electronic polarization, which is introduced in Sect. 6.7.2, owing to thephonon contribution to the total potential. Neglecting this phonon contributionto the polarization function is one of the ingredients of the adiabatic approxi-mation [38]. This approximation reduces the dielectric function and the screenedinteraction in Eq. (6.156) to the purely electronic quantities, which are introducedin Eqs. (6.148) and (6.149). The next step is the calculation of the lattice contribu-tion. Similar to the Green’s function for electrons, one can consider the expectationvalue of the density of nuclei

h�n.1/i D hTC f OS extC �n.1/gi

hTC OS extC i : (6.158)

With similar steps for deriving Eq. (6.133), the density response of the nuclei underthe action of OH ext can be calculated as


ıh�n.1/iıU.2/

D � i

„hTC f OS ext

C Œ�el.2/C �n.2/ �n.1/gihTC OS ext

C i

C i

„hTC f OS ext

C �n.1/gihTC f OS extC Œ�el.2/C �n.2/ gi

hTC OS extC i2

;

D � i

„hTC f OS ext

C Œ��el.2/C��n.2/ ��n.1/gihTC OS ext

C i :

(6.159)

In the last step, the deviation operator �� D � � h�i is introduced. Furthermore,the relation hABi � hAihBi D h.A � hAi/.B � hBi/i is used. Now the additionalexternal field J comes into play, which allows us to eliminate the mixed electron-nuclei contribution. By steps completely analogous to those used before, one finds

ıh�el.1/iıJ.2/

D � i

„hTC f OS ext

C ��el.1/��n.2/gihTC OS ext

C i ; (6.160)

which together with Eq. (6.159), yields the result

ıh�n.1/iıU.2/

D ıh�el.1/C �n.1/iıJ.2/

;

D D.12/C ıh�el.1/iıJ.2/

;

(6.161)

where the density-density correlation function of the nuclei is defined as

D.12/ D � i

„hTC f OS ext

C ��n.2/��n.1/gihT extC

OSC i ;

D ıh�n.1/iıJ.2/

:

(6.162)

One can again apply the chain rule to Eq. (6.161) to eliminate the ıh�eli=ıJcontribution:

ıh�n.1/iıU.2/

D D.12/CZC

d3ZC

d4ıh�el.2/iıUeff.3/

V .3�4/ıh�el.4/C �n.4/iıJ.1/

: (6.163)

Making use of the relation Eq. (6.161) once more, one can solve the resultingequation with respect to ıh�ni=ıU and express the solution in terms of the dielectricfunction. After insertion in Eq. (6.156), this yields the total dielectric screeningfunction as

ıUeff.1/

ıU.2/D ��1.12/C

ZC

d3ZC

d4 W.13/D.34/��1.42/ : (6.164)


The desired effective electron-electron interaction induced by lattice vibrations isthus finally given by [41]

Weff.12/ D W.12/CZC

d3ZC

d4 W.13/D.34/W.42/„ ƒ‚ …

Wph

: (6.165)

Therefore, the problem of electron-phonon interaction is reduced to the replacementof the electronically screened interaction by the effective interaction.

6.7.4 The Phonon Green’s Function

The density-density correlation function of the nuclei is reduced in the following toa quantity of more practical interest, namely, the phonon Green’s function within theharmonic approximation. One can expand Eq. (6.153) up to first-order in the latticedisplacement u˛.R0

i / with respect to the equilibrium positions of ions [41]

�n.r; t / DXR0i

�.r � R0i /C

X˛R0i

r˛�.r � R0i /u˛.R

0i ; t / ; (6.166)

where ˛ denotes the Cartesian components u D P˛ e˛u˛ with e˛ representing the

unit vectors of the ˛th Cartesian coordinate. This expansion reduces Eq. (6.162) to

D˛ˇ.r; t I r0; t 0/ DXR0i R0j

r˛�.r � R0i /D˛ˇ.R0

i ; t I R0j ; t

0/rˇ�.r0 � R0j / ; (6.167)

where the phonon Green’s function in real space is

D˛ˇ.R0i ; t I R0

j ; t0/ D � i

„hTC fu˛.R0i ; t /uˇ.R

0j ; t

0/gi : (6.168)

By means of Eq. (4.28), the Fourier transformation of Eq. (6.168) is given by

D˛ˇ.q; t I q0; t 0/ D � i

„hTC fOu˛.q; t /Ou�ˇ.q0; t 0/giıq;q0 : (6.169)

By diagonalizing the dynamical matrix, one obtains the eigenvectors "q� andeigenfrequencies !q� of the lattice vibrations, see Eqs. (4.30) and (4.36). Thiseigenvector expansion allows one to write Eq. (6.168) for each phonon mode as

D�.qI t; t 0/ D � i

„hTC f OAq�.t/ OA�q�.t 0/gi ; (6.170)


where OAq�.t/ D Obq�.t/ C Ob��q�.t/. This factorization allows one to evaluate thecoupling for any combination of phonon branch indices.

6.7.5 The Phonon Self-Energy

In the previous sections, helpful forms of the electron-phonon interaction arederived. One can link up with the many-particle theory and introduce phononcontributions into the quantum kinetic equations. As shown in the derivation ofthe general result Eq. (6.165), one has to add the phonon induced contribution tothe electronically screened interaction. Together with Eq. (6.142), this defines thephonon self-energies, which enter the quantum kinetic equations. As the phononinduced interaction is not a functional of the single-particle Green’s function, theproblem is slightly less complicated than the electron-electron interaction. As in thepurely electronic case, one is dependent on approximation schemes. An expansionof the self-energy in powers of the phonon-induced interaction is easily generatedby means of Eq. (6.142). The contribution linear in the phonon induced interaction,i.e., the single-phonon self-energy, takes the form

˙1.12/ D i„Wph.21/G.12/ : (6.171)

By performing eigenfunction expansion (see Sect. 6.7.4), one obtains

˙1.kI t1; t2/ D �i„X

q

Wph.qI t1; t2/G.k C qI t1; t2/ ; (6.172)

Wph.qI t1; t2/ DX�

jM�.q/j2D�.qI t1; t2/ ; (6.173)

where M�.q/ are the coupling constants for the different phonon branches.

6.7.6 Approximation of the Self-Energy

Depending on the problem, one can either attempt a summation of a selected classof dominant contributions or perform an expansion with respect to the screenedinteraction. As shown in Sect. 6.7.2, the self-energy itself and the polarizationpropagator, which determines the screened interaction, depend sensitively onthe vertex corrections. Thus there is a complicated functional dependence ofsingle-particle properties on two-particle properties and vice versa, and one has toconsider the consistency of approximations at the single-particle and two-particlelevels. In principle, there are two different approaches to handle the coupled


system of equations derived in Sect. 6.7.2: an iterative procedure and self-consistentapproximations.

Starting such an iterative solution, one can first neglect vertex corrections inEq. (6.151) and obtain an approximation for the self-energy by means of Eq. (6.142)together with Eqs. (6.149) and (6.150). Making use of this approximation, onecalculates ı˙=ıG, and includes vertex corrections in the next step. The iterationof such a procedure generates an expansion in terms of the screened interaction andthe Green’s function defined as a self-consistent solution of the Dyson equation.

For the iterative procedure, the sequence of steps can be defined by the vertexfunction Eq. (6.151), which yields by means of the chain rule the recurrence relation

�nC1.123/ D �ı1;2ı1;3 � ı˙n.12/

ıUeff.3/: (6.174)

One starts with the Hartree-approximation, i.e., ˙0 D 0, which delivers G0, �1 D�ı1;2ı1;3 and the screened interaction W1. In the subsequent step, one obtains ˙1,G1 and �2 and so forth. The effect of this interaction is two-fold. In the n-th step,the Green’s functions contributing to ˙nŒGn�1 become dressed by an additionalinteraction line and additionally new types of diagrams are generated.

For the Self-consistent approximations, one selects a certain class of self-energydiagrams ˙ŒG . The Dyson equation becomes a non-linear functional equationof the Green’s functions, which has to be solved self-consistently. The selectioncorresponds to the summation of a certain class of diagrams up to infinite-order inthe interaction, whereas others, which contribute even in lower order, are neglected.The difficulty is in finding the correct way to choose a subset of diagrams for eachorder. In order to deliver physically meaningful results, any approximation shouldguarantee certain macroscopic conservation laws. This condition can be imposedby the postulate that all diagrams contributing to the self-energy are obtained fromthe functional derivative of a functional ˚ŒG with respect to G. Solving the Dysonequation self-consistently with a ˚-derivable self-energy yields a Green’s function,which conserves particle number, energy, and momentum [24].

6.8 Relation to Observables

Observables, such as particle and current densities, are directly linked to the greaterand lesser Green’s functions. In this section, some of the most important observablesand their relations to the Green’s functions are discussed.

6.8.1 Electron and Hole Density

The electron and hole concentration are respectively given by

6.8 Relation to Observables 153

n.r; t / D h O �.r; t / O .r; t /i D �i„G<.r; t I r; t / ; (6.175)

p.r; t / D h O .r; t / O �.r; t /i D Ci„G>.r; t I r; t / : (6.176)

Under steady-state condition, see Sect. 6.6.3, these relations read [12]

n.r/ D �iZ

dE

2�G<.r; E/ ; (6.177)

p.r/ D CiZ

dE

2�G>.r; E/ : (6.178)

The total space charge density is therefore given by

%.r/ D e .p.r/ � n.r// : (6.179)

6.8.2 Spectral Function and Local Density of States

The spectral function is defined as

A.r; r0IE/ D i�Gr.r; r0IE/ �Ga.r; r0IE/� : (6.180)

The spectral function provides information about the nature of the allowed elec-tronic states, regardless of whether they are occupied or not, and can be consideredas a generalized density of states. The diagonal elements of the spectral functiongive the local density of states

�.rIE/ D 1

2�A.r; rIE/ D � 1

�=m ŒGr.r; rIE/ : (6.181)

The trace of the spectral function represents the density of states

N.E/ D Tr ŒA.E/ DZ

drA.r; rIE/ : (6.182)

6.8.3 Current Density

To derive an equation for the current density, one uses the conservation law ofquantum mechanical variables [24]. The starting point is the subtraction of theequations in Eq. (6.116) from each other:


�i„ .@t1 C @t2/C „2

2mŒ.r r1 C r r2 /.r r1 � r r2 / � ŒU.1/ � U.2/

�G<.12/ D

Zd3 Œ˙ r.13/G<.32/C˙<.13/Ga.32/CGr.13/˙<.32/CG<.13/˙ a.32/ ;

(6.183)

where H0.1/ D �„2=2mr 21 C U.1/ has been assumed. By taking the limit 1 !

2 (r2 ! r1 and t2 ! t1) and assuming that the right-hand-side of Eq. (6.183)approaches zero in this limit, one obtains

i„ limt2!t1

Œ@t1G<.12/C @t2G

<.12/ C r �� „22m

limr2!r1

.r r1 � r r2G<.12/

�D 0 :

(6.184)

By multiplying both sides by �e and recalling the definition of the charge density,one recovers the continuity equation

@t1%.r1; t1/C r � J.r1; t1/ D 0 ; (6.185)

where the current density is defined as

J.r1; t1/ D � i„2e2m

limr2!r1

.r r1 � r r2 / G<.r1; t1I r2; t1/ : (6.186)

Under steady-state condition, the current density takes the form [12]

J.r1/ D � i„e

2m

ZdE

2�lim

r2!r1.r r1 � r r2 / G

<.r1; r2; E/ : (6.187)

The current is conserved as long as the right-hand-side of Eq. (6.183) approacheszero as 2 ! 1:

lim2!1

Zd3�˙ r.13/G<.32/C˙<.13/Ga.32/CGr.13/˙<.32/CG<.13/˙ a.32/

� D 0 :(6.188)

This relation holds if there is no interaction, whereas the situation is different inthe interacting case. As described in Sect. 6.5.5, the interactions are described interms of appropriate self-energies. However, self-energies can often be obtainedonly approximately. Therefore, one could choose an approximation which violatesthe continuity equation, which is not physical. It is straightforward to show thatthe approximated self-energy due to electron-phonon interaction within the self-consistent Born approximation Eq. (6.114) preserves the current continuity.

References 155

References

1. Anantram, M.P., Lundstrom, M.S., Nikonov, D.E.: Modeling of nanoscale devices. Proc. IEEE96(9), 1511–1550 (2008)

2. Bandy, W.R., Glick, A.J.: Tight-binding Green’s-function calculation of electron tunneling. I.One-dimensional two-band model. Phys. Rev. B 13(8), 3368–3380 (1976)

3. Bandy, W.R., Glick, A.J.: Tight-binding Green’s-function calculations of electron tunneling.II. Diagonal disorder in the one-dimensional two-band model. Phys. Rev. B 16(6), 2346–2349(1977)

4. Binder, R., Koch, S.W.: Nonequilibrium semiconductor dynamics. Prog. Quantum Electron.19(4–5), 307–462 (1995)

5. Bruus, H., Flensberg, K.: Many-Body Quantum Theory in Condensed Matter Physics: AnIntroduction. Oxford University Press, Oxford (2004)

6. Caroli, C., Combescot, R., Lederer, D., Nozieres, P., Saint-James, D.: A direct calculation ofthe tunnelling current. II. Free electron description. J. Phys. C: Solid State Phys. 4(16), 2598–2610 (1971)

7. Caroli, C., Combescot, R., Nozieres, P., Saint-James, D.: Direct calculation of the tunnelingcurrent. J. Phys. C: Solid State Phys. 4(8), 916–929 (1971)

8. Caroli, C., Combescot, R., Nozieres, P., Saint-James, D.: A direct calculation of the tunnellingcurrent: IV. Electron-phonon interaction effects. J. Phys. C: Solid State Phys. 5(1), 21–42(1972)

9. Cini, M.: Time-dependent approach to electron transport through junctions: general theory andsimple applications. Phys. Rev. B 22(12), 5887–5899 (1980)

10. Combescot, R.: A direct calculation of the tunnelling current. III. Effect of localized impuritystates in the barrier. J. Phys. C: Solid State Phys. 4(16), 2611–2622 (1971)

11. Danielewicz, P.: Quantum theory of nonequilibrium processes, I. Ann. Phys. 152(2), 239–304(1984)

12. Datta, S.: Electronic Transport in Mesoscopic Systems. Cambridge University Press,Cambridge (1995)


14. Dick, R.: Advanced Quantum Mechanics: Materials and Photons. Springer, New York (2012)15. Dirac, P.A.M.: The quantum theory of the emission and absorption of radiation. Proc. Phys.

Soc. A 114(769), 710–728 (1927)16. Fetter, A.L., Walecka, J.D.: Quantum Theory of Many-Particle Systems. McGraw-Hill,

San Francisco (1971)17. Feynman, R.P.: Space-time approach to quantum electrodynamics. Phys. Rev. 76(6), 769–789

(1949)18. Fujita, S.: Thermodynamic evolution equation for a quantum statistical gas. J. Math. Phys.

6(12), 1877–1885 (1965)19. Fujita, S.: Resolution of the hierarchy of Green’s functions for fermions. Phys. Rev. A 4(3),

1114–1122 (1971)20. Gell-Mann, M., Low, F.: Bound states in quantum field theory. Phys. Rev. 84(2), 350–354

(1951)21. Goldstone, J.: Derivation of the Brueckner many-body theory. Proc. Phys. Soc. A 239(12173),

267–279 (1957)22. Hall, A.G.: Non-equilibrium Green’s functions: generalized Wick’s theorem and diagrammatic

perturbation theory with initial correlations. J. Phys. A: Math. Gen. 8(2), 214–224 (1975)23. Haug, H., Jauho, A.P.: Quantum Kinetics in Transport and Optics of Semiconductors. Springer,

Berlin (1996)24. Kadanoff, L.P., Baym, G.: Quantum Statistical Mechanics: Green’s Function Methods in

Equilibrium and Non-equilibrium Problems. W.A. Benjamin, New York (1962)


25. Keldysh, L.V.: Diagram technique for nonequilibrium processes. Sov. Phys. JETP 20(4), 1018–1026 (1965)

26. Kim, G., Arnold, G.B.: Theoretical study of tunneling phenomena in double-barrier quantum-well heterostructures. Phys. Rev. B 38(5), 3252–3262 (1988)

27. Klimeck, G., Ahmed, S.S., Kharche, N., Korkusinski, M., Usman, M., Prada, M., Boykin, T.B.:Atomistic simulation of realistically sized nanodevices using NEMO 3-D part I: models andbenchmarks. IEEE Trans. Electron Devices 54(9), 2079–2089 (2007)

28. Korenman, V.: Nonequilibrium quantum statistics: application to the laser. Ann. Phys. 39(1),72–126 (1966)

29. Kukharenkov, Y.A., Tikhodeev, S.G.: A diagram technique in the theory of relaxationprocesses. Sov. Phys. JETP 56(4), 831–838 (1982)

30. Lake, R., Datta, S.: Nonequilibrium Green’s-function method applied to double-barrierresonant-tunneling diodes. Phys. Rev. B 45(12), 6670–6685 (1992)

31. Lake, R., Klimeck, G., Bowen, R.C., Jovanovic, D.: Single and multiband modeling of quantumelectron transport through layered semiconductor devices. J. Appl. Phys. 81(12), 7845–7869(1997)

32. Lake, R., Pandey, R.R.: Non-equilibrium Green functions in electronic device modeling. In:Handbook of Semiconductor Nanostructures and Devices, vol. 3, pp. 409–443. AmericanScientific Publishers, Los Angles (2006)

33. Langreth, D.C.: Linear and Non-linear Electron Transport in Solids. NATO Advanced StudyInstitutes Series : Series B, Physics, vol. 17, pp. 3–18. Plenum Press, New York (1976)

34. Luisier, M.: Quantum transport for nanostructures. Technical report, Integrated SystemsLaboratory, ETH Zürich (2006). https://nanohub.org/resources/1792

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36. Martin, P.C., Schwinger, J.: Theory of many-particle systems. I. Phys. Rev. 115(6), 1342–1373(1959)

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43. Schwinger, J.: Brownian motion of a quantum oscillator. J. Math. Phys. 2(3), 407–432 (1961)

https://nanohub.org/resources/1792

Chapter 7Implementation

This chapter describes the techniques for efficient numerical solution of the outlinedNEGF equations for nanoelectronic devices. The elements of the kinetic equationsare the Hamiltonian, contact and scattering self-energies. Selection of appropriatebasis function for these elements are discussed. A fast converging method for theevaluation of contact self-energies and computationally efficient approximations forthe calculating of scattering self-energies are described. For an accurate analysisit is essential to solve the coupled system of transport and Poisson equations self-consistently. The convergence of this coupled equation system is discussed next.Finally, a summary of all required equations and techniques are presented.

7.1 Basis Functions and Matrix Representation

To formulate the theory of quantum transport, one has a choice of what representa-tion to use and the optimum choice depends on the problem at hand. For a numericalsolution of the Green’s functions, one should convert the Green’s functions andthe related kinetic equations from a spatial grid to a matrix representation. Forthis purpose, one can employ a set of localized functions, such as atomic s-and p-orbitals [3, 18, 21], Wannier functions [35, 39], and so forth. Assuming anorthonormal basis set 'm and using Eqs. (3.52) and (3.55), the field operators canbe written as

O .r/ DXm

'm.r/ Oam ;

O �.r/ DXm

'�m.r/ Oa�m :

(7.1)


157

158 7 Implementation

The Green’s functions in this representation are given by

Gr;a;?.r1; t1; r2; t2/ DXmn

Gr;a;?mn .t1; t2/'m.r1/'�

n .r2/ : (7.2)

By Fourier transforming Eq. (7.2), the Green’s functions under steady state read as

Gr;a;?.r1; r2; E/ DXmn

Gr;a;?mn .E/'m.r1/'�

n .r2/ : (7.3)

A similar transformation can be applied to self-energies, thus the quantum kineticequations (6.123) and (6.124) take the form

E Grmm.E/ �

Xp

Hmp Grpn.E/ �

Xp

˙ rmp.E/ G

rpn.E/ D ım;n ; (7.4)

G?mn.E/ D

Xpq

Grmp.E/ ˙

?pq.E/ G

aqn.E/ : (7.5)

It is straightforward to show the following relations between the Green’s functions

Gamn D ŒGr

nm � ; (7.6)

Grmn.E/ �Ga

mn.E/ D G>mn.E/ �G<

mn.E/ : (7.7)

The matrix representation of Eqs. (7.4) and (7.5) can be written as

A Gr D I ; (7.8)

A G? D ˙? ŒGr � ; (7.9)

where the underlined quantities represent matrices Ga D ŒGr �, and A is

A D ŒEI �H �˙ r : (7.10)

For many applications one needs to employ non-orthogonal basis functions. In thisbasis representation, one should include matrix S , which represents the overlapbetween the basis functions:

Smn DZ

dr '�m.r/'n.r/ : (7.11)

For orthogonal bases, Smn D ım;n so that S is the identity matrix, as stated earlier.One can use the standard approach to transform H , ˙ , and G into an orthogonalrepresentation QH , Q̇ , and QG:

7.1 Basis Functions and Matrix Representation 159

QH D S�1=2 H S�1=2 ;

Q̇ D S�1=2 ˙ S�1=2 ;

QG D SC1=2 G SC1=2 :

(7.12)

Equation (7.8) with a non-orthogonal basis set can be rewritten as [19]

ŒES �H �˙ r Gr D I : (7.13)

7.1.1 Free Transverse-Direction

A representation based on eigenstates, which is referred to as mode-space rep-resentation, is often convenient for analytical calculations since the Hamiltonianis diagonal. On the other hand, a real-space representation is intuitively moreappropriate for electronic devices. In many systems there is one or two transversedirections that can be modeled by applying either a periodic or a closed boundarycondition. It is therefore convenient to use the mode-space representation for thetransverse directions while employing a real-space representation for the longitudi-nal direction. One can separate the overall Hamiltonian into a longitudinal OHl and atransverse part OHt [7]:

OH D OHl C OHt ;

OHt� D "t� :

(7.14)

For devices with a large cross-section, it is common to assume periodic boundaryconditions in that direction since the real boundary conditions are believed to haveminimal effect on the observed properties. The transverse eigenstates are then givenby plane waves:

�.kt/ D 1

Aeikt�rt ; (7.15)

where A is the transverse cross-sectional area and kt is the wave vector along thetransverse direction. The overall basis functions can be therefore represented by.m;kt/. Thus the Hamiltonian matrix elements can be written as [7]

ŒHl CHt mkt;nk0

tD ŒHl mn C Œ".kt/I ıkt;k0

t: (7.16)

As transverse wave vectors kt are the eigenstates of the Hamiltonian, the off-diagonal matrix elements connecting two different transverse modes kt and k0

t arezero. As long as one neglects elastic or inelastic scattering processes that coupledifferent transverse modes, one can think of the transverse modes kt as separate


Fig. 7.1 The sketch of a three-dimensional structure. The transport of carrier is along thex-direction, carriers are confined along the y-direction, and they are free to move along the z-direction. The device can be considered a layered structure with Nx and Ny points along the x andy-direction, respectively

parallel channels. Each transverse mode kt has an extra transverse energy ".kt/

that should be added to the longitudinal energy. In this representation, Eqs. (7.4)and (7.5) are given by

.E � ".kt// Grmm.kt; E/ �

Xp

Hmp Grpn.kt; E/

�Xp

˙ rmp.kt; E/ G

rpn.kt; E/ D ım;n ;

(7.17)

G?mn.kt; E/ D

Xpq

Grmp.kt; E/ ˙

?pq.kt; E/ G

aqn.E/ : (7.18)

7.1.2 Real-Space Representation

To discuss real-space and mode-space representations, we follow the approachdescribed in Ref. [21]. Figure 7.1 shows a three-dimensional structure where thetransport of carrier is along the x-direction, carriers are confined along the y-direction, and they are free to move along the z-direction. This structure coversall possible boundary conditions that can occur in a device. In general this modelcan be easily extended to other structures, such as devices with two confinementdirections or two directions for free movement of electrons. As shown in Fig. 7.1, thedevice can be considered as a layered structure withNx points along the x-directionand Ny points along the y-direction. A common approximation used to describethe Hamiltonian of layered structures consists of non-vanishing interactions onlybetween nearest neighbor layers. That is, each layer i interacts only with itself andits nearest neighbor layers i � 1 and i C 1. Then the single particle Hamiltonianof the layered structure is a block tridiagonal matrix, where the diagonal block ˛iis a Ny � Ny tridiagonal matrix representing the interaction between the elementsof layer i , and the off-diagonal block ˇ

i;iC1 is a Ny � Ny block diagonal matrixrepresenting the interaction between the elements of layers i and i C 1 withˇiC1;i D ˇ�

i;iC1. Therefore, the total Hamiltonian is a .NxNy/ � .NxNy/ blocktridiagonal matrix [21]:


H D

266666664

˛1 ˇ120 : : : 0

ˇ21˛2 ˇ

23: : : 0

:::: : :

: : :: : :

:::

0 0 ˇNx�1Nx�2 ˛Nx�1 ˇ

Nx�1Nx0 0 0 ˇ

NxNx�1 ˛Nx

377777775; (7.19)

˛i D

2666664

hii11 hii12 0 : : : 0

hii21 hii22 hii23 : : : 0:::

: : :: : :

: : ::::

0 0 hiiNy�1Ny�2 hiiNy�1Ny�1 hiiNy�1Ny0 0 0 hiiNyNy�1 hiiNyNy

3777775; (7.20)

ˇi1i2

D

2666664

hi1i211 0 0 : : : 0

0 hi1i222 0 : : : 0:::

: : :: : :

: : ::::

0 0 0 hi1i2Ny�1Ny�1 0

0 0 0 0 hi1i2NyNy

3777775; (7.21)

where hi1i2j1j2 is the Hamiltonian matrix element expressed in terms of localizedbasis functions centered at some lattice points .xi1 ; yj1/ and .xi2 ; yj2/. In the tight-binding method, one can take the basis functions to be any set of localized basisfunctions, such as atomic orbitals. In this model, diagonal Hamiltonian matrix ele-ments represent on-site potentials whereas off-diagonal elements represent hoppingparameters. In the effective mass approximation, however, the basis function can beconsidered as a product of delta functions centered at some lattice point .xi ; yj /:ı.x � xi /ı.y � yj /. Based on this model and first nearest neighbor approximation,the Hamiltonian matrix elements are given by [21]

hi1i2j1j2 D

8̂ˆ̂̂<ˆ̂̂̂:

ti1i1j1j1C1 C ti1i1j1j1�1 C ti1i1�1j1j1 C ti1i1C1j1j1 ; i2 D i1 ; j2 D j1

�ti1i1˙1j1j1 ; i2 D i1 ˙ 1 ; j2 D j1

�ti1i1j1j1˙1 ; i2 D i1 ; j2 D j1 ˙ 1

0 otherwise ;(7.22)

ti1i2j1j2 D

8̂ˆ̂̂̂<ˆ̂̂̂:̂

2„2.m�

y i1j1Cm�

y i1j1˙1/�yj̇1 .�yCj1

C�y�j1/; i2 D i1 ; j2 D j1 ˙ 1

2„2.m�

x i1j1Cm�

x i1˙1j1/�xi̇1 .�xCi1

C�x�i1/; i2 D i1 ; j2 D j1 ˙ 1

0 otherwise ;(7.23)


where m�x ij is the effective mass along the x-direction at some lattice point .xi ; yj /

and �xi̇ D jxi˙1 � xi j. In this approach, Gi1i2;j1j2 .kz; E/ is the Green’s functionat position .j1; j2/ in the Ny � Ny block located at .i1; i2/. In real-space approach,the numerical solution of the Green’s functions for each transverse mode kz requiresinversion of .NxNy/ � .NxNy/ matrices, which can be computationally expensive.Based on relations explained in Sect. 7.5, the carrier concentration and currentdensity for real-space representation are given by

nij D � 2i

Lz�x�y

Xkz

ZdE

2�G<iijj .kz; E/ ; (7.24)

JxiC1=2j D � 2e

„Lz�y

Xkz

ZdE

2�2<eŒhiC1ijjG<

iiC1jj .kz; E/ ; (7.25)

Jy ijC1=2 D � 2e

„Lz�x

Xkz

ZdE

2�2<eŒhiijC1jG<

iijjC1.kz; E/ : (7.26)

7.1.3 Coupled Mode-Space Approach

The coupled mode-space approach is based on expressing the transverse confineddirection in eigenstate rather than real-space representation. This method cansignificantly reduce the computational cost of real-space approach whereas it keepsall the relevant physics of the problem. The nth eigenstate of the confined transversedirection is indicated by 'n.x; y/ where its dependence on x is due to the fact thatconfinement is not the same all along the transport direction. The Green’s functionin real-space can be expanded in terms of these eigenstates as [21]

Gi1i2j1j2.kz; E/ DXmn

Gi1i2mn.kz; E/'mi1 .yj1/'�ni2 .yj2/ : (7.27)

One can construct vectors with Ny components as

�mi

D

26664

'mi.y1/

'mi.y2/:::

'mi.yNy /

37775 : (7.28)

As �mi

are the eigenvectors of the Hamiltonian ˛i , they form a complete orthogonalbasis, which satisfies the following relations:

˛i�miD Em�

mi; (7.29)


Zdy 'mi.y/'

�ni.y/ D ımn ; (7.30)

Xn

'ni.y1/'�ni.y2/ D ı.y1 � y2/ : (7.31)

The total number of modes corresponds to the dimension of ˛i , which is Ny . Thusthe size of the Green’s function is not changed in the coupled mode-space. In deviceswith strong confinement, however, only a few low energy modes are populated andcontribute to transport phenomena. As a result, only a reduced number of modesNm � Ny need to be considered. The Green’s function matrices in coupled mode-space GCMS have the size .NxNm/ � .NxNm/ instead of .NxNy/ � .NxNy/ for itsreal-space counter part GRS. It should be noted that, in general, Nm can vary alongthe channel [21]. To relate the Green’s function matrices in real-space and mode-space, one can form matrices with size Ny �Nm as

ui D Œ�1i: : : �

Nmi D

26664

'1i .y1/ '2i .y1/ : : : 'Nmi .y1/

'1i .y2/ '2i .y2/ : : : 'Nmi .y2/:::

: : :: : ::::

'1i .yNy / '2i .yNy / : : : 'Nmi .yNy /

37775 : (7.32)

Next, one can generate a transformation matrix U , with size .NxNy/ � .NxNm/,which includes Nx ui s

U D

26664

u1 0 : : : 0

0 u2 : : : 0:::: : :

: : ::::

0 0 : : : uNx

37775 : (7.33)

In coupled mode-space, the Hamiltonian is a block tridiagonal matrix:

HCMS D U � HRS U ; (7.34)

where the blocks ˛i from Eq. (7.20) are replaced by

˛iCMS D u�i ˛iRS ui ; (7.35)

which is a diagonal Nm �Nm matrix. ˇii˙1 from Eq. (7.21) are replaced by

ˇii˙1CMS

D u�i ˇii˙1RSui : (7.36)

As the modes do not necessarily have the same shape along the channel, one has


u�i ui˙1 ¤ I ; (7.37)

thus the blocks of ˇii˙1CMS

may be full, implying that the modes are coupled. Thereal-space and coupled mode-space Green’s functions are therefore related by thefollowing transformations:

GCMS D U � GRS U : (7.38)

Using the mentioned transformations, the equation of motion for the coupled mode-space Green’s function can be obtained as [21]

Œ.E � ".kz//I �HCMS �˙ rCMS.kz; E/ G

rCMS.kz; E/ D I ; (7.39)

G?CMS.kz; E/ D Gr

CMS.kz; E/ ˙?CMS.kz; E/ G

aCMS.kz; E/ : (7.40)

The carrier concentration and current density in the coupled mode-space read as

nij D � 2i

Lz�x�y

Xkz

Xmn

ZdE

2�G<iimn.kz; E/'mi.yj /'

�ni.yj / ; (7.41)

JxiC1=2j D � 2e

„Lz�y

Xkz

Xmn

ZdE

2�2<eŒhiC1ijjG

<iiC1mn.kz; E/'mi.yj /'

�niC1.yj / ;

(7.42)

Jy ijC1=2 D � 2e

„Lz�x

Xkz

Xmn

ZdE

2�2<eŒhiijC1jG<

iimn.kz; E/'mi.yj /'�ni.yjC1/ :

(7.43)

7.1.4 Decoupled Mode-Space

If the shape of each transverse mode remains constant along the channel

@'n.x; y/

@xD 0 ; (7.44)

the coupling between modes disappears, which infers

Gi1i2mn D 0 for m ¤ n : (7.45)

Mathematically, both the block ˛iCMS and ˇii˙1CMS

can be simultaneously diago-nalized. After diagonalizing these blocks and then reordering the Hamiltonian, oneobtains [14]

7.2 Contacts 165

HDMS D

2666664

Hn 0 0

0 H2 0 0

:::: : :

: : ::::

0 0 0 Hm

3777775; (7.46)

where Hm is the Hamiltonian matrix for the mth mode

Hm D

266664

˛1m ˇm 0 0

ˇ�m ˛2m ˇm 0

:::: : :

: : ::::

0 0 ˇ�m ˛Nxm

377775 : (7.47)

Under this condition, the Green’s functions can be defined for the mth mode asGi1i2m. Therefore, one can solve the equation of motion for each mode indepen-dently:

Œ.E � ".kz//I �Hm.kz; E/ �˙ rm.kz; E/ G

rm.kz; E/ D I : (7.48)

G?m.kz; E/ D Gr

m.kz; E/ ˙?m.kz; E/ G

am.kz; E/ : (7.49)

The carrier concentration and current density can be evaluated for each modeseparately from the following relations:

nim D � 2i

Lz�x�y

Xkz

Xm

ZdE

2�G<

iim.kz; E/j'mi.yj /j2 ; (7.50)

JxiC1=2j D � 2e

„Lz�y

Xkz

Xm

ZdE

2�2<eŒhiC1imG

<iiC1m.kz; E/'mi.yj /'

�miC1.yj / :

(7.51)

It should be noted that as the modes are decoupled, the current between modes iszero.

7.2 Contacts

As shown in Fig. 7.2, one can partition the layered structure into left contact withindex L, device region with index D, and right contact with index R. The devicecorresponds to the region where one solves the transport equations and the contactsare the highly conducting regions connected to the device. While the device regionconsists of onlyN layers, the matrix equations corresponding to Eqs. (7.8) and (7.9)


Fig. 7.2 Partitioning of the simulation domain into device region, and left and right contacts

are infinite-dimensional due to the semi-infinite contacts. It is shown next that theinfluence of the semi-infinite contacts can be folded into the device region, wherethe semi-infinite contacts only affect layers 1 and N of the device region. This canbe viewed as additional self-energy due to the transitions between the device andthe contacts [4, 15, 18]. Here, the method described in [2] is followed.

7.2.1 Matrix Truncation

By defining A D ŒEI � H � ˙S , where ˙S is the self-energy due to variousscattering processes, one can write A Gr D I (Eq. (7.8)) as

24ALL ALD 0

ADL ADD ADR

0 ARD ARR

3524G

rLL G

rLD Gr

LR

GrDL G

rDD G

rDR

GrRL G

rRD Gr

RR

35 D

24 I 0 00 I 0

0 0 I

35 ; (7.52)

where ALL, ARR, and ADD correspond to the left semi-infinite contact, right semi-infinite contact, and the device region, respectively:

ALL D

26666664

: : :: : :

: : :: : :

: : : �t �L3;2 AL2 �tL2;1 0

: : : 0 �t �L2;1 AL1 �tL1;00 0 �t �L1;0 AL0

37777775; (7.53)

ARR D

26666664

AR0 �tR0;1 0 0

�t �R0;1 AR1 �tR1;2 0: : :

0 �t �R1;2 AR2 �tR2;3: : :

: : :: : :

: : :: : :

37777775; (7.54)

7.2 Contacts 167

ADD D

266666666664

A1 �t 1;2 0 0 : : : 0

�t �1;2 A2 �t 2;3 0 : : : 0

:::: : :

: : :: : :

: : ::::

0 : : : 0 �t �N�2;N�1 AN�1 �tN�1;N

0 : : : 0 0 �t �N�1;N AN

377777777775

: (7.55)

The coupling between the left and right contacts and device are given by

ALD D

264

0 : : : 0:::

: : ::::

�tLD : : : 0

375 ; ARD D

2640 : : : �tRD:::: : :

:::

0 : : : 0

375 : (7.56)

It should be noted that ADL D A�LD, ADR D A

�RD, and ALD and ADL (ARD, and ADR)

are sparse matrices. Their only non-zero entry represents the coupling of the left(right) contact and device. From Eq. (7.52), one obtains

GrLD D �A�1

LL ALD GrDD ; (7.57)

GrRD D �A�1

RR ARD GrDD ; (7.58)

ADL GrLD C ADD G

rDD C ADR G

rRD D I : (7.59)

Substituting Eqs. (7.57) and (7.58) in Eq. (7.59), one obtains a matrix equation witha dimension corresponding to the total number of grid points in device layers:

ŒADD � ADL A�1LL ALD � ADR A

�1RR ARD G

rDD D I : (7.60)

The second and third terms of Eq. (7.60) are self-energies due to coupling of thedevice region to left and right contacts, respectively. The Green’s functions of theisolated semi-infinite contacts are defined as

ALLgrL

D I ; ARRgrR

D I : (7.61)

The surface Green’s function of the left and right contacts are the Green’s functionelements corresponding to the first edge layer of the respective contact:

grL0;0

D A�1LL0;0 ; gr

R0;0D A�1

RR0;0 : (7.62)


7.2.2 Surface Green’s Function

The main information needed to solve Eq. (7.81) is the surface Green’s functions ofgrL

and grR

. Using the recursive relation Eq. (7.121), Eq. (7.61) can be written as

ŒALi � tLi;iC1grLiC1;iC1

t�LiC1;i

grLi;i

D I ;

ŒARi � t�Ri;i�1

grRi�1;i�1

tRi�1;i grRi;i

D I :

(7.63)

If the potential does not vary in the left and right contacts and if the couplingbetween different layers are equal, then ALL and ARR become semi-infinite periodicmatrices with

AL0 D AL1 D AL2 D : : : D AL ;

AR0 D AR1 D AR2 D : : : D AR ;

tL0;1 D tL1;2 D tL2;3 D : : : D tL ;

tR0;1 D tR1;2 D tR2;3 D : : : D tR :

(7.64)

Under this condition one obtains

grL1;1

D grR2;2

D : : : D grL;

grR1;1

D grR2;2

D : : : D grR:

(7.65)

Therefore, the surface Green’s functions can be obtained by solving the quadraticmatrix equations

ŒAL � tL grLt�L g

rL

D I ;

ŒAR � t �R grRtR g

rR

D I :

(7.66)

These equations can be solved iteratively by

ŒAL � tL grL

hm�1i t �L grL

hmi D I ;

ŒAR � t �R grR

hm�1i tR grR

hmi D I ;

(7.67)

where m represents the iteration number. It should be noted that the solution toEq. (7.66) is analytic if the dimension of AR is one.

7.2 Contacts 169

7.2.3 Sancho-Rubio Iterative Method

A conventional iterative method described in Eq. (7.67) for evaluating the sur-face Green’s function converges very slowly. Sancho-Rubio proposed an efficientmethod in Ref. [31]. They reformulated decimation techniques with the help of theeffective layer concept. The method involves replacing the original chain by aneffective one of twice the lattice constant, where each layer plus its two nearestneighbors in the original chain are replaced by an effective layer in the new chain.These effective layers interact through energy-dependent residual interactions whichare weaker than those of the original chain. This replacement can be repeatediteratively until the residual interactions between effective layers are as small as onewishes. After m iterations, one has a chain of lattice constant 2m times the originalone, each effective layer replacing 2m original layers. The effective interlayerinteractions become vanishingly small after a few iterations. Each new iterationdoubles the number of original layers included in the new effective layer, making ita fast converging method.

We follow the approach described in Ref. [31] to describe this method. First, oneshould consider a semi-infinite layered-structure, where layer 0 is the one whichis connected to only one neighboring layer. The Hamiltonian of this structure istherefore given by

h D

2666666666664

h0;0 h0;1 0 0 0

h1;0 h1;1 h1;2 0 0

0 h2;1 h2;2 h2;3 0

0 0 h3;2 h3;3: : :

0 0 0: : :

: : :

3777777777775

: (7.68)

The Green’s for this layered-structure is

h D

2666666666664

g0;0g0;1g0;2g0;3: : :

g1;0g1;1g1;2g1;3: : :

g2;0g2;1g2;2g2;3: : :

g3;0g3;1g3;2g3;3

: : :

::::::

:::: : :

: : :

3777777777775

; (7.69)

where the dependence of the Green’s functions on energy and transverse momentumare omitted for compactness g

i1;i2� g

i1;i2.kt; E/. For a column g

i;0one has


.EI � h0;0/g0;0 � h0;1g1;0 D I ;

�h1;0g0;0 C .EI � h1;1/g1;0 � h1;2g2;0 D 0 ;

�h2;1g1;0 C .EI � h2;2/g2;0 � h2;3g3;0 D 0 ;

:::

(7.70)

which can be generalized as

.EI � hn;n/gn;0 D hn;n�1gn�1;0 C hn;nC1gnC1;0 : (7.71)

One can make the simplifying but not essential assumption that h00 D h11 D : : :

and h01 D h12 D : : :. Thus Eq. (7.70) and Eq. (7.71) can be rewritten as

.EI � "s0/g0;0

D I C ˛0g1;0;

.EI � "0/gn;0 D ˇ0gn�1;0 C ˛0gnC1;0 ;

(7.72)

where

"s0 D h0;0 ; "0 D h0;0 ; ˛0 D h0;1 ; ˇ

0D h1;0 : (7.73)

Equation (7.72) describes the relationship between the neighboring layer. Therelations between the second neighboring layer corresponding to Eq. (7.72) read as

.EI � "s0/g0;0

D I C ˛0.EI � "0/�1.ˇ0g0;0 C ˛0g2;0/ ;

.EI � ."s0 C ˛0.EI � "s

0/�1ˇ

0//g

0;0D I C ˛0.EI � "0/�1˛0g2;0 ;

.EI � "s1/g0;0

D I C ˛1g2;0;

(7.74)and

.EI � "0/gn;0 D ˇ0.EI � "0/�1.ˇ0gn�2;0 C ˛0g

n;0/

C˛0.EI � "0/�1.ˇ0gn;0 C ˛0gnC2;0/ ;�

EI � ."0 C ˇ0.EI � "0/�1˛0

C˛0.EI � "0/�1ˇ0/�gn;0

D ˇ0.EI � "0/�1ˇ0gn�2;0

C˛0.EI � "0/�1˛0gnC2;0 ;

.EI � "1/gn;0 D ˇ1gn�2;0 C ˛1gnC2;0 ;

(7.75)

7.2 Contacts 171

where

"s1 D "s

0 C ˛0.EI � "0/�1ˇ0;

"1 D "0 C ˇ0.EI � "0/�1˛0 C ˛0.EI � "0/�1ˇ

0;

˛1 D ˛0.EI � "0/�1˛0 ;

ˇ1

D ˇ0.EI � "0/�1ˇ

0;

(7.76)

which define an effective Hamiltonian describing a chain of effective layers of latticeconstant twice the original one. Each effective layer contains implicitly the effect ofits nearest neighbors in the original chain. Now, when considering the subset formedby taking only even values for n Eqs. (7.74) and (7.75), one obtains

.EI � "s1/g0;0

D I C ˛1g2;0;

.EI � "1/g2n;0 D ˇig2.n�1/;0 C ˛ig2.nC1/;0 :

(7.77)

These equations define a chain which couples the Green-function matrix elementswith even indices only (g

2n;0) through effective nearest-neighbor interactions.

Repeating the argument i times, we have the iterative sequence between 2i thneighboring layer:

"si D "s

i�1 C ˛i�1.EI � "i�1/�1ˇi�1 ;

"i D "i�1 C ˇi�1.EI � "i�1/�1˛i�1 C ˛i�1.EI � "i�1/�1ˇ

i�1 ;

˛i D ˛i�1.EI � "i�1/�1˛i�1 ;

ˇi

D ˇi�1.EI � "i�1/�1ˇ

i�1 :

(7.78)

Starting with Eq. (7.73), Eq. (7.78) defines an effective Hamiltonian for a chain oflattice constant 2i times the original one with nearest-neighbor matrix elements˛i and ˇ

i, and on-site matrix elements "i and "s

i . After i th iterations, the Green’sfunction take the form

.EI � "si /g0;0

D I C ˛ig2i n;0;

.EI � "i /g2i n;0 D ˇig2i .n�1/;0 C ˛ig2i .nC1/;0 :

(7.79)

Each layer of the i th chain contains implicitly the effect of the nearest neighborsof the previous chain i � 1. After m iterations, the zeroth layer is equivalent tothe original zeroth layer coupled to 2m layers. The iteration is to be repeated until


˛m and ˇm

are as small as one wishes. Obviously, "sm ' "s

m�1 and "m ' "m�1,therefore, one obtains an approximation for the surface Green’s function

g0;0

' .EI � "sm/ : (7.80)

As a summary, the following steps should be followed for the evaluation of thesurface Green’s functions by the Sancho-Rubio method:

• Specifying the Hamiltonian matrix of the contact unit cell and the couplingmatrices to the neighboring unit cells.

• Initializing Eq. (7.73) with the Hamiltonian and coupling matrices.• The set of equations in Eq. (7.78) is iterated until a convergence criterion

condition is satisfied.• The surface Green’s function is evaluated from Eq. (7.80).

These procedures are applied to obtain the surface Green’s function of an AGNR(Sect. 8.1). Figure 7.3 compares the convergence behavior of the Sancho-Rubiomethod with that of the simple iterative method. All results indicate that the Sancho-Rubio method converges more rapidly in comparison with the simple iterativemethod, which results in significant computation time reduction.

7.2.4 Contact Self-Energies

The surface Green’s functions defined in Eq. (7.62) enable us to rewrite Eq. (7.60)in a form very similar to Eq. (7.8),

ŒEI �H �˙ rS �˙ r

C GrDD D I ; (7.81)

where

˙ rC1;1

D tDL grL1;1

tLD D ˙ rL ;

˙ rCN;N

D tDR grR1;1

tRD D ˙ rR :

(7.82)

All other elements of ˙ rC are zero. ˙ r

L and ˙ rR are self-energies due to the left and

right contacts, respectively, and tDL D t�LD and tDR D t

�RD. By following the same

procedure, one obtains the equation of motion for the lesser and greater Green’sfunctions as [18]

G?DD D Gr

DD

h˙

7S C ˙ r

C

iGa

DD ; (7.83)

7.2 Contacts 173

a b

c

Fig. 7.3 Relative residual error of the surface Green’s function (defined as log.jgn�gn�1j=jgnj/)as a function of iteration number for (a) Sancho-Rubio method and (b) simple iterative method.The evaluated surface Green’s function is for an AGNR withN D 18, see Sect. 8.1 at E D 3:6 eV.(c) The required number of iterations needed for calculating the surface Green’s function at variousenergies. The same convergence criterion is used for the Sancho-Rubio (solid line) and simpleiterative method (dashed line). Due to the presence of van-hove singularities, the convergencerequires more iterations at energies close to the bottom of each subband. For a better comparison,the iteration number of the Sancho-Rubio method is multiplied by a factor of 50

where,

˙?C1;1

D tDL g?L1;1

tLD D ˙?L ;

˙?CN;N

D tDR g?R1;1

tRD D ˙?R :

(7.84)

Since the contacts are by definition in equilibrium, one obtains (Sect. 6.4.6)

g<1;1

D i a1;1fL ;

g<1;1

D i a1;1fR ;

(7.85)

where a D i.gr � ga/ D �2=mŒgr is the spectral function and fL.R/ is the Fermifactor of the left (right) contact. By defining the broadening function as


� C1;1D i .˙ r

1;1� ˙ a

1;1/ D tDL a1;1 tLD D � L ;

� CN;ND i .˙ r

1;1� ˙ a

1;1/ D tDR a1;1 tRD D � R ;

(7.86)

Eq. (7.84) can be rewritten as

˙<L D Ci � L fL ;

˙<R D Ci � R fR :

(7.87)

In a similar manner, one can show that

˙>L D �i � L .1 � fL/ ;

˙>R D �i � R .1 � fR/ :

(7.88)

7.2.5 Wide-Band Limit

The wide-band limit (WBL) approximation is usually applied to model contacts [26,27, 38]. The contact self-energy, similar to scattering self-energy, is a complexfunction, the real-part describes the energy shift of the level, and the imaginary partdescribes broadening. The finite imaginary part appears as a result of the continuousspectrum in the leads. In WBL approximation, the real-part of the self-energy isneglected and only the level broadening effect of the contact is considered.

The WBL approximation amounts to neglecting the electronic structure of thedevice leads by assuming constant density of states (DOS) and constant couplingto the central device. In particular, transport is often dominated by states close tothe Fermi level, and since DOS are generally slowly varying functions of energy,see Fig. 7.4, the WBL for this case is an excellent approximation. The advantage ofthis approximation is that it focuses on the electronic structure of the contact by fullyneglecting the atomistic structure of the contacts and the details of its geometry [15].If the coupling between device and contacts is parameterized by coupling strengtht , WBL replaces the evaluation of Eq. (7.82) with

˙ rSB � �i

�

2; (7.89)

� D t 2N .EF/ ; (7.90)

where N .EF/ is the surface density of states of the metallic contact that is evaluatedat the Fermi energy. To calculate the surface density of states, one can employthe tight-binding method [29], density functional theory [26], or extended Hückeltheory [17].

7.3 Scattering Self-Energies 175

Fig. 7.4 Calculated surfacedensity of states for Au along[111] direction (Reproducedwith permission fromRef. [17]. Copyright (2005),Springer)

7.3 Scattering Self-Energies

In Green’s function formalism, scattering processes are modeled by self-energies.As discussed in Sect. 6.5.5, self-energies are commonly approximated by the firstor second-order terms. In addition, depending on the scattering type and the devicegeometry, other approximations can be employed to reduce the computation cost ofself-energy calculations. In this section the self-energies for acoustic, optical, andpolar-optical phonon scattering are derived and common approximations for eachself-energy are discussed. Self-energies for other sources of scattering are discussedin Ref. [18]. We assume here that the transport of electrons occurs along the x-direction and two transverse directions are infinitely extended. The transverse wavevector for electrons and phonons are denoted by kt and qt, respectively.

7.3.1 Electron-Phonon Scattering

The electron-phonon self-energies in the self-consistent Born approximation areexpressed in terms of the full electron and phonon Green’s functions. One shouldtherefore first study the influence of the bare electron states on the phonons, andthen calculate the effect on the electrons of the renormalized phonon states [23]. Inthis work we assume that the phonon renormalization can be neglected. By doingso we miss capturing a possible reduction of the phonon lifetime. The aboveconsiderations also appeal to the Migdal theorem [25], which states that the phonon-induced renormalization of the electron-phonon vertex (see Sect. 6.7.3) scales withthe ratio of the electron mass to the ion mass [12]. Therefore, one can assume thatthe phonon bath is in thermal equilibrium so that the full phonon Green’s functionDcan be replaced by the non-interacting Green’s functions D0 from Eq. (6.89). Thusthe self-energies in Eq. (6.125) can be written as


˙<i1i2.ktIE/ D 1

V

Xq

eiqx.xi1�xi2 /jMqj2

� �.nq C 1/G<

i1i2.kt � qt; EC „!q/C nqG

<i1i2.kt � qt; E � „!q/

�;

(7.91)

where the first term on the right hand side is due to phonon emission and the secondterm due to phonon absorption. In a similar way, the greater self-energy reads as

˙>i1i2.ktIE/ D 1

V

Xq

eiqx.xi1�xi2 /jMqj2

� �.nq C 1/G>

i1i2.kt � qt; E � „!q/C nqG

>i1i2.kt � qt; E C „!q/

�:

(7.92)

As described in Sect. 6.6.3, the retarded self-energy can be evaluated from thebroadening function:

�i1i2 .kt; E/ D 2=mŒ˙>i1i2.kt; E/ ; (7.93)

˙ ri1i2.kt; E/ D � i

2�i1i2 .kt; E/C P

ZdE 0

2�

�i1i2 .kt; E0/

E �E 0 : (7.94)

Upon the evaluation of the lesser and greater self-energies, the imaginary part ofthe retarded self-energy can be easily obtained. The calculation of the real-part,however, can be rather time consuming. The imaginary part of the self-energy givesthe finite life-time of the state whereas the real-part causes an energy shift. In manycases, the energy shift due to scattering can be neglected in comparison with theelectrostatic potential. By neglecting the real-part of the retarded self-energy, thecomputation time can be significantly reduced.

˙ ri1i2.kt; E/ � � i

2�i1i2 .kt; E/ : (7.95)

The summation over wave vector q in self-energies can be generally transformedinto an integral over the first Brillouin zone:

Xq

D V

.2�/3

Zdq : (7.96)

To calculate electron-phonon self-energies, the integral in Eq. (7.96) must be eval-uated. Approximations for each phonon scattering process, which can significantlysimplify the calculations, are discussed next.


7.3.2 Acoustic Phonon

Acoustic phonon (AP) scattering dominates in non-polar materials, such as Silicon.A common deformation-potential model uses a linear phonon dispersion !q D qvsand the interaction potential in Eq. (4.43). Since „!q is small compared to kBT ,

nq D

exp

� „!kBT

�� 1

��1� kBT

„˝q� nq C 1 : (7.97)

In addition,G.E˙„˝q/ � G.E/ such that the scattering is essentially elastic. Theself-energy in Eq. (7.91) is therefore reduced to

˙<i1i2.ktIE/ D 1

�i1i2

�2kBT

V �v2s

Xqt

G<i1i2.kt � qt; E/ ; (7.98)

1

�i1i2

DXqx

eiqx.xi1�xi2 / D Lx

�

sin .�.xi1 � xi2/=�x/xi1 � xi2

; (7.99)

where the function 1=�i1i2 peaks around xi1 D xi2 with a full width at halfmaximum of 1:2�x, where �x is the lattice spacing along the transport direction.It is reasonable to assume

1

�i1i2

� ıi1i2Lx

�x: (7.100)

Based on this approximation, only diagonal elements of the self-energy need to beconsidered:

˙?;r;ai1i2

.ktIE/ D ıi1;i2�2kBT

�v2s�x

Zdqt

.2�/2G

?;r;ai1i2

.kt � qt; E/ : (7.101)

The Hamiltonian matrix is usually sparse, for example, (block) tridiagonal. Bykeeping only diagonal elements of the electron-phonon self-energy, the matricesremain sparse. Therefore, an efficient recursive method (Sect. 7.4) can be used tocalculate the inverse matrices. This implies considerable reduction of computationtime and memory requirement. Due to the made approximations, the retarded andadvanced self-energy for scattering with AP is simplified and directly related tothe retarded Green’s function. Therefore, one does not need to evaluate integralslike Eq. (7.94), which implies a considerable saving of computational cost.

7.3.3 Optical Phonons

The interaction with optical phonons (OP) is generally inelastic and the energy ofOPs are usually larger than kBT , thus the approximation used in case of AP can


not be applied. However, the matrix elements of OP scattering are assumed to beindependent of the wave vector (see Eq. (4.45)):

˙<i1i2.ktIE/ D 1

�i1i2

„D20

2V �!q

Xqt

� �.nq C 1/G<i1i2.kt � qt; E C „!q/C nqG

<i1i2.kt � qt; E � „!q/

�:

(7.102)

As a result, OP self-energies, similar to Eq. (7.101), can be approximated by theirdiagonal elements:

˙<i1i2.ktIE/ D ıi1;i2

„D20

2�!q�x

Zdqt

.2�/2

� �.nq C 1/G<i1i2.kt � qt; E C „!q/C nqG

<i1i2.kt � qt; E � „!q/

�:

(7.103)

The greater self-energy for OP is obtained in a similar way. Knowing the lesser andthe greater self-energies, the retarded self-energy can be evaluated. As mentionedearlier, to reduce computation time, one can ignore the real-part of the retarded self-energy and approximate it only with the imaginary part.

7.3.4 Polar Optical Phonons

The interaction potential of polar optical phonon (POP) depends on the phononwave vector, see Eq. (4.59). As a result, the integration over the longitudinal termswill not result in the delta function and localized scattering. Here we follow theapproach described in Ref. [19] for evaluating the self-energy of POP scattering.First we change the dummy indices such that the Green’s function in the self-energyappears only as a function of qt rather than kt � qt:

˙>i1i2.ktIE/ D 1

V

Xq

e�iqx.xi1�xi2 /jMkt�qj2

� �.nq C 1/G>

i1i2.qt; E � „!q/C nqG

>i1i2.qt; E C „!q/

�:

(7.104)

Furthermore, we assume that G>.qtIE/ is independent of the angle of qt inthe transverse-direction, which allows for performing of the angular integrationanalytically. This approximation is justified near the conduction band-edge wherethe bands are spherical. By converting the sum over q to an integral in cylindricalcoordinates, one obtains


Zdqt

.2�/2G>.qt; E ˙ „!q/

Z ��x

��x

dqx2�

e�iqx.xi1�xi2 /Z 2�

0

d�jkt � qtj2�jkt � qtj2 C q2D

�2„ ƒ‚ …

I�

:

(7.105)

The integral over � is

I� DZ 2�

0

d�k2t C q2t C q2x � 2ktqt cos ��

k2t C q2t C q2x C q2D � 2ktqt cos ��2 : (7.106)

This integral is evaluated by making the substitution

z D ei� ; (7.107)

which converts the integral over � into a contour integral around the unit circle. Theintegral is then evaluated using the residue theorem. By substituting Eq. (7.107) inEq. (7.106), one obtains

I� D i

ktqt

Idzn 1

z2 � bz C 1C q2Dz

ktqt Œz2 � bz C 1 2

o; (7.108)

where

b D k2t C q2t C q2x C q2Dktqt

: (7.109)

Evaluating Eq. (7.108) with the residue theorem, we obtain

I� .q2z ; q

2t ; k

2/ D 2�

(1q

.q2x C q2t C k2t C q2D/2 � 4k2t q2t

� q2D.q2x C q2t C k2t C q2D/�

.q2x C q2t C k2t C q2D/2 � 4k2t q2t

�3=2):

(7.110)

The integral over qx in Eq. (7.105) can be written as

2

Z �=�

0

dqx2�

cos Œqx.xi1 � xi2/ I� .q2x; q2t ; k2t / ; (7.111)

thus the self energy for POP scattering takes the form


˙<i1i2.kt; E/ D e2„!o

2�"p

Zdqt

4�2Ii1i2

�k2t ; q

2t

�

� �nqG>i1i2.qt; E C „!o/C .nq C 1/G>

i1i2.qt; E � „!o/

�:

(7.112)

where

Ii1i2�k2t ; q

2t

� DZ �=�x

0

dqxcos Œqx.xi1 � xi2/ q

.q2x C q2t C k2t C q2D/2 � 4k2t q

2t

�Z �=�

0

dqxcos Œqx.xi1 � xi2/ q2D.q2x C q2t C k2t C q2D/�

.q2x C q2t C k2t C q2D/2 � 4k2t q

2t

�3=2 :

(7.113)

As the integral over the angle was evaluated analytically, the integrand in Eq. (7.112)is only a function of the magnitude qt D jqtj. Using an effective mass model, onecan we make the following change of variables in Eq. (7.113):

Zdqt

.2�/2!

Zd"qt�2D."qt/ ; (7.114)

where �2D is the two-dimensional density of states and "qt . For a parabolic disper-sion, "qt D „2q2t =.2m�/ and �2D D m�=.2�„2/. This approximation simplifies theevaluation of the self-energy.

The lesser self-energy is evaluated in a similar way to Eq. (7.113). After theevaluation of the lesser and greater self-energies, the imaginary part of the retardedself-energy is simply obtained. The real-part can be computed by the Hilberttransformation of the imaginary part, or it can be completely ignored for reducingthe computation time.

7.4 Recursive Method for Calculating Green’s Functions

For evaluating the Green’s functions, matrix inversions are required. However, sincethe matrices are block tridiagonal and also most of the observables are related to thediagonal blocks of the Green’s functions, one can employ a recursive method tocalculate the observables efficiently [18]. We follow here the approach described inRef. [34].

7.4 Recursive Method for Calculating Green’s Functions 181

7.4.1 Retarded and Advanced Green’s Functions

The Dyson equation for the retarded Green’s function and the left-connectedGreen’s function [18] are employed to calculate the diagonal blocks of the fullGreen’s function recursively. The solution to the matrix equation

AZ;Z AZ;Z0

AZ0;Z AZ0;Z0

�"GrZ;Z Gr

Z;Z0

GrZ0;Z G

rZ0;Z0

#DI 0

0 I

�; (7.115)

is

Gr D Gr0 CGr0UGr D Gr0 CGr UGr0 ; (7.116)

where

Gr D"GrZ;Z Gr

Z;Z0

GrZ0;Z G

rZ0;Z0

#; (7.117)

Gr0 D"Gr0Z;Z 0

0 Gr0Z0;Z0

#DA�1Z;Z 0

0 A�1Z0;Z0

�; (7.118)

U D

0 �AZ;Z0

�AZ0;Z 0

�: (7.119)

The left-connected retarded Green’s function grLq

is defined by the first q blocks of

Eq. (7.8) by

A1Wq;1Wq

grLq

D I1Wq;1Wq

: (7.120)

grLqC1

is defined in a manner identical to grLq

except that the left-connected system is

comprised of the first qC1 blocks of Eq. (7.8). In terms of Eq. (7.115), the equationgoverning gr

LqC1follows by setting Z D 1 W q and Z0 D q C 1. Using Eq. (7.116),

one obtains

grLqC1;qC1

D�AqC1;qC1

� AqC1;q

grLq;qAq;qC1

�1: (7.121)

It should be noted that the last block grLN;N

is equal to the fully connected Green’s

function GrLN;N

, which is the solution to Eq. (7.8). The full Green’s function can beexpressed in terms of the left-connected Green’s function by considering Eq. (7.115)such that AZ;Z D A

1Wq;1Wq, AZ0;Z0 D A

qC1WN;qC1WNand AZ;Z0 D A

1Wq;qC1WN. By noting

that the only non-zero block ofA1Wq;qC1WN

isAq;qC1

and using Eq. (7.116), one obtains


Grq;q

D grLq;q

C grLq;q

�Aq;qC1

GrqC1;qC1

AqC1;q

grLq;q

D grLq;q

� grLq;qAq;qC1

GrqC1;q

:

(7.122)

Both Grq;q

and GrqC1;q

are used for the calculation of the electron density, and thusstoring both sets of matrices will be useful. In view of the above equations, thealgorithm to compute the diagonal blocks Gr

q;qis given by the following steps:

• grL1;1

D A�11

• For q D 1; 2; : : : ; N � 1, Eq. (7.121) is computed• For q D N � 1; N � 2; : : : ; 1, Eq. (7.122) is computed

7.4.2 Lesser and Greater Green’s Functions

The algorithm to calculate the electron density (diagonal elements of G<) isdiscussed in terms of the Dyson equation for the lesser and the left-connectedGreen’s functions. The solution to the matrix equation

AZ;Z AZ;Z0

AZ0;Z AZ0;Z0

�"G<Z;Z G<

Z;Z0

G<Z0;Z G

<Z0;Z0

#D"˙<Z;Z ˙<

Z;Z0

˙<Z0;Z ˙

<Z0;Z0

#"GaZ;Z Ga

Z;Z0

GaZ0;Z G

aZ0;Z0

#

(7.123)

can be written as

G< D Gr0 U G< CGr0˙<Ga ; (7.124)

where Gr0 and U have been defined in Eqs. (7.118) and (7.119), and G< and Ga

are readily identifiable from Eq. (7.123). Using the relation Ga D Gr� D Ga0 CGa0U �Ga, Eq. (7.124) can be written as

G< D G<0 CG<0U � Ga CGr0UG< D G<0 CGrUG<0 CG<U �Ga0 ;

(7.125)

where

G<0 D Gr0 ˙< Ga0 : (7.126)

The left-connected lesser Green’s function g<Lq

is defined by the first q blocks of

Eq. (7.9):

A1Wq;1Wq

g<Lq

D ˙<1Wq;1Wq

gaL1Wq;1Wq

: (7.127)

7.4 Recursive Method for Calculating Green’s Functions 183

g<LqC1

is defined in a manner identical to g<Lq

except that the left-connected system is

comprised of the first qC1 blocks of Eq. (7.9). In terms of Eq. (7.123), the equationgoverning g<

LqC1follows by setting Z D 1 W q and Z0 D q C 1. Using the Dyson

equations for Gr and G<, g<LqC1;qC1

can be recursively obtained as [34]

g<LqC1;qC1

D grLqC1;qC1

h˙<

qC1;qC1C �<

qC1

igaLqC1;qC1

CgrLqC1;qC1

˙<qC1;q

gaLq;qC1

C grLqC1;q

˙<q;qC1

gaLqC1;qC1

;(7.128)

which can be written in a more intuitive form as

g<LqC1;qC1

D grLqC1;qC1

h˙<

qC1;qC1C �<

qC1�˙<

qC1;qgaLq;qA�q;qC1

� AqC1;q

grLq;q˙<

q;qC1

igaLqC1;qC1

;

(7.129)

where �<qC1

D AqC1;q

g<Lq;qA�q;qC1

. Equation (7.129) has the physical meaning that

g<LqC1;qC1

has contributions due to three injection functions: (i) an effective self-

energy due to the left-connected structure that ends at q, which is represented by�<qC1

, (ii) the diagonal self-energy component at grid point qC1 that enters Eq. (7.9),and (iii) the two off-diagonal self-energy components involving grid points q andq C 1.

To express the full lesser Green’s function in terms of the left-connected Green’sfunction, one can consider Eq. (7.123) such that AZ D A

1Wq;1Wq, A0

Z D AqC1WN;qC1WN

and AZ;Z0 D A1Wq;qC1WN

. Noting that the only non-zero block of A1Wq;qC1WN

is Aq;qC1

and using Eq. (7.125), one obtains

G<q;q

D g<Lq;q

� g<Lq;qA�q;qC1

GaqC1;q

� g<0q;qC1

A�qC1;q

Gaq;q

� grLq;qAq;qC1

G<qC1;q

:

(7.130)

Using Eq. (7.125), G<qC1;q

can be written in terms of G<qC1;qC1

and other knownGreen’s functions as

G<qC1;q

D g<0qC1;q

�GrqC1;q

Aq;qC1

g<0qC1;q

�GrqC1;qC1

AqC1;q

g<Lq;q

�G<qC1;qC1

A�q;qC1

gaLq;q

:

(7.131)

Substituting Eq. (7.131) in Eq. (7.130) and using Eq. (7.116), one obtains

G<q;q

D g<Lq;q

C grLq;q

�Aq;qC1

G<qC1;qC1

A�qC1;q

gaLq;q

�hg<Lq;qA�q;qC1

GaqC1;q

CGrq;qC1

AqC1;q

g<Lq;q

i�hg<0q;qC1

A�qC1;q

Gaq;q

CGrq;qAq;qC1

g<0qC1;q

i;

(7.132)


where

g<0q;qC1

D gr0q;q˙<

q;qC1ga0qC1;qC1

;

g<0qC1;q

D gr0qC1;qC1

˙<qC1;q

ga0q;q:

(7.133)

The terms inside the square brackets of Eq. (7.132) are Hermitian conjugates of eachother. In view of the above equations, the algorithm to compute the diagonal blocksof G< is given by the following steps:

• g<L1;1

D gr01;1˙<L g

a01;1

• For q D 1; 2; : : : ; N � 1, Eq. (7.129) is computed• For q D N � 1; N � 2; : : : ; 1, Eqs. (7.132) and (7.133) are computed

7.5 Evaluation of Observables

To solve the Poisson equation in a self-consistent scheme, one has to know thecarrier density profile in the device. To study device characteristics, the currentthrough the device needs to be calculated. In this section the numerical evaluation ofthese two observables is discussed. Here we assume x-axis as the transport-directionand also that carriers are free to move along two transverse-directions. The Green’sfunctions are therefore defined asG?

i1i2.kt; E/. It is straightforward to generalize this

model to other configurations.

7.5.1 Carrier Concentration

The diagonal elements of the Green’s function correspond to the spectrum ofcarrier occupation Eq. (6.177) at a given energy E. So the total electron and holeconcentration at some site i is given by

ni D � 2i

A�x

Xkt

ZdE

2�G<i;i .kt; E/ ; (7.134)

pi D 2i

A�x

Xkt

ZdE

2�G>i;i .kt; E/ ; (7.135)

where �x is the mesh spacing along the transport direction. The factor 2in Eqs. (7.134) and (7.135) is due to double spin degeneracy. To evaluate theseintegrals numerically, the energy grid should be selected such that the numericalerror of the calculation can be controlled. This issue is discussed in Sect. 7.6.

7.5 Evaluation of Observables 185

7.5.2 Current Density

By expanding the Green’s function in terms of the basis functions, the continuityequation (Eq. (6.185)) reads as

� 2i„e

A�x

Xkt

limt2!t1

�@t1G

<i;i .kt; t1; t2/ C @t2G

<i;i .kt; t1; t2/

�„ ƒ‚ …

@t %

C JiC1=2.t1/ � Ji�1=2.t1/�x„ ƒ‚ …r �J

D 0 ;

(7.136)

where JiC1=2 represents the current passing through a point between iC1 and i . Thetime derivative of the Green’s functions can be replaced by the relation Eq. (6.116):

@t%i D � 2e

A�x

Xj

nŒHi;jG

<j;i .kt; t; t / � G<

i;j .kt; t; t /Hj;i

CZdt 0 Œ˙ r

i;j .kt; t; t0/ G<

j;i .kt; t0; t / C ˙<

i;j .kt; t; t0/ Ga

j;i .kt; t0; t /

C Gri;j .kt; t; t

0/ ˙<j;i .kt; t

0; t / C G<i;j .kt; t; t

0/ ˙ aj;i .kt; t

0; t / o;

D �JiC1=2.t/ � Ji�1=2.t/�x

;

(7.137)

where the term inside the integral is zero due to the condition stated in Eq. (6.188).The next step is separating J

iC1=2from Ji�1=2 by decomposing Eq. (7.136). Caroli

proposed the following ansatz in [5]. The current J is the difference between theflow of particles from left to right and from right to left. This leads to the followingexpression for Ji [5]

JiC1=2.t/ D � 2e

A

Xkt

Xj�iC1

Xk�i

�H

j;kG<k;j .kt; t; t / �G<

j;k.kt; t; t /Hk;j

;

(7.138)

where the factor 2 comes from spin-degeneracy. It is straightforward to showthat Eq. (7.138) along with an expression for Ji�1 satisfies Eq. (7.137). Understeady-state condition, one can transform the time difference coordinate to energyto obtain


JiC1=2 D � 2e

A „X

kt

Xj�nC1

Xk�n

ZdE

2�

�Hj;kG

<k;j .kt; E/ �G<

j;k.kt; E/Hk;j

;

D � 2e

A „X

kt

Xj�iC1

Xk�i

ZdE

2�2 <eŒHj;kG

<k;j .kt; E/ :

(7.139)

If interactions up to the first nearest-neighbors are considered, the current densitywill be simplified as

JiC1=2 D � 2e

A „X

kt

ZdE

2�

�HiC1;iG<

i;iC1.kt; E/ �G<iC1;i .kt; E/Hi;iC1

�;

D � 2e

A „X

kt

ZdE

2�2 <eŒG<

iC1;i .kt; E/Hi;iC1 :

(7.140)

The carrier concentration is related to the diagonal elements of the Green’s function.The calculation of the current requires only the nearest off-diagonal elements ofthe Green’s function. Furthermore, the Hamiltonian matrix is tridiagonal. Con-sidering these factors, one can employ an efficient method, such as the recursiveGreen’s method, to calculate only the required elements of the Green’s functions,see Sect. 7.4.1. The operations required to solve for all elements of Gr with a sizeof N � N scales as N3. However, the required operations for the recursive methodscale linearly with N [34].

7.5.3 Transmission Probability

In phase-coherent transport, one can derive an alternative expression for currentdensity which is based on the transmission probability. We follow the approachdescribed in Ref. [1]. By using the recursive relations obtained in Sect. 7.4.2, onecan expand both terms of Eq. (7.140) and express the current density from the leftcontact as

JL D 2e

„Z

dE

2�

�ŒtLDG

r1;1.E/tDLg

<L0;0

.E/C tLDG<1;1.E/tDLg

aL0;0.E/

�ŒtDLG<L 0;0

.E/tLDGa1;1.E/C tDLg

aL0;0.E/tLDG

<1;1.E/

;

D 2e

„Z

dE

2�

�ŒGr

1;1.E/ �Ga1;1.E/ tDLg

<L0;0

.E/tLD

�G<1;1.E/tDLŒg

rL0;0.E/ � ga

L0;0.E/ tLD

�;

D 2e

„Z

dE

2�.ŒGr

1;1.E/ �Ga1;1.E/ ˙

<L .E/C iG<

1;1.E/�L.E// ; (7.141)

7.6 Selection of the Energy Grid 187

where for transition from the second to third line, the relations˙<L D tDLg

<L0;0

tLD Di�LfL and �L D itDLŒg

rL0;0

� gaL0;0 tLD have been used. In phase-coherent transport,

˙S D 0 and the only non-zero self-energies are in layers 1 and n due to the contacts.One can define matrices Q�Lj1;1 D �L and Q�Rjn;n D �R, which consist of n blockscorresponding to the n device layers. By left multiplying ADDG

rDD D I by Ga and

right multiplying the Hermitian conjugate of this relation by G, and subtracting theresulting two equations, one can show that

Gr �Ga D Ga.˙ r �˙ a/Gr , (7.142)

where ˙ r D ˙ rC C ˙ r

S is the total self-energy due to scattering processes and thecontacts. In the absence of scattering, Eq. (7.142) reads as

ŒGr �Ga D �iGa. Q� L C Q� R/Gr . (7.143)

It also follows from the definition of self-energies that

iG< D �Gr . Q� LfL C Q� RfR/Ga . (7.144)

Now using Eqs. (7.143) and (7.144) in Eq. (7.141), the current in phase-coherentlimit reads as

JL D 2e

„Z

dE

2�T .E/ŒfL.E/ � fR.E/ . (7.145)

The total transmission at energy E is identified from Eq. (7.145) to be

T .E/ D TraceŒ Q� L.E/Gr.E/ Q� R.E/G

a.E/ . (7.146)

7.6 Selection of the Energy Grid

For a numerical solution of the transport equations, one has to discretize the Green’sfunctions in the energy domain. However, due to the presence of narrow resonancesat some energies, one has to be careful about the selection of the energy grid. Anapproximation for the electron concentration due to a confined state is derived. Thisanalytical function is used as a reference for comparing the results of the non-adaptive and adaptive methods of selecting the energy grid.

7.6.1 Confined States

The non-interacting Green’s functions for electrons are given by Eq. (6.81). Fora bound state with the well-defined energy E0, the Green’s function is given by


Gr0.E/ D ŒE �E0 C i� �1, where � D 0C is a small positive number. The

renormalized retarded Green’s function in the presence of interaction is given bythe Dyson equation:

Gr.E/ D 1

ŒGr0.E/

�1 �˙ r.E/D 1

E �E0 �˙ r.E/: (7.147)

The self-energy has imaginary and real-parts:

˙ r D �C i�=2 : (7.148)

The imaginary part is interpreted as damping of the particle motion. It is related tothe finite life-time of the state. The real-part causes an energy shift, which may alsochange the effective mass or group velocity. Under equilibrium condition, the lesserGreen’s function is given by (see Eq. (6.85))

G<.E/ D if .E/A.E/ D if .E/ � �2=mŒGr0.E/ D if .E/

�

.E �E0 C�/2 C .� =2/2:

(7.149)

The lesser Green’s function is of Lorentzian shape [6]. The peak of the resonanceis shifted by � and is broadened by � , as shown in the inset of Fig. 7.5. In opensystems, localized states broaden due to the coupling to contacts (� > 0), even inthe absence of scattering processes.

The electron concentration for each of the confined states can be calculated as

n DZ Emax

Emin

dE

2�f .E/

�

.E �E0 C�/2 C .� =2/2: (7.150)

We assume that the Fermi level is far above the resonanceEF E0 ��, so that theFermi factor can be replaced by 1. Equation (7.150) is used as a reference functionfor comparing the results of different numerical integration methods.

7.6.2 Non-adaptive Energy Grid

One can straightforwardly divide the integration domain into NE equidistantintervals �E D .Emax � Emin/=NE. A disadvantage of this method is that thenumerical error can not be pre-defined. This problem is more pronounced whenthe integrand is not smooth. To evaluate Eq. (7.150) numerically, a trapezoidal ruleand an equidistant grid spacing are used. The dependence of the accuracy on thefollowing two parameters is studied, namely, the grid spacing, �E, and the relativedistance between the peak and the nearest grid point, ıE. These parameters arenormalized as ˛ D �E=� and ˇ D ıE=� . The relative error in calculating thecarrier concentration, .n�Qn.˛//=n, as a function of grid spacing is shown in Fig. 7.5.


Fig. 7.5 The relative error in evaluating the carrier concentration .n� Qn.˛//=n, with respect to thegrid spacing, is shown. The inset shows the normalized Lorentzian shape of the density of states ofa bound state. The peak of the resonance is shifted to the zero point. At E D ˙�=2 the functionis half of its maximum. The solid line shows the exact function and the dashed curve shows theapproximation of the function based on the Trapezoidal rule. The grid spacing is �E and the shiftof energy grids from the reference point is ıE. These parameters are normalized as ˛ D �E=�

and ˇ D ıE=� . The reference ˇ D 0 implies that the one of the grid points aligns with the peakof the resonance. The parameters in this figure are ˛ D 1=3 and ˇ D 0

Fig. 7.6 The relative variation of the calculated carrier concentration .Qn.0/ � Qn.ˇ//=Qn.0/ withrespect to the normalized position ˇ of energy grid points

Here, n is the analytically exact value of the carrier concentration Eq. (7.150) and Qnrefers to the numerically calculated carrier concentration as a function of ˛ and ˇ.

The variation of the calculated carrier concentration . Qn.0/ � Qn.ˇ//= Qn.0/ withrespect to the shift of energy points is shown in Fig. 7.6. The reference ˇ D 0 impliesthat one of the grid points aligns with the peak of the resonance. The oscillatorybehavior depends on the grid spacing. A shift equal to the grid spacing gives thesame result. As a measure of the sensitivity of the calculated carrier concentrationwith respect to grid positions, @ Qn= Qn@ˇ is shown in Fig. 7.7. To reduce this sensitivity,very fine grid spacing has to be adopted. This quantity is characteristic of the


Fig. 7.7 The relativesensitivity of the calculatedcarrier concentration @Qn=Qn@ˇwith respect to the position ofenergy grid points. This termoriginates from the numericalerror in the evaluation of thecarrier concentration. Forcoarse grid spacing ˛ > 1,this quantity increasesconsiderably

numerical error, and needs to be controlled to avoid a convergence problem in theself-consistent iteration loop (see Sect. 7.7.2).

In summary, the accuracy of the non-adaptive method strongly depends on thegrid spacing and the position of grid points. If the grid spacing is sufficiently fine,˛ < 1, the numerical error is small, but it increases considerably for coarser gridspacing, ˛ > 1. For accurate results, a grid spacing smaller than � has to beemployed. For example, to resolve a resonance of � � 1eV width in an energyrange of 1 eV, more than 106 energy grid points are required, which would severelyincrease the computational cost. For even narrower resonances, (e.g., � � 1 n eV),an equidistant grid is no longer feasible. To avoid these problems an adaptive methodneeds to be employed.

7.6.3 Adaptive Energy Grid

There are a variety of methods available for numerical adaptive integration [8].Adaptive strategies divide the integration interval into sub-intervals and typicallyemploy a progressive formula in each sub-interval with some fixed upper limit onthe number of points. If the required accuracy is not achieved by the progressiveformula, the sub-interval is bisected and a similar procedure carried out on eachhalf. This sub-division process is carried out recursively until the desired accuracyis achieved. An obvious way to obtain an error estimate is based on the comparisonbetween two quadrature approximations [22]. However, due to the dependence ofsuch procedures on the underlying integration formula, this method may not bereliable [10]. An error estimation with sequences of null rules has been proposed asa simple solution [9]. In adaptive quadrature algorithms, the error estimate governsthe decision on whether to accept the current approximation and terminate or tocontinue. Therefore, both the efficiency and the reliability depend on the errorestimation algorithm. The decision to further subdivide a region may be based oneither local or global information. Local information refers only to the region being


a b

Fig. 7.8 (a) shows the number of required energy grid points versus the maximum desired relativeerror, �, in the adaptive integration method; (b) shows the number of required energy grid pointsversus the width of the resonance, � . The Lorentzian function Eq. (7.150) is used as a reference

currently processed, while global information refers obviously to data concerningall regions. Integration programs based on global subdivision strategies are moreefficient and reliable [24].

In this work a global error estimator based on the null rules method has beenemployed [10]. The efficiency of this method is studied for the Lorentzian functionEq. (7.150). Figure 7.8a shows the number of required energy grid points for aninterval Œ�1; 1 eV versus the relative error � of the integration. The required numberof energy grid points versus the width of the resonance, � , is shown in Fig. 7.8b.To resolve a very narrow resonance (� � 10�9 eV) with very high accuracy(� D 10�6), only a few hundred grid points (NE � 500) are required. Figure 7.9shows the normalized spectrum of the carrier concentration in a Schottky typeCNT-FET. The length of the device is 50 nm. Energy barriers at the metal-CNTinterfaces cause longitudinal confinement in the tube. Since the device is quite long,the spacing between confined states is very small. In CNTs the electron-phononinteraction is rather weak and the confining Schottky barriers are thick, such thatresonances are only weakly broadened. Due to phonon absorption and emissionprocesses, there will be more resonances compared to the ballistic case. In this case,if a non-adaptive method is employed, the numerical error in the calculation of thecarrier concentration can be large. The right part of Fig. 7.9 compares the resultsachieved from the adaptive and non-adaptive methods. The relative error in theelectron density of the non-adaptive method reaches up to 53 % in the middle ofthe device.

In Ref. [11] the resonant states have been determined by an eigenvalue solverfor finding the roots of the characteristic equation. However, this method hasseveral drawbacks. Due to the non-linearity introduced by the self-energies, a non-linear eigenvalue solver has to be employed. Usually non-linear solvers are based


Fig. 7.9 The left figure shows the normalized spectrum of the carrier concentration in a Schottkytype CNT-FET. The right figure shows the spectrum of the carrier concentration in the middle of thedevice for the energy range shown by the arrow. The results achieved from the adaptive and non-adaptive method are compared. With the aid of the adaptive method, narrow and close resonancesare resolved with a total number of NE � 1;000 energy grid points, whereas the non-adaptivemethod misses some resonances with the same number of energy grid points

on Newton’s method. Using a non-linear solver for each iteration can increasethe simulation time severely and introduce additional convergence problems. Forexample, most solvers fail to find narrow resonances located closely to each other.The output of the solver is the energy position and the width of the resonance butdoes not give any information about the shape of the resonance. In general the shapeof resonances deviates from the ideal Lorentzian shape. The grid has to be allocatedbased on an initial guess. This implies that the accuracy of the calculated carrierconcentration can not be predefined and strongly depends on the how close the initialguess is to the actual solution. With the adaptive method, the discussed problems donot occur.

7.7 Self-Consistent Simulations

For an accurate analysis it is essential to solve the coupled system of transportand Poisson equations self-consistently [13]. The iterative method for solving thiscoupled system is presented. Thereafter, the convergence behavior of the self-consistent iteration is studied.

7.7 Self-Consistent Simulations 193

Fig. 7.10 Block diagram of the iterative procedure employed to solve the coupled system oftransport and Poisson equations. For the first step, an initial guess for the scattering self-energy isrequired, here we assume ˙S D 0

7.7.1 Self-Consistent Iteration Scheme

Figure 7.10 depicts the block diagram of the iterative procedure employed to solvethe coupled system of transport and Poisson equations. One should solve the kinetic


equations to obtain the Green’s functions. The required elements for calculating theGreen’s functions are the Hamiltonian and the scattering and contact self-energies.Diagonal elements of the Hamiltonian are potential energies, which can be obtainedfrom the solution of the Poisson equation, and off-diagonal elements representthe hopping parameters between neighbors. Given the contact properties and thecontact-device coupling parameters, the contact self-energy ˙ r

C can be calculatedonce at the start of the simulation (see Sect. 7.2.4).

The calculation of the scattering self-energies˙ rS is presented in Sect. 7.3. Within

the self-consistent Born approximation of the electron-phonon self-energy, the non-interacting Green’s functionG0 is replaced by the full Green’s functionG. However,the full Green’s is the not known and has to be calculated. As a result, a coupledsystem of equations is achieved, which can be solved by iteration

Grhmi D ŒEI �H �˙ rhm�1i

S �˙ rC ; (7.151)

wherem denotes the iteration number. For the first step, the scattering self-energy isassumed to be zero and the Green’s function is calculated from the kinetic equation.The next iteration starts with the calculation of the scattering self-energy basedon the Green’s function from the previous iteration. The updated self-energy isthen used for the calculation of the Green’s function. This iteration continues tilla convergence criterion is satisfied. Finally, the total charge density is calculated.

In semi-classical simulations, the coupled system of the transport and Poissonequations is solved by Gummel’s iterative or Newton’s method [32]. Both Gummel’smethod [30] and a variation of Newton’s method [20] can be employed in self-consistent quantum mechanical simulations. While Gummel’s method has a fastinitial error reduction, for Newton’s method it is very important that the initial guessbe close to the solution. However, the computational cost per iteration of Newton’smethod can be much higher than that for Gummel’s method. After convergenceof the scattering self-energies, the Poisson equation is solved once. Based on theupdated electrostatic potential, the Green’s functions and the scattering self-energiesare iterated again. These two iterations continue until a convergence criterion issatisfied. Finally, the total current through the device is calculated.

7.7.2 Convergence of the Self-Consistent Simulations

The coupled system of transport and Poisson equation can be solved by iterationwith appropriate numerical damping, which terminates if a convergence criterionis satisfied [33]. In this work, the maximum element of the potential update,corresponding to L1 D j�k � �k�1j1, is considered as a measure of convergence.

One of the reasons for convergence problems [16, 36] is the exponential depen-dence of the carrier concentration on the electrostatic potential, n / exp.q�=kBT /.A small potential variation causes large variation in the carrier concentration:

7.7 Self-Consistent Simulations 195

@n

n@�� e

kBT: (7.152)

As a result, a strong damping is required in many cases, which increases thesimulation time. To avoid this problem, a non-linear Poisson equation is generallyemployed [37]. Solving a non-linear Poisson equation takes such an exponentialdependence into account. Compared to the linear Poisson equation, it leads to fasterconvergence in both semi-classical [37] and quantum mechanical [28, 36] transportsimulations. In this work, the Gummel method along with a non-linear Poissonequation is employed.

However, we show that an inappropriate energy grid for the discretization ofthe transport equations can be another reason for convergence problems in quantumtransport simulations. It is demonstrated that with adaptive energy grids, the iterativesolution can converge very fast and the simulation time can decrease considerably.

In Sect. 7.6.2 it was shown that with a shift of an equidistant grid, the calcu-lated carrier concentration can change sharply. This sensitivity resulting from thenumerical error causes convergence problems in the self-consistent loop. In all non-adaptive methods, some fixed energy grid is adopted. In successive iterations of thePoisson and transport equations, the electrostatic potential changes and this in turnaffects the relative distance between resonance energies and the energy grid points.As a result, the evaluated carrier concentration can vary sharply in one iterationstep, which affects the calculated electrostatic potential for the next iteration. Fora quantitative analysis, one can assume that the shift of energy grid is due to thevariation of the electrostatic potential e@� D ıE D � @ˇ:

@ QnQn@� D q

�

@ QnQn@ˇ : (7.153)

The sensitivity of the calculated carrier concentration with respect to energy gridshifts defined by Eq. (7.153) is shown in Fig. 7.7. For a relatively coarse grid, ˛ �2�3, the sensitivity Eq. (7.153) can be approximated as

@ QnQn@� � q

�: (7.154)

In this case the contribution of a resonance of width � kBT will be larger thanEq. (7.152). To reduce the effect of this term, fine grid spacing must be used (˛ 1). The non-adaptive method requires many grid points to resolve fine resonances,while the adaptive method puts many energy grid points close to resonances and fewones away from resonances. Therefore, the adaptive method keeps the total numberof grid points quite low and maintains a high accuracy.

The convergence of the self-consistent loop using the adaptive and non-adaptivemethods is studied. With the non-adaptive method, 104 energy grid points are used.For the adaptive method, relative errors of � D 10�3 and � D 10�6 are assumed.


Fig. 7.11 The infinity normof the potential update aftereach iteration

23 24 25 26 27Position [nm]

10-1

100

Nor

mal

ized

ele

ctro

n co

ncen

trat

ion

It. 1

It. 2

It. 3

It. 4

Fig. 7.12 Explanation of theoscillation in the calculatedcarrier concentration insuccessive self-consistentiterations. The results are fora non-adaptive method with104 grid points. Thesensitivity of the calculationsfor each of these iterations isshown in Fig. 7.7. The firstand third iterations are in ahigh sensitive region, whilethe second and the fourthiterations are in a lowsensitive region

Figure 7.11 shows the infinity norm of the potential update after each iteration.With the adaptive method the norm of the potential update decreases exponentiallyand finally reaches a limit, which depends on the error tolerance of the integration.With the non-adaptive method, the norm of the potential update oscillates and noconvergence is achieved. Figure 7.12 shows the calculated carrier concentration dueto several confined states, based on four successive iterations of the non-adaptivemethod. From the first to the second iteration, the carrier concentration changesvery sharply. Therefore, at the first iteration one is close to the highly sensitiveregion (see Fig. 7.7). From the second to the third iteration, the carrier concentrationchanges only a little, which can be mapped to the low sensitive region. Fromthe third to the fourth iteration, the variation is large, which implies that we areagain close to the highly sensitive region. This sequence continues and preventsconvergence. To avoid this problem, a fine grid spacing can be used, which decreasessensitivity in all regions. As was shown in Fig. 7.5, the non-adaptive method requiresa grid spacing smaller than � for accurate result.

References 197

Fig. 7.13 The infinity normof the potential update versusCPU-time. Simulations basedon the adaptive methodconverge fast and theminimum achievable norm ofthe potential update dependson the accuracy of theintegration. The non-adaptivemethod does not convergence

By reducing � D 10�3 to � D 10�6 in the adaptive method, the self-consistentiteration yields more accurate results but the number of required energy grid pointsincreases, which increases the simulation time of each iteration. Figure 7.13 showsthe infinity norm of the potential update versus CPU-time. A suitable criterion forthe termination of the self-consistent loop was found as qj�k � �k�1j1 < kBT=10.If the maximum potential update in the device is much smaller than kBT, the carrierconcentration will change only weakly during the next iteration. For most of thesimulations performed, such a criterion was satisfied for � � 10�3.

References

1. Anantram, M.P., Lundstrom, M.S., Nikonov, D.E.: Modeling of nanoscale devices. Proc. IEEE96(9), 1511–1550 (2008)

2. Anantram, M.P., Svizhenko, A.: Multidimensional modelling of nanotransistors. IEEE Trans.Electron Devices 54(9), 2100–2115 (2007)

3. Bowen, R.C., Klimeck, G., Lake, R., Frensley, W.R., Moise, T.: Quantitative simulation of aresonant tunneling diode. J. Appl. Phys. 81(7), 3207–3213 (1997)

4. Caroli, C., Combescot, R., Lederer, D., Nozieres, P., Saint-James, D.: A direct calculation ofthe tunnelling current. II. Free electron description. J. Phys. C Solid State Phys. 4(16), 2598–2610 (1971)

5. Caroli, C., Combescot, R., Nozieres, P., Saint-James, D.: Direct calculation of the tunnelingcurrent. J. Phys. C Solid State Phys. 4(8), 916–929 (1971)

6. Datta, S.: Electronic Transport in Mesoscopic Systems. Cambridge University Press, New York(1995)

7. Datta, S.: Nanoscale device modeling: the Green’s function method. Superlattices Microstruct.28(4), 253–278 (2000)

8. Davis, P., Rabinowitz, P.: Methods of Numerical Integration, 2nd edn. Academic, Orlando(1984)

9. Espelid, T.: DQAINT: An Algorithm for Adaptive Quadrature over a Collection of FiniteIntervals. NATO Advanced Study Institutes Series: Series C, Mathematical and PhysicalSciences, vol. 357, pp. 341–342. Kluwer Academic, Boston (1992)


10. Espelid, T.: Adaptive doubly quadrature routines based on Newton-Cotes Rules. BIT 43(2),319–337 (2003)

11. Fernando, C., Frensley, W.: An efficient method for the numerical evaluation of resonant states.J. Appl. Phys. 76(5), 2881–2886 (1994)

12. Fetter, A.L., Walecka, J.D.: Quantum Theory of Many-Particle Systems. McGraw-Hill,San Francisco (1971)

13. Ghosh, A.W., Rakshit, T., Datta, S.: Gating of a molecular transistor: electrostatic andconformational. Nano Lett. 4(4), 565–568 (2004)

14. Guo, J., Datta, S., Lundstrom, M., Anantram, M.: Multi-scale modeling of carbon nanotubetransistors. Int. J. Multiscale Comput. Eng. 2(2), 257–278 (2004)

15. Jauho, A., Wingreen, N.S., Meir, Y.: Time-dependent transport in interacting and noninteract-ing resonant-tunneling systems. Phys. Rev. B 50(8), 5528–5544 (1994)

16. Kerkhoven, T., Galick, A., Ravaioli, U., Arends, J., Saad, Y.: Efficient numerical simulation ofelectron states in quantum wires. J. Appl. Phys. 68(7), 3461–3469 (1990)

17. Kienle, D., Ghosh, A.W.: Atomistic modeling of metal-nanotube contacts. J. Comput. Electron.4(1–2), 97–100 (2005)

18. Lake, R., Klimeck, G., Bowen, R.C., Jovanovic, D.: Single and multiband modeling of quantumelectron transport through layered semiconductor devices. J. Appl. Phys. 81(12), 7845–7869(1997)

19. Lake, R., Pandey, R.R.: Non-equilibrium Green functions in electronic device modeling. In:Handbook of Semiconductor Nanostructures and Devices, vol. 3, pp. 409–443. AmericanScientific Publishers, Los Angles (2006)

20. Laux, S., Kumar, A., Fischetti, M.: Analysis of quantum ballistic electron transport inultrasmall silicon devices including space-charge and geometric effects. J. Appl. Phys. 95(10),5545–5582 (2004)

21. Luisier, M., Schenk, A., Fichtner, W., Klimeck, G.: Atomistic simulation of nanowires in thesp3d5s� tight-binding formalism: from boundary conditions to strain calculations. Phys. Rev.B 74, 205323 (2006)

22. Lyness, J.: Notes on the adaptive Simpson quadrature routine. J. ACM 16(3), 483–495 (1969)23. Mahan, G.D.: Many-Particle Physics. Physics of Solids and Liquids, 2nd edn. Plenum Press,

New York (1990)24. Malcolm, M., Simpson, R.: Local versus global strategies for adaptive quadratures. ACM

Trans. Math. Softw. 1(2), 129–146 (1975)25. Migdal, A.B.: Interaction between electrons and lattice vibrations in a normal metal. Sov. Phys.

JETP 7(6), 996–1001 (1958)26. Nemec, N., Tománek, D., Cuniberti, G.: Contact dependence of carrier injection in carbon

nanotubes: an ab initio study. Phys. Rev. Lett. 96, 076802 (2006)27. Nemec, N., Tomanek, D., Cuniberti, G.: Modeling extended contacts for nanotube and

Graphene devices. Phys. Rev. B 77, 125420 (2008)28. Pacelli, A.: Self-consistent solution of the Schrödinger equation in semiconductor devices by

implicit iteration. IEEE Trans. Electron Devices 44(7), 1169–1171 (1997)29. Palacios, J., Louis, E., Pérez-Jiménez, A.J., Fabián, E.S., Vergés, J.: An ab initio approach to

electrical transport in molecular devices. Nanotechnology 13(3), 378–381 (2002)30. Pinaud, O.: Transient simulations of a resonant tunneling diode. J. Appl. Phys. 92(4), 1987–

1994 (2002)31. Sancho, M.P.L., Rubio, J.M.L., Rubio, L.: Highly convergent schemes for the calculation of

bulk and surface Green functions. J. Phys. F Met. Phys. 15(4), 851–858 (1985)32. Selberherr, S.: Analysis and Simulation of Semiconductor Devices. Springer, Vienna (1984)33. Stern, F.: Iteration methods for calculating self-consistent fields in semiconductor inversion

layers. J. Comput. Phys. 6(1), 56–67 (1970)34. Svizhenko, A., Anantram, M.P., Govindan, T.R., Biegel, B., Venugopal, R.: Two-dimensional

quantum mechanical modeling of nanotransistors. J. Appl. Phys. 91(4), 2343–2354 (2002)35. Ting, D.Z.Y., Change, Y.C.: � -X mixing in GaAs/AlxGa1�xAs and AlxGa1�xAs/AlAs

superlattices. Phys. Rev. B 36(8), 4359–4374 (1987)

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36. Trellakis, A., Galick, A.T., Pacelli, A., Ravaioli, U.: Iteration scheme for the solution of thetwo-dimensional Schrödinger-Poisson equations in quantum structures. J. Appl. Phys. 81(12),7880–7884 (1997)

37. Venturi, F., Smith, R., Sangiorgi, E., Pinto, M., Ricco, B.: A general purpose device simulatorcoupling Poisson and Monte Carlo transport with applications to deep submicron MOSFETs.IEEE Trans. Comput.-Aided Des. 8(4), 360–369 (1989)

38. Verzijl, C.J.O., Seldenthuis, J.S., Thijssen, J.M.: Applicability of the wide-band limit in DFT-based molecular transport calculations. J. Chem. Phys. 138, 094102 (2013)

39. Wacker, A.: Semiconductor superlattices: a model system for nonlinear transport. Phys. Rep.357(1), 1–111 (2002)

Chapter 8Applications

In this chapter we apply the formalisms and techniques that have been introducedto graphene, nanotube, and nanoribbon-based devices with different operatingprinciples. Charge, spin, and phonon transport are studied. Line-edge roughness andsubstrate surface corrugation are modeled and their effects on transport propertiesare investigated. Finally, the characteristics of tunneling transistors and photodetec-tors are evaluated, employing the non-equilibrium Green’s function formalism.

8.1 Introduction

The continued miniaturization of Si integrated devices in CMOS technology isapproaching the physical limits. To meet the ITRS roadmap [98], novel nano-electronic devices are expected to be utilized. Carbon-based materials have beenextensively studied in recent years due to their exceptional electronic, opto-electronic, and mechanical properties [5,9]. Carbon materials are found in a varietypf forms, such as fullerenes, graphite, graphene, carbon nanotubes, and diamond.The reason why carbon assumes many structural forms is that a carbon atom canform several distinct types of orbital hybridization. Carbon atoms have a total ofsix electrons: two in the inner shell and four in the outer shell. The four outershell electrons in an individual carbon atom are available for chemical bonding.The spn-hybridization of these outer electrons is essential for determining thedimensionality of not only carbon based molecules but also carbon based solids.Carbon is the only element in the periodic table that has isomers from zero-dimensions to three-dimensions. In spn-hybridization, .n C 1/ � bonds per carbonatom are formed, which form a skeleton for the local structure of the n-dimensionalstructure. In sp-hybridization, two � bonds make a one-dimensional chain structure,which is known as a carbyne. Interestingly, sp2-hybridization, which is known asgraphene, also forms a planar local structure in the closed polyhedra of the (zero-dimensional) fullerene family and the one-dimensional cylinders called carbon


201

202 8 Applications

Fig. 8.1 The energydispersion relations forgraphene are shown throughthe whole region of theBrillouin zone. Thecoordinates of high symmetrypoints are � D .0; 0/,KD .0; 2�=3a/,MD .2�=

p3a; 0/. The

energy value for the K, M,and � are 0, t , and 3t ,respectively

nanotubes (CNTs). Four � bonds defining a regular tetrahedron are sufficient toform a three-dimensional structure known as the diamond structure.

Among various forms of carbon, graphene, a one-atomic carbon sheet with ahoneycomb structure, has attracted significant attention due to its unique proper-ties [34,35]. In graphene, 2s, 2px , and 2py orbitals hybridize such that each carbonatom is bonded to its three neighbors by strong sp2 or � bonds. The remaining pz or� orbital determines the low-energy electronic structure of graphene. The unit cellof graphene contains two � orbitals, which disperse to form two � bands that maybe thought of as bonding (the lower energy valence band) and anti-bonding (thehigher energy conduction band) in nature. The bonding-anti-bonding gap closesat the corners of the Brillouin zone, or the K points, see Fig. 8.1. As a result,the �-band dispersion is approximately linear around the K points: E D „vFjkj,where k is the wave vector measured from K-point and vF is the Fermi velocityin graphene, which is approximately 106 m/s. Due to the unique linear dispersion,carriers in graphene move at a constant speed of vF that does not depend on theirkinetic energy. This is similar to the behavior of photons, which always travel atthe speed of light. From a physical point of view, it is an excellent platform forthe study of the massless Dirac fermion system [35, 60, 89]. This material shows anextraordinarily high carrier mobility of more than 2 � 105 cm2=Vs [14, 19, 25, 83]and is considered to be a major candidate for future high speed transistor materials.In addition, graphene has shown its ability to transport charge carriers with spincoherence even at room temperature and is regarded as a pivotal material in theemerging field of spintronics [20, 107].

One of the problems which limits the application of graphene in electronicdevices is the absence of an intrinsic bandgap. To induce an electronic bandgap onecan confine electrons. Structures based on graphene that realize this behavior arecarbon nanotubes (CNTs) and graphene nanoribbons (GNRs) with, respectively,

8.1 Introduction 203

Armchair

Zigzag

Chiral

x

ba

period

ic b

ound

ary

cond

itio

nFig. 8.2 (a) The vector Ch D na1 C ma2 describes how to roll up the graphene sheet to makethe CNT. T denotes the tube axis, and a1 and a2 are the unit vectors of graphene in real space. Thechiral angle � is measured with respect to the zigzag axis (� D 0). � D 30-direction denotes anarmchair .n; n/ CNT, � D 0-direction indicates a zigzag .n; 0/ CNT, and a general � -direction,with 0 < � < 30ı, depicts a chiral .n;m/ CNT. (b) Schematic model of a rolled and unrolled.n; 0/ zigzag CNT. The first nearest neighbor hoping parameters are represented by t

periodic and zero boundary conditions for the transverse electron wave vector. ACNT can be viewed as a rolled-up sheet of graphene with a diameter of a fewnano-meters. As depicted in Fig. 8.2, depending on the chiral angle, the CNTcan either be metallic or semiconducting. Semiconducting CNTs can be used aschannels for transistors. CNT-FETs have been the subject of intensive researchfor the last decade [9, 17, 53]. Recently, graphene sheets have been patterned intonarrow nanoribbons [13]. GNRs have recently attracted much interest as theyare recognized as promising building blocks for nanoelectronic devices [31]. Theelectronic properties of GNRs exhibit a dependence on the ribbon direction andwidth. In GNRs with zigzag edges, transport is dominated by edge states. Owingto their high degeneracy, these states are spin polarized, making zigzag GNRsattractive for spintronics [102]. In principle, GNRs can be patterned directly intodevice structures and even into integrated circuits by a single patterning process ofa graphene sheet, as has been demonstrated by recent experiments [18, 44, 105].As shown in Fig. 8.3, with respect to the crystallographic direction, GNRs canbe categorized as armchair (AGNR), zigzag (ZGNR), or a combination of thesetwo [64]. This chapter continues with the application of the NEGF to CNT andGNR-based devices to study charge, spin, and phonon transport in such structures.

204 8 Applications

1A 1B

3A 3B

2B 2A

NA NB

unit cell

hard

-wal

l bo

unda

ry c

ondi

tion

x

y

armchair edge1A

2A

NA

unit cell

zigzag edge

1B

2B

NB

a b

Fig. 8.3 The structure of GNRs with (a) armchair and (b) zigzag edges along the x-direction.Each unit-cell consists of N numbers of A and B sublattices. The additional hard wall boundaryconditions are imposed on both edges (p D 0 and p D N C 1) [124]

8.2 Electronic Transport

The GNR lattice consists of two sublattices, A and B (see Fig. 8.3a). The unitcell of an armchair GNR contains A A-type and A B-type carbon atoms and thewidth is given by W D .N C 1/.

p3=2/acc, where N is the number of carbon

atoms along the width direction and acc D 1:42Å is the distance between nearestneighbor carbon atoms. A first nearest neighbor tight binding model is commonlyused to model the bandstructure of GNRs [124]. This model predicts that one-third of all armchair GNR configurations are metallic [15, 28]. However, bothexperimental data [18, 44, 69] and ab-initio calculations [12, 101, 113] show that allnarrow armchair GNRs have a finite bandgap. A tight binding model can accuratelydescribe the electronic bandstructure of GNRs only if interactions up to third-nearestneighbors are considered [40, 113]:

OH D OH1 C OH3 : (8.1)

Here OH1 and OH3 represent the first and third nearest neighbor Hamiltonians, respec-tively. These Hamiltonians can be expressed in terms of creation and annihilationoperators c� and c, respectively, which act on the � state on each site:

OH1 DXhi;j i

ti;j Oc�i Ocj ; OH3 DXhi;ki

ti;k Oc�i Ock : (8.2)

Here ti;j � �3:2 eV and ti;k � �0:3 eV are the hopping parameter between thefirst and the third nearest neighbor, respectively. The summation of i runs over theentire nanoribbon lattice, while j and k are restricted to the first and the third nearestneighbor to site i , respectively.

8.2 Electronic Transport 205

8.2.1 Transport Models

The transmission probability of carriers through the device, in the absence of phase-incoherent scattering processes, can be evaluated as [80], see Sect. 7.5.3:

T .E/ D TraceŒ� LGrD� RG

aD ; (8.3)

where � L=R is the broadening function of the left or right contact. In the linearresponse regime the spectrum of conductance is given by

G.E/ D G0T .E/

�� @f@E

�; (8.4)

with G0 D 2q2=h. In the absence of scattering, carrier transport is in the ballisticregime where the conductance is independent of the device length. In the presenceof scattering, transport of carriers is in the diffusive regime where the transmissioncan be written as

hT .E/i D Nch

1C L=�.E/: (8.5)

Here �.E/ is the mean free path, Nch is the number of active conduction chan-nels, and L is the length of the ribbon. Therefore, the conductance in the diffusiveregime is inversely proportional to the device length:

G.E/ � G0Nch

1C L=�.E/

�� @f@E

�: (8.6)

For phase coherent transport, in the presence of disorder, the carrier wave functioncan be scattered back and forth between potential barriers, and standing waves alongthe device form. In this regime, referred to as the localization regime, the transportof carriers is due to tunneling between localized states, and the average transmissionprobability decreases exponentially with the device length [4]:

hlnŒT .E/ i � �L=�.E/ ; (8.7)

where �.E/ is the localization length. In the localization regime, the conductancecan be obtained as [22]

G.E/ � G0 exp

� L

�.E/

�� @f@E

�: (8.8)

The mean free path can be obtained by fitting a linear function to the inverse ofthe average transmission in the diffusive regime. In the localization regime, thetransmission probability decreases exponentially with the length. Therefore, the

206 8 Applications

slope of the transmission probability as a function of length in the logarithmic scaleis inversely proportional to the localization length.

8.2.2 Line-Edge Roughness

In armchair GNRs, the bandgap is inversely proportional to the width. A bandgapsuitable for electronic applications can be achieved by making sufficiently narrowGNRs W < 10 nm, [44]. Both experimental data [116] and theoretical predic-tions [8, 27, 41, 84] indicate that the LER is the dominant source of scattering innarrow GNRs. LER is a statistical phenomenon which can be well described bymeans of an auto-correlation function (ACF) [39]. Assuming that the deviation ofthe width from its ideal value at some position x1 is represented by ıW.x1/, thenthe correlation between ıW.x1/ and ıW.x2/ can be described by an ACF as

R.x1; x2/ D hıW.x1/ıW.x2/i ; (8.9)

where h: : :i denotes statistical ensemble average. For stationary processes,R.x1; x2/depends only on the relative distance between the points, R.x1; x2/ D R.x1 � x2/.The Fourier transform of the ACF is called the power spectral density:

R.q/ DZ

dxR.x/ exp.�iqx/ : (8.10)

It is common to use a Gaussian or an exponential ACF to describe line-edgeroughness. In Ref. [39] an exponential ACF has been employed to model Si=SiO2

interface roughness. Short wave length fluctuations of the edge arise from the highfrequency tail of the exponential power spectrum [90]. Here an exponential ACF isused [118, 119]:

R.x/ D �W 2 exp

�� jxj�L

�; x D n�x ; (8.11)

where�W is the root mean square of the fluctuation amplitude,�L is the roughnesscorrelation length, which is a measure of smoothness, and �x is the samplinginterval chosen equal to acc=2. To create line-edge roughness in real space, we firstevaluate the Fourier transform of the ACF, which gives the power spectrum of theroughness:

R.q/ D �W 2�L

1C q2�L2: (8.12)

By applying a random phase to the power spectrum followed by an inverse Fouriertransform, roughness in real space is achieved [115]. We create many samples


Fig. 8.4 (a) The average transmission probability (solid-black line) over different samples andthe transmission probability of each sample (gray lines) as a function of energy at L D 2 nmand L D 10 nm. (b) Comparison between the average transmission probabilities as a functionof energy at various lengths. For all devices �L D 3 nm and �W=W D 2% (Reported withpermission from Ref. [119]. Copyright (2011), IEEE)

with the same roughness parameters and evaluate their electronic properties,employing the NEGF. Finally, a statistical ensemble averaging on these samplesis performed. The transmission probability for many samples and the averagetransmission probability for different lengths are shown in Fig. 8.4. With increasingthe ribbon’s length, the average transmission probability is reduced. Furthermore,as the length increases, the steps in the transmission probability are smoothed out.To discriminate the diffusive and the localization transport regimes, the averagetransmission probability hT .E/i and hlnŒT .E/ i as a function of length at E D0:6 eV are shown in Fig. 8.5. In short ribbons, the average transmission is inverselyproportional to the length (diffusive regime), but for longer ribbons, the transmissionprobability decreases exponentially with the length. In the diffusive regime one canobtain the mean free path �.E/ by fitting a curve similar to Eq. (8.5) to the averagetransmission probability, see Fig. 8.5a. If L � �, the transport is ballistic.

In the localization regime, where the transmission probability decreases expo-nentially with the length, one can obtain the localization length �.E/ by fitting acurve similar to Eq. (8.7), see Fig. 8.5b. For L � � , transport will be diffusive. AtE D 0:6 eV the mean free path can be estimated as � � 10 nm and the localizationlength is � � 30 nm. It can be shown that the ratio of the localization length to themean free path is proportional to the number of available subbands [106]:

�

�/ Nch.E/ : (8.13)

For the discussed sample at E D 0:6 eV, this ratio is 3, which is exactly the numberof available subbands at this energy. Figure 8.6 shows the estimated mean free pathand the localization length as a function of energy for W D 5 nm and W D 7:5 nm.The elastic mean free path in the energy interval of each subband increases. But itdecreases at the beginning of the next subband because the scattering rate increases

208 8 Applications

0 100 200 3000

1

2

3

L[nm]

T(E

)

aE=0.6eV

0 100 200 300

−10

−5

0

L[nm]

Ln[

T(E

)]

bE=0.6eV

Fig. 8.5 (a) The average transmission probability as a function of length. (b) The average of thelogarithm of the transmission probability as a function of length. W D 5 nm, �W=W D 2%,and�L D 3 nm. The solid lines show the fitted curves to the average transmission probability andlogarithm of the transmission probability. According to fitted data, the mean free path is � � 10 nmand the localization length is � � 30 nm. The ratio of localization length and mean free path isequal to the number of available channels at this energy (Nchannel D 3) (Reported with permissionfrom Ref. [119]. Copyright (2011), IEEE)

−2 −1 0 1 210

0

101

102

l[nm

]

E [eV]

a

W=5nm

−1 −0.5 0 0.5 110

0

101

102

103

x[nm

]

E [eV]

b

Fig. 8.6 (a) The mean free path (solid-curve) and the ballistic transmission probability (dashed-curve) as a function of energy and (b) the localization length as a function of energy forW D 5 nm.�L D 3 nm and �W=W D 2% (Reported with permission from Ref. [119]. Copyright (2011),IEEE)

at the edge of subbands as result of the van-Hove singularities in the density ofstates. According to this figure, we find that the localization length becomes shorter,as the width of the ribbon decreases. This width dependence of the localizationlength suggests that edge disorder plays an important role in the carrier localizationin narrow GNRs. Figure 8.7 shows that the ratio (Eq. (8.13)) is more or less equal tothe number of the conducting channels at the respective energies.

Figure 8.8a shows the transmission probability as a function of length and energy.Due to a shorter localization length at low energies, the transmission probabilityis strongly suppressed, especially at long channel lengths. To quantify the roleof localization on the transport properties, we define an effective transport gap(EGeff D �EG C EG), where the transmission probability drops to values below


−0.5 0 0.50

1

2

3

4

5

x Ωl

E [eV]

W=5nm

Fig. 8.7 The ratio of the localization length and the mean free path as a function of energy (solid-curve). The dashed-curve shows the available conducting channel at each energy.�L D 3 nm and�W=W D 2% (Reported with permission from Ref. [119]. Copyright (2011), IEEE)

0 100 200 3000

0.5

1

1.5

EG

eff[e

V]

L [nm]

∝L0.5

W=5nmba

Fig. 8.8 (a) The average transmission probability as a function of energy and length. The dashedlines indicate the border of the region where the transmission is smaller than 10�2, where wedefined it as the effective bandgap in presence of edge-roughness. (b) Comparison between theeffective transport bandgap (rectangles) and the bandstructure gap (circles). The dashed linerepresents the scaling of the effective bandgap versus length. �L D 3 nm and �W=W D 2%(Reported with permission from Ref. [119]. Copyright (2011), IEEE)

10�2, indicated by the white dashed lines in Fig. 8.8a. The effective transport gapfor a 5 nm wide GNR as a function of length is compared with the bandstructure gapin Fig. 8.8b. Apparently the effective transport gap increases with the length of thesample, where transport takes place in the localization regime (L � 20 nm). In thisregime the effective transport gap is proportional to L1=2, which is consistent withthe analytical model proposed in [118].

The conductance of GNR versus length for different widths, but with the sameroughness percentage, is plotted in Fig. 8.9 in logarithmic scale. As the device lengthbecomes larger than the localization length, one gets into the localization regime,where the conductance decreases exponentially with the length. This figure showsthat by increasing the width, localization occurs at longer lengths. As the width ofthe ribbon increases, the number of available channels and as a result the localizationlength increases, see Eq. (8.13).

210 8 Applications

0 50 100 150 200 250

100

L[nm]L

n[G

/G0]

W=2, 5, 7.5nm

EF=1.25eV

ΔW/W=2%

Fig. 8.9 The logarithm of average conductance as a function of length at various widths butwith the same roughness amplitude percent. The dashed lines are fitted exponential curves to thesimulation results. �L D 3 nm (Reported with permission from Ref. [119]. Copyright (2011),IEEE)

−0.4 −0.2 0 0.2 0.4 0.610−2

10−1

100

101

102

VG[V]

I D[¹

A]

Ion

Ioff

a

−0.4 −0.2 0 0.2 0.4 0.60

5

10

15

20

25

30

35

VG[V]

I D[ ¹

A]

Ideal

ΔW/W=2%

ΔW/W=3%

ΔW/W=5%

ΔW/W=10%

b

Fig. 8.10 The ensemble average of the transfer characteristics in (a) logarithmic and (b) linearscale. W D 1:6 nm, L D 20 nm, and �L D 10 nm (Reported with permission from Ref. [119].Copyright (2011), IEEE)

Next we investigate the role of roughness on the transfer characteristics of GNR-FETs. The source-drain current of the GNR-FET can be calculated as

I D 2e

h

ZdE T .E/ ŒfS.E/ � fD.E/ : (8.14)

Here fS and fD are the source and the drain Fermi functions, respectively. Tocapture the statistical nature of roughness for given geometrical and roughnessparameters, 200 samples are generated and simulated and an ensemble average oftheir characteristics is performed. For all simulations, a supply voltage of VD D0:5V and room temperature operation are assumed. Figure 8.10a, b compare thetransfer characteristics at various roughness amplitudes in logarithmic and linearscale, respectively. At small roughness the off current increases with the roughnessamplitude. This behavior is related to the formation of some localized states in thebandgap [75]. The band to band tunneling of carriers is strongly enhanced in thepresence of such states and as a result the off-current increases. With the increase


Fig. 8.11 3D sketch of a corrugated AGNR (Reported with permission from Ref. [108]. Copyright(2013), AIP)

of the off-current, the device performance in terms of the Ion=Ioff ratio and thesubthreshold swing is degraded. As roughness increases further, however, the off-current decreases. This behavior is due to the increase of the transport bandgap.As we mentioned before, the effective transport gap increases with the roughnessamplitude and as result the current decreases. In the case of strong roughness, theincrease of the transport gap dominates the effects of gap states and both the on andoff-currents decrease.

8.2.3 Substrate Corrugation

To use graphene for building electronic devices, it should be placed on a substrate.Scattering of carriers due to charged impurities and surface corrugation of the sub-strate surface, however, can degrade the electronic properties of graphene [14,59]. Inparticular, microscopic corrugations have been observed both on suspended [81] andsupported [24,36,52,74] graphene sheets. This rippling has been invoked to explainthe thermodynamic stability of free-standing graphene sheets [29]. In principle, theunderlying substrate has always had a degree of surface corrugation, which dependson the material type and the applied polishing method. Surface corrugation affectsthe surface morphology and can reduce mobility even further [83]. When grapheneis placed on a substrate, it follows the ripples of the substrate surface [63] (seeFig. 8.11), where such ripples induce significant stress in the graphene sheet [59].The height of the surface corrugation approximately varies between 25 pm for mica[74] to 300 pm for SiO2 [36, 52]. Although the surface of mica is much smootherthan that of SiO2, SiO2 is a more common material for semiconductor industries.

Surface corrugations change the bonding lengths between carbon atoms, whichsignificantly modulate the hopping parameters. Based on the Harrison’s model, thehopping parameter is inversely proportional to the square of the bonding length [46]:

t / 1

l2: (8.15)

Using this model, the effect of the substrate corrugation – which affects the bondinglengths between carbon atoms – on the hopping parameter and as a result the

212 8 Applications

electronic properties of GNRs is investigated. Small bending of pz-orbitals due tocorrugation need to be considered but it has been shown that the modulation of thehopping parameters due to the bonding length variation is much stronger than thatof orbital bending [62].

Surface corrugation of the substrate is a statistical phenomena which can bemodeled by a Gaussian ACF [36, 39, 52, 74]:

R.x; y/ D ıh2 exp

�� x2

Lx2

� y2

Ly2

�; (8.16)

where Lx and Ly are the roughness correlation lengths along the x and y-direction,respectively, which indicate the distance at which corrugation is relatively repeated.ıh is the root mean square of the fluctuation amplitude and is an indication of thefluctuations height. To generate surface corrugation in spatial domain, the ACFis Fourier transformed to obtain the spectral function. A random phase with evenparity is applied and followed by an inverse Fourier transformation, see Sect. 8.2.2.For the given geometrical and roughness parameters, many samples are created andthe electronic characteristics of each sample are evaluated. By taking an ensembleaverage, the role of corrugation parameters on the average device characteristics isinvestigated.

Depending on the material type and cleaning process, the corrugation amplitudeand correlation length can approximately vary between 24–340 pm and 2–32 nm,respectively [24, 36, 52, 74]. The average transmission probability as a functionof energy at various ıh for an AGNR and a ZGNR are shown in Fig. 8.12a andb, respectively. As ıh increases, the transmission probability decreases due toincreased carrier scattering rate. Subbands can be recognized for small values ofıh (small perturbation). At large values of ıh, steps in the transmission probabilityare smoothed out. On the other hand, the average transmission probability increaseswith the correlation length, see Fig. 8.12c for AGNR and Fig. 8.12d for ZGNR. Asthe correlation length increases, surface corrugation becomes smoother, leading tothe reduction of the carrier scattering rate.

In the absence of scattering, the transmission probability is independent of thechannel length. However, the transmission has an inverse proportionality to thechannel length in the presence of scattering, see Eq. (8.5). To quantify the role ofsurface corrugation on the electronic properties of AGNRs, the MFPs as functionsof the corrugation amplitude and correlation length are extracted. For this purpose,at each energy, a curve based on Eq. (8.5) is fitted to the average transmissionprobability as a function of the channel length (Fig. 8.13a) and the respective MFPat that particular energy is extracted, see Fig. 8.13b. The MFP increases at eachsubband with energy but it falls down at the edge of the next subband. A van-hovesingularity appears at the edge of each subband, resulting in a significant increaseof the density of state and scattering rate of carriers; therefore, the MFP decrease atthese edges. Because of the absence of such van-hove singularities in the electronic


a b

c dδ

δ

δ

δ

Fig. 8.12 The average transmission probability as a function of energy at various corrugationamplitudes and Lx D Ly D 25 nm for the (a) AGNR and (b) ZGNR. The average transmissionprobability as a function of energy at various correlation lengths for the (c) AGNR with ıh D50 pm and (d) ZGNR with ıh D 150 pm. W D 2:5 nm for both the AGNR and ZGNR andL D 100 nm for the AGNR and L D 50 nm for the ZGNR (Reported with permission fromRef. [108]. Copyright (2013), AIP)

a b

Fig. 8.13 (a) The average transmission probability as a function of length at E D 1 eV forıh D 250 pm. The dashed-line is a fitted curve based on Eq. (8.3) for extracting the MFP. (b) TheMFP as a function of energy for ıh D 150 pm. All results are for the AGNR with W D 2:5 nmand Lx D Ly D 15 nm (Reported with permission from Ref. [108]. Copyright (2013), AIP)

bandstructure of graphene, one should expect surface corrugation to play a moreimportant role in the electronic properties of AGNRs than that of graphene.

Figure 8.14a, b indicate the MFP scales with corrugation amplitude as ıh�4 forboth the AGNR and ZGNR, respectively, whereas MFP scales linearity with the

214 8 Applications

a b

c d

Fig. 8.14 The dependency of the MFP with the corrugation amplitude at Lx D Ly D 15 nmfor the (a) AGNR with W D 5 nm and the (b) ZGNR with W D 3 nm. The symbols indicatesimulation results and the dashed curves are fitted lines based on � / ıh�4. The inset of (a)shows the MFP as a function of h�4. The inset of (b) depicts the MFP as a function of energy forthe ZGNR. Relatively large MFPs close to the Dirac point are due to the robustness of the edgestates of ZGNR against disorder. The dependency of the MFP with the correlation length for the (c)AGNR with W D 2:5 nm, ıh D 150 pm and (d) ZGNR with W D 3 nm and ıh D 150 pm. Thesymbols are the simulation results and the dashed curves are fitted to � / Lx;Ly . All MFPs in(a)–(d) are extracted at E D 1 eV (Reported with permission from Ref. [108]. Copyright (2013),AIP)

correlation length, see Fig. 8.14c, d for AGNR and ZGNR, respectively. It shouldbe noted that the relatively larger MFP of the ZGNR, in comparison with thatof the AGNR, is due to the robustness of edge-states against disorder in zigzagconfiguration [8] (see the inset of Fig. 8.14b). As the correlation length increases,surface corrugation becomes smoother and the scattering rate is reduced. Thescaling of MPF with the corrugation amplitude can be approximated from the Fermigolden rule. The bonding length in the presence of surface corrugation can beexpressed as l2 D a2cc C �h2ij where �hij is the height difference between twonearest neighbor atoms and acc is the distance between nearest neighbor carbonatoms in graphene. Assuming hij � acc, one can employ Harrison’s [46] model toapproximate the hopping parameters with

tij D t01

.lij=acc/2D t0

1C�h2ij=a2cc

' t0

1 � �h2ij

a2cc

!: (8.17)


Therefore, the modulation of the hopping parameter is quadratically proportional tocorrugation amplitude ıtij / �h2ij and one obtains hi jıH jj i D ıtij / �h2ij. Takingan ensemble average over hopping parameters yields

hıtiji / h�h2iji D hjh.x; y/ � h.x C acc; y/j2i ;

D hh.x; y/2i C hh.x C acc; y/2i � 2hh.x; y/h.x C acc; y/i ;

(8.18)

where hh.x; y/h.x C x0; y C y0/i is the definition of the surface corrugation ACF:

hh.x; y/2i D hh.x C acc; y/2i D ıh2 ; (8.19)

hh.x; y/h.x C acc; y/i D ıh2 exp��a2cc=L

2x

�: (8.20)

As a result,

hıtiji / ıh2 : (8.21)

Based on the Fermi golden rule, the scattering rate due to surface roughness isgiven by

��1 / jhi jıH jj ij2 D jhıtijij2 / ıh4 : (8.22)

The MFP therefore scales with corrugation amplitude as

� / 1

ıh4: (8.23)

Next we investigate the role of surface corrugation on the mobility of AGNRs.Mobility can be evaluated from [68]

�.E/ D �.E/=n ; (8.24)

in which n is the electron concentration and �.E/ is the conductivity that can beobtained from Eq. (8.4). Figure 8.15a shows mobility as a function of width atvarious corrugation amplitudes. The results indicate that the mobility is considerablydegraded for corrugation amplitudes around 250 pm, which is a typical value forSiO2 substrates. Figure 8.15b compares the role of edge roughness and surfacecorrugation on the mobility of AGNR placed on a SiO2 substrate. The details of edgeroughness limited mobility evaluation is described in Ref. [118,119]. Although edgeroughness is the most detrimental scattering mechanism on the electronic propertiesof narrow AGNRs, surface corrugation appears as a dominant scattering mechanismfor AGNRs with widths wider than approximately 3 nm placed on SiO2 substrates.

216 8 Applications

ba

Fig. 8.15 (a) The mobility of the AGNR as a function of width at various corrugation amplitudes.(b) A comparison between the role of surface corrugation amplitude and that of line-edgeroughness on the mobility as a function of the ribbon’s width. Lx D Ly D 15 nm and L D 50 nm(Reported with permission from Ref. [108]. Copyright (2013), AIP)

8.3 Spin Transport

Due to a low atomic number of carbon, graphene has a relatively small spin-orbitinteraction. Theoretical studies predict a spin relaxation time of in the range ofmicro- to milli-second [26]. However, experimentally measured data indicate avalue of 100–200 ps [107]. It has been suggested that extrinsic effects cause thisdiscrepancy [109], i.e., substrate and adatoms [26], ripples [51], and impurities[85]. As discussed in Sect. 8.2.3, substrate surface corrugation is an extrinsicsource of scattering that can significantly affect electronic transport. In this sectionwe investigate the role of surface corrugation on spin-transport in nanoribbons,employing the NEGF along with a multi-orbital TB model including spin-orbitcoupling.

8.3.1 Multi-orbital Model

The electronic bandstructure of graphene can be described by px , py , and pz

orbitals, see Fig. 8.16. The multi-orbital TB model is given by (see Eq. (4.19))

OH DXli�

"l Oc�li� Ocli� �Xlmij�

tlmij Oc�li� Ocmj� C OHSO; (8.25)

where Ocmj� annihilates an electron on atomic orbital with label m at some site j

with spin index � D" or � D# and Oc�li� create one electron at atomic orbital l andsite i with spin index � . The on-site potential for p orbitals is assumed to be zero("p D 0) and the hopping parameters between orbital are tpp� D 5:037 eV and

8.3 Spin Transport 217

Left-contact Right-contact

Spin Flip bySpin-orbit Interaction

Channel

a

b

Fig. 8.16 (a) Spin transport and the related transmission probabilities in the presence of spin-flipmechanisms. (b) Sketch of px , py , and pz orbitals and spin transitions due to spin-orbit interactionand hopping

tpp� D 3:033 eV [82]. The Hamiltonian matrix size is therefore 3 � N , where Nis the total number of carbon atoms in the device. By including spin, however, theHamiltonian size increase to 6 � N . The Hamiltonian due to spin-orbit interactionis (see Eq. (2.79))

OHSO D � OL � OS ; (8.26)

where � D 12meV is the spin-orbit coupling constant [50] and OL and OS are theatomic angular momentum operator and spin operator, respectively. By defining

SC ��0 1

0 0

�; s� �

�0 0

1 0

�; sz �

�120

0 � 12

�; (8.27)

LC �0@0

p2 0

0 0p2

0 0 0

1A ; L� �

0@ 0 0 0p

2 0 0

0p2 0

1A ; Lz �

0@1 0 0

0 0 0

0 0 �1

1A ;

jpzi � jL D 1;Lz D 0i ;

218 8 Applications

jpxi � 1p2.jL D 1;Lz D 1i C jL D 1;Lz D �1i/ ;

jpyi � Cip2.jL D 1;Lz D 1i � jL D 1;Lz D �1i/ ; (8.28)

the spin-orbit Hamiltonian becomes

OHSO D �

LCS� C L�SC

2C LzSz

�; (8.29)

which can be written in second quantization language as [50]

OHSO D �h

Oc�z" Ocx# � Oc�z# Ocx" C i Oc�z" Ocy# � i Oc�z# Ocy" C i Oc�x# Ocy# � i Oc�

x" Ocy" C h:c:i;

(8.30)

where the operators Oc�z;x;yI� and Ocz;x;yI� refer to the corresponding pz, px andpy atomic orbitals. Intrinsic spin-orbit coupling in graphene can be completelydetermined from the symmetry properties of the honeycomb lattice. In an idealflat graphene sheet, spin-orbit coupling vanishes for the first nearest neighborsand just second nearest neighbors have a negligible contribution to this intrinsicspin-orbit interaction [56]. When this symmetry is broken, for example, by surfacecorrugation, an effective spin-orbit interaction is induced. A similar effect has beenpredicted for curved graphene sheets in Ref. [50].

8.3.2 Transport Model

The Green’s function matrix for spin-polarized transport is written as [117]

Gr

"".E/ Gr

"#.E/

Gr#".E/ Gr

##.E/

!D"EI �

H

""H

"#

H#"

H##

!� ˙ r

L"0

0 ˙ rL#

!� ˙ r

R"0

0 ˙ rR#

!#�1

;

(8.31)

where ˙ rL=R;� are the self-energies for the left (L) and right (R) contacts, respec-

tively, for up-spin � D" or down-spin � D#. The transmission probabilities can beobtained by

T"" D TraceŒ� L"Gr""�

rR"G

a"" ; T"# D TraceŒ� L"Gr

"#� R#Ga"# ; (8.32)

where T"" denotes transmission probability of carriers with up-spin in the left con-tact to up-spin in the right-contact, and T"# represents the transmission probability

8.3 Spin Transport 219

0 0.5 1 1.5 210

−6

10−4

10−2

100

102

E[eV]

2 pm

0 0.5 1 1.5 20

1

2

3

4

5

6x 10

−3

E[eV]

pm

0 0.5 1 1.5 210

−4

10−3

10−2

10−1

E[eV]

pm

0 50 100 150 2000.92

0.94

0.96

0.98

1

E=.9eV

pm

a b

c d

Fig. 8.17 Ensemble averages of (a) T"", (b) T"#, (c) 1 � P as a function of energy at varioussubstrate surface corrugation amplitudes, and (d) polarization versus substrate surface corrugationamplitude at E D 0:6 eV. All results are for AGNRs with W D 2 nm, L D 15 nm and Lx;Ly D10 nm

of carriers with up-spin in the left contact to down-spin in the right contact as shownin Fig. 8.16. T"# is an indication of spin-flip due to spin-orbit interaction along thechannel.

8.3.3 Results

We follow the same approach described in Sect. 8.2.3 to generate surface corruga-tion and modulate hopping parameters. In addition we modulate orbital directionsusing the model described in [92]. In Fig. 8.17a, b, T"" and T"#, respectively, areplotted as a function of energy at various corrugation amplitudes. T"" decreases

220 8 Applications

E[eV]E[eV]

0 0.5 1 1.5 20

1

2

3

4

5

6

x 10−4

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

a b

Fig. 8.18 Ensemble averages of (a) T"" and (b) T"# as functions of energy at various channellengths

with the increase of the corrugation amplitude due to increased spin-flipping rate.On the one hand, the increased spin-flipping rate should increase T"#. On the otherhand, the increased scattering rate can reduce transmission probability. For the datashown in Fig. 8.17b, the latter effect dominates, which results in the reduction of T"#with the increase of corrugation amplitude. To quantify the role of the spin-flippingprocess on spin transport, one can define a polarization efficiency as [95]

P D T"" � T"#T"" C T"#

: (8.33)

Figure 8.17c shows 1 � P as a function of energy at various roughness amplitudes.For a better comparison, Fig. 8.17d depicts the polarization as a function of rough-ness amplitude at E D 0:6 eV. Apparently, as roughness increases, polarizationdecreases. Figures 8.18a and 8.18b show that T"" and T"# also decrease as thechannel length increases due to increased scattering rate.

8.4 Phonon Transport

Thermal properties of nanostructures have been recently investigated as theyare of interest for nanoelectronic and thermoelectric applications. High thermalconductivity is beneficial for thermal management and nanoelectronic devices, inwhich hot spots, caused by heat dissipation in a relatively small volume, need to becooled down [38,77]. On the other hand, the performance of thermoelectric devicesis inversely proportional to the thermal conductivity. The thermoelectric figure of

8.4 Phonon Transport 221

merit for materials is defined as

ZT D S2GT

.Kel CKph/; (8.34)

where S is the Seebeck coefficient,G the electrical conductance, T the temperature,Kel and Kph the electrical and lattice contributions to the thermal conductivity,respectively [88]. The numerator of Z is called power factor. The figure of meritdetermines the efficiency of a thermoelectric material (device) and can be improvedby increasing the power factor and decreasing thermal conductivity. Hence, ther-moelectric materials must simultaneously have a high Seebeck coefficient, highelectrical conductivity, and low thermal conductivity [45, 110].

Experimental studies have reported a high Seebeck coefficient in graphene-based devices [112, 126]. In addition, a high thermoelectric performance has beentheoretically predicted in some graphene-based structures [57, 99] by degrading theextraordinary ability of pristine graphene to conduct heat [38]. The high thermalconductivity of graphene is mostly due to the lattice contribution, whereas theelectronic contribution to the thermal conduction is negligibly small [10,11,48,86].Therefore, by proper engineering of phonon transport, relatively high or a lowthermal conductivity, as needed by a specific application, can be achieved.

In this section, the methodology used for calculating thermal conductivity andas a result thermoelectric coefficients, is discussed. Two steps are needed in thesimulation procedure: (i) material properties, a step including the calculation ofelectronic (see Sect. 8.2) and phononic bandstructures and (ii) transport properties,using the NEGF formalisms. This section continues with the application of thesetheories for the evaluation of GNR thermal properties.

8.4.1 Phonon Bandstructure

The phonon bandstructure can be described by first principle models [55, 120],the valence force field (VFF) method [66, 71], and the force constant method(FCM). The latter has the lowest computation time requirements. In this model,the dynamics of atoms are simply described by a few force springs connecting anatom to its surroundings up to a given number of neighbors. In contrast, the VFFmethod is based on the evaluation of the force constants [93], which requires muchlarger computational times. The FCM uses a small set of empirical fitting parametersand can be easily calibrated to experimental measurements. Despite its simplicity, itcan provide accurate and transferable results [111, 114]. Thus it is a convenient androbust method to investigate the thermal properties of crystals and in particular ofgraphene nanostructures.

The FCM model we employ involves a fourth nearest-neighbor approximation.The tensor describing the coupling between the i th and the j th carbon atoms, whichare the N th nearest-neighbor of each other, is given by

222 8 Applications

Table 8.1 The fittingparameters of the forceconstant tensor in N/m forgraphene [97]

N ˚r ˚ti ˚to

1 365:0 245:0 98:2

2 88:0 �32:3 �4:03 30:0 �52:5 1:5

4 �19:2 22:9 �5:8

˚0ij D

0B@˚.N/r 0 0

0 ˚.N/ti 0

0 0 ˚.N/to

1CA ; (8.35)

where ˚r , ˚ti and ˚to are the radial, the in-plane transverse, and the out-of-planetransverse components of the force constant tensor, respectively. Their values arepresented in Table 8.1 [97]. In Cartesian coordinates, it is given by

˚ D U�1˚0U ; (8.36)

where U is a unitary matrix defined as

U D0@ cos�ij sin�ij 0

� sin�ij cos�ij 0

0 0 1

1A : (8.37)

The motion of the atoms can be described by a dynamical matrix, see Eq. (4.29):

D D ŒDij D

2664 1p

MiMj

�

8̂<̂ˆ̂:

˚ij ; i ¤ j

�Xl¤i

˚il ; i D j

3775 ; (8.38)

where Mi is the atomic mass of the i th carbon atom, and ˚ij is a 3�3 force constanttensor describing the coupling between the i th and the j th carbon atom.

8.4.2 Phonon Green’s Function

The FCM can also be effectively coupled to NEGF for the investigation of coherentphonon transport in low-dimensional systems [122]. The Green’s function forphonons reads as

Gr.E/ D �E2I �D �˙ r

S �˙ rD

��1; (8.39)


where D is device dynamic matrix and E D „! is the phonon energy [78]. Notethat this is a Green’s function of a classical wave equation, which is second order intime. Therefore, the square of the energy appears in Eq. (8.39). This is in contrast tothe Schrödinger equation for the quantum treatment of electrons, which is first orderin time, such that E appears linearly in that case. Similar to the electron transport,the transmission probability of phonons through the channel can then be calculatedas (see Sect. 7.5.3)

Tph.E/ D TraceŒ� SGr� DG

a : (8.40)

Here the phononic broadening and contact self-energy matrices are obtained in asimilar way as the electronic ones with the substitutions H ! D and E ! „2!2.

8.4.3 Phonon Thermal Conductivity

In semiconductors, the largest component of the heat current is due to the phonontransport. Under the condition of ballistic phonon transport, the heat flow is propor-tional to the phonon transmission function Tph.!/, the Bose-Einstein distributionfunction nS=D.!/ of source and drain contacts, and the phonon energy/frequency! as

Iph D 1

„Z

d.„!/2�

Tph.!/„! ŒnS.!/ � nD.!/ : (8.41)

In the linear response regime, the heat current is proportional to the temperaturedifference by

Iph D 1

„Z

d.„!/2�

Tph.!/„!�T ŒnS.!/ � nD.!/

�T;

D �T

„Z

d.„!/2�

Tph.!/„! @n.!/@T

:

(8.42)

This can alternatively be written as

Iph D ��T kB2T �2

3„Z

d.„!/2�

Tph.!/Wph.„!/ ; (8.43)

where the phononic window function is given by [54]

224 8 Applications

Wph.„!/ D � 3

�2

� „!kBT

�2@n

@.„!/ ;

D 3

4�2kBT

� „!kBT

�2 sinh

� „!2kBT

��2:

(8.44)

As Iph D �Kph�T , one can express the lattice contribution to the thermalconductance as [91]

Kph D kB2T �2

3„Z.d„!/2�

Tph.!/Wph.!/ : (8.45)

The lattice thermal conductivity is given by

�l D Kl

L

WH; (8.46)

where L is the channel length, W is the ribbon’s width, and H D 0:335 nm is theeffective thickness of the graphene monolayer [3, 67].

8.4.4 Ballistic Phonon Transport

In this section, the role of geometrical and roughness parameters on the phonontransport in AGNRs is studied. The ribbon’s width varies between 1 and 10 nm.This corresponds to AGNRs with 7, 16 25, 31, 40, 49, and 80 indices. Channellengths up to 40 nm are studied. The maximum roughness amplitude is chosen tobe 10 % of ribbons’ width, and the correlation lengths varies between 1 and 10 nm.The diffusive thermal conductivity is extracted from the results as well.

The ballistic transmission function of phonons, which is the number of phononicchannels at some energy „!, is shown in Fig. 8.19. As expected, the transmissionfunction increases with the ribbon’s width. In addition, the phononic windowfunction is shown in the inset of Fig. 8.19 at various temperatures. This function,which qualifies the contribution of different phonon frequencies in the thermalconductance, increases with temperature. Therefore, at higher temperatures, highenergy phonons contribute to the thermal transport as well.

Figure 8.20 shows the ballistic thermal conductance of AGNRs as a functionof temperature and the ribbon’s width. By increasing the temperature and thusthe phononic window function, thermal conductance increases. However, as theenergy spectrum of AGNRs is limited to 0:2 eV (Fig. 8.19), the thermal conductancesaturates at very high temperatures (�T > 800K). As shown in Fig. 8.20b, thethermal conductance increases linearly with the ribbon’s width. In fact, the ballisticthermal conductance divided by the width is constant. This is due to the fact thatby increasing the width and thus the number of carbon atoms in the unit cell,


0.05 0.1 0.15 0.20

10

20

30

40

50

Tra

nsm

ission

h̄ω [eV]

W=1, 3, 6nm0.05 0.1 0.15

Wph

h̄ω [eV]

T=150,300,500K

Fig. 8.19 Ballistictransmission of phonons inAGNRs of widths 1, 3, and6 nm. The inset shows thephononic window function atT D 150, 300, and 500K(Reported with permissionfrom Ref. [58]. Copyright(2013), IEEE)

300 600 9000

2

4

6

8

10

12

Kl [

nW/K

]

Temperature [K]

a

~1nm

~2nm

~3nm

~4nm

~5nm

~6nm

0 2 4 6Width [nm]

b

T=500 K

T=300 K

T=150 K

Fig. 8.20 Ballistic latticethermal conductance ofAGNRs at various widths as afunction of (a) temperatureand (b) width (Reported withpermission from Ref. [58].Copyright (2013), IEEE)

the number of modes increases almost linearly in the whole energy spectrum, andtherefore the ballistic transmission function is directly proportional to the width.

On the other hand, a high thermal conductivity in the range of �2,000–5,300 W/mK has been reported for suspended single-layer graphene [11,87], whichis only weakly affected by the boundary and substrate scatterings. Using thissuperior thermal conductivity, a phonon mean-free-path (MFP) of �775 nm hasbeen extracted [38]. Due to this relatively large MFP, no saturation has beenobserved in the thermal conductivity of short ribbons with smooth edges [43]. Whenthe MFP is large enough, the conductivity is mostly determined by the channellength (Eq. (8.46)) rather than phonon scattering. In the next section we examinehow the LER scattering in very narrow AGNRs can drastically decrease the MFP,which can significantly affect the length dependence of the thermal conductivity.

To investigate the effect of LER on thermal conductivity and the MFP, we per-formed simulations on a statistical sample of 16-AGNR with roughness parametersof �W D 0:1 nm and �L D 2 nm. The statistical average of the transmissionfunction is shown in Fig. 8.21 for channel lengths of 5 and 40 nm. For reference, the

226 8 Applications

0 0.05 0.1 0.15 0.20

5

10

15

Tra

nsm

ission

h̄! [eV]

Ballistic

L=5nm

L=40nm

W = 2nm

ΔW = 0:1nm

ΔL = 2nm

Fig. 8.21 The transmission function for 16-AGNR: Ballistic result (black); in the presence ofLER with �W D 0:1 nm and �L D 2 nm for channel lengths of 5 nm (red) and 40 nm (green).Increasing the length decreases the transmission function of the rough ribbons (Reported withpermission from Ref. [58]. Copyright (2013), IEEE)

0.05 0.1 0.15 0.20

10

20

30

40

50

60

l ph

[nm

]

h̄! [eV]

a

0 10 20 30 400

5

10

15

20

25

30

35

κl[W

/mK

]

Length [nm]

lph = 13 nm

lph = 14 nm

lph = 23 nm

b

T=500

T=300

T=150

10 20 30 400

2

4

6

Tra

nsm

ission

Length [nm]

h̄! = 50 meVl = 7 nm

Fig. 8.22 (a) LER-limited MFP of 16-AGNR for �W D 0:1 nm and �L D 2 nm as a functionof phonon energy. The inset shows the transmission function at „! D 50meV as a function ofthe channel length. Its corresponding MFP is about 7 nm. (b) Lattice thermal conductivity as afunction of the channel length. Using Eq. (8.49) with L1 D 5 nm and L2 D 20 nm, the effectiveMFPs are extracted as 23, 14, and 13 nm at T D 150, 300, and 500K, respectively. The dashedlines are plotted based on Eq. (8.49), L1 D 5 nm, and varying L2 as the channel length (Reportedwith permission from Ref. [58]. Copyright (2013), IEEE)

ballistic transmission of 16-AGNR is also shown in black. Figure 8.21 shows thetransmission functions of 16-AGNR assuming perfect edges and rough edges withroughness parameters of �W D 0:1 nm and �L D 2 nm. Apparently, increasingthe length decreases the transmission function. To quantify the dependence of thetransmission function on the channel length, the phonon MFP is defined as [23, 54]

T ph.„!/ D Nph.„!/1C L=�ph.„!/ ; (8.47)


where Nph.„!/ is the ballistic transmission function and �ph.„!/ is the phononMFP at energy „!. In the inset of Fig. 8.22a, the transmission function of 16-AGNR at „! D 50meV is shown as a function of the channel length. The extractedMFP at this phonon energy is about 7 nm. Figure 8.22a indicates that the MFP issmaller than 30 nm in most of the spectrum, except at very low frequencies. Thelength dependence of the lattice thermal conductivity of this ribbon is presentedin Fig. 8.22b. In contrast to the ribbons with smooth edges, here the thermalconductivity increases with length and starts to saturate above L D 40 nm. Thesymbols in Fig. 8.22b are ensemble average values, however, the numerical resultsshow a standard deviation of �0.5–1.0 W/mK in the thermal conductivity. As arough estimate, the standard deviations of various quantities calculated in this workare about 10 % of the corresponding average value for short and narrow channels,whereas they decrease to �5 % of the average values for long and wide channel.To study the dependence of the thermal conductance on the channel length, one candefine an effective MFP �ph as [99]

Kl D Kl;B

�ph

LC �ph

; (8.48)

which covers the contribution of phonons of different frequencies. Here theballistic thermal conductance of AGNRs with perfect edges is denoted by Kl;B .Alternatively, the effective MFP can be expressed as

�l .L1/

�l .L2/D L1

L2

�ph C L2

�ph C L1; (8.49)

which makes the numerical calculation more tractable. In Fig. 8.22b, usingEq. (8.49) with L1 D 5 nm and L2 D 20 nm, the effective MFPs are extractedas 23, 14, and 13 nm at T D 150, 300, and 500K, respectively. The dashedlines are plotted based on Eq. (8.49), L1 D 5 nm, and varying L2 as the channellength. The effective MFP is high at low temperatures because at low temperaturesthermal transport is dominated by low frequency phonons, which have longer MFPs(see Fig. 8.22a). Low frequency phonons with long-wavelengths undergo mostlyspecular scattering on the boundaries [125]. As shown in Fig. 8.23, the ratio ofthe effective MFPs at T D 150K and T D 300K increases with the roughnessamplitude, indicating that short-length roughness affects the transport of short-wavelength phonons more than that of long-wave phonons. However, in the rest ofthis work we consider only the room-temperature operation.

The lattice thermal conductance as a function of width in the presence of LERwith �W D 0:1 nm and �L D 2 nm is depicted in Fig. 8.24a. The ballisticconductance is proportional to the ribbon’s width. In the presence of roughness,the thermal conductance is smaller than the ballistic one. It increases quadraticallywithW for narrow ribbons and then a linear increase is observed. This behavior canbe understood by considering the fact that an effective MFP increases with ribbon’s

228 8 Applications

0 0.1 0.2 0.31

1.2

1.4

1.6

1.8

2

Δ W [nm]

¸ph(1

50)=¸

ph(3

00)

16−AGNR

Δ L=0.2 nm

Fig. 8.23 The ratio ofeffective MFPs at T D 150

and T D 300K (Reportedwith permission fromRef. [58]. Copyright (2013),IEEE)

a b

Fig. 8.24 (a) Thermal conductance and (b) thermal conductivity as a function of the ribbon’swidth at room temperature. The inset shows that the phonon MFP scales linearly with the ribbon’swidth. The parameters are �W D 0:1 nm, �L D 2 nm, and L D 20 nm. The dashed lines areguides to the eye (Reported with permission from Ref. [58]. Copyright (2013), IEEE)

width (inset of Fig. 8.24b). Therefore, in wide ribbons, the MFP is larger than thechannel lengthL D 20 nm. According to Eq. (8.48), the conductance is proportionalto the ballistic conductance, which scales linearly with the ribbon’s width. Onthe other hand, for narrow ribbons, the MFP in the denominator of Eq. (8.48) isnegligible in comparison with the channel length L, such thatKl � Kl;B�ph, and asa resultKl � W 2. Therefore, the thermal conductivity (�Kl=W ) saturates for wideribbons and the feature of constant ballistic thermal conductance per unit width (seeFig. 8.20b) is observed in the thermal conductivity.

When the channel length is larger than the MFP, purely diffusive thermalconductivity, which is length independent, can be extracted using Eq. (8.49). Thediffusive thermal conductivity as a function of ribbon’s width is shown in Fig. 8.25for two cases of constant roughness amplitude and constant relative roughness. Theresults indicate that a relative roughness between �0.5 % and �5 % can cover therange of the experimental data. However, as shown in Ref. [2], phonon transport innarrow GNRs (W < 130 nm) is limited by LER, indicating that the distinction isexpected to be negligible. In the case of fixed roughness amplitude, the diffusivethermal conductivity is proportional to the ribbon’s width, similar to the effectiveMFP (inset of Fig. 8.24b), implying that the LER relaxation time is proportionalto W as proposed in the conventional formula for boundary scattering [1, 125].On the other hand, at fixed relative roughness, the diffusive thermal conductivity


0 5 10 150

50

100

150

200

250

300

l[W

/mK

]

Width [nm]

Exp.l ∼ W 0.3

ΔWW = 5%

ΔW = 0:1 nm

l ∼ W

Fig. 8.25 Room temperature diffusive lattice thermal conductivity as a function of width. Aconstant roughness amplitude (�W D 0:1 nm) and a constant relative roughness (�W

WD 5%)

are considered. The roughness correlation length is �L D 2 nm. The dashed lines are fitted basedon the least mean square error. Experimental results for rough GNRs are supported by SiO2 andare adopted from Ref. [70] (Reported with permission from Ref. [58]. Copyright (2013), IEEE)

Fig. 8.26 The room temperature effective phonon MFP as a function of the ribbon’s width. Botha constant roughness amplitude (�W D 0:1 nm) and a constant relative roughness (�W

WD 5%)

are considered. The roughness correlation length is �L D 2 nm. The dashed lines are fitted basedon the least mean square error (Reported with permission from Ref. [58]. Copyright (2013), IEEE)

is only weakly dependent on the width (�W 0:3). This behavior can be understoodby considering the dependence of the effective MFP on the roughness amplitude.As shown in Fig. 8.26, the effective MFP is weakly related to the ribbon’s width,�W 0:3, at fixed relative roughness amplitude, whereas at fixed roughness amplitude,it scales linearly with the width.

Figure 8.27 shows that both the effective MFP and diffusive thermal conduc-tivity are inversely proportional to the relative roughness amplitude, and scalelinearly with the correlation length. It is worth mentioning that here the roughnessparameters change at fixed width, in contrast to the Fig. 8.26. Although the relativeroughness amplitude affects the thermal conductivity more than the roughness

230 8 Applications

a b

Fig. 8.27 (a) The room temperature effective MFP and (b) diffusive thermal conductivity of 25-AGNR as a function of relative roughness and correlation length. For �L-varying curves �W

WD

4% and for �WW

-varying curves �L D 3 nm. The dashed lines are fitted based on the least meansquare error (Reported with permission from Ref. [58]. Copyright (2013), IEEE)

correlation length, at large roughness amplitude, the role of correlation length isas important as the relative roughness amplitude.

8.5 Graphene-Based Tunneling Transistors

In conventional FETs, the gate controls the thermionic emission current, therefore asub-threshold swing of about 64 mV=dec can be achieved at room temperature [6].However, due to short channel effects, much larger values for this quantity havebeen reported. On the other hand, tunneling FETs (TFETs) have attracted theattention of scientists for their improved performance in short channel devices.In these structures, tunneling between source and drain is controlled by the gate-source voltage. In comparison with conventional FETs, tunneling FETs have someadvantages, such as having sub-threshold swings smaller than 60mV/dec and higherIon=Ioff ratios [7].

Graphene-based heterostructures, such as graphene-hexagonal boron nitride(hBN), have attracted the attention of scientists [21, 79]. Recently, a grapheneTFET based on a vertical graphene heterostructure has been proposed [16]. Inthis structure, source and drain are composed of a monolayer of graphene andhBN or MoS2 are used as a tunneling barrier. VTGFETs exhibit room-temperatureswitching ratios of �50 and �10;000 for hBN and MoS2, respectively [16]. Thesketch of a VTGFET/VTGNRFET is shown in Fig. 8.28a where the source and draincontacts are made from mono-layers of graphene/GNR. A few layers of hBN (3–7layers) serve as the tunneling barrier between the source and drain contacts. The gateoxide is a 300 nm layer of SiO2. To avoid the degradation of the electronic propertiesof the source and drain graphene/GNR sheets, two thick layers of hBN (20–50 nm)are assumed as the buffer layer. The energy barrier between the graphene Dirac

8.5 Graphene-Based Tunneling Transistors 231

point and the top of the hBN valence band .�1:4 eV/ is much smaller than that fromthe bottom of the conduction band .�3:34 eV/ [61]. This implies that the tunnelingbarrier for holes is smaller than that for electrons. Since the tunneling barrier is muchlarger than thermal energy, the device characteristics are approximately independentof temperature.

It is assumed that at zero bias voltages, the Dirac points of the graphene sheetsare aligned with the work-function of the gate. By applying a gate voltage, theFermi level and the carrier concentration of the source contact are modulated. Dueto the atomic thickness and also relatively small carrier concentration of carrierin graphene/GNR, the source contact only weakly screens the gate electric field.As a result, the Fermi level and the carrier concentration of the drain contact arealso modulated by the gate voltage. A positive gate-source voltage increases theconcentration of electrons and results in n-type operation corresponding to off state,see Fig. 8.28b, whereas a negative gate-source voltage results in a p-type deviceoperation and switches on the device, see Fig. 8.28c. The applied gate voltage islimited by the gate oxide breakdown, which is about 1V/nm for SiO2. The drain-source bias voltage VB gives rise to the tunneling current through the hBN layer.

Operation of transistors based on one- and two-dimensional materials, such asCNTs, GNRs and graphene, allows access to the so-called quantum capacitancelimit wherein the potential within the channel is determined mostly by the gatepotential rather than the charge in the channel [76]. As a result, strong modulation ofthe Fermi levels of graphene/GNR sheets and therefore the change in barrier heightslead to relatively large tunneling currents and steep sub-threshold slopes [30].

In VTGNRFETs, the base materials for the source and drain contacts are GNRs.As shown in Fig. 8.28d, a similar operation principle holds for such devices exceptin the case of the presence of an energy gap along with a parabolic dispersionrelation. Due to the presence of an energy bandgap in GNRs, one expects the Ioff

of VTGNRFETs to be smaller than that of VTGFETs, which can result in a higherIon=Ioff ratio.

8.5.1 Modeling

An atomistic TB model is used to describe the electronic bandstructure of theheterostructure of graphene and hBN (Table 8.2). The NEGF formalism is employedfor a quantum mechanical description of carrier transport in VTGFET and VTGN-RFET. In a recently fabricated VTGFET, the graphene sheets of the source anddrain are in contact with several metallic electrodes [16], providing better electricalcharacteristics. As shown in Fig. 8.29a, a similar structure is used here. The atom-istic diagram of the studied structure is sketched in Fig. 8.29b. Two graphene/GNRlayers at the top and the bottom are separated by hBN, all arranged in the Bernal(AB) stacking.

To study VTGFET where the source and drain graphene sheets are infinitelyextended along the transverse direction, Bloch periodic boundary condition with a

232 8 Applications

Fig. 8.28 (a) The sketch of a VTGFET and VTGNRFET. The bandstructure and operationprinciple for a VTGFET in the (b) off state and (c) on state. (d) The bandstructure and operationprinciple for a VTGNRFET in the on state. The source and drain are denoted by S and D,respectively (Reported with permission from Ref. [37]. Copyright (2014), IEEE)

Fig. 8.29 (a) The sketch of the simulated structures (VTGFET and VTGNRFET) which consistof two source and drain contacts for improved electrical characteristics. (b) The atomistic diagramof the studied structure. All layers are stacked in Bernel (AB) order. In a VTGFET the graphenesheet is infinitely extended along the transverse direction (Reported with permission from Ref. [37].Copyright (2014), IEEE)

period equal top3acc, where acc D 1:42Å is the carbon-carbon bonding length,

is imposed [72]. The Hamiltonian of a single layered graphene/hBN in the deviceregion can be written as


Table 8.2 TB parameters for heterostructure of graphene and hBN [96, 100]. All parameters areexpressed in terms of electron-volt

Consite Bonsite Nonsite tCC tBN t?GrBN t?BNBN

0 3.34 �1.4 2.64 2.79 0.43 0.6

H L D

0BBBB@

˛ ˇ 0 0

ˇ� ˛� ˇ 0

0 ˇ� ˛ ˇ

0 0 ˇ�: : :

1CCCCAC

0BBB@

U1 0 0 0

0 U2 0 0

0 0 U3 0

0 0 0: : :

1CCCA : (8.50)

with

ˇ D�

0 0

tCC=BN 0

�; ˛ D

0 ty

t�y 0

!: (8.51)

Ui is the potential of the i -th atom and is equal to the sum of on-site energy and theelectro-static potential energy, tCC=BN is the hopping parameter between two nearest-neighbor carbon atoms on the graphene sheet or that between two nearest-neighborboron and nitrogen atoms on the hBN sheet, see Table 8.2, and ty is

ty D t C teikya0 : (8.52)

For an armchair ribbon configuration, a box-boundary condition is imposed to theDirac equation. This results in a transverse wave vector, which reads as [15]

ky D .2�

3a0C 2�n

2W/˙ 2�

3a0; (8.53)

whereW is the width of the device and n is an integer. The last term, which accountsfor the momentum of the Dirac points K and K0, is used with a plus/minus signwhen n is even/odd, respectively. The device Hamiltonian can be formed as a blockmatrix withH L blocks as diagonal elements and interlayer hopping parameter t? asoff diagonal elements. Therefore, the device Hamiltonian is expressed as

H D

0BBBBBBBBBBB@

˛ ˇ 0 0 � 0 � � �ˇ� ˛� ˇ 0 0 � � � �0 ˇ� ˛ ˇ 0 0 � � �0 0 ˇ� ˛� ˇ 0 � � �� 0 0 ˇ� ˛ ˇ � � �0 �� 0 0 ˇ� ˛� � � �::::::::::::::::::: : :

1CCCCCCCCCCCA

C U ; (8.54)

234 8 Applications

a bFig. 8.30 (a) The devicestructure and (b) thecorresponding equivalentcapacitive circuit model(Reported with permissionfrom Ref. [37]. Copyright(2014), IEEE)

with

� D�0 0

0 t?

�; U D

0B@U1 0 0

0 U2 0

0 0: : :

1CA : (8.55)

The retarded Green’s function of the device with four contacts can be written as

G.ky;E/ D ŒEI �H.ky/ �˙ rS1.ky; E/ �˙ r

S2.ky; E/ �˙ rD1.ky; E/

�˙ rD2.ky; E/

�1 ; (8.56)

where ˙ rS1;S2.ky; E/ and ˙ r

D1;D2.ky; E/ are self-energies of the source and draincontacts. Knowing the retarded Green’s function, the transmission function betweensome contacts j and k is evaluated as

Tjk.ky; E/ DXky

TraceŒ� j .ky; E/Gr.ky; E/� k.ky; E/G

a.ky; E/ ; (8.57)

with j D S1; S2; k D D1;D2 and � .ky;E/ D iŒ˙ r.ky; E/ � ˙ a.ky; E/ asthe broadening function of the respective contact. The summation in Eq. (8.57) isperformed on 64 equidistant grid points for the transverse wave vector ky . Finally,the current between the some contacts j and k is given by

Ijk D 2e2

„Z

dE

2�Tjk.E/.fj .E/ � fk.E// : (8.58)

The total current between the source and drain contact is I D Pj;k Ijk.


8.5.2 Self-Consistent Potential

For an accurate analysis of VTGFETs and VTGNRFETs, it is necessary to solve thetransport equation self consistently with the Poisson equation. For this purpose, weuse the capacitive device model described in Ref. [123]. The equivalent capacitivecircuit model of the studied structures is shown in Fig. 8.30. Cox and Ct are the gateinsulator and tunneling barrier capacitances, respectively. These two capacitancesare simply given by Cox D "=tox and Ct D "=tBN, where tox is the sum of thethickness of the SiO2 and hBN buffer layer, and tBN is the thickness of the hBNdielectric. The dielectric constants of both the gate insulator and the tunnelingbarrier are assumed to be the same and equal to " � 4"0. CQS and CQD represent thequantum capacitances of the source and drain graphene sheets, respectively:

CQS=D D @QS=D

@VS=D; (8.59)

where QS=D D e.pS=D � nS=D/ is the charge density of the source/drain contacts.One can write the electron density in graphene or GNR sheets of the source, anddrain contacts as

nS=D DZf .E �EFS=D � eVS=D/g.E/dE ; (8.60)

where g.E/ is the density of states of graphene or GNR, depending on the contactbase material. EFS D 0 and EFD D EFS � qVB are Fermi levels of the source anddrain contacts, respectively. �qVS and �qVD are the Dirac point potential energy ofthe source and drain graphene/GNR sheets, respectively, and VB is the drain-sourcebias voltage. Assuming a charge neutrality condition, VS and VD can be obtainedfrom the following relations:

Cox.VG � VS/ D Ct.VS � VD/C CQS

2.VS C EFS

e/ ;

Ct.VS � VD/ D CQD

2.VD C EFD

e/ :

(8.61)

One can self-consistently solve Eq. (8.61) along with Eq. (8.59) and obtain thesource and drain potentials. The potential distribution along the gate oxide and hBNlayers can be approximated by a linear function. Figure 8.31 compares the carrierconcentrations of the source and drain contacts of a VTGFET at zero drain-sourcebias voltage evaluated from the capacitive model with that from experimental results[16], where excellent agreement verifies the accuracy of our model. The resultsindicate that the screening by the source graphene sheet results in a lower carrierconcentration in the drain graphene sheet, especially at high carrier concentration ofthe source contact.

236 8 Applications

Fig. 8.31 The carrierconcentrations of the sourceand drain contacts extractedfrom the capacitive modeland experimental data ofRef. [16] (Reported withpermission from Ref. [37].Copyright (2014), IEEE)

8.5.3 Device Characteristics

The structure shown in Fig. 8.29 is studied for both VTGFET and VTGNRFETstructures. VTGNRFETs are investigated at various GNR widths. For all devices,the 3, 5 and 7 layers of hBN in Bernel (AB) stacking are examined. Assuming aninterlayer distance of 3:5Å[96], the thickness of the hBN dielectric lies between1–3 nm. For a fair comparison with experimental results [16], a 300 nm layer ofSiO2 is used as a gate oxide. The applied gate voltage is limited by the gate oxidebreakdown, which is about 1V/nm for SiO2. The bias voltage is applied betweendrain and source electrodes and is limited to about 1:5V.

Figure 8.32a shows the drain current as a function of the drain-source bias voltageat various gate voltages in a VTGFET based on the experimental data of Ref. [16].Unlike conventional FETs, the current-voltage characteristics of VTGFETs are notsaturated. In this structure the drain-source bias voltage significantly modulatesthe tunneling barrier, resulting in a lack of saturation in the drain current [65].Figure 8.32b, c are our numerical results for the same structure with differentinter-layer hopping parameters. The results shown in Fig. 8.32b are obtained byconsidering the hopping parameter between the graphene and hBN sheet as a fittingparameter, where by using t?GrBN D 0:054 eV, excellent quantitative and qualitativeagreement with experimental data can be achieved. Theoretical studies [100],however, predict a much larger value for the inter-layer hopping parameter. Thisdiscrepancy can be attributed to the miss-alignment of graphene over the hBNsheet. A number of causes can also contribute to this discrepancy, such as parasiticresistances, defects, imperfections, and various scattering mechanisms. As shownin Fig. 8.32c, using a larger value for t?GrBN mostly affects the magnitude ofthe tunneling current, whereas the trend of the current remains nearly the same.Throughout this work, the inter-layer hopping parameter of Ref. [100] has beenemployed, see Table 8.2.

Figure 8.32d depicts the drain current as a function of the drain-source biasvoltage for a VTGNRFET with a GNR width of 5 nm and 7 layers of hBN.Apparently, the Ion of the VTGNRFET is smaller than that of the VTGFET because


a b

c d

Fig. 8.32 Drain current as a function of the drain-source bias (VB) based on (a) experimental dataof Ref. [16], (b) atomistic simulation results for a VTGFET with t?GrBN D 0:054 eV regarded asa fitting parameter with experimental results, (c) atomistic simulation results for a VTGFET witht?GrBN from Ref. [100], and (d) atomistic simulation results for a VTGNRFET with a GNR width ofabout 5 nm and 7 layers of hBN. FG D VG=tox is the gate electric field (Reported with permissionfrom Ref. [37]. Copyright (2014), IEEE)

of the presence of a non-zero bandgap of the GNR. This can be well understoodby considering the transmission probability as a function of energy at variousGNR widths (Fig. 8.33a). In a VTGNRFET, because of the presence of an energybandgap, a sharp drop of the transmission probability can be observed. The widthof this region depends on the energy bandgap of the GNR, which is inverselyproportional to the ribbon’s width. The decrease of the transmission probabilitywithin the energy gap results in a smaller tunneling current of VTGNRFETs incomparison with VTGFETs. On the other hand, in a VTGFET the only degreeof freedom which can affect the transmission probability is the thickness of thetunneling barrier. Figure 8.33b shows the transmission probability of the VTGFETwith 3, 5 and 7 layers of hBN at flat band condition. The valence band andconduction band-edges of hBN are at energies of �1:4 and 3:34 eV, respectively.The current of carriers with energies smaller than �1:4 eV and larger than 3:34 eV isdue to thermionic emission and is nearly independent of the number of hBN layers.However, at energies between these two limits, the current is due to the tunneling ofcarriers and exponentially decreases as the number of hBN layers increases.

238 8 Applications

ba

Fig. 8.33 The transmission probability of (a) VTGNRFET with GNR widths of 1, 2 and 5 nmcompared with that of a VTGFET with 7 layers of hBN and (b) VTGFET with 3, 5 and 7 layers ofhBN (Reported with permission from Ref. [37]. Copyright (2014), IEEE)

a b

c d

Fig. 8.34 (a) Drain current as a function of the applied gate electric field in VTGFET with 3, 5,and 7 layers of hBN. Drain current as a function of the applied gate electric field in VTGNRFETat various GNR widths in comparison with that of VTGFET with (b) 7 (c) 5, and (d) 3 layers ofhBN. VB is equal to 1:5V (Reported with permission from Ref. [37]. Copyright (2014), IEEE)

VTGNRFET shows a smaller tunneling current in both the on and off state incomparison with the VTGFET, however, it exhibits a significantly larger Ion=Ioff

ratio. Figure 8.34a–d compares the drain current as a function of the applied gate


Table 8.3 The intrinsic sub-threshold swings for VTGFET and VTGNRFET at different numbersof hBN layers and various GNR widths. All parameters expressed in terms of mV/dec (Reportedwith permission from Ref. [37]. Copyright (2014), IEEE)

7 Layers of 5 Layers of 3 Layers ofhBN hBN hBN

VTGFET 198 558 1,5344.3 nm 184 484 1,297

VTGNRFET 3.2 nm 178 463 1,0731.7 nm 170 388 1,038

electric field for VTGFET and VTGNRFET at different numbers of hBN layers andvarious GNR widths. As can be seen in these figures for an electric field with amagnitude of about 1V/nm, the dependency of the tunneling current on the numberof hBN layers and the width of the ribbon is weaker than that for smaller electricfields, indicating that the contribution of the thermionic current increases in thisregime. Interestingly, the Ion=Ioff ratios for VTGNRFETs are significantly largerthan that of the VTGFET. This can be understood by comparing the transmissionprobabilities of the VTGFET with that of the VTGNRFET (Fig. 8.33). While thetransmission probability of a VTGFET is nearly flat close to 0 eV, the transmissionprobability of a VTGNRFET undergoes a sharp variation in this region due to thepresence of an energy bandgap. Therefore, as the GNR width decreases, the Ion=Ioff

ratio increases due to the increase of the energy gap.Another important figure of merit for FETs is the sub-threshold swing defined as

SS D dVG

d.log.ID//: (8.62)

In available experimental data, a gate oxide of 300 nm is used, resulting in arelatively large sub-threshold swing. By employing a thin and high-� dielectric, onecan significantly reduce the sub-threshold swing. To gain insight into the intrinsiclimit of the VTGNRFET and VTGFETs, the variation of the source instead of thevoltage is considered in Eq. (8.62). The intrinsic sub-threshold swings for VTGFETand VTGNRFET at different numbers of hBN layers and various GNR widthsare listed in Table 8.3. It should be noted that large Ion=Ioff ratios occur at largegate voltages, which are much larger than the supply voltage required for next-generation devices .VDD < 0:7V/. The partial screening of the gate bias voltageby the graphene source contact leads to poor electrostatic control of gate voltageon the tunneling barrier. In comparison with the sub-threshold swing of about90mV/dec in modern MOSFETs, and the ideal value of 60mV/dec, this structurehas a larger sub-threshold swing. Table 8.3 shows that the sub-threshold swing of theVTGNRFET is smaller than that of the VTGFET. The lower density of states andtherefore smaller quantum capacitance of VTGNRFET result in a weaker screeningeffect in VTGNRFET structures. The increase of the numbers of hBN layers, which

240 8 Applications

100

102

104

100

10−2

102

ION

/IOFF

τ [p

S]

7 Layers of hBN

5 Layers of hBN

3 Layers of hBN

ION

/IOFF

=3

τ=0.007pS

ION

/IOFF

=12

ION

/IOFF

=441

τ=4.2pS

τ=0.7pS

100

101

102

10−3

10−2

ION

/IOFF

τ [p

S]

W = 1.3 nm

W = 1.7 nm

W = 3.2 nm

W = 4.3 nm

Graphene

3 Layers of hBN

τ=0.008pS

ION

/IOFF

=3

τ=0.007pS

ION

/IOFF

=50

105

100

102

ION

/IOFF

τ [p

S]

W = 1.3 nm

W = 2 nm

W = 3.2 nm

W = 4.3 nm

Graphene

7 Layers of hBNτ=0.9pS

ION

/IOFF

=441

τ=4.2pS ION

/IOFF

=23270

102

101

103

10−1

100

ION

/IOFF

τ [p

S]

W = 1.3 nm

W = 3.2 nm

W = 3.9 nm

W = 4.3 nm

Graphene

5 Layers of hBN

ION

/IOFF

=667

τ=0.1pS

ION

/IOFF

=12

τ=0.7pS

a b

c d

Fig. 8.35 (a) The intrinsic gate-delay time of a VTGFET with 3, 5, and 7 layers of hBN. Thecomparison between the intrinsic gate-delay time of VTGNRFETs at various GNR widths and thatof a VTGFET with (b) 7, (c) 5, and (d) 3 layers of hBN. VB is equal to 1:5V (Reported withpermission from Ref. [37]. Copyright (2014), IEEE)

corresponds to a wider barrier, leads to a smaller sub-threshold swing. Furtherimprovement in the sub-threshold swing is observed as the ribbon’s width decreases.

For fair comparison between VTGFET and VTGNRFET, one can investigate thegate-delay time with respect to the Ion=Ioff ratio [42]. The intrinsic gate-delay timecan be obtained as [42]

� D CG�VG

Ion: (8.63)

CG in Eq. (8.63) is extracted from the slope of the total charge in the channel withrespect to the top gate voltage. Further, Ion is the on-state current and �VG is equalto the difference between the on- and off-state gate voltages. Figure 8.35a comparesthe intrinsic gate-delay time as a function of the Ion=Ioff ratio of VTGFET structureswith different numbers of hBN layers. VTGFETs with thinner hBN dielectric havea much larger Ion, see Fig. 8.34a. As a result, smaller intrinsic gate-delay time at thecost of a lower Ion=Ioff ratio can be achieved for devices with three layers of hBN. Inconventional MOSFET transistors, as the Ion=Ioff ratio increases, the intrinsic gate-

8.6 CNT and GNR-Based Photodetectors 241

delay time increases due to the saturation of the drain current [42]. However, inVTGFET the drain current is not saturated and therefore an increase in Ion=Ioff ratioleads to a smaller and more desirable intrinsic gate-delay time. This is an importantadvantage of this structure in comparison with conventional MOSFET transistors.Figure 8.35b–d depicts the intrinsic gate-delay time of VTGFET and VTGNRFETwith different numbers of hBN layers and various GNR widths. In comparison withVTGFET, VTGNRFET has a smaller Ion and a smaller gate capacitance. In mostcases the effect of these two terms in Eq. (8.63) are cancelled and an intrinsic gate-delay time similar to that of VTGFET is observed for VTGNRFETs, except forthose with ribbons narrower than 2 nm.

For analog applications, the cut-off frequency fT is an important figure of merit.Assuming a quasi-static condition, the cut-off frequency is given by [30]

fT D gm

2�CG; (8.64)

with gm D @ID=@VG and CG D CGS C CGD, where CGS and CGD are the gate-source and gate-drain capacitances. To focus on the intrinsic response, parasiticcapacitances are neglected and only the gate insulator, tunneling barrier andquantum capacitances are considered [30]; thus CG D @Q=@VG, where Q is thetotal charge in the channel.

The cut-off frequency of VTGFET with a different number of hBN layers isshown in Fig. 8.36a. The cut-off frequency is inversely proportional to the hBNlayer thickness. As the hBN layer becomes thinner, the control of gate voltage overthe channel increases. As a result, gm and fT increase. The cut-off frequency ofVTGFET with three layers of hBN can reach to about 60 GHz. Figure 8.36b–dcompare the cut-off frequency of VTGNRFETs at various GNR widths and differentnumbers of hBN layers with that of a VTGFET. Smaller intrinsic capacitances ofGNRs result in larger cut-off frequencies of VTGNRFETs in comparison with thatof VTGFET. The highest cut-off frequency can reach to 130GHz for a devicewith GNR width of 2 nm and three layers of hBN. Because of the presence ofsubbands and their dependence on GNR width, non-smooth behavior is observed inthe transconductance and, as a result, in the cut-off frequency, see Fig. 8.36b–d. Dueto strong modulation of the Fermi levels and carrier concentration in VTGNRFETswith three layers of hBN, the curves for such devices are more non-smooth. As seenin Fig. 8.36d, the cut-off frequency has a local peak at a gate electric field of about�0:8V/nm, where EFS moves from the valence to the conduction band, which inturn increases the transconductance and cut-off frequency.

8.6 CNT and GNR-Based Photodetectors

The direct bandgap and the tunability of the bandgap with the tube diameter orribbon’s width render CNTs and GNRs as suitable candidates for opto-electronic

242 8 Applications

−1 −0.9 −0.8 −0.7

100

10−2

102

FG

[V/nm]

f T [

GH

z]3 Layers of hBN

5 Layers of hBN

7 Layers of hBN

58 GHz

1.5 GHz

0.5 GHz

−1 −0.9 −0.8 −0.7

102

FG

[V/nm]

f T [

GH

z]

W = 2 nm

W = 3.9 nm

W = 5 nm

Graphene

58 GHz

132 GHz

3 Layers of hBN

−1 −0.9 −0.8 −0.7

100

10−1

FG

[V/nm]

f T [

GH

z]

Graphene

7 Layers of hBN

0.5 GHz

1.8 GHz

W = 5, 3.2, 1.3 nmGNR

−1 −0.9 −0.8 −0.7

100

101

FG

[V/nm]

f T[G

Hz]

Graphene

W = 5, 3.9, 1.7, 1.3 nm

1.5 GHz

20 GHz

GNR

5 Layers of hBN

a b

c d

Fig. 8.36 (a) The cut-off frequency of VTGFET with 3, 5, and 7 layers of hBN. The cut-offfrequency of VTGNRFETs at various GNR widths in comparison with that of a VTGFET with(b) 7, (c) 5, and (d) 3 layers of hBN. VB is equal to 1:5V (Reported with permission from Ref. [37].Copyright (2014), IEEE)

devices, especially for infra-red (IR) applications [32, 33], due to the relativelynarrow bandgap [32, 73, 121]. Here we employ the NEGF method along with a TBmodel to study CNT and GNR based IR photodetectors. For simplicity, we assumethe interaction of electrons takes place with a monochromatic light that is polarizedalong the CNT and GNR axis (z-direction), see Fig. 8.37.

8.6.1 Electron-Photon Self-Energy

By using dipole approximation for a set of localized basis functions 'm, the electron-photon interaction Hamiltonian is given by (see Sect. 3.2.8)

OHe-� DXmn

Mmn

� Obq�.t/C Ob�q�.t/

Oc�m.t/ Ocn.t/ ; (8.65)

where the matrix elements are given by [47]


a b

Fig. 8.37 (a) The sketch of the simulated device. (b) The process of electron-hole generationby photo-absorption. Incident photons generate electron-hole pairs and the electric field driveselectrons and holes towards the drain and source contacts, respectively. EG D 0:6 eV, „! D0:65 eV

Mmn D �e

� „2V �!

�1=2h'mj Opz

mj'ni ;

D �e

�„p�r�r

2N!�cI�

�1=2h'mj i

„ ŒOH0; z j'ni ;

D .zm � zn/ie

„�„p

�r�r

2N!�cI�

�1=2Hmn ;

(8.66)

where I� is the photon flux, defined as the number of photons per unit time per unitarea:

I� � n�c

Vp�r�r

; (8.67)

where �r and �r are the relative dielectric and magnetic susceptibility. The numberof incident photon per unit area is given by

n� D P�=.„!/ ; (8.68)

where P� is the incident power per unit area. For transition from the first line to thesecond line, the commutator relation between Hamiltonian and position operator isused. It should be noted that the Hamiltonian here is a non-interacting single-particleone. As shown in the third line of Eq. (8.66), this relation significantly simplifiesthe evaluation of matrix elements. However, it should be noted that intra-atomictransitions are neglected based on the relation [94]. By using a similar procedure for

244 8 Applications

a b

Fig. 8.38 (a) The calculated photo-current as a function of the included off-diagonal elements ofthe retarded self-energy (˙R). The current is normalized to the value with full matrix elements.The full matrix size is 60�60. (b) The retarded Green’s function in two coordinate representation.The existence of relatively strong off-diagonal elements indicate the non-locality of the interactionand the need to include the full matrix

obtaining phonon self-energy, the lesser self-energy for electron-photon interactioncan be written as [47]

˙<lp .E/ D

Xmn

MlmMnp�n�G

<mn.E � „!/C .n� C 1/G<

mn.E C „!/� ; (8.69)

where the first term corresponds to the excitation of an electron by the absorptionof a photon and the second term corresponds to the emission of a photon byde-excitation of an electron. The greater self-energy can be derived in a similarway. After the evaluation of the lesser and greater, the retarded self-energy can becalculated (see Sect. 7.3). Under intense illumination, where the number of photonsis relatively large n� � n� C 1, one can approximate the self-energy as

˙<lp .E/ D

Xmn

.zl � zm/.zp � zn/e2

p�r�rI�

2„!�c�G<

mn.E � „!/CG<mn.E C „!/� :

(8.70)

When scattering via a self-energy is introduced, the determination of the Green’sfunction requires inversion of a matrix of huge rank. To reduce the computationalcost, the local scattering approximation is frequently used. In this approximationthe scattering self-energy terms are diagonal in coordinate representation. It allowsone to employ the recursive algorithm for computing the Green’s functions [104].The local approximation is well justified for electron-phonon scattering causedby deformation potential interaction, see Sect. 7.3. However, the self-energy of


Fig. 8.39 The quantumefficiency of the CNT as afunction of the incidentphoton energy. The numberof included off-diagonalelements of the self-energyhas a strong influence on thecalculated quantum efficiency

electron-photon interaction is apparently non-local. For the given structure Fig. 8.37,the calculated photo current is shown in Fig. 8.38a. The results are indicated as afunction of the number of included off-diagonal elements of the retarded self-energy,which includes the effects of electron-photon interaction. By including only thediagonal elements of the self-energy (local scattering approximation), the calculatedcurrent is only 4 % of its value in case of full matrix consideration. This behaviorcan be well understood by considering that electron-photon self-energy is in generalnon-local and depends on neighboring Green’s functions. The off-diagonal elementsof the Green’s function indicate the correlation between different sites. Figure 8.38bshows the Green’s function in two coordinate representation. Off-diagonal elementsare relatively strong, which indicates the need for a full matrix description.

8.6.2 Quantum Efficiency

To investigate CNT and GNR photodetectors, we study the quantum efficiency,defined as

˛ D I�=e

P�=.„!/ : (8.71)

I� is the photo current and P� is the incident optical power. This quantitycorresponds in fact to the energy conversion efficiency of a photodetector.

Figure 8.39 shows the quantum efficiency of the CNT as a function of the incidentphoton energy. The efficiency is maximized when the photon energy matches thebandgap of the CNT; however, at the energy the inclusion of off-diagonal elementsbecomes more important. This can be understood by considering the fact that at thatpeak, the carrier energy is close to the conduction and valence band energy, where it

246 8 Applications

a b

Fig. 8.40 (a) The density of states of an .12; 0/ armchair GNR. Some of the most importanttransitions are marked: Eij denotes a transition from the i th valence band to the j th conductionband. (b) The calculated quantum efficiency as a function of the incident photon energy

has a longer wave-length. The result is in agreement with experimental data wherethe maximum quantum efficiency is estimated to be between 10 and 20 % [32].

Figure 8.40a shows the density of states of an .12; 0/ armchair GNR for the firstthree subbands. Van-hove singularities in the density of states result in large photon-assisted transitions from the valence to the conduction band [49]. Some of themost important transitions are marked. Figure 8.40b shows the calculated quantumefficiency of the investigated device as a function of the incident photon energy. Theefficiency is maximized when the photon energy matches the bandgap of the GNR.The maximum quantum efficiency ranges from 9 to 11 % and is fairly independentof the bandgap [103]. Due to periodic boundary conditions, the subbands of CNTsappear as double degenerate. However, in GNRs this symmetry is removed andsubbands are no longer degenerate. As a result, the current capacity in GNRs isroughly half of that of their CNT counterparts. It is therefore reasonable to expect amaximum quantum efficiency of 10% in GNR devices.

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Index

A

Acoustic phonon 66, 177Adaptive integration 190Analytical continuation 121Anti-commutator 20Anti-time-ordered 122

B

Band index 58Basis function 157Bloch’s theorem 57, 59Boltzmann transport equation 96Bose-Einstein statistics 76, 80, 94Bosons 31

non-interacting 127Bra-ket 12Bravais lattice 55, 57Brillouin zone 57

C

Carbon nanotube 202, 241Carrier concentration 152, 184Chemical equilibrium 81Chemical potential 80, 83Closed system 11Collective excitations 46Collision operator 97Commutator 20, 22Continuity equation 186

Convergence 169, 194Crystal 55Current density 153, 184, 185Cut-off frequency 241

D

Deformation potential 67–69, 177Density matrix 91Density operator 91Dipole approximation 53Dirac

delta function 13, 19, 161equation 10notation 12, 14

Disconnected diagrams 134Distinguishability 79Dynamical matrix 64, 222Dyson equation 134, 181

E

Effective mass approximation 59Elementary excitation 47Energy grid 187, 188, 190Ensemble 78

average 111non-equilibrium 118

canonical 88grand 37, 88, 116

micro-canonical 88theory 76

Entropy 80

M. Pourfath, The Non-Equilibrium Green’s Function Method for Nanoscale DeviceSimulation, Computational Microelectronics, DOI 10.1007/978-3-7091-1800-9,© Springer-Verlag Wien 2014

253

254 Index

Equilibrium 81, 111Ergodicity 79Error estimation 190Extensive variables 80

F

Fermi-Dirac statistics 80, 93Fermi golden rule 96, 99Fermions 31

non-interacting 124Feynman diagrams 131First quantization 29, 34Fock space 37, 38Force constant 221Fourier transform 57

G

Gate-delay time 240Grand thermodynamic potential 84Graphene 201, 230Graphene nanoribbon 202, 230, 241

armchair 203zigzag 203

Green’s functionadvanced 122, 181contour-ordered 119equilibrium 113finite temperature 115greater 122, 182lesser 122, 182Matsubara 117non-equilibrium 118non-interacting 124perturbation expansion 128phonon 150real-time 125recursive 181, 182retarded 122, 181time-ordered 122zero temperature 113

H

Harmonicapproximation 63, 150oscillator 47, 48

Helmholtz free energy 83Hermitian

matrix 60, 91operator 14, 18, 21, 24, 38, 91

Hilbert space 12, 14, 18, 31, 37Hilbert transformation 180Hopping 61, 62, 64Hubbard model 61

I

Identical particles 30Indistinguishability 30Infra-red 242Intensive variables 80Interaction

electron-electron 61, 131, 137, 143electron-phonon 67, 72, 137, 147, 175electron-photon 52, 241

Internal energy 80Iterative method 169Iterative scheme 193

K

Kadanoff-Baym formalism 139Keldysh

contour 121formalism 139

Kinetic equations 138steady-state 140

Kramers’ theorem 59

L

Langreth theorem 123Larmor interaction 26Lattice 55

vector 57Laws of thermodynamics 80Legendre transformation 84Line-edge roughness 206Liouville 98Local density of states 153

M

Many-body 10, 29, 113Matrix truncation 166

Index 255

Maxwell-Boltzmann statistics 79, 95Mechanical equilibrium 81Mode-space 159, 162

decoupled 164Monte Carlo 97

N

Non-equilibrium statistical mechanics 77

O

Observables 152, 184Occupation number 37, 38On-site

energy 61repulsion 61

Open system 83, 89, 188Operator 16

annihilation 38contour-ordering 120creation 38density 111, 113Hermitian 18momentum 20orbital angular momentum 19projection 17quantum field 43time evolution 108

imaginary 111time-ordering 122

Optical phonon 66

P

Partition function 89Pauli

equation 26exclusion principle 24, 31, 33, 39master equation 99matrices 24

Perturbation expansion 131Phase-incoherent 205Phase space 76Phonon 31, 56, 62, 66

transport 220Photodetector 241Photon 31, 49

flux 243Picture

Heisenberg 106interaction 107Schrödinger 106

Poisson equation 194Polar interaction 178Polarization 69, 144

efficiency 220vector 50, 65, 242

Power factor 221Pure state 91

Q

Quantizationelectromagnetic field 49phonon 66

Quantum field theory 10, 37Quasi particles 46

R

Real-space 159, 160Real-time

formalism 121Green’s function 125integral 121

Reciprocallattice vector 57space 57

Relativistic effects 26Reversible process 81

S

Sancho-Rubio method 169Scattering

acoustic phonon 177deformation potential 67elastic 68inelastic 68line-edge roughness 206optical phonon 177piezoelectric 68polar optical 68polar optical phonon 178umklapp 73

256 Index

Screened interaction 144Second quantization 10, 29, 37, 40Secular equation 60Seebeck coefficient 221Self-consistent 194, 235Self-consistent Born approximation 136, 154Self-energy 151

acoustic phonon 177approximation 136contact 165, 172, 174electron-phonon 137, 147, 175optical phonon 177polar optical phonon 178proper 134scattering 175variational derivation 142

Slaterdeterminant 31, 33permanent 31, 34

Spectral function 153Spin 19, 24

transport 216Spin-orbit

coupling 26, 217interaction 20, 217

Spin-statistics 30Spinor 24Statistical

ensemble 75equilibrium 77mechanics 75, 76

Sub-threshold swing 239Surface corrugation 211Surface Green’s function 168, 169

T

Thermal conductivity 220

Thermal equilibrium 81Thermodynamic limit 80Thermoelectric 220Thomas precession 26Tight-binding 59, 157, 204, 216, 230Time-reversal 59Transmission probability 186, 205, 223Transverse

direction 159wave vector 159

Tunneling transistor 230

U

Uncertainty principle 22, 95, 97Unit cell 55

V

Vacuum 39Valence force field 221Vertex function 144

W

Wick’s theorem 120, 129Wide-band limit 174

Z

Zeeman 26Zone boundary 67

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Mahdi˜Pourfath The Non-Equilibrium Green's Function …