Basic Semiconductor Physics

Basic Semiconductor Physics

Springer-Verlag Berlin Heidelberg GmbH Physics and Astronomy ONLINE LlBRARY http://www.springer.de/phys/

Chihiro Hamaguchi Basic Semiconductor Physics With 168 Figures and 25 Tables, Springer

Professor Chihiro Hamaguchi Osaka University Graduated School ofengineering Department of Electronic Engineering Suita City, Osaka 565-0871 Japan E-Mail: hamaguti@ele.eng.osaka-u.ac.jp ISBN 978-3-642-07492-9 ISBN 978-3-662-04656-2 (ebook) DOI 10.1007/978-3-662-04656-2 Library of Congress Cataloging-in-Publication Data Hamaguchi, Chihiro- Basic semiconductor physics / Chihiro Hamaguchi. p. cm. Includes bibliographieal references and index. 1. Semiconductors. I. Title. QC611.H26 2001 537.6' 22-dc21 2001031177 Die Deutsche Bibliothek - CIP-Einheitsaufnahme Hamaguchi, Chihiro: Basic semiconductor physies: with 25 tables / Chihiro Hamaguchi. - Berlin ; Heidelberg; New York; Barcelona; Hong Kong ; London ; Milan; Paris; Singapore ; Tokyo : Springer, 2001 (Physics and astronomy online library) This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions ofthe German Copyright Law ofseptember 9, '965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. http://www.springer.de Springer-Verlag Berlin Heidelberg 2001 Originally published by Springer-Verlag Berlin Heidelberg New York in 2001. Softcover reprint ofthe hardcover 1 st edition 2001 The use of general descriptive names, 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. Typesetting: Camera ready by the author using aspringer Tpcmacro package Cover concep!: estudio Calamar Steinen Cover production: design & production GmbH, Heidelberg SPIN: 10785783 57/3'4,/ad - 54321 0 - Printed on acid-free paper

Preface More than 50 years have passed since the invention of the transistor in December 1947. The study of semiconductors was initiated in the 1930s but we had to wait for 30 years (till the 1960s) to understand the physics of semiconductors. When the transistor was invented, it was still unclear whether germanium had a direct gap or indirect gap. The author started to study semiconductor physics in 1960 and the physics was very difficult for a beginner to understand. The best textbook of semiconductors at that time was "Electmns and Holes in Semiconductors" by W. Shockley, but it required a detailed knowledge of solid state physics to understand the detail of the book. In that period, junction transistors and Si bipolar transistors were being produced on a commercial basis, and industrialization of semiconductor technology was progressing very rapidly. Later, semiconductor devices were integrated and applied to computers successfully, resulting in a remarkable demand for semiconductor memories in addition to processors in the late 1970s to 1980s. Now we know that semiconductors play the most important role in information technology as the key devices and we cannot talk about the age of information technology without semiconductor devices. On the other hand, the physical properties of semiconductors such as the electrical and optical properties wcre investigated in detail in the 1950s, leading to the understanding of the energy band structures. Cyclotron resonanee experiments and their detailed analysis first reported in 1955, were the most important contribution to the understanding of the energy band structures of semiconductors. From this work it was revealed that the valence bands consist of degenerate heavy-hole and light-hole bands. Another important contribution comes from energy band calculations. Energy band ealculations based on the empirie al pseudopotential method and the k. p perturbation method reported in 1966 enabled us to understand the fundamental properties of semiconductors. In this period high-field transport and current instabilities due to the Gunn effect and the acoustoelectric effeet attracted great interest. In addition, modulation spectroscopy and light scattering were developed and provided detailed information of the optical properties of semiconductors. These contributions enabled us to understand the physical properties of bulk semiconductors almost eompletely.

VI Preface At the same time, late in the 1960s and early 1970s, Leo Esaki and his co-workers developed a new crystal growth method, molecular beam epitaxy, and initiated studies of semiconductor heterostructures such as quantum wells and su:perlattices. This led to a newage of semiconductor research which demonstrated phenomena predicted from quantum mechanics. This approach is completely different from past research in that new crystals and new structures are being created in the laboratory. This field is therefore called "band gap engineering". It should be noted here that such research did not set up to fabricate devices for real applications but to investigate new physics. The proposal of modulation doping in the late 1970s and the invention of the high electron mobility transistor (HEMT) in 1980 triggered a wide variety of research work related to this field. Later HEMTs have been widely used in such applications as the receivers for satellite broadcasting. Although the commercial market for LSI memories based on Si technologies is huge, metal semiconductor field-effect transistors (MESFETs) based on GaAs have become key devices for mobile phones (cellular phones) in the 1990s and it is believed that their industrialization will playa very important role in the 21st century. Klaus von Klitzing et al. discovered the quantum Hall effect (later called the integer quantum Hall effect) in the two-dimensional electron gas system of a Si MOSFET in 1980, and this discovery changed semiconductor research dramatically. The discovery of the fractional quantum Hall effect followed the integer quantum Hall effect and many papers on these subjects have been reported at important international conferences. At the same time attempts to fabricate microstructures such as quantum wires and metal rings were carried out by using semiconductor microfabrication technologies and led to the discovery of new phenomena. These are the Aharonov-Bohm effect, ballistic transport, electron interference, quasi-one-dimensional transport, quantum dots, and so on. The sampies used for these studies have a size between the microscopic and macroscopic regions, which is thus called the "mesoscopic region". The research in cmesoscopic structures is still progressing. The above overview is baed on the private view of the author and very incomplete. Those who are interested in semiconductor physics and in device applications of new phenomena require a deep understanding of semiconductor physics. The situation is quite different for the author who had to grope his own way in semiconductor physics in the 1960s, while the former are requested to begin their own work after understanding the established semiconductor physics. There have been published various textbooks in the field of semiconductors, but only few cover the field from the fundamentals to new phenomena. The author has published several textbooks in Japanese, but they do not cover such a wide range of semiconductor physics. In order to supplement the textbooks he has used printed texts for graduate students in the last 10 years, revising and including new parts.

Preface VII This textbook is not intended to give an introduction to semiconductors. Such introductions to semiconductors are given in courses on solid-state physics and semiconductor devices at many universities in the world. It is clear from the contents of this textbook that electron statistics in semiconductors, pn junctions, pnp or npn bipolar transistors, MOSFETs and so on are not dealt with. This textbook is written for graduate students or researchers who have finished the introductory courses. Readers can understand such device-oriented subjects easily after reading this textbook. A large part of this book has been used in lectures several times for the solid-state physics and semiconductor physics courses for graduate students at the Electronic Engineering Department of Osaka University and then revised. In order to understand semiconductor physics it is essential to learn energy band structures. For this reason various methods for energy band calculations and cyclotron resonance are described in detail. As far as this book is concerned, many of the subjects have been carried out as research projects in our laboratory. Therefore, many figures used in the textbook are those reported by us in scientific journals and from new data obtained recently by carrying out experiments so that digital processing is possible. It should be noted that the author does not intend to disregard the priorities of the outstanding papers written by many scientists. Important data and their analysis are referred to in detail in the text, and readers who are interested in the original papers are advised to read the references. This book was planned from the beginning to be prepared by ~1EX and the figures are prepared in EPS files. Figures may be prepared by using a scanner but the quality is not satisfactory compared to the figures drawn by software such as PowerPoint. This is the main reason why we used our own data much more than those from other groups. Numerical calculations such as energy band structures were carried out in BA SIC and FORTRAN. Theoretical curves were calculated using Mathematica and equations of simple mathematical functions were drawn by using SMA4 Windows. The final forms of the figures were then prepared using PowerPoint and transformed into EPS files. However, some complicated figures used in Chap.8 were scanned and then edited using PowerPoint. The author would like to remind readers that this book is not written for those interested in the theoretical study of semiconductor physics. He believes that it is a good guide for experimental physicists. Most of the subjects are understood within the framework of the one-electron approximation and the book requires an understanding of the Schrödinger equation and perturbation theory. All the equations are written using SI units throughout so that readers can easily estimate the values. In order to understand solid-state physics it is essential to use basic theory such as the Dirac delta function, Dirac identity, Fourier transform and so on. These are explained in the appendices. In addition, abrief introduction to group theoretical analysis of strain tensors, random phase approximations, boson operators and the density matrix is

VIII Preface given in the appendices. With this background the reader is expected to understand all the equations derived in the text book. The author is indebted to many graduate students for discussions and the use of their theses. There is not enough space to list all the names of the students. He is also very thankful to Prof. Dr. Nobuya Mori for his critical reading of the manuscript and valuable comments. He thanks Dr. Masato Morifuji for his careful reading of the text. Dr. Hideki Momose helped the author to prepare the Jb.1E;X format and Prof. Dr. Nobuya Mori revised it. Also, thanks are due to Mr. Hitoshi Kubo, who took the Raman scattering data in digital format. He is also very thankful to Prof. Dr. Laurence Eaves and Prof. Dr. Klaus von Klitzing for their encouragement from the early stage of 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 he would like to thank Prof. Dr. Erich Gornik for providing this opportunity and for 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 horne and thc author wants to thank his wife Wakiko for her patience. Osaka, Japan, March 2001 Chihiro Hamaguchi

Contents 1. Energy Band Structures of Semiconductors............... 1 1.1 Free-Electron Model.................................... 1 1. 2 Bloch Theorem......................................... 3 1.3 Nearly Free Electron Approximation...................... 4 1.4 Reduced Zone Scheme... 8 1.5 Free-Electron Bands (Empty-Lattice Bands)............... 9 1.6 Pseudopotential Method................................. 12 1. 7 k p Perturbation.................................... 17 2. Cyclotron Resonance and Energy Band Structures... 25 2.1 Cyclotron Resonance... 25 2.2 Analysis of Valence Bands............................... 33 2.3 Spin-Orbit Interaction.................................. 37 2.4 Non-parabolicity of the Conduction Band.................. 46 2.5 Electron Motion in a Magnetic Field and Landau Levels..... 48 2.5.1 Landau Levels... 48 2.5.2 Landau Levels of a Non-parabolic Band............. 55 2.5.3 Landau Levels of the Valence Bands................ 59 3. Wannier Function and Effective Mass Approximation..... 65 3.1 Wannier Function...................................... 65 3.2 Effective-mass Approximation............................ 67 3.3 Shallow Impurity Levels................................. 71 3.4 Impurity Levels in Ge and Si............................. 74 3.4.1 Valley-Orbit Interaction... 77 3.4.2 Central Cell Correction... 79 4. Optical Properties 1...................................... 81 4.1 Reflection and Absorption............................... 81 4.2 Direct Transition and Absorption Coefficient............... 85 4.3 Joint Density of States.................................. 87 4.4 Indirect Transition...................................... 92 4.5 Exciton... 97 4.5.1 Direct Exciton................................... 97

X Contents 4.5.2 Indirect Exciton... 106 4.6 Dielectric Function... 108 4.6.1 Eo, Eo + Llo Edge... 111 4.6.2 EI and EI + Ll I Edge... 113 4.6.3 E 2 Edge... 113 4.6.4 Exciton... 114 4.7 Piezobirefringence... 116 4.7.1 Phenomenological Theory of Piezobirefringence... 116 4.7.2 Deformation Potential Theory... 117 4.7.3 Stress-Induced Change in Energy Band Structure... 120 5. Optical Properties 2... 125 5.1 Modulation Spectroscopy... 125 5.1.1 Electro-optic Effect... 125 5.1.2 Franz-Keldysh Effect... 126 5.1.3 Modulation Spectroscopy... 130 5.1.4 Theory of Electroreflectance and Third-Derivative Form of Aspnes... 134 5.2 Raman Scattering... 139 5.2.1 Selection Rule of Raman Scattering... 144 5.2.2 Quantum Mechanical Theory of Raman Scattering... 149 5.2.3 Resonant Raman Scattering... 154 5.3 Brillouin Scattering... 158 5.3.1 Scattering Angle... 159 5.3.2 Brillouin Scattering Experiments... 163 5.3.3 Resonant Brillouin Scattering... 167 5.4 Polaritons... 170 5.4.1 Phonon Polaritons... 170 5.4.2 Exciton Polaritons... 174 5.5 Free-Carrier Absorption and Plasmon... 177 6. Electron-Phonon Interaction and Electron Transport.... 183 6.1 Lattice Vibrations... 183 6.1.1 Acoustic Mode and Optieal Mode... 183 6.1.2 Harmonie Approximation... 187 6.2 Boltzmann Transport Equation... 196 6.2.1 Collision Term and Relaxation Time... 198 6.2.2 Mobility and Electrical Conductivity... 201 6.3 Scattering Probability and Transition Matrix Element... 205 6.3.1 Transition Matrix Element... 205 6.3.2 Deformation Potential Scattering (Acoustic Phonon Scattering)...................... 208 6.3.3 Ionized Impurity Scattering... 210 6.3.4 Piezoelectric Potential Scattering... 214 6.3.5 Non-polar Optical Phonon Scattering... 216

Contents XI 6.3.6 Polar Optical Phonon Scattering... 217 6.3.7 Inter-Valley Phonon Scattering... 222 6.3.8 Deformation Potential in Degenerate Bands... 223 6.3.9 Theoretical Calculation of Deformation Potentials... 225 6.3.10 Electron-Electron Interaction and Plasmon Scattering 230 6.3.11 Alloy Scattering... 237 6.4 Scattering Rate and Relaxation Time... 238 6.4.1 Acoustic Phonon Scattering... 242 6.4.2 Non-polar Optical Phonon Scattering... 245 6.4.3 Polar Optical Phonon Scattering... 246 6.4.4 Piezoelectric Potential Scattering... 248 6.4.5 Inter-Valley Phonon Scattering... 248 6.4.6 Ionized Impurity Scattering... 250 6.4.7 Neutral Impurity Scattering... 251 6.4.8 Plasmon Scattering... 252 6.4.9 Alloy Scattering... 252 6.5 Mobility... 252 6.5.1 Acoustic Phonon Scattering... 253 6.5.2 Non-Polar Optical Phonon Scattering... 254 6.5.3 Polar Optical Phonon Scattering... 256 6.5.4 Piezoelectric Potential Scattering... 256 6.5.5 Inter-Valley Phonon Scattering... 257 6.5.6 Ionized Impurity Scattering... 258 6.5.7 Neutral Impurity Scattering... 259 6.5.8 Alloy Scattering... 259 7. Magnetotransport Phenomena.... 261 7.1 Phenomenological Theory of the Hall Effect... 261 7.2 Magnetoresistance Effects... 267 7.2.1 Theory of Magnetoresistance... 267 7.2.2 General Solutions for a Weak Magnetic Field... 268 7.2.3 Case of Scalar Effective Mass... 269 7.2.4 Magnetoresistance... 271 7.3 Shubnikov-de Haas Effect... 275 7.3.1 Theory of Shubnikov-de Haas Effect... 275 7.3.2 Longitudinal Magnetoresistance Configuration... 279 7.3.3 Transverse Magnetoresistance Configuration... 281 7.4 Magnetophonon Resonance... 285 7.4.1 Experiments and Theory of Magnetophonon Resonance 285 7.4.2 Various Types of Magnetophonon Resonance... 292 7.4.3 Magnetophonon Resonance under High Electric and High Magnetic Fields... 297 7.4.4 Polaron Effect... 302

XII Contents 8. Quantum Structures... 307 8.1 Historical Background... 307 8.2 Two-Dimensional Electron Gas Systems... 308 8.2.1 Two-Dimensional Electron Gas in MOS Inversion Layer... 308 8.2.2 Quantum Wells and HEMT... 317 8.3 Transport Phenomena in a Two-Dimensional Electron Gas... 324 8.3.1 Fundamental Equations... 324 8.3.2 Scattering Rate... 327 8.3.3 Mobility of a Two-Dimensional Electron Gas... 342 8.4 Superlattices... 347 8.4.1 Kronig-Penney Model... 347 8.4.2 Effect of Brillouin Zone Folding... 349 8.4.3 Tight Binding Approximation... 351 8.4.4 sp3 s* Tight Binding Approximation... 354 8.4.5 Energy Band Calculations for Superlattices... 355 8.4.6 Second Nearest-Neighbor sp3 Tight Binding Approximation... 361 8.5 Mesoscopic Phenomena... 368 8.5.1 Mesoscopic Region... 368 8.5.2 Definition of Mesoscopic Region... 370 8.5.3 Landauer Formula and Büttiker-Landauer Formula... 372 8.6 Research in the Mesoseopie Region... 378 8.7 Aharonov-Bohm Effeet (AB Effeet)... 378 8.8 Ballistie Eleetron Transport... 379 8.9 Quantum Hall Effeet... 381 8.10 Coulomb Bloekadeand Single Electron Transistor... 393 Appendices... 401 A Delta Function and Fourier Transform... 401 A.1 Dirae Delta Function... 401 A.2 Cyclie Boundary Condition and Delta Function... 403 A.3 Fourier Transform... 405 B Uniaxial Stress and Strain Components in Cubie Crystals... 407 C Boson Operators... 410 DRandom Phase Approximation and Rindhard Dieleetrie Function... 414 E Density Matrix... 416 References.... 419 Index... 429