Optical Properties of Crystalline and Amorphous Semiconductors:

Materials and Fundamental Principles

Optical Properties of Crystalline and Amorphous Semiconductors:

Materials and Fundamental Principles

By

Sadao Adachi

Department of Electronic Engineering Gunma University

w Springer Science+Business Media, LLC

Library of Congress Cataloging-in-Publication Data

Adachi, Sadao, 1950-Optical properties of crystalline and amorphous semiconductors :

materials and fundamental principles / by Sadao Adachi p. cm.

Includes bibliographical references and index. ISBN 978-1-4613-7389-6 ISBN 978-1-4615-5241-3 (eBook) DOI 10.1007/978-1-4615-5241-3 1. Semiconductors-Optical properties. I. Title

QC611.6.06 A26 1999 537.6'226»dc21 99-23738

CIP

Copyright © 1999 Springer Science+Business Media New York Originally published by Kluwer Academic Publishers in 1999 Softcover reprint of the hardcover 1st edition 1999

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, mechanical, photo­copying, recording, or otherwise, without the prior written permission of the publisher, Springer Science+Business Media, L L C .

Printed on acid-free paper.

To

Yuki, Mai, and Koya

CONTENTS

Preface xi

Acknowledgments xiii

Contents of the Companion Book xv

Abbreviations and Acronyms xvii

i Materials, Properties, and Basic Formulas 1 1.1 Materials and Their Structures, 1

1.1.1 Crystalline Materials, 1 (a) Diamond-, zinc-blende-, and wurtzite-type crystals, 1 (b) Silicon carbide, 5 (c) Rocksalt-type crystals, 5

1.1.2 Amorphous Materials, 6 1.2 Brillouin Zones and Electronic Energy-Band Structure, 7

1.2.1 Brillouin Zone, 7 (a) Face-centered cubic lattice, 7 (b) Hexagonal lattice, 8 (c) Rhombohedral lattice, 10

1.2.2 Electronic Energy-Band Structure, 10 (a) Diamond-type crystal: Si, 10 (b) Zinc-blende-type crystal: GaAs, 13 (c) Wurtzite-type crystal: CdS, 16

1.3 Optical Constants and Some Dispersion Relations, 20 1.3.1 Dielectric Constant: Tensor Representation, 20 1.3.2 Optical Dispersion Relations and Sum Rules, 21 1.3.3 Optical Spectrum and Its Classification into Several Regions, 25

(a) Crystalline materials, 25 (b) Amorphous materials, 28

vii

viii CONTENTS

References, 30

2 The Reststrahlen Region 33 2.1 Static and High-Frequency Dielectric Constants, 33

2.1.1 General Consideration, 33 2.1.2 Alloys, 38 2.1.3 Temperature and Pressure Effects, 38

2.2 Reststrahlen Spectra, 41 2.2.1 Crystalline Materials, 41

(a) Zinc-blende-type crystals, 41 (b) Wurtzite-type crystals, 48 (c) Alloys, 48 (d) Temperature effect, 56

2.2.2 Amorphous Materials, 56 References, 61

3 The Interband Transition Region: Crystalline Materials 63 3.1 Optical Dispersion Model, 63

3.1.1 Critical Points and Dispersion Theory, 63 (a) Harmonic oscillator approximation, 64 (b) Standard critical-point model, 66 (c) Model dielectric junction, 69

1. Eo and Eo+Llo transitions, 72 2. E} and E}+l!.} transitions, 74 3. Eo' and E2 transitions, 77 4. Indirect-band-gap transitions, 79

3.1.2 Plasma and d-Band Effects, 81 3.2 Temperature, Pressure, and Doping Effects, 84

3.2.1 Temperature Effect, 84 3.2.2 Pressure Effect, 95 3.2.3 Doping Effect, 102

3.3 Fresnel Reflection Coefficient, Ellipsometry Definition, and Surface Oxide Effect, 106 3.3.1 Fresnel Reflection Coefficient, 106

(a) Isotropic medium, 106 (b) Anisotropic medium, 109 (c) Reflection at Multiple Interfaces, 111

3.3.2 Ellipsometry Definition and Optical Constants, 113 3.3.3 Effect of Surface Oxide, 114

3.4 Effect of Surface Roughness: Effective-Medium Approximation, 120 3.4.1 Effective-Medium Model, 120 3.4.2 Roughened Surface, 122

References, 126

CONTENTS ix

4 The Interband Transition Region: Amorphous and Microcrystalline Materials 131 4.1 Amorphous Materials, 131

4.1.1 Optical Band Gap, 131 (a) Definition, 131 (b) Temperature and pressure effects, 136

1. Temperature effect, 136 2. Pressure effect, 138

4.1.2 Optical Dispersion Model, 140 (a) Density-ot-states model, 140 (b) Plasma-frequency (Campi-Coriasso) model, 141 (c) Penn-gap (Forouhi-Bloomer) model, 142 (d) Model dielectric function, 144

4.1.3 Temperature, Pressure, and Doping Effects on Optical Constants, 154 4.2 Microcrystalline Materials, 155

4.2.1 General Remarks, 155 4.2.2 Amorphization of Crystalline Si by Ion Implantation, 156

(a) As-implanted samples, 156 (b) Implanted and annealed samples, 159

4.2.3 Recrystallization of Amorphous Si by Thermal Annealing, 162 (a) LRA-EMA analysis, 162 (b) MDF analysis, 168

References, 175

5 At or Below the Fundamental Absorption Edge 5.1 Optical Absorption, 179

5.1.1 One-Electron Approximation, 179 (a) Direct-gap edge, 179 (b) Indirect-gap edge, 181

5.1.2 Exciton Effect, 183 (a) Direct exciton, 183 (b) Indirect exciton, 192

5.1.3 Urbach Tail, 194 5.1.4 Temperature, Pressure, and Doping Effects, 195

(a) Temperature effect, 195 (b) Pressure effect, 198 (c) Doping effect, 202

5.1.5 Amorphous Materials, 206 5.2 Refractive Index, 207

5.2.1 Theoretical Expression, 207 (a) Sellmeier equation, 207 (b) Ketteler-Helmholtz formula, 208 (c) Single-oscillator model, 208

179

x

(d) Modified single-oscillator model, 208 (e) Pikhtin-Yas'kov formula, 209 (f) Exponential band-edge model, 209 (g) Simplified interband-transition model, 210 (h) Quantum-density-matrix formulation, 211

5.2.2 Empirical Fonnula, 211 5.2.3 Experimental Data, 213

(a) Comparison with calculated results, 213 (b) Alloy-composition variation, 216 (c) Temperature, pressure, and doping effects, 218

1. Temperature effect, 218 2. Pressure effect, 225 3. Doping effect, 227

5.3 Free-Carrier Absorption and Related Phenomena, 232 5.3.1 Free-Carrier Absorption, 232 5.3.2 Interband Absorption, 238

(a) Interconduction-band absorption, 238 (b) Intervalence-band absorption, 242

5.3.3 Carrier-Induced Change in Refractive Index, 246 References, 247

6 Concluding Remarks References, 256

Index

CONTENTS

251

257

PREFACE

Knowledge of the refractive indices and absorption coefficients of semiconductors is especially important in the design and analysis of optical and optoelectronic devices. The determination of the optical constants of semiconductors at energies beyond the fundamental absorption edge is also known to be a powerful way of studying the electronic energy-band structures of the semiconductors.

The purpose of this book is to present an introduction to the fundamental optical properties of semiconductors. The aim is to develop an understanding of the optical response of crystalline and amorphous semiconductors over the entire spectral range.

This book presents tutorial articles in the categories of materials and fundamental principles (Chapter 1), optical properties in the reststrahlen region (Chapter 2), those in the interband transition region (Chapters 3 and 4) and at or below the fundamental absorption edge (Chapter 5).

This material is, for the most part, in a form which could serve to teach the underlying concepts of semiconductor optical properties and its implementation.

The bird's-eye view of optical properties over the wide spectral range afforded by the tables and graphs in the companion book "Optical Constants of Crystalline and Amorphous Semiconductors: Numerical Data and Graphical Information", serves as a good introduction to understanding a wide variety of absorption process and dispersion relation and, consequently, optical properties.

The extensive bibliography is included for those who wish to find additional information if required. It is hoped that the book will attract attention of not only device engineers, but also solid-state physicists and material scientists, and par­ticularly students specializing in the fields of semiconductor physics and device engineering.

xi

ACKNOWLEDGMENTS

The author wishes to thank the editors and authors of the following journals and books for permission to reproduce previously published figures: John Wiley & Sons for Fig. 1.1; Physical Review for Figs. 1.8, 1.11, 1.13, 1.15,2.1,2.4, 2.12(a), 2.16, 3.11-3.21, 3.24-3.26, 3.28, 3.29, 4.5, 4.12-4.14, 5.7(a), 5.8, 5.11, 5.15, 5.26, and 5.38; North-Holland Publishing Company for Fig. 1.14; Journal of Applied Physics for Figs. 1.17, 2.2, 3.22, 3.42-3.44, 4.6, 4.15, 4.16, 5.3, 5.7(b), 5.9, 5.14, 5.17, 5.18, 5.21-5.25, 5.27, 5.29, 5.30, and 5.36; World Scientific for Figs. 2.5, 2.6, 2.8, 2.9, and 2.12-2.14; Solid State Communications for Fig. 2.10; Applied Optics for Fig. 2.15; Physica Status Solidi for Fig. 2.17; Journal de Physique for Fig. 3.23; Applied Physics Letters for Figs. 3.27, 5.31, and 5.37; Japanese Journal of Applied Physics for Figs. 4.17, 4.18, 4.20-4.29, 4.31-4.38, and 5.19; Academic Press for Fig. 4.4; Soviet Physics Semiconductors for Fig. 5.20; Journal of Physics: Con­densed Matters for Fig. 5.28; Journal of Crystal Growth for Fig. 5.35; Journal of Physics and Chemistry of Solids for Fig. 5.40; and IEEE Journal of Quantum Electronics for Fig. 5.41.

xiii

CONTENTS OF THE COMPANION BOOK

OPTICAL CONSTANTS OF CRYSTALLINE AND AMORPHOUS SEMICONDUCTORS: Numerical Data and Graphical Information

Sadao Adachi, Author

Introductory Remarks

A Group-IV Semiconductors

B III-V Binary Semiconductors

C m -V Alloy Semiconductors

D II-VI Semiconductors

E IV -VI Semiconductors

F Amorphous Semiconductors

xv

ABBREVIATIONS AND ACRONYMS

a­AFM

BZ

c-c-CP CPA CVD

DFT DHO DOS DP

EMA EMA ENPM EPM ER ETBT

fcc

h-HH HOA HREELS

IR

KK KSM

LA LCAO LDA

- amorphous-- Atomic Force Microscopy

- Brillouin Zone

- crystalline-- cubic-- Critical Point - Coherent-Potential Approximation - Chemical-Vapor Deposition

- Density-Functional Theory - Damped Harmonic Oscillator - Density of States - Deformation Potential

- Effective-Mass Approximation - Effective-Medium Approximation - Empirical Nonlocal Pseudopotential Method - Empirical Pseudopotential Method - Electroreflectance - Empirical Tight-Binding Theory

- face-centered cubic

- hexagonal-- Heavy-Hole - Harmonic Oscillator Approximation - High-Resolution Electron-Energy Loss Spectroscopy

- Infrared

- Kramers-Kronig - Koster-Slater Method

- Longitudinal Acoustic - Linear Combination of Atomic Orbitals - Local-Density Approximation

xvii

xviii

LEED LH LMfO LO LPE LRA

MBE MDF MOCVD MOVPE

PL PR

RCA rf rms RS

SCP SC1(2) SE SO STM

TA TO

UHV UV

VCA VLSI

w-

XPS

Jlc-

1D 2D 3D

ABBREVIATIONS AND ACRONYMS

- Low-Energy Electron Diffraction - Light-Hole - Linear Muffin-Tin Orbitals - Longitudinal Optical - Liquid-Phase Epitaxy - Linear Regression Analysis

- Molecular Beam Epitaxy - Model Dielectric Function - Metalorganic Chemical Vapor Deposition - Metalorganic Vapor Phase Epitaxy

- Photoluminescence - Photoreflectance

- Radio Corporation of America - radio frequency - root mean square - Raman Scattering

- Standard Critical Point - Standard Cleaning 1(2) - Spectroscopic Ellipsometry - Spin-Orbit - Scanning Tunneling Microscopy

- Transverse Acoustic - Transverse Optical

- Ultrahigh Vacuum - Ultraviolet

- Virtual-Crystal Approximation - Very-Large-Scale Integration

- wurtzite-

- X-Ray Photoelectron Spectroscopy

- microcrystalline-

- One-Dimensional - Two-Dimensional - Three-Dimensional