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11 Springer Series in Solid-State Sciences Edited by Hans-Joachim Queisser
Springer Series in Solid-State Sciences Editors: M. Cardona P. Fulde H.-J. Queisser
Volume 1 Principles of Magnetic Resonance 2nd Edition 2nd Printing By C. P. Slichter
Volume 2 Introduction to Solid-State Theory By O. Madelung
Volume 3 Dynamical Scattering of X-Rays in Crystals By Z. G. Pinsker
Volume 4 Inelastic Electron Tunneling Spectroscopy Editor: T. Wolfram
Volume 5 Fundamentals of Crystal Growth I. Macroscopic Equilibrium and Transport Concepts By F. Rosenberger
Volume 6 Magnetic Flux Structures in Superconductors By R. P. Huebener
Volume 7 Green's Functions in Quantum Physics By E. N. Economou
Volume 8 Solitons and Condensed Matter Physics Editors: A. R. Bishop and T. Schneider
Volume 9 Photoferroelectrics By V. M. Fridkin
Volume 10 Phonon Dispersion Relations in Insulators By H. Bilz and W. Kress
Volume 11 Electron Transport in Compound Semiconductors By B. R. Nag
Volume 12 The Physics of Elementary Excitations By S. Nakajima
Volume 13 ~e Physics of Selenium and Tellurium Editors: E. Gerlach and P. Grosse
Volume 14 Magnetic Bubble Technology By A. H. Eschenfelder
Volume 15 Modem Crystallography I. Crystal Symmetry, Method of Structural Crystallography By B. K. Vainshtein
Volume 16 Electronic States in Organic Molecular Crystals By E. Silinsh
Volume 17 The Theory of Magnetism I. Ground State and Elementary Excitations ByD. Mattis
B.R.Nag
Electron Transport in Compound Semiconductors
With 148 Figures
Springer-Verlag Berlin Heidelberg New York 1980
Professor Dr. Biswaranjan Nag
Institute of Radiophysics and Electronics University College of Science and Technology 92, Acharya Prafulla Chandra Road, 700 009 Calcutta, India
Series Editors:
Professor Dr. Manuel Cardona Professor Dr. Peter Fulde Professor Dr. Hans-Joachim Queisser
Max-Planck-Institut flir Festk6rperforschung Heisenbergstrasse 1, D-7000 Stuttgart 80, Fed. Rep. of Germany
ISBN-13 :978-3-642-81418-1 e-ISBN-13 :978-3-642-81416-7 DOT: 10.1007/978-3-642-81416-7
Library of Congress in Publication Data. Nag, B. R. Electron transport in compound semiconductors. (Springer series in solid-state sciences; v. 11) Bibliography: p. Includes index. I. Compound semiconductors. 2. Electron transport. I. Title. II. Series. QC611.8.C64N34 537.6'22 79-26983
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law, where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin Heidelberg 1980 Softcover reprint of the hardcover 1st edition 1980
The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
2153/3130-5432 10
To Mriduchandra, Biswadeep, and Mridula
Preface
Discovery of new transport phenomena and invention of electron devices through exploitation of these phenomena have caused a great deal of interest in the properties of compound semiconductors in recent years. Extensive re
search has been devoted to the accumulation of experimental results, par
ticularly about the artificially synthesised compounds. Significant advances have also been made in the improvement of the related theory so that the values of the various transport coefficients may be calculated with suf
ficient accuracy by taking into account all the complexities of energy band
structure and electron scattering mechanisms. Knowledge about these developments may, however, be gathered only from original research contributions, scattered in scientific journals and conference proceedings. Review articles have been published from time to time, but they deal with one particular
material or a particular phenomenon and are written at an advanced level. Available text books on semiconductor physics, do not cover the subject
in any detail since many of them were written decades ago. There is, there
fore, a definite need for a book, giving a comprehensive account of electron transport in compound semiconductors and covering the introductory material as well as the current work. The present book is an attempt to fill this gap
in the literature. The first chapter briefly reviews the history of the developement of compound
semiconductors and their applications. It is also an introduction to the
contents of the book. The fundamentals of crystal structure, energy band
structure and electron statistics are discussed in Chaps.2-5. Care has been taken to include all the recent developments when presenting these in
troductory subjects. The k'E perturbation theory and the statistics related to nonparabolic band structure and quantizing magnetic fields have been discussed at great length.
The scattering theory and the Boltzmann equation are presented, respectively, in Chaps.6,7. Special attention has been given to those methods
which were developed in the past few years for calculating the transport
VIII
coefficients including the effects of the nonparabolicity of the band structure and the polar character of the materials. Low-field DC transport coefficients, i.e., drift mobility, Hall mobility, magnetoresistance, diffusivity, thermoelectric power and electronic thermal conductivity are covered in Chap.8. A major part of the discussion in this chapter is based on the author's own research and is not available in the literature. Chapter 9 deals with the AC transport coefficients. Low-frequency AC conductivity, AC magnetoconductivity, and the phenomena of cyclotron resonance, Faraday rotation and free-carrier absorption are all discussed in detail.
Electron transport in quantizing magnetising fields has been a subject of special interest in relation to compound semiconductors. Phenomena associated with this aspect of transport are presented in Chap.10. Hot-electron transport is discussed in Chap.11. The study of this aspect of transport has been particularly fruitful for the realisation of new devices and it is a subject of continued interest. The discussion in this chapter has been aimed at acquainting the reader with early research and at the same time making him aware of current trends.
Chapter 12 gives a review of the properties of some important compound semiconductors. Chapter 13 is the concluding chapter and it deals with problems of current interest.
The book has been written at an introductory level giving all the mathematical details and computer programs to suit the needs of students and newcomers in the field. Enough current information on the properties and physical constants of all kinds of semiconducting compounds and up-to-date surveys of research results are also included to make the book useful for specialists.- It is hoped that the book will be of use to all physicists and engineers who have a broad interest in the properties and applications of compound semiconductors and that it will stimulate further research in the area.
Calcutta, February 1980 Biswaranjan Nag
Acknowledgments
It is almost impossible to name and acknowledge individually the contributions of all the colleagues and students whose discussions and exchange
of ideas helped the author in writing the book. He would, however, like to record his indebtedness to P.N. Robson, P.N. Butcher, R.A. Stradling, J.C. Woolley, J. Frey, P.J. Price, A.K. Das, P.K. Basu, A.N. Chakravarti,
D. Mukherji and P. Fentem with whom he had discussed parts of the book at different times. He is particularly indebted to D. Chattopadhyay for carefully reading the manuscript, suggesting improvements, and sharing with the author the labour of preparing the final version. He is also indebted to A. Choudhury, N. Purkait, S. Dhar, S. Sen, N.R. Saha, D. Roy Choudhury, P.C. Rakshit, G. Ghosh, D. Das Gupta and a few other colleagues for their help in the preparation of the book. Thanks are also due to A.K. Chakrabarty, A. Saha and A.M. Banerjee for carefully typing the manuscript and for drawing the figures.
Some figures presented in the book are reproduced from earlier publications and the original sources are mentioned in each case. The author takes this opportunity to acknowledge the generosity of the authors and publishers for permitting him to use this material.
The book was partly written when the author was a Jawaharlal Nehru Fellow and he takes this opportunity to record his appreciation of the support provided by the Jawaharlal Nehru Memorial Fund for this work.
Biswaranjan Nag
Contents
1. Introduction .................................................... 1
1.1 Hi stori ca 1 Note ............................................. 1
1.2 App 1 i ca ti ons .....•.......................................... 2
1.3 Transport Coefficients of Interest .......................... 5
1.4 Scope of the Book .................................•........ 7
2. Crystal Structure ............................................... 9
2.1 Zinc-Blende Structure....................................... 9
2.2 Wurtzi te Structure .......................................... 14
2.3 Rock-Salt Structure......................................... 17
2.4 Chalcopyrite Structure 18
3. Energy Band Structure .....................................•..... 21
3.1 Electron Wave Vector and Brillouin Zone..................... 21
3.2 Brillouin Zone for the Sphalerite and Rock-Salt Crystal
Structure ................................................... 23
3.3 Brillouin Zone for the Wurtzite Structure ................... 24
3.4 Brillouin Zone for the Chalcopyrite Structure ............... 25
3.5 E-~ Diagrams................................................ 25
3.5.1 Energy Bands for the Sphalerite Structure ............. 26
3.5.2 Energy Bands for the Wurtzite Structure ............... 30
3.5.3 Energy Bands for the Rock-Salt Structure .............. 32
3.5.4 Band Structure of Mixed Compounds..................... 35
3.6 Conclusion.................................................. 38
4. Theory of Energy Band Structure ................................. 40
4.1 Models of Band Structure .................................... 40
4.2 Free-Electron Approximation Model 41
4.3 Tight-Binding Approximation Model........................... 45
XII
4.4 Energy Bands in Semiconductor Superlattices ................ 50
4.5 The ~'E Perturbation Method for Derivating E-~ Relation ..... 53 4.5.1 Single-Band Perturbation Theory....................... 55
4.5.2 Two-Band Approximation ................................ 57
4.5.3 Effect of Spin-Orbit Interaction ...................... 61
4.5.4 Nonparabolic Relation for Extrema at Points Other than
the r Point........................................... 63
4.6 External Effects on Energy Bands............................ 64
4.6.1 Effects of Doping..................................... 64
4.6.2 Effects of Large Magnetic Fields 69
5. EZeatron Statistias ............................................. 74
5.1 Fermi Energy for Parabolic Bands............................ 75
5.2 Fermi Energy for Nonparabolic Bands......................... 78
5.3 Fermi Energy in the Presence of a Quantising Magnetic Field 80
5.3.1 Density of States..................................... 80
5.3.2 Fermi Level............. .............................. 84
5.4 Fermi Energy and Impurity Density........................... 88
5.4.1 General Considerations ................................ 88
5.4.2 General Formula....................................... 89
5.4.3 Discussion of Parabolic Band ....................•..... 90
5.4.4 Effect of Magnetic Field .............................. 92
5.5 Conclusions................................................. 92
6. Saattering Theory ............................................... 93
6.1 Collision Processes......................................... 94
6.2 Transition Probability...................................... 100
6.3 Matrix Elements............................................. 102
6.4 Free-C~rrier Screening ...................................... 104
6.5 Overlap Integrals........................................... 107
6.6 Scattering Probability S(~) ................ ................. 112
6.6.1 S(~) for Ionised Impurity Scattering.................. 113
6.6.2 S(~) for Piezoelectric Scattering ..................... 116
6.6.3 S(~) for Deformation-Potential Acoustic Phonon
Scattering ............................................ 117
6.6.4 S(~) for Polar Optic Phonon Scattering ................ 118
6.6.5 S(~) for Intervalley and Nonpolar Optic Phonon
Scattering ............................................ 119
XIII
6.7 Scattering Probabilities for Anisotropic Bands .............. 120
6.7.1 Herring-Vogt Transformation ........................... 121
6.7.2 Scattering Integrals.................................. 122
6.8 S(~) for Neutral Impurity, Alloy, and Crystal-Defect
Scattering .................................................. 124
6.8.1 Neutral-Impurity Scattering ........................... 125
6.8.2 Alloy Scattering...................................... 126
6.8.3 Defect Scattering..................................... 126
6.9 Conclusions................................................. 127
7. The BoZtzmann Transport Equation and Its SoZution ...•.....•..... 129
7.1 The Liouville Equation and the Boltzmann Equation ........... 129
7.2 The Boltzmann Transport Equation ............................ 131
7.3 The Collision Integral...................................... 134
7.4 Linearised Boltzmann Equation ............................... 135
7.5 Simplified Form of the Collision Terms ...................... 139
7.5.1 Collision Terms for Elastic Scattering ................ 140
7.5.2 Collision Terms for Inelastic Scattering .............. 141
7.6 Solution of the Boltzmann Equation.......................... 143
7.6.1 Relaxation-Time Approximation ......................... 143
7.6.2 Variational Method.................................... 145
7 .6.3 ~latrix Method......................................... 150
7.6.4 Iterati on Method ...................................... 153
7.6.5 Monte Carlo Method.................................... 154
7.7 Method of Solution for Anisotropic Coupling Constants and
Anisotropic Electron Effective Mass ......................... 160
7.7.1 Solution for Elastic Collisions ....................... 164
7.7.2 Solution for Randomising Collisions ................... 168
7.7.3 Snlution for Nonrandomising Inelastic Collisions ...... 168
7.8 Conclusions................................................. 170
8. Low-FieZd DC Transport Coeffiaients ..••.•.......•.......•.....•. 171
8.1 Evaluation of Drift Mobility................................ 172
8.1.1 Formulae for Relaxation-Time Approximation ............ 173
8.1.2 Evaluation by the Variational Method.................. 177
8.1.3 Evaluation by Matrix and Iteration Methods ............ 179
8.1.4 Evaluation by the Monte Carlo Method .................. 179
XIV
8.2 Drift Mobility for Anisotropic Bands........................ 180
8.2.1 Ellipsoidal Band...................................... 180
8.2.2 Warped Band........................................... 183
8.3 Galvanomagnetic Transport Coefficients...................... 183
8.3.1 Formulae for Hall Coefficient, Hall Mobility, and
Magnetoresi stance ..................................... 185
8.3.2 Reduced Boltzmann Equation for the Galvanomagnetic
Coefficients .......................................... 186
8.3.3 Solution Using the Relaxation-Time
Approximation Method .................................. 188
8.3.4 A Simple Formula for the Low-Field Hall Mobil ity ...... 191
8.3.5 Numerical Methods for the Galvanomagnetic Coefficients
for Arbitrary Magnetic Fields......................... 192
8.3.6 Evaluation of the Galvanomagnetic Transport
Coefficients for Anisotropic Effective Mass ........... 195
8.4 Transport Coefficients for Nonuniform conditions ............ 209
8.4.1 Diffusion............................................. 210
8.4.2 Thermal Transport Coefficients........................ 215
8.4.3 Formula for Thermoelectric Power...................... 218
8.4.4 Electronic Thermal Conductivity....................... 225
8.5 Conclusions................................................. 229
9. Low-FieZd AC Transport Coefficients ..•.•...•.•.•...•.•.......... 230
9.1 Classical Theory of AC Transport Coefficients ............... 231
9.1.1 Solution for the Relaxation-Time Approximation
9.1.2 Solution for Polar Optic Phonon and Mixed
233
Scattering ............................................ 238
9.1.3 Solution for Nonparabolic and Anisotropic Bands ....... 239
9.2 AC Galvanomagnetic Coefficients ............................. 242
9.3 Cyclotron Resonance and Faraday Rotation .................... 246
9.3.1 Propagation of Electromagnetic Waves in a Semiconductor
in the Presence of a Magnetic Field ................... 246
9.3.2 Cyclotron Resonance Effect ............................ 248
9.3.3 Faraday Rotation...................................... 256
9.4 Free-Carrier Absorption (FCA) ............................... 260
9.4.1 Classical Theory of FCA ............................... 261
9.4.2 Quantum-Mechani ca 1 Theory of FCA .. ".................... 263
9.5 Concluding Remarks.......................................... 283
xv
10. Electron Transport in a Strong Magnetic Field .................. 284
10.1 Scattering Probabilities .... .......... .... ..... .... ....... 285
10.2 Mobility in Strong Magnetic Fields........................ 290
10.3 El ectron ~10bil ity in the Extreme Quantum Limit (EQL) 295
10.3.1 Electron Mobility for Polar Optic Phonon Scattering in the EQL ....... ....... ......... ....... 297
10.4 Oscillatory Effects in the Magnetoresistance ....... ....... 298
10.4.1 Shubnikov-de Haas Effect. ...... ... ... ....... ....... 299
10.4.2 Magnetophonon Oscillations... ...... .... ....... ..... 303
10.5 Experimental Results on Magnetophonon Resonance .... ....... 307
10.6 Conclusions............................................... 310
11. Hot-Electron Transport 311
11.1 Phenomenon of Hot El ectrons ............................... 311
11.2 Experimental Characteristics ......... .......... ..... ...... 314
11.3 Negative Differential Mobility and Electron Transfer Effect 323
11.4 Analytic Theories......................................... 327 11.4.1 Differential Equation Method. ..... ................. 328
11.4.2 Maxwellian Distribution Function Method ............ 332
11.4.3 Displaced Maxwellian Distribution Function Method.. 334
11 .5 Numeri ca 1 Methods ......................................... 337
11.5.1 Iteration Method................................... 338
11.5.2 Monte Carlo Method ................. ....... ......... 344
11.6 Hot-Electron AC Conductivity.............................. 350
11.6.1 Phenomenological Theory for Single-Valley ~laterials 351
11.6.2 Phenomenological Theory for Two-Valley Materials ... 355
11.6.3 Large-Signal AC Conductivity....................... 358
11.7 Hot-Electron Diffusion .... .... .......... ......... ......... 359
11.7.1 Einstein Relation for Hot-Electron Diffusivity ..... 359
11.7.2 Electron Diffusivity in Gallium Arsenide 363
11.7.3 Monte Carlo Calculation of the Diffusion Constant.. 365
11.8 Conclusion................................................ 368
12. Review of Experimental Results 370
12.1 Transport Coefficients of III-V Compounds. ............ .... 373
12.1.1 Indium Antimonide.................................. 373
12.1.2 Gallium Arsenide .... .............. ................. 378
XVI
12.1.3 Indium Phosphide................................... 384
12.1.4 Indium Arsenide.................................... 386
12.1.5 Indirect-Band-Gap III-V Compounds .................. 387
12.2 II-VI Compounds........................................... 388
12.2.1 Cubic Compounds of Zinc and Cadmium ................ 389
12.2.2 Wurtzite Compounds of Zinc and Cadmium ............. 390
12.2.3 Mercury Compounds.................................. 391
12.3 IV-VI Compounds 392
12.4 Mixed Compounds........................................... 394
12.5 Chalcopyrites ............................................. 397
12.6 Conclusion................................................ 397
13. Conclusions .................................................... 399
13.1 Problems of Current Interest .............................. 401
13.1.1 Heavily Doped Materials............................ 401
13.1. 2 All oy Semi conductors ............................... 402
13.1.3 Transport Under Magnetically Quantised Conditions.. 403
13.1. 4 I nvers i on Layers ................................... 404
13.1.5 Superlattices and Heterostructures ................. 406
13.2 Scope of Further Studies ............ ...... ...... .... ...... 407
Appendix A: Table of Fermi Integrals ......... ...... ................ 409
Appendix B: Computer Program for the Evaluation of Transport Coefficients by the Iteration t1ethod ................... 410
Appendix C: Values of ai and bi for Gaussian Quadrature Integration. 417
Appendix D: Computer Program for the Monte Carlo Calculation of
Hot-Electron Conductivity and Diffusivity ......... ..... 418
List of Symbols 423
References ...........•......................................•...... 431
SUbject Index. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 451