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Lecture Notes on Condensed Matter Physics (A Work in Progress)

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Lecture Notes on Condensed Matter Physics (A Work in Progress) 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.
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