Solid-State Physics for Electronics by André Moliton, Slides of Solid State Physics

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Solid-State Physics

for Electronics

André Moliton

Series Editor

Pierre-Noël Favennec

Solid-State Physics for Electronics

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First published in France in 2007 by Hermes Science/Lavoisier entitled : Physique des matériaux pour l’électronique © LAVOISIER, 2007 First published in Great Britain and the United States in 2009 by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

ISTE Ltd John Wiley & Sons, Inc. 27-37 St George’s Road 111 River Street London SW19 4EU Hoboken, NJ 07030 UK USA

www.iste.co.uk www.wiley.com

© ISTE Ltd, 2009

The rights of André Moliton to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Cataloging-in-Publication Data

Moliton, André. [Physique des matériaux pour l'électronique. English] Solid-state physics for electronics / André Moliton. p. cm. Includes bibliographical references and index. ISBN 978-1-84821-062-

1. Solid state physics. 2. Electronics--Materials. I. Title. QC176.M5813 2009 530.4'1--dc 2009016464

British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN: 978-1-84821-062-

Cover image created by Atelier Istatis. Printed and bound in Great Britain by CPI Antony Rowe, Chippenham and Eastbourne.

Table of Contents

Foreword.......................................... xiii

Introduction........................................ xv

Chapter 1. Introduction: Representations of Electron-Lattice Bonds.... 1 1.1. Introduction.................................... 1 1.2. Quantum mechanics: some basics...................... 2 1.2.1. The wave equation in solids: from Maxwell’s to Schrödinger’s equation via the de Broglie hypothesis........... 2 1.2.2. Form of progressive and stationary wave functions for an electron with known energy (E)..................... 4 1.2.3. Important properties of linear operators................. 4 1.3. Bonds in solids: a free electron as the zero order approximation for a weak bond; and strong bonds......................... 6 1.3.1. The free electron: approximation to the zero order.......... 6 1.3.2. Weak bonds................................. 7 1.3.3. Strong bonds................................. 8 1.3.4. Choosing between approximations for weak and strong bonds... 9 1.4. Complementary material: basic evidence for the appearance of bands in solids......................................... 10 1.4.1. Basic solutions for narrow potential wells............... 10 1.4.2. Solutions for two neighboring narrow potential wells........ 14

Chapter 2. The Free Electron and State Density Functions.......... 17 2.1. Overview of the free electron......................... 17 2.1.1. The model.................................. 17 2.1.2. Parameters to be determined: state density functions in k or energy spaces................................. 17

Table of Contents vii

3.2. Mathieu’s equation............................... 59 3.2.1. Form of Mathieu’s equation........................ 59 3.2.2. Wave function in accordance with Mathieu’s equation....... 59 3.2.3. Energy calculation............................. 63 3.2.4. Direct calculation of energy when k (^) a S  r.............. 64 3.3. The band structure................................ 66 3.3.1. Representing E f (k) for a free electron: a reminder........ 66 3.3.2. Effect of a cosinusoidal lattice potential on the form of wave function and energy........................... 67 3.3.3. Generalization: effect of a periodic non-ideally cosinusoidal potential............................... 69 3.4. Alternative presentation of the origin of band systems via the perturbation method............................. 70 3.4.1. Problem treated by the perturbation method.............. 70 3.4.2. Physical origin of forbidden bands.................... 71 3.4.3. Results given by the perturbation theory................ 74 3.4.4. Conclusion.................................. 77 3.5. Complementary material: the main equation................ 79 3.5.1. Fourier series development for wave function and potential.... 79 3.5.2. Schrödinger equation............................ 80 3.5.3. Solution.................................... 81 3.6. Problems..................................... 81 3.6.1. Problem 1: a brief justification of the Bloch theorem......... 81 3.6.2. Problem 2: comparison of E(k) curves for free and semi-free electrons in a representation of reduced zones........ 84

Chapter 4. Properties of Semi-Free Electrons, Insulators, Semiconductors, Metals and Superlattices............................... 87 4.1. Effective mass (m*)............................... 87 4.1.1. Equation for electron movement in a band: crystal momentum.. 87 4.1.2. Expression for effective mass....................... 89 4.1.3. Sign and variation in the effective mass as a function of k..... 90 4.1.4. Magnitude of effective mass close to a discontinuity......... 93 4.2. The concept of holes.............................. 93 4.2.1. Filling bands and electronic conduction................. 93 4.2.2. Definition of a hole............................. 94 4.3. Expression for energy states close to the band extremum as a function of the effective mass......................... 96 4.3.1. Energy at a band limit via the Maclaurin development (in k = kn = n^ a

S ).................................. 96 4.4. Distinguishing insulators, semiconductors, metals and semi-metals.. 97

viii Solid-State Physics for Electronics

4.4.1. Required functions............................. 97 4.4.2. Dealing with overlapping energy bands................. 97 4.4.3. Permitted band populations........................ 98 4.5. Semi-free electrons in the particular case of super lattices........ 107 4.6. Problems..................................... 116 4.6.1. Problem 1: horizontal tangent at the zone limit (k | S/a) of the dispersion curve.............................. 116 4.6.2. Problem 2: scale of m* in the neighborhood of energy discontinuities............................. 117 4.6.3. Problem 3: study of EF(T)......................... 122

Chapter 5. Crystalline Structure, Reciprocal Lattices and Brillouin Zones 123 5.1. Periodic lattices................................. 123 5.1.1. Definitions: direct lattice......................... 123 5.1.2. Wigner-Seitz cell.............................. 125 5.2. Locating reciprocal planes........................... 125 5.2.1. Reciprocal planes: definitions and properties............. 125 5.2.2. Reciprocal planes: location using Miller indices........... 125 5.3. Conditions for maximum diffusion by a crystal (Laue conditions)... 128 5.3.1. Problem parameters............................ 128 5.3.2. Wave diffused by a node located by (^) U m n p (^) , , m ˜ a  n ˜ b  p ˜ c

G G^ G^ G

5.4. Reciprocal lattice................................ 133 5.4.1. Definition and properties of a reciprocal lattice............ 133 5.4.2. Application: Ewald construction of a beam diffracted by a reciprocal lattice............................... 134 5.5. Brillouin zones................................. 135 5.5.1. Definition.................................. 135 5.5.2. Physical significance of Brillouin zone limits............. 135 5.5.3. Successive Brillouin zones........................ 137 5.6. Particular properties............................... 137 5.6.1. Properties of Gh k l , ,

G

and relation to the direct lattice........ 137 5.6.2. A crystallographic definition of reciprocal lattice........... 139 5.6.3. Equivalence between the condition for maximum diffusion and Bragg’s law................................... 139 5.7. Example determinations of Brillouin zones and reduced zones..... 141 5.7.1. Example 1: 3D lattice........................... 141 5.7.2. Example 2: 2D lattice........................... 143 5.7.3. Example 3: 1D lattice with lattice repeat unit (a) such that the base vector in the direct lattice is

G

a.......................... 145 5.8. Importance of the reciprocal lattice and electron filling of Brillouin zones by electrons in insulators, semiconductors and metals.. 146 5.8.1. Benefits of considering electrons in reciprocal lattices....... 146

x Solid-State Physics for Electronics

7.1.3. V- and S-bonds............................... 209 7.1.4. Conclusion.................................. 210 7.2. Form of the wave function in strong bonds: Floquet’s theorem..... 210 7.2.1. Form of the resulting potential...................... 210 7.2.2. Form of the wave function........................ 212 7.2.3. Effect of potential periodicity on the form of the wave function and Floquet’s theorem............................... 213 7.3. Energy of a 1D system............................. 215 7.3.1. Mathematical resolution in 1D where x { r............... 215 7.3.2. Calculation by integration of energy for a chain of N atoms.... 217 7.3.3. Note 1: physical significance in terms of (E 0 – D) and E...... 220 7.3.4. Note 2: simplified calculation of the energy.............. 222 7.3.5. Note 3: conditions for the appearance of permitted and forbidden bands................................ 223 7.4. 1D and distorted AB crystals......................... 224 7.4.1. AB crystal.................................. 224 7.4.2. Distorted chain............................... 226 7.5. State density function and applications: the Peierls metal-insulator transition.............................. 228 7.5.1. Determination of the state density functions.............. 228 7.5.2. Zone filling and the Peierls metal–insulator transition........ 230 7.5.3. Principle of the calculation of Erelax (for a distorted chain)...... 232 7.6. Practical example of a periodic atomic chain: concrete calculations of wave functions, energy levels, state density functions and band filling. 233 7.6.1. Range of variation in k........................... 233 7.6.2. Representation of energy and state density function for N = 8... 234 7.6.3. The wave function for bonding and anti-bonding states....... 235 7.6.4. Generalization to any type of state in an atomic chain........ 239 7.7. Conclusion.................................... 239 7.8. Problems..................................... 241 7.8.1. Problem 1: complementary study of a chain of s-type atoms where N = 8..................................... 241 7.8.2. Problem 2: general representation of the states of a chain of V–s-orbitals (s-orbitals giving V-overlap) and a chain of V–p-orbitals. 243 7.8.3. Problem 3: chains containing both V–s- and V–p-orbitals...... 246 7.8.4. Problem 4: atomic chain with S-type overlapping of p-type orbitals: S–p- and S*–p-orbitals..................... 247

Chapter 8. Strong Bonds in Three Dimensions: Band Structure of Diamond and Silicon................................. 249 8.1. Extending the permitted band from 1D to 3D for a lattice of atoms associated with single s-orbital nodes (basic cubic system, centered cubic, etc.).................................. 250

Table of Contents xi

8.1.1. Permitted energy in 3D: dispersion and equi-energy curves..... 250 8.1.2. Expression for the band width...................... 255 8.1.3. Expressions for the effective mass and mobility............ 257 8.2. Structure of diamond: covalent bonds and their hybridization...... 258 8.2.1. The structure of diamond......................... 258 8.2.2. Hybridization of atomic orbitals..................... 259 8.2.3. sp 3 Hybridization.............................. 262 8.3. Molecular model of a 3D covalent crystal (atoms in sp 3 -hybridization states at lattice nodes)..................... 268 8.3.1. Conditions.................................. 268 8.3.2. Independent bonds: effect of single coupling between neighboring atoms and formation of molecular orbitals........... 272 8.3.3. Coupling of molecular orbitals: band formation............ 273 8.4. Complementary in-depth study: determination of the silicon band structure using the strong bond method................... 275 8.4.1. Atomic wave functions and structures.................. 275 8.4.2. Wave functions in crystals and equations with proper values for a strong bond approximation........................ 278 8.4.3. Band structure............................... 282 8.4.4. Conclusion.................................. 287 8.5. Problems..................................... 287 8.5.1. Problem 1: strong bonds in a square 2D lattice............ 287 8.5.2. Problem 2: strong bonds in a cubic centered or face centered lattices................................ 294

Chapter 9. Limits to Classical Band Theory: Amorphous Media...... 301 9.1. Evolution of the band scheme due to structural defects (vacancies, dangling bonds and chain ends) and localized bands.............. 301 9.2. Hubbard bands and electronic repulsions. The Mott metal–insulator transition........................................ 303 9.2.1. Introduction................................. 303 9.2.2. Model..................................... 304 9.2.3. The Mott metal–insulator transition: estimation of transition criteria.................................. 307 9.2.4. Additional material: examples of the existence and inexistence of Mott–Hubbard transitions.................... 309 9.3. Effect of geometric disorder and the Anderson localization....... 311 9.3.1. Introduction................................. 311 9.3.2. Limits of band theory application and the Ioffe–Regel conditions. 312 9.3.3. Anderson localization........................... 314 9.3.4. Localized states and conductivity. The Anderson metal-insulator transition....................................... 319 9.4. Conclusion.................................... 322

Foreword

A student that has attained a MSc degree in the physics of materials or electronics will have acquired an understanding of basic atomic physics and quantum mechanics. He or she will have a grounding in what is a vast realm: solid state theory and electronic properties of solids in particular. The aim of this book is to enable the step-by-step acquisition of the fundamentals, in particular the origin of the description of electronic energy bands. The reader is thus prepared for studying relaxation of electrons in bands and hence transport properties, or even coupling with radiance and thus optical properties, absorption and emission. The student is also equipped to use by him- or herself the classic works of taught solid state physics, for example, those of Kittel, and Ashcroft and Mermin.

This aim is reached by combining qualitative explanations with a detailed treatment of the mathematical arguments and techniques used. Valuably, in the final part the book looks at structures other than the macroscopic crystal, such as quantum wells, disordered materials, etc., towards more advanced problems including Peierls transition, Anderson localization and polarons. In this, the author’s research specialization of conductors and conjugated polymers is discernable. There is no doubt that students will benefit from this well placed book that will be of continual use in their professional careers.

Michel SCHOTT

Emeritus Research Director (CNRS), Ex-Director of the Groupe de Physique des Solides (GPS), Pierre and Marie Curie University, Paris, France

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xvi Solid-State Physics for Electronics

The last four chapters touch on theories which are rather more complex. Chapter 7 is dedicated to the description of the strong bond in 1D media. Floquet’s theorem, which is a sort of physical analog for the Hückel’s theorem that is so widely used in physical chemistry, is established. These results are extended to 3D media in Chapter 8, along with a simplified presentation of silicon band theory. The huge gap between the discovery of the working transistor (1947) and the rigorous establishment of silicon band theory around 20 years later is highlighted. Chapter 9 is given over to the description of energy levels in real solids where defaults can generate localized levels. Amorphous materials are well covered, for example, amorphous silicon is used in non-negligible applications such as photovoltaics. Finally, Chapter 10 contains a description of the principal quasi -particles in solid state, electronic and optical physics. Phonons are thus covered in detail. Phonons are widely used in thermics; however, the coupling of this with electronic charges is at the origin of phonons in covalent materials. These polarons, which often determine the electronic transport properties of a material, are described in all their possible configurations. Excitons are also described with respect to their degree of extension and their presence in different materials. Finally, the coupling of an electromagnetic wave with electrons or with (vibrating) ions in a diatomic lattice is studied to give a classical description of quasi -particles such as plasmons and polaritons.

Chapter 1

Introduction: Representations

of Electron-Lattice Bonds

1.1. Introduction

This book studies the electrical and electronic behavior of semiconductors, insulators and metals with equal consideration. In metals, conduction electrons are naturally more numerous and freer than in a dielectric material, in the sense that they are less localized around a specific atom.

Starting with the dual wave-particle theory, the propagation of a de Broglie wave interacting with the outermost electrons of atoms of a solid is first studied. It is this that confers certain properties on solids, especially in terms of electronic and thermal transport. The most simple potential configuration will be laid out first (Chapter 2). This involves the so-called flat-bottomed well within which free electrons are simply thought of as being imprisoned by potential walls at the extremities of a solid. No account is taken of their interactions with the constituents of the solid. Taking into account the fine interactions of electrons with atoms situated at nodes in a lattice means realizing that the electrons are no more than semi-free, or rather “ quasi -free”, within a solid. Their bonding is classed as either “weak” or “strong” depending on the form and the intensity of the interaction of the electrons with the lattice. Using representations of weak and strong bonds in the following chapters, we will deduce the structure of the energy bands on which solid- state electronic physics is based.