Quantum Theory of Materials

667 Pages·2019·16.992 MB·English

by  Efthimios Kaxiras, John D. Joannopoulos

Quantum Theory of Materials
This accessible new text introduces the theoretical concepts and tools essential for
graduate-level courses on the physics of materials in condensed matter physics, physical
chemistry, materials science and engineering, and chemical engineering.
Topics covered range from fundamentals such as crystal periodicity and symmetry, and
derivation of single-particle equations, to modern additions including graphene, two-
dimensional solids, carbon nanotubes, topological states, and Hall physics. Advanced
topics such as phonon interactions with phonons, photons, and electrons, and magnetism,
are presented in an accessible way, and a set of appendices reviewing crucial fundamental
physics and mathematical tools makes this text suitable for students from a range of
backgrounds.
Students will benefit from the emphasis on translating theory into practice, with worked
examples explaining experimental observations, applications illustrating how theoretical
concepts can be applied to real research problems, and 242 informative full-color illustra-
tions. End-of-chapter problems are included for homework and self-study, with solutions
and lecture slides for instructors available online.
EfthimiosKaxiras
is the John Hasbrouck Van Vleck Professor of Pure and Applied Physics at
Harvard University. He holds joint appointments in the Department of Physics and the
School of Engineering and Applied Sciences, and is an affiliate of the Department of
Chemistry and Chemical Biology. He is the Founding Director of the Institute for Applied
Computational Science, a Fellow of the American Physical Society, and a Chartered
Physicist and Fellow of the Institute of Physics, London.
John D. Joannopoulos
is the Francis Wright Davis Professor of Physics at MIT, where he is
Director of the Institute for Soldier Nanotechnologies. He is a member of the National
Academy of Sciences and American Academy of Arts and Sciences, a Fellow of the
American Association for the Advancement of Science, a Fellow of the American Physical
Society, and a Fellow of the World Technology Network. His awards include the MIT
School of Science Graduate Teaching Award (1991), the William Buechner Teaching Prize
of the MIT Department of Physics (1996), the David Adler Award (1997) and Aneesur
Rahman Prize (2015) of the American Physical Society, and the Max Born Award of the
Optical Society of America.
Quantum Theory of Materials
EFTHIMIOS KAXIRAS
Harvard University
JOHN D. JOANNOPOULOS
Massachusetts Institute of Technology
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www.cambridge.org
Information on this title: www.cambridge.org/9780521117111
DOI: 10.1017/9781139030809
© Efthimios Kaxiras and John D. Joannopoulos 2019
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published 2019
Printed in Singapore by Markono Print Media Pte Ltd
A catalogue record for this publication is available from the British Library .
ISBN 978-0-521-11711-1 Hardback
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accurate or appropriate.
Contents
List of Figures page x
List of Tables xviii
Preface xxi
Acknowledgments xxiii
1 From Atoms to Solids
1
1.1 Electronic Structure of Atoms 2
1.2 Forming Bonds Between Atoms 5
1.2.1 The Essence of Metallic Bonding: The Free-Electron Model 6
1.2.2 The Essence of Covalent Bonding 10
1.2.3 Other Types of Bonding in Solids 14
1.3 The Architecture of Crystals 15
1.3.1 Atoms with No Valence Electrons 16
1.3.2 Atoms withs Valence Electrons 21
1.3.3 Atoms withs andp Valence Electrons 23
1.3.4 Atoms withs andd Valence Electrons 32
1.3.5 Atoms withs,d, and f Valence Electrons 33
1.3.6 Solids with Two Types of Atoms 33
1.3.7 Hydrogen: A Special One-s-Valence-Electron Atom 37
1.3.8 Solids with More Than Two Types of Atoms 39
1.4 Bonding in Solids 41
Further Reading 45
Problems 46
2 Electrons in Crystals: Translational Periodicity
52
2.1 Translational Periodicity: Bloch States 52
2.2 Reciprocal Space: Brillouin Zones 59
k
2.2.1 Nature of Wave-Vector 59
2.2.2 Brillouin Zones and Bragg Planes 60
2.2.3 Periodicity in Reciprocal Space 65
2.2.4 Symmetries Beyond Translational Periodicity 66
2.3 The Free-Electron and Nearly Free-Electron Models 69
k·p
2.4 Effective Mass, “ ” Perturbation Theory 75
2.5 The Tight-Binding Approximation 76
2.5.1 Generalizations of the TBA 96
2.6 General Band-Structure Methods 99
v
vi Contents
2.6.1 Crystal Pseudopotentials 103
2.7 Localized Wannier Functions 105
2.8 Density of States 107
2.8.1 Free-Electron Density of States 110
2.8.2 Local Density of States 111
2.8.3 Crystal DOS: Van Hove Singularities 112
Further Reading 115
Problems 116
3 Symmetries Beyond Translational Periodicity
119
3.1 Time-Reversal Symmetry for Spinless Fermions 119
3.2 Crystal Groups: Definitions 121
3.3 Symmetries of 3D Crystals 124
3.4 Symmetries of the Band Structure 129
k
3.5 Application: Special -Points 135
3.6 Group Representations 137
3.7 Application: The N-V-Center in Diamond 155
Further Reading 159
Problems 159
4 From Many Particles to the Single-Particle Picture
161
4.1 The Hamiltonian of the Solid 162
4.1.1 Born–Oppenheimer Approximation 163
4.2 The Hydrogen Molecule 166
4.3 The Hartree and Hartree–Fock Approximations 173
4.3.1 The Hartree Approximation 173
4.3.2 The Hartree–Fock Approximation 176
4.4 Hartree–Fock Theory of Free Electrons 178
4.5 Density Functional Theory 182
4.5.1 Thomas–Fermi–Dirac Theory 182
4.5.2 General Formulation of DFT 183
4.5.3 Single-Particle Equations in DFT 186
4.5.4 The Exchange–Correlation Term in DFT 189
4.5.5 Time-Dependent DFT 193
4.6 Quasiparticles and Collective Excitations 195
4.7 Screening: The Thomas–Fermi Model 197
4.8 Quasiparticle Energies: GW Approximation 199
4.9 The Pseudopotential 202
4.10 Energetics and Ion Dynamics 208
4.10.1 The Total Energy 208
4.10.2 Forces and Ion Dynamics 218
Further Reading 221
Problems 222
vii Contents
5 Electronic Properties of Crystals
227
5.1 Band Structure of Idealized 1D Solids 227
5.1.1 A Finite “1D Solid”: Benzene 227
5.1.2 An Infinite “1D Solid”: Polyacetylene 231
5.2 2D Solids: Graphene and Beyond 235
5.2.1 Carbon Nanotubes 239
5.3 3D Metallic Solids 244
5.4 3D Ionic and Covalent Solids 248
5.5 Doping of Ideal Crystals 256
5.5.1 Envelope Function Approximation 256
5.5.2 Effect of Doping in Semiconductors 261
5.5.3 The p–n Junction 265
5.5.4 Metal–Semiconductor Junction 273
Further Reading 276
Problems 276
6 Electronic Excitations
280
6.1 Optical Excitations 280
6.2 Conductivity and Dielectric Function 284
6.2.1 General Formulation 285
6.2.2 Drude and Lorentz Models 286
6.2.3 Connection to Microscopic Features 291
6.2.4 Implications for Crystals 293
6.2.5 Application: Optical Properties of Metals and Semiconductors 299
6.3 Excitons 302
6.3.1 General Considerations 303
6.3.2 Strongly Bound (Frenkel) Excitons 308
6.3.3 Weakly Bound (Wannier) Excitons 311
Further Reading 316
Problems 317
7 Lattice Vibrations and Deformations
319
7.1 Lattice Vibrations: Phonon Modes 319
7.2 The Born Force-Constant Model 324
7.3 Applications of the Force-Constant Model 328
7.4 Phonons as Harmonic Oscillators 340
7.5 Application: Specific Heat of Crystals 343
7.5.1 The Classical Picture 343
7.5.2 The Quantum-Mechanical Picture 344
7.5.3 The Debye Model 346
7.5.4 Thermal Expansion Coefficient 349
7.6 Application: Mo¨ssbauer Effect 350
7.7 Elastic Deformations of Solids 353
viii Contents
7.7.1 Phenomenological Models of Solid Deformation 354
7.7.2 Elasticity Theory: The Strain and Stress Tensors 356
7.7.3 Strain Energy Density 361
7.7.4 Isotropic Solid 363
7.7.5 Solid with Cubic Symmetry 368
7.7.6 Thin Plate Equilibrium 369
7.8 Application: Phonons of Graphene 371
Further Reading 375
Problems 375
8 Phonon Interactions
379
8.1 Phonon Scattering Processes 379
8.1.1 Scattering Formalism 381
8.2 Application: The Debye–Waller Factor 385
8.3 Phonon–Photon Interactions 387
8.3.1 Infrared Absorption 389
8.3.2 Raman Scattering 390
8.4 Phonon–Electron Interactions: Superconductivity 392
8.4.1 BCS Theory of Superconductivity 394
8.4.2 The McMillan Formula forTc 411
8.4.3 High-Temperature Superconductors 412
Further Reading 415
Problems 415
9 Dynamics and Topological Constraints
417
9.1 Electrons in External Electromagnetic Fields 417
9.1.1 Classical Hall Effect 419
9.1.2 Landau Levels 421
9.1.3 Quantum Hall Effect 424
9.1.4 de Haas–van Alphen Effect 431
9.2 Dynamics of Crystal Electrons: Single-Band Picture 434
9.3 Time-Reversal Invariance 438
9.3.1 Kramers Degeneracy 440
9.4 Berry’s Phase 440
9.4.1 General Formulation 441
9.4.2 Berry’s Phase for Electrons in Crystals 446
9.5 Applications of Berry’s Phase 454
9.5.1 Aharonov–Bohm Effect 454
9.5.2 Polarization of Crystals 456
9.5.3 Crystal Electrons in Uniform Electric Field 458
9.6 Chern Numbers 459
9.7 Broken Symmetry and Edge States 463
9.7.1 Broken Symmetry in Honeycomb Lattice 463
9.7.2 Edge States of Honeycomb Lattice 465
9.8 Topological Constraints 469
ix Contents
Further Reading 476
Problems 476
10 Magnetic Behavior of Solids
480
10.1 Overview of Magnetic Behavior of Insulators 481
10.2 Overview of Magnetic Behavior of Metals 486
10.2.1 Free Fermions in Magnetic Field: Pauli Paramagnetism 487
10.2.2 Magnetization in Hartree–Fock Free-Electron Model 490
10.2.3 Magnetization of Band Electrons 493
10.3 Classical Spins: Simple Models on a Lattice 497
10.3.1 Non-interacting Spins on a Lattice: Negative Temperature 497
10.3.2 Interacting Spins on a Lattice: Ising Model 502
10.4 Quantum Spins: Heisenberg Model 509
10.4.1 Motivation of the Heisenberg Model 510
10.4.2 Ground State of Heisenberg Ferromagnet 514
10.4.3 Spin Waves in Heisenberg Ferromagnet 516
10.4.4 Heisenberg Antiferromagnetic Spin Model 520
10.5 Magnetic Domains 522
Further Reading 525
Problems 525
Appendices
529
Appendix A Mathematical Tools 531
Appendix B Classical Electrodynamics 549
Appendix C Quantum Mechanics 565
Appendix D Thermodynamics and Statistical Mechanics 610
Index 646
List of Figures
1.1 Different forms of carbon-based solids page 2
1.2 Filling of electronic shells for the elements in the Periodic Table with atomic
numbers Z =1–18 3
1.3 Filling of electronic shells for elements with atomic numbersZ = 19–36 4
1.4 Periodic behavior of the properties of the elements 4
1.5 Schematic representation of the core and valence electrons in an atom 5
1.6 Illustration of metallic bonding between many atoms 6
1.7 A simple 1D model of the energy levels associated with the free-electron
model for a system of six “atoms” 9
1.8 Schematic representation of the formation of a bond between two atoms with
one valence electron each 10
+ −
1.9 Symmetric( ), left panel, and antisymmetric( ), right panel, linear
combinations of single-particle orbitals 12
1.10 Illustration of covalent bonding between many atoms 14
1.11 Shapes of the unit cells in some lattices that appear in the Periodic Table 16
1.12 Illustration of the face centered cubic (FCC) crystal structure 17
1.13 Illustration of the hexagonal close packed (HCP) crystal structure 20
1.14 Illustration of two interpenetrating FCC or HCP lattices 20
1.15 Three different views of the regular icosahedron 21
1.16 Illustration of the body centered cubic (BCC) crystal structure 22
1.17 Representation of the character ofs,p,d atomic orbitals 23
1.18 Illustration of covalent bonding in a single plane of graphite, called
“graphene” 25
1.19 Illustration of atomic arrangement in graphite in a top view (left) and a
perspective view (right) 27
1.20 The structure of the C molecule or “buckyball” (left) and of carbon
60
nanotubes (right) 27
1.21 Illustration of covalent bonding in the diamond lattice 29
1.22 The diamond crystal 30
1.23 The structure of the diamond lattice as revealed by the transmission electron
microscope 30
1.24 Typical layer structures that group-V elements form 31
1.25 Illustration of cubic crystals formed by ionic solids 34
1.26 The zincblende lattice in which every atom is surrounded by four neighbors
of the opposite type 34
x