Foundations Of One Electron Theory Of Solids

Description

In this book the behavior of electrons in a crystal is described in terms of the one-electron model. Clearly, such a system would be best approached by a many-body formalism but because of mathe­ matical complexities only a very few real problems can be solved in this way. In the one-electron model, each electron is considered separately as moving in the averaged field of the nuclei and all the other electrons. The resulting picture is simpler conceptually, much more tractable mathematically and, at the same time, is quite comprehensive, which makes the one-electron model the most popu­ lar approach currently in use in solid state physics. In presenting the material, we have tried to show the logic the theory followed as it was developed and, to some extent at least, to show the “scaffolding” used to build it. Whether or not we have succeeded is for the reader to judge. The concept of pseudopotential, as we believe the reader will become convinced, is a direct consequence of quantum-mechanical scattering theory and we hope therefore that the book will be of interest not only for solid state specialists but also for a reader in­ volved in other “quantum” activities. The book should be read more than once, though some places can perhaps be skipped over. Some of the topics were chosen be­ cause they arc likely to be important in the future development of the one-electron model (these topics include scattering outside iso- energetic surfaces, the concept of effective medium, and higher- order perturbation corrections). Some problems are left, unobtrusive­ ly, for the reader to solve. For example, will a self-consistent cal­ culation yield wider or narrower energy bands than those resulting from the non-self-consistent treatment? To answer this question, the reader will have to master the material in Chapter 3. Our hope is that in the course of the book the reader will himself pose problems like this and we foresee the satisfaction he (or she) will take in solv­ ing them. We hope that this book will be helpful for both theorists and ex­ perimentalists. Contents Preface 7 Introduction 11 Chapter 1. Principles of the one-electron theory Part 1 Theoretical principles of the pseudopotential method Chapter 2. Scattering theory for “solid-state people” 2.1. Mathematical formalism 23 2.2. Scattering on an isolated potential 31 2.3. Pseudism and scattering 46 2.4. Bound states, pseudopotentials and the convergence of series 54 2.5. Scattering theory and potential form factors 61 Chapter 3. Theory of potential 3.1. Potential seen by an atomic electron 69 3.2. Dielectric screening 83 3.3. The self-consistency of pseudopotential and additive screening 99 3.4. Muffin-tin potential 107 3.5. Average value of the screened potential 124 Chapter 4. Theory of pseudopotential form factors 4.1. Nonlocality, the energy dependence of form factors and per­turbation theory 132 4.2. The OPW formfactor 144 4.3. Phase-shift form factors 157 4.4. Effective medium and pseudopotential form factors 173 Chapter 5. Pseudism and the secular equations of band theory 5.1. The Green’s function (or KKR) method 181 5.2. Pseudopotential secular equations 196 Part 2 The use of pseudopotential theory for crystal-structure calculations Chapter 6. Formalism of crystal-structure energy calculations 6.1. Basic assumptions 205 6.2. Band structure energy of pure metals and binary alloys 205 6.3. Electrostatic energy 223 6.4. The total internal energy of an alloy: second-order pertur­bation theory and the locality approximation 227 6.5. Higher-order perturbation analysis 232 6.6. OPW nonlocal alloy theory 236 Chapter 7. Pseudopotential theory of alloys. Structure stabili­ty application 7.1. Phase boundaries in terms of pseudopotential theory 241 7.2. Ordered phases, their structures, and existence conditions 245 7.3. Short-range order problems 256 7.4. Crystal structure stability in the OPW approach 261 Chapter 8. Pseudopotential theory and imperfections in crystals 8.1. Introductory remarks 267 8.2. Crystal lattice vibrations 267 8.3. Static imperfections 279 Chapter 9. Principles of pseudopotential calculations of the properties of metals 9.1. Genera 287 9.2. Calculation of the atomic properties of crystalline metals and alloys 287 9.3. Transport properties of noncrystalline metals and alloys 297 References 317 Index 331 Title