OPTICAL PROPERTIES OF SOLIDS OPTICAL PROPERTIES OF SOLIDS Frederick. Wooten Department of Applied Science University of California Davis, California 1972 ACADEMIC PRESS New York and London soJjfc ia Copyright © 1972, by Academic Press, Inc. all rights reserved no part of this book may be reproduced in any form, by photostat, microfilm, retrieval system, or any other means, without written permission from the publishers. ACADEMIC PRESS, INC. Ill Fifth Avenue, New York, New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London NW1 Library of Congress Catalog Card Number: 72-187257 PRINTED IN THE UNITED STATES OF AMERICA To My Father CONTENTS Preface A cknowledgments Chapter 1 Introduction l.i Band Theory of Solids 1.2 Optical Reflectivity 1.3 Photoemission 1.4 Characteristic Energy Loss Spectra 14 Chapter 2 Maxwell's Equations and the Dielectric Function 2.1 Maxwell's Microscopic Equations 16 2.2 Maxwell's Macroscopic Equations 16 2.3 Formal Solutions of Maxwell's Equations 18 2.4 Analysis of Charge and Current Densities 22 2.5 Properties of the Medium 25 2.6 Interaction of Light with the Medium 26 2.7 External Sources and Induced Responses 29 2.8 Fourier Analysis of Maxwell's Equations 31 2.9 The Dielectric Tensor 34 Problems 39 Further Reading 4° Chapter 3 Absorption and Dispersion 3.1 The Lorentz Oscillator 42 3.2 The Drude Model for Metals 52 3.3 A Qualitative Look at Real Metals 55 3.4 Photoemission from Copper 66 3.5 Quantum Theory of Absorption and Dispersion 67 vii 1 Vlll Contents 3.6 Oscillator Strengths and Sum Rules 72 3.7 Applications of Sum Rules 75 3.8 The Absorption Coefficient, Optical Conductivity, and Dielectric Function 80 Problems g2 Further Reading 84 Chapter 4 Free-Electron Metals 4.1 Classical Theory of Free-Electron Metals 86 4.2 The Classical Skin Effect 90 4.3 The Anomalous Skin Effect 94 4.4 Optical Properties and the Fermi Surface 99 Problems j Qg References and Further Reading 107 Chapter 5 Interband Transitions 5.1 Periodic Perturbation 108 5.2 Direct Interband Transitions 110 5.3 Joint Density of States and Critical Points 116 5.4 Direct Transitions in Germanium 122 5.5 Direct Transitions in Silver: Effects of Temperature and Alloying 5.6 Indirect Transitions j 34 5.7 The Absorption Edge in Ge, AgBr, and AgBr(Cl) 144 5.8 Excitons 149 5.9 Direct and Indirect Transitions in Photoemission 153 5.10 Nondirect Transitions: Photoemission from Cs 3 Bi 156 5. 1 Transport and Escape Effects Cone on Photoemission 1 59 5.12 Photoemission and Electron Transport in Al and GaAs 164 Problems 1 (39 Further Reading 1 70 References 171 Chapter 6 Dispersion Relations and Sum Rules 6. 1 Linear Response Functions and Kramers-Kronig Relations 173 6.2 Reflectivity and Phase Shift Dispersion Relations 181 6.3 Sum Rules 182 Problems 184 Further Reading 1 85 Contents IX Chapter 7 Self-Consistent Field Approximation 7.1 Self-Consistent Field Approximation 186 7.2 Special Cases and Applications 194 Problems 201 References and Further Reading 202 Chapter 8 Current-Current Correlations and the Fluctuation- Dissipation Theorem 8.1 Transition Rate and Current-Current Correlations 204 8.2 Current Fluctuations 206 8.3 The Fluctuation-Dissipation Theorem and the Conductivity 207 Problem 209 References and Further Reading 209 Chapter 9 Plasmons and Characteristic Energy Losses 9.1 Single-Electron Excitations in Metals 211 9.2 Plasmons in Simple Metals 214 9.3 The Plasmon Cutoff Wave Vector 215 9.4 Characteristic Energy Loss Spectra 218 9.5 Surface Plasmons 220 Problems 223 References and Further Reading 224 Appendix A Decomposition of a Vector Field into Longitudinal 225 and Transverse Parts Appendix B The Local Field B.l Insulators 227 B.2 Nonlocalized Electrons 230 Further Reading 231 Appendix C Reflection at Normal Incidence 232 Appendix D The /-Sum Rule for a Crystal 234 Appendix E Interaction of Radiation with Matter 236 Contents Appendix F M Critical Points 240 t Appendix G Reflectance and Phase-Shift Dispersion Relations G. 1 The Phase-Shift Dispersion Relation 244 G.2 Numerical Integration of the Phase-Shift Equation 248 Further Reading 250 Appendix H k • p Perturbation Theory 251 Index 255 PREFACE The present book attempts to fill a need for a fundamental textbook which explains the optical properties of solids. It is based on two short courses I gave in the Department of Applied Science and a series of fifteen lectures at Chalmers Tekniska Hogskola, Goteborg, Sweden presented at the invitation of Professors Stig Hagstrom, Gosta Brogren, and H. P. Myers. This book is meant to explain a number of important concepts rather than present a complete survey of experimental data. Its emphasis is almost entirely on intrinsic optical properties and photoelectric emission. Little is said concerning imperfections, color centers, etc. However, the principles are general, so the book serves as a stepping stone to the more advanced review articles and papers on a wide variety of topics. The book assumes a background in quantum mechanics, solid state physics, and electromagnetic theory at about the level of a senior- year undergraduate or first-year graduate student in physics. Problems and exercises have been included to elaborate more fully on some aspects of the physics, to gain familiarity with typical characteristics of optical properties, and to develop some skills in mathematical techniques. The central theme of the book is the dielectric function (a macroscopic quantity) and its relationship to the fundamental microscopic electronic properties of solids. The emphasis is on basic principles, often illustrated by simple models. The necessary mathematics needed to understand the models is generally carried through to completion, with no steps missing, and no "it can be shown" statements. Thus, the text is intended to be suitable for self-study, as well as for use in a one-semester first-year graduate course. XI ACKNOWLEDGMENTS There are many people who have contributed in some way to the making of this book. I thank especially those who have permitted the use of their figures from published work. In a more direct personal way, Dr. Louis F. Wouters first aroused in me a latent interest in light and the photoelectric effect and provided the opportunity for my initial research on photoelectric emission. I am grateful to Dr. James E. Carothers and Dr. Jack N. Shearer for their encouragement and support over the years. Dr. Tony Huen, my friend and colleague, aided substantially with many helpful discussions in our day-to-day collaboration. Others whom I thank for helpful discus- sions are Professor William E. Spicer, Dr. Erkki Pajanne, Dr. Birger Bergersen, Professor Lars Hedin, Professor Stig Lundquist, Dr. Per-Olov Nilsson, Major L. P. Mosteller, Dr. Geoffrey B. Irani, Captain Harry V. Winsor, Lt. George Fuller, Professor Ching-Yao Fong, and Dr. David Brust. Thanks also to Mrs. Peggy Riley, Mrs. Kathryn Smith, and Mrs. Donna Marshall who typed parts of the manuscript. xu Chapter 1 INTRODUCTION This book presents an introduction to the fundamental optical spectra relationship of of solids. The aim is to develop an understanding of the measurable optical properties to the dielectric function and the microscopic electronic structure of solids. The usual way to determine the optical properties of a solid is to shine monochromatic light onto an appropriate sample and then to measure the reflectance or transmittance of the sample as a function of photon energy. Other methods, such as ellipsometry, are sometimes used. However, these methods are of no concern here. The choice of experimental technique obtained. We is largely one of convenience, not of the basic information shall concentrate on reflectivity. Details of experimental technique and methods of data analysis are left to other monographs and papers, some of which are included in the references throughout the book. One exception incidence reflectance is the inclusion of a discussion of the analysis of normal data with the use of the Kramers-Kronig equations; but, the importance Kramers- here lies in the physics and great generality contained in the Kronig equations, not in the experimental techniques for measuring the reflectance at normal incidence. In recent years, photoelectric emission and characteristic energy loss experiments have proven useful as methods of studying electronic band in structure. These experiments are closely related to optical experiments terms of the kind of information they provide. They are discussed at various points throughout the book. There are experimental techniques other than the optical types which * Chapter 1 Introduction provide information on band structure. These include cyclotron resonance, de Haas-van Alphen effect, galvanomagnetic effects, and magnetoacoustic resonance. However, even though often of high accuracy, these experiments yield information pertaining to energy levels only within a few kT of the Fermi surface. Ion neutralization spectroscopy and soft x-ray emission provide information over a wide energy range, but they have not proven as useful as optical methods. This chapter discusses briefly the kinds of experiments that are most typical and indicates the kind of information that can be obtained. To provide a framework for discussions of optical measurements, photo- electric emission, and characteristic energy loss spectra, we begin in Section 1.1 with a reminder of some of the ideas of band theory, but no more than that. The reader is assumed to have an adequate understanding of the basic ideas of band theory. Next is a brief introduction to optical reflectivity. This is followed by a discussion of photoelectric emission. Since even an elementary discussion of the physics of photoelectric emission is not included in most textbooks on solid-state physics, it is included here.