《材料科学导论 Introduction to Materials Science》参考书籍：Solid-State Physics for Electronics（André Moliton）

Solid-State Physics for Electronics Andre molton SE WWILEY

Solid-state Physics for electronics Andre molitor Series editor Pierre-Noel favennec WWILEY

Solid-State Physics for Electronics André Moliton Series Editor Pierre-Noël Favennec

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This page intentionally left blank

Solid-State Physics for Electronics

Solid-State Physics for Electronics

This page intentionally left blank

This page intentionally left blank

Solid-state Physics for electronics Andre molitor Series editor Pierre-Noel favennec WWILEY

Solid-State Physics for Electronics André Moliton Series Editor Pierre-Noël Favennec

First published in France in 2007 by Hermes Science/Lavoisier entitled: Physique des materiaux pour Electronique O 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 USA www.wiley.com he rights of Andre 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 Molton, Andre. Physique des materiaux pour electronique. English Solid-state physics for electronics/Andre Molton Includes bibliographical references and index. ISBN978-1-84821-0622 Solid state physics. 2. Electronics--Materials. I. Title. QC176Ms8132009 530.4"l-dc22 ation Data A CIP for this book is available from the British Library ISBN 34821-062-2 Cover image created by Atelier Istatis Printed and bound in Great Britain by CPI Antony Rowe, Chippenham and Eastbourne

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 978-1-84821-062-2 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

Table of Contents Foreword trod ctio Chapter 1 Introduction: Representations of Electron-Lattice Bonds L 1. Introduction 1. 2. Quantum mechanics: some basics 1. 1. The wave equation in solids: from Maxwells to Schrodinger'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 6 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 solic 1.4.1. Basic solutions for narrow potential wells 1.4.2. Solutions for two neighboring narrow potential wells 14 Chapter 2. The Free Electron and State Density Functions 2.1. Overview of the free electron 17 2.1.1. The model 2. 2. Parameters to be determined: state density functions in k or energy spaces

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

vi Solid-State Physics for Electronics 2.2. Study of the stationary regime of small scale(enabling the establishment of nodes at extremities) symmetric wells(ID 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 s 22 3. Study of the stationary regime for asymmetric wells(ID model) with L s a favoring the establishment of a stationary regime with nodes at extremities 23 2.4. Solutions that favor propagation: wide potential wells where L=l mm, i.e. several orders greater than inter-atomic distances 24 2 41. 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 ID system 2.5.1. Stationary solution for FBC 2.5.2. Progressive solutions for progressive boundary conditions(PBC) 790 2.5.3. Conclusion: comparing the number of calculated states for fbc and pbc 2.6. From electrons in a 3D system(potential box) 32 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 ID 41 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 90555 3. 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

vi Solid-State Physics for Electronics 2.2. Study of the stationary regime of small scale (enabling the establishment of nodes at extremities) symmetric wells (1D model) . . 19 2.2.1. Preliminary remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 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) . . . . . . . . . . . . . . 19 2.2.3. Study of energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.4. State density function (or “density of states”) in k space . . . . . . . 22 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4. Solutions that favor propagation: wide potential wells where L § 1 mm, i.e. several orders greater than inter-atomic distances . . . 24 2.4.1. Wave function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4.2. Study of energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4.3. Study of the state density function in k space . . . . . . . . . . . . . 27 2.5. State density function represented in energy space for free electrons in a 1D system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.5.1. Stationary solution for FBC . . . . . . . . . . . . . . . . . . . . . . . . 29 2.5.2. Progressive solutions for progressive boundary conditions (PBC) . 30 2.5.3. Conclusion: comparing the number of calculated states for FBC and PBC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.6. From electrons in a 3D system (potential box) . . . . . . . . . . . . . . . 32 2.6.1. Form of the wave functions . . . . . . . . . . . . . . . . . . . . . . . . 32 2.6.2. Expression for the state density functions in k space . . . . . . . . . 35 2.6.3. Expression for the state density functions in energy space . . . . . . 37 2.7. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.7.1. Problem 1: the function Z(E) in 1D . . . . . . . . . . . . . . . . . . . 41 2.7.2. Problem 2: diffusion length at the metal-vacuum interface . . . . . 42 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 . . . 44 2.7.4. Problem 4: Fermi energy of a 3D conductor . . . . . . . . . . . . . . 47 2.7.5. Problem 5: establishing the state density function via reasoning in moment or k spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.7.6. Problem 6: general equations for the state density functions expressed in reciprocal (k) space or in energy space . . . . . . . . . . . . . 50 Chapter 3. The Origin of Band Structures within the Weak Band Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.1. Bloch function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.1.1. Introduction: effect of a cosinusoidal lattice potential . . . . . . . . 55 3.1.2. Properties of a Hamiltonian of a semi-free electron . . . . . . . . . . 56 3.1.3. The form of proper functions . . . . . . . . . . . . . . . . . . . . . . . 57

Table of Contents vii 3.2. Mathieu's equation 3.2. 1. Form of Mathieu's equation 39 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 =t- 33. 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 71 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 7y90 3.5.2. Schrodinger 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 Chapter 4. Properties of Semi-Free Electrons, Insulators, Semiconductors, Metals and Superlattices 87 4. 1. Effective mass(m") 4.1.1. Equation for electron movement in a band: crystal momentum 4. 1.2. Expression for effectiv ve mass 4.1.3. Sign and variation in the effective mass as a function of k 4. 4. Magnitude of effective mass close to a discontinuity 93 4.2. The concept of holes 4.2.1. Filling bands and electronic conduction 93 4. 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-) 4.4. Distinguishing insulators, semiconductors, metals and semi-metal

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 a k 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 = a n S ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.4. Distinguishing insulators, semiconductors, metals and semi-metals . . 97