Tai ngay!!! Ban co the xoa dong c This page intentionally left blank Advanced Condensed Matter Physics This graduate textbook includes coverage of important topics that are not commonly featured in other textbooks on condensed matter physics, such as treatments of surfaces, the quan- tum Hall effect, and superfluidity. It avoids complex formalism, such as Green’s functions, which can obscure the underlying physics, and instead emphasizes fundamental physical reasoning. Intended for classroom use, it features plenty of references and extensive prob- lems for solution based on the author’s many years of teaching in the Physics Department at the University of Michigan. This textbook is suitable for physics, chemistry and engineer- ing graduate students, and as a reference for research students in condensed matter physics.
Engineering students will find the treatment of the fundamentals of semiconductor devices and the optics of solids of particular interest. Sander is Professor of Physics at the University of Michigan. His research interests are in theoretical condensed matter physics and non-equilibrium statistical physics, especially the study of growth patterns. Advanced Condensed Matter Physics Leonard M.
Sander Department of Physics, The University of Michigan CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.org Information on this title: www. Sander 2009 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2009 ISBN-13 978-0-511-50679-6 eBook (EBL) ISBN-13 978-0-521-87290-4 hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
To Mae & Evelyn Contents Preface page xi 1 The nature of condensed matter 1 1.1 Some basic orders of magnitude 1 1.2 Quantum or classical 3 1.4 The exchange interaction 5 Suggested reading 6 Problems 6 2 Order and disorder 8 2.3 Other ordered states 21 2.5 Disordered condensed matter 22 Suggested reading 23 Problems 23 3 Crystals, scattering, and correlations 25 3.2 Fourier analysis and the reciprocal lattice 32 3.4 Correlation functions 46 Suggested reading 50 Problems 51 4 Surfaces and crystal growth 53 4.1 Observing surfaces: scanning tunneling microscopy 53 4.2 Surfaces and surface tension 54 4.4 Equilibrium crystal shapes 62 4.5 Crystal growth 64 Suggested reading 71 Problems 71 viii Contents 5 Classical and quantum waves 73 5.1 Lattice vibrations and phonons 73 5.2 Spin waves and magnons 102 5.5 Two dimensions 111 Suggested reading 112 Problems 112 6 The non-interacting electron model 114 6.2 Thermally excited states and heat capacity 120 6.3 Band theory 122 Suggested reading 135 Problems 135 7 Dynamics of non-interacting electrons 139 7.2 Transport in Sommerfeld theory 141 7.3 Semiclassical theory of transport 143 7.4 Scattering and the Boltzmann equation 146 7.5 Donors and acceptors in semiconductors 151 7.8 Large magnetic fields 156 Suggested reading 168 Problems 169 8 Dielectric and optical properties 172 8.2 The fluctuation-dissipation theorem 174 8.3 Self-consistent response 177 8.4 The RPA dielectric function 181 8.5 Optical properties of crystals 187 Suggested reading 189 Problems 189 9 Electron interactions 193 9.1 Fermi liquid theory 193 9.2 Many-electron atoms 198 9.3 Metals in the Hartree–Fock approximation 202 9.4 Correlation energy of jellium 205 9.5 Inhomogeneous electron systems 210 9.6 Electrons and phonons 216 ix Contents 9.7 Strong interactions and magnetism in metals 220 Suggested reading 224 Problems 224 10 Superfluidity and superconductivity 226 10.1 Bose–Einstein condensation and superfluidity 227 10.5 Ginsburg–Landau theory 253 10.6 Josephson effect 259 Suggested reading 261 Problems 261 References 263 Index 269 Preface This book is intended as a textbook for a graduate course in condensed matter physics. It is based on many years’ experience in teaching in the Physics department at The University of Michigan. The material here is more than enough for a one-semester course. Usually I teach two semesters, and in the second, I add material such as the renormalization group.
In this book advanced techniques such as Green’s functions are not used. I have tried to introduce as many of the concepts of modern condensed matter physics as I could without them. As a result, some topics that are of central importance in modern research do not appear. The problems are an integral part of the book.
Some concepts that are used in later chapters are introduced as problems. Students are expected to have a good background in statistical physics, non-relativistic quantum theory, and, ideally, know undergraduate Solid State physics at the level of Kittel (2005). I decided to write this book as a result of coming back to teaching Condensed Matter after a number of years covering other subjects. I had hoped to find a substitute for the grand old standards like Ziman (1972) or Ashcroft & Mermin (1976) which I used at the beginning of my teaching career.
Though there are newer texts that are interesting in many ways, I found that none of them quite fit my needs as an instructor. It is for the reader to decide how well I have succeeded in giving a modern alternative to the classics – they are very hard acts to follow. Many people have helped me in writing this book. Craig Davis and Cagilyan Kurdak have been remarkably generous with their time, and found many errors.
Jim Allen and Michal Zochowski have given valuable advice. I would like to particularly thank Brad Orr, Andy Dougherty, Dave Weitz, Jim Allen, Roy Clarke, and Meigan Aronson for figures. And, of course, my students have given invaluable feedback over more than three decades. 1 The nature of condensed matter Condensed matter physics is the study of large numbers of atoms and molecules that are “stuck together.” Solids and liquids are examples.
In the condensed state many molecules interact with each other. The physics of such a system is quite different from that of the individual molecules because of collective effects: qualitatively new things happen because there are many interacting particles. The behavior of most of the objects in our everyday experience is dominated by collective effects. Examples of materials where such effects are important are crystals and magnets.
This is a vast field: the subject matter could be taken to include traditional solid state physics (basically the study of the quantum mechanics of crystalline matter), magnetism, fluid dynamics, elasticity theory, the physics of materials, aspects of polymer science, and some biophysics. In fact, condensed matter is less a field than a collection of fields with some overlapping tools and techniques. Any course in this area must make choices. This is my personal choice.
In this chapter I will discuss orders of magnitude that are important, review ideas from quantum mechanics and chemistry that we will need, outline what holds condensed matter together, and discuss how order arises in condensed systems. The discussion here will be qualitative. Later chapters will fill in the details.1 Some basic orders of magnitude To fix our ideas, consider a typical bit of condensed matter, a macroscopic piece of solid copper metal. As we will see later it is best to view the system as a collection of cuprous (Cu+ ) ions and conduction electrons, one per atom, that are free to move within the metal.
We discuss some basic scales that will be important for understanding the physics of this piece of matter. Lengths A characteristic length that will be important is the distance between the Cu atoms. In a solid this distance will be of order of a chemical bond length: L ≈ 3 Å ≈ 3 × 10−8 cm.1) Note that this is very tiny on the macroscopic scale. The whole art of condensed matter physics consists in bridging the gap between the atomic scale and the macroscopic properties of condensed matter.
2 The nature of condensed matter Energies We can ask about the characteristic energy scales for the sample. One important energy scale is the binding energy of the material per atom. A closely related quantity is the melting temperature in energy units: 1357 K = 0.2) This is a typical scale to break up the material. If we probe at much larger energies (KeV, for example) we will be probing the inner shells of Cu, namely the domain of atomic physics, or at MeV, the Cu nucleus, i.
Cu has an interesting color (it is copper colored, in fact), so we might expect something interesting at the scale of the energy of ordinary light, namely, E ≈ ωopt ∼ 3 eV (1.3) which is also the strength of a typical chemical bond. A somewhat larger, but comparable scale is that of the Coulomb interaction of two electrons a distance L apart: E ≈ e2 /L ≈ 5 eV.4) These energies are low even for atomic physics. This means that in our study of condensed matter we will always be interested only in the outer (valence electrons) which are least bound. Speeds When a piece of Cu carries an electrical current of density, j, the conduction electrons move at a drift velocity vd : j = ne vd (1.5) where n is the number density of conduction electrons and e is the charge on the electron.
For ordinary sized currents we find a very small speed, vd ≈ 0. There is another characteristic speed, the mean thermal speed, vT of the Cu ions when they vibrate at finite temperature. We estimate vT as follows. From the Boltzmann equipartition theorem the mean kinetic energy of an ion is: MvT2 /2 ∼ kB T .6) Here T is the absolute temperature, kB is Boltzmann’s constant, M is the mass of a Cu ion, and vT is the mean thermal velocity.
At room temperature we get vT ∼ 3 × 104 cm/sec. There is a larger speed associated with the electrons, namely the quantum mechanical speed of the valence electrons. We estimate this speed as [frequency of an optical transition] x length: v ∼ (E/)(L) ≈ 107 cm/sec.7) As we will see below, there is another relevant speed, the magnitude of the Fermi velocity, which is of the same order.3 Chemical bonds In any case, all of these speeds are small compared to the speed of light. Thus, we seldom need the theory of relativity in condensed matter physics.
(An exception is the spin-orbit interaction of heavy elements.) Large numbers and collective effects The essential point of the subject is that we deal with very large numbers of ions and electrons, ≈ 1027 in a macroscopic sample. In a famous essay P. Anderson (1972) pointed out the significance of this fact. When many things interact we often generate new phenomena, sometimes called emergent phenomena.
Or, as Anderson put it, “more is different.” Some examples of collective effects that we will emphasize in this book are the existence of order of various types, e. crystalline order, magnetic order, and superconducting order.2 Quantum or classical We have seen that we are interested in non-relativistic physics. We can go further: for the case of Cu there are conduction electrons and Cu+ ions. What type of physics is applicable to each? In particular, do we need quantum mechanics? A useful criterion is to compare the de Broglie wavelength of the relevant particle, λ = h/mv, to the interparticle spacing.
For the ions, the relevant speed is vT which we estimated above.8) This is smaller than the spacing by an order of magnitude. For all ions in solids (except for He and H at very low temperatures) we can use classical mechanics. (As we will see, for vibrations of ions at low T , we need quantum mechanics too.