www.pdfgrip.com Solid-State Physics for Electronics André Moliton Series Editor Pierre-Noël Favennec www.pdfgrip.com This page intentionally left blank www.pdfgrip.com Solid-State Physics for Electronics www.pdfgrip.com This page intentionally left blank www.pdfgrip.com Solid-State Physics for Electronics André Moliton Series Editor Pierre-Noël Favennec www.pdfgrip.com 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 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 Solid state physics 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 www.pdfgrip.com Table of Contents Foreword xiii Introduction xv Chapter 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 1 Chapter 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 www.pdfgrip.com 4 6 10 10 14 17 17 17 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 (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 § 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 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 www.pdfgrip.com 19 19 19 21 22 23 24 24 26 27 27 29 30 30 32 32 35 37 40 41 42 44 47 49 50 55 55 55 56 57 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 S a 59 59 59 63 64 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 66 66 67 69 70 70 71 74 77 79 79 80 81 81 81 84 Chapter 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 S 87 87 87 89 90 93 93 93 94 96 ) 96 4.4 Distinguishing insulators, semiconductors, metals and semi-metals 97 a www.pdfgrip.com Chapter Strong Bonds in One Dimension This chapter starts with a concise description of the origin and construction of atomic and molecular orbitals found in a covalent solid (molecular films and polymers included) The results are then applied to energy levels in 1D covalent materials (notably molecular wires) 7.1 Atomic and molecular orbitals 7.1.1 s- and p-type orbitals In the approximation for an atomic configuration (that gives the quantum numbers n, l, m…), we assume that each electron of an atom moves in a potential that has a spherical symmetry The result is that: – the potential of the nucleus varies with respect to 1/r; – this spherical potential gives a first approximation to the action of the other electrons The electronic state is thus represented by a wave function denoted \n,l,m that is dependent on three quantum numbers n, l and m, while the energy is only dependent on n and l (the degree of degeneration is equal to the number of values that m can take on) More details on this can be found in most basic courses on wave and atomic physics that use hydrogen as an example of a system with a spherical potential symmetry www.pdfgrip.com 200 Solid-State Physics for Electronics When l = 0, the atomic orbitals are denoted by the letter s and the wave functions only depend on n: