Basic Semiconductor Physics

Chihiro Hamaguchi

Basic Semiconductor Physics

Second Edition

With 245 figures

123

Prof. Chihiro HamaguchiProfessor Emeritus of Osaka UniversityGraduate School of EngineeringDept. Electronic EngineeringSuita, Osaka565-0871 Japanhamaguchi-c@r6.dion.ne.jp

ISBN 978-3-642-03302-5 e-ISBN 978-3-642-03303-2DOI 10.1007/978-3-642-03303-2Springer Heidelberg Dordrecht London New York

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Preface

When the first edition of Basic Semiconductor Physics was published in 2001,there were already many books, review papers and scientific journals deal-ing with various aspects of semiconductor physics. Since many of them weredealing with special aspects of newly observed phenomena or with very fun-damental physics, it was very difficult to understand the advanced physics ofsemiconductors without the detailed knowledge of semiconductor physics. Forthis purpose the author published the first edition for the readers who areinvolved with semiconductor research and development. Basic SemiconductorPhysics deals with details of energy band structures, effective mass equa-tion and k ·p perturbation, and then describes very important phenomena insemiconductors such as optical, transport, magnetoresistance, and quantumphenomena. Some of my friends wrote to me that the textbook is not onlybasic but advanced, and that the title of the book does not reflect the con-tents. However, I am still convinced that the title is appropriate, because theadvanced physics of semiconductor may be understood with the knowledgeof the fundamental physics. In addition new and advanced phenomena ob-served in semiconductors at an early time are becoming well-known and thusclassified in basic physics.

After the publication of the first edition, many typographical errors havebeen pointed out and the corrected version was published in 2006. The pub-lisher and my friends persuade me to revise the book adding new chapters,keeping the subject at the appropriate level. When I started writing the firstedition, I decided not to include physics of semiconductor devices such as p-njunction diode, bipolar transistor and MOSFET (Metal Oxide SemiconductorField Effect Transistor). This is because the large numbers of books dealingwith the subjects are available and a big or bulky volume is not accepted byreaders. On the other hand many researchers are involved with opto-electronicdevices such as LED (Light Emitting Diode) and LD (Laser Diode) becausememory devices such as DVD and Blue Ray Disks are becoming importantfor writing and reading memory devices. In such devices semiconductor laserdiodes are used. In addition the communication system based on the optical

VI Preface

fiber plays a very important role in network, where laser diode is the keydevice. Although many books on semiconductor physics and technology havebeen published, the basic physics of semiconductor laser is not properly de-scribed. When the readers of my book understand the characteristics of twodimensional electron gas and strain effect of semiconductors, they feel easyto understand double heterostructure lasers and strained quantum well lasers,but it is easier if they study some more detailed discussion on the laser action.Another subject is physics of low dimensional semiconductors. Basic Semicon-ductor Physics deals with two dimensional electron gas, but zero dimensionalor quantum dot structure is not included. The physics of quantum dot includesvery important physics of artificial atoms, and gives a good information of fewelectron systems.

In this revised version I included three main topics. The first one isSect. 3.5, where electron motion in an external field is discussed with thederivation of effective mass. The most important relation for transport equa-tion is the velocity (group velocity) of an electron in a periodic crystal. In thissection the expectation value of the velocity operator is evaluated and shownto be proportional to the gradient of the electron energy with respect to thewave vector. Then the classical motion of equation is proved to be valid for anelectron in a crystal when we use the effective mass. In Sect. 8.8 the physics ofquantum dots is discussed in connection with the charging energy (additionenergy) required to add an extra electron in a quantum dot. The treatment isvery important to understand Coulomb interaction of many electron system.In this section the exact diagonalization method based on Slater determinantsis discussed in detail. Chapter 9 is devoted to the discussion on the physicsof semiconductor laser, where Einstein coefficients A and B, spontaneous andstimulated emission, luminescence, double heterostructure, and quantum welllasers are discussed. The strain effect of the quantum well laser is described indetail because it is well known that the effect is very important to understandthe modes (TE and TM modes) of quantum well laser oscillations.

I would like to express my special thanks to Professor Nobuya Mori forhelping me to clarify the subject and providing me his calculated results usedin Chap. 9, and also to my colleagues at Sharp Corporation with whom Ihave had many stimulated discussions on the basic physics of semiconductorlasers. It was very sad that Professor Tatsuya Ezaki of Hiroshima Universitydied very recently, who made the detailed analysis of quantum dot physics forhis Ph.D thesis (see Sect. 8.8).

Finally I want to thank Dr. Claus E. Ascheron and the staff of the SpringerVerlag for their help and for the valuable suggestions for clarification of thisbook.

Osaka, September 2009 Chihiro Hamaguchi

Preface to the First Edition

More than 50 years have passed since the invention of the transistor in Decem-ber 1947. The study of semiconductors was initiated in the 1930s but we had towait for 30 years (till the 1960s) to understand the physics of semiconductors.When the transistor was invented, it was still unclear whether germaniumhad a direct gap or indirect gap. The author started to study semiconductorphysics in 1960 and the physics was very difficult for a beginner to understand.The best textbook of semiconductors at that time was “Electrons and Holes inSemiconductors” by W. Shockley, but it required a detailed knowledge of solidstate physics to understand the detail of the book. In that period, junctiontransistors and Si bipolar transistors were being produced on a commercialbasis, and industrialization of semiconductor technology was progressing veryrapidly. Later, semiconductor devices were integrated and applied to comput-ers successfully, resulting in a remarkable demand for semiconductor memoriesin addition to processors in the late 1970–1980s. Now we know that semicon-ductors play the most important role in information technology as the keydevices and we cannot talk about the age of information technology withoutsemiconductor devices.

On the other hand, the physical properties of semiconductors such as theelectrical and optical properties were investigated in detail in the 1950s, lead-ing to the understanding of the energy band structures. Cyclotron resonanceexperiments and their detailed analysis first reported in 1955, were the mostimportant contribution to the understanding of the energy band structures ofsemiconductors. From this work it was revealed that the valence bands consistof degenerate heavy-hole and light-hole bands. Another important contribu-tion comes from energy band calculations. Energy band calculations basedon the empirical pseudopotential method and the k · p perturbation methodreported in 1966 enabled us to understand the fundamental properties ofsemiconductors. In this period high-field transport and current instabilitiesdue to the Gunn effect and the acoustoelectric effect attracted great inter-est. In addition, modulation spectroscopy and light scattering were developedand provided detailed information of the optical properties of semiconductors.

VIII Preface to the First Edition

These contributions enabled us to understand the physical properties of bulksemiconductors almost completely.

At the same time, late in the 1960s and early 1970s, Leo Esaki and hisco-workers developed a new crystal growth method, molecular beam epitaxy,and initiated studies of semiconductor heterostructures such as quantum wellsand superlattices. This led to a new age of semiconductor research whichdemonstrated phenomena predicted from quantum mechanics. This approachis completely different from the past research in that new crystals and newstructures are being created in the laboratory. This field is therefore called“band gap engineering”. It should be noted here that such a research wasnot carried out up to fabricate devices for real applications but to investigatenew physics. The proposal of modulation doping in the late 1970s and theinvention of the high electron mobility transistor (HEMT) in 1980 triggereda wide variety of research work related to this field. Later HEMTs have beenwidely used in such applications as the receivers for satellite broadcasting.Although the commercial market for LSI memories based on Si technologies ishuge, metal semiconductor field-effect transistors (MESFETs) based on GaAshave become key devices for mobile phones (cellular phones) in the 1990s andit is believed that their industrialization will play a very important role in the21st century.

Klaus von Klitzing et al. discovered the quantum Hall effect (later calledthe integer quantum Hall effect) in the two-dimensional electron gas systemof a Si MOSFET in 1980, and this discovery changed semiconductor researchdramatically. The discovery of the fractional quantum Hall effect followed theinteger quantum Hall effect and many papers on these subjects have beenreported at important international conferences. At the same time attemptsto fabricate microstructures such as quantum wires and metal rings were car-ried out by using semiconductor microfabrication technologies and led to thediscovery of new phenomena. These are the Aharonov–Bohm effect, ballis-tic transport, electron interference, quasi-one-dimensional transport, quantumdots, and so on. The samples used for these studies have a size between themicroscopic and macroscopic regions, which is thus called the “mesoscopicregion”. The research in cmesoscopic structures is still progressing.

The above overview is baed on the private view of the author and veryincomplete. Those who are interested in semiconductor physics and in deviceapplications of new phenomena require a deep understanding of semiconduc-tor physics. The situation is quite different for the author who had to gropehis own way in semiconductor physics in the 1960s, while the former arerequested to begin their own work after understanding the established semi-conductor physics. There have been published various textbooks in the fieldof semiconductors, but only few cover the field from the fundamentals to newphenomena. The author has published several textbooks in Japanese, but theydo not cover such a wide range of semiconductor physics. In order to supple-ment the textbooks he has used printed texts for graduate students in the last10 years, revising and including new parts.

Preface to the First Edition IX

This textbook is not intended to give an introduction to semiconduc-tors. Such introductions to semiconductors are given in courses on solid-statephysics and semiconductor devices at many universities in the world. It is clearfrom the contents of this textbook that electron statistics in semiconductors,pn junctions, pnp or npn bipolar transistors, MOSFETs and so on are notdealt with. This textbook is written for graduate students or researchers whohave finished the introductory courses. Readers can understand such device-oriented subjects easily after reading this textbook. A large part of this bookhas been used in lectures several times for the solid-state physics and semi-conductor physics courses for graduate students at the Electronic Engineer-ing Department of Osaka University and then revised. In order to understandsemiconductor physics it is essential to learn energy band structures. For thisreason various methods for energy band calculations and cyclotron resonanceare described in detail. As far as this book is concerned, many of the subjectshave been carried out as research projects in our laboratory. Therefore, manyfigures used in the textbook are those reported by us in scientific journals andfrom new data obtained recently by carrying out experiments so that digitalprocessing is possible. It should be noted that the author does not intend todisregard the priorities of the outstanding papers written by many scientists.Important data and their analysis are referred to in detail in the text, andreaders who are interested in the original papers are advised to read the ref-erences. This book was planned from the beginning to be prepared by LATEXand the figures are prepared in EPS files. Figures may be prepared by usinga scanner but the quality is not satisfactory compared to the figures drawnby software such as PowerPoint. This is the main reason why we used ourown data much more than those from other groups. Numerical calculationssuch as energy band structures were carried out in BASIC and FORTRAN.Theoretical curves were calculated using Mathematica and equations of sim-ple mathematical functions were drawn by using SMA4 Windows. The finalforms of the figures were then prepared using PowerPoint and transformed intoEPS files. However, some complicated figures used in Chap. 8 were scannedand then edited using PowerPoint.

The author would like to remind readers that this book is not written forthose interested in the theoretical study of semiconductor physics. He believesthat it is a good guide for experimental physicists. Most of the subjects areunderstood within the framework of the one-electron approximation and thebook requires an understanding of the Schrodinger equation and perturbationtheory. All the equations are written using SI units throughout so that read-ers can easily estimate the values. In order to understand solid-state physicsit is essential to use basic theory such as the Dirac delta function, Diracidentity, Fourier transform and so on. These are explained in the appendices.In addition, a brief introduction to group theoretical analysis of strain ten-sors, random phase approximations, boson operators and the density matrixis given in the appendices. With this background the reader is expected tounderstand all the equations derived in the text book.

X Preface to the First Edition

The author is indebted to many graduate students for discussions and theuse of their theses. There is not enough space to list all the names of thestudents. He is also very thankful to Prof. Dr. Nobuya Mori for his criticalreading of the manuscript and valuable comments. He thanks Dr. MasatoMorifuji for his careful reading of the text. Dr. Hideki Momose helped theauthor to prepare the LATEX format and Prof. Dr. Nobuya Mori revised it.Also, thanks are due to Mr. Hitoshi Kubo, who took the Raman scatteringdata in digital format. He is also very thankful to Prof. Dr. Laurence Eavesand Prof. Dr. Klaus von Klitzing for their encouragement from the early stageof the preparation of the manuscript. A large part of the last chapter, Chap. 8,was prepared during his stay at the Technical University of Vienna and hewould like to thank Prof. Dr. Erich Gornik for providing this opportunity andfor many discussions. Critical reading and comments from Prof. L. Eaves, Prof.K. von Klitzing, Prof. G. Bauer and Prof. P. Vogl are greatly appreciated.Most of the book was prepared at home and the author wants to thank hiswife Wakiko for her patience.

Osaka, Japan, Chihiro HamaguchiMarch 2001

Contents

1 Energy Band Structures of Semiconductors . . . . . . . . . . . . . . . . 11.1 Free-Electron Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Bloch Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Nearly Free Electron Approximation . . . . . . . . . . . . . . . . . . . . . . . 41.4 Reduced Zone Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5 Free-Electron Bands (Empty-Lattice Bands) . . . . . . . . . . . . . . . . 91.6 Pseudopotential Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.7 k · p Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.8 Density of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2 Cyclotron Resonance and Energy Band Structures . . . . . . . . 292.1 Cyclotron Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.2 Analysis of Valence Bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.3 Spin–Orbit Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.4 Non-parabolicity of the Conduction Band. . . . . . . . . . . . . . . . . . . 512.5 Electron Motion in a Magnetic Field and Landau Levels . . . . . . 54

2.5.1 Landau Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542.5.2 Landau Levels of a Non-parabolic Band . . . . . . . . . . . . . . 612.5.3 Landau Levels of the Valence Bands . . . . . . . . . . . . . . . . . 65

3 Wannier Function and Effective Mass Approximation . . . . . . 733.1 Wannier Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.2 Effective-Mass Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.3 Shallow Impurity Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803.4 Impurity Levels in Ge and Si . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.4.1 Valley–Orbit Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 863.4.2 Central Cell Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.5 Electron Motion Under an External Field . . . . . . . . . . . . . . . . . . 893.5.1 Group Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 903.5.2 Electron Motion Under an External Force . . . . . . . . . . . . 933.5.3 Electron Motion and Effective Mass . . . . . . . . . . . . . . . . . 96

XII Contents

4 Optical Properties 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.1 Reflection and Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.2 Direct Transition and Absorption Coefficient . . . . . . . . . . . . . . . . 1034.3 Joint Density of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.4 Indirect Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104.5 Exciton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

4.5.1 Direct Exciton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1164.5.2 Indirect Exciton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

4.6 Dielectric Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1274.6.1 E0, E0 + Δ0 Edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1304.6.2 E1 and E1 + Δ1 Edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1324.6.3 E2 Edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1334.6.4 Exciton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

4.7 Piezobirefringence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1354.7.1 Phenomenological Theory of Piezobirefringence . . . . . . . 1354.7.2 Deformation Potential Theory . . . . . . . . . . . . . . . . . . . . . . 1364.7.3 Stress-Induced Change in Energy Band Structure . . . . . . 140

5 Optical Properties 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1475.1 Modulation Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

5.1.1 Electro-Optic Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1475.1.2 Franz–Keldysh Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1495.1.3 Modulation Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . 1535.1.4 Theory of Electroreflectance

and Third-Derivative Form of Aspnes . . . . . . . . . . . . . . . . 1575.2 Raman Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

5.2.1 Selection Rule of Raman Scattering . . . . . . . . . . . . . . . . . . 1675.2.2 Quantum Mechanical Theory of Raman Scattering . . . . 1725.2.3 Resonant Raman Scattering . . . . . . . . . . . . . . . . . . . . . . . . 177

5.3 Brillouin Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1805.3.1 Scattering Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1825.3.2 Brillouin Scattering Experiments . . . . . . . . . . . . . . . . . . . . 1875.3.3 Resonant Brillouin Scattering . . . . . . . . . . . . . . . . . . . . . . . 190

5.4 Polaritons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1935.4.1 Phonon Polaritons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1935.4.2 Exciton Polaritons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

5.5 Free–Carrier Absorption and Plasmon . . . . . . . . . . . . . . . . . . . . . . 200

6 Electron–Phonon Interaction and Electron Transport . . . . . . 2076.1 Lattice Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

6.1.1 Acoustic Mode and Optical Mode . . . . . . . . . . . . . . . . . . . 2076.1.2 Harmonic Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

6.2 Boltzmann Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 2216.2.1 Collision Term and Relaxation Time . . . . . . . . . . . . . . . . . 2236.2.2 Mobility and Electrical Conductivity . . . . . . . . . . . . . . . . . 225

Contents XIII

6.3 Scattering Probability and Transition Matrix Element . . . . . . . . 2306.3.1 Transition Matrix Element . . . . . . . . . . . . . . . . . . . . . . . . . 2306.3.2 Deformation Potential Scattering

(Acoustic Phonon Scattering) . . . . . . . . . . . . . . . . . . . . . . . 2336.3.3 Ionized Impurity Scattering . . . . . . . . . . . . . . . . . . . . . . . . . 2356.3.4 Piezoelectric Potential Scattering . . . . . . . . . . . . . . . . . . . . 2406.3.5 Non-Polar Optical Phonon Scattering . . . . . . . . . . . . . . . . 2426.3.6 Polar Optical Phonon Scattering . . . . . . . . . . . . . . . . . . . . 2436.3.7 Inter-Valley Phonon Scattering . . . . . . . . . . . . . . . . . . . . . . 2486.3.8 Deformation Potential in Degenerate Bands . . . . . . . . . . . 2496.3.9 Theoretical Calculation of Deformation Potentials . . . . . 2516.3.10 Electron–Electron Interaction and Plasmon Scattering . 2566.3.11 Alloy Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

6.4 Scattering Rate and Relaxation Time . . . . . . . . . . . . . . . . . . . . . . 2646.4.1 Acoustic Phonon Scattering . . . . . . . . . . . . . . . . . . . . . . . . 2686.4.2 Non-Polar Optical Phonon Scattering . . . . . . . . . . . . . . . . 2716.4.3 Polar Optical Phonon Scattering . . . . . . . . . . . . . . . . . . . . 2736.4.4 Piezoelectric Potential Scattering . . . . . . . . . . . . . . . . . . . . 2746.4.5 Inter-Valley Phonon Scattering . . . . . . . . . . . . . . . . . . . . . . 2756.4.6 Ionized Impurity Scattering . . . . . . . . . . . . . . . . . . . . . . . . . 2766.4.7 Neutral Impurity Scattering . . . . . . . . . . . . . . . . . . . . . . . . 2776.4.8 Plasmon Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2786.4.9 Alloy Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

6.5 Mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2796.5.1 Acoustic Phonon Scattering . . . . . . . . . . . . . . . . . . . . . . . . 2806.5.2 Non-Polar Optical Phonon Scattering . . . . . . . . . . . . . . . . 2806.5.3 Polar Optical Phonon Scattering . . . . . . . . . . . . . . . . . . . . 2836.5.4 Piezoelectric Potential Scattering . . . . . . . . . . . . . . . . . . . . 2836.5.5 Inter-Valley Phonon Scattering . . . . . . . . . . . . . . . . . . . . . . 2846.5.6 Ionized Impurity Scattering . . . . . . . . . . . . . . . . . . . . . . . . . 2856.5.7 Neutral Impurity Scattering . . . . . . . . . . . . . . . . . . . . . . . . 2866.5.8 Alloy Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

7 Magnetotransport Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2877.1 Phenomenological Theory of the Hall Effect . . . . . . . . . . . . . . . . . 2877.2 Magnetoresistance Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

7.2.1 Theory of Magnetoresistance . . . . . . . . . . . . . . . . . . . . . . . . 2937.2.2 General Solutions for a Weak Magnetic Field . . . . . . . . . 2947.2.3 Case of Scalar Effective Mass . . . . . . . . . . . . . . . . . . . . . . . 2967.2.4 Magnetoresistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

7.3 Shubnikov–de Haas Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3017.3.1 Theory of Shubnikov–de Haas Effect . . . . . . . . . . . . . . . . . 3017.3.2 Longitudinal Magnetoresistance Configuration . . . . . . . . 3057.3.3 Transverse Magnetoresistance Configuration . . . . . . . . . . 308

7.4 Magnetophonon Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

XIV Contents

7.4.1 Experiments and Theory of Magnetophonon Resonance 3127.4.2 Various Types of Magnetophonon Resonance . . . . . . . . . . 3197.4.3 Magnetophonon Resonance

Under High Electric and High Magnetic Fields . . . . . . . . 3247.4.4 Polaron Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

8 Quantum Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3338.1 Historical Background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3338.2 Two-Dimensional Electron Gas Systems . . . . . . . . . . . . . . . . . . . . 334

8.2.1 Two-Dimensional Electron Gasin MOS Inversion Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

8.2.2 Quantum Wells and HEMT. . . . . . . . . . . . . . . . . . . . . . . . . 3438.3 Transport Phenomena in a Two-Dimensional Electron Gas . . . . 351

8.3.1 Fundamental Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3518.3.2 Scattering Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3538.3.3 Mobility of a Two-Dimensional Electron Gas . . . . . . . . . . 378

8.4 Superlattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3868.4.1 Kronig–Penney Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3868.4.2 Effect of Brillouin Zone Folding . . . . . . . . . . . . . . . . . . . . . 3888.4.3 Tight Binding Approximation . . . . . . . . . . . . . . . . . . . . . . . 3918.4.4 sp3s∗ Tight Binding Approximation . . . . . . . . . . . . . . . . . 3938.4.5 Energy Band Calculations for Superlattices . . . . . . . . . . . 3958.4.6 Second Nearest–Neighbor sp3 Tight Binding

Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4008.5 Mesoscopic Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406

8.5.1 Mesoscopic Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4068.5.2 Definition of Mesoscopic Region . . . . . . . . . . . . . . . . . . . . . 4108.5.3 Landauer Formula and Buttiker–Landauer Formula . . . . 4128.5.4 Research in the Mesoscopic Region . . . . . . . . . . . . . . . . . . 4178.5.5 Aharonov–Bohm Effect (AB Effect) . . . . . . . . . . . . . . . . . . 4178.5.6 Ballistic Electron Transport . . . . . . . . . . . . . . . . . . . . . . . . 419

8.6 Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4218.7 Coulomb Blockade and Single Electron Transistor . . . . . . . . . . . 4338.8 Quantum Dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

8.8.1 Addition Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4408.8.2 Exact Diagonalization Method . . . . . . . . . . . . . . . . . . . . . . 4438.8.3 Hamiltonian for Electrons in a Quantum Dot . . . . . . . . . 4458.8.4 Diagonalization of N Electrons Hamiltonian Matrix . . . 4478.8.5 Electronic States in Quantum Dots . . . . . . . . . . . . . . . . . . 4498.8.6 Quantum Dot States in Magnetic Field . . . . . . . . . . . . . . 4518.8.7 Electronic States in Elliptic and Triangular Quantum

Dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452

Contents XV

9 Light Emission and Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4559.1 Einstein Coefficients A and B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4569.2 Spontaneous Emission and Stimulated Emission . . . . . . . . . . . . 4589.3 Band Tail Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4649.4 Luminescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468

9.4.1 Luminescence due to Band-to-Band Transition . . . . . . . . 4699.4.2 Luminescence due to Excitons . . . . . . . . . . . . . . . . . . . . . . 4709.4.3 Luminescence via Impurities . . . . . . . . . . . . . . . . . . . . . . . . 4729.4.4 Luminescence in GaP and GaAsP via N Traps . . . . . . . . 4779.4.5 Luminescence from GaInNAs . . . . . . . . . . . . . . . . . . . . . . . 4799.4.6 Light Emitting Diodes (LEDs) in Visible Region . . . . . . 481

9.5 Heterostructure Optical Waveguide . . . . . . . . . . . . . . . . . . . . . . . . 4829.5.1 Wave Equations for Planar Waveguide . . . . . . . . . . . . . . . 4849.5.2 Transverse Electric Modes . . . . . . . . . . . . . . . . . . . . . . . . . . 4879.5.3 Transverse Magnetic Modes . . . . . . . . . . . . . . . . . . . . . . . . . 4899.5.4 Effective Refractive Index . . . . . . . . . . . . . . . . . . . . . . . . . . 4909.5.5 Confinement Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4929.5.6 Laser Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

9.6 Stimulated Emission in Quantum Well Structures . . . . . . . . . . . 4969.6.1 Confinement in Quantum Well . . . . . . . . . . . . . . . . . . . . . 4999.6.2 Optical Transition in Quantum Well Structures . . . . . . . 5039.6.3 Reduced Density of States and Gain . . . . . . . . . . . . . . . . . 5079.6.4 Strain Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510

9.7 Wurtzite Semiconductor Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515

Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525A Delta Function and Fourier Transform . . . . . . . . . . . . . . . . . . . . . 525

A.1 Dirac Delta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525A.2 Cyclic Boundary Condition and Delta Function . . . . . . . 527A.3 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530

B Uniaxial Stress and Strain Components in Cubic Crystals . . . . 531C Boson Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535D Random Phase Approximation

and Lindhard Dielectric Function . . . . . . . . . . . . . . . . . . . . . . . . . . 539E Density Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541F Spontaneous and Stimulated Emission Rates . . . . . . . . . . . . . . . . 543

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563