Solid state physics for electronics

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Solid-State Physics
for Electronics

André Moliton
Series Editor
Pierre-Noël Favennec

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Solid-State Physics for Electronics

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Solid-State Physics
for Electronics

André Moliton
Series Editor
Pierre-Noël Favennec

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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
27-37 St George’s Road
London SW19 4EU
UK

John Wiley & Sons, Inc.
111 River Street
Hoboken, NJ 07030
USA

www.iste.co.uk

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© 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 978-1-84821-062-2
1. Solid state physics. 2. Electronics--Materials. I. Title.
QC176.M5813 2009
530.4'1--dc22
2009016464
British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN: 978-1-84821-062-2
Cover image created by Atelier Istatis.
Printed and bound in Great Britain by CPI Antony Rowe, Chippenham and Eastbourne.

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Table of Contents

Foreword. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xv

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

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

k or energy spaces . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.2. Study of the stationary regime of small scale (enabling
the establishment of nodes at extremities) symmetric wells (1D model) . .
2.2.1. Preliminary remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2. Form of stationary wave functions for thin symmetric wells
with width (L) equal to several inter-atomic distances (L | a),
associated with fixed boundary conditions (FBC) . . . . . . . . . . . . . .
2.2.3. Study of energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.4. State density function (or “density of states”) in k space . . . . . . .
2.3. Study of the stationary regime for asymmetric wells (1D model)
with L § a favoring the establishment of a stationary regime
with nodes at extremities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4. Solutions that favor propagation: wide potential wells
where L § 1 mm, i.e. several orders greater than inter-atomic distances . . .
2.4.1. Wave function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2. Study of energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.3. Study of the state density function in k space . . . . . . . . . . . . .
2.5. State density function represented in energy space for free electrons
in a 1D system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1. Stationary solution for FBC . . . . . . . . . . . . . . . . . . . . . . . .
2.5.2. Progressive solutions for progressive boundary conditions (PBC) .
2.5.3. Conclusion: comparing the number of calculated states
for FBC and PBC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6. From electrons in a 3D system (potential box) . . . . . . . . . . . . . . .
2.6.1. Form of the wave functions . . . . . . . . . . . . . . . . . . . . . . . .
2.6.2. Expression for the state density functions in k space . . . . . . . . .
2.6.3. Expression for the state density functions in energy space. . . . . .
2.7. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.1. Problem 1: the function Z(E) in 1D . . . . . . . . . . . . . . . . . . .
2.7.2. Problem 2: diffusion length at the metal-vacuum interface . . . . .

2.7.3. Problem 3: 2D media: state density function and the behavior
of the Fermi energy as a function of temperature for a metallic state . . .
2.7.4. Problem 4: Fermi energy of a 3D conductor . . . . . . . . . . . . . .
2.7.5. Problem 5: establishing the state density function via reasoning
in moment or k spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.6. Problem 6: general equations for the state density functions
expressed in reciprocal (k) space or in energy space . . . . . . . . . . . . .
Chapter 3. The Origin of Band Structures within the Weak Band
Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1. Bloch function . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1. Introduction: effect of a cosinusoidal lattice potential . . .
3.1.2. Properties of a Hamiltonian of a semi-free electron . . . . .
3.1.3. The form of proper functions . . . . . . . . . . . . . . . . . .

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Table of Contents vii

3.2. Mathieu’s equation . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1. Form of Mathieu’s equation. . . . . . . . . . . . . . . . .
3.2.2. Wave function in accordance with Mathieu’s equation
3.2.3. Energy calculation . . . . . . . . . . . . . . . . . . . . . .
3.2.4. Direct calculation of energy when k  r

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3.3. The band structure . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1. Representing E f (k) for a free electron: a reminder . . . .

3.3.2. Effect of a cosinusoidal lattice potential on the form
of wave function and energy . . . . . . . . . . . . . . . . . . . . . . .
3.3.3. Generalization: effect of a periodic non-ideally
cosinusoidal potential . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4. Alternative presentation of the origin of band systems
via the perturbation method . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1. Problem treated by the perturbation method . . . . . . . . . .
3.4.2. Physical origin of forbidden bands. . . . . . . . . . . . . . . .
3.4.3. Results given by the perturbation theory . . . . . . . . . . . .
3.4.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5. Complementary material: the main equation . . . . . . . . . . . .
3.5.1. Fourier series development for wave function and potential
3.5.2. Schrödinger equation . . . . . . . . . . . . . . . . . . . . . . . .
3.5.3. Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.1. Problem 1: a brief justification of the Bloch theorem . . . . .
3.6.2. Problem 2: comparison of E(k) curves for free
and semi-free electrons in a representation of reduced zones . . . .

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Chapter 4. Properties of Semi-Free Electrons, Insulators, Semiconductors,
Metals and Superlattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1. Effective mass (m*) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1. Equation for electron movement in a band: crystal momentum . .
4.1.2. Expression for effective mass . . . . . . . . . . . . . . . . . . . . . . .
4.1.3. Sign and variation in the effective mass as a function of k . . . . .
4.1.4. Magnitude of effective mass close to a discontinuity . . . . . . . . .
4.2. The concept of holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2.1. Filling bands and electronic conduction. . . . . . . . . . . . . . . . .
4.2.2. Definition of a hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3. Expression for energy states close to the band extremum
as a function of the effective mass . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1. Energy at a band limit via the Maclaurin development
(in k = kn = n

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4.4. Distinguishing insulators, semiconductors, metals and semi-metals . .

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viii Solid-State Physics for Electronics

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

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Chapter 5. Crystalline Structure, Reciprocal Lattices and Brillouin Zones
5.1. Periodic lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1. Definitions: direct lattice . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.2. Wigner-Seitz cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2. Locating reciprocal planes . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1. Reciprocal planes: definitions and properties . . . . . . . . . . . . .
5.2.2. Reciprocal planes: location using Miller indices . . . . . . . . . . .
5.3. Conditions for maximum diffusion by a crystal (Laue conditions) . . .
5.3.1. Problem parameters . . . . . . . . . . . . . . . . . . G. . . . .G . . . . G.
G
5.3.2. Wave diffused by a node located by Um ,n , p m ˜ a  n ˜ b  p ˜ c
5.4. Reciprocal lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1. Definition and properties of a reciprocal lattice . . . . . . . . . . . .
5.4.2. Application: Ewald construction of a beam diffracted
by a reciprocal lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5. Brillouin zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.2. Physical significance of Brillouin zone limits . . . . . . . . . . . . .
5.5.3. Successive Brillouin zones . . . . . . . . . . . . . . . . . . . . . . . .
5.6. Particular properties
G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.1. Properties of G h ,k ,l and relation to the direct lattice . . . . . . . .

5.6.2. A crystallographic definition of reciprocal lattice . . . . . . . . . . .
5.6.3. Equivalence between the condition for maximum diffusion
and Bragg’s law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7. Example determinations of Brillouin zones and reduced zones . . . . .
5.7.1. Example 1: 3D lattice . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7.2. Example 2: 2D lattice . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7.3. Example 3: 1D lattice with lattice repeat unit (a) such that the base
G
vector in the direct lattice is a . . . . . . . . . . . . . . . . . . . . . . . . . .
5.8. Importance of the reciprocal lattice and electron filling
of Brillouin zones by electrons in insulators, semiconductors and metals . .
5.8.1. Benefits of considering electrons in reciprocal lattices . . . . . . .

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Table of Contents ix

5.8.2. Example of electron filling of Brillouin zones in simple structures:
determination of behaviors of insulators, semiconductors and metals . . .
5.9. The Fermi surface: construction of surfaces and properties . . . . . . .
5.9.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.9.2. Form of the free electron Fermi surface . . . . . . . . . . . . . . . .
5.9.3. Evolution of semi-free electron Fermi surfaces . . . . . . . . . . . .
5.9.4. Relation between Fermi surfaces and dispersion curves . . . . . . .
5.10. Conclusion. Filling Fermi surfaces and the distinctions
between insulators, semiconductors and metals . . . . . . . . . . . . . . . . .
5.10.1. Distribution of semi-free electrons at absolute zero . . . . . . . . .
5.10.2. Consequences for metals, insulators/semiconductors
and semi-metals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.11. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.11.1. Problem 1: simple square lattice . . . . . . . . . . . . . . . . . . . .
5.11.2. Problem 2: linear chain and a square lattice . . . . . . . . . . . . .
5.11.3. Problem 3: rectangular lattice . . . . . . . . . . . . . . . . . . . . . .

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Chapter 6. Electronic Properties of Copper and Silicon . . . . .
6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2. Direct and reciprocal lattices of the fcc structure . . . . . . .
6.2.1. Direct lattice . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.2. Reciprocal lattice . . . . . . . . . . . . . . . . . . . . . . .
6.3. Brillouin zone for the fcc structure . . . . . . . . . . . . . . .
6.3.1. Geometrical form . . . . . . . . . . . . . . . . . . . . . . .
6.3.2. Calculation of the volume of the Brillouin zone . . . . .
6.3.3. Filling the Brillouin zone for a fcc structure . . . . . . .
6.4. Copper and alloy formation . . . . . . . . . . . . . . . . . . .
6.4.1. Electronic properties of copper . . . . . . . . . . . . . . .
6.4.2. Filling the Brillouin zone and solubility rules . . . . . .

6.4.3. Copper alloys . . . . . . . . . . . . . . . . . . . . . . . . .
6.5. Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.1. The silicon crystal . . . . . . . . . . . . . . . . . . . . . .
6.5.2. Conduction in silicon. . . . . . . . . . . . . . . . . . . . .
6.5.3. The silicon band structure . . . . . . . . . . . . . . . . . .
6.5.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.6. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.6.1. Problem 1: the cubic centered (cc) structure . . . . . . .
6.6.2. Problem 2: state density in the silicon conduction band

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Chapter 7. Strong Bonds in One Dimension .
7.1. Atomic and molecular orbitals . . . . .
7.1.1. s- and p-type orbitals . . . . . . . . .
7.1.2. Molecular orbitals . . . . . . . . . .

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x Solid-State Physics for Electronics

7.1.3. V- and S-bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2. Form of the wave function in strong bonds: Floquet’s theorem . . . . .
7.2.1. Form of the resulting potential . . . . . . . . . . . . . . . . . . . . . .

7.2.2. Form of the wave function . . . . . . . . . . . . . . . . . . . . . . . .
7.2.3. Effect of potential periodicity on the form of the wave function
and Floquet’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3. Energy of a 1D system . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.1. Mathematical resolution in 1D where x { r. . . . . . . . . . . . . . .
7.3.2. Calculation by integration of energy for a chain of N atoms . . . .
7.3.3. Note 1: physical significance in terms of (E0 – D) and E . . . . . .
7.3.4. Note 2: simplified calculation of the energy . . . . . . . . . . . . . .
7.3.5. Note 3: conditions for the appearance of permitted
and forbidden bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4. 1D and distorted AB crystals . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.1. AB crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.2. Distorted chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5. State density function and applications: the Peierls
metal-insulator transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5.1. Determination of the state density functions . . . . . . . . . . . . . .
7.5.2. Zone filling and the Peierls metal–insulator transition . . . . . . . .
7.5.3. Principle of the calculation of Erelax (for a distorted chain). . . . . .
7.6. Practical example of a periodic atomic chain: concrete calculations
of wave functions, energy levels, state density functions and band filling .
7.6.1. Range of variation in k. . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6.2. Representation of energy and state density function for N = 8 . . .
7.6.3. The wave function for bonding and anti-bonding states . . . . . . .
7.6.4. Generalization to any type of state in an atomic chain . . . . . . . .
7.7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.8. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.8.1. Problem 1: complementary study of a chain of s-type atoms
where N = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 .

7.8.3. Problem 3: chains containing both V–s- and V–p-orbitals . . . . . .
7.8.4. Problem 4: atomic chain with S-type overlapping of
p-type orbitals: S–p- and S*–p-orbitals . . . . . . . . . . . . . . . . . . . . .
Chapter 8. Strong Bonds in Three Dimensions: Band Structure
of Diamond and Silicon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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.). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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209
210
210
210
212
213
215
215
217
220
222
223
224
224
226
228
228
230
232

233
233
234
235
239
239
241
241
243
246
247
249
250


Table of Contents xi

8.1.1. Permitted energy in 3D: dispersion and equi-energy curves . .
8.1.2. Expression for the band width . . . . . . . . . . . . . . . . . . .
8.1.3. Expressions for the effective mass and mobility . . . . . . . . .
8.2. Structure of diamond: covalent bonds and their hybridization . . .
8.2.1. The structure of diamond . . . . . . . . . . . . . . . . . . . . . .
8.2.2. Hybridization of atomic orbitals . . . . . . . . . . . . . . . . . .
8.2.3. sp3 Hybridization . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3. Molecular model of a 3D covalent crystal (atoms in
sp3-hybridization states at lattice nodes) . . . . . . . . . . . . . . . . . .
8.3.1. Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3.2. Independent bonds: effect of single coupling between
neighboring atoms and formation of molecular orbitals . . . . . . . .
8.3.3. Coupling of molecular orbitals: band formation . . . . . . . . .

8.4. Complementary in-depth study: determination of the silicon
band structure using the strong bond method . . . . . . . . . . . . . . . .
8.4.1. Atomic wave functions and structures. . . . . . . . . . . . . . .
8.4.2. Wave functions in crystals and equations with proper values
for a strong bond approximation . . . . . . . . . . . . . . . . . . . . .
8.4.3. Band structure . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.5. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.5.1. Problem 1: strong bonds in a square 2D lattice . . . . . . . . .
8.5.2. Problem 2: strong bonds in a cubic centered or
face centered lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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250
255
257
258
258
259
262

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268
268

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272
273

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275
275

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278
282

287
287
287

. . .

294

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

301

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303
303
304
307
309
311

311
312
314
319
322


xii Solid-State Physics for Electronics

9.5. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.5.1. Additional information and Problem 1 on the Mott transition:
insulator–metal transition in phosphorus doped silicon . . . . . . . . . . .
9.5.2. Problem 2: transport via states outside of permitted bands
in low mobility media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter 10. The Principal Quasi-Particles in Material Physics . . . . . .
10.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2. Lattice vibrations: phonons . . . . . . . . . . . . . . . . . . . . . . . .
10.2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2.2. Oscillations within a linear chain of atoms . . . . . . . . . . . .
10.2.3. Oscillations within a diatomic and 1D chain . . . . . . . . . . .
10.2.4. Vibrations of a 3D crystal . . . . . . . . . . . . . . . . . . . . . .
10.2.5. Energy of a vibrational mode . . . . . . . . . . . . . . . . . . . .
10.2.6. Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2.7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3. Polarons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3.1. Introduction: definition and origin . . . . . . . . . . . . . . . . .
10.3.2. The various polarons . . . . . . . . . . . . . . . . . . . . . . . . .
10.3.3. Dielectric polarons . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3.4. Polarons in molecular crystals . . . . . . . . . . . . . . . . . . .
10.3.5. Energy spectrum of the small polaron in molecular solids . . .

10.4. Excitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.4.1. Physical origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.4.2. Wannier and charge transfer excitons . . . . . . . . . . . . . . .
10.4.3. Frenkel excitons . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.5. Plasmons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.5.1. Basic definition . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.5.2. Dielectric response of an electronic gas: optical plasma . . . .
10.5.3. Plasmons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.6. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.6.1. Problem 1: enumeration of vibration modes (phonon modes) .
10.6.2. Problem 2: polaritons . . . . . . . . . . . . . . . . . . . . . . . . .

324
331

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347
348
350
351
352
352
352
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357
361
364

364
365
367
368
368
368
372
373
373
375

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

385

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

387

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324


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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Introduction

This volume proposes both course work and problems with detailed solutions. It
is the result of many years’ experience in teaching at MSc level in applied, materials
and electronic physics. It is written with device physics and electronics students in

mind. The book describes the fundamental physics of materials used in electronics.
This thorough comprehension of the physical properties of materials enables an
understanding of the technological processes used in the fabrication of electronic
and photonic devices.
The first six chapters are essentially a basic course in the rudiments of solid-state
physics and the description of electronic states and energy levels in the simplest of
cases. The last four chapters give more advanced theories that have been developed
to account for electronic and optical behaviors of ordered and disordered materials.
The book starts with a physical description of weak and strong electronic bonds
in a lattice. The appearance of energy bands is then simplified by studying energy
levels in rectangular potential wells that move closer to one another. Chapter 2
introduces the theory for free electrons where particular attention is paid to the
relation between the nature of the physical solutions to the number of dimensions
chosen for the system. Here, the important state density functions are also
introduced. Chapter 3, covering semi-free electrons, is essentially given to the
description of band theory for weak bonds based on the physical origin of permitted
and forbidden bands. In Chapter 4, band theory is applied with respect to the electrical
and electronic behaviors of the material in hand, be it insulator, semiconductor or
metal. From this, superlattice structures and their application in optoelectronics is
described. Chapter 5 focuses on ordered solid-state physics where direct lattices,
reciprocal lattices, Brillouin zones and Fermi surfaces are good representations of
electronic states and levels in a perfect solid. Chapter 6 applies these representations
to metals and semiconductors using the archetypal examples of copper and silicon
respectively. An excursion into the preparation of alloys is also proposed.

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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.

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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 solidstate electronic physics is based.

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2

Solid-State Physics for Electronics

1.2. Quantum mechanics: some basics
1.2.1. The wave equation in solids: from Maxwell’s to Schrödinger’s equation via
the de Broglie hypothesis
In the theory of wave-particle duality, Louis de Broglie associated the

wavelength (O) with the mass (m) of a body, by making:

O

h
mv

[1.1]

.

For its part, the wave propagation equation for a vacuum (here the solid is
thought of as electrons and ions swimming in a vacuum) is written as:

's 

1 w ²s

[1.2]

0.

c ² wt ²

If the wave is monochromatic, as in:
s

A (x , y , z )e i Zt

then 's = 'A e i Zt and

A (x , y , z )e i 2SQt

w ²s

Z² Ae i Zt (without modifying the result we can

wt ²

interchange a wave with form
introducing O

2S

c

s

A (x , y , z )e i Zt

A (x , y , z )e i 2SQt ).

By

(length of a wave in a vacuum), wave propagation

Z

equation [1.2] can be written as:

'A 

Z²

c²

A

[1.3]

0

R
'A 

4S ²
O²

A

[1.3’]

0.

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Representations of Electron-Lattice Bonds

3

A particle (an electron for example) with mass denoted m, placed into a timeindependent potential energy V(x, y, z), has an energy:

1

E

m v²  V

2

(in common with a wide number of texts on quantum mechanics and solid-state
physics, this book will inaccurately call potential the “potential energy” – to be
denoted V ).
The speed of the particle is thus given by
2  E V

v

m




[1.4]

.

E

The de Broglie wave for a frequency Q

h

can be represented by the function

< (which replaces the s in equation [1.2]):

<

ȥe i 2SQt

ȥe

E

2 Si

h

t

\e

i

E
=

t

\ e  i Zt .

[1.5]

Accepting with Schrödinger that the function \ (amplitude of < ) can be used in
an analogous way to that shown in equation [1.3’], we can use equations [1.1] and
[1.4] with the wavelength written as:
O

h

h

mv

2m  E  V




,

[1.6]

so that:
'ȥ 

2m
=²

(E  V )ȥ

0.

[1.7]

This is the Schrödinger equation that can be used with crystals (where V is
periodic) to give well defined solutions for the energy of electrons. As we shall see,

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4

Solid-State Physics for Electronics

these solutions arise as permitted bands, otherwise termed valence and conduction
bands, and forbidden bands (or “gaps” in semiconductors) by electronics specialists.
1.2.2. Form of progressive and stationary wave functions for an electron with
known energy (E)
G
In general terms, the form (and a point defined by a vector r ) of a wave
function for an electron of known energy (E) is given by:
G
< (r , t )

G
\(r )e  j Zt

G j
\ (r )e

E
=

t

,

G
where \ (r ) is the wave function at amplitudes which are in accordance with
Schrödinger’s equation [1.7]:
G
– if the resultant wave < (r , t ) is a stationary wave, then \ (r ) is real;
G
G
– if the resultant wave < (r , t ) is progressive, then \ (r ) takes on the form
G 2S G
G
G GG
G
\ (r ) f (r )e jk .r where f (r ) is a real function, and k
u is the wave vector.
O

1.2.3. Important properties of linear operators

1.2.3.1. If the two (linear) operators H and T are commutative, the proper functions
of one can also be used as the proper functions of the other

For the sake of simplicity, non-degenerate states are used. For a proper function
\ of H corresponding to the proper non-degenerate value (D), we find that:
H\

D\

Multiplying the left-hand side of the equation by T gives:
TH \

T D\

DT \.

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