Lecture Notes on Condensed Matter Physics (A Work in Progress) Daniel Arovas Department of Physics University of California, San Diego March 14, 2010 Contents 0.1 Preface . vi 0 Introductory Information 1 0.1 References . 2 1 Boltzmann Transport 5 1.1 References . 5 1.2 Introduction . 5 1.3 Boltzmann Equation in Solids . 6 1.3.1 Semiclassical Dynamics and Distribution Functions . 6 1.3.2 Local Equilibrium . 10 1.4 Conductivity of Normal Metals . 11 1.4.1 Relaxation Time Approximation . 11 1.4.2 Optical Reflectivity of Metals and Semiconductors . 14 1.4.3 Optical Conductivity of Semiconductors . 16 1.4.4 Optical Conductivity and the Fermi Surface . 18 1.5 Calculation of the Scattering Time . 19 1.5.1 Potential Scattering and Fermi's Golden Rule . 19 1.5.2 Screening and the Transport Lifetime . 23 1.6 Boltzmann Equation for Holes . 25 1.6.1 Properties of Holes . 25 1.7 Magnetoresistance and Hall Effect . 28 i ii CONTENTS 1.7.1 Boltzmann Theory for ραβ(!; B) ................... 28 1.7.2 Cyclotron Resonance in Semiconductors . 31 1.7.3 Magnetoresistance: Two-Band Model . 32 1.7.4 Hall Effect in High Fields . 34 1.8 Thermal Transport . 36 1.8.1 Boltzmann Theory . 36 1.8.2 The Heat Equation . 40 1.8.3 Calculation of Transport Coefficients . 41 1.8.4 Onsager Relations . 43 1.9 Electron-Phonon Scattering . 45 1.9.1 Introductory Remarks . 45 1.9.2 Electron-Phonon Interaction . 45 1.9.3 Boltzmann Equation for Electron-Phonon Scattering . 48 2 Mesoscopia 51 2.1 References . 51 2.2 Introduction . 51 2.3 The Landauer Formula . 51 2.3.1 Example: Potential Step . 54 2.4 Multichannel Systems . 56 2.4.1 Transfer Matrices: The Pichard Formula . 60 2.4.2 Discussion of the Pichard Formula . 62 2.4.3 Two Quantum Resistors in Series . 64 2.4.4 Two Quantum Resistors in Parallel . 67 2.5 Universal Conductance Fluctuations in Dirty Metals . 75 2.5.1 Weak Localization . 78 2.6 Anderson Localization . 80 2.6.1 Characterization of Localized and Extended States . 82 2.6.2 Numerical Studies of the Localization Transition . 83 CONTENTS iii 2.6.3 Scaling Theory of Localization . 85 2.6.4 Finite Temperature . 89 3 Linear Response Theory 91 3.1 Response and Resonance . 91 3.1.1 Energy Dissipation . 93 3.2 Kramers-Kronig Relations . 93 3.3 Quantum Mechanical Response Functions . 95 3.3.1 Spectral Representation . 97 3.3.2 Energy Dissipation . 99 3.3.3 Correlation Functions . 100 3.3.4 Continuous Systems . 101 1 3.4 Example: S = 2 Object in a Magnetic Field . 101 3.4.1 Bloch Equations . 102 3.5 Electromagnetic Response . 104 3.5.1 Gauge Invariance and Charge Conservation . 106 3.5.2 A Sum Rule . 106 3.5.3 Longitudinal and Transverse Response . 107 3.5.4 Neutral Systems . 107 3.5.5 The Meissner Effect and Superfluid Density . 108 3.6 Density-Density Correlations . 110 3.6.1 Sum Rules . 112 3.7 Dynamic Structure Factor for the Electron Gas . 113 3.7.1 Explicit T = 0 Calculation . 114 3.8 Charged Systems: Screening and Dielectric Response . 119 3.8.1 Definition of the Charge Response Functions . 119 3.8.2 Static Screening: Thomas-Fermi Approximation . 120 3.8.3 High Frequency Behavior of (q;!) . 121 3.8.4 Random Phase Approximation (RPA) . 122 iv CONTENTS 3.8.5 Plasmons . 125 4 Magnetism 127 4.1 References . 127 4.2 Introduction . 127 4.2.1 Absence of Orbital Magnetism within Classical Physics . 129 4.3 Basic Atomic Physics . 129 4.3.1 Single electron Hamiltonian . 129 4.3.2 The Darwin Term . 130 4.3.3 Many electron Hamiltonian . 130 4.3.4 The Periodic Table . 132 4.3.5 Splitting of Configurations: Hund's Rules . 133 4.3.6 Spin-Orbit Interaction . 135 4.3.7 Crystal Field Splittings . 136 4.4 Magnetic Susceptibility of Atomic and Ionic Systems . 137 4.4.1 Filled Shells: Larmor Diamagnetism . 138 4.4.2 Partially Filled Shells: van Vleck Paramagnetism . 139 4.5 Itinerant Magnetism of Noninteracting Systems . 141 4.5.1 Pauli Paramagnetism . 141 4.5.2 Landau Diamagnetism . 143 4.6 Moment Formation in Interacting Itinerant Systems . 145 4.6.1 The Hubbard Model . 145 4.6.2 Stoner Mean Field Theory . 146 4.6.3 Antiferromagnetic Solution . 150 4.6.4 Mean Field Phase Diagram of the Hubbard Model . 151 4.7 Interaction of Local Moments: the Heisenberg Model . 153 4.7.1 Ferromagnetic Exchange of Orthogonal Orbitals . 153 4.7.2 Heitler-London Theory of the H2 Molecule . 155 4.7.3 Failure of Heitler-London Theory . 157 CONTENTS v 4.7.4 Herring's approach . 157 4.8 Mean Field Theory . 158 4.8.1 Ferromagnets . 161 4.8.2 Antiferromagnets . 161 4.8.3 Susceptibility . 162 4.8.4 Variational Probability Distribution . 163 4.9 Magnetic Ordering . 166 4.9.1 Mean Field Theory of Anisotropic Magnetic Systems . 168 4.9.2 Quasi-1D Chains . 169 4.10 Spin Wave Theory . 170 4.10.1 Ferromagnetic Spin Waves . 171 4.10.2 Static Correlations in the Ferromagnet . 173 4.10.3 Antiferromagnetic Spin Waves . 173 4.10.4 Specific Heat due to Spin Waves . 178 vi CONTENTS 0.1 Preface This is a proto-preface. A more complete preface will be written after these notes are completed. These lecture notes are intended to supplement a graduate level course in condensed matter physics. Chapter 0 Introductory Information Instructor: Daniel Arovas Contact : Mayer Hall 5671 / 534-6323 / [email protected] Lectures: Tu Th / 9:30 am - 10:50 am / Mayer Hall 5301 Office Hours: W 2:00 pm - 3:30 pm / Mayer Hall 5671 A strong emphasis of this class will be on learning how to calculate. I plan to cover the following topics this quarter: Transport: Boltzmann equation, transport coefficients, cyclotron resonance, magnetore- sistance, thermal transport, electron-phonon scattering Mesoscopic Physics: Landauer formula, conductance fluctuations, Aharonov-Bohm ef- fect, disorder, weak localization, Anderson localization Magnetism: Weak vs. strong, local vs. itinerant, Hubbard and Heisenberg models, spin wave theory, magnetic ordering, Kondo effect Other: Linear response theory, Fermi liquid theory (time permitting) There will be about four assignments and a take-home final examination. I will be following my own notes, which are available from the course web site. 1 2 CHAPTER 0. INTRODUCTORY INFORMATION 0.1 References • D. Feng and G. Jin, Introduction to Condensed Matter Physics (I) (World Scientific, Singapore, 2005) New and with a distinctly modern flavor and set of topics. Looks good. • N, Ashcroft and N. D. Mermin, Solid State Physics (Saunders College Press, Philadelphia, 1976) Beautifully written, this classic text is still one of the best comprehensive guides. • M. Marder, Condensed Matter Physics (John Wiley & Sons, New York, 2000) A thorough and advanced level treatment of transport theory in gases, metals, semi- conductors, insulators, and superconductors. • D. Pines, Elementary Excitations in Solids (Perseus, New York, 1999) An advanced level text on the quantum theory of solids, treating phonons, electrons, plasmons, and photons. • P. L. Taylor and O. Heinonen, A Quantum Approach to Condensed Matter Physics (Cambridge University Press, New York, 2002) A modern, intermediate level treatment of the quantum theory of solids. • J. M. Ziman, Principles of the Theory of Solids (Cambridge University Press, New York, 1979). A classic text on solid state physics. Very readable. 0.1. REFERENCES 3 • C. Kittel, Quantum Theory of Solids (John Wiley & Sons, New York, 1963) A graduate level text with several detailed derivations. • H. Smith and H. H. Jensen, Transport Phenomena (Oxford University Press, New York, 1989). A detailed and lucid account of transport.