Condensed Matter Physics I

Peter S. Riseborough

February 5, 2015

Contents

1 Introduction 101.1 The Born-Oppenheimer Approximation . . . . . . . . . . . . . . 10

2 Crystallography 182.0.1 Exercise 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Structures 203.1 Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2 Crystalline Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3 The Direct Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3.1 Primitive Unit Cells . . . . . . . . . . . . . . . . . . . . . 303.3.2 The Wigner-Seitz Unit Cell . . . . . . . . . . . . . . . . . 31

3.4 Symmetry of Crystals . . . . . . . . . . . . . . . . . . . . . . . . 333.4.1 Symmetry Groups . . . . . . . . . . . . . . . . . . . . . . 343.4.2 Group Multiplication Tables . . . . . . . . . . . . . . . . 353.4.3 Point Group Operations . . . . . . . . . . . . . . . . . . . 373.4.4 Exercise 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.4.5 Limitations Imposed by Translational Symmetry . . . . . 413.4.6 Exercise 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.4.7 Point Group Nomenclature . . . . . . . . . . . . . . . . . 44

3.5 Bravais Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.5.1 Exercise 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.5.2 Cubic Bravais Lattices. . . . . . . . . . . . . . . . . . . . 533.5.3 Tetragonal Bravais Lattices. . . . . . . . . . . . . . . . . . 583.5.4 Orthorhombic Bravais Lattices. . . . . . . . . . . . . . . . 593.5.5 Monoclinic Bravais Lattice. . . . . . . . . . . . . . . . . . 603.5.6 Triclinic Bravais Lattice. . . . . . . . . . . . . . . . . . . . 613.5.7 Trigonal Bravais Lattice. . . . . . . . . . . . . . . . . . . . 633.5.8 Hexagonal Bravais Lattice. . . . . . . . . . . . . . . . . . 663.5.9 Exercise 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.6 Point Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723.6.1 Exercise 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.7 Space Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

1

3.7.1 Exercise 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.8 Crystal Structures with Bases. . . . . . . . . . . . . . . . . . . . 78

3.8.1 Diamond Structure . . . . . . . . . . . . . . . . . . . . . . 783.8.2 Exercise 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.8.3 Graphite Structure . . . . . . . . . . . . . . . . . . . . . . 793.8.4 Exercise 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . 823.8.5 Hexagonal Close-Packed Structure . . . . . . . . . . . . . 833.8.6 Exercise 10 . . . . . . . . . . . . . . . . . . . . . . . . . . 863.8.7 Exercise 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 863.8.8 Other Close-Packed Structures . . . . . . . . . . . . . . . 863.8.9 Sodium Chloride Structure . . . . . . . . . . . . . . . . . 883.8.10 Cesium Chloride Structure . . . . . . . . . . . . . . . . . 903.8.11 Fluorite Structure . . . . . . . . . . . . . . . . . . . . . . 923.8.12 The Copper Three Gold Structure . . . . . . . . . . . . . 933.8.13 Rutile Structure . . . . . . . . . . . . . . . . . . . . . . . 943.8.14 Exercise 12 . . . . . . . . . . . . . . . . . . . . . . . . . . 953.8.15 Zinc Blende Structure . . . . . . . . . . . . . . . . . . . . 953.8.16 Zincite Structure . . . . . . . . . . . . . . . . . . . . . . . 963.8.17 Exercise 13 . . . . . . . . . . . . . . . . . . . . . . . . . . 983.8.18 The Perovskite Structure . . . . . . . . . . . . . . . . . . 983.8.19 Exercise 14 . . . . . . . . . . . . . . . . . . . . . . . . . . 99

3.9 Lattice Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003.9.1 Exercise 15 . . . . . . . . . . . . . . . . . . . . . . . . . . 1033.9.2 Exercise 16 . . . . . . . . . . . . . . . . . . . . . . . . . . 1033.9.3 Exercise 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 104

3.10 Quasi-Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4 Structure Determination 1124.1 X Ray Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.1.1 The Bragg condition . . . . . . . . . . . . . . . . . . . . . 1124.1.2 The Laue conditions . . . . . . . . . . . . . . . . . . . . . 1144.1.3 Equivalence of the Bragg and Laue conditions . . . . . . . 1184.1.4 The Ewald Construction . . . . . . . . . . . . . . . . . . . 1194.1.5 X-ray Techniques . . . . . . . . . . . . . . . . . . . . . . . 1204.1.6 Exercise 18 . . . . . . . . . . . . . . . . . . . . . . . . . . 1234.1.7 The Structure and Form Factors . . . . . . . . . . . . . . 1254.1.8 Exercise 19 . . . . . . . . . . . . . . . . . . . . . . . . . . 1334.1.9 Exercise 20 . . . . . . . . . . . . . . . . . . . . . . . . . . 1344.1.10 Exercise 21 . . . . . . . . . . . . . . . . . . . . . . . . . . 1364.1.11 Exercise 22 . . . . . . . . . . . . . . . . . . . . . . . . . . 1364.1.12 Exercise 23 . . . . . . . . . . . . . . . . . . . . . . . . . . 1374.1.13 Exercise 24 . . . . . . . . . . . . . . . . . . . . . . . . . . 1374.1.14 Exercise 25 . . . . . . . . . . . . . . . . . . . . . . . . . . 140

4.2 Neutron Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . 1414.2.1 Exercise 26 . . . . . . . . . . . . . . . . . . . . . . . . . . 143

4.3 Theory of the Differential Scattering Cross-section . . . . . . . . 145

2

4.3.1 Time Dependent Perturbation Theory . . . . . . . . . . . 1464.3.2 The Fermi Golden Rule . . . . . . . . . . . . . . . . . . . 1484.3.3 The Elastic Scattering Cross-Section . . . . . . . . . . . . 1504.3.4 The Condition for Coherent Scattering . . . . . . . . . . . 1524.3.5 Exercise 27 . . . . . . . . . . . . . . . . . . . . . . . . . . 1554.3.6 Exercise 28 . . . . . . . . . . . . . . . . . . . . . . . . . . 1554.3.7 Exercise 29 . . . . . . . . . . . . . . . . . . . . . . . . . . 1564.3.8 Anti-Domain Phase Boundaries . . . . . . . . . . . . . . . 1564.3.9 Exercise 30 . . . . . . . . . . . . . . . . . . . . . . . . . . 157

4.4 Elastic Scattering from Quasi-Crystals . . . . . . . . . . . . . . . 1584.5 Elastic Scattering from a Fluid . . . . . . . . . . . . . . . . . . . 162

5 The Reciprocal Lattice 1645.0.1 Exercise 31 . . . . . . . . . . . . . . . . . . . . . . . . . . 165

5.1 The Reciprocal Lattice as a Dual Lattice . . . . . . . . . . . . . . 1655.1.1 Exercise 32 . . . . . . . . . . . . . . . . . . . . . . . . . . 169

5.2 Examples of Reciprocal Lattices . . . . . . . . . . . . . . . . . . . 1695.2.1 The Simple Cubic Reciprocal Lattice . . . . . . . . . . . . 1695.2.2 The Body Centered Cubic Reciprocal Lattice . . . . . . . 1695.2.3 The Face Centered Cubic Reciprocal Lattice . . . . . . . 1705.2.4 The Hexagonal Reciprocal Lattice . . . . . . . . . . . . . 1715.2.5 Exercise 33 . . . . . . . . . . . . . . . . . . . . . . . . . . 172

5.3 The Brillouin Zones . . . . . . . . . . . . . . . . . . . . . . . . . 1725.3.1 The Simple Cubic Brillouin Zone . . . . . . . . . . . . . . 1735.3.2 The Body Centered Cubic Brillouin Zone . . . . . . . . . 1755.3.3 The Face Centered Cubic Brillouin Zone . . . . . . . . . . 1775.3.4 The Hexagonal Brillouin Zone . . . . . . . . . . . . . . . . 1785.3.5 The Trigonal Brillouin Zone . . . . . . . . . . . . . . . . . 180

6 Electrons 183

7 Electronic States 1837.1 Many-Electron Wave Functions . . . . . . . . . . . . . . . . . . . 184

7.1.1 Exercise 34 . . . . . . . . . . . . . . . . . . . . . . . . . . 1907.2 Bloch’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1907.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 1957.4 Plane Wave Expansion of Bloch Functions . . . . . . . . . . . . . 1987.5 The Bloch Wave Vector . . . . . . . . . . . . . . . . . . . . . . . 2027.6 The Density of States . . . . . . . . . . . . . . . . . . . . . . . . 203

7.6.1 Exercise 35 . . . . . . . . . . . . . . . . . . . . . . . . . . 2067.7 The Fermi Surface . . . . . . . . . . . . . . . . . . . . . . . . . . 207

3

8 Approximate Models 2118.1 The Nearly-Free Electron Model . . . . . . . . . . . . . . . . . . 211

8.1.1 Perturbation Theory . . . . . . . . . . . . . . . . . . . . . 2128.1.2 Non-Degenerate Perturbation Theory . . . . . . . . . . . 2138.1.3 Degenerate Perturbation Theory . . . . . . . . . . . . . . 2158.1.4 Empty Lattice Approximation Band Structure . . . . . . 2228.1.5 Exercise 36 . . . . . . . . . . . . . . . . . . . . . . . . . . 2308.1.6 Degeneracies of the Bloch States . . . . . . . . . . . . . . 2318.1.7 Exercise 37 . . . . . . . . . . . . . . . . . . . . . . . . . . 2418.1.8 Exercise 38 . . . . . . . . . . . . . . . . . . . . . . . . . . 2428.1.9 Brillouin Zone Boundaries and Fermi Surfaces . . . . . . . 2428.1.10 The Geometric Structure Factor . . . . . . . . . . . . . . 2488.1.11 Exercise 39 . . . . . . . . . . . . . . . . . . . . . . . . . . 2588.1.12 Exercise 40 . . . . . . . . . . . . . . . . . . . . . . . . . . 2608.1.13 Exercise 41 . . . . . . . . . . . . . . . . . . . . . . . . . . 2618.1.14 Exercise 42 . . . . . . . . . . . . . . . . . . . . . . . . . . 2628.1.15 Exercise 43 . . . . . . . . . . . . . . . . . . . . . . . . . . 263

8.2 The Pseudo-Potential Method . . . . . . . . . . . . . . . . . . . . 2648.2.1 The Pseudo-Potential Theorem . . . . . . . . . . . . . . . 2678.2.2 The Cancellation Theorem . . . . . . . . . . . . . . . . . 2698.2.3 The Scattering Approach . . . . . . . . . . . . . . . . . . 2728.2.4 The Ziman-Lloyd Pseudo-potential . . . . . . . . . . . . . 2748.2.5 Exercise 44 . . . . . . . . . . . . . . . . . . . . . . . . . . 2768.2.6 Exercise 45 . . . . . . . . . . . . . . . . . . . . . . . . . . 2768.2.7 Exercise 46 . . . . . . . . . . . . . . . . . . . . . . . . . . 277

8.3 The Tight-Binding Model . . . . . . . . . . . . . . . . . . . . . . 2788.3.1 Tight-Binding s Band Metal . . . . . . . . . . . . . . . . . 2868.3.2 Tight-Binding Bands of Diamond Structured Semicon-

ductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2898.3.3 Exercise 47 . . . . . . . . . . . . . . . . . . . . . . . . . . 2938.3.4 Exercise 48 . . . . . . . . . . . . . . . . . . . . . . . . . . 2948.3.5 Exercise 49 . . . . . . . . . . . . . . . . . . . . . . . . . . 2968.3.6 Exercise 50 . . . . . . . . . . . . . . . . . . . . . . . . . . 2968.3.7 Exercise 51 . . . . . . . . . . . . . . . . . . . . . . . . . . 2978.3.8 Exercise 52 . . . . . . . . . . . . . . . . . . . . . . . . . . 2978.3.9 Wannier Functions . . . . . . . . . . . . . . . . . . . . . . 2978.3.10 Exercise 53 . . . . . . . . . . . . . . . . . . . . . . . . . . 3018.3.11 Exercise 54 . . . . . . . . . . . . . . . . . . . . . . . . . . 3018.3.12 Example of Tight-Binding: Graphene . . . . . . . . . . . 301

9 Electron-Electron Interactions 3079.1 The Landau Fermi Liquid . . . . . . . . . . . . . . . . . . . . . . 307

9.1.1 The Scattering Rate . . . . . . . . . . . . . . . . . . . . . 3099.1.2 The Quasi-Particle Energy . . . . . . . . . . . . . . . . . 3169.1.3 Exercise 55 . . . . . . . . . . . . . . . . . . . . . . . . . . 319

9.2 The Hartree-Fock Approximation . . . . . . . . . . . . . . . . . . 320

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9.2.1 The Free Electron Gas. . . . . . . . . . . . . . . . . . . . 3249.2.2 Exercise 56 . . . . . . . . . . . . . . . . . . . . . . . . . . 342

9.3 The Density Functional Method . . . . . . . . . . . . . . . . . . . 3449.3.1 Hohenberg-Kohn Theorem . . . . . . . . . . . . . . . . . . 3459.3.2 Functionals and Functional Derivatives . . . . . . . . . . 3469.3.3 The Variational Principle . . . . . . . . . . . . . . . . . . 3509.3.4 The Electrostatic Terms . . . . . . . . . . . . . . . . . . . 3529.3.5 The Kohn-Sham Equations . . . . . . . . . . . . . . . . . 3539.3.6 The Local Density Approximation . . . . . . . . . . . . . 355

9.4 Static Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3599.4.1 The Thomas-Fermi Approximation . . . . . . . . . . . . . 3619.4.2 Linear Response Theory . . . . . . . . . . . . . . . . . . . 3649.4.3 Density Functional Response Function . . . . . . . . . . . 3689.4.4 Exercise 57 . . . . . . . . . . . . . . . . . . . . . . . . . . 3719.4.5 Exercise 58 . . . . . . . . . . . . . . . . . . . . . . . . . . 372

10 Stability of Structures 37710.1 Momentum Space Representation . . . . . . . . . . . . . . . . . . 37710.2 Real Space Representation . . . . . . . . . . . . . . . . . . . . . . 384

11 Metals 39211.1 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 392

11.1.1 The Sommerfeld Expansion . . . . . . . . . . . . . . . . . 39311.1.2 The Specific Heat Capacity . . . . . . . . . . . . . . . . . 39511.1.3 Exercise 59 . . . . . . . . . . . . . . . . . . . . . . . . . . 39811.1.4 Exercise 60 . . . . . . . . . . . . . . . . . . . . . . . . . . 39811.1.5 Pauli Paramagnetism . . . . . . . . . . . . . . . . . . . . 39811.1.6 Exercise 61 . . . . . . . . . . . . . . . . . . . . . . . . . . 40211.1.7 Exercise 62 . . . . . . . . . . . . . . . . . . . . . . . . . . 40211.1.8 Landau Diamagnetism . . . . . . . . . . . . . . . . . . . . 40211.1.9 Landau Level Quantization . . . . . . . . . . . . . . . . . 40311.1.10The Diamagnetic Susceptibility . . . . . . . . . . . . . . . 405

11.2 Transport Properties . . . . . . . . . . . . . . . . . . . . . . . . . 40811.2.1 Electrical Conductivity . . . . . . . . . . . . . . . . . . . 40811.2.2 Scattering by Static Defects . . . . . . . . . . . . . . . . . 40811.2.3 Exercise 63 . . . . . . . . . . . . . . . . . . . . . . . . . . 41511.2.4 The Hall Effect and Magneto-resistance. . . . . . . . . . . 41611.2.5 Multi-band Models . . . . . . . . . . . . . . . . . . . . . . 424

11.3 Electromagnetic Properties of Metals . . . . . . . . . . . . . . . . 42711.3.1 The Longitudinal Response . . . . . . . . . . . . . . . . . 43011.3.2 Electron Scattering Experiments . . . . . . . . . . . . . . 44011.3.3 Exercise 64 . . . . . . . . . . . . . . . . . . . . . . . . . . 44511.3.4 Exercise 65 . . . . . . . . . . . . . . . . . . . . . . . . . . 44711.3.5 The Transverse Response . . . . . . . . . . . . . . . . . . 45211.3.6 Optical Experiments . . . . . . . . . . . . . . . . . . . . . 45711.3.7 Kramers-Kronig Relation . . . . . . . . . . . . . . . . . . 459

5

11.3.8 Exercise 66 . . . . . . . . . . . . . . . . . . . . . . . . . . 46011.3.9 Exercise 67 . . . . . . . . . . . . . . . . . . . . . . . . . . 46111.3.10The Drude Conductivity . . . . . . . . . . . . . . . . . . . 46211.3.11Exercise 68 . . . . . . . . . . . . . . . . . . . . . . . . . . 46611.3.12Exercise 69 . . . . . . . . . . . . . . . . . . . . . . . . . . 46811.3.13The Anomalous Skin Effect . . . . . . . . . . . . . . . . . 46811.3.14 Inter-Band Transitions . . . . . . . . . . . . . . . . . . . . 472

11.4 The Fermi Surface . . . . . . . . . . . . . . . . . . . . . . . . . . 47411.4.1 Semi-Classical Orbits . . . . . . . . . . . . . . . . . . . . 47411.4.2 de Haas - van Alphen Oscillations . . . . . . . . . . . . . 47811.4.3 Exercise 70 . . . . . . . . . . . . . . . . . . . . . . . . . . 48111.4.4 The Lifshitz-Kosevich Formulae . . . . . . . . . . . . . . . 48111.4.5 Geometric Resonances . . . . . . . . . . . . . . . . . . . . 48711.4.6 Cyclotron Resonances . . . . . . . . . . . . . . . . . . . . 489

11.5 The Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . . 49411.5.1 The Integer Quantum Hall Effect . . . . . . . . . . . . . . 49511.5.2 Exercise 71 . . . . . . . . . . . . . . . . . . . . . . . . . . 50411.5.3 Exercise 72 . . . . . . . . . . . . . . . . . . . . . . . . . . 50511.5.4 The Fractional Quantum Hall Effect . . . . . . . . . . . . 50611.5.5 Quasi-Particle Excitations . . . . . . . . . . . . . . . . . . 50811.5.6 Skyrmions . . . . . . . . . . . . . . . . . . . . . . . . . . . 51211.5.7 Composite Fermions . . . . . . . . . . . . . . . . . . . . . 520

12 Insulators and Semiconductors 52312.1 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 528

12.1.1 Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52912.1.2 Intrinsic Semiconductors . . . . . . . . . . . . . . . . . . . 53112.1.3 Extrinsic Semiconductors . . . . . . . . . . . . . . . . . . 53312.1.4 Exercise 73 . . . . . . . . . . . . . . . . . . . . . . . . . . 536

12.2 Transport Properties . . . . . . . . . . . . . . . . . . . . . . . . . 53612.3 Optical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 537

13 Phonons 538

14 Harmonic Phonons 53814.1 Lattice with a Basis . . . . . . . . . . . . . . . . . . . . . . . . . 54814.2 A Sum Rule for the Dispersion Relations . . . . . . . . . . . . . . 549

14.2.1 Exercise 74 . . . . . . . . . . . . . . . . . . . . . . . . . . 55114.3 The Nature of the Phonon Modes . . . . . . . . . . . . . . . . . . 552

14.3.1 Exercise 75 . . . . . . . . . . . . . . . . . . . . . . . . . . 55414.3.2 Exercise 76 . . . . . . . . . . . . . . . . . . . . . . . . . . 55514.3.3 Exercise 77 . . . . . . . . . . . . . . . . . . . . . . . . . . 55614.3.4 Exercise 78 . . . . . . . . . . . . . . . . . . . . . . . . . . 55614.3.5 Exercise 79 . . . . . . . . . . . . . . . . . . . . . . . . . . 55614.3.6 Exercise 80 . . . . . . . . . . . . . . . . . . . . . . . . . . 55714.3.7 Exercise 81 . . . . . . . . . . . . . . . . . . . . . . . . . . 558

6

14.4 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 55914.4.1 The Specific Heat . . . . . . . . . . . . . . . . . . . . . . 56214.4.2 The Einstein Model of a Solid . . . . . . . . . . . . . . . . 56314.4.3 The Debye Model of a Solid . . . . . . . . . . . . . . . . . 56414.4.4 Exercise 82 . . . . . . . . . . . . . . . . . . . . . . . . . . 56614.4.5 Exercise 83 . . . . . . . . . . . . . . . . . . . . . . . . . . 56714.4.6 Exercise 84 . . . . . . . . . . . . . . . . . . . . . . . . . . 56714.4.7 Exercise 85 . . . . . . . . . . . . . . . . . . . . . . . . . . 56814.4.8 Lindemann Theory of Melting . . . . . . . . . . . . . . . . 56814.4.9 Thermal Expansion . . . . . . . . . . . . . . . . . . . . . 57214.4.10Thermal Expansion of Metals . . . . . . . . . . . . . . . . 573

14.5 Anharmonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57414.5.1 Exercise 86 . . . . . . . . . . . . . . . . . . . . . . . . . . 575

15 Phonon Measurements 57715.1 Inelastic Neutron Scattering . . . . . . . . . . . . . . . . . . . . . 57715.2 The Scattering Cross-Section . . . . . . . . . . . . . . . . . . . . 578

15.2.1 The Zero-Phonon Scattering Process . . . . . . . . . . . . 58215.3 The Debye-Waller Factor . . . . . . . . . . . . . . . . . . . . . . 583

15.3.1 The One-Phonon Scattering Processes. . . . . . . . . . . . 58515.3.2 Multi-Phonon Scattering . . . . . . . . . . . . . . . . . . . 59015.3.3 Exercise 87 . . . . . . . . . . . . . . . . . . . . . . . . . . 59215.3.4 Exercise 88 . . . . . . . . . . . . . . . . . . . . . . . . . . 59315.3.5 Exercise 89 . . . . . . . . . . . . . . . . . . . . . . . . . . 593

15.4 Raman and Brillouin Scattering of Light . . . . . . . . . . . . . . 593

16 Phonons in Metals 59916.1 Screened Ionic Plasmons . . . . . . . . . . . . . . . . . . . . . . . 600

16.1.1 Kohn Anomalies . . . . . . . . . . . . . . . . . . . . . . . 60116.2 Dielectric Constant of a Metal . . . . . . . . . . . . . . . . . . . . 60216.3 The Retarded Electron-Electron Interaction . . . . . . . . . . . . 60416.4 Phonon Renormalization of Quasi-Particles . . . . . . . . . . . . 60516.5 Electron-Phonon Interactions . . . . . . . . . . . . . . . . . . . . 60816.6 Electrical Resistivity due to Phonon Scattering . . . . . . . . . . 609

16.6.1 Umklapp Scattering . . . . . . . . . . . . . . . . . . . . . 61416.6.2 Phonon Drag . . . . . . . . . . . . . . . . . . . . . . . . . 615

17 Phonons in Semiconductors 61617.1 Resistivity due to Phonon Scattering . . . . . . . . . . . . . . . . 61617.2 Polarons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61717.3 Indirect Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . 617

7

18 Impurities and Disorder 61918.1 Scattering by Impurities . . . . . . . . . . . . . . . . . . . . . . . 625

18.1.1 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63218.2 Virtual Bound States . . . . . . . . . . . . . . . . . . . . . . . . . 633

18.2.1 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63618.3 Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63718.4 Coherent Potential Approximation . . . . . . . . . . . . . . . . . 638

18.4.1 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . 64018.5 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641

18.5.1 Anderson Model of Localization . . . . . . . . . . . . . . . 64218.5.2 Scaling Theories of Localization . . . . . . . . . . . . . . . 644

19 Magnetic Impurities 64719.1 Localized Magnetic Impurities in Metals . . . . . . . . . . . . . . 64719.2 Mean-Field Approximation . . . . . . . . . . . . . . . . . . . . . 64719.3 The Atomic Limit . . . . . . . . . . . . . . . . . . . . . . . . . . 65219.4 The Schrieffer-Wolf Transformation . . . . . . . . . . . . . . . . . 656

19.4.1 The Kondo Hamiltonian . . . . . . . . . . . . . . . . . . . 65919.5 The Resistance Minimum . . . . . . . . . . . . . . . . . . . . . . 660

20 Collective Phenomenon 668

21 Itinerant Magnetism 66821.1 Stoner Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668

21.1.1 Exercise 90 . . . . . . . . . . . . . . . . . . . . . . . . . . 67121.1.2 Exercise 91 . . . . . . . . . . . . . . . . . . . . . . . . . . 671

21.2 Linear Response Theory . . . . . . . . . . . . . . . . . . . . . . . 67121.3 Magnetic Instabilities . . . . . . . . . . . . . . . . . . . . . . . . 67321.4 Spin Fluctuations near Ferromagnetic Instabilities . . . . . . . . 677

21.4.1 Ferromagnetic Spin Waves . . . . . . . . . . . . . . . . . . 68021.5 The Slater-Pauling Curves . . . . . . . . . . . . . . . . . . . . . . 68521.6 The Heisenberg Model . . . . . . . . . . . . . . . . . . . . . . . . 687

22 Localized Magnetism 68822.1 Holstein-Primakoff Transformation . . . . . . . . . . . . . . . . . 69022.2 Spin Rotational Invariance . . . . . . . . . . . . . . . . . . . . . . 694

22.2.1 Exercise 92 . . . . . . . . . . . . . . . . . . . . . . . . . . 69722.3 Anti-ferromagnetic Spinwaves . . . . . . . . . . . . . . . . . . . . 697

22.3.1 Exercise 93 . . . . . . . . . . . . . . . . . . . . . . . . . . 700

23 Spin Glasses 70223.1 Mean-Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 70623.2 The Sherrington-Kirkpatrick Solution. . . . . . . . . . . . . . . . 708

8

24 Magnetic Neutron Scattering 71124.1 The Inelastic Scattering Cross-Section . . . . . . . . . . . . . . . 711

24.1.1 The Dipole-Dipole Interaction . . . . . . . . . . . . . . . . 71124.1.2 The Inelastic Scattering Cross-Section . . . . . . . . . . . 711

24.2 Time-Dependent Spin Correlation Functions . . . . . . . . . . . . 71524.3 The Fluctuation - Dissipation Theorem . . . . . . . . . . . . . . 71824.4 Magnetic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . 720

24.4.1 Neutron Diffraction . . . . . . . . . . . . . . . . . . . . . 72024.4.2 Exercise 94 . . . . . . . . . . . . . . . . . . . . . . . . . . 72224.4.3 Exercise 95 . . . . . . . . . . . . . . . . . . . . . . . . . . 72224.4.4 Spin Wave Scattering . . . . . . . . . . . . . . . . . . . . 72324.4.5 Exercise 96 . . . . . . . . . . . . . . . . . . . . . . . . . . 72424.4.6 Critical Scattering . . . . . . . . . . . . . . . . . . . . . . 724

25 Superconductivity 72625.1 Experimental Manifestation . . . . . . . . . . . . . . . . . . . . . 727

25.1.1 The London Equations . . . . . . . . . . . . . . . . . . . . 72925.1.2 Thermodynamics of the Superconducting State . . . . . . 730

25.2 The Cooper Problem . . . . . . . . . . . . . . . . . . . . . . . . . 73325.3 Pairing Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 738

25.3.1 The Pairing Interaction . . . . . . . . . . . . . . . . . . . 73825.3.2 The B.C.S. Variational State . . . . . . . . . . . . . . . . 74025.3.3 The Gap Equation . . . . . . . . . . . . . . . . . . . . . . 74225.3.4 The Ground State Energy . . . . . . . . . . . . . . . . . . 745

25.4 Quasi-Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74825.4.1 Exercise 97 . . . . . . . . . . . . . . . . . . . . . . . . . . 752

25.5 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 75225.6 Perfect Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . 75625.7 The Meissner Effect . . . . . . . . . . . . . . . . . . . . . . . . . 75825.8 Landau-Ginzburg Theory . . . . . . . . . . . . . . . . . . . . . . 760

25.8.1 Extremal Configurations . . . . . . . . . . . . . . . . . . . 76425.8.2 Characteristic Length Scales . . . . . . . . . . . . . . . . 76525.8.3 The Surface Energy . . . . . . . . . . . . . . . . . . . . . 76825.8.4 The Little-Parks Experiment . . . . . . . . . . . . . . . . 77025.8.5 The Critical Current . . . . . . . . . . . . . . . . . . . . . 772

9

1 Introduction

Condensed Matter Physics is the study of materials in Solid and Liquid Phases.It encompasses the study of ordered crystalline phases of solids, as well as disor-dered phases such as the amorphous and glassy phases of solids. Furthermore,it also includes materials with short-ranged order such as conventional liquidsand liquid crystals which show unconventional order intermediate between thoseof a crystalline solid and a liquid. Condensed matter has the quite remarkableproperty that, due to the large number of particles involved, the behavior ofthe materials may be qualitatively distinct from those of the individual con-stituents. The behavior of the incredibly large number of particles is governedby (quantum) statistics which, through the chaotically complicated motion ofthe particles, produces new types of order. These emergent phenomena are bestexemplified in phenomenon such as magnetism or superconductivity where thecollective behavior results in transitions to new phases.

In surveying the properties of materials, it is convenient to separate theproperties according to two (usually) disparate time scales. One time scale isa slow time scale which governs the structural dynamics and the other is afaster time scale that governs the electronic motion. The large difference be-tween the time scales is due to the large ratio of the nuclear masses to theelectronic mass, MN/me ∼ 103. The long-ranged electromagnetic force bindsthese two constituents of different mass into electrically neutral material. Theslow moving nuclear masses can be considered to be quasi-static, and are re-sponsible for defining the structure of matter. In this approximation, the fastmoving electrons equilibrate in the quasi-static potential produced by the nuclei.

1.1 The Born-Oppenheimer Approximation

The difference in the relevant time scales for electronic and nuclear motionallows one to make the Born-Oppenheimer Approximation1. In this approxi-mation, the electronic states are treated as if the nuclei were at rest at fixedpositions. However, when treating the slow motions of the nuclei, the electronsare considered as adapting instantaneously to the potential of the charged nu-clei, thereby minimizing the electronic energies. Thus, the nuclei charges aredressed by a cloud of electrons forming ionic or atomic-like aggregates.

A qualitative estimate of the relative energies of nuclear versus electronicmotion can be obtained by considering metallic hydrogen. The electronic en-ergies are calculated using only the Bohr model of the hydrogen atom. Theequation of motion for an electron of mass me has the form

− Z e2

a2= − me v

2

a(1)

1M. Born and R. Oppenheimer, Ann. Phys. (Leipzig), 84, 457 (1927).

10

where Z is the nuclear charge and a is the radius of the atomic orbital. The stan-dard semi-classical quantization condition due to Bohr and Sommerfeld restrictsthe angular momentum to integral values of h

me v a = n h (2)

These equations can be combined to find the Bohr radius as

a = n2 h2

me Z e2(3)

and also the quantized total electronic energy of the hydrogen atom

Ee = −Z e2

a+

me v2

2

= −Z e2

2 a

= −me Z2 e4

2 n2 h2 (4)

which are standard results from atomic physics. Note that the kinetic energyterm and the electrostatic potential term have similar magnitudes.

Now consider the motion of the nuclei. The forces consist of Coulomb forcesbetween the nuclei and electrons and the quantum mechanical Pauli forces. Theelectrostatic repulsions and attractions have similar magnitudes since the inter-nuclear separations are of the same order as the Bohr radius. In equilibrium,the sum of the forces vanish identically. Furthermore, if an atom is displacedfrom the equilibrium position by a small distance equal to r, the restoring forceis approximately given by the dipole force

− αZ2 e2

a3r (5)

where α is a dimensionless constant. Hence, the equation of motion for thedisplacement of a nuclei of mass MN is

− αZ2 e2

a3r = MN

d2r

dt2(6)

which shows that the nuclei undergo harmonic oscillations with frequency

ω2 = αZ2 e2

MN a3(7)

The semi-classical quantization condition

MN

∮dr . v = 2 π n h (8)

11

yields the energy for nuclear motion as

EN = n h ω

= nme Z

2 e4

h2 α12

(Z me

MN

) 12

(9)

where we have substituted the expression for the Bohr radius for a in the ex-pression for ω. Thus, the ratio of the energies of nuclear motion to electronicmotion are given by the factor

EN

Ee∼(

me

MN

) 12

(10)

Since the ratio of the mass of electron to the proton mass is 12000 , the nuclear

kinetic energy is negligible when compared to the electronic kinetic energy. Amore rigorous proof of the validity of the Born-Oppenheimer approximation wasgiven by Migdal2.

—————————————————————————————————-

Example: Beyond the Born-Oppenheimer Approximation

An example of a correction to the Born-Oppenheimer approximation is givenby (incoherent) inelastic neutron-atom scattering. The scattering occurs onlyvia the nuclear force between the neutrons and the nucleus. The electrons arebound to the nucleus and are only excited via an indirect process. We intend toshow that the probability that an electronic transition occurs is governed by heratio of the mass of the electron me to the mass of the nucleus MN . First, weshall consider the elastic-scattering process in the Born-Oppenheimer approxi-mation and then we shall consider inelastic-scattering.

In the consideration of the elastic-scattering process, the electronic statesmay be ignored since the electrons are not excited. Furthermore, consistent withthe Born-Oppenheimer approximation, the mass of the electron is neglected incomparison with the mass of the nuclei when we consider the kinetics of thescattering. Therefore, in an incoherent scattering process, a neutron of massmn is scattered from an individual nucleus of mass MN . The elastic-scatteringprocess satisfies conservation of energy and momentum

mn

2v2

i +MN

2V 2

i =mn

2v2

f +MN

2V 2

f

mn vi + MN V i = mn vf + MN V f (11)

The scattering results in a transfer of energy and momentum between the twoparticles. The energy and momentum transferred between the neutron and the

2A. B. Migdal, Sov. Phys. J.E.T.P. 7, 996 (1958).

12

Nuclear Compton Scattering

θpi

pf

Pi PfMN

mn

Figure 1: The scattering of a neutron of mass mn by an atom with a nucleus ofmass MN .

nucleus is given by

h q = mn ( vi − vf )

h ω =mn

2( v2

i − v2f )

= h q . vi −h2

2 mnq2 (12)

The neutron’s energy and momentum loss can be determined by experiment.If the nucleus is initially at rest, the magnitude of the final momentum of theneutron is related to the initial momentum via the scattering angle θ, via

vf =cos θ +

√(MN

mn)2 − sin2 θ

1 + (MN

mn)

vi (13)

It is seen that, if MN

mn 1, the scattering is elastic. The positioning of the

detector selects the neutrons which are scattered through the angle θ. The scat-tering of the neutrons by the nuclei is characterized by the correlation functionS(q, ω) which embodies the above conservation laws in a delta function factor

S(q, ω) ∝ δ

(hω − h

mnq . p

i+

h2

2mnq2)

(14)

13

The correlation function can be directly expressed in terms of the initial prop-erties of the nuclei via

S(q, ω) =∫

d3P i n(P i) δ(hω − h

MNq . P i −

h2

2MNq2)

(15)

where n(P ) is the initial momentum distribution of the nuclei, which is givenby

n(P ) = | φ(P ) |2 (16)

and where φ(P ) is the nuclear wave function in the momentum representation

φ(P ) =1

( 2 π h )32

∫d3R exp

[− i

hP . R

]χ(R) (17)

in which χ(R) is the nuclear wave function in the real space representation.Since the scattering potential is represented by a Fermi point-scattering pseudo-potential,

V (rn −R) =2 π h2

mnb δ(rn −R) (18)

where b is the scattering length, its Fourier transform is given by

V (q) =2 π h2

mnb exp

[i q . R

](19)

Therefore, the neutron scattering cross-section can be entirely expressed interms of the nuclear density-density correlation function(

d2σ

dΩdω

)=

kf

ki

(mn

2 π h2

)2

| V (q) |2 S(q, ω) (20)

From this, we conclude that measurements of the scattering cross-section yieldsinformation about the kinetics of the transition and the matrix-elements of theinteraction potential.

We shall now consider the effect of the neutron scattering from a neutralatom. Since the interaction between the neutron and the atom is identicallyzero in the asymptotic initial and final states, the wave function of the atomΨ(r,R) can be analyzed in terms of its center of mass and relative coordinates.If the atomic Hamiltonian for the asymptotic in and out states is expressed as

H =P 2

2 MN+

p2

2 me− Z e2

| r −R |(21)

then it decouples in the center of mass and relative coordinates

RCM =MN R + me r

MN + me

rrel = r − R (22)

14

When expressed in terms of these coordinates, the atomic Hamiltonian decouplesas

H = − h2

2 ( MN + me )∇2

CM − h2 ( MN + me )2 MN me

∇2rel −

Z e2

| rrel |(23)

Hence, the asymptotic atomic wave functions have the form of products

Ψ(r,R) = φ(rrel) χ(RCM ) (24)

If we neglect the mass of the electron compared with the mass of the nucleus,me/MN 1, we expect to recover the Born-Oppenheimer approximation. Inthis approximation, the center of mass coordinate reduces to the nuclear coor-dinate RCM → R. In the (incoherent) inelastic scattering process, the electronsin the atom are excited, hence the initial and final states are represented asproduct states involving the electron and the atomic nucleus

Ψi,n(r,R) = φi(r −R) χn(R)Ψj,m(r,R) = φj(r −R) χm(R) (25)

In the impulse approximation, the nuclear wave functions are represented byplane waves

χn(R) =1√V

exp[i kn . R

](26)

The inelastic scattering cross-section for the neutron beam is represented as(d2σ

dΩdω

)=

kf

ki

(mn

2 π h2

)2 ∑i,n;j,m

| < Ψj,m | V (q) |Ψi,n > |2 δ(hω + Ei,n − Ej,m

)(27)

since h ω and h q are the energy and momentum gained by the atom. Theintegration over the vector R yields the condition for the conservation of mo-mentum. Hence, the scattering cross-section is proportional to the overlap ofthe initial and final electronic states∫

d3r φ∗j (r −R) φi(r −R) (28)

expressed in terms of the relative coordinates. On using the orthonormalitycondition for the electronic energy eigenstates, one obtains a factor of δi,j , sothat the initial and final electronic states are identical. Therefore, in the Born-Oppenheimer approximation, the scattering is purely elastic.

In going beyond the Born-Oppenheimer approximation, one needs to includethe mass of the electrons, me. The conservation of momentum and energy refersto the center of mass motion of the atom, where the atomic mass is the totalmass MN + me. The center of mass-coordinate of the atom is defined by

RCM =MN R + me r

MN + me(29)

15

The center of mass part of the initial and final state atomic wave function,respectively, contains the factor of

exp[i kn .

MN R + me r

MN + me

](30)

and

exp[i km .

MN R + me r

MN + me

](31)

where kn and km are the initial and final momenta of the atom. We shall expressthe electronic position r relative to the nuclear position as r′ = r − R. Withthis notation, the integration over the nuclear coordinate still yields conservationof momentum. However, the electronic part of the scattering matrix elementsnow involves the factor which is a function of the momentum q gained by thenucleus ∫

d3r′ φ∗j (r′) exp

[− i q .

me r′

MN + me

]φi(r′) (32)

The exponential factor allows inelastic transitions to take place. The scatteringamplitude for inelastic transitions is proportional to

q

(me

MN + me

)= q

me

MN

1 + me

MN

(33)

and also to the “dipole” matrix element∫d3r′ φ∗j (r

′) r′ φi(r′) (34)

Hence, we have shown that the probability amplitude that the electrons areexcited by the nuclear motion is controlled by the ratio

me

MN(35)

The Born-Oppenheimer approximation is valid whenever me

MN 1.

—————————————————————————————————-

In the first part of the course it is assumed that the Born-Oppenheimer ap-proximation is valid.

First, the subject of Crystallography shall be discussed, and the charactersof the equilibrium structures of the dressed nuclei in matter will be described.An important class of such materials are those which posses long-ranged peri-odic translational order and other symmetries. It shall be shown how these longrange ordered and amorphous structures can be effectively probed by variouselastic scattering experiments in which the wave length of the scattered particles

16

is comparable to the distance between the nuclei.

In the second part, the properties of the Electrons shall be discussed. Onassuming the validity of the Born-Oppenheimer approximation, the nature ofthe electronic states that occur in the presence of the potential produced bythe static nuclei shall be discussed. One surprising result of this approach isthat, even though the strength of the ionic potential is quite large (of the or-der of Rydbergs), in some metals the highest occupied electronic states bear aclose resemblance to the states expected if the ionic potential was very weak ornegligible. In other materials, the potential due to the ionic charges can pro-duce gaps in the electronic energy spectrum. Using Bloch’s theorem, it shall beshown how periodic long-ranged order can produce gaps in the electronic spec-trum. Another surprising result is that in most metals, it appears as thoughthe electron-electron interactions can be neglected or, more precisely, that theexcitations of the interacting electron system are similar to those of a non-interacting electron gas, albeit with renormalized masses or magnetic moments.

The thermodynamic properties of electrons in these Bloch states shall betreated using Fermi-Dirac statistics. Furthermore, the concepts of the Fermienergy and Fermi surface of metals will be introduced. It shall be shown howthe electronic transport properties of metals are dominated by states with en-ergies close to the Fermi surface, and how the Fermi surface can be probed.

The third part concerns the motion of the ions or nuclei. In particular, itwill be considered how the fast motion of the electrons dresses or screens theinter-nuclear potentials. The low-energy excitations of the dressed nuclear orionic structure of matter give rise to harmonic-like vibrations. The elementaryexcitation of the quantized vibrations are known as Phonons. It shall be shownhow these phonon excitations manifest themselves in experiments, in thermo-dynamic properties and, how they participate in limiting electrical transport.

The final part of the course concerns some of the more striking examplesof the Collective Phenomenon such as Magnetism and Superconductivity.These phenomena involve the interactions between the elementary excitationsof the solid which through collective action, spontaneously break the symmetryof the Hamiltonian. In many cases, the spontaneously broken symmetry is ac-companied by the formation of a new branch of low-energy excitations.

17

2 Crystallography

Crystallography is the study of the structure of ordered solids, disordered solidsand also liquids. In this section, it shall be assumed that the nuclei are static,frozen into their average positions. Due to the large nuclear masses and stronginteractions between the nuclei (dressed by their accompanying clouds of elec-trons), one may assume that the nuclear or ionic motion can be treated classi-cally. The most notable failure of this assumption occurs with the very lightestof nuclei, such as He. In the anomalous case of He, where the separation be-tween ions, d, is of the order of angstroms, the uncertainty of the momentumis given by h

d and so the kinetic energy EK for this quantum zero point motioncan be estimated as

EK ≈ h2

2 M d2(36)

The kinetic energy is large since the mass M of the He atom is small. Themagnitude of the kinetic energy of the zero point fluctuations is larger than theweak van der Waals or London force between the He ions. Thus, the inter-ionicforces are insufficient to bind the He ion into a solid and the material remains ina liquid-like state until the lowest attainable temperatures. For these reasons,He behaves like a quantum fluid. However, for the heavier nuclei, the quantumnature of the particles only manifest themselves in more subtle ways.

First, the various types of structures and the symmetries that can be foundin Condensed Matter will be described and then the various experimental meth-ods used to observe these structures will be discussed.

——————————————————————————————————

2.0.1 Exercise 1

Consider the interaction potential between two electrically neutral atoms po-sitioned at R1 and R2. The positions of the electrons located on atom 1 aredenoted by the set of ri and the positions of the electrons associated with thesecond atom are denoted by the set rj . If the atoms are sufficiently far apart,the sets of electrons belonging to each atom may be thought of as being distin-guishable. In this case, the interaction between the two atoms can be expressedas

Hint =Z2e2

| R1 − R2 |+∑i,j

e2

| ri − rj |−∑

i

Ze2

| ri − R2 |−∑

j

Ze2

| R1 − rj |(37)

This interaction can be expanded in inverse powers of | R1 − R2 |.

If the eigenstates with energy eigenvalue En of the individual atoms aredenoted by | Ψ(1)

n > and | Ψ(2)n >, then second-order perturbation theory

18

yields the shift of the ground state energy due to the interaction between thepair of atoms as

∆E = < Ψ(1)0 Ψ(2)

0 | Hint |Ψ(1)0 Ψ(2)

0 > +∑n,m

| < Ψ(1)0 Ψ(2)

0 | Hint | Ψ(1)n Ψ(2)

m > |2

2 E0 − En − Em

(38)Show that the first term is just the classical electrostatic interaction due to thecharge density distributions around each atom. Using the hydrogenic-like 1sone-electron wave functions

φ1s(r) =

√κ3

πexp

[− κ r

](39)

for the ground state and the 2s and 2p wave functions

φ2s(r) =

√κ3

π

(1 − κ r

)exp

[− κ r

]φ2p,0(r) =

√κ3

πcos θ κ r exp

[− κ r

]φ2p,±1(r) =

√κ3

2 πsin θ exp

[± i ϕ

]κ r exp

[− κ r

](40)

etc. for the excited states, estimate the sign and magnitude of the energy shiftfor atoms with completely filled shells3.

Estimate the Z dependence of the strengths of the London interaction4 forthe inert gases and compare your results with the values obtained from a partof the semi-empirical Lennard-Jones potential

Vint(R) = V0

[ (a

R

)12

−(a

R

)6 ](41)

where the values of V0 and a, respectively, are given in units of eV and Angstromsby

element Ne Ar Kr Xe

V0 0.0124 0.0416 0.056 0.080a 2.74 3.40 3.65 3.98

3R. Eisenschitz and F. London, Z. fur Physik, 60, 491 (1930), J. C. Slater and J. G.Kirkwood, Phys. Rev. 37, 682 (1931).

4F. London, Z. fur Physik, 63, 245 (1930).

19

——————————————————————————————————

3 Structures

The structure of condensed materials is usually thought about in terms of den-sity of either electrons or nuclear matter. To the extent that the regions ofnon-zero density of the nuclear matter are highly localized in space, with lin-ear dimensions of 10−15 meters, the nuclei can be discussed in terms of pointobjects. The electron density is more extended and varies over length scales of10−10 meters. The length scale for the electronic density in solids and fluids isvery similar to the length scale over which the electron density varies in isolatedatoms. The similarity of scales occurs as electrons are partially responsible forthe bonding of atoms into a solid. That is, the characteristic atomic lengthscale is almost equal to the characteristic separation of the nuclei in condensedmatter. Due to the near equality of these two length scales, the electron den-sity in solids definitely cannot be represented in terms of a superposition of thedensity of well defined atoms. However, the electron density does show a signifi-cant variation that can be interpreted in terms of the electron density of isolatedatoms, subject to significant modifications when brought together5. As the elec-tron density for isolated atoms is usually spherically symmetric, the structurein the electronic density may, for convenience of discussion, be approximatelyrepresented in terms of a set of spheres of finite radius.

3.1 Fluids

Both liquids and gases are fluids. The microscopic structure of a fluid varieslocally from position to position and in time. The macroscopic characteristicsof fluids are that they are spatially uniform and isotropic, which means that theaverage environment of any atom is identical to the average environment of anyother atom.

The density is defined by the function

ρ(r) =∑

i

δ3( r − ri ) (42)

in which ri is the instantaneous position of the i-th atom. A measurement ofthe density usually results in the time average of the density which corresponds

5An example of the change in the charge density of the valence electrons of Si, caused bysolid formation, can be found in the contour plot in the article authored by D. R. Hamann,Phys. Rev. Lett. 42, 622 (1979).

20

Figure 2: Contour plot of the valence charge density of Si (in atomic units).The positions of the atoms are denoted by dots. [After Hamann (1979).]

to the time averaged positions of the atoms.

In particular, for a fluid, spatial homogeneity ensures that the time averageddensity ρ(r) at position r is equal to the average density at a displaced positionr +R,

ρ(r) = ρ(r +R) (43)

The value of the displacement R is arbitrary, so the average density is inde-pendent of r and can be expressed as ρ(0). This just means that the averageposition of an individual atom is undetermined.

The operations which leave the system unchanged are the symmetry oper-ations. For a fluid, the symmetry operations consist of the continuous transla-tions through an arbitrary displacement R, rotations through an arbitrary angleabout an arbitrary axis, and also reflections in arbitrary mirror planes.

The set of symmetry operations form a group called the symmetry group.For a fluid, the symmetry group is the Euclidean group. Fluids have the largestpossible number of symmetry operators and have the highest possible symmetry.All other materials are invariant under a smaller number of symmetry opera-tions.

Nevertheless, fluids do have microscopic short-ranged structure which is ex-emplified by locating one atom and then examining the positions of the neigh-boring atoms. The local spatial correlations are expressed by the density -

21

density correlation function which is expressed as an average

C(r, r′) = ρ(r) ρ(r′)

=∑i,j

δ3( r − ri ) δ3( r′ − rj )

(44)

Since fluids are homogeneous, the correlation function is only a function ofthe difference of the positions r − r′. Furthermore, since fluids are isotropicand invariant under rotations, the correlation function is only a function of thedistance separating the two regions of space | r − r′ |. At sufficiently largeseparation distances, the positions of the atoms become uncorrelated, thus,

limr−r′ → ∞

C(r, r′) → ρ(r) ρ(r′)

→ ρ(0) ρ(r − r′) (45)

That is, at large spatial separations, the density - density correlation functionreduces to the product of the independent average of the density at the originand the average density at a position r. From the homogeneity of the fluid, ρ(r)is identical to the average of ρ(0).

The density - density correlation function contains the correlation betweenthe same atom, that is, there are terms with i = j. This leads to a contributionwhich shows up at short distances,∑

i=j

δ3( r − ri ) δ3( r′ − ri ) = δ3( r − r′ )∑

i

δ3( r − ri )

= δ3( r − r′ ) ρ(r) (46)

which is proportional to the density.

The pair distribution function g(r− r′) is defined as the contribution to thedensity - density correlation function which excludes the correlation between anatom and itself,

g( r − r′ ) = C( r − r′ ) − δ3( r − r′ ) ρ(r) (47)

For a system which possesses continuous translational invariance, the pair dis-tribution function can be evaluated as the temporal and spatial average

g( r − r′ ) =∑i 6=j

δ3( r − ri ) δ3( r′ − rj )

=1V

∫d3R

∑i 6=j

δ3( r − ri − R ) δ3( r′ − rj − R )

=1V

∑i 6=j

δ3( r − r′ − ri + rj ) (48)

22

Since the sum over i runs over all of the inter-atomic separations rj − ri for eachfixed value of j, spatial homogeneity demands that the contribution from everyj value is identical. There are N such terms, and this leads to an expression forthe pair distribution function involving an atom at the central site r0 and theothers at sites i in the form

g(r) = ρ(0)∑

i

δ3( r − ri + r0 ) (49)

whereρ(0) =

N

V(50)

As this only depends on the radial distance | r |, this is also called the radialdistribution function g(r). The radial distribution function for liquid Argon6 is shown in fig(3). For large r, the pair distribution function, like C(r, 0),

Figure 3: The radial distribution function g(r) for Argon at T = 85 K. [AfterYarnell et al. (1973).]

approaches ρ(0)2, or

limr → ∞

g(r) → ρ(0)2

(51)

For all fluids, g(r) vanishes as r → 0

limr → 0

g(r) → 0 (52)

6J. L. Yarnell, M. J. Katz, R. G. Wenzel, and S. H. Koenig, Phys. Rev. A 7, 2130 (1973).

23

since the presence of an atom at the origin excludes other atoms from residingat this position. Because correlations in fluids are strongest at short distances,g(r) usually exhibits a large peak at a radial distance greater than the diameterof two atoms. The largest peak in g(r) is usually associated with the shellof atoms nearest to the one at the origin. The integral over the peak in g(r)approximately yields the number of the atoms in the nearest shell (Nnn) timesthe density

4 π∫ R+

R−

dr r2 g(r) ≈ Nnn ρ(0) (53)

Liquids are defined as the fluids that have high densities. The liquid phaseis not distinguished from the higher temperature gaseous phase by a changein symmetry. In the liquid phase, the density is higher and the inter-atomicforces play a more important role than in the low density gaseous phase. Theinteraction forces are responsible for producing the short-ranged correlation inthe density - density function. A model potential that is representative of typicalinter-atomic force between two neutral atoms is the Lennard-Jones potential.

V (r) = 4 V0

[ (a

r

)12

−(a

r

)6 ](54)

The potential has a short-ranged repulsion between the atoms caused by the

The Lennard Jones Potential for Neon

-0.005

-0.0025

0

0.0025

0.005

0 1 2 3 4 5 6

r [ Angstroms ]

V(r

) [ e

V ]

Figure 4: The radial dependence of the Lennard-Jones potential V (r).

24

overlap of the electronic states, and the long-ranged van der Waals attractioncaused by fluctuation-induced electric polarizations of the atoms. The resultingpotential falls to zero at r = a and has a minimum at r = 2

16 a. The potential

at the minimum of the well is given by − V0. Given the form of the potential,the atomic positions can be calculated from Newton’s laws to yield a computersimulated structure of a fluid7. These molecular dynamics calculations revealmuch more information about the structure than do averages. For example, theposition of the peak in g(r) only gives information about the average nearestneighbor separation, whereas the molecular dynamics simulation also yields thestatistical variation of these distances.

Another model potential that is often used to describe liquids is the hardsphere potential which excludes the center of another atom from the region ofradius 2 a centered on the central atom. These mutually impenetrable spheresare then irregularly packed within some volume such that the resulting structurecontains no cavities large enough to contain another sphere8. As the repulsionbetween atoms dominates the structure of liquids, the Bernal model of randomclose packing of hard spheres is responsible for most of the structure of a liq-uid. This model of randomly close-packed spheres produces a structure whichis irregular but densely packed. The structure does contain small regions inwhich the arrangement of atoms has near-perfect hexagonal symmetry. Thepacking fraction is defined as the total volume of the hard spheres divided bythe (minimum) volume that contains all the spheres. On randomly packingspheres, one finds a limiting upper bound on the packing fraction which is givenby 0.638. However, there do exist regular (non-random) close-packed structureswith packing fractions of 0.7405. Bernal also showed that, on averaging over therandom close-packed structure, each sphere was in contact with approximately8.5 other spheres. The randomly close-packed structure may also be analyzedin terms of Voronoi polyhedra. A Voronoi polyhedron of a specific sphere in thestructure is constructed by first joining the centers of the sphere to the otherspheres by a set of straight line segments. Next, a bisecting plane is constructedfor each line segment. The set of closest planes which completely enclose thecenter of the specific sphere forms the Voronoi polyhedron for that sphere. Forthe hard sphere model, the Voronoi polyhedron completely encloses the sphere.The resulting set of Voronoi polyhedra can then be analyzed in terms of theirvolumes, their number of faces and edges. For the randomly packed structure,the average number of faces was found to be 14.25. A difference between theaverage number of faces and the average number of contacts is expected in arandom system, since one can pack more spheres around the central sphere ifthe separations between the centers are allowed to vary. The shell of closestneighboring spheres are defined as the set of spheres which, when joined to thecentral sphere by line segments, have bisecting planes that form the faces of theVoronoi polyhedron. The radial distances between the central sphere and the

7A. Rahman, J. Chem. Phys. 45, 2585 (1966).8J. D. Bernal, Nature, 183, 141 (1959).

25

Figure 5: A random packing of non-overlapping discs in two-dimensions.

spheres in the shell of closest neighbors are found to have values in the range2 a < r < 2.3 a. The average number of edges per face of the Voronoi poly-hedra are found to be 5.16.

Random packings of hard spheres can be used to calculate the radial distribu-tion function g(r). The random close-packing model shows that there are strongshort-ranged correlations between the “closest” atoms. The model also showsthat, in three dimensions, the radial distribution function is zero for r < 2 a.The radial distribution function then peaks up at a radius slightly greater than2 a with an intensity which corresponds to the average number of faces of theVoronoi polyhedra. The short-ranged correlations also show up as other peaksin the radial distribution function at greater distances which correspond to thenext few shells of neighboring atoms. Due to the larger variation in the radialseparation between the central atom and the atoms in the more distant neighborshells, these other peaks are significantly broader than the peak correspondingto the first shell.

It is noteworthy that the hard sphere model does not yield structures withpacking fractions intermediate between the limiting value of 0.638 (found forrandom close packing) and the value of 0.7405 (found for perfectly ordered

26

close-packed structures). One might have expected that there would be a seriesof structures with continuously varying packing fractions. The discontinuity inthe densities of the hard sphere structures may be correlated with the signif-icant but discontinuous change in density that occurs as a liquid freezes andtransforms into a crystalline solid.

27

3.2 Crystalline Solids

A perfect crystal can be partitioned into identical, non-overlapping, structuralunits that completely fill the volume of the (infinite) crystal. When the iden-tical structural units are packed together, they form a periodic structure. If apoint is chosen in one structural unit, the set of equivalent points in the otherunits forms a periodic lattice. There are many different ways of partitioning ofthe crystal and, therefore, there are many alternate forms of the structural unit.The structural unit is called the unit cell. The unit cells which have the smallestpossible volume are called primitive unit cells. A unit cell may contain one ormore atoms. The crystal is only specified if the lattice is specified and the typesand positions of all the atoms in the unit cell are specified. The locations andtypes of atoms in the unit cell forms the basis of the crystal.

3.3 The Direct Lattice

The set of equivalent points, one taken from each unit cell in a perfect crystal,form a periodic lattice. The points are called lattice points. Any lattice pointcan be reached from any other by a translation R that is a combination of aninteger multiple of three primitive lattice vectors a1, a2, a3,

R = n1 a1 + n2 a2 + n3 a3 (55)

Here, n1, n2 and n3 are integers that determine the magnitudes of three com-ponents of a three-dimensional vector. The set of integers (n1, n2, n3) can beused to represent a lattice point in terms of the primitive lattice vectors. Theset of values of the ni run through all positive and negative integers. Any twolattice vectors R can be combined by addition to produce another lattice vector.As will be seen later, this is responsible for the set of translations being closedunder addition and, therefore, the translation operations form a group.

Given any lattice, there are many choices for the primitive lattice vectorsa1, a2, a3.

The array of lattice points have arrangements and orientations which areidentical in every respect when viewed from origins centered on different latticepoints. For example, on translating the origin of the unprimed reference framethrough a lattice vector Rm, one obtains a new reference frame (the primedreference frame). The displacements in the primed reference frame are relatedto displacements in the unprimed reference frame via

r′ = r + m1 a1 + m2 a2 + m3 a3 (56)

and the integers labelling the lattice points in the two frames are related via

n′i = ni + mi (57)

28

and as the numbers ni and n′i take on all possible integer values, the set of alllattice points are identical in the two reference frames.

A crystal structure is composed of a lattice in which a basis of atoms isattached to each lattice point. That is, a complete specification of a crystalstructure requires specifying the lattice and the distribution of the various atomsaround each lattice point. The basis is specified by giving the number of atomsand types of the atoms in the basis (j) together with their positions relative tothe lattice points. The position of the j-th atom relative to the lattice point rj ,is denoted by

rj =(

xj a1 + yj a2 + zj a3

)(58)

where the set (xi, yi, zi) may be non-integer numbers.

The choice of lattice and, therefore, the basis, is non-unique for a crystalstructure. An example of this is given by a two-dimensional crystal structure

a1

a2a'1

a'2

a"1

a"2

Figure 6: A two-dimensional lattice, and some choices for the primitive latticevectors.

which can be represented many different ways including the possibilities of arepresentation either as a lattice with a one atom basis or as a lattice with atwo atom basis.

29

a1

a2 r2

r1

Figure 7: A two-dimensional lattice, and with a choice of the (non-primitive)lattice vectors corresponding to a lattice with a two-site basis.

3.3.1 Primitive Unit Cells

The parallelepiped defined by the primitive lattice vectors forms a primitiveunit cell. When repeated, a primitive unit cell will fill all space. The primitiveunit cell is also a unit cell with the minimum volume. Although there are anumber of different ways of choosing the primitive lattice vectors and unit cells,the number of basis atoms in a primitive cell is unique for each crystal structure.No basis contains fewer atoms than the basis associated with a primitive unitcell.

There is always just one lattice point per primitive unit cell.

If the primitive unit cell is a parallelepiped with lattice points at each of theeight corners, then each corner is shared by eight cells, so that the total numberof lattice points per cell is unity as 8 × 1

8 = 1.

The volume of the parallelepiped is given in terms of the primitive latticevectors via

Vc = | a1 . ( a2 ∧ a3 ) | (59)

The primitive unit cell is a unit cell of minimum volume.

30

A Primitive Unit Cell

a1a2

a3

Figure 8: A primitive unit cell.

3.3.2 The Wigner-Seitz Unit Cell

An alternate method of constructing a unit cell was proposed by Wigner andSeitz. The Wigner-Seitz cell has the important property that there are no arbi-trary choices made in defining the unit cell. The absence of any arbitrary choicehas the consequence that the Wigner-Seitz unit cell always has the same sym-metry as the lattice. The Wigner-Seitz unit cell is constructed by forming a setof planes which bisect the lines joining a central lattice point to all other latticepoints. The region of space surrounding the central lattice point, of minimumvolume, which is completely enclosed by a set of the bisecting planes consti-tutes the Wigner-Seitz cell. Thus, the Wigner-Seitz cell consists of the volumecomposed of all the points that are closer to the central lattice site than to anyother lattice site.

The equation of the plane bisecting the vector from the central point to thei-th lattice point is given by(

r − 12Ri

). Ri = 0 (60)

where Ri is the lattice vector. The planes which bound a volume closer to theorigin than any other lattice site form the surface of the Wigner Seitz-cell.

31

A two atom basis

a1

a2

a3

z2 a3

y2 a2

x2 a1

Figure 9: A primitive unit cell, with a two-atom basis. One basis atom is locatedat the origin r1 = (0, 0, 0). The second atom is located at r2 = (x2, y2, z2).

As the definition does not involve any arbitrary choice of primitive latticevectors, the Wigner-Seitz cell possesses the full symmetry of the lattice. Fur-thermore, the Wigner-Seitz cell is space filling since every point in space mustlie closer to one lattice site than any other.

32

Two-dimensional Wigner-Seitz cell

Figure 10: The Wigner-Seitz unit cell of a two-dimensional lattice. The sym-metry of the Wigner-Seitz cell is independent of the arbitrary choice of theprimitive lattice vectors (a1, a2).

3.4 Symmetry of Crystals

A symmetry operation acts on a crystal producing a new crystal by shiftingthe constituent particles to new positions such that the new crystal is identicalin appearance to the original crystal. That is, after the symmetry operation,the positions of the particles in the new crystal coincide with the positions ofsimilar particles in the original crystal. Each particle can be assigned a unique(non-physical) label, irrespective of whether the particles are physically distin-guishable or not. The uniquely labelled particles are called points. The symme-

33

Wigner-Seitz construction

O

r

Ri/2 Ri

r-Ri/2

Figure 11: The plane bisecting the lattice vector Ri used in the construction ofthe Wigner-Seitz cell.

try operations can then be described by their actions on the set of points. Thesymmetry operations may consist of :

(i) Translation operations which leave no point unchanged.

(ii) Symmetry operations which leave one point unchanged.

(iii) Combinations of the above two types of operations.

3.4.1 Symmetry Groups

A set of symmetry operations form a group if, when the symmetry operationsare combined, the following properties are satisfied :

(I) The combination of any two symmetry operators from the set, say A andB, defined by A B = C has a product C which is also in the set. That is, theset of symmetry operations is closed under composition.

(II) The composition of any three elements is associative, which means thatthe symmetry operation is independent of whether the first and second operatorsare combined before they are combined with the third, or whether the second

34

and third operators are combined before they are combined with the first.

A ( B C ) = ( A B ) C (61)

(III) There exists a symmetry operator which leaves all the atoms in theiroriginal places called the identity operator E. The product of any symmetryoperator arbitrarily chosen from the group with the identity gives back thearbitrarily chosen operator.

A E = E A = A (62)

(IV) For each operator in the group, there exists a unique inverse operatorsuch that when the operator is combined with its inverse operator, they producethe identity.

A A−1 = A−1 A = E (63)

A group of symmetry operators may contain a sub-set of symmetry operatorswhich also form a group. That is, the group laws are obeyed for all the elementsof the sub-set. This sub-set of elements forms a sub-group of the group, but isonly a sub-group if the elements are combined with the same law of compositionas the group.

The symmetry group of the direct lattice contains at least two sub-groups.These are the sub-group of translations and the point group of the lattice. Undera translation which is not the identity, no point remains invariant. The pointgroup of the lattice consists of the set of symmetry operations under which atleast one point of the lattice is invariant.

3.4.2 Group Multiplication Tables

The properties of a group are concisely represented by the group multiplicationtable. The number of elements in the group is called the order of the group, sothe general group with n operators is of order n. The group multiplication tableof a group of order n consists of an n by n array. The group multiplication tablehas the convention that if A × B = C, then the operator A which is the firstelement of the product is located on the left most column of the table, and theoperator B which is the second element is located in the uppermost row. Theproduct C is entered in the same row as the element A and the same column as

35

element B.E . . B .. . . . .A . . C .. . . . .. . . . .

In general, the symmetry operations do not commute, that is, A × B 6= B × A.The identity operator is placed as the first element of the series of symmetryoperators, so the first row and first column play a dual role. The first columnand first row play one role as the list of the groups elements. The second rolethat they play is as a record of the products found by compounding the elementswith the identity. Every operator appears once, and only once, in each row orcolumn of the group table. The fact that each operator occurs only once in anyrow or in any column, is a consequence of the uniqueness of the inverse.

As an example, consider the point group for a single H2O molecule. The

The symmetry elements of an H2O molecule

C2σv

σv'

Figure 12: The symmetry operations of an H2O molecule. The symmetry ele-ments are composed of a two-fold axis of rotation, and two mirror planes.

group contains a symmetry operation which is a rotation by π about an axis inthe plane of the molecule that passes through the O atom and bisects the linejoining the two H atoms. This is a two-fold axis, since if the rotation aboutthe axis is followed by a second rotation about the same axis then the com-bined symmetry operation is equivalent to the identity. The two-fold rotation

36

is labelled as C2. In this case, the two-fold axis is the rotation axis of highestorder and thus, is considered to define the vertical direction. In addition to thetwo-fold axis, there are two mirror planes. It is conventional to denote a mirrorplane that contains the n-fold axis of rotation (Cn) with highest n as a verticalplane. The H2O molecule is symmetric under reflection in a mirror plane pass-ing through the two-fold axis in the plane which contains the molecule. That is,the mirror plane is the plane passing through all three atoms. This is a verticalmirror symmetry operation and is denoted by σv. The second mirror symmetryoperation is a reflection in another vertical plane passing through the C2 axisbut this time, the mirror plane is perpendicular to the plane of the moleculeand is denoted by σ′v. The symmetry group contains the elements E, C2, σv,σ′v. The group is of order 4. The group table is given by

E C2 σv σ′v

C2 E σ′v σv

σv σ′v E C2

σ′v σv C2 E

Since all the operations in this group commute (i.e. A B = B A ), the groupis known as an Abelian group. Inspection of the table immediately shows thatσv × C2 = σ′v.

The symmetry group of a crystal has at least two sub-groups. One sub-groupis the group of translations through the set lattice vectors R. In the case of thetranslations, the law of composition is denoted as addition. A general transla-tion which is not the identity leaves no point unchanged. A second sub-groupis formed by the set of all transformations which leave a particular point of thecrystal untransformed. This sub-group is the point group.

3.4.3 Point Group Operations

The crystallographic point group consists of the symmetry operations that leaveat least one point untransformed. Furthermore, the various symmetry opera-tions that belong to one point group always leave the same point unchanged.The possible symmetry elements of the point group are:

Rotations around an axis through angles which are integer multiples of 2 πn .

An n-fold rotation is denoted as Cn. A rotation by 2 π mn can be expressed

as the composition of m successive rotations about the same axis (Cn)m. Acombination of n rotations by 2 π

n about the same axis leads to the identity(Cn)n = E, and n is known as the order of the axis.

37

C3

2ππππ/3

Figure 13: A molecule with an n-fold rotation axis Cn (with n = 3).

Reflections which take every point into its mirror image with respect to aplane known as the mirror plane. Reflections are denoted by σ.

Inversions which take every point r, as measured from an origin, into the point− r. The inversion operator is denoted by I.

Inversion I

r

- r

Figure 14: The inversion operator I transforms the point r into the point −r.

Rotation Reflections which are rotations about an axis through integer mul-tiples of 2 π

n followed by reflection in a plane perpendicular to the axis. The

38

n-fold rotation reflections are denoted by Sn. For even n, (Sn)n = E, whilefor odd n, (Sn)n = σ.

Roto-Reflection S2

2π/2

Figure 15: A two-fold rotation-reflection axis.

Rotation Inversions which are rotations about an axis through integer mul-tiples of 2 π

n followed by an inversion through an origin. The Internationalnotation for a rotation reflection is n. The rotation inversion and rotation re-flection operations are related for example, 3 = S−1

6 , 4 = S−14 and 6 = S−1

3 .

Since at least one particular point is invariant under all the transformationsof the point group, the rotation axes and mirror planes must all intersect at thispoint.

Symmetry operations A and B are defined to be equivalent if the symmetrygroup contains an element C such that

A = C−1 B C (64)

The set of elements which are all equivalent to each other form an equivalenceclass. It can be seen that equivalent symmetry operations are of the same type,but may involve different orientations of the translations, rotation axes or mirrorplanes. For example, two rotations through the same angle but about differentaxes may be equivalent to each other. Likewise, two reflections in different mir-ror planes may be equivalent to each other.

——————————————————————————————————

39

Roto-Reflection S3

2π/3

Roto-Reflection S4

2π/4

Figure 16: The n-fold rotation-reflection axis, with n = 3 and n = 4.

3.4.4 Exercise 2

Consider a structure with a symmetry group that contains a principal axis oforder n (Cn) and also contains one vertical mirror plane (σ1). Any atom atpoint P1 in the structure is related to an identical atom at Q1 by the mirrorsymmetry

Q1 = σ1 P1 (65)

Likewise, atoms at points P1 and Q1, respectively, are related to identical atomsat Pm and Qm via the repeated actions of Cn,

Pm = ( Cn )m P1

Qm = ( Cn )m Q1 (66)

40

Show that the symmetry group must contain symmetry operations σm whichare equivalent to σ1 via

σm = ( Cn )m σ1 ( Cn )−m (67)

Identify the symmetry operations σm.

P1

P2

P3

Q3

Q2

Q1

C3

σ3 σ2

σ1

Figure 17: A group with an n-fold rotation axis Cn and one vertical mirrorplane σ1 has a set of equivalent vertical mirror planes σn.

——————————————————————————————————

3.4.5 Limitations Imposed by Translational Symmetry

Although all point group operations are allowable for isolated molecules, certainpoint group symmetries are not allowed for periodic lattices. The limitations onthe possible types of point group symmetry operations can be seen by examiningthe effect of an n-fold axis in a plane perpendicular to a line through latticepoints A − B . . . C − D, with 1 + m1 lattice points on it. The directionof the line will be chosen as the direction of the primitive lattice vector a1, andthe line is assumed to have a length m1 a1. A clockwise rotation of 2 π

n aboutthe n-fold axis of rotation through point A will generate a new line A − B′.Likewise, a counter clockwise rotation of 2 π

n about the n-fold axis of rotationthrough point D will generate a new line D − C”. The line constructed throughB′ − C” is parallel to the initial line A − D. The length of the line B′ − C”is equal to m1 a1 − 2 cos 2 π

n a1 and must be equal to an integer multiple ofa1, say m′

1. Then

cos2 πn

=m1 − m′

1

2(68)

Thus, cos 2 πn must be an integer or a half odd integer which is in the set

±1, 0,± 12. This restriction limits the possible n-fold rotation axis to be of

41

The limitations on rotational symmetry imposed by periodic translational invariance

A B C D

B' C"θ = 2π/n θ

m1

m1'

Figure 18: The limitations on rotational symmetry elements imposed by peri-odicity.

order n = 1 , 2 , 3 , 4 , 6 and allows no others. Thus, a crystalline latticecan only contain two-fold, three-fold, four-fold or six-fold axes of rotation. How-ever, there do exist solids that possess five-fold symmetry, such as quasi-crystals.Quasi-crystals are not crystals as they do not possess periodic translational in-variance.

——————————————————————————————————

3.4.6 Exercise 3

When point group symmetry operations are combined, new symmetry elementsmay arise. These new operations are also point group operations, since theyleave a specific point invariant.

42

θ2/2

θ1/2

φ3

φ2

φ1

P1

P2

P3

Figure 19: The Euler construction shows how a rotation by θ1 about the axesP1 can be combined with a rotation through angle θ2 about the axis P2. Theserotations when combined in the correct order are identical to a rotation aboutthe axis P3.

Prove that a rotation about an axis Cn1 , followed by a rotation about dif-ferent axis Cn2 , is identical to a rotation Cn3 about a third axis. The anglebetween the Cn1 axis and the Cn2 axis is denoted by ϕ3. Show that, if there areno other axes of rotations present, the angle between the axes and the orders ofthe axes must satisfy the condition

cosϕ3 =cos π

n3+ cos π

n1cos π

n2

sin πn1

sin πn2

(69)

Hence, for a periodic crystal, show that the only allowed non-trivial combina-tions of three rotational axes have orders n1 n2 n3 given by 2 2 2, 2 2 3, 2 2 4,2 2 6, 2 3 3 and 2 3 4.

43

(n1, n2, n3) θ1 =(

2πn1

)θ2 =

(2πn2

)θ3 =

(2πn3

)ϕ1 ϕ2 ϕ3

(2,2,2) π π π π2

π2

π2

(2,2,3) π π 2π3

π2

π2

π3

(2,2,4) π π π2

π2

π2

π4

(2,2,6) π π π3

π2

π2

π6

(2,3,3) π 2π3

2π3 cos−1 1

3 cos−1 1√3

cos−1 1√3

(2,3,4) π 2π3

π2 cos−1 1√

3π4 cos−1

√23

——————————————————————————————————

3.4.7 Point Group Nomenclature

The point groups are frequently referred to by using either one or the other oftwo different notation schemes: the Schoenflies and the International notation.In the following examples, the groups are first labelled by their Schoenflies des-ignation and then by their International designation.

In enumerating the point groups, we shall first list the rotational groups andthen list the rotational groups which are adjoined by mirror planes or improperrotations that do not introduce any further proper rotational symmetry opera-tions.

The point groups are:

Cn The groups Cn only contain an n-fold rotation axis. The number ofelements in the group Cn is equal to the order of the axis. The group is knownas the cyclic group of order n and it is Abelian. The international symbol des-ignating this group is n.

Cn,v The groups Cn,v contain the n-fold rotation axis and have vertical mir-ror planes which contain the axis of rotation. The effect of the n-fold axis, if n

44

The symmetry elements of Cnv

C4

σv

σv'

Figure 20: The group Cn,v has an n-fold rotation axis and equivalent verticalmirror planes.

is odd, is such that it produces a set of n equivalent mirror planes. This yields2 n symmetry operations, which are the n rotations and the reflections in then mirror planes. If n is even, the effect of repeating Cn only produces n

2 equiv-alent mirror planes. The other n

2 rotations merely bring the mirror plane intocoincidence with itself, but with the two surfaces of the mirror interchanged.A mirror plane is equivalent to its partner mirror plane found by rotating itthrough π since by definition, a mirror plane is two-sided. However, for evenn, the effect of the compounded operation Cn σv acting on an arbitrary pointP produces a point P ′ which is identical to the point P ′ produced by reflectionof P in the mirror plane that bisects the angle between two equivalent mirrorplanes σv. Thus, the symmetry element given by the product Cn σv is identicalto a mirror reflection in the bisecting (vertical) mirror plane. The effect of Cn

is to transform these bisecting mirror planes into a set of n2 equivalent bisect-

ing mirror planes. Thus, if n is even, there are also 2 n symmetry operations.These 2 n symmetry operations are the set of n rotations and the two sets ofn2 reflections. Mirror planes which are not perpendicular to the rotation axisare recorded as m without any special marking. For even n, the Internationalsymbol for Cn,v is nmm. The two m’s refer to two distinct sets of mirror planes:one from the original vertical mirror plane and the second m refers to the verti-cal mirror planes which bisect the first set. For odd n, the international symbolis just nm, as the group only contain one set of mirror planes and does not

45

contain a set of bisecting mirror planes.

P1Q1

Q2C4

σv

σv

σv'

σv' = C4 σ v

Figure 21: A group with an n-fold rotation axis Cn, where n is even, and onevertical mirror plane σv has two sets of equivalent vertical mirror planes σv andσ′v.

Cn,h The groups Cn,h contain the n-fold rotation, and have a horizontalmirror plane which is perpendicular to the axis of rotation. The group contains2 n elements and, if n is even, the group contains C

n2

n . σh = C2 . σh = Iwhich is the inversion operator. The International notation usually refers tothe group Cn,h as n/m. The diagonal line indicates that the symmetry planeis perpendicular to the axis of rotation. The only exception is C3h or 6. Theinternational symbol signifies that C3h is relegated to the group of rotation re-flections which are, in general, designated by n.

Sn The groups Sn only contain the n-fold rotation - reflection axis. Thesegroups are obtained by attaching an m′-fold rotation - reflection axis to therotation group containing Cm, such that S2

m′ = Cm. That is, the rotation -reflection axis must coincide with the axis of the rotation group. For even n,the group contains only n elements since (Sn)n = E. For odd n, (Sn)n = σ,so the group must contain 2 n elements. The International notation is givenby the equivalent rotation inversion group n. For example, S6 ≡ 3, S4 ≡ 4,S3 ≡ 6.

Dn The groups Dn contain an n-fold axis of rotation and a two-fold axiswhich is perpendicular to the n-fold axis. The effect of the n-fold rotation isto produce a set of equivalent two-fold axes. If n is odd there are n equivalenttwo-fold axes. If n is even, the n-fold rotation produces n

2 equivalent two-foldaxes which are two sided. When n is even, the action of a two-fold rotationfollowed by an n-fold rotation is equivalent to a new two-fold rotation about anaxis that bisects the original sets of two-fold axes. This can be seen by following

46

Symmetry elements of D3

C2

C2

C2

C3

Figure 22: The dihedral group Dn, for odd n, has an n-fold rotation axis and aset of n equivalent two-fold rotation axes.

the action of an arbitrary point P with coordinates (x, y, z) under the two-foldrotation about a horizontal axis, say the x axis. The rotation by π about the xaxis sends z → − z and y → − y. A further rotation of 2 π

n about the z axis,sends the point (x,−y,−z) to the final image point P ′. Note that the z coordi-nate of point P ′ is − z. Construct the line joining P and P ′. The mid-point ofP − P ′ lies on the plane z = 0, and subtends an angle of π

n with the x axisand, therefore, lies on the bisecting rotation axis. As the bisecting axis passesthrough the mid-point of line P − P ′, this shows that the arbitrary point Pcan be sent to P ′ via a π rotation about the bisecting axis. Thus, for even n,there are n

2 bisecting two-fold axes, and the original n2 two-fold axes. In case of

either even or odd n, the group contains 2 n elements consisting of the n-foldrotations and a total of n two-fold rotations. The International designation forDn is either n22 or just n2, depending on whether n is even or odd. Thesetwo designations occur for similar reasons as to why there are two Internationaldesignations for Cn,v. For odd n the designation n2 indicates that there is onen-fold axis and one set of equivalent two-fold axes. For even n, the symbol n22indicates the existence of an n-fold axes and two inequivalent sets of two-foldaxes.

Dnh The groups Dnh contain all the elements of Dn and also contain a hor-izontal mirror plane perpendicular to the n-fold axis. The effect of a rotation

47

Symmetry Elements of D4

C4

C2

C2

C2'

C2'

Figure 23: The dihedral group Dn, for even n, has an n-fold rotation axis andtwo sets of equivalent two-fold rotation axes.

about a two-fold axis followed by the reflection σh is equivalent to a reflectionabout a vertical plane σv passing through the two-fold axis. Since rotating σv

about the Cn axis produces a set of n vertical mirror planes, adding a horizontalmirror plane to Dn produces n vertical mirror planes σv. The group has 4 nelements which are formed from the 2 n rotations of Dn, the n reflections in thevertical mirror planes, and n rotation reflections Ck

n σh. For even n, the Inter-national symbol is n

m2m

2m which is often abbreviated to n/mmm. The symbol

indicates that the n-fold axis has a perpendicular mirror plane, and also thatthe two sets of two-fold axes have perpendicular mirror planes. For odd n, theInternational symbol for the group acknowledges the 2n-fold rotation inversionsymmetry and is labelled as 2n2m.

Dnd The groups Dnd contain all the elements of Dn and mirror planes whichcontain the n-fold axis and bisect the two-fold axes. The action of the two-foldrotations generate a total of n vertical reflection planes. There are 4 n elements,the 2 n rotations of Dn, n mirror reflections σd in the n vertical planes. The re-maining n elements are rotation reflections about the principle axis of the formS2k+1

2n where k = 0 , 1 , 2 , . . . , ( n − 1 ). The principle axis is, therefore, a2n-fold rotation reflection axis. The International symbol is n2m indicating an-fold axis, a perpendicular two-fold axis and a vertical mirror plane.

48

T The tetrahedral group corresponds to the group of rotations of the reg-ular tetrahedron. The elements are comprised of four three-fold rotation axespassing through one vertex and through the center of the opposite face of thetetrahedron. The compound action of two of the three-fold rotations yields a ro-

Rotational Symmetries of a Tetrahedron

C3

C2

Figure 24: The rotational symmetry elements of a tetrahedron. The tetrahedronhas four three-fold axes C3 passing through each vertex, and has three two-foldaxes which pass through the mid-points of opposite edges.

tation about a two-fold axis. There are three such two-fold axes passing throughthe mid-points of opposite edges of the tetrahedron. The tetrahedral group hastwelve elements. The symmetry operations can also be found in a cube, if thethree four-fold rotation axes present in the cube are discarded. The group hasthe International symbol of 23.

Td The group Td corresponds to the tetrahedral group adjoined by a reflec-tion plane passing through one edge and the mid-point of the opposite edge ofthe tetrahedron. The reflection planes bisect a pair of two-fold axes of T . Sincethe tetrahedron has six edges, there are six mirror planes σd. For the cube,these six mirror planes are the diagonal planes which motivates the use of thesubscript d. The mirror planes convert the two-fold axes to produce four-foldrotation reflection axes S4. The group Td contains twentyfour elements. Thegroup Td has the International designation as 43m.

Th The group Th consists of the tetrahedral group adjoined by a mirror

49

A tetrahedron inscribed in a cube

Figure 25: The rotational symmetry operations of a tetrahedron are also foundamongst symmetry elements of a cube. The tetrahedron can be inscribed in acube.

plane which bisects the angle between the three-fold axes. For the tetrahedron,these mirror planes are equivalent to the mirror planes of Td. However, forthe cube, the mirror plane is parallel to opposite faces and is mid-way betweenthem. There are only three such horizontal planes for the cube. The planesbisect the angles between the three-fold axes, and, therefore, convert them intosix-fold rotation reflection axes. Since the group contains S6, it also contains I.Hence, Th = T × Ci. The group has twentyfour elements. The Internationaldesignation for the group Th is either 2

m3 or m3.

O The octahedral group has three mutually perpendicular four-fold axes.There are four three-fold axes, and six two-fold axes. It has twentyfour ele-ments. It has an International designation of 432.

Oh On adjoining a mirror plane to the octahedral group, one obtains Oh.Adding a vertical mirror plane generates three other mirror planes. The effectof a reflection in the vertical mirror plane followed by a rotation C4 is equivalentto a reflection in a diagonal mirror plane. There are six of these diagonal mirrorplanes. The C3 axes becomes S6 axes, just as in the case of Th. The group con-tains fortyeight elements. The International designation is either 4

m3 2m or m3m.

50

C4

C3

C2

Figure 26: The rotational symmetry operations of an octahedron.

3.5 Bravais Lattices

There are an infinite number of possible lattices since there are no restrictionson the primitive lattice vectors. However, only a few special types of lattices areinvariant under point group operations. These special types of lattices are calledBravais lattices. The Bravais lattices can be categorized by their symmetries.In three dimensions, there are fourteen Bravais lattices. The fourteen Bravaislattices are organized according to seven crystal systems. These seven crystalsystems are classified according to the relations between their rotation axes.

The conventional unit cells have lattice vectors a, b, and c, of length a, band c, as shown in the figure. The angles between the vectors are denoted byα, β and γ, such that α is the angle between b and c, etc. That is, α ( 6 b , c ),β ( 6 a , c ), and γ ( 6 a , b ).

——————————————————————————————————

51

Octahedron inscribed in a cube

Figure 27: The rotational symmetry operations of an octahedron are equivalentto the rotational symmetry elements of a cube. The octahedron can be inscribedin a cube.

3.5.1 Exercise 4

Show that the volume of a conventional unit cell, Vc is given by

Vc = a b c

[1 + 2 cos α cos β cos γ − cos2 α − cos2 β − cos2 γ

] 12

(70)

——————————————————————————————————

If the point group contains four three-fold axes C3 or (3), the system is cu-bic. It is possible to choose three coordinate axes which are orthogonal to eachother and are perpendicular to the faces of a cube that has the four three-foldaxes as the body diagonals.

52

A conventional unit cell

a

b

c

αβ

γ

Figure 28: A conventional unit cell is specified by the lengths of the sides (a, b, c)and the angles (α, β, γ) subtended by the pairs of sides.

3.5.2 Cubic Bravais Lattices.

The cubic Bravais lattices have the highest symmetry. The simple cube (P) hasthree four-fold rotation axes and four three-fold axes, along with six two-foldaxes. There are three mirror planes that can be adjoined to the set of rotationalsymmetry operations. The three four-fold rotation axes (C4) are mutually per-pendicular and pass through the centers of opposite faces of the cube. Anyrotation which is an integer multiple of 2 π

4 will bring the cube into coincidencewith itself. The four three-fold axes (C3) pass through pairs of opposite ver-tices of the cube. A rotation of any multiple of 2 π

3 will bring the cube intocoincidence with itself, as can be seen by inspection of the three edges at thevertex which the rotation axis passes through. The six two-fold axes (C2) jointhe mid-points of opposite edges. The highest symmetry group when mirrorsymmetry is not included is the octahedral group O. The octahedral group Ocontains twenty four symmetry operations. On adjoining a mirror plane to theset of rotations of the octahedral group, one has the highest symmetry pointgroup which is labelled as Oh or m3m and has fortyeight symmetry elements.

The reason that the cubic group is called the octahedral group is explainedby the following observation. The group of symmetry operations of the cubeis equivalent to the group of symmetry operations on the regular octahedron.This can be seen by inscribing an octahedron inside a cube, where each vertex

53

Rotational symmetry elements of a cube

C4

C3

C2

Figure 29: The rotational symmetry elements of a cube.

of the octahedron lies on the center of the faces of the cube. Thus, the cubicpoint group is called the octahedral group O.

There are three types of cubic Bravais lattices: the simple cubic (P), thebody centered cubic (I) and face centered cubic (F) Bravais lattices.

The primitive lattice vectors for the simple cubic lattice (P) can be takenas the three orthogonal vectors which form the smallest cube with the latticepoints as vertices. The three primitive lattice vectors have equal length, a, andare orthogonal. The vertices of the cube can be labelled as (0, 0, 0), (0, 0, 1),(0, 1, 0), (1, 0, 0), (1, 1, 0), (1, 0, 1), (0, 1, 1) and (1, 1, 1). The primitive cell is thecube which contains one lattice point and has a volume a3.

The body centered cubic Bravais lattice (I) has a lattice point at the ver-tices of the cube and also one at the central point. The central point is locatedat a

2 (1, 1, 1) when specified in terms of the Cartesian coordinates formed bythe edges of the conventional non-primitive unit cell (which is the cube). Theprimitive lattice vectors are given in terms of the Cartesian coordinates bya1 = a

2 (1, 1,−1), a2 = a2 (−1, 1, 1), a3 = a

2 (1,−1, 1). These are the threevector displacements originating at any lattice point which end up at one ofthree neighboring body centers. The conventional unit cell contains two latticesites and has a volume a3, where a is the length of the side of the cube. The

54

Conventional unit cell for the b.c.c. lattice

(1/2,1/2,1/2)

(0,0,0)

Figure 30: The conventional unit cell of a b.c.c. lattice.

primitive unit cell is a rhombohedron of edge a√

32 which contains one lattice

site and has a volume 18 4 a3. The angles between the primitive lattice vectors

is given by cos γ = − 13 . In the primitive cell, each body center of the conven-

tional unit cell is connected by three primitive lattice vectors to three verticesof the conventional cell.

The Wigner-Seitz cell for the body centered cubic lattice is a truncated oc-tahedron. It is made of eight hexagonal planes which are bisectors of the linesjoining the body center to the vertices. These eight planes are truncated by theplanes of the cube which coincide with the bisectors of the lines between theneighboring body centers. The truncation produces the six square faces of thebody centered cubic Wigner-Seitz cell.

The face centered cubic Bravais lattice (F) consists of the lattice points atthe vertices of the cube and lattice points at the centers of the six faces. Thelattice points at the face centers are located at a

2 (1, 1, 0), a2 (1, 0, 1), a

2 (0, 1, 1),a2 (1, 1, 2), a

2 (1, 2, 1) and a2 (2, 1, 1). The primitive lattice vectors point from the

vertex centered at (0, 0, 0) to the three closest face centers, a1 = a2 (1, 1, 0),

a2 = a2 (1, 0, 1), a3 = a

2 (0, 1, 1). Since each face is shared by two adjacentnon-primitive unit cells there are 4 lattice sites in the conventional non-primitivecubic unit cell. The primitive unit cell is a rhombohedron with side a√

2. The

edges of the primitive unit cell connect two opposite vertices of the cube via

55

The primitive lattice vectors for a b.c.c. Bravais lattice

a1

a2

a3

Figure 31: The primitive lattice vectors of a b.c.c. Bravais lattice.

the six face centers. The edges of the primitive cell are found by connecting thevertex to the three neighboring face centers. The volume of the primitive unitcell is found to be 1

4 a3. The angles between the primitive lattice vectors are π

3 .

The Wigner-Seitz cell for the face centered cubic lattice is best seen by trans-lating the conventional unit cell by a

2 along one axis. After the translation hasbeen performed, the unit cell has the appearance of being a cube which haslattice sites at the body center and at the mid-points of the twelve edges ofthe cube. The Wigner-Seitz cell can then be constructed by finding the twelveplanes bisecting the lines from the body center to the mid-points of the edges.The resulting figure is a rhombic dodecahedron.

——————————————————————————————————

The presence of either one four-fold C4, (4) or four-fold inversion rotationaxes S4, (4), makes it possible to choose three vectors such that a = b 6= c,α = β = γ = π

2 and c is parallel to the C4 or S4 axes. This is the tetragonalsystem.

56

The conventional unit cell for an f.c.c. lattice

(1,0,1)

(0,1,1)

(1,0,1)(0,0,0)

Figure 32: The conventional unit cell for an f.c.c. lattice.

The primitive unit cell for a f.c.c. Bravais lattice

a1

a2

a3

Figure 33: The primitive lattice vectors for an f.c.c. lattice.

57

body centered tetragonal unit cell

aa

c

α = β = γ = π/2

Figure 34: The body centered tetragonal lattice.

3.5.3 Tetragonal Bravais Lattices.

The tetragonal Bravais lattice can be considered to be formed from the cubicBravais lattices by deforming the cube, by stretching it, or contracting it alongone axis. This special axis is denoted as the c axis. Thus, the conventional unitcell can be constructed, starting with a square base of side a, by constructingedges of length c 6= a parallel to the normals of the base, from each corner.

The simple tetragonal Bravais lattice has a four-fold rotational axis and twoorthogonal two-fold axes. These symmetry elements generate the group D4.On adding a horizontal mirror plane to D4, one obtains the highest symmetrytetragonal point group which isD4h or 4/mmm with sixteen symmetry elements.

There are two tetragonal Bravais lattices: the simple tetragonal Bravais lat-tice (P) and the body centered tetragonal Bravais lattice (I).

The face centered tetragonal lattice is equivalent to the body centered tetrag-onal lattice. This can be seen by considering a body centered tetragonal latticein which the conventional unit cell can be described in terms of a side of length cperpendicular to the square base of side a and area a2. Consider the view alongthe c axis which is perpendicular to the square base. By taking a new base ofarea 2 a2 and sides

√2 a which are the diagonals of the original base, one finds

that the body centers can now be positioned as the face centers. That is, the

58

body centered tetragonal is equivalent to the face centered tetragonal unit cell.

The equivalence between the body centered and face centered structures doesnot apply to the cubic system. However, the conventional body centered cubicunit cell is equivalent to a face centered tetragonal unit cell in which the heightalong the c-axis has a special relation to the side of the base. The equivalentface centered tetragonal unit cell has a height of a along the c-axis and the sideof the base is

√2 a. Using the converse construction, the face centered cubic

unit cell can be shown to be equivalent to the body centered tetragonal latticewith a particular length of the c-axis.

——————————————————————————————————

The orthorhombic system has three mutually perpendicular two-fold rota-tion axes. The existence of the three mutually perpendicular two-fold rotationaxes is compatible with the point groups D2, C2v and D2h. It is possible toconstruct a unit cell α = β = γ = π

2 .

3.5.4 Orthorhombic Bravais Lattices.

The conventional orthorhombic unit cell can be considered to be formed by de-forming the tetragonal unit cell by stretching the base along an axis in the basalplane. Thus, the base can be viewed as consisting of a rectangle of side a 6= b.The unit cell has another set of edges which are parallel with the normal to thebase and have lengths c. Thus, the conventional unit cell has edges which areparallel to three orthogonal unit vectors.

The simple orthorhombic lattice (P) has only two-fold rotation axes. Thetwo-fold axes are perpendicular, so the rotational group is D2. The effect of ad-joining a horizontal mirror plane converts D2 into the orthorhombic point groupwith highest symmetry which is D2h or 2/mmm with four symmetry operations.

There are four inequivalent orthorhombic Bravais lattices. These are the sim-ple orthorhombic lattice (P), the body centered orthorhombic (I), face centeredorthorhombic (F) and a new type of lattice, the c-side centered orthorhombiclattice (C).

The c-side centered orthorhombic lattice (C) can be constructed from thetetragonal lattice in the following manner9. View the square net of side a, whichforms the bases of the tetragonal unit cells, in terms of a non-primitive unit cellwith a square base of side

√2 a with sides along the diagonal. This larger

non-primitive unit cell contains one extra lattice site at the center of the base9The nomenclature used is that, if the two faces of the unit cell which have the same normal

as the a − b plane are centered, then the structure is designated as C. Likewise, if the facesparallel to the a− c plane are centered, the structure is designated by B, etc.

59

Primitive monoclinic unit cell

γ

mP

c

a

bπ/2 π/2

Figure 35: The primitive monoclinic lattice.

and the top. When this base centered tetragonal structure is then stretchedalong one of its sides (one of the diagonal sides of length

√2 a), one obtains the

orthorhombic base-centered lattice.

——————————————————————————————————

The monoclinic lattice system requires a minimum of one two-fold rotationaxis. Due to the conditions imposed by the two-fold rotation symmetry, it ispossible to choose α = β = π

2 6= γ. The monoclinic systems are compatiblewith the point groups C2, Cs and C2h.

3.5.5 Monoclinic Bravais Lattice.

The monoclinic Bravais lattice is obtained from the orthorhombic Bravais lat-tices by distorting the rectangular base perpendicular to the c axis into a par-allelogram. The base is a parallelogram, and the two basal lattice vectors areperpendicular to the c axis.

The simple monoclinic lattice (P) has a two-fold axis parallel to the c axis.The rotational group is C2. If a horizontal mirror plane is added to C2, thenone finds that the most symmetric monoclinic point group is C2h or 2/m whichhas four elements.

60

Body-centered monoclinic unit cell

γ

mI

Figure 36: The body centered monoclinic lattice.

There are two types of monoclinic Bravais lattices: the simple monoclinic (P)and the body centered monoclinic Bravais lattice (I). The two types of mono-clinic Bravais lattices correspond to the two types of tetragonal Bravais lattices.The four orthorhombic lattices collapse onto two lattices in the tetragonal andmonoclinic systems, as the centered square net is not distinct from a square net.Likewise, the centered parallelogram is not distinct from a parallelogram.

——————————————————————————————————

The groups C1 and Ci impose no specific restrictions on the lattice. This isthe triclinic lattice system.

3.5.6 Triclinic Bravais Lattice.

The triclinic Bravais lattice is obtained from the monoclinic lattice by tiltingthe c axis so that it is no longer orthogonal to the base. There is only the simpletriclinic Bravais lattice (P). The three axes are not orthogonal and the sides areall different.

Apart from inversion which is required by the periodic translational invari-ance of the lattice, the triclinic lattice has no special symmetry elements. The

61

Primitive monoclinic unit cell

γ

mC ≡ mP

Body-centered monoclinic unit cell

γ

mF ≡ mI

Body-centered monoclinic unit cell

γ

mC ≡ mI

Figure 37: The base centered monoclinic lattice is equivalent to the primitivemonoclinic lattice. The face centered monoclinic lattice is equivalent to the bodycentered monoclinic lattice. The side centered monoclinic lattice is equivalentto the body centered monoclinic lattice.

62

The Triclinic Lattice

aP

α

β

γ a b

c

Figure 38: The primitive triclinic lattice.

point group of highest symmetry is Ci or 1 which has two elements.

——————————————————————————————————

The presence of only one three-fold axes, either C3 (3) or S6 (3), producesthe trigonal system. There are two types of trigonal systems. In one of thetrigonal systems, a primitive unit cell may be chosen with a = b = c andα = β = γ such that the three-fold axes is along the body diagonal. Theother trigonal system has a = b 6= c and α = β = π

2 and γ = 2 π3 . This

latter system is denoted as the hexagonal system.

3.5.7 Trigonal Bravais Lattice.

The trigonal Bravais lattice is a deformation of the cube produced by stretchingit along the body diagonal. The lengths of the sides remain the same and thethree angles between the sides are all identical. There is only one trigonal Bra-vais lattice. The point symmetry group is D3h or 62m with twelve symmetryoperations.

The body centered cubic and face centered cubic Bravais lattices can be con-sidered to be special cases of the trigonal lattice. For these cubic systems, thesides of the primitive unit cells are all equal and the angles are 109.47 degrees

63

The Triclinic Lattice

aC ≡ aP

The Triclinic Lattice

aI ≡ aP

The Triclinic Lattice

aF ≡ aP

Figure 39: The base centered, body centered and face centered triclinic latticeare equivalent to primitive lattices.

64

The Trigonal unit cell

α

a = b = cα = β = γ

Figure 40: The primitive trigonal unit cell.

Conventional Trigonal unit cell

R

Figure 41: The conventional trigonal unit cell.

65

A

B

A

C

Figure 42: The relationship between the conventional unit cell of the trigonalBravais lattice and the primitive unit cell.

for the b.c.c. structure and 60 degrees for the f.c.c. structure.

The trigonal unit cell has two vertices on the extremes of the body diagonal.The remaining six vertices are arranged on two equilateral triangles. These twotriangles are related by a translation along the body diagonal and a rotation by2π6 around the body diagonal. In the trigonal lattice, all the vertices of the unit

cell are arranged on equilateral triangles which form two-dimensional hexagonalnets. The difference between the trigonal lattice and the hexagonal lattice ismerely due to the different stacking of the hexagonal planes.

——————————————————————————————————

The presence of either a six-fold axes C6 (6) or a rotation - inversion axesS3 (6), indicates that the system is hexagonal. The hexagonal unit cell hasa = b 6= c and α = β = π

2 and γ = 2 π3 .

3.5.8 Hexagonal Bravais Lattice.

The hexagonal Bravais lattice has a unit cell in which the base has sides ofequal length, inclined at an angle of 2 π

3 with respect to each other. The c axisis perpendicular to the base.

The hexagonal system has a point group D6h or 6/mmm which has twenty-

66

The Hexagonal Bravais Lattice

a1

a2

a3

hP

Figure 43: The hexagonal Bravais lattice and the primitive lattice vectors.

four symmetry elements.

There is only one hexagonal Bravais lattice. The primitive unit cells arerhombic prisms which can be stacked to build the hexagonal non-primitive unitcell. The six-fold rotational symmetry of the hexagonal Bravais lattice is mostevident by inspection of the non-primitive unit cell.

The primitive lattice vector are given in terms of Cartesian coordinates by

a1 = a ex

a2 =a

2

(ex +

√3 ey

)a3 = c ez (71)

Possible types of unit cells for a Bravais lattice.

67

Type of unit cell and Symbol Location of Non-origin basis point No. of Lattice Points

Primitive (P) - 1Body-centered (I) The cell center 2Side-centered (A) Centers of the A face (1, 0, 0) 2Side-centered (B) Centers of the B face (0, 1, 0) 2Side-centered (C) Centers of the C face (0, 0, 1) 2Face-centered (F) Centers of all faces 4

68

Relation between conventional and primitive lattice vectors.

Ia1 = 1

2 ( − a + b + c )a2 = 1

2 ( a − b + c )a3 = 1

2 ( a + b − c )

Fa1 = 1

2 ( b + c )a2 = 1

2 ( a + c )a3 = 1

2 ( a + b )

Ra1 = 1

3 ( 2 a + b + c )a2 = 1

3 ( − a + b + c )a3 = 1

3 ( − a − 2 b + c )

Aa1 = aa2 = b

a3 = 12 ( b + c )

Ba1 = aa2 = b

a3 = 12 ( c + a )

Ca1 = a

a2 = 12 ( a + b )

a3 = c

69

In summary the following structures were found:

The Fourteen Bravais Lattices.

Bravais Lattice Lattice Symbol Characteristic Symmetry

Cubic cP, cI, cF a = b = c Four three-fold axesα = β = γ = π

2 along body-diagonals

Tetragonal tP, tI a = b 6= c One four-fold axesα = β = γ = π

2

Orthorhombic oP, oF, oC, oI a 6= b 6= c Three two-fold axesα = β = γ = π

2 mutually perpendicular

Monoclinic mP, mI a 6= b 6= c One two-fold axisα = β = π

2 6= γ

Triclinic aP a 6= b 6= c one-fold axis onlyα 6= β 6= γ 6= π

2 (identity or inversion)

Trigonal hR a = b = c One three-fold axis onlyα = β = γ < 2π

3 , 6= π2

Hexagonal hP a = b 6= c One six-fold axis onlyα = β = π

2 , γ = 2π3

This completes the discussion of the set of fourteen Bravais lattices. In orderto specify crystal structures, it is necessary to associate a basis along with theunderlying Bravais lattice. The addition of a basis can reduce the symmetry ofthe crystal from the symmetry of the Bravais lattices. This results in thirty twopoint groups, and by adjoining the translations and combined operations, onefinds the two hundred and thirty space groups.

——————————————————————————————————

3.5.9 Exercise 5

Form a table of the number of the n-th nearest neighbors and the distancesto the n-th neighbors for the face centered cubic (f.c.c.), body centered cubic

70

(b.c.c.) and simple cubic (s.c.) lattices, for n = 1, n = 2 and n = 3.

——————————————————————————————————

Distribution of types of unit cells amongst the Crystal Systems.

Type of unit cell and Symbol Crystal System Distribution No. of Lattices

Primitive (P) One in each of the seven crystal systems 7Body-centered (I) Cubic, Tetragonal, Orthorhombic, Monoclinic 4

Side-centered (A-B-C) Orthorhombic 1Face-centered (F) Cubic and Orthorhombic 2

Total 14

Having just used symmetry to enumerate all the possible Bravais lattices,we shall now discuss the possible symmetries of crystals. Due to the addition ofthe basis, the point group symmetry of a crystal can be different from the pointgroup symmetry of the Bravais lattice.

71

3.6 Point Groups

The addition of a basis to a lattice can result in a reduction of the symmetry ofthe point group. Here, the point groups are enumerated according to the Bra-vais lattice types and by the Schoenflies designation followed by the appropriate(International) symbol.

The cubic system with a basis can have the point symmetry group of eitherOh (m3m), O (43), Td (43m), Th (m3) or T (23).

The tetragonal system can have point group symmetry of D4h (4/mmm),D4 (42), C4v (4mm), C4h (4/m) or C4 (4).

The orthorhombic system can have point group symmetry of either D2h

(mmm), D2 (222) or C2v (2mm).

The monoclinic system can exist with point group symmetry of either C2h

(2/m), C2 (2) and Cs (m). The group Cs only consists of the identity and areflection operation.

The triclinic system only contains C1 (1) and Ci (m). The group Ci onlyconsists of the identity and the inversion operation.

The trigonal system has the point groups D3h (62m), D3 (32), C3v (3m), S6

(3), or C3 (3).

The hexagonal system has the point groups D6h (6/mmm), D6 (62), C6v

(6mm), C6h (6/m), or C6 (6).

The other four remaining groups are: The groups C3h (6) and D3d (3m)which are usually included in the hexagonal system; and finally, there are thegroups S4 (4) and D2d (42m) which are included with the tetragonal systems.

This completes the enumeration of the thirtytwo point groups.

——————————————————————————————————

3.6.1 Exercise 6

When point group symmetry operations are adjoined to the translations throughlattice vectors, new symmetry elements arise. These new operations are like thepoint group operations but involve other invariant points.

Prove that a rotation about an axis Cn, followed by a translation through avector R which is perpendicular to the axis, is equivalent to a rotation Cn about

72

a parallel axis located on the bisecting plane of R at a distance d perpendicularto the axis. Show that the distance d is given by

d =12|R | cot

π

n(72)

A consequence of this theorem is that, if there is a two-fold rotational axisC2 which is perpendicular to a primitive lattice vector a, then there must beanother two-fold axis of rotation passing through 1

2 a.

——————————————————————————————————

3.7 Space Groups

On combining the point group symmetry operations with lattice translations,one can generate 230 space groups. Often, the space group is composed fromsymmetry operations of the point group and symmetry operations that aretranslations by the vectors of the direct lattice. These space groups are calledsymmorphic groups. Lattices with symmorphic space groups can be constructedby attaching bases with the various point group symmetries on the various Bra-vais lattices. For example, a basis which has the symmetry of any one of the fivecubic point groups can be placed on the three cubic Bravais lattices, yieldinga total of fifteen cubic space groups. Likewise, bases with the symmetries ofthe seven tetragonal point groups can be placed on the two tetragonal Bravaislattices, yielding fourteen tetragonal space groups. This process only leads to61 different space groups. In the other cases, the space groups contain two newtypes of symmetry operations that cannot be compounded from translationsby Bravais lattice vectors and operations contained in the point groups. Thesegroups are non-symmorphic. The new types of symmetry operations occur whenthere is a special relation between the basis dimensions and the size of the Bra-vais lattice. These new symmetry elements include :

Screw Axes. A screw operation is a translation by a vector, not in theset of Bravais lattice vectors, which is followed by a rotation about the axisdefined by the translation vector. A screw symmetry is denoted by nm, wheren represents the rotations 2 π

n , where n = 2 , 3 , 4 , 6 and m represents thenumber of translations by lattice vectors which produce one complete rotationby 2 π. Thus, n screw operations nm, each producing a rotation of 2 π

n , resultin a total translation through m lattice spacings. In other words, the subscriptm indicates a translation of m

n lattice spacings produced by one screw operation.

Glide Planes. A glide operation is composed of a translation by a vector,not in the Bravais lattice, which is followed by a reflection in a plane containingthe translation vector. Glide planes are denoted by either a, b or c (according to

73

21 Screw Axis

cc/2

21

π

31 Screw Axis

cc/3

31

2π/3

2π/3

32 Screw Axis

c

2c/3

32

2π/3

2π/3

Figure 44: Screw operations nm, consists of rotations by 2πn and a translation

by mn lattice spacings.

74

whether the translation is along the a, b or c axis). Also, there are glide planeswhich are denoted n or d (the diagonal or diamond glides) that are special casesinvolving translations along more than one axis.

——————————————————————————————————

3.7.1 Exercise 7

An axial glide reflection is one in which the translation is parallel to a primitiveunit lattice vector, say c. Prove that in this case, the only allowable axial glidesinvolve a translation through c/2.

——————————————————————————————————

The hexagonal close-packed lattice structure has both glide and screw typesof non-symmorphic symmetry operations. The hexagonal close-packed struc-ture can be described by a three-dimensional unit cell which contains a centeredhexagonal base, and which has an identical centered hexagonal top located atvertical distance c directly above the base. If one considers the base hexagon tobe formed by six equilateral triangles, then there are lattice sites at the vertexof each triangle. These lattice sites form a triangular net in the basal planeand there is a similar triangular net in the upper plane. These lattice sites aredesignated as the A sites. There is a second net of triangles at a distance c

2vertically over the base. The centers of the mid-plane equilateral triangles arelocated directly over the (central) lattice sites of the base. There are two pos-sible orientations for these triangles. On choosing any one orientation, the setof lattice sites on this mid-plane are located such that they lie directly over thecenters of every other equilateral triangle in the base. These mid-plane latticesites are designated as the B sites.

Consider a line, parallel to the c axis. The line is equidistant to two neigh-boring B lattice sites and is equidistant to the two A lattice sites that formthe section of the perimeter of the basal hexagon which is parallel to the lineconnecting the above two B lattice sites. Viewed from the c axis, the verticalline passes through the center of the rectangle formed by the two A and twoB lattice sites. This line is the screw axis. The screw operation 21 consists ofa translation by c

2 followed by rotation of π, and brings the A hexagons intocoincidence with the sites of the B hexagons.

The glide planes can also be found by considering the projection of the latticealong the c axis. A line can be constructed which connects any two of the threeB sites inside the hexagonal unit cell. Then a parallel line can be constructedwhich connects a pair of neighboring A sites that forms part of the perimeterof the hexagonal base. Since this line is on the perimeter of the unit cell, itis equivalent to the parallel line segment connecting A sites at the opposite

75

Hexagonal close-packed lattice

C3 21-screw axis

Figure 45: The screw-axis in the hexagonal close-packed structure.

Hexagonal close-packed lattice

σc-glide l

Figure 46: The glide-plane in the hexagonal close-packed structure.

76

boundary. Consider the pair of parallel lines, one which connects the B sites,and the other which is the closest line segment that connects the A sites on theperimeter of the base hexagon. The projection of the glide plane along the caxis is parallel to and equidistant from the above pair of lines. The glide op-eration is a translation by c

2 along the c axis followed by a reflection in the plane.

There are two different systems of nomenclature for space groups: one is dueto Schoenflies and the other is due to Hermann and Mauguin. The Hermann -Maugin space group nomenclature consists of a letter P , I , F , R , C whichdescribes the Bravais lattice type followed by a statement of the essential sym-metry elements that are present. As an example, the space group P63/mmc hasa primitive (P ) hexagonal Bravais lattice with point group symmetry 6/mmm.Another example is given by the space group Pba2 which represents a primitive(P ) orthorhombic Bravais lattice and has a point group of mm2 (the a and bglide planes being simple mirror planes in point group symmetry).

There is some arbitrariness in the distinction between the trigonal and hexag-onal crystal systems. While it is true that the trigonal point group is a sub-group of the hexagonal point group, the trigonal lattice cannot be obtained byinfinitesimal distortion of the hexagonal lattice. As a result, the space groupsof the hexagonal and trigonal lattices are listed together.

Distribution of Space Groups.

Crystal System Number of Space Groups

Cubic 36Tetragonal 68

Orthorhombic 59Monoclinic 13Triclinic 2

Hexagonal - Trigonal 52

The International Tables for Crystallography10 includes listings of crystalstructures for each space group.

10T. Hahn, editor, International Tables for Crystallography, Kluwer publishers, Utrecht,Holland (1996).

77

3.8 Crystal Structures with Bases.

Crystal structures are specified by giving a basis and a Bravais lattice. Thebasis is specified by the positions of and types of atoms in the unit cell.

Sometimes it is also useful to specify the local coordination polyhedronaround each inequivalent site in the lattice. This provides information aboutthe local environment of the atom which is important for bonding. Small de-formations in the positions of the atoms can lower the symmetry of a crystalstructure but usually does not affect the connectivity or topology of the atoms.Therefore, slight deformations of the local environment are often specified bythe same local coordination polyhedra. The local coordination polyhedra havebeen enumerated by W. B. Jensen11, and by Villars and Daams12.

3.8.1 Diamond Structure

The diamond lattice is formed by the carbon atoms in a diamond crystal. Thestructure is cubic, and has the space group Fd3m. The underlying Bravais lat-tice is the face centered cubic lattice and has a two atom basis. In the diamondstructure, both atoms are identical. They are located on the f.c.c. Bravais lat-tice site (0, 0, 0) and at a second site displaced from the Bravais lattice site by adistance a( 1

4 ,14 ,

14 ) in terms of the Cartesian coordinates of the conventional unit

cell. There are four lattice points corresponding to the sites of the conventionalf.c.c. unit cell. There are also four interior points which are displaced from theBravais lattice points by the basis vector a( 1

4 ,14 ,

14 ). Thus, the diamond struc-

ture consists of two interpenetrating face centered cubic lattices with C atomson each lattice site. Diamond possesses a center of inversion located half-waybetween the origins of the two interpenetrating f.c.c. lattices. This is a glide-like inversion operation. The center of inversion is located at a( 1

8 ,18 ,

18 ). When

this is chosen as the origin, the crystal is symmetric under the transformationr → − r.

Each atom is covalently bonded to four other atoms. The bonds on the twoinequivalent basis sites point in different directions. The four neighboring atomsform a tetrahedron centered on each atom. The diamond lattice is most stablefor compounds in which the bonds are highly directional. Directional covalentbonding is often found in the elements of column IV of the periodic table. Inparticular, Carbon, Silicon and Germanium can crystallize in the diamond struc-ture. The great strength of diamond is a consequence of the three-dimensionalnetwork of strong covalent bonds. The diamond structure is relatively open asthe packing fraction is only 0.34.

11W. B. Jensen, The Structures of Binary Compounds, North Holland publishers (1988).12P. Villars and J. L. C. Daams, Journal of Alloys and Compounds, 197, 177 (1993).

78

Diamond Structure

a

cF

Figure 47: The diamond structure can be considered as being composed of twointerpenetrating f.c.c. lattices.

——————————————————————————————————

3.8.2 Exercise 8

Find the angles between the tetrahedral bonds of diamond.

——————————————————————————————————

3.8.3 Graphite Structure

Graphite is the stable form of Carbon. Graphite has a hexagonal unit cell andhas the space group P63/mmc. The primitive lattice vectors may be representedby

a1 =√

3 a ex

a2 =√

32

a ex +32a ey

a3 = c ez (73)

where a is the distance between adjacent atoms in the x− y plane. The atoms

79

Diamond with bonds oriented along the (1,1,1) direction

Figure 48: The diamond structure has atoms on two inequivalent sites whichhave bonds that are oriented in different directions.

a1

a2

Figure 49: A vertical projection of the unit cell of graphite. The primitive latticevectors a1, a2 and the vertices of the unit cell in the basal plane are drawn ingreen. The positions of the other C atoms in the basal plane are drawn in blue.

are located at [0, 0, z] and [0, 0, 12 + z] where z ≈ 0, where the coordinates

are given in terms of the primitive lattice vectors. Another two atoms are lo-cated at the positions [ 23 ,

23 , z] and [13 ,

13 ,

12 + z], where z ≈ 0. The structure is

80

Crystal Structure of Graphite

c

a

Figure 50: The primitive unit cell of the graphite structure.

formed in layers, where each atom is bonded to three other atoms in the plane,thereby forming a two-dimensional hexagonal network. The central site of thetwo-dimensional hexagonal ring is open. The side of the hexagon is of lengtha. The stacking sequence of the layers just corresponds to a translation of onelayer by [ 13 ,

13 ,

12 ] with respect to the other, such that one C atom lies above the

hexagonal hollow in the layer below. The layers are relatively far apart and asis expected, there is only weak van der Waals bonding between the layers. Thisstructure explains the cleavage and other characteristic properties of graphite.

Carbon may crystallize into either as a diamond lattice or as graphite, underdifferent conditions. This is an example of polymorphism which is quite commonamong the elements. Diamonds are not forever as they actually are an unstableform of C under ambient conditions, although the rate of transformation to thestable form (graphite) is exceedingly slow.

Boron and Nitrogen, which occur on either side of Carbon in the periodictable, form compounds which have properties that are strikingly similar to Car-bon. The Boron and Nitrogen atoms can be bonded in either planar structures

81

Crystal Structure of Graphite

c

a

ex

ey

Figure 51: The crystal structure of graphite can be considered as being con-structed by successively stacking open hexagonal layers.

like graphite, or tetrahedral structures, like diamond. The tetrahedral bondedBoron - Nitrogen materials have extremely high melting points and hardness,and have great importance in materials engineering.

——————————————————————————————————

3.8.4 Exercise 9

There are two forms of graphite. The most common form is hexagonal graphitewhich has a stacking sequence A − B − A − B. The other form of graphite isrhombohedral graphite. This is based on a trigonal form which has a stackingsequence A−B −C −A−B −C. Describe the primitive unit cells for the twoforms of graphite. How many atoms are in the primitive unit cells of graphite?

——————————————————————————————————

82

3.8.5 Hexagonal Close-Packed Structure

The hexagonal close-packed structure has hexagonal symmetry and the spacegroup is P63/mmc. It is described as a hexagonal Bravais lattice with a two

The unit cell of the hexagonal close-packed structure

ex

ey

c

a

Figure 52: The hexagonal close-packed structure can be considered as a hexag-onal Bravais lattice with a two-atom basis.

atom basis. The two basis atoms are identical and one is positioned at [0, 0, 0]which is at the vertex of the primitive lattice cell, and the other atom is locatedat [ 13 ,

13 ,

12 ] as expressed in terms of the primitive lattice vectors. (The square

brackets indicate that the direction in the direct lattice are specified with respectto the primitive lattice vectors.) The primitive lattice vectors are

a1 = a ex

a2 =a

2

(ex +

√3 ey

)a3 = c ez (74)

Thus, the hexagonal close-packed structure has a basis of two atoms: one atr1 = (0, 0, 0) and the other at

r2 =13

(a1 + a2

)+

12a3

=a

2ex +

a

2√

3ey +

c

2ez (75)

83

The Hexagonal Bravais Lattice

a1

a2

a3

Figure 53: The hexagonal Bravais lattice.

Since a1 and a2 are inclined at an angle π3 , the structure can be considered to be

formed by two interpenetrating simple hexagonal Bravais lattices. Alternately,the structure may be viewed as being formed by stacking two-dimensional tri-angular lattices above one another with a separation between the layers of halfthe height of the unit cell. Each atom has twelve nearest neighbors: six withinthe hexagonal plane and three in each of the planes above and below the atom.

The name hexagonal close-packed comes from thinking of this structure asbeing formed from hard spheres with radii a

2 and forming a close-packed hexag-onal layer. The second layer is formed by stacking a second hexagonal layer ofatoms above the first. However, the centers of the atoms in the second layerare positioned above the dimples in the first layer. There are two sets of dim-ples between the atoms so there are two different choices for placing the secondlayer of atoms. The third layer is stacked such that the centers of the atomsare directly above the centers of the atoms of the first layer, and the fourth isstacked directly over the second layer, etc. Thus, there are two interpenetratinghexagonal lattices displaced by

13a1 +

13a2 +

12a3 (76)

or [ 13 ,13 ,

12 ].

84

B B

B

C

C C

A

A A

A A

A A

Figure 54: The vertical projection of the hexagonal close-packed stacked struc-ture. The close-packed layer of spheres in the basal plane are centered at the Asites. The next layer of atoms are centered over the sites B (or equivalently C),and the third layer is centered over the A sites. Hence, the layers are stackedin the sequence A−B −A−B etc.

There are a total of twelve nearest neighbor atoms which are distributed as6 neighbors in the plane, 3 in the plane above, and 3 in the plane below. Thisgives a total of twelve nearest neighbor atoms.

On assuming that the atoms are hard spheres with radii r and that thespheres are touching, the lattice constants satisfy a = b = 2 r and c = 4

√2√3r.

This yields the hexagonal close-packed structure and has the particular ratio ofthe c to the a axis lengths of

c

a=

√83

= 1.633 (77)

This is the ideal c to a ratio. Hexagonal close-packed systems with the idealratio have a packing fraction of 0.74. As atoms are not hard spheres, there isno reason for this value of the c to a ratio to be found in naturally occurringcrystals, and deviations from the ideal value are found most frequently. OnlyHe has the ideal c to a ratio.

The most frequently occurring structures are the close-packed structures.These are the hexagonal close-packed, face centered cubic and body centeredcubic structures, which have packing fractions of 0.74, 0.74 and 0.68, respec-tively. Both simple and transition metals frequently form in the hexagonalclose-packed structure, or other close-packed structures.

85

——————————————————————————————————

3.8.6 Exercise 10

Show that the ca ratio for an ideal hexagonal close-packed lattice structure is

c

a=(

83

) 12

(78)

——————————————————————————————————

3.8.7 Exercise 11

Na transforms from b.c.c. to h.c.p. at 23 K via a Martensitic transition. Onassuming that the density remains constant and the h.c.p. structure is ideal,find the h.c.p. lattice constant a in terms of the b.c.c. value a′.

——————————————————————————————————

3.8.8 Other Close-Packed Structures

One can form other close-packed structures by altering the sequence of stackingof the close-packed layers. The hexagonal close-packed can be characterizedby the repeated stacking sequence A - B - A - B etc. That is, the atoms inthe planes above and below the triangular lattice have centers directly over thedimples and each other, thereby creating a two layer unit cell.

Another stacking sequence is given by A - B - C in which the unit cellconsists of three layers. The A and C layers have the atoms centered on the twoinequivalent sets of triangular dimples of the B layer. This close-packed stackingcorresponds to the face centered cubic lattice. The packing fraction of the facecentered cubic lattice and hexagonal close-packed lattice are identical. Thetriangular close-packed nets are the planes perpendicular to the body diagonalof the conventional f.c.c. unit cell. There are two such planes which pass throughthe conventional unit cell and two further planes that each just graze one vertexof the unit cell. The intercepts of the planes with the conventional (Cartesian)axes are (1, 0, 0), (0, 1, 0) and (0, 0, 1). The next plane has intercepts (2, 0, 0),(0, 2, 0) and (0, 0, 2). The sets of planes are known as 1, 1, 1 planes and arecomposed of triangular arrays of atoms, where the sides of the triangles havelengths a√

2. The normal to the planes are in the direction [1, 1, 1] i.e.

n =1√3

(ex + ey + ez

)(79)

86

Conventional face-centered cubic unit cell

Figure 55: The conventional unit cell of the f.c.c. structure. The diagonal planesare seen to consist of close-packed layers.

where e are the orthogonal unit vectors of the conventional cell. The equationsof the planes are (

r − m a ex

). n = 0 (80)

where m is an integer that labels the plane by the intercept with the x axis.The quantity m is related to the perpendicular distance, s, between the planeand the origin by

s = ma√3

(81)

for integer m.

It is convenient to introduce three new orthogonal unit vectors to describethe positions of the atoms in the planes. The first is n the normal to the planes

n =1√3

(ex + ey + ez

)(82)

The other vectors e1 and e2 are chosen to be vectors in the planes. These forma new set of Cartesian non-primitive lattice vectors which are defined by

e1 =1√2

(ex − ey

)(83)

87

which corresponds to the face diagonal of the conventional unit cell that lies inthe triangular plane and

e2 =1√6

(ex + ey − 2 ez

)(84)

which is the “lateral” direction in the triangular plane. The lateral displace-ments of atoms between one triangular plane, say the plane which passes throughthe atom at ( 1

2 ,12 , 0), and the atoms on the next plane (centered on the origin

(0, 0, 0)) can be written as

∆r =a

2

(ex + ey

)− n

a√3

=a

6

(ex + ey − 2 ez

)=

1√3

a√2e2 (85)

This can be re-written as23

√3

2a√2e2 (86)

as a√2

is the triangular lattice constant and√

32

a√2

is the height of the triangle.Thus, the atoms in consecutive planes are displaced “laterally” by 0, 2

3 , and 43

and this sequence then repeats. The resulting structure has layers which havea stacking sequence A−B − C −A−B − C etc.

There are other possible stacking sequences, with longer periodicities. Theearlier lanthanides and late actinides have a stacking sequence A - B - A - Cwith four layers per unit cell, however, the Sm structure only repeats itself af-ter nine layers. The longest known periodicity is 594 layers which is found ina polytype of SiC. The long-ranged crystallographic order is not due to long-ranged forces, but is caused by spiral steps caused by dislocations in the growthnucleus. There is also the possibility of random stacking sequences.

3.8.9 Sodium Chloride Structure

The Sodium Chloride or NaCl structure is cubic. The space group is Fm3m.It has an ordered array of Na and Cl ions located on the sites of a simple cubiclattice of linear dimension a

2 . Each type of ion is surrounded by six ions of theopposite charge located at a distance a

2 away. The twelve next nearest neigh-bors have like charge and are located at a distance 1√

2a away along the face

diagonals of the cubic unit cell. There are four units of NaCl in the unit cell.The structure may be most efficiently visualized as having the Na+ ions locatedon the sites of a face centered cubic lattice which has its origin at (0, 0, 0) andthe Cl− ions are located on the sites of an interpenetrating face centered cubic

88

The Sodium Chloride Structure

cF

a

Figure 56: The Sodium Chloride structure can be considered as an f.c.c. Bravaislattice with a two-atom basis.

lattice which has its origin at the center of the cubic unit cell ( 12 ,

12 ,

12 ).

The Sodium Chloride structure is favored by many ionic compounds. In thisstructure, the electrostatic interactions are balanced by the short-ranged repul-sive interactions due to the finite size of the ions. The short-ranged repulsionsare due to the Pauli exclusion principle. The sizes of the ions are importantin determining the stability of this structure. If the ions of opposite charge areenvisaged as just touching, then the ionic radii must satisfy the equality

a = 2[r(Na+) + r(Cl−)

](87)

Ions of the same type are closest along the face diagonals, so if they do nottouch, the lattice constant satisfies the inequality

1√2a > 2 r(Cl−) (88)

Combining the above two equations yields an inequality for the ratio of the ionicradii of the ions

r(Cl−)r(Na+)

≤ 1 +√

2 (89)

If this inequality is not obeyed, the Pauli forces render the structure unstable.

89

Examples of materials that form in the NaCl structure are the alkali halidesmade from the alkaline elements Li, Na, K, Rb or Cs with a halide elementF , Cl Br or I. Alternatively, one can go to the neighboring columns of theperiodic table and combine Mg, Ca, Sr or Ba with a chalcogen O, S, Se or Teto form the NaCl structure.

3.8.10 Cesium Chloride Structure

The ionic compound Cesium Chloride or CsCl has a cubic structure. The spacegroup is Pm3m. The Cs+ ion is located at (0, 0, 0) and the Cl− ion at the bodycenter of the cube (1

2 ,12 ,

12 ). Thus, the CsCl structure resembles a body centered

cubic structure in which one type of atom is at the simple cubic sites and theother type of atom is at the body center. Each ion is surrounded by eight atoms

Cesium Chloride Structure

cP

Figure 57: The Cesium Chloride structure can be considered as a simple cubicBravais lattice with a two-atom basis.

of opposite charge located at a distance√

32 a away, which corresponds to half

the length of the body diagonal of the cube. Each atom has six neighbors ofsimilar charge located a distance a away. The ratio of the ionic radii requiredfor this structure to be possible is

r(Cl−)r(Cs+)

≤ (√

3 + 1 )2

(90)

90

If the radii ratio is greater than 1.366, but less than 2.42, ionic compounds pre-fer the NaCl structure.

Examples of compounds that form the CsCl structure are the Cs halides,T l halides, CuZn (beta brass), CuPd, AgMg and LiHg.

Linus Pauling has produced a set of empirical rules which determine thecoordination numbers in terms of the ionic radii of the ions. If one assumesthat the anion adopts the cubic close-packed structure (f.c.c.), there are threetypes of holes between the close-packed spheres and each type of hole has adifferent size. It is assumed that the cations fit into one set of holes. Thecentral void of the conventional f.c.c. unit cell is surrounded by an octahedronand, therefore, has a coordination number of six. There are also tetrahedralholes with coordination number four. The tetrahedral holes are located nearthe 8 corners of the f.c.c. cube, and the vertices of the tetrahedra are located atthe corner and the three neighboring face centers. Alternatively, the tetrahedralholes can be seen by considering an octant of the f.c.c. cube. The tetrahedralhole site is at the center of the octant, and the four vertices of the tetrahedronare located at four of the octant’s eight corners. There are twelve trigonalholes which are located near the eight vertices of the conventional unit cell. Thetrigonal sites reside in the planes formed by the vertex and any two of the closestface centers. The radius ratio rule suggests that the structure is determined bymaximizing the coordination numbers while keeping ions of opposite charge incontact. This procedure seems likely to maximize the electrostatic attractionenergy. By considering the geometry of the holes, one expects that certainstructures will be stable for different values of the radius ratio

rr =r(X−)r(R+)

(91)

For the tetragonal sites, by considering the body diagonal of the octant, oneexpects that

2[r(X−) + r(R+)

]=

√3

2a (92)

and by considering the face diagonal

2 r(X−) <a√2

(93)

Hence, we find the tetragonal hole has the limiting radius ratio of

r(X−)r(R+)

> 2( √

32

+ 1)

(94)

In particular, the radius ratio rules suggest that the range of radii ratios where

91

the various f.c.c. based configurations are stable are given by

Name Coordination No.

6.45 > rr > 4.45 trigonal 34.45 > rr > 2.41 tetrahedral 42.41 > rr > 1.37 octahedral 6

If the atoms have comparable sizes, then it is necessary to consider more openstructures with higher coordination numbers such as simple cubic. For the sim-ple cubic structure, the coordination number of the central hole is eight andthe hole size is larger, so 1.37 > rr. Thus, since rr ∼ 1.8 for Na and Cl,it fits the radius ratio rules as being octahedrally coordinated like in the NaClstructure. On the other hand, for Cs and Cl where the ions have comparablesizes, the radius ratio is rr ∼ 1.07 which is compatible with the cubic holestructure found in CsCl.

3.8.11 Fluorite Structure

Fluorite or CaF2 has a cubic structure. The space group is Fm3m. Ioniccompounds of the form RX2, in which the ratio of the ionic radii r satisfy theinequality

r(X−)r(R2+)

≤ (√

3 + 1 )2

(95)

can form the fluorite structure. The unit cell has four Ca2+ ions, one at theorigin and the others are located at the face centers of the cube. The eight F−

ions are interior to the cube. The F− ions are located at the vertices of simplecubes which are concentric with the unit cells, but the simple cubes have onlyhalf the lattice spacing of the unit cell. Alternatively, the eight F− ions can beconsidered to lie on two interpenetrating f.c.c. lattices with origins ( 3

4 ,14 ,

14 ) and

( 34 ,

34 ,

34 ). Since the two F sites are symmetrically displaced from the Ca site by

one quarter of the f.c.c. body diagonal, there is an inversion symmetry abouteach of the Ca sites. Each F anion occupies a site at the center of a tetrahedronformed by the Ca cations.

Materials such as LiO2, form an anti-fluorite structure. The anti-fluoritestructure is the same as the fluorite structure except that the positions of theanions and cations are reversed. The O anions are in the f.c.c. positions andthe Li cations form a simple cubic array.

92

The Fluorite Structure

a

cF12

Figure 58: The Fluorite structure can be considered as an f.c.c. Bravais latticewith an interpenetrating simple cubic lattice.

3.8.12 The Copper Three Gold Structure

The Cu3Au structure is cubic, and has the space group Pm3m. The Bravaislattice corresponds to a primitive cubic structure. There are three Cu atomsand one Au per unit cell. All the atoms are located on the sites of a facecentered cubic unit cell. The Au atom can be envisaged as being positioned onthe corners of the cube, whereas the three Cu atoms sit on the faces centers ofthe cube, thereby forming an octahedron inside the cube. Thus, the basis of thestructure consists of the position of the Au atom

r0 = 0 (96)

and the three Cu atoms are located at

r1 =a

2( ey + ez )

r2 =a

2( ex + ez )

r3 =a

2( ex + ey ) (97)

The Au atoms have twelve Cu nearest neighbors located at a distance a√2,

whereas the Cu atoms only have four Au nearest neighbors.

93

Cu3Au Structure

cP4

Figure 59: The Cu3Au structure is equivalent to a simple cubic structure witha four atom basis.

Other compounds with the Cu3Au structure are Ni3Al, TiP t3 and themetastable compound Al3Li.

3.8.13 Rutile Structure

The structure possessed by rutile (TiO2), by cassiterite (SnO2) and by nu-merous other substances with small cations, is tetragonal. The space group isP42/mnm. The Ti4+ ions occupy positions : (0, 0, 0) ; ( 1

2 ,12 ,

12 ) while the O2−

ions occupy the four positions ± (x, x, 0) ; ( 12±x,

12∓x,

12 ) where x ≈ 3

10 . Thus,the titanium atoms occupy the sites of a body centered tetragonal lattice. Theoxygen atoms are located on lines which are oriented along one set of face diago-nals of the base. Oxygen atoms are also located on horizontal lines through thebody centers, and are orthogonal to the lines in the base. Thus, the titaniumion is surrounded by six O atoms which form a slightly distorted octahedron.

——————————————————————————————————

94

Rutile Structure

c

a tP6

Figure 60: A unit cell of the Rutile structure.

3.8.14 Exercise 12

Consider the Rutile structure for TiO2. What conditions must hold for the Oatoms to form a hexagonal close-packed structure?

——————————————————————————————————

3.8.15 Zinc Blende Structure

The Zinc Blende structure or ZnS structure is cubic. This is also known as theSphalerite structure. The space group is F43m. The Zn2+ ions are positionedat (0, 0, 0) and at the face centers of the cube. The S2− are positioned onthe sites of an interpenetrating face centered cubic lattice with origin ( 1

4 ,14 ,

14 ).

There are four units of ZnS in the unit cell. The Zinc Blende structure is similarto the diamond structure. The main difference between the Zinc Blende anddiamond structures is that the Zinc Blende structure involves two different typesof atoms, while the diamond structure only involves one type of atom. Eachatom in ZnS is surrounded by a regular tetrahedron of atoms of the oppositetype. Unlike diamond, Zinc Blende has no center of inversion, as the diamondinversion operator interchanges the two different types of atoms. The radius

95

Zinc Blende Structure

a

cF

Figure 61: The Zinc Blende structure can be considered as an f.c.c. Bravaislattice with a two-atom basis.

ratio rules suggest that the Zinc Blende structure will be adopted whenever

2( √

32

+ 1)

> rr > 1 +√

2 (98)

The Zinc Blende structure is often found for binary compounds formed frompairs of elements from either the II - VI columns, III - V columns or the I - VIIcolumns of the periodic table.

3.8.16 Zincite Structure

Zincite (ZnO) has a hexagonal structure. This structure is also known as theWurtzite structure (the hexagonal form of ZnS). The space group is P63mc.The primitive lattice vectors are given by

a1 = a ex

a2 =a

2( ex +

√3 ey )

a3 = c ez (99)

The Zn and O atoms occupy the positions [ 0, 0, z ]; [ 23 ,

23 ,

12 + z ] where

z = 0 for Zn and about z ≈ 38 for O. Since ZnS also is found in this form

96

above 1300 K, it is not surprising that Zincite structure has a local coordina-tion similar to that of the low-temperature Zinc Blende structure. Each atomis surrounded by a tetrahedron of atoms of the opposite type. The tetrahedraform continuous interconnected networks. However, symmetry does not requirethat the tetrahedra are regular.

Wurtzite Structure

Figure 62: The Zincite structure can be considered as two interpenetratinghexagonal close-packed lattices.

The cubic Zinc Blende and the hexagonal Wurtzite structures are closelyrelated. They merely differ by the stacking sequence of the Zn (S) close-packedplanes. The structure consists of alternate close-packed planes which eithercontain only Zn or only S ions. The set of planes form layers consisting of apair of planes. Within a layer, the Zn atoms in one plane and the S atoms inthe other plane are bonded by vertical tetrahedral bonds. The remaining threetetrahedral bonds join the layer with atoms in the successive layers. Due to theorientation of the inter-layer tetrahedral bonds, successive pairs of planes aredisplaced horizontally. Thus, the successive sets of vertical bonds are displacedhorizontally.

In the cubic Zinc Blende sequence, the tetrahedra of the S atom bonds havethe same rotational orientation in each layer so that each S layer is displaced inthe same direction. The net horizontal displacement produced in three verticalS layers is equal to the periodicity in the direction of the displacement. This

97

can be considered as having a stacking sequence A - B - C which repeats.

In the hexagonal Wurtzite sequence, the tetrahedra of bonds are rotated byπ between successive S layers. Thus, the horizontal displacement that occursbetween one S layer and the next are cancelled by the opposite displacementthat occurs by going to the very next S layer. This stacking sequence is A - Bwhich repeats.

——————————————————————————————————

3.8.17 Exercise 13

Consider the Wurtzite structure in which one type of atom can be regarded asoccupying some of the tetrahedral interstitial sites. Show that in the ideal case,the lattice parameters are given by

c

a=

2√

63

z =38

Also show that, if the nearest neighbor distances are the same in the Zinc Blendeand Wurtzite lattices, then the lattice constants are related via az =

√2aw.

——————————————————————————————————

3.8.18 The Perovskite Structure

The perovskite structure, as exemplified by BaTiO3, is cubic at high tempera-tures but becomes slightly tetragonal on cooling below a ferro-electric transitiontemperature. The cubic structure has the space group Pm3m. The structure iscomposed of the Ti atoms positioned on the simple cubic lattice sites (0, 0, 0),and the Ba atoms positioned at the body center sites ( 1

2 ,12 ,

12 ). The three O

atoms are located at the mid-points of the edges of the cube, i.e. at (0, 0, 12 ),

(0, 12 , 0) and ( 1

2 , 0, 0). An alternate representation of the unit cell is found bycentering the lattice on the Ba ions and by translating the origin via 1

2 (1, 1, 1).In this representation, the Ti atoms are located at the body centers, and the Oatoms lie on the face centers. The TiO2 form a set of parallel planes separatedby planes of BaO. Each Ti atom is surrounded by an octahedron of O atomswhich have corners that are shared with the octahedron surrounding the neigh-boring Ti atoms.

——————————————————————————————————

98

Perovskite Structure

(1/2,0,0)(0,1/2,0)

(0,0,1/2)

(1/2,1/2,1/2)

Figure 63: The cubic perovskite structure, as found in BaTiO3.

3.8.19 Exercise 14

The density of the face centered cubic structure is highest, body centered cubicis the next largest, followed by simple cubic and then diamond has the lowestdensity. This correlates with the coordination numbers. The coordination num-ber is defined to be the number of nearest neighbors. The coordination numbersare 12 for the f.c.c. lattice, 8 for b.c.c., 6 for s.c. and 4 for diamond. Assumethat the atoms are hard spheres that just touch. Find the packing fraction ordensity of these materials.

——————————————————————————————————

99

3.9 Lattice Planes

A Bravais lattice plane, by definition, passes through three non-collinear Bravaislattice points. Since these points are connected by combinations of multiplesof the primitive lattice vectors, and due to the periodic translational symmetryof the lattice, the lattice planes must contain an infinite number of lattice points.

Given one such lattice plane, there exists a family consisting of an infiniteset of parallel lattice planes with the same normal. One such lattice plane mustpass through each Bravais lattice point, since the lattice viewed from any latticepoint is identical to the lattice when viewed from any other lattice point. Thus,the family of parallel planes contain all the points of the Bravais lattice.

Each member of the set of lattice planes must intersect the axes given bythe primitive lattice vectors a1, a2 and a3. The planes need not intersect anyparticular axis at a lattice point, however, every lattice point located on anyaxis will have one member of the family pass through it. In particular, one planemust pass through the origin O.

A Family of Lattice Planes

a1

(3,2,1)

2 a1 3 a11/3 a1

3 a22 a2

a21/2 a2

a3

2 a3

3 a3

O

Figure 64: Members of a family of lattice planes. Each plane is uniquely specifiedby its intercepts with the axes.

Each plane is uniquely specified by the three intercepts of the plane with theaxes formed by three primitive lattice vectors directed from the origin to theBravais lattice points a1, a2 and a3. The positions of the intercepts are denoted

100

by the numbers x1, x2 and x3, where the x’s are measured in units of the lengthof the primitive lattice vectors. That is, the intercepts are x1 a1, x2 a2 and x3 a3.

The three points of intersection of one lattice plane with the three primi-tive axes can be represented as κ [ 1

h1, 0, 0], κ [0, 1

h2, 0] and κ [0, 0, 1

h3], where κ

is a positive or negative integer, and (h1, h2, h3) are also positive or negativeintegers. The integers (h1, h2, h3) are chosen such that they have no commonfactors. The index κ serves to distinguish between the different members of thesame family of planes. The plane that passes through the origin has κ = 0,whereas the plane that passes next closest to the origin has κ = 1. The planesthat are at successively further distances from the origin have larger magnitudesof κ.

The indices (h1, h2, h3) are found by locating the intercepts of the plane withthe three primitive axes, say x1 a1, x2 a2 and x3 a3, inverting the intercepts1x1, 1

x2, 1

x3, and then finding the smallest three integers which have the same

ratio1x1

:1x2

:1x3

= h1 : h2 : h3 (100)

The set of integers (h1, h2, h3) are enclosed in round brackets and denote theMiller indices of the plane. A negative valued integer, such as − h1, is denotedby an overbar such as h1.

The Miller indices label the direction of the normal to the family of planes.Since the vectors between pairs of intercepts are parallel to the plane, the threevectors

1h1

a1 − 1h2

a2

1h2

a2 − 1h3

a3

1h3

a3 − 1h1

a1 (101)

are parallel to the plane. Any two of these vectors span the plane, so the thirdvector is not independent. The normal to the plane is parallel to the vectorproduct of any two non-collinear vectors in the plane

n ∝ κ2

(1h1

a1 − 1h2

a2

)∧(

1h2

a2 − 1h3

a3

)=

κ2

h1 h2 h3

(h3 a1 ∧ a2 + h2 a3 ∧ a1 + h1 a2 ∧ a3

)(102)

Thus, the direction of the normal to the plane is given in terms of the compo-nents hi in the three directions defined by aj ∧ ak. The three vectors have the

101

same directions as the primitive “reciprocal lattice vectors”.

The primitive reciprocal lattice vectors are defined by

b1 = 2 πa2 ∧ a3

a1 . ( a2 ∧ a3 )(103)

and cyclic permutations of the set (1, 2, 3). These primitive reciprocal latticevectors are, in general, not orthogonal. The normal to the plane is then givenby the direction of the reciprocal lattice vector Bh

Bh = h1 b1 + h2 b2 + h3 b3 (104)

where (h1, h2, h3) are the Miller indices. The length of this reciprocal latticevector is defined as

| Bh |2 =(h1 b1 + h2 b2 + h3 b3

)2

=(

2 πdh

)2

(105)

The quantity dh is the minimum distance which separates the two closest mem-bers of the family of planes. This is seen through the following consideration:

Interplanar spacings

a

bd110

d130

Figure 65: the spacing between successive members of a family of planes isrelated to the length of the reciprocal lattice vector.

102

The equation for the points r on the plane which intercept the primitive latticevectors ai at distances xi = κ

hiis given by

r . Bh =κ

h1a1 . Bh

=κ

h1a1 .

(h1 b1

)= 2 π κ (106)

The minimum distance s, between the origin and the plane is given by

s = r .Bh

| Bh |

=(dh

2 π

)r . Bh

= κ dh (107)

Thus, it is found that the spacing between successive planes in the family isgiven by s = dh, and the planes are equidistant.

Sets of families of planes that are equivalent in a given crystal structureare denoted by h, k, l. For example, in a cubic crystal the families of planes(1, 0, 0), (0, 1, 0) and (0, 0, 1) are equivalent and are denoted by 1, 0, 0.

A direction of a vector in the direct lattice is specified by three integers insquare brackets [n1, n2, n3] and specify a vector

n1 a1 + n2 a2 + n3 a3 (108)

A negative value for a component is also denoted by an overbar. The set of direc-tions which are equivalent for a crystal structure are denoted by < n1, n2, n3 >.

——————————————————————————————————

3.9.1 Exercise 15

Consider the (1, 0, 0) and (0, 0, 1) planes of an f.c.c. lattice with axes describedby the conventional unit cell. What are the indices of the planes when referredto the primitive axes?

——————————————————————————————————

3.9.2 Exercise 16

The angles α1 ( 6 a2 , a3 ), α2 ( 6 a3 , a1 ) and α3 ( 6 a1 , a2 ) between thethree primitive lattice vectors of the direct lattice, ai, are related to the angles

103

A primitive unit cell

a1

α1

α2

α3

a2

a3

Figure 66: The primitive unit cell and the angles between the primitive latticevectors.

βi between the three primitive lattice vectors of the reciprocal lattice, bi. Theangles βi are defined as β1 ( 6 b2 , b3 ), β2 ( 6 b3 , b1 ) and β3 ( 6 b1 , b2 ).Show that

cos α1 =cos β2 cos β3 − cos β1

| sin β2 sin β3 |(109)

and also find the inverse relation.

——————————————————————————————————

3.9.3 Exercise 17

Complete the table which shows the values of the inter-planar spacings (dh1,h2,h3)in terms of the Miller indices for the seven primitive Bravais lattices.

——————————————————————————————————

104

System dh1,h2,h3

Cubic[

1a2 (h2

1 + h22 + h2

3)]− 1

2

Tetragonal[(

h21+h2

2a21

)+ h2

3a23

]− 12

Orthorhombic[

h21

a21

+ h22

a22

+ h23

a23

]− 12

Monoclinic[( h2

1a21+

h22

a22− 2h1h2 cos α3

a1a2

sin2 α3

)+ h2

3a23

]− 12

Triclinic ?

Trigonal[

1a2

(h21+h2

2+h23) sin2 α+2(h1h2+h2h3+h3h1)(cos

2 α−cos α)1−3 cos2 α+2 cos3 α

]− 12

Hexagonal[

43

(h21+h2

2+h1h2

a21

)+ h2

3a23

]− 12

105

3.10 Quasi-Crystals

Quasi-crystals have symmetries intermediate between a crystal and a liquid.Quasi-crystals are usually intermetallic alloys. The quasi-crystal is space filling,but unlike a regular Bravais lattice, does not have just one unit cell. (Theyusually have two types of unit cell.) These different “unit cells” are stacked in away such that there is no long-ranged positional order but nevertheless, retainorientational order. The absence of long-ranged positional order lifts the restric-tion on the symmetry of the lattice but puts a restriction on the vectors thatdescribe the “unit cells”. For example, an Al−Mn quasi-crystal13 has icosahe-dral symmetry with two, three and five-fold axes. The structure is made fromblocks consisting of a central Mn atom surrounded by twelve Al atoms arrangedat the corners of an icosahedron. This type of icosahedral structure is often thearrangement of thirteen atoms which has the lowest energy14. The icosahedraare stacked together with the same orientation. The voids between the icosahe-dra are the second structural unit. The five-fold symmetry of the icosahedra isnot allowed for a regular Bravais lattice. The five-fold point group symmetryimposes a restriction on the lengths of the “lattice vectors” of a quasi-crystalto have certain irrational ratios. Thus, the reciprocal lattice contains reciprocallattice vectors of arbitrary small magnitude which show up as an extremely highdensity of reflections in x-ray scattering15.

A way of obtaining quasi-crystal structures is by projecting a periodic Bra-vais lattice structure in higher dimensions (six or more) onto three dimensions16.To illustrate this, consider a square two-dimensional lattice, with lattice con-stant a. On any unit cell, construct two parallel lines with slope tan θ passingthrough opposite corners. The equations of the lower line is given by

y = x tan θ (110)

and the upper line is determined by

y = a + ( x + a ) tan θ (111)

For rational values of the slope, tan θ = pq , the lattice points cross the line

periodically, with repeat distance q a along the x direction and have periodicityp a along the y direction. Lines with irrational values of the slope cannot crossmore than one lattice point and, therefore, do not have periodic long-rangedorder. The points (na,ma) contained in the area between the two lines mustsatisfy the inequality

1 + ( n + 1 ) tan θ > m > n tan θ (112)

13D. Shechtman, I. Blech, D. Gratais and J. W. Cahn, Phys. Rev. Lett. 53, 1951 (1984).14F. C. Frank, Proc. Roy. Soc. London, 215, 43 (1952).15D. Levine and P. Steinhart, Phys. Rev. Lett. 53, 2477 (1984).16P. Kramer and R. Neri, Acta. Crystallogr. Sec. A 40, 580 (1984).

106

Construction of a one-dimensional Quasicrystal

x-axis

y-ax

is θ

Figure 67: The one-dimensional quasi-crystal is constructed by projecting thelattice points on a strip of a two-dimensional lattice.

Project the lattice points contained within the strip onto one of the lines. Thedistance s along the lower line is given by

s = n a cos θ + m a sin θ (113)

wherem =

∑m′

m′ Θ(1 + (n+ 1) tan θ −m′) Θ(m′ − n tan θ) (114)

For irrational values of the slope, the resulting array of points is a quasi-periodicarray. The spacing between adjacent points of the quasi-periodic array is eithergiven by cos θ or sin θ. The spacings are not distributed periodically but nev-ertheless, are distributed according to some irregular or more complex pattern.If the slopes of the lines are equal to 1

2 (√

5 − 1 ), the spacings between theprojected points forms a Fibonacci series. For a Fibonacci series of numbers,the first term can be chosen in any way but the next term is given by the sum

107

of the preceding two numbers, i.e., Fn+1 = Fn + Fn−1. Thus, both series 1, 1 , 2 , 3 , 5 , 8 , 13 etc. or 3 , 3 , 6 , 9 , 15 etc. are Fibonacci series. Thegolden mean 1

2 (√

5 + 1 ) is the limit of the ratio of the successive terms. Inour example, the sequences of spacings is given by s c s c c s c s c . . .. The firstelement of the Fibonacci series is s, the second element is c, the third elementcomprises of s c, the next element is c s c, which is followed by s c c s c etc.If this type of analysis is applied to high-dimensional Bravais lattices, one canfind three-dimensional quasi-crystal structures with five-fold symmetry.

A five-fold symmetry is also found when tiling a two-dimensional plane withtwo types of tiles, both having the same length of edge s, but with angles ofπ5 or 2 π

5 . The “diameter” to side ratios of these two types of tiles satisfyds = s

d′ =√

5 − 12 . The sides of the tiles are marked and the tiles are ad-

joined so that the markings match17. The result is a tiling without long-ranged

Figure 68: The two types of Penrose tiles. The tiles are decorated by arrowswhich indicate the matching rules.

periodic order although every finite area segment repeats an infinite number oftimes in the plane. These types of tilings are known as Penrose tilings. ThePenrose tiling has long-ranged orientational order, as can be seen by decoratingeach tile with lines. The lines on the tiles join up to form five families of parallellines (Ammann lines). The five sets of lines make an angle of 2 π

5 with respect toeach other. The spacing between the successive members of a family of parallellines form Fibonacci series.

Most of the theoretical studies of real quasi-crystals have considered projec-tions of a d-dimensional lattice onto three-dimensions. The d-dimensional hy-percubic lattice is spanned by d orthogonal unit vectors and contains d(d−1)/2!different orientations of its two-dimensional areas and d(d− 1)(d− 2)/3! orien-tations of its three-dimensional volumes etc18. The projection of a hypercubeonto a three-dimensional space is an isohedral polyhedron whose faces are par-allelograms. The faces of the polyhedron have d(d−1)/2! different orientations,

17M. Gardner, Scientific American, 236, 110 (1977).18The number Nk of k-dimensional sub-units of the d-dimensional hypercube is given by

the coefficient of xk in the expansion of (2x + 1)d. That is

Nk = 2d−k(

dk

)108

Figure 69: A two-dimensional quasi-crystal or Penrose tiling.

Figure 70: If the Penrose tiles are decorated with lines as shown, the Penrosetiling reveals sets of parallel lines.

and the volume can be partitioned into d(d−1)(d−2)/3! rhombohedra. The 24

The generalization of Euler’s formula to d dimensions is given by

d−1∑k=0

( − 1 )k Nk = 1 − ( − 1 )d (115)

109

Figure 71: Ammann lines on a Penrose tiling.

vertices of the four-dimensional hypercube can be projected onto the verticesof a rhombic dodecahedron. A rhombic dodecahedron has fourteen vertices andtwelve faces which have identical shapes in the form of a parallelogram. Theparallelograms have fixed angles tan α1

2 =√

2 and tan α22 = (

√2)−1. The rhom-

bic dodecahedron can be partitioned into four equivalent rhombohedra. The 26

vertices of the six-dimensional hypercube can be projected onto the vertices of atriacontahedron. The triacontahedron has the full symmetry of the icosahedralgroup. The five-fold symmetry can be regarded as originating from a five-foldrotation symmetry about a principal axis of the six-dimensional lattice. Underthis rotation, the projections of the remaining five orthogonal unit vectors areinterchanged. Due to the choice of the projection, the sixty four vertices ofthe six-dimensional hypercube project onto thirty two vertices of the triaconta-hedron. Of the thirty two vertices, twelve vertices form a regular icosahedronand the twenty other vertices form a regular dodecahedron. Five faces meet ateach vertex of the icosahedron and three faces meet at each vertex of the do-decahedron. Hence, the axes of the five-fold and the three-fold rotations of theicosahedral point group pass through these vertices. The edges join a vertex ofthe icosahedron to a vertex of the dodecahedron (E = 5× 12 = 3× 20). Thus,these thirty two vertices are joined by sixty edges which form thirty identicalfaces (V −E+F = 2). The faces are parallelograms which contain two five-foldvertices and two three-fold vertices (F = 5×12

2 = 3×202 ). It can be shown that

the projection of the hypercube from the six-dimensional space results in thirty

110

Triacontahhedron

Figure 72: A triacontahedron.

equivalent parallelograms with fixed angles α1 and α2, where cosα1 = 1√5

and α2 = π − α1. The triacontahedron can be partitioned into twentyrhombohedra, and these twenty rhombohedra fall into two sets of ten equiv-alent rhombohedra. That is, the triacontahedron can be partitioned into tenequivalent prolate rhombohedra and ten equivalent oblate rhombohedra. Thesetwo types of rhombohedra are known as the Ammann rhombohedra, and onlythese two types of rhombohedra can be formed from the two-dimensional par-allelograms. The three-dimensional Penrose tilings are space filling packings ofthe Ammann rhombohedra.

111

4 Structure Determination

Structure can be determined by experiments in which beams of particles arescattered from the material. Elastic scattering experiments are usually preferredsince the underlying material is not dynamically deformed by these processes.In order that the results be easily interpretable in terms of the structure, it isnecessary that the wave length associated with the beam of particles has thesame order of magnitude as the spacing between atoms in the structure andsecondly, the beam of particles should only interact weakly with the structure.The first condition allows for a clear resolution of diffraction peaks caused bythe atomic structure. The second condition ensures that the beam is scatteredprimarily in the bulk or interior of the material, and not just the surface. Italso allows for an easy interpretation of the data via second order perturbationtheory.

4.1 X Ray Scattering

X-rays are usually used in the determination of the atomic structure of solids.The strength of the interaction is measured by the deviation of the dielectricconstant from its vacuum value ( 1 ). At energies of about 10 keV, the wavelength of the x-rays λ is ∼ 10−10 m, and at these high energies, the refractiveindex is almost unity.

In x-ray diffraction, the x-rays are elastically scattered from the charge den-sity of the electrons. The formal theory of x-ray scattering shows that theintensity of the reflected waves is given by the Fourier Transform of the electrondensity - density correlation function. For a solid which possesses long-rangedorder, the resulting expression for the intensity can be simplified down to involvethe square of the Fourier transform of the electron density. In order to eluci-date the role of the Bravais lattice and the coherent nature of x-ray scattering,the atoms shall first be considered to be point-like objects. Later, the spatialdistribution of the electrons around the nuclei shall be re-introduced.

4.1.1 The Bragg condition

Bragg considered the specular reflection of a beam of x-rays from successiveplanes of atoms separated by distances d. If the angle between the beam of x-raysand the planes (not the normal to the plane) is θ

2 , then the difference in opticalpath lengths for a beam specularly reflected at the lower of two consecutivelayers is

2 d sinθ

2(116)

112

Bragg scattering by planes of atoms

θ/2

θ/2

d

d sin θ/2 d sin θ/2

Figure 73: Bragg reflections from successive planes of atoms. The angle θ is thescattering angle.

In this expression, θ is the scattering angle of the particles in the beam. Thereflected beams superimpose with a phase difference of

4 πd

λsin

θ

2(117)

and constructive interference occurs whenever

n λ = 2 d sinθ

2(118)

This is Bragg’s law19. The value of n is called the order of the Bragg reflection.Since the successive planes are equi-spaced, the scattering for an entire familyof planes is constructive when the scattering from two neighboring planes in thefamily is constructive. Since there are a large number of planes in a family, andsince the solid is almost transparent to x-rays, the scattering amplitude fromeach member of the family adds coherently, giving rise to a very high intensityof the scattered beam whenever Bragg’s condition is satisfied.

In the application of Bragg’s law to x-ray scattering, not only must one con-sider the different coherent scattering conditions from a single family of planes,

19W. L. Bragg, Proc. Cambridge Phil. Soc. 17, 43 (1913).W. L. Bragg, Nature, 90, 402 (1913).

113

but one must also consider scattering from all the different families of planes inthe solid. Different families of planes of atoms in a solid have different orien-tations of their common normal. Since a plane of every family passes througheach lattice point, the spacing between the members of a family of planes maydepend on the orientation of the normal. That is, the minimum spacing betweenlattice planes d can vary from family to family. The different Bragg reflectionsare usually indexed by the Miller indices (m1,m2,m3) of the planes that theyare reflected from.

4.1.2 The Laue conditions

Laue’s condition is more general than that of Bragg. The Laue condition20 isderived by considering scattering from the basis atoms in each of the primitiveunit cells in the solid. The individual cells scatter the x-rays almost isotropi-cally, however, the scattering in a specific direction will only be coherent at wavelengths for which the scattered waves from each unit cell add constructively.

e'e

θ'/2θ/2

d

The Laue Condition

Figure 74: The Laue condition is equivalent to requiring that the scatteringfrom any two unit cells interfere constructively.

20W. Friedrich, P. Knipping and M. von Laue, Proc. Bavarian Acad. Sci. 303 (1912).

114

The wave vector of the incident beam is expressed as k, where

k =2 πλ

e (119)

and the reflected wave has wave vector k′

k′ =2 πλ

e′ (120)

where e and e′ are two unit vectors. Let us consider two identical scatteringcenters separated by a vector displacement d. Consider two rays in the incidentbeam, each of which is scattered from one of the centers. The difference inoptical path lengths of the two x-rays is composed of two non-equal segments

d sinθ

2= d . e

− d sinθ′

2= d . e′ (121)

where the angle between the wave-front of the incident beam and d is given byθ2 , so the angle between d and e is π

2 −θ2 . The optical path difference between

the two waves is given by the difference

d sinθ

2+ d sin

θ′

2= d . ( e − e′ ) (122)

Thus, constructive interference of the scattered waves from two unit cells occurswhenever

d .

(e − e′

)= m λ (123)

holds for integer m. This condition can be re-expressed in terms of the wavevectors of the incident and scattered x-rays as

d .

(k − k′

)= 2 π m (124)

If this condition is fulfilled for the set of vectors d that consists of all the Bravaislattice vectors R, one finds the Laue condition for coherent scattering

R .

(k − k′

)= 2 π m (125)

or alternatively

exp[i

(k − k′

). R

]= 1 (126)

If this condition is satisfied for all R in a solid with N unit cells, constructiveinterference will occur between all pairs of unit cells, giving rise to coherentscattering. The cross-section will have N2 such contributions, and the scattered

115

wave will be extremely intense.

If the scattering vector q is defined as

q = k − k′ (127)

the Laue condition is satisfied for the special set of q values Q, which satisfy

exp[i Q . R

]= 1 ∀ R (128)

These special q values can be used to obtain the k values for which coherentscattering will occur. The expression for the momentum transfer is

k′ = k − Q (129)

which can be squared to yield

k'

k

θ

q

q = k - k'

The Scattering Geometry

Figure 75: The scattering angle θ is determined by the momentum transfer qand the momentum of the beam of incident particles.

k′2 = k2 − 2 k . Q + Q2 (130)

This equation may be combined with the condition for elastic scattering | k | =| k′ |, to result in a condition on the incident k values for coherent scattering ofthe form

Q2 = 2 k . Q (131)

116

O

kk-Q/2

Q/2 Q

Figure 76: The Laue condition is satisfied when k resides on the Bragg plane.

Thus, k will satisfy the Laue condition for coherent scattering when the compo-nent of k along Q bisects Q. Thus, the projection of k along Q must be equal tohalf the length of Q. The incident wave vector k must lie on the plane bisectingthe origin and Q, which is called the Bragg plane.

The Laue condition is satisfied if Q . R = 2 π m for all lattice vectors R.In particular, if the Laue condition is satisfied, one can choose R to be any oneof three primitive lattice vectors. The three choices of primitive lattice vectorsyields the three equations,

a1 . Q = 2 π m1

a2 . Q = 2 π m2

a3 . Q = 2 π m3

(132)

Since any lattice vector R can be expressed as integer multiples of the primitivelattice vectors, these three Laue equations are equivalent to the Laue condition.The three Laue equations have a geometrical interpretation. Namely, Q lies ona cone around the direction of a1 with projection 2 π m1. Similarly, Q alsolies on a certain cone around a2, and also on a cone around a3. Thus, Q mustlie on the common intersection of the three cones. This is a severe constraint:the values of k for which this is satisfied can only be found by systematicallysweeping the magnitude of k or by rotating the direction of k which is equivalentto systematically re-orienting the crystal.

However, once Qivalues have been found which satisfy the Laue conditions,

117

other Q values can be found which are integral multiples of the initial Qi’s.

General considerations show that there are three basis vectors bi which satisfythe set of equations

ai . bj = 2 π δi,j (133)

The three basis vectors bi can be used to construct the general values of Q whichsatisfy the Laue condition. These special values of Q have the form of reciprocallattice vectors

Q =3∑

i=1

mi bi (134)

where the mi are integers.

4.1.3 Equivalence of the Bragg and Laue conditions

The Laue condition makes it clear that each Q value defines a normal to a set oflattice planes indexed by the Miller indices (m1,m2,m3). Since a plane belong-ing to each family of planes passes through each lattice point, it can be shownthat the Laue condition is equivalent to the Bragg condition.

Let Q = k − k′ be a scattering wave vector such that Q . R = 2 π m forall lattice vectors R. Since it can be shown that

Q2 = − 2 k′ . Q= + 2 k . Q (135)

and as k and k′ have the same magnitude, the incident and scattered wave vec-tors make the same angle θ

2 with the Bragg plane.

Due to the elastic scattering condition, one has |Q| = 2 k sin θ2 and, if

the scattering is coherent, the magnitude of Q can be written as |Q| = 2 π nd ,

where n is the order of the reflection and d is the minimum distance betweenplanes in the family. Combining the elastic and Laue conditions, one has

k sinθ

2=

π n

d

2 d sinθ

2= n λ (136)

Thus, the Laue diffraction peak associated with the change in k given byk − k′ = Q, just corresponds to a Bragg reflection by an effective familyof planes which have Q as their normal.

The order n of the Bragg reflection just corresponds to the magnitude of| Q | divided by 2 π

d , where d is the separation between the closest members ofthe family of planes.

118

The Elastic Scattering Condition

k

- k'

θ/2θ/2q

k = k'

Figure 77: For elastic scattering, the magnitude of the momentum transfer isgiven by q = 2 k sin θ

2 .

4.1.4 The Ewald Construction

Since the Laue condition is very restrictive, the vectors k which produce co-herent scattering are relatively few and far between. The Ewald construction21

provides a convenient way of visualizing how the Laue condition may be fulfilled.

The incident wave vector k is centered on the origin O. A sphere of radiusk′ ( = k ) is constructed which is centered on the tip of k. This is the Ewaldsphere. The scattered wave vectors have the magnitude k′ and may be repre-sented by vectors k′ directed from points on the sphere’s surface towards thecenter of the sphere. The vector − k′ is also represented by a vector centeredon the tip of k with an end on the Ewald sphere. The scattering wave vectorsq = k − k′ are directed from the origin towards the points on the surface ofthe sphere.

The scattering wave vectors Q which are solutions of

Q . R = 2 π m (137)

form a lattice of points (the reciprocal lattice) which include Q = 0. Whenthe lattice of Q points is imposed on the Ewald sphere, a lattice point has tobe centered on the origin. This lattice point corresponds to the un-scatteredbeam for which k = k′, hence q = 0. The lattice of Q points is indexed bythree integers (m1,m2,m3) corresponding to the components along three prim-itive (reciprocal) lattice vectors. When a second point of the lattice of Q pointsresides on the surface of the Ewald sphere, say at the tip of − k′, it producesa Bragg reflected beam. In this case, the Laue condition is satisfied and the

21P. P. Ewald, Z. Krist. 56, 129 (1921).

119

The Ewald Sphere Construction

kk'

-k'QO

Figure 78: Coherent scattering occurs when a reciprocal lattice vector Q resideson the surface of the Ewald sphere.

incident beam will be Bragg reflected at this k′ value. In general, it is expectedthat the Ewald sphere will not have a second lattice point on the surface. WhenBragg reflections occur, they are indexed by the integers (m1,m2,m3) whichdescribe the family of planes associated with the momentum transfer Q.

4.1.5 X-ray Techniques

There are various techniques which can be used to obtain diffracted beams.

In the Laue Method, a beam of x-rays with a continuum of wave lengthsin the range between λ0 and λ1 is used, and the incident beam has a fixeddirection. Thus, it is only appropriate to use this method for a single crystal,as a polycrystalline sample would correspond to an average over the relativeorientation with the incident beam. In the Laue method, the continuous wave-length of the beam broadens the surface of the Ewald sphere into a finite volumeenclosed between two Ewald spheres with the limiting wave lengths. For a largeenough mismatch between the wave length of the interior Ewald sphere λ0 andthe exterior sphere λ1, it is quite likely that at least one Bragg reflection willoccur. This method provides the simplest method for orienting a single crystalrelative to the direction of the incident beam. If the incident beam is along adirection of high symmetry of the lattice of Q points, the pattern of reflected

120

Ewald Construction for the Laue method

k1

k2

|k1|

|k2|

O

Figure 79: In the Laue method, coherent scattering occurs whenever a reciprocallattice vector is located within the volume enclosed between the Ewald spheresof radius k2 and k1

beams should exhibit the same symmetry. It should be noted that the x-raypattern will always show a center of symmetry, even if the crystal does not haveone. This discovery is due to Friedel22.

The Rotating Crystal Method uses a monochromatic beam of x rays, andin the experiment, the relative direction of the incident beam and the crystal isvaried. If one considers the lattice of points Q as being fixed, then the Ewaldsphere rotates around the origin and, for large enough k, will sweep some latticepoints through the surface of the sphere. This experiment produces a set ofBragg reflected beams that are recorded on a photographic film. In practice,the crystal is rotated about a crystallographic axis, say a1, while the incidentbeam has a fixed direction perpendicular to a1. The photographic film is bentinto a cylinder with an axis which is chosen to coincide with the axis of rotationof the crystal. Since the incident beam is perpendicular to the rotation axis, thenthe Bragg reflected beams occur within cones of fixed angle. That is, the b2 andb3 components of the lattice of Q points form planes which are perpendicularto a1. Therefore, under the rotation, these two components of the Q vectors

22G. Friedel, Comptes Rendus, Acad. Sci. (Paris) 157, 1533 (1913).

121

Ewald Sphere for the Rotating Crystal Method

k

Spheres with radii |Q|Ewald Sphere with radius |k|

Figure 80: In the rotating crystal method, Bragg scattering occurs wheneverspheres with radii Q centered on the origin cut through the Ewald sphere.

a1

k

k'

- Q . a1

Figure 81: In the rotating crystal method, the scattering occurs within conesaround the crystallographic axis a1.

122

are rotated in the planes. However, the components of Q parallel to a1 remaininvariant and are governed by m1, since

Q . a1 = 2 π m1 (138)

Furthermore, since k and a1 are perpendicular

k′ . a1 = − Q . a1 (139)

and the reflected beams produce a series of Bragg spots which exist in ringswrapped around the photographic film cylinder. Each ring corresponds to adifferent value of m1. Direct observation of the angle between k′ and a1 allowsthe magnitude of a1 to be obtained with ease.

The Debye-Scherrer Method uses a polycrystalline or powdered sample.Each grain of the sample has a random orientation, therefore, this method isequivalent to the rotating crystal method in which the sample is rotated overall possible orientations. Each reciprocal lattice point will generate a sphere ofradius equal to the magnitude of the reciprocal lattice vector. If this sphericalshell of reciprocal lattice vectors intersects with the Ewald sphere, it producesBragg reflections. Each lattice vector with length less than 2 k will produce acone of Bragg reflections with an angle θ relative to the un-scattered beam. Themagnitude of the reciprocal lattice vector is given by Q = 2 k sin θ

2 . Thus,a measurement of θ will give the lengths of the smallest reciprocal lattice vectors.

These methods can be used to determine the reciprocal lattice vectors andhence, the Bravais lattice associated with the crystal. In order to completelydetermine the crystal structure, one must determine the basis. This can be doneby examining the structure and form factors.

——————————————————————————————————

4.1.6 Exercise 18

The icosahedral molecule C60 molecule has been proposed as having the formof a truncated icosahedron with twenty six-membered rings and twelve five-membered rings23. These molecules can be condensed into solid phases, inwhich the C60 molecules are rotationally disordered. Fleming and co-workersdescribe the structure of various alkaline metal C60 compounds in their Naturearticle, Nature 352, 701, (1991).

In figure (2.a) of their paper, Fleming et al. state that the solid has an f.c.c.structure. Indicate the axes of the conventional unit cell on their unit cell.

23H. W. Kroto, J. R. Heath, S. C. O’Brien, R. F. Curl and R. E. Smalley, Nature, 318, 162(1985).

123

Powder Diffraction

Sample Film

Figure 82: The diffraction of x-rays by a powdered sample gives rise to a patternof concentric circles on the photographic film.

Figure 83: Unit cells of crystals of doped C60. [After Fleming et al. (1991).]

If a powder x -ray diffraction experiment is performed on Rb doped C60 withx-rays of wavelength λ = 0.9 A, for the dopings 3, 4 and 6 in the paper, whatare the scattering angles 2 θ for the first five diffraction peaks for the observedstructures?

——————————————————————————————————

124

4.1.7 The Structure and Form Factors

If the lattice has a basis, the scattered wave from each unit cell must be com-posed from the scattered waves from each atom in the basis. This means thatthe scattering from each type of atom in the basis must be determined andthen superimposed to find the scattered wave. The scattering from the electrondensity of each atom can be expressed in terms of the form factor24. The form

0

4

8

12

16

20

24

28

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

sin (θ/2) / λ [ θ/2) / λ [ θ/2) / λ [ θ/2) / λ [ Angstroms-1 ] ] ] ]

FF

e(q) (110)

(200)(211)

(220)

(310)

(222)

(321)

(411)

Hartree-Fock

(330)

λ = 0.709 Αλ = 0.709 Αλ = 0.709 Αλ = 0.709 Α

Figure 84: The experimentally determined and calculated atomic form factorsFFe(q) for Fe. The angle θ is defined as the scattering angle. [After Batterman,Chipman and DeMarco, Phys. Rev. 122, 69 (1961).]

factor for atoms in a solid differ only slightly from the form factors of isolatedatoms25, and are mainly determined by the atomic charge number Z. Althoughthere are differences due to the bonding, the form factors are determined by allthe electrons and not just those involved in bonding. The form factor of thej-th atom in the basis is denoted by Fj(q). It is conventional to use a scale suchthat the forward scattering (θ = 0) atomic form factor equals the number ofelectrons in the atom. Since the coherent scattering is restricted to scatteringvectors Q that satisfy the Laue condition, the form factor only needs to beevaluated at these values of Q. The amplitude of the scattered wave from theatoms in the basis of the unit cell can be expressed in terms of the structurefactor S(Q) which is given by

S(Q) =∑

j

exp[i Q . rj

]Fj(Q) (140)

24C. G. Darwin, Phil. Mag. 43, 800 (1922).25The covalently bonded semiconductors provide notable exceptions to this statement. For

example, in diamond, the electronic charge density has maxima intermediate between theatomic positions (N. A. W. Holzwarth, S. G. Louie and S. Rabii, Phys. Rev. B 26, 5382(1982).). This is a manifestation of the covalent bonding, and has a result that the structurefactor does not vanish at Q = 2π

a(2, 2, 2). Hence, this is seen as an extra peak in x-ray

diffraction spectra. (S. Gottlicher and E. Wolfel, Z. Electrochemie, 63, 891 (1959).)

125

Figure 85: The calculated and experimentally determined valence electroncharge densities of graphite. [After Holzwarth et al. (1982).]

This is just the component of Fourier Transform of the electron density fromone unit cell. The intensities of the Bragg peaks are proportional to the factor

| S(Q) |2 (141)

The Q dependence of the intensity can be used to determine the basis of thecrystal. Unfortunately, only the modulus of S(Q) and not its phase, can befound from diffraction experiments which produce only one scattered Braggbeam26. Therefore, indirect methods have to be used to discover the crystalstructure. However, if the crystal is centro-symmetric, then if there is an atomat the basis point rj then there is another atom of the same type at − rj andS(Q) is purely real. In this case, the phase problem just simplifies to the ques-tion as to whether S(Q) is positive or negative.

26In the case that more than one Bragg is produced in the scattering experiment, saycorresponding to reciprocal lattice vectors Q

1and Q

2, then the relative phase of the structure

factor S(Q1) and S(Q

2) can be determined. (L. D. Chapman, D. R. Yoder, and R. Colella

Phys. Rev. Lett. 46, 1578 (1981), B. Post, Phys. Rev. Lett. 39, 760 (1977)).

126

If the basis of a crystal structure is mono-atomic, the atomic form factorcan be factorized out, and the amplitude of the scattered wave is partiallydetermined by the geometric structure factor

SG(Q) =∑

j

exp[i Q . rj

](142)

The geometric structure factor expresses the interference between identical atomsin the basis. The intensity of the Bragg peak is still determined by the productof the modulus of the form factor with the modulus of the geometric structurefactor. The vanishing or variation of the Bragg peak intensities due to interfer-ence can be used to determine the positions of the basis atoms.

An example of the ambiguity imposed by the non-measurability of the phaseof the Structure Factor is given by Friedel’s law for non-centrosymmetric crys-tals. The structure factor S(Q) is a complex number, and can be written as

S(Q) = A + i B (143)

For each Q that satisfies the Laue condition, there is a vector −Q which corre-sponds to the negative integers (−m1,−m2,−m3). The structure factor S(−Q)is just the complex conjugate of S(Q)

S(−Q) = A − i B (144)

Since the structure factor for both the vectors Q and −Q have the same magni-tude, the Bragg peaks have the same intensity. Thus, the diffraction pattern hasa center of inversion symmetry, even if the crystal structure does not. Excep-tions to Friedel’s law only occur if the crystal has anomalous dispersion. Thishappens when the x-rays are highly absorbed by the crystal.

Face Centered Cubic Structures.

A structure with a face centered cubic lattice and a one atom basis can alsobe represented in terms of a simple cubic lattice with a four atom basis. Thescattering from this structure can be expressed in terms of the Laue condition forthe simple cubic lattice but modulated by the geometric structure factor. Thefour atom basis of the non-primitive (conventional) unit cell of a face centeredcubic structure with a monoatomic basis consists of the atomic positions

r1 = 0

r2 =a

2

(ex + ey

)r3 =

a

2

(ez + ex

)r4 =

a

2

(ez + ey

)(145)

127

The Bragg vectors for the conventional simple cubic cell are easily found to be

bx =(

2 πa

)ex

by =(

2 πa

)ey

bz =(

2 πa

)ez (146)

so a general simple cubic Bragg scattering vector is given by

Q =(

2 πa

) (m1 ex + m2 ey + m3 ez

)(147)

The geometric structure factor for the conventional f.c.c. unit cell is found tohave four contributions

SG(Q) =∑

j

exp[i Q . rj

]

=

[1 + exp

[+ i π ( m1 + m2 )

]+

+ exp[

+ i π ( m1 + m3 )]

+ exp[

+ i π ( m2 + m3 )] ]

(148)

one contribution is provided by each basis atom. When evaluated at the Braggvectors, the geometric structure factor adds coherently

SG(Q) = 4 (149)

if the integers (m1,m2,m3) are either all even or are all odd. The geometricstructure factor interferes destructively

SG(Q) = 0 (150)

if only one integer is different from the other two. That is, if one integer is ei-ther even or odd, while the other two respectively, are odd or even, then SG(Q)vanishes. Thus, the f.c.c. lattice has the same pattern of Bragg reflections asthe simple cubic lattice, but has missing Bragg spots. The resulting lattice ofBragg spots is cubic with twice the dimensions (in q space) but has missingBragg spots at the mid-points of the edges and at the face centers. Thus, it isfound that the diffraction pattern has the form of a body centered cubic lattice.

Body Centered Cubic Structures.

128

A structure with a body centered cubic lattice and a one atom basis can alsobe viewed as a simple cubic lattice with a two atom basis

r0 = 0

r1 =a

2

(ex + ey + ez

)(151)

Then, the geometric structure factor for the conventional b.c.c. unit cell onlycontains two terms

SG(Q) = 1 + exp[i Q .

a

2( ex + ey + ez )

]= 1 + exp

[ia

2( Qx + Qy + Qz )

](152)

Each basis atom provides one contribution to the geometric structure factor.Now the Bragg vectors for the simple cubic structure are just

Q =2 πa

( m1 ex + m2 ey + m3 ez ) (153)

therefore, at these Q values the geometric structure factor simplifies to

SG(Q) = 1 + exp[i π ( m1 + m2 + m3 )

]= 1 +

(− 1

)( m1 + m2 + m3 )

= 2 for ( m1 + m2 + m3 ) even

= 0 for ( m1 + m2 + m3 ) odd(154)

Thus, the body centered cubic lattice has Bragg spots that form a cubic lattice.However, the intensity of the odd indexed Bragg spots vanish, leading to a facecentered cubic lattice of Bragg spots.

The Diamond Structure.

The diamond structure is described by an f.c.c. lattice with a two atombasis. The positions of the basis atoms are given by

r0 = 0

r1 =a

4

(ex + ey + ez

)(155)

where the conventional f.c.c. unit cell has sides of length a.

129

From the discussion of scattering from an f.c.c. structure, one finds that theQ vectors of the Bragg spots can be expressed in terms of the set of primitivevectors for a b.c.c. lattice

Q =∑

i

mi bi (156)

The primitive lattice vectors are given by

b1 =2 πa

(ey + ez − ex

)b2 =

2 πa

(ez + ex − ey

)b3 =

2 πa

(ex + ey − ez

)(157)

The geometric structure factor of the diamond lattice relative to the lattice ofBragg spots of the real space f.c.c. lattice, is given by

SG(Q) = 1 + exp[iπ

2( m1 + m2 + m3 )

](158)

From this it is found that the geometric structure factor not only gives riseto extinctions but also modulates the intensity of the non-zero Bragg spotsaccording to the rule

SG(Q) = 2 for ( m1 + m2 + m3 ) 2 × even

SG(Q) = 0 for ( m1 + m2 + m3 ) 2 × odd

SG(Q) = 1 ± i for ( m1 + m2 + m3 ) odd(159)

As the f.c.c. lattice has Bragg spots arranged on a b.c.c. lattice, it is con-venient to transform the Bragg vectors into the coordinates system used for aconventional b.c.c. unit cell

Q =4 πa

[ex

(12

( m1 + m2 + m3 ) − m1

)+ ey

(12

( m1 + m2 + m3 ) − m2

)+ ez

(12

( m1 + m2 + m3 ) − m3

) ](160)

130

The rule for the modulation of intensities is expressed directly in terms of thequantity ∑

i

Qi a

4 π=

12

( m1 + m2 + m3 ) (161)

Thus, one can describe the system of Bragg spots as residing on a b.c.c. latticewhere the cubic cell has sides with dimensions of 4 π

a . The b.c.c. lattice can bere-interpreted in terms of two interpenetrating simple cubic lattices. Thus, theBragg spots with non-equal intensities reside on two interpenetrating simple cu-bic lattices with dimensions of 4 π

a . This scale is twice as large as the reciprocallattice spacing of the (simple cubic) lattice constructed from the conventionalunit cell.

One simple cubic lattice contains the origin Q = 0, and the Bragg spotshave integer coefficients for the unit vectors ex, ey and ez. This means that( m1 + m2 + m3 ) is even for this simple cubic lattice. On dividing by afactor of 2, the resulting number is odd and even at consecutive lattice points.When ( m1 + m2 + m3 )/2 is an even integer, S = 2 and the intensitiesare finite. However, when ( m1 + m2 + m3 )/2 is odd then S = 0 so theintensities of the Bragg peaks vanish. Thus, the non-zero intensities on this sim-ple cubic reciprocal lattice actually forms a face centered cubic reciprocal lattice.

The second interpenetrating simple cubic lattice has Bragg points with half(odd) integer coefficients for the unit vectors ex, ey and ez. This means thatthe sum ( m1 + m2 + m3 ) is odd for this simple cubic lattice. These latticepoints are the body center points of the underlying b.c.c. lattice. The geometricstructure factor is simply SG(Q) = 1 ± i and thus, the Bragg spots on thissimple cubic lattice all have the same intensities.

Extinctions due to Glide Planes and Screw Axes.

The symmetry and metric properties of an X-ray diffraction pattern can beused to determine the point group symmetry of the Bravais lattice of a sample.The observation of systematic absences of Bragg spots due to any centering ofunit cells and other symmetry operations with translational components, maybe extremely useful in the identification of the space group of the sample.

Consider an orthorhombic solid with a glide symmetry where the translationis along the ez axis and the reflection occurs in a plane with a normal along theey axis. Thus, if there is an atom at (x, y, z) in units of the lattice parameters,there is an equivalent atom at (x, y, z + 1

2 ). The pairs of basis atoms eachcontribute a term

SG(Q) = exp[

2 π i ( x m1 + y m2 + z m3 )]

+ exp[

2 π i ( x m1 − y m2 + ( z +12

)m3 )]

(162)

131

Glide Operation

-2

-1

0

1

2

-2 -1 0 1 2

ez

ey

ex

(x,y,z)

(x,-y,z+1/2)x

Figure 86: A crystal is symmetric under the glide operation consists of a trans-lation by c

2 in the ez direction, followed by a reflection in the x − z plane.Therefore, if an atom is located at (x, y, z), then an identical atom is located at(x,−y, z + 1

2 ).

to the geometric structure factor. One can see that for the special case m2 = 0the structure factor is composed of terms with the form

SG(Q) = exp[

2 π i ( x m1 + z m3 )] (

1 + exp[π i m3

] )= exp

[2 π i ( x m1 + z m3 )

] (1 + ( − 1 )m3

)= 0 if m3 is odd

= 2 exp[

2 π i ( x m1 + z m3 )]

if m3 is even

(163)

Thus, reflections of the type (m1, 0,m3) will be missing unless m3 is an evennumber.

Similar extinctions occur for screw axes. Consider a two-fold screw axisparallel to ey. The equivalent positions are (x, y, z) and (x, 1

2 + y, z). Thus, the

132

Figure 87: The structure of a C60 molecule.

structure factor at m1 = 0 and m3 = 0 has contributions of the form

SG(Q) = exp[

2 π i y m2

] (1 + ( − 1 )m2

)= 0 if m2 is odd

= 2 exp[

2 π i y m2

]if m2 is even

(164)

Thus, reflections of the type (0,m2, 0) will be missing unless m2 is an even in-teger.

——————————————————————————————————

4.1.8 Exercise 19

Experiments on solid AxC60 show that the C60 molecules are located on a facecentered cubic lattice with lattice spacing a = 14.11 A, and that the (2, 0, 0)x-ray diffraction peak is very weak when compared to the (1, 1, 1) Bragg peak.

133

Fleming et al. Nature 352, 701 (1991). Calculate the structure factor for these

Figure 88: Data from x-ray diffraction experiments on crystals of doped C60.[After Fleming et al. (1991).]

reflections in an approximation which assumes that the electron distribution ofeach fullerene molecule is uniformly spread over a spherical shell of radius 3.5 A.

——————————————————————————————————

4.1.9 Exercise 20

The Hendricks-Teller model27 for x-ray diffraction from a disordered systemconsiders a one-dimensional line of molecules. The probability that a pair ofatoms is separated by a distance a is given by p and the probability that theyare separated by a + da is given by 1 − p. The random system has an infiniteunit cell. Calculate the averaged scattering intensity for this model, and show

27S. B. Hendricks and E. Teller, J. Chem. Phys. 10, 147 (1942).

134

that

I(q) = N I0

p ( 1 − p )[

1 − cos q da]

1− p(1− p)− p cos qa− (1− p) cos[ q(a+ da) ] + p(1− p) cos qda(165)

Due to the random phases introduced by the distribution of distances, the scat-tering is not fully coherent. However, for small amounts of disorder, the coher-ence can be almost fully recovered.

——————————————————————————————————

Polyatomic Crystals.

For a polyatomic crystal, the structure factor has both the geometric con-tribution and the contribution from the atomic form factors of the basis atoms

S(Q) =∑

j

exp[i Q . rj

]Fj(Q) (166)

The atomic form factor Fj(Q) is determined by the internal structure of theatom that occupies the position rj in the basis.

The atomic form factor is normalized to the electronic charge of the atom.For a single atom, the form factor is given by

F (Q) =∫

d3r ρ(r) exp[− i Q . r

](167)

where ρ(r) is the atomic electron density. If the charge density is sphericallysymmetric, then the form factor can be reduced to a radial integral

F (Q) = 2 π∫ ∞

0

dr r2 ρ(r)∫ 1

−1

d cos θ exp[− i Q r cos θ

]= 2 π

∫ ∞

0

dr r2 ρ(r)2 sin Q r

Q r

= 4 π∫ ∞

0

dr r2 ρ(r)sin Q r

Q r(168)

For forward scattering, Q = 0, the form factor reduces to

F (0) = 4 π∫ ∞

0

dr r2 ρ(r)

= Z (169)

where Z is the atomic number. Typically, F (Q) decreases monotonically withincreasing Q, falling off as a power of 1

Q2 for large Q.

——————————————————————————————————

135

4.1.10 Exercise 21

Calculate the x-ray scattering intensities for the close-packed structures formedby stacking hexagonal layers in the following sequences:

(a) The sequence ABAB... (the h.c.p. sequence).

(b) The sequence ABCABC... (the f.c.c. sequence).

(c) The random sequence in which all the consecutive layers are different,but given one layer (say A), there is an equal probability that it will be followedby either one of the two other layers.

——————————————————————————————————

4.1.11 Exercise 22

Find the atomic form factor for the hydrogen atom using the electron density

ρ(r) =1

π a3exp

[− 2 r

a

](170)

where a is the Bohr radius.

——————————————————————————————————

Sodium Chloride.

An example of a diatomic crystal with a basis is provided by NaCl. This hasa face centered cubic lattice and has Na+ ions at the positions (0, 0, 0), ( 1

2 ,12 , 0),

( 12 , 0,

12 ) and (0, 1

2 ,12 ). The Cl− ions reside at (1

2 , 0, 0), (0, 12 , 0), (0, 0, 1

2 ) and( 12 ,

12 ,

12 ). The structure can be viewed as a simple cubic lattice with a six atom

basis. In this case, we can use the simple cubic representation of the Braggvectors Q. Thus, the structure factor is given by

S(Q) = FNa(Q)(

1 + exp[i π ( m1 + m2 )

]+ exp

[i π ( m2 + m3 )

]+ exp

[i π ( m3 + m1 )

] )+ FCl(Q)

(exp

[i π m1

]+ exp

[i π m2

]+ exp

[i π m3

]+ exp

[i π ( m1 + m2 + m3 )

] )(171)

136

Since exp[ i π m ] = ( − 1 )m, the structure factor can be factorized as

S(Q) =(FNa(Q) +

(− 1

)m1

FCl(Q))

×[

1 +(− 1

)(m1+m2)

+(− 1

)(m2+m3)

+(− 1

)(m3+m1) ](172)

The structure factor is 0 unless the indices are either all odd or all even. This ischaracteristic of face centering. The intensities of the Bragg spots with all evenindices and all odd indices are different as the atomic form factors either add orsubtract.

——————————————————————————————————

4.1.12 Exercise 23

Potassium Chloride has the same structure as NaCl. However, unlike NaCl,K+ and Cl− are iso-electronic and so have very similar form factors. Determinethe indices (m1,m2,m3) of the allowed Bragg reflections.

——————————————————————————————————

4.1.13 Exercise 24

Calculate the structure factor for the Zinc Blende structure. The Zinc Blendestructure is a face centered cubic lattice of side a, with a positively charged ionat the origin and a negatively charged ion at a

4 ( ex + ey + ez ).

——————————————————————————————————

Since the differences between the atomic form factors show up in the exper-imentally observed structure factor of compounds, it is possible to distinguishbetween ordered binary compounds and binary compounds with site disorder.The order-disorder transition in Cu3Au has been observed by x-ray scattering.At high temperatures, the atoms in this material are randomly distributed suchthat there is one atom at each site of an f.c.c. lattice. In this disordered state,the apriori probability that a Au atom will be found on any particular site isroughly 1

4 , and the probability that a Cu atom will be found at any particu-lar site is 3

4 . However, there is a transition from the disordered phase, whichoccurs above a critical temperature of Tc ≈ 660 K, to an ordered phase atlower temperatures. In the completely disordered phase, the structure factor isthat pertaining to an f.c.c. crystal in which the form factor is replaced by the

137

statistically averaged value

Fav(Q) =34FCu(Q) +

14FAu(Q) (173)

Thus, at high temperatures, the structure factor is given by

S(Q) = Fav(Q)(

1 + exp[i π ( m1 + m2 )

]+ exp

[i π ( m2 + m3 )

]+ exp

[i π ( m3 + m1 )

] )(174)

Hence, the peaks have intensity of either 16 | Fav(Q) |2 or zero depending onwhether the indices are all even or all odd, or whether they are mixed. In theordered phase, the Cu atoms reside on the face center sites and the Au on thevertices of the cubes. In this ordered phase, “super-lattice” peaks appear inthe spectra for mixed indices. For the completely ordered phase, the structurefactor is given by

S(Q) = FAu(Q) + FCu(Q)(

exp[i π ( m1 + m2 )

]+ exp

[i π ( m2 + m3 )

]+ exp

[i π ( m3 + m1 )

] )(175)

The “super-lattice” peaks occur for mixed indices. The relative intensity of the“super-lattice” peaks are approximately given by

I(1, 0, 0)I(2, 0, 0)

∼(FAu(0) − FCu(0)FAu(0) + 3 FCu(0)

)2

(176)

which leads to a relative intensity of about 0.09.

At very low temperatures, CuZn exists as an ordered compound of the CuCltype. The structure consists of two interpenetrating simple cubic sub-latticeswhich have a relative displacement of [ 12 ,

12 ,

12 ]. The Cu atoms occupy the sites

of one sub-lattice, say the A sub-lattice, and the Zn atoms are located on theother sub-lattice, say the B sub-lattice. For an infinite solid the A and B sub-lattices are equivalent thus, the compound may also form with the Cu atomson the B sub-lattice and the Zn atoms on the A sub-lattice. Since the x-rayform factors are FCu(0) = 29 and FZn(0) = 30, the relative intensity of the“super-lattice” peaks of CuZn, or beta brass, are of the order of 0.0003. Thus,the super-lattice peaks are difficult to observe in x-ray scattering. However, theorder-disorder transition in CuZn and related compounds is easily observableby neutron diffraction28.

28C. G. Shull and S. Siegel, Phys. Rev. 75, 1008 (1949).

138

Ordered Phase

T < Tc

Figure 89: The ordered phase of CuZn.

Disordered Phase

T > Tc

Figure 90: The disordered phase of CuZn.

At temperatures above the order-disorder transition temperature of CuZn,Tc ≈ 741 K, the material exists in a disordered phase in which the Cu and Znatoms are randomly positioned on the sites of the A and B sub-lattices. At thetransition temperature, a phase transition occurs between the high temperaturedisordered phase and the low-temperature ordered phase. The order parameterfor the phase transition is given by the scalar quantity, φ(T ), where

φ(T ) = n(Cu)A − n(Cu)B

= n(Zn)B − n(Zn)A (177)

139

Figure 91: The neutron diffraction pattern of the ordered and disordered phasesof FeCo [After Shull and Siegel (1949).].

where n(Cu)A and n(Cu)B are, respectively, the number of Cu atoms on the Aand B sub-lattices. The second line follows from the fact that an atom of onetype or the other exists at each site. In particular, if the total number of sitesis 2 N , the numbers of Zn atoms at the sites of the A and B sub-lattices are,respectively, given by

n(Zn)A = N − n(Cu)A

n(Zn)B = N − n(Cu)B (178)

Above the transition temperature, the Cu atoms are equally probable to befound on the A and B sublattices and so the order parameter is zero, φ = 0.Below the transition temperature, the order parameter has a non-zero magni-tude φ0(T ) which is temperature dependent, and has either a positive or negativesign depending on whether the Cu atoms spontaneously select to occupy the Aor B sites, φ(T ) = ± φ0(T ). In the ordered state, the temperature dependenceof the order parameter is given by

φ0(T ) ∝ ( Tc − T )β (179)

where β ≈ 0.32. As the Hamiltonian is symmetric under interchange of the Aand B sub-lattices, this order-disorder transition provides an example of spon-taneous symmetry breaking.

——————————————————————————————————

4.1.14 Exercise 25

Express the inelastic x-ray scattering intensity for CuZn in terms of the atomicform factors FCu(Q), FZn(Q), and the order parameter φ(T ). Assume that the

140

Temperature-dependence of the order parameter

0

0.2

0.4

0.6

0.8

1

1.2

0 50 100 150 200 250

T [ K ]

φ φφφ(T

)

Tc

Figure 92: The ordered parameter of the order-disorder transition of CuZn.

deviations of the site occupancies from the average values at different sites areun-correlated.

——————————————————————————————————

4.2 Neutron Diffraction

Elastic neutron scattering from the nuclei of a solid involves a change in themomentum of the neutron from the initial value h k to the final value h k′ of

q = k − k′ (180)

Conservation of momentum requires that the transferred momentum must beequal to a momentum component of the interaction potential. This momentumis ultimately transferred to the solid. Experimentally accessible ranges of q forneutrons are in the range of 0.01 < q < 30 A, which covers the range that isuseful to determine crystalline structures.

The interaction between the neutron and one nucleus is short-ranged andcan be modelled by a point contact interaction,

Hint =2 π h2

mnb δ3( r − R ) (181)

141

Figure 93: The neutron diffraction pattern of ZnFe2O4 (top) and NiFe4O2

(bottom). [After Hastings and Corliss (1953).]

where b is the scattering amplitude of the order of 10−14 m. The differentialscattering cross-section represents the number of particles per unit time whichare scattered into solid angle dΩ per incident flux. The differential scatteringcross-section for one nucleus is assumed to be isotropic and is given by

dσ

dΩ= | b |2 (182)

Hence, the total cross-section for the one nucleus is given by

σ =∫dΩ

dσ

dΩ= 4 π | b |2 (183)

where the scattering takes place over all solid angles.

For a crystalline lattice of nuclei, as it shall be shown, the scattering cross-section is given by

dσ

dΩ=∑i,j

exp[i q . ( Ri − Rj )

]bi b

∗j (184)

where bi is the scattering amplitude from the i-th nucleus. The value of bi de-pends on what isotope exists at the lattice site and also on the direction of thenuclear spin.

In general, the different isotopes are randomly distributed so they must beaveraged over. Thus, bi and bj are independent or uncorrelated if they belongto different sites, and the average for i 6= j is given by the product of theaverages

bi b∗j = bi b∗j = | b |2 (185)

142

while, if i = j, one has the average of the squared amplitude

bi b∗i = | b |2 (186)

In general, the average of bi b∗j has the form

bi b∗j = | b |2 + δi,j

(| b |2 − | b |2

)(187)

The scattering cross-section can be written as the sum of two parts, a coherentpart where i 6= j and an incoherent part which has i = j.

The coherent cross-section is given by

dσ

dΩ=∑i,j

exp[i q . ( Ri − Rj )

]| b |2 (188)

For coherent scattering from every nuclei in the solid, the momentum transfermust satisfy the Laue condition and so q must be equal to Q, where Q satisfies

Q . R = 2 π m (189)

for all lattice vectors R and m is any integer. When this condition is satis-fied, the scattering produces Bragg reflections similar to those observed in x-rayscattering. When the Bragg scattering condition is satisfied, the coherentlyscattered beam has an intensity which is proportional to N2.

The incoherent scattering cross-section comes from the terms with i = jand is given by

dσ

dΩ= N

(| b |

2− | b |2

)(190)

The incoherent scattering is proportional to the number of nuclei N and is in-dependent of the direction of q. It is obvious that the coherent and incoherentcontributions are profoundly different. Only the coherent contribution can beutilized to determine the crystalline structure. Examples of structural determi-nation from neutron diffraction patterns are provided by Zinc Ferrite and NickelFerrite29.

——————————————————————————————————

4.2.1 Exercise 26

Consider a beam of unpolarized neutrons being diffracted from a material whichhas nuclei with spin I. The scattering length bJ depends on the total angular

29J. M. Hastings and L. M. Corliss, Rev. Mod. Phys. 25, 114 (1953).

143

momentum J of the combined neutron and nucleus. The scattering length is b+for J = I + 1

2 and b− for J = I − 12 . Show that

b =[

I + 12 I + 1

b+ +I

2 I + 1b−

](191)

and

| b |2 =[

I + 12 I + 1

| b+ |2 +I

2 I + 1| b− |2

](192)

The scattering from iron, Fe, is almost entirely coherent since the most abun-dant isotope has I = 0. On the other hand, the scattering from vanadium, V ,is almost completely incoherent. Other materials such as those containing Eu,have nuclei which are strong neutron absorbers and, therefore, neutron diffrac-tion is not a suitable technique to determine their structure.

——————————————————————————————————

144

4.3 Theory of the Differential Scattering Cross-section

By definition, the differential scattering cross-section dσdΩ is the ratio of the

number of particles scattered dNscatt (per unit time) into a solid angle dΩ =sin θ dθ dϕ to the incident flux of particles F (number of particles crossing unitarea per unit time) times the solid angle element

dNscatt = Fdσ

dΩdΩ (193)

Consider a beam of particles collimated to have a momentum k which fallsincident on a crystal. The particles are assumed to interact with either the elec-

Scattering Geometry

θθθθ

Scattering angle

Sample

Detector

Incident Beam

k

k'Scattered Beam

dΩΩΩΩ

Figure 94: A collimated beam of particles with momentum k is elastically scat-tered by the sample. The scattered particles are observed at the detector.

trons or nuclei of the solid. One example is x-ray diffraction in which the beamof photons interacts elastically with the electron density,. A second exampleis neutron diffraction experiments in which the beam of neutrons interacts, viashort-ranged nuclear forces, with the nuclei of the solid.

The interaction Hamiltonian between a particle in the beam and the relevantparticles of the solid can be represented as the sum of single-particle interactions

Hint =∑

j

Vj(r − rj) (194)

145

Here, r represents the position of the beam particle and rj is the position of thej-th particle in the solid.

For x-rays in which the energy of the photon is in the keV range, the photonenergy is much greater than the electronic energy scale. This has the effectthat only certain terms of the interaction Hamiltonian between the x-rays andthe electron need be considered. The non-relativistic form of the interactionbetween the electromagnetic field represented by a vector potential A(r, t) andparticles of charge q and mass m is given by

Hint = −∑

j

[q

2 m c

(p

j. A(rj , t) +A(rj , t) . pj

)− q2

2 m c2A(rj , t) . A(rj , t)

](195)

where rj and pj

are the position and momentum of the j-th particle. Thefirst pair of terms involve processes in which a single photon is absorbed oremitted, whereas the last term involves the interaction of two photons with thecharged particle. To calculate the cross-section for light scattering, one needs toconsider terms of fourth order in the vector potential A(r, t), as both the initialand final states each involve a photon. In principle, this requires includingthe first pair of terms in fourth order as well as the last term in second order.However, as the fourth order processes involve intermediate states in whicha very high energy photon has either been absorbed or emitted, the energydenominator involving the intermediate state is large. Thus, these contributionscan safely be ignored and only the last term in the interaction need be consideredexplicitly in the calculation of the elastic scattering cross-section. Therefore, inthis approximation, the x-rays couple to the density of the charged particles,ρ(r) =

∑j δ( r − rj ). For electrons, the coupling constant is proportional

to the length scale given by

e2

me c2=(e2

h c

)h

me c∼ 2.82 × 10−15 m (196)

which involves the fine structure constant and the Compton wave length. Theresulting length scale is the so-called classical radius of the electron.

4.3.1 Time Dependent Perturbation Theory

We shall assume that, asymptotically as t → − ∞ before the interaction isturned on, the scattered particles have the asymptotic form of a momentumeigenstate with eigenvalue h k and energy E(k)

limt → − ∞

Ψk(r, t) →(

1V

) 12

exp[

+ i k . r

]exp

[− i

E(k) th

](197)

It is expected that close to the source of the particles, the incident beam can bedescribed in terms of asymptotically free particles at all times. The choice of

146

normalization corresponds to one particle per volume V . The time-independentpart of the asymptotic initial state will be denoted by | k > in Dirac notation.The scattered wave at the detector has an asymptotic form of a momentumeigenstate | k′ > with momentum eigenvalue h k′.

The matrix elements of the interaction potential are given as

< k′ | Hint | k > =1V

∑j

∫V

d3r exp[− i k′ . r

]Vj(r − rj) exp

[+ i k . r

]

=1V

∑j

∫V

d3r′ exp[

+ i

(k − k′

). r′

]Vj(r′) exp

[+ i

(k − k′

). rj

](198)

where we have substituted r′ = r − rj in the second line. The integrationover R′ yields the Fourier transform of the interaction potential between thescattered particle and the j-th particle of the system

Vj(q) =∫

V

d3r′ exp[

+ i q . R′]Vj(r′)

≈∫d3r′ exp

[+ i q . r′

]Vj(r′) (199)

Hence, we find

< k′ | Hint | k > =1V

∑j

Vj(q) exp[

+ i q . rj

](200)

where the sum over j runs over all the particles of the target material.

Given one incident particle in the state | Ψk (t) >, which is initially in anenergy eigenstate | k > before the interaction Hint is turned on adiabaticallyat t → − ∞, the state of this particle evolves according to the Schrodingerequation

i h∂

∂t| Ψk (t) > =

(H0 + Hint

)| Ψk (t) > (201)

As the interaction is weak, the Schrodinger equation can be solved perturba-tively using the interaction representation. In the interaction representation,the states are transformed through a unitary operator in a manner such that

| Ψk (t) > = exp[

+i

hH0 t

]| Ψk (t) > (202)

This unitary transformation would make the eigenstate of the non-interactingparticle time-independent. However, the presence of a non-zero interaction termleads to the time-dependent equation of motion

i h∂

∂t| Ψk (t) > = ˆHint(t) | Ψk (t) > (203)

147

where the new interaction operator is time-dependent and is given by

ˆHint(t) = exp[

+i

hH0 t

]Hint exp

[− i

hH0 t

](204)

The equation of motion in the interaction representation can be solved by iter-ation. The equation is integrated to yield

| Ψk (t) > = | k > − i

h

∫ t

−∞dt′ ˆHint(t′) | Ψk (t′) > (205)

On iterating once, it is found that the state is given to first order in the inter-action by

| Ψk (t) > = | k > − i

h

∫ t

−∞dt′ ˆHint(t′) | k > + . . . (206)

This shows that, if wave function is started in an initial state which is an en-ergy eigenstate of the unperturbed Hamiltonian, the time evolution caused bythe interaction will admix other states into the wave function. In this sense,the particle described by the wave function may be considered as undergoingtransitions between the unperturbed energy eigenstates.

4.3.2 The Fermi Golden Rule

The rate at which the particle makes a transition from the initial state | k >to state | k′ >, due to the effect of Hint, is given in second order perturbationtheory by the Fermi Golden rule. The probability that the system has made atransition at time t is given by the squared modulus of the transition amplitude

< k′ | Ψk(t) > (207)

However, it is more convenient to calculate the probability based on the matrixelements evaluated in the interaction representation

< k′ | Ψk(t) > (208)

These two quantities are equivalent, as they are simply related via

< k′ | Ψk(t) > = exp[− i

hE(k′) t

]< k′ | Ψk(t) > (209)

and the phase factor cancels out in the squared modulus.

To first order in the perturbation, the transition amplitude is given by

< k′ | Ψk(t) > = − i

h

∫ t

−∞dt′ < k′ | ˆHint(t′) | k >

= − i

h

∫ t

−∞dt′ exp

[i

h( E(k′) − E(k) − i η ) t′

]< k′ | Hint | k >

(210)

148

where E(k) and E(k′) are the unperturbed (non-interacting) energies of theinitial and final states of the beam particles. The factor η corresponds to adia-batically switching on the interaction at t′ → − ∞. The probability that thetransition has occurred at time t is given by

1h2

∣∣∣∣ ∫ t

−∞dt′ exp

[i

h( E(k′) − E(k) − i η ) t′

]< k′ | Hint | k >

∣∣∣∣2(211)

The rate at which the transition occurs is given by the time derivative of thetransition probability

P (k → k′, t) =1h2

∂

∂t

∣∣∣∣ ∫ t

−∞dt′ exp

[i

h( E(k′) − E(k) − i η ) t′

]< k′ | Hint | k >

∣∣∣∣2(212)

The transition rate is evaluated as

P (k → k′, t) = | < k′ | Hint | k > |2 ∂

∂t

( exp[

2 η th

]( E(k′) − E(k) )2 + η2

)=

2h| < k′ | Hint | k > |2 exp

[2 η th

]η

( E(k′) − E(k) )2 + η2

(213)

Then, in the limit η → 0, the transition rate becomes time-independent and theenergy-dependent factor reduces to π times an energy conserving delta functionsince

limη → 0

η

( E(k′) − E(k) )2 + η2= π δ( E(k′) − E(k) ) (214)

Hence, we have obtained the Fermi Golden rule

limη → 0

P (k → k′, t) =2 πh

| < k′ | Hint | k > |2 δ( E(k′) − E(k) )

(215)

This expression represents the probability per unit time for a transition to oc-cur from the initial state to a very specific final state with a precisely known k′

that exactly conserves energy. As the rate contains a Dirac delta function, it isnecessary, for the rate to be mathematically meaningful, to introduce a distri-bution of final states. Thus, one must sum over all states with k′ in the solidangle subtended by the detector, irrespective of the magnitude of k′. Thus, theDirac delta function is to be replaced by the density of final states with energyE = E(k) which are travelling in the direction of the detector. That is, forthe particles to be detected, the particles must have been scattered through an

149

angle θ into the solid angle dΩ which is subtended by the detector.

The probability that a particle makes the transition from state k to stateswith final momentum in a solid angle dΩ distributed around k′, per unit time,is given by summing over the number of allowed final states

P (k → dΩ) =V

( 2 π )3

∫ ∞

0

dk′ k′2 dΩ2 πh

| < k′ | Hint | k > |2 δ( E − E(k′) )

=2 πh

| < k′ | Hint | k > |2 ρdΩ(E, k′) (216)

where ρdΩ(E, k′) is the density of final scattering states per unit energy range,defined as

ρdΩ(E, k′) =V

( 2 π )3dΩ

∫ ∞

0

dk′ k′2 δ( E − E(k′) ) (217)

The matrix elements of the interaction operator are to be evaluated with k′

that have the magnitude of k and are headed in the direction of the detector.That is, the particles which are detected must be travelling towards the detector(θ, ϕ) such that their velocities fall within the solid angle dΩ subtended by thedetector to the target.

4.3.3 The Elastic Scattering Cross-Section

The scattering cross-section is defined by(dσ

dΩ

)dΩ = P (k → dΩ) / F (218)

where the incident flux F is the density of particles (which is one per unitvolume, i.e. 1

V ) times the velocity. For massive particles, the velocity is justh km . Thus, for particles of mass mn, the flux is given by

F =h k

mn V(219)

On changing the variable of integration from dk′ to dE′, the density of finalstates is evaluated by integrating over the energy conserving delta function

ρdΩ(E, k′) =V

( 2 π )3

∫ ∞

0

dE′ dΩdk′ k′2

dE′δ( E − E(k′) )

=V

( 2 π )3dΩ

dk′ k′2

dE′(220)

where the magnitude of k′ is determined by the solution of E = E(k′), hencek′ = k. For massive particles, one has the energy momentum relation

dE′ =h2 k′

mndk′ (221)

150

and so, the density of final states can be written as

ρdΩ(E, k′) =V

( 2 π )3mn k′

h2 dΩ (222)

On inserting the Fermi golden rule expression for P (k → dΩ)

P (k → dΩ) =2 πh

| < k′ | Hint | k > |2 ρdΩ(E, k′) (223)

the final density of states ρdΩ(E, k) and the flux F into the expression eqn(218)for the scattering cross-section, one finds that the elastic scattering cross-sectionfor massive particles such as neutrons is calculated as

dσ

dΩ=

(V mn

2 π h2

)2 ∣∣∣∣ ∫V

d3r Ψ∗k′(r) Hint(r) Ψk(r)

∣∣∣∣2=

(mn

2 π h2

)2 ∑j,j′

Vj(q) V ∗j′(q) exp[i q .

(rj − rj′

) ](224)

where q is the scattering vector

k − k′ = q (225)

The magnitude of the scattering vector is related to the scattering angle θ via

q = 2 k sinθ

2(226)

On substituting the point contact interaction appropriate for nuclear scattering,and noting that the Fourier transform of the delta function is q independent,one finds the expression for the Fourier component of the potential

Vj(q) =2 π h2

mnbj (227)

For crystals with a mono-atomic basis that are in static equilibrium, the posi-tions of the nuclei rj can be identified with the lattice vectors Rj . Substitutionof Vj(q) in the above expression for the cross-section, yields the formula for theelastic neutron scattering cross-section

dσ

dΩ=∑j,j′

bj b∗j′ exp

[i q .

(Rj − Rj′

) ](228)

previously discussed.

For massless particles such as photons, the incident flux is just

F =c

V(229)

151

if the incident vector potential is normalized to yield one photon per volume V .The appropriately normalized vector potential can be expressed as

A(r, t) = eα c

√4 π h2 ω V

exp[i ( k . r − ω t )

]+ c.c. (230)

With this normalization, the vector potential represents one incident photonper volume V , with frequency ω and incident polarization eα. The density offinal states (for polarization eβ) is just

ρdΩ(E, k′) =V

( 2 π )3k′2

h cdΩ (231)

Thus, it is found that the cross-section for elastic x-ray scattering is simplygiven by

dσ

dΩ=

2 πh2 c

V 2 ω2

( 2 π c )3

(e2

2 me c2

)2 ∣∣∣∣ ∫V

d3r A∗k′(r) . ρ(r) Ak(r)∣∣∣∣2

=∣∣∣∣ eα . eβ

∣∣∣∣2 ( e2

me c2

)2 ∑j,j′

S(q) S∗(q) exp[i q .

(Rj − Rj′

) ](232)

where the structure factor S(q) is the contribution of a unit cell to the Fouriertransform of the electron density. The vectors Ri are the lattice vectors. Thus,the factors of V and ω cancel, leading to a scattering cross-section that onlydepends on the Fourier transform of the electronic density and has a couplingconstant which is the square of the classical radius of the electron

r2e =(

e2

me c2

)2

(233)

From the form of this coupling constant, it can be seen that the scattering ofx-rays from the density of charged nuclei is entirely negligible compared withthe scattering from the electron density.

4.3.4 The Condition for Coherent Scattering

Consider scattering from a crystal which has a mono-atomic basis and has afinite spatial extent. In this case, the subscript on the atomic potential can bedropped, and the summation over j and j′ run over all the lattice sites. Forconvenience, it shall be assumed that the crystal has the same shape as theprimitive unit cell but has overall dimensions ( N1 − 1 ) a1, ( N2 − 1 ) a2

and ( N3 − 1 ) a3 along the various primitive lattice directions. The solid,therefore, contains a total of N1 N2 N3 primitive unit cells and since the basisconsists of one atom, the solid contains a total of N = N1 N2 N3 atoms.

152

The summation over Rj in the scattering cross-section can be performed byexpressing the general reciprocal lattice vector in terms of the primitive latticevectors,∑

j

exp[i q . Rj

]=

∑n1,n2,n3

exp[i n1 q . a1

]exp

[i n2 q . a2

]exp

[i n3 q . a3

](234)

The sums over n1 runs from 0 to N1−1, and similarly for n2 and n3. This givesthe products of three factors, each of the form

n1=N1−1∑n1=0

exp[i n1 a . a1

]=

1 − exp[i N1 q . a1

]1 − exp

[i q . a1

]= exp

[+ i

( N1 − 1 )2

q . a1

]×

×

(exp

[i N1

2 q . a1

]− exp

[− i N1

2 q . a1

] )exp

[i 1

2 q . a1

]− exp

[− i 1

2 q . a1

]

= exp[

+ i( N1 − 1 )

2q . a1

] (sin N1

2 q . a1

sin 12 q . a1

)(235)

This function exhibits the effect of the constructive and destructive interferencebetween the scattered waves emanating from the various atoms forming thesolid. The numerator of the function falls to zero at

q . a1 =2 m π

N1(236)

for general integer values of m. The numerator has maximum magnitude at

q . a1 =( 2 m + 1 ) π

N1(237)

The overall q dependence is dominated by the denominator which falls to zerowhen q . a1 = 2 m π for integer m. At these special q values, the function hasto be evaluated by l’hopital’s rule and has the limiting value of N1. This occurssince, for these q values, the exponential phase factors are all in phase (andequal to unity) and so the sum over the N1 terms simply yields N1. Thus, thescattering cross-section is proportional to the product of the modulus square ofthree of these factors

dσ

dΩ=

∣∣∣∣ eα . eβ

∣∣∣∣2 r2e | F (q) |2 ×

153

×

(sin N1

2 q . a1

sin 12 q . a1

)2 (sin N2

2 q . a2

sin 12 q . a2

)2 (sin N3

2 q . a3

sin 12 q . a3

)2

(238)

Since for a macroscopic solid the numbers N1, N2 and N3 are of the order of107, the three factors rapidly vary with the magnitudes of q . ai. The maxima

Diffraction Pattern for a finite one-dimensional crystal

0

100

200

300

400

500

-1.5 -1 -0.5 0 0.5 1 1.5

q a /ππππ

d σ σσσ/d

Ω ΩΩΩ(q

)

N=20

Figure 95: The scattering cross-section of a one-dimensional lattice with N unitcells, as a function of the momentum transfer.

occur when the three conditions

q . a1 = 2 π m1

q . a2 = 2 π m2

q . a3 = 2 π m3

(239)

are satisfied. These special values of q are denoted by Q. In this case, one findsthat the scattering cross-section is simply proportional to

dσ

dΩ∼∣∣∣∣ eα . eβ

∣∣∣∣2 r2e | F (Q) |2 N2 (240)

which is just equal to the square of the number of atoms in the solid. Thecoherent scattering from an ordered solid should be contrasted with incoherent

154

scattering from the atoms of a gas. Due to the positional disorder in the gas, thephase factors may be considered to be random. The net scattering intensity forscattering of a gas of N atoms is then approximately equal to just N times thescattering intensity for an isolated atom. The coherent scattering from atoms ina solid possessing long-ranged order is a factor of N2 larger than the scatteringintensity for an isolated atom. In summary, the condition that there is completeconstructive interference between all the atoms in the solid is given by

exp[i Q . Ri

]= 1 ∀ i (241)

The intensity of the scattered beam is exceptionally large at these special valuesQ, compared with all other q values. Thus, coherent scattering is the dominantfeature of diffraction from crystalline solids but occurs only infrequently, as itonly occurs when the scattered wave length and scattering angle satisfy theabove stringent condition. These special values of Q are the lattice vectors ofthe reciprocal lattice.

——————————————————————————————————

4.3.5 Exercise 27

Consider a sample with N unit cells arranged in M micro-crystals that areoriented parallel with respect to each other, but their positions are random.Calculate the width and height of the Bragg peak.

——————————————————————————————————

4.3.6 Exercise 28

At finite temperatures, the atoms of a crystal undergo thermal vibrations. Dueto the vibrations, the intensity of the Bragg peaks are reduced by a Debye-Waller factor which involves the spectrum of lattice vibrations. However, thissituation can be approximately modelled by assuming that each atom undergoesa small random displacement δR from its equilibrium position R. Assume thatthe displacements are small compared with the separation between neighbor-ing atoms, | δR | a, and are Gaussian distributed. Also assume that thedisplacements of different atoms are entirely uncorrelated δi,R δj,R′ = 0 forR 6= R′ . Calculate the diffraction peak intensity, and show that the largestreduction in the intensity occurs for large Q values.

——————————————————————————————————

155

4.3.7 Exercise 29

Evaluate the effect of a significant number of thermally induced vacancies (miss-ing atoms) in the elastic scattering cross-section from a crystal.

——————————————————————————————————

4.3.8 Anti-Domain Phase Boundaries

The order-disorder transition usually starts at several nucleation centers in acrystal. For CuZn, the underlying CsCl lattice can be divided into two inter-penetrating simple cubic sub-lattices: the A and B sub-lattice. In several re-gions, the nucleation may start with the Cu atoms condensing on the A sub-lattice, whereas the nucleation may occur in other regions where the Cu atomscondense on the B sub-lattices. These distinct domains of nucleation grow andspread through the crystal until they meet and the entire crystal is ordered.The interfaces of the different domains meet at anti-domain phase boundariesat which there is a mismatch of the long-ranged ordering of the atoms. Due tothe mismatch, two planes containing similar atoms form the anti-domain phaseboundary30. The effect of anti-domain phase boundaries is to smear out the“super-lattice” Bragg peaks. This can be seen by considering the amplitudeof the scattered x-rays as a superposition of the scattering from the variousdomains. For simplicity, let us consider the scattering from two domains ofidentical shape and size. If the scattering amplitude from one domain is de-noted by A1(q) and the scattering from the second domain is denoted by A2(q)then, as the scattering amplitudes are additive, one obtains

A(q) = A1(q) + A2(q) (242)

where

A2(q) = exp[i q . δR

]A1(q) (243)

δR is the vector displacements of the origins of the two domains. The scatteringamplitude A1(q) is given by

A1(q) ∝

(sin N1 qx a

2

sin qx a2

) (sin N2 qy a

2

sin qy a2

) (sin N3 qz a

2

sin qz a2

)(244)

For a domain wall in the y − z plane, the displacement between the origins ofthe Cu sub-lattices in the two domains is given by

δR = ( N1 +12

) a ex +a

2ey +

a

2ez (245)

30F. W. Jones and C. Sykes, Proc. Roy. Soc. A, 166, 376 (1938).

156

Hence, for a CsCl-type structure and if q is close to Q, the total scatteringamplitude is given by the expression

A(q) ∼ A1(q)(

1 + ( − 1 )m1+m2+m3 exp[i N1 qz a

] )(246)

The total intensity of the scattered wave is proportional to

I(q) ∝ 2 | A1(q) |2(

1 + ( − 1 )m1+m2+m3 cosN1 qx a

)(247)

Thus, if m1 +m2 +m3 is even, the intensity is modulated by the factor

4 cos2N1 qx a

2(248)

whereas if m1 +m2 +m3 is odd, the intensity is modulated by the factor

4 sin2 N1 qx a

2(249)

This factor is due to the interference of the scattering from the two domains.The destructive interference causes an exact cancellation of the intensity at theexact Bragg wave vector at odd m1 +m2 +m3. However, for qx slightly off theBragg position

qx =2 πa

m1 + δqx

δqx ∼ π

N1 a(250)

the scattered intensity is finite and large. That is, the single anti-domain phaseboundary between identical domains of identical shapes and sizes produces ahole in the Bragg peak with odd m1 +m2 +m3.

For a crystal with a CuCl type structure which contains several anti-domainphases, one expects there to be three sets of anti-domain phase boundaries, andone expects that each domain has a different size. On averaging over the distri-bution of domains, one expects the small oscillations in the scattered intensityfrom the single domain S1(q) to be washed out. Furthermore, one expects thatthe intensities of the “super-lattice” peaks to be smeared out in q space.

4.3.9 Exercise 30

Consider the scattering produced by a CuCl type material with anti-domainwalls. For simplicity, only consider the component of the scattering ampli-tude associated with a single primitive lattice vector. That is, consider a one-dimensional model. Let p be the probability of not crossing a domain wall on

157

traversing one step a along the primitive lattice vector, and let q be the proba-bility of crossing a domain wall, where q ∼ 1

N1. Show that the average scattered

intensity near the “super-lattice” peaks is proportional to the factor

| A(qx) |2 ∝ N1 +N1∑

m1=1

( N1 − m1 ) ( p − q )m1 2 cosm1 qx a (251)

Evaluate the summation. Hence, show that as the number of domain walls in-creases, the intensities of the “super-lattice” Bragg peaks are diminished andacquire low amplitude tails.

4.4 Elastic Scattering from Quasi-Crystals

The scattering intensity from three-dimensional quasi-crystals show ten-fold,six-fold and five-fold symmetric diffraction patterns which can be understoodas arising from a space of six or more dimensions. Icosahedral symmetry can befound in a six dimensional hyper-cubic lattice. An icosahedron has 20 identicalfaces made of equilateral triangles. Five of the faces meet at each of the twelvevertices of the icosahedron. These vertices are responsible for the five-fold sym-metry.

Figure 96: Selected electron diffraction patterns from a single grain of an Al −Mn quasi-crystal. [After Gratais et al. (1984).]

The x-ray scattering amplitude A(q) from a one-dimensional quasi-crystalcan be found by a projection from a two-dimensional lattice. The amplitude is

158

a linear superposition from the scattered amplitudes from the sites sn, where

sn = n a cos θ + m′ a sin θ (252)

and where the points (na,m′a) are restricted to lie in a two-dimensional strip.The amplitude is given by

A(q) =∑

n

exp[i q sn

]=

∑n,m′

exp[i q a ( cos θ n + sin θ m′ )

]

=∑n,m

exp[i q a ( cos θ n + sin θ m )

]Θ(1 + (n+ 1) tan θ −m) Θ(m− n tan θ)

(253)

This can be expressed as an integral over a two-dimensional space

A(q) =∑n,m

exp[i q a ( cos θ n + sin θ m )

]Θ(1 + (n+ 1) tan θ −m) Θ(m− n tan θ)

=∫dx

∫dy exp

[i q ( cos θ x + sin θ y )

] ∑n,m

δ(x− na) δ(y −ma)×

× Θ(a+ (x+ a) tan θ − y) Θ(y − x tan θ)

This is a two-dimensional Fourier transform

A(q) =∫d2r exp

[i q . r

] ∑n,m

δ(x− na) δ(y −ma)×

× Θ(a+ (x+ a) tan θ − y) Θ(y − x tan θ)(254)

which is to be evaluated with q restricted to have values on the one-dimensionalline

q = q ( cos θ , sin θ ) (255)

The two-dimensional Fourier transform is recognized as the Fourier transformof a product

A(q) =∫d2r exp

[i q . r

]B(r) C(r) (256)

where B(r) is non-zero on the sites of a two-dimensional array

B(r) =∑m,n

δ(x− na) δ(y −ma) (257)

and the function C(r) projects onto a two-dimensional strip

C(r) = Θ(a+ (x+ a) tan θ − y) Θ(y − x tan θ) (258)

159

The Fourier transform of the product of functions can be evaluated using theconvolution theorem. The result is given by the convolution of the product ofFourier-transformed functions

A(q) =∫

d2q′

( 2 π )2B(q − q′) C(q′) (259)

The function B(q) is the scattering amplitude from the two-dimensional lattice

B(q) =∑n,m

exp[i ( qx n a + qy m a )

](260)

while the function C(q′) is evaluated as

C(q′) =∫

dx exp[i ( q′x + q′y tan θ ) x

] (exp[ i q′y a (1 + tan θ) ] − 1

i q′y

)

= ( 2 π ) δ( q′x + q′y tan θ )

(exp[ i q′y a (1 + tan θ) ] − 1

i q′y

)(261)

The scattering amplitude for the two-dimensional lattice is only non-zero at thetwo-dimensional reciprocal lattice vectors q = Q. Thus, the scattering fromthe the two-dimensional lattice is represented by the factor

B(q) =(

2 πa

)2 ∑Q

δ2(q −Q) (262)

Hence, we find that the amplitude in the two-dimensional space is given by

A(q) =1a2

∑Q

C(q −Q)

=2 πa2

∑Q

δ( qx − Qx + ( qy − Qy ) tan θ ) ×

×

(exp[ i ( qy − Qy ) a (1 + tan θ) ] − 1

i ( qy − Qy )

)(263)

Evaluating this on the line in q space yields the amplitude for scattering fromthe one-dimensional quasi-crystal

A(q) = 2 π∑Q

cos θ δ( q a − Qx a cos θ − Qy a sin θ ) ×

160

×

(exp[ i ( q sin θ − Qy ) a (1 + tan θ) ] − 1

i ( q a sin θ − Qy a )

)= 2 π

∑Q

δ( q a − Qx a cos θ − Qy a sin θ ) ×

×

(exp[ i ( Qx sin θ − Qy cos θ ) a (cos θ + sin θ) ] − 1

i ( Qx a sin θ − Qy a cos θ )

)(264)

This has delta function-like peaks at the wave vectors given by

q a = 2 π ( m1 cos θ + m2 sin θ ) (265)

where m1 and m2 are integers. In contrast to the scattering from a one-dimensional periodic crystal, the peaks in the scattering cross-section of a one-dimensional quasi-crystal are indexed by two integers. The intensities of thepeaks are proportional to

| A(q) |2 ∝sin2

(π ( m1 sin θ − m2 cos θ ) ( cos θ + sin θ )

)( m1 sin θ − m2 cos θ )2

(266)

Thus, the inelastic scattering spectra consists of a dense set of sharp peaks, butwith varying intensities. The intensities are large when the ratios of m2 and m1

are close to the value of tan θ. The quasi-crystal diffraction pattern collapses

0

0.2

0.4

0.6

0.8

1

-10 -5 0 5 10(qa/π)

I(q)/I(0)

Figure 97: Calculated Intensities of the Diffraction Peaks for a one-dimenionalquasi-crystal.

into the diffraction pattern of a one-dimenional crystal with a non-trivial basis,

161

in the limit when tan θ is a rational number.

Data from high resolution x-ray diffraction experiments on a three-dimensionalquasi-crystal31 are shown in fig(98).

Figure 98: High resolution x-ray diffraction spectra of quenched (top) and an-nealed (bottom) Al −Mn powder. [After Bancel et al. (1995).]

4.5 Elastic Scattering from a Fluid

The structure of a fluid, as expressed by the pair correlation function, can beinferred from elastic scattering experiments. The intensity of a beam of particlesscattered from a liquid can be considered as analogous to the scattering from asolid with an infinite unit cell. First, we shall consider the atoms of the fluidas static point particles. The amplitude of the beams scattered from each atomadd, giving a total amplitude which is proportional to

S(q) =∑

j

exp[i q . rj

]

=∫

d3r exp[i q . r

] ∑j

δ3(r − rj)

(267)

The scattering intensity is given by the square of the scattered amplitude

I(q) ∝ | S(q) |2

=∑i,j

exp[

+ i q . ri

]exp

[− i q . rj

]31P. A. Bancel, P. A. Heiney, P. W. Stephens, A .I. Goldman and P. Horn, Phys. Rev. Lett.

54, 2422 (1995).

162

=∫

d3r

∫d3r′ exp

[i q . ( r − r′ )

] ∑i,j

δ3(r − ri) δ3(r′ − rj)

(268)

On considering the long time average of the atomic positions, one obtains

I(q) ∝∫

d3r

∫d3r′ exp

[i q . ( r − r′ )

] ∑i,j

δ3(r − ri) δ3(r′ − rj)

=∫

d3r

∫d3r′ exp

[i q . ( r − r′ )

]C(r, r′)

(269)

The scattering intensity can be expressed in terms of the radial distributionfunction g(r), since

C(r − r′) = δ3(r − r′) ρ(0) + g(r − r′) (270)

Hence, the

I(q) ∝∫

d3r ρ(0) +∫d3r

∫d3r′ exp

[i q . ( r − r′ )

]g(r − r′)

= N + V

∫d3r exp

[i q . r

]g(r)

(271)

However, the integral over g(r) can be split into two parts

I(q) ∝ N + V

∫d3r exp

[i q . r

]ρ(0)

2+ V

∫d3r exp

[i q . r

]( g(r) − ρ(0)

2)

= N + V ( 2 π )3 ρ(0)2δ3(q) + V

∫d3r exp

[i q . r

]( g(r) − ρ(0)

2)

= N + N2 ( 2 π )3

Vδ3(q) + V

∫d3r exp

[i q . r

]( g(r) − ρ(0)

2)

= N + N2 ( 2 π )3

Vδ3(q) + V

4 πq

∫ ∞

0

dr r sin q r ( g(r) − ρ(0)2

)

(272)

The first term represents the incoherent scattering. The second term representscoherent forward scattering. The integral in the last term is convergent andyields non-trivial information about the structure of the fluid. As an example,g(r) has been derived from measurements of the intensity of elastically-scatteredneutrons I(q), for liquid argon near its triple point32.

32J. L. Yarnell, M. J. Katz, R. G. Wenzel, and S. H. Koenig, Phys. Rev. A, 7, 2130 (1973).

163

Figure 99: A comparison between the theoretical and the experimentally deter-mined structure factor of liquid argon. [After Yarnell et al. (1975).]

5 The Reciprocal Lattice

The reciprocal lattice vectors play an important role in describing the propertiesof a solid that has periodic translational invariance. Any property of the solid,whether scalar, vector or tensor, should have the same periodic translationalinvariance as the potential due to the charged nuclei. This means that, due tothe translational invariance, physical properties only need to be specified in afinite volume, and this volume can then be periodically continued over all space.The vectors of the reciprocal lattice play an important and special role in theFourier transform of the physical quantity.

The reciprocal lattice vectors have dimensions of inverse distance and aredefined in terms of the direct primitive lattice vectors a1, a2 and a3. Theprimitive reciprocal lattice vectors b(i), are defined via the scalar product

ai . b(j) = 2 π δj

i (273)

where the Kronecker delta function δji has the value 1 if i = j and is zero

if i 6= j. Thus, the primitive reciprocal lattice vectors are orthogonal to twoprimitive lattice vectors of the direct lattice.

164

The primitive reciprocal lattice vectors can be constructed via

b(1) = 2 πa2 ∧ a3

a1 . ( a2 ∧ a3 )

b(2) = 2 πa3 ∧ a1

a1 . ( a2 ∧ a3 )

b(3) = 2 πa1 ∧ a2

a1 . ( a2 ∧ a3 )(274)

where the last two expressions are found from the first by cyclic permutation ofthe labels (1, 2, 3). The denominator is just the volume of the primitive unit cell.

The reciprocal lattice consists of the points given by the set of vectors Qwhere

Q = m1 b(1) + m2 b

(2) + m3 b(3) (275)

and (m1,m2,m3) are integers. This set of vectors are the reciprocal lattice vec-tors. The reciprocal lattice vectors denote directions in the reciprocal lattice orare the normals to a set of planes in the direct lattice. In the latter case, asit shall be seen, the numbers (m1,m2,m3) are equivalent to Miller indices and,hence, are enclosed in round brackets.

——————————————————————————————————

5.0.1 Exercise 31

Find the volume of the primitive unit cell of the reciprocal lattice.

——————————————————————————————————

5.1 The Reciprocal Lattice as a Dual Lattice

The reciprocal lattice vectors can be considered to be the duals of the directlattice vectors. This relation can be seen by expressing the primitive latticevectors aj in terms of the primitive reciprocal lattice vectors bi, via

aj =1

2 π

∑i

gj,i b(i) (276)

If the quantity gi,j is identified as the metric, then the above relation definesthe set of b(i) to be basis vectors which are dual to the set of basis vectors ai.The quantity gi,j is given by the metric, since

aj . ak =1

2 π

∑i

gj,i b(i) . ak (277)

165

and sinceb(i) . ak = 2 π δi

k (278)

one hasgj,k = aj . ak (279)

Hence, gj,k is defined to be the metric tensor. The metric tensor has the propertythat it expresses the length s of a vector r in terms of its components xi alongthe basis vectors ai. That is, if one expresses the vector r in terms of the basisvectors ai and the components xi via

r =∑

i

xi ai (280)

then, for a constant metric, the length is given in terms of the covariant com-ponents33 xi via

s2 =∑i,j

gi,j xi xj (281)

The metric tensor, when evaluated in terms of the parameters of the primitiveunit cell, is given by the matrix

( gi,j ) =

a21 a1a2 cosα3 a1a3 cosα2

a1a2 cosα3 a22 a2a3 cosα1

a1a3 cosα2 a2a3 cosα1 a23

(282)

The inverse transform is given by

b(i) = 2 π∑

k

gi,k ak (283)

where the quantity gi,k is identified as the metric for the dual vectors. Since

aj =1

2 π

∑i

gj,i b(i)

=∑

i

gj,i

∑k

gi,k ak (284)

33The components xi which occur in the decomposition

r =∑

i

xi ai

are usually known as the covariant components of r, whereas if one decomposes the vector rin terms of the set of basis vectors b(i) via

r =∑

i

xi b(i)

then the xi are known as the contra-variant components of r.

166

and asaj =

∑k

δkj ak (285)

one infers thatδkj =

∑i

gj,i gi,k (286)

Hence, the metric tensor (gi,j) is the inverse of the metric tensor (gi,j) for thedual vectors.

The volume of the unit cell, Vc, is given by

V 2c = det ( gi,j ) (287)

or

V 2c = a2

1 a22 a

23

(1− cos2 α1 − cos2 α2 − cos2 α3 + 2 cosα1 cosα2 cosα3

)(288)

The dual metric tensor is given by the inverse of the metric tensor. The dualmetric tensor is expressed as the matrix

(gi,j)

=

a22a2

3(1−cos2 α1)V 2

c

a23a1a2(cos α1 cos α2−cos α3)

V 2c

a22a1a3(cos α1 cos α3−cos α2)

V 2c

a23a1a2(cos α1 cos α2−cos α3)

V 2c

a21a2

3(1−cos2 α2)V 2

c

a21a2a3(cos α2 cos α3−cos α1)

V 2c

a22a1a3(cos α1 cos α3−cos α2)

V 2c

a21a2a3(cos α2 cos α3−cos α1)

V 2c

a21a2

2(1−cos2 α3)V 2

c

(289)

This dual metric is also defined as

bi . bj = ( 2 π )2 gi,j (290)

From this, one can immediately find that the length of the reciprocal latticevectors are given by

b1 = 2 πa2 a3

Vc| sinα1 | (291)

etc., and the angle β3 between b(1) and b(2) is given by

cosβ3 =( cosα1 cosα2 − cosα3 )

| sinα1 sinα2 |(292)

etc. On using the inverse transformation, the reciprocal lattice vectors are givenin terms of the primitive direct lattice vectors by

b(1) = 2πa21a

22a

23

V 2c

(a1

(1− cos2 α1)a21

+ a2

(cosα1 cosα2 − cosα3)a1a2

+ a3

(cosα1 cosα3 − cosα2)a1a3

)b(2) = 2π

a21a

22a

23

V 2c

(a1

(cosα1 cosα2 − cosα3)a1a2

+ a2

(1− cos2 α2)a22

+ a3

(cosα2 cosα3 − cosα1)a2a3

)b(3) = 2π

a21a

22a

23

V 2c

(a1

(cosα1 cosα3 − cosα2)a1a3

+ a2

(cosα2 cosα3 − cosα1)a2a3

+ a3

(1− cos2 α3)a23

)(293)

167

These expressions for the b(i) are equivalent to the expressions in terms of thevector products of the primitive lattice vectors ai, and they also satisfy thedefinition of the primitive reciprocal lattice vectors

ai . b(j) = 2 π δj

i (294)

Any vector of the direct Bravais lattice can be expressed as

R = n1 a1 + n2 a2 + n3 a3 (295)

A reciprocal lattice vector Q can also be written as

Q = m1 b(1) + m2 b

(2) + m3 b(3) (296)

where (m1,m2,m3) are integers. Any vector k in the reciprocal lattice can berepresented as a superposition of the reciprocal lattice vectors

k = µ1 b(1) + µ2 b

(2) + µ3 b(3) (297)

where the µi are non-integer. Thus, the scalar product of an arbitrary vector kin the reciprocal lattice and a Bravais lattice vector R is evaluated as

k . R = 2 π(µ1 n1 + µ2 n2 + µ3 n3

)(298)

If k is a reciprocal lattice vector Q then the set of µi’s take on integer valuesmi, so that the scalar product reduces to

Q . R = 2 π(m1 n1 + m2 n2 + m3 n3

)(299)

As the sum of the products of integers is still an integer ( say M ), the Lauecondition can be expressed as

Q . R = 2 π M (300)

for all R. Thus, the Reciprocal Lattice vectors satisfy the Laue condition. Thisrequirement is equivalent to the condition that the exponential phase factorgiven by

exp[i Q . R

]= 1 (301)

is unity for all Bravais lattice vectors R.

The vectors Q form a Bravais lattice in which the primitive lattice vectorscan be expressed in terms of the vectors b(i). Also, the reciprocal lattice of areciprocal lattice is the original direct lattice.

——————————————————————————————————

168

5.1.1 Exercise 32

Determine the primitive lattice vectors of the lattice that is reciprocal to the re-ciprocal lattice. How are they related to the vectors of the original direct lattice?

——————————————————————————————————

5.2 Examples of Reciprocal Lattices

Now some examples of reciprocal lattices are examined.

5.2.1 The Simple Cubic Reciprocal Lattice

In terms of Cartesian coordinates, the primitive lattice vectors of the simplecubic direct lattice are

a1 = a ex

a2 = a ey

a3 = a ez (302)

The primitive reciprocal lattice vectors are determined to be

b(1) =2 πa

ex

b(2) =2 πa

ey

b(3) =2 πa

ez (303)

These are three orthogonal vectors which are oriented parallel to the directlattice vectors. The reciprocal lattice of the simple cubic direct lattice is alsosimple cubic.

5.2.2 The Body Centered Cubic Reciprocal Lattice

In terms of Cartesian coordinates, the primitive lattice vectors of the bodycentered cubic direct lattice are

a1 =a

2

(ex + ey − ez

)a2 =

a

2

(− ex + ey + ez

)a3 =

a

2

(ex − ey + ez

)(304)

169

The volume of the unit cell is Vc = | a1 . ( a2 ∧ a3 ) | = a3

2 .

The primitive reciprocal lattice vectors are determined to be

b(1) =2 πa

(ex + ey

)b(2) =

2 πa

(ey + ez

)b(3) =

2 πa

(ex + ez

)(305)

The three primitive reciprocal lattice vectors span the three-dimensional recip-rocal lattice, but have different orientations from the direct lattice vectors. Thereciprocal lattice has cubic symmetry as can be seen by combining the threeprimitive reciprocal lattice vectors (adding any two and subtracting the third)to yield three orthogonal reciprocal lattice vectors of equal magnitude. Thereciprocal lattice of the body centered cubic direct lattice is face centered cubic,with a conventional cell of side 4 π

a .

5.2.3 The Face Centered Cubic Reciprocal Lattice

In terms of Cartesian coordinates, the primitive lattice vectors of the face cen-tered cubic direct lattice are

a1 =a

2

(ex + ey

)a2 =

a

2

(ex + ez

)a3 =

a

2

(ey + ez

)(306)

The primitive reciprocal lattice vectors are determined to be

b(1) =2 πa

(ex + ey − ez

)b(2) =

2 πa

(ex − ey + ez

)b(3) =

2 πa

(− ex + ey + ez

)(307)

These are three non co-planar vectors, but have different orientations from thedirect lattice vectors. The reciprocal lattice has cubic symmetry. This can beseen by combining pairs of the primitive reciprocal lattice vectors, which yieldthree orthogonal reciprocal lattice vectors of equal magnitude. The reciprocallattice of the face centered cubic direct lattice is body centered cubic, with aconventional unit cell of side 4 π

a .

170

5.2.4 The Hexagonal Reciprocal Lattice

The Hexagonal Bravais Lattice

a1

a2

a3

Figure 100: The primitive unit cell for the hexagonal Bravais lattice.

The hexagonal lattice has primitive lattice vectors which can be chosen as

a1 =a

2

( √3 ex + ey

)a2 =

a

2

(−√

3 ex + ey

)a3 = c ez (308)

The volume of the primitive unit cell is

Vc =√

32

a2 c (309)

The primitive reciprocal lattice vectors are

b(1) =2 πa

(+

1√3ex + ey

)b(2) =

2 πa

(− 1√

3ex + ey

)b(3) =

2 πc

ez (310)

171

Hexagonal reciprocal lattice

b2b1

b3

(2π/c)

Figure 101: The primitive unit cell for the reciprocal lattice of the hexagonalBravais lattice.

Thus, the reciprocal lattice of the hexagonal lattice is its own reciprocal lattice,but is rotated about the z axis.

——————————————————————————————————

5.2.5 Exercise 33

A trigonal lattice is defined by three primitive lattice vectors a1, a2 and a3, allof equal length a where the angle α between any pair of these lattice vectorsis a constant. Show that the three vectors a1 = [r, s, t], a2 = [t, r, s] anda3 = [s, t, r], referenced to an orthonormal basis, are primitive lattice vectorsfor a trigonal lattice. Prove that the reciprocal lattice of the trigonal lattice isanother trigonal lattice.

——————————————————————————————————

5.3 The Brillouin Zones

The first Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice. Thatis, the first Brillouin zone is a volume of a primitive unit cell in the reciprocallattice. This cell is found by first connecting a central reciprocal lattice pointO to all the other reciprocal lattice points via the reciprocal lattice vectors Q

i.

172

Secondly, these connecting lines are bisected by planes. The equations for theset of these planes are given by(

k − 12Q

i

). Q

i= 0 (311)

for each i. The smallest volume around the origin O enclosed by these planesis the first Brillouin zone. That is, the first Brillouin zone consists of all theregions of space that can be reached from O without crossing any of the planes.

The regions of the entire reciprocal lattice can be partitioned off into Bril-louin zones of higher order. The planes defined by eqn(311) form a set ofboundaries for the set of Brillouin zones. The n-th order Brillouin zone consistsof the regions of k space that is accessed from the origin by crossing a minimumof n− 1 boundaries. Although the n-th order Brillouin zone exists in the form

Brillouin Zones for a Square Lattice

(1)

(2)

(2) (2)

(2)

(3)

(3)

(3) (3)

(3)

(3)

(3) (3)

(4) (4)

(4)(4)

(4) (4)

(4)

(4)

(4) (4)

(4)

(4)

(5)(5) (5)

(5)

(5)

(5)

(5) (5)(5)

(5)

(5)

(5)

(5)

(5)

(5)

(5) (5)

(5)

(5)

(5)

Figure 102: The higher-order Brillouin zones of a square lattice.

of isolated regions of k space, these regions can be brought together to makea contiguous volume by translating the isolated regions through appropriatelychosen reciprocal lattice vectors Q

i.

5.3.1 The Simple Cubic Brillouin Zone

The first Brillouin zone of the simple cubic direct lattice is a simple cube cen-tered at the origin O. It is bounded by lattice planes with normals that aremembers of the set 1, 0, 0, and thus are parallel to mirror planes of the direct

173

First Brillouin Zone for a simple cubic lattice

(2π/a)

Figure 103: The first Brillouin zone of the simple cubic lattice.

lattice. The sides of the cubic Brillouin zone are of length 2 πa and the Brillouin

zone has a volume of ( 2 πa )3 which, when given in terms of the volume of the

unit cell of the direct lattice, is equal to 8 π3

Vc.

Second Brillouin Zone of the simple cubic lattice

Figure 104: The second Brillouin zone of the simple cubic lattice.

Points of high symmetry are usually given special names. Points interior tothe first Brillouin zone are designated by Greek letters and those on the surfaceare designated by Roman letters. The center of the zone (0, 0, 0) is denoted byΓ, the vertex of the cube 2 π

a ( 12 ,

12 ,

12 ) is called R. The center of the x face lo-

174

cated at 2 πa ( 1

2 , 0, 0) is called X, and the mid-points of the edges at 2 πa ( 1

2 ,12 , 0)

are denoted by M .

Points on high-symmetry lines are also given special designations. The pointsbetween M and X are denoted by Z. The points on the lines between R andX are denoted by S, the points on the lines between R and M are denoted byT . The points on high-symmetry lines in the interior have the following desig-nations: the points between Γ and M are denoted by Σ, the points between Γand X by ∆, the points on lines between Γ and R are denoted by Λ.

5.3.2 The Body Centered Cubic Brillouin Zone

The first Brillouin zone for the body centered direct lattice is a rhombic dodec-ahedron34. It is bounded by lattice planes that are in the set 1, 1, 0, and thusare parallel to planes of reflection symmetry of the direct lattice. The Brillouinzone can be visualized by capping each square face of a cube with a squarepyramid, and noting that the triangular faces which are joined along one edgeof the cube actually form a single face. The vertices of the cube connect threeedges, whereas four edges intersect at each apex of the square pyramids. Thecell is centered at the origin Γ = (0, 0, 0). The vertices are located either on the

34The rhombic dodecahedron is the dual of the Archimedean solid which is known as thecuboctahedron. The Archimedean dual of a solid is formed from an Archimedean solidby interchanging the numbers of faces with the number of vertices. Therefore, the dualArchimedean solid has equivalent faces and congruent edges. The dual of an Archimedeansolid can be characterized by listing the number of edges per vertex in cyclic order about agiven face. In this scheme, the rhombic dodecahedron would be labelled by ( 3 . 4 )2.

Third Brillouin Zone for a s.c. lattice

Figure 105: The third Brillouin zone of the simple cubic lattice.

175

The Brillouin zone of the body-centered cubic lattice

Q=(2π/a)[(m1+m3)ex+(m1+m2)ey+(m2+m3)ez]

ez

ex

ey

(0,1,0)

(1,0,0)

(0,0,1)

Figure 106: Brillouin zone of the b.c.c. lattice.

positive or negative Cartesian axes at H = 2 πa (1, 0, 0) or at diagonal points

P = 2 πa ( 1

2 ,12 ,

12 ). The centers of the faces are denoted by N = 2 π

a ( 12 ,

12 , 0).

The Brillouin zone of the body-centered cubic lattice

N H

P

Γ

H

N

Figure 107: High symmetry points and lines of the b.c.c. Brillouin zone.

176

Points on the high-symmetry lines joining P and H are denoted by F . Otherspecial points are: G which are on the high-symmetry line between N and H, orD between P and N . The names of interior points on high-symmetry lines areΣ which are intermediate between Γ and N , ∆ which are intermediate betweenΓ and H, and Λ which are intermediate between Γ and P .

5.3.3 The Face Centered Cubic Brillouin Zone

The Brillouin zone for a face-centered cubic latticeez

ex

ey

1/2(1,1,1)

1/2(1,1,0)

Figure 108: Brillouin zone of the f.c.c. lattice.

The Brillouin zone for the face centered cubic lattice has the form of a trun-cated octahedron. The truncated octahedron is an Archimedean solid as it hasfaces which are regular polygons and as it has equivalent vertices and congruentedges35. The f.c.c. Brillouin zone is bounded by square faces with normals inthe set 2, 0, 0 and hexagonal faces with normals in the set 1, 1, 1. Onlythe 2, 0, 0 faces are parallel to mirror planes of the point group. The Bril-louin zone has twenty four vertices located at W = 2 π

a (1, 12 , 0). The centers

of the square faces are denoted by X and are located at 2 πa (1, 0, 0). These

squares are connected to eight hexagonal faces with centers at the L pointsL = 2π

a ( 12 ,

12 ,

12 ). The mid-points of the edges joining two hexagonal faces are

at 2 πa ( 3

4 ,34 , 0), and are denoted by K. The mid-points of the edges between

35The are thirteen Archimedean solids which are convex polyhedra. Their faces are com-posed of regular polygons. Their vertices are equivalent and their edges are congruent. TheArchimedean solids are enumerated by listing the number of edges per face in cyclic orderabout a given vertex. The truncated octahedron is labelled by 4 . 62.

177

the square and hexagonal faces are denoted by U , where U = 2πa ( 1

4 ,14 , 1).

The Brillouin zone for a face-centered cubic lattice

L

X

XK

UΓ

W

Figure 109: High symmetry points and lines of the f.c.c. Brillouin zone.

The points on the lines between X and U contained on the square faces aredenoted by S while those between X and W are denoted by Z. The points onthe high-symmetry lines between L and W on the hexagonal faces are denotedby Q. The points on the high-symmetry lines between Γ and K are denoted byΣ, the points on the lines between Γ and X are denoted by ∆, and the pointson the line running through Γ and L are known as Λ.

5.3.4 The Hexagonal Brillouin Zone

The Brillouin zone for the hexagonal lattice is hexagonal. The upper and lowerfaces are hexagons. The hexagonal face centers are at A = 2 π

a (0, 0, a2 c ). The

vertices are at the H points, H = 2 πa ( 1√

3, 1

3 ,a

2 c ). The centers of the verticalrectangular faces are denoted by M and M = 2 π

a ( 1√3, 0, 0). The mid-points

of the horizontal edges are denoted by L where L = 2 πa ( 1√

3, 0, a

2 c ) and themid-points of the vertical edges are denoted by K where K = 2 π

a ( 1√3, 1

3 , 0).

The interior high-symmetry points is Γ the zone center. Points on the in-terior high-symmetry lines are denoted as follows: Σ are points located on thehigh-symmetry lines Γ M , ∆ are the points on the lines Γ A, and Λ are thepoints on the lines between Γ and K.

178

The hexagonal Brillouin Zone

b1 b2

b3

Figure 110: The Brillouin zone for the hexagonal Bravais lattice.

The Hexagonal Brillouin Zone

Γ

A

H

KM

L

Figure 111: High symmetry points of the hexagonal Brillouin zone.

Points on the high-symmetry lines on the surface of the hexagonal Brillouinzone are enumerated as: T the points which are located on the horizontal linesK M , the points U which reside on the vertical line M L. The high-symmetrylines on the hexagonal faces are: A L which contain points that are denoted byR, and the line A H which contains points that are denoted by S.

179

5.3.5 The Trigonal Brillouin Zone

The trigonal unit cell has sides of length a, and the angles between any twoprimitive lattice vectors are all equal and denoted by α. If the unit cell isoriented such that the three primitive lattice vectors subtend the same angleθ with the z axis, then the primitive lattice vectors can be expressed in theCartesian coordinate system as

a1 = a ( sin θ ex + cos θ ez )

a2 = a ( − 12

sin θ ex +√

32

sin θ ey + cos θ ez )

a3 = a ( − 12

sin θ ex −√

32

sin θ ey + cos θ ez ) (312)

The relation between the angle θ and α is found from the scalar product as

cosα = − 12

sin2 θ + cos2 θ (313)

which yields

cos θ =

√1 + 2 cosα

3(314)

Since the reciprocal lattice is also trigonal, the primitive reciprocal lattice vec-tors are given by

b1 = b ( sin Θ ex + cos Θ ez )

b2 = b ( − 12

sinΘ ex +√

32

sinΘ ey + cos Θ ez )

b3 = b ( − 12

sinΘ ex −√

32

sinΘ ey + cos Θ ez ) (315)

where Θ is related to β, the angle between any two primitive reciprocal latticevectors, by a relation analogous to that between θ and α

cos Θ =

√1 + 2 cosβ

3(316)

Since the length of the primitive reciprocal lattice vector is given by

b =2π a2

Vcsinα (317)

and sinceV 2

c = a6 ( 1 − 3 cos2 α + 2 cos3 α ) (318)

180

one finds that b is given by

b =2πa

√1 + cosα

( 1 − cosα ) ( 1 + 2 cos α )(319)

Furthermore, since

cosβ =( cos2 α − cosα )

sin2 α(320)

after simplification, one finds that the angle β is given by

cosβ = − cosα1 + cosα

(321)

The reciprocal lattice vectors can be expressed as the sum of integer multiplesof the primitive reciprocal lattice vectors

Q = b

(m1 − 1

2( m2 + m3 )

)sinΘ ex

+ b

√3

2( m2 − m3 ) sinΘ ey

+ b ( m1 + m2 + m3 ) cos Θ ez (322)

The types of planes which bound the Brillouin zone depend on the angle β in

Trigonal Brillouin Zone

(1,1,1)

(1,0,0)(0,0,1)

(0,-1,0)

(1,0,1)

(0,-1,-1)(-1,-1,0)

Figure 112: The Brillouin zone of the trigonal lattice.

a non-trivial way. That is, even the number of faces depends on the value ofβ. For example, in Bi, one finds that α = 0.497 radians and β = 0.960

181

radians. The Brillouin zone of Bi has fourteen bounding planes. One set ofboundaries correspond to the six planes of the type (1, 0, 0) and (1, 0, 0), whichare located at a distance b

2 from the origin. The other boundaries are formedby six planes of the type (0, 1, 1) and (0, 1, 1). These planes occur at a distancefrom the origin equal to b 1√

2

√1 + cosβ. The top and bottom boundaries

of the Brillouin zone are given by the planes (1, 1, 1) and (1, 1, 1), which arelocated at a distance from the origin equal to b

√3

2

√1 + 2 cosβ.

182

6 Electrons

The types of states of single electrons in the potentials produced by the crys-talline lattice are discussed in the next three chapters. For simplicity, we shallfirst implicitly assume that the effect of the Coulomb interactions between elec-trons can be neglected. The neglect of electron - electron interactions is un-justified, as can be seen by considering the electrical neutrality of solids. Thecondition of electrical neutrality leads to the electron charge density being com-parable with the charge density due to the lattice of nuclei or ions. Thus, thestrength of the interactions between the electrons is expected to be comparableto the strength of the potential due to the nuclei. A simple order of magni-tude estimate, based upon the typical linear dimensions of a unit cell a0 ∼ 2Angstroms, leads to the average value of e2

r ∼ 3 eV for both these interactionenergies. Nevertheless, as a discussion of the effect of pseudo-potentials reveals,for most metals, the effect of the periodic potential of the lattice may be consid-ered as small. The small value of the effective potential (or pseudo-potential)leads to a useful approximation namely, that of nearly-free electrons. The effectof the finite strength of electron-electron interactions is a more complex issue,and is not yet fully understood. In principle, density functional theory providesa method of evaluating the ground state electron density including the effect ofelectron-electron interactions. However, the density functional method does notdescribe the excited states. The effect of the electron-electron interactions isthat of disturbing the electron density around any excited electron. On assum-ing that the interactions can be treated as a small perturbation, it can be shownthat most of the effects of electron-electron interactions on the low-energy ex-cited electrons merely involve the dressing of the single excited electron thereby,forming a quasi-particle excitation. That is, the effects of the excitation inducedmodifications of the surrounding gas of electrons can be absorbed as renormal-izations of the properties of the single-electron excitation. This feature can leadto the low-temperature properties of the electronic system being determined bythe gas of quasi-particles, which has the same form as a non-interacting gas ofelectrons. Systems where this simplification occurs are known as Landau Fermiliquids. The effect of electron-electron interactions will be delayed to a laterchapter.

7 Electronic States

In describing electronic states in metals first, the nature of the many-electronwave function and its decomposition into the sum of anti-symmetric productsof one-electron wave functions shall be described. Then, the general propertiesof the one-electron basis wave functions shall be discussed. The one-electronwave functions, or Bloch functions, are taken to be eigenfunctions of a suitablenon-interacting Hamiltonian in which the potential has the periodicity of theunderlying Bravais lattice.

183

7.1 Many-Electron Wave Functions

The energy of the electrons in a solid can be written as the sum of the kineticenergies, the ionic potential energy acting on the individual electrons, and theinteraction potential between pairs of electrons. Thus, for a system with Ne

electrons, the Hamiltonian can be written as the sum

H =i=Ne∑i=1

(− h2

2 m∇2

i + Vions(ri))

+12

∑i 6=j

e2

| ri − rj |(323)

where ri denotes the position of the i-th electron, Vions is the potential due tothe lattice of ions, and the last term is the pair-wise interaction between theelectrons. This Hamiltonian can be separated into two sets of terms,

H = H0 + Hint (324)

where

H0 =i=Ne∑i=1

(− h2

2 m∇2

i + Vions(ri))

(325)

is the sum of one-body Hamiltonians acting on the individual electrons, and theinteraction term is given by the sum of two-body terms

Hint =12

∑i 6=j

e2

| ri − rj |(326)

Since electrons are indistinguishable, the Hamiltonian must be symmetric un-der all permutations of the indices i labelling the electrons. Also, the modulussquared wave function must be invariant under all possible permutations of theelectron labels. An arbitrary permutation of the labels can be built up throughsequentially permuting pairs of labels.

The permutation operator Pi,j is defined as the operator which interchangesthe indices i and j labelling a pair of otherwise indistinguishable particles. Thus,if

Ψ(r1, . . . ri, . . . rj , . . . rNe)

is an arbitrary Ne particle wave function, the permutation operator can bedefined as

Pi,j Ψ(r1, . . . ri, . . . rj , . . . , rNe) = Ψ(r1, . . . rj , . . . ri, . . . rNe

)(327)

184

Since the Hamiltonian is symmetric under interchange of the indices i and jlabelling any two identical particles, the permutation operators commute withthe Hamiltonian

[ Pi,j , H ] = 0 (328)

Likewise, the permutation operators must also commute with any physical op-erator A

[ Pi,j , A ] = 0 (329)

otherwise measurements of the quantity A could lead to the particles being dis-tinguished.

Since the Hamiltonian commutes with all the permutation operators, one canfind simultaneous eigenstates of the Hamiltonian H and all the permutationoperators Pi,j . The energy eigenstates Ψ corresponding to physical states ofindistinguishable particles must satisfy the equations

H Ψ = E ΨPi,j Ψ = pi,j Ψ ∀ i, j (330)

where pi,j are the eigenvalues of the permutation operators Pi,j . As permut-ing the same pair of particle indices twice always reproduces the initial wavefunction, one has

Pi,j2 = I (331)

where I is the identity operator. Thus, the eigenstates of the permutationoperators satisfy the two equations

Pi,j2 Ψ = p2

i,j Ψ= Ψ (332)

Hence, the eigenvalues of the permutation operators must satisfy

pi,j2 = 1 (333)

orpi,j = ± 1 (334)

Thus, the Ne particle wave functions have the property that, under any permu-tation of a single pair of identical particles which are labelled by i and j, theun-permuted and permuted wave functions are related by

Ψ(r1, . . . rj , . . . ri, . . . rNe) = ± Ψ(r1, . . . ri, . . . rj , . . . rNe

)(335)

The upper sign holds for boson particles and the lower sign holds for fermions.Also, since pi,j is a constant of motion, the nature of the particles does notchange with respect to time. Electrons are fermions and, thus, the wave func-tion must always be anti-symmetric with respect to the interchange of any pair

185

of electron labels. Furthermore, the modulus square of the many-electron wavefunction must be invariant under all possible permutations of the electron labels.

The energy eigenfunction for the system of Ne electrons can be written as

Ψ(r1, r2, . . . rNe) (336)

The many-electron energy eigenstates Ψ usually cannot be found exactly. How-ever, they can be expressed in terms of a superposition formed from a com-plete set of many-electron eigenfunctions Φα1,α2,...,αNe

(r1, r2, . . . , rNe) of the

one-particle Hamiltonian H0. The subscript αi represents the complete set ofquantum numbers (including spin) which completely describes the state of asingle fermion state.

H0 Φα1,α2,...αNe(r1, r2, . . . rNe

) = E0 Φα1,α2,...αNe(r1, r2, . . . rNe

)(337)

This many-electron eigenfunction is interpreted as representing the state inwhich the Ne electrons are distributed in the set of single-electron states withthe specific quantum numbers α1, α2, . . . αNe

. The basis states are orthonormaland so satisfy the relations

Ne∏j=1

(∫V

d3rj

)Φ∗β1,β2,...(r1, . . . , rNe

) Φα1,α2,...(r1, . . . , rNe) = δα1,β1 δα2,β2 . . .

(338)where we have assumed that the sets of single-electron eigenvalues have beenenumerated and ordered. Since the basis is complete, the exact many-bodyeigenstates of the full Hamiltonian H can be written as a linear superpositionof the complete set of basis functions

Ψ(r1, r2, . . . , rNe) =

∑α1,α2,...,αNe

Cα1,α2,...αNeΦα1,α2,...,αNe

(r1, r2, . . . rNe)

(339)where the sum over the set of αi runs over all possible distributions of the Ne

electrons in the set of all the single-electron states. The coefficients Cα1,α2,...,αNe

have to be determined. The coefficients represent the probability amplitudesthat electrons occupy the set of single-electron states labelled by α1, α2, . . . , αNe

.

The set of many-electron basis functions Φα1,α2,...αNe(r1, r2, . . . rNe

) can beexpressed directly in terms of the one-electron wave functions φα(r). First,note that the non-interacting Hamiltonian H0 can be decomposed as the sumof Hamiltonians which only act on the individual electrons

H0 =i=Ne∑i=1

Hi (340)

186

where the one-particle non-interacting Hamiltonian is given by

Hi = − h2

2 m∇2

i + Vions(ri) (341)

This one-particle Hamiltonian has eigenstates, φβ(ri), which satisfy the eigen-value equation

Hi φβ(ri) = Eβ φβ(ri) (342)

The many-particle non-interacting Hamiltonian H0 has eigenfunctions whichare the products of Ne one-particle eigenfunctions φβ(r)

χ(r1, α1; r2, α2; . . . rNe, αNe

) = φα1(r1) φα2(r2) . . . φαNe(rNe

) (343)

and the non-interacting energy eigenvalue E0 for the many-particle state is givenas the sum of the one-electron energy eigenvalues Eαi

that are occupied by theelectrons

E0 =i=Ne∑i=1

Eαi(344)

However, the wave functions χ(r1, α1; r2, α2; . . . rNe, αNe

) do not represent phys-ical wave functions since each of the single-particle states with quantum numbersα1, α2, . . . αNe

are occupied by the respective electron labelled by r1, r2, . . . rNe

and, hence, the electrons have been labelled. As the electrons are indistinguish-able, it is impermissible to distinguish them by this type of labelling. Thus,physical wave functions should contain terms which are related by all the pos-sible relabelling of the indices of the particles.

Electrons are fermions and, therefore, they have wave functions which areanti-symmetric under the interchange of any pair of particles. The proper ba-sis set of the many-electron wave function Φ must correspond to all possiblepermutations of the single-particle indices. The proper anti-symmetrized wavefunction Φα1,α2,...,αNe

is given by the Slater determinant

Φα1...αNe= 1√

Ne!

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

φα1(r1) φα1(r2) . . . φα1(ri) . . . φα1(rNe)

φα2(r1) φα2(r2) . . . φα2(ri) . . . φα2(rNe)

......

......

φαi(r1) φαi

(r2) . . . φαi(ri) . . . φαi

(rNe)

......

......

φαNe(r1) φαNe

(r2) . . . φαNe(ri) . . . φαNe

(rNe)

∣∣∣∣∣∣∣∣∣∣∣∣∣∣The normalization is 1√

Ne!as there are Ne! terms in the determinant, corre-

sponding to the Ne! permutations of the electron labels.

The anti-symmetric wave function has the property that if there are two ormore particles in the same one-particle eigenstate, say αi = αj , then the wavefunction vanishes. This can be seen by noting that two rows of the determinant

187

are identical and, hence, the determinant vanishes. As the wave function van-ishes if two or more electrons occupy the same single-particle eigenstate, there isno state in which a one-particle eigenstate is occupied by two or more electrons.The anti-symmetric nature of the fermion wave function directly leads to thePauli exclusion principle. The Pauli exclusion principle can be stated as “nounique single-particle state can be occupied by two or more electrons.” For elec-trons which have spin one-half, a single-particle state is uniquely specified onlyif the spin quantum number is also specified. The single-particle wave functionφα(r) should be supplemented by the spinor χσ. That is, the single-electronwave function should be replaced by the product

φα(r) → φα(r) χσ (345)

where χσ is a spinor or a normalized two-component column vector. The spinindex σ can be considered to label an eigenstate of a component of an arbitrarysingle-electron spin operator, and the label σ should be considered as analogousto the single-particle eigenvalue α. A complete set of labels for the single-electron state are given by α and σ. An arbitrary spinor χσ can be decomposedas the linear superposition of two basis spinors χ±

χσ =∑±

γ± χ± (346)

where the normalization condition is given by∑±

| γ± |2 = 1 (347)

The two basis spinors χ± are usually denoted by the two component columnvectors

χ+ =(

10

)(348)

corresponding to an eigenstate of the Pauli matrix σz with spin-up and

χ− =(

01

)(349)

corresponding to the eigenstate with spin-down. Thus, the arbitrary spin statecan be written as

χσ =(γ+

γ−

)(350)

In this representation, the two components of an arbitrary spinor, χσ, representthe internal degree of freedom of the spin and, thus, are analogous to the de-gree of freedom represented by r in the position representation. The complexconjugate wave function should be replaced by

φ∗α′(r) → χTσ′ φ

∗α′(r) (351)

188

which contains χTσ′ which is the complex conjugated transpose of the spinor

states given by the two-dimensional row matrices

χTσ′ =

(γ′+

∗ γ′−∗ ) (352)

In the situations where the electron spin has to be explicitly considered, thesereplacements lead to the inner product of two one-electron states not only in-volving the integration of the product φ∗α′(r) φα(r) over the electron’s position rbut also automatically involves evaluating the matrix elements of the individualelectron’s row spinor state χT

σ′ with the column spinor state χσ.

The probability density ρ(r) for finding an electron at position r can be ob-tained from the matrix elements of the many-electron wave function Ψ(r1; r2, . . . rNe

)with the one-electron density operator ρ. The one-electron density operator isgiven by a Dirac delta function

ρ(r) =i=Ne∑i=1

δ3( r − ri ) (353)

The density ρ(r) is evaluated as

ρ(r) = Ne

∫V

d3r1

∫V

d3r2 . . .

∫V

d3rNeδ( r − r1 ) | Ψ(r1; r2, . . . rNe

) |2

(354)Thus, the trace over the positions particles can be evaluated by integrating overall but one of the particles positions

ρ(r) = Ne

∫V

d3r2

∫V

d3r3 . . .

∫V

d3rNe| Ψ(r; r2, . . . rNe

) |2 (355)

The matrix elements of the spin states has also to be taken. The resulting elec-tron density is normalized to Ne.

The probability density for finding an electron at position r and anotherelectron at r′ is a correlation function ρ(r, r′) which is given by the matrixelements of the operator

ρ(r, r′) =∑

i

∑j 6=i

δ3( r − ri ) δ3( r′ − rj ) (356)

The resulting expression for the two-particle density is found by integrating overthe positions of all the electrons except two

ρ2(r, r′) = Ne ( Ne − 1 )∫

V

d3r3

∫V

d3r4 . . .

∫V

d3rNe| Ψ(r; r′; . . . rNe

) |2

(357)This two-particle density correlation function is normalized to twice the numberof pairs of electrons, Ne ( Ne − 1 ).

——————————————————————————————————

189

7.1.1 Exercise 34

Evaluate the single-particle density and two-particle density correlation functionfor a many-particle basis wave function Φα1,α2,...αNe

given by a single Slater de-terminant of single-particle wave functions φα(r).

——————————————————————————————————

The properties of the single-electron wave functions, φα(ri), that are to beused in forming the many-particle basis functions Φα1,α2...αNe

(r1, r2, . . . rNe) as

Slater determinants, are discussed in the next chapter. In the following, theelectron labels i in the one-electron wave functions are omitted.

7.2 Bloch’s Theorem

Bloch’s theorem describes the properties of the one-electron states φα(r) whichare eigenstates of the one-electron Hamiltonian with a periodic potential. Anelectron in the solid experiences a periodic potential that has the periodicityof the underlying lattice of ions. In particular, the potential is invariant under

Periodic potential of a set of ions

-4 -3 -2 -1 0 1 2 3 4

r/a

Vio

ns(r

)

εεεεF

Figure 113: The periodic potential Vions(r) must satisfy the conditionVions(r −R) = Vions(r), for all Bravais lattice vectors R.

190

translation through any Bravais lattice vector Ri

Vions(r −Ri) = Vions(r) (358)

General properties of the solution of the Schrodinger equation for a singleelectron in a solid can be found by considering the periodicity of Vions(r). If theelectron-electron interactions are neglected, the independent electrons obey theone-particle Schrodinger equation with the periodic potential,

H φα(r) =(− h2

2 m∇2 + Vions(r)

)φα(r) = Eα φα(r) (359)

For an infinite solid, the physically acceptable solutions of this equation areknown as the Bloch wave functions. The energies of the Bloch states are usuallylabelled by two quantum numbers n and k, instead of by α. The one-dimensionalcase, where the values of k were restricted to real values, was investigated byKramers36.

Bloch’s theorem applies to the eigenstates of the one-particle Hamiltonian,

H =(− h2

2 m∇2 + Vions(r)

)(360)

in which the potential has the symmetry

Vion(r −Ri) = Vions(r) (361)

for all lattice vectors Ri in the Bravais lattice. Bloch theorem states that theeigenfunctions can be found in the form

φn,k(r) =1√V

exp[i k . r

]un,k(r) (362)

where the function un,k is invariant under the translation through any Bravaislattice vector

un,k(r −Ri) = un,k(r) (363)

Bloch’s theorem asserts that the periodic translational symmetry manifestsitself in the transformation of the wave function

φn,k(r −Ri) = exp[− i k . Ri

]φn,k(r) (364)

Thus, a translation of the wave function through a Bravais lattice vector onlyshows up through the presence of an exponential factor. Furthermore, if the

36H. A. Kramers, Physica 2, 483 (1935).

191

wave vector k is real, then the electron density for the Bloch state is identicalfor each unit cell in the crystal. This prevents the wave function from divergingat the boundaries of the solid.

The proof of Bloch’s theorem is based on the consideration of the translationoperator TR which, when acting on an arbitrary function f(r), has the effect oftranslating it through a Bravais lattice vector R

TR f(r) = f(r −R) (365)

This translation operator can be applied to the wave function H φ(r) which

The Translation Operator TR

0

0.2

0.4

0.6

0.8

1

f(r) f(r-R)

r0 r0+R

0 01 2 23 1 3

r r

TR f(r) = = = = f(r-R)

Figure 114: The effect of the translation operator TR on an arbitrary functionf(r).

yields

TR H φ(r) = H(r −R) φ(r −R)

= H(r) φ(r −R)= H TR φ(r) (366)

Thus, the Hamiltonian commutes with the translation operator which producesa translation through a Bravais lattice vector,

[ H , TR ] = 0 (367)

192

This means that it is possible to find simultaneous eigenstates of both TR and H.

Furthermore, the translation operators corresponding to translations throughdifferent lattice vectors commute. This can be shown by successive translationsTR and TR′ , which yields

TR TR′ φ(r) = φ(r −R′ −R)

φ(r −R−R′) = TR′ TR φ(r) (368)

Thus, the translation operators commute

[ TR , TR′ ] = 0 (369)

This proves that the wave functions can be chosen to be simultaneous eigenstatesof the Hamiltonian and all the translation operators that produce translationsthrough Bravais lattice vectors. The Bloch functions are chosen such that theysatisfy

H φ(r) = E φ(r)

TR φ(r) = c(R) φ(r) (370)

and, thus, are the simultaneous eigenstates of H and all the TR.

The translation operators can be compounded as

TR′ TR φ(r) = φ(r −R−R′)

= TR+R′ φ(r) (371)

When two translation operators are successively applied to the simultaneouseigenfunctions of the translation operators, it may be re-interpreted in terms ofthe compound translation

TR′ TR φ(r) = c(R′) c(R) φ(r)

= TR+R′ φ(r) = c(R+R′) φ(r) (372)

This shows that the products of two eigenvalues of different translation operatorsgives the eigenvalue of the compound translation

c(R′) c(R) = c(R+R′) (373)

Since a general Bravais lattice vector can be expressed as the sum

R = n1 a1 + n2 a2 + n3 a3 (374)

where (n1, n2, n3) are integers, a general eigenvalue can be decomposed in termsof products

c(R) = c(a1)n1 c(a2)

n2 c(a3)n3 (375)

193

Hence, on introducing the notation

c(a1) = exp[− i 2 π x1

]

c(a2) = exp[− i 2 π x2

]

c(a3) = exp[− i 2 π x3

](376)

for the primitive eigenvalues, one can define a vector k via

k =(x1 b

(1) + x2 b(2) + x3 b

(3)

)(377)

With these definitions, the eigenvalue of the translation operator can be ex-pressed in terms of the k vector as

c(R) = exp[− i k . R

](378)

Thus, the eigenvalue equation for the translation operator is expressed as

TR φ(r) = φ(r −R)= c(R) φ(r)

= exp[− i k . R

]φ(r) (379)

which completes the proof of Bloch’s theorem.

The wave functions which are simultaneous eigenfunctions of the energyand the periodic translation operators are the Bloch functions37. The Blochfunctions, φn,k(r), are labelled by the translation quantum number k and aquantum number n that pertains to the single-particle energy eigenvalue En,k.It should be noted that Bloch’s theorem does not guarantee that the quantityk is real. Since k is the quantum number associated with the eigenvalue of theoperator which translates through a Bravais lattice vector

TR φn,k(r) = φn,k(r −R)

= exp[− i k . R

]φn,k(r) (380)

then it should be clear that as

exp[i Q . R

]= 1 (381)

37F. Bloch, Zeit. fur Physik, 52, 555 (1928).

194

the eigenvalue labelled by k is identical to the eigenvalue labelled by k+Q. Thismeans that the two wave vectors can be identified, i.e., k + Q ≡ k. Thus, theBloch wave vector when translated through a reciprocal lattice vector Q leadsto an equivalent wave vector. Furthermore, if the convention

φn,k+Q(r) = φn,k(r) (382)

is adopted, then the eigenvalues must be related through

En,k+Q = En,k (383)

Thus, if k is real, any k value can be restricted to lie within one unit cell ofreciprocal space38.

φk(r)

Vions(r)

Figure 115: The spatial variation of the real part of a Bloch function (schematic).

7.3 Boundary Conditions

Bloch’s theorem does not ensure that the wave vector k is real. In fact, forsurface states or impurity states, k may become imaginary. However, for bulkstates the wave vector is real, as can be ascertained by applying appropriateboundary conditions.

Consider a crystalline solid of finite size which has the same shape as theprimitive unit cell of the Bravais lattice but with dimensions L1 = N1 | a1 |,L2 = N2 | a2 | and L3 = N3 | a3 | along the three primitive axes. The solidthen contains a total number of N = N1 N2 N3 lattice points.

38L. Brillouin, J. Phys. Radium, 1, 377 (1930).

195

a1a2

a3

N1 a1

N2 a2

N3 a3

Figure 116: A hypothetical crystal of the same shape as the primitive unit cellis constructed by stacking N1 ×N2 ×N3 unit cells together.

Born-von Karman or periodic boundary conditions39 are imposed on thewave function

φn,k(r −Ni ai) = φn,k(r) for i = 1 , 2 or 3 . (384)

The periodic boundary conditions ensure that the electronic states are homoge-neous bulk states and are unmodified in the vicinity of the surface of the solid.Therefore, application of Bloch’s theorem yields the condition

φn,k(r) = φn,k(r −Ni ai)

= exp[− i Ni k . ai

]φn,k(r) for i = 1 , 2 or 3

(385)

Thus, the periodic boundary conditions are fulfilled if the wave vectors k satisfythe conditions

exp[− i Ni k . ai

]= 1 (386)

Since k can be written in terms of the primitive reciprocal lattice vectors, b(i),via

k =i=3∑i=1

xi b(i) (387)

39M. Born and Th. von Karman, Zeit. fur Physik, 13, 297 (1912), M. Born and Th. vonKarman, Zeit. fur Physik, 14, 15 (1913).

196

and as ai . b(j) = 2 π δj

i , then the periodic boundary conditions require that

exp[− i 2 π Ni xi

]= 1 for i = 1 , 2 or 3

(388)

Thus, the components xi must be in the form of ratios

xi =mi

Ni(389)

where mi are integers. This proves that the general Bloch wave vector k is areal vector, and the k vectors have the general form

k =i=3∑i=1

mi

Nib(i) (390)

Since Ni 1, the k vectors form a dense set of points in reciprocal space.

The properties of a solid can be expressed in terms of summations over theelectronic states. Since each state can be expressed in terms of the discretek quantum number, the summation are over a dense set of k vectors. Thesummation over a dense set of k vectors can be represented in terms of anintegral over the energy, weighted by the density of states. From the form of k,the volume of k space per allowed k value is

∆3k =b(1)

N1.

(b(2)

N2∧ b(3)

N3

)=

1N

b(1) .

(b(2) ∧ b(3)

)(391)

As the volume of the Brillouin zone is given by

b(1) .

(b(2) ∧ b(3)

)(392)

the volume of one state is 1N times the volume of the Brillouin zone. This implies

that the number of allowed k values within the Brillouin zone is equal to thenumber of unit cells in the crystal. The volume ∆3k associated with a Blochstate is given by

∆3k =1N

( 2 π )3

a1 . ( a2 ∧ a3 )

=1N

( 2 π )3

Vc(393)

Now, since the volume of the solid V is N times the volume of the cell Vc,

V = N Vc (394)

197

b1/N1b2/N2

b3/N3

b1

b2

b3∆∆∆∆3k

Figure 117: The volume of k-space associated with a one-electron state, ∆3k, isa factor of 1/N smaller than the volume of a primitive unit cell of the reciprocallattice.

then the volume of k space associated with each Bloch state is

∆3k =( 2 π )3

V(395)

Hence, in the continuum limit, the number of one-electron states (per spin) inan infinitesimal volume of phase d3k is given by

V

( 2 π )3d3k (396)

7.4 Plane Wave Expansion of Bloch Functions

Any function obeying Born-von Karman boundary conditions can be expandedas a Fourier series. This implies that the Bloch functions can also be expandedas

φn,k(r) =∑

q

Cq1√V

exp[i q . r

](397)

where the wave vectors q are to be related to k. From Bloch’s theorem, theBloch functions can also expressed as

φn,k(r) =1√V

exp[i k . r

]un,k(r) (398)

198

Since un,k(r) has periodic translational invariance, it only contains reciprocallattice vectors Q. The Fourier series expansion of the periodic function is

un,k(r) =∑Q

un,k(Q) exp[i Q . r

](399)

and the inverse transform is given by the integral

un,k(Q) =1V

∫V

d3r un,k(r) exp[− i Q . r

](400)

On comparing the above two forms for the Bloch functions, one has

φn,k(r) =∑

q

Cq1√V

exp[i q . r

]

=∑Q

un,k(Q)1√V

exp[i ( k + Q ) . r

](401)

Thus, the allowed q values in the Bloch wave functions are equal to k, modulo areciprocal lattice vector. Furthermore, the Cq are equal to the Fourier compo-nents un,k(Q). Next, it shall be shown how the Cq can be determined directlyfrom the Schrodinger equation which contains the periodic potential Vions(r).

The Bloch functions can be found by solving the Schrodinger equation wherethe Hamiltonian contains the periodic potential Vions(r). The periodic potentialalso has a Fourier series expansion

Vions(r) =∑Q

Vions(Q) exp[i Q . r

](402)

and the inverse transform is given by the integral

Vions(Q) =1V

∫V

d3r Vions(r) exp[− i Q . r

](403)

Furthermore, since Vions(r) is real, the Fourier transform of the potential hasthe symmetry

Vions(−Q) = V ∗ions(Q) (404)

This follows from taking the complex conjugate of the Fourier series expansionof Vions(r). A second condition on the Fourier expansion coefficients existsfor crystals which have an inversion symmetry around a suitable origin. Theinversion symmetry implies that the potential is symmetric

Vions(r) = Vions(−r) (405)

199

-1

-0.8

-0.6

-0.4

-0.2

0

0 1 2 3 4 5 6

(Qa/2π)

Vio

ns(Q

) a/4πZ

e2

(2,0,0)(2,2,0)

(1,1,1)

(2,2,2)

(3,1,1)

(4,0,0)

(3,3,1)

(4,2,0)

Figure 118: The first few Fourier components Vions(Q) for a model potential ofan f.c.c. solid.

and this implies that the Fourier transform of the potential has the property

Vions(Q) = Vions(−Q) = V ∗ions(Q) (406)

The expansion coefficients Cq in the Bloch function are found by substitutingthe Fourier series into the energy eigenvalue equation. The kinetic energy termis evaluated from

p2

2 mφn,k(r) = − h2

2 m∇2 φn,k(r)

=∑

q

h2 q2

2 mCq

1√V

exp[i q . r

](407)

The potential term in the energy eigenvalue equation has the form of a convo-lution when expressed in terms of the Fourier Transforms

Vions(r) φn,k(r) =∑q′

∑Q′

Vions(Q′) Cq′1√V

exp[i

(q′ + Q′

). r

](408)

The form of the energy eigenvalue equation is simplified if q′ is expressed asq′ = q − Q′, so

Vions(r) φn,k(r) =∑

q

∑Q′

Vions(Q′) Cq−Q′1√V

exp[i q . r

](409)

200

Then, the energy eigenvalue equation takes the form

∑q

( [h2 q2

2 m− E

]Cq +

∑Q′

Vions(Q′) Cq−Q′

)exp

[i q . r

]= 0 (410)

The wave vectors q are expressed as q = k − Q so that k is always locatedwithin the first Brillouin zone. On equating the coefficients of the plane waveswith zero, one finds the matrix eigenvalue equation(

h2 ( k − Q )2

2 m− E

)Ck−Q +

∑Q′

Vions(Q′) Ck−Q−Q′ = 0 (411)

The reciprocal lattice vector is transformed as Q′ → Q” = Q′ + Q in thesecond term, leading to an infinite set of coupled equations(

h2 ( k − Q )2

2 m− E

)Ck−Q +

∑Q”

Vions(Q”−Q) Ck−Q” = 0

(412)

Thus, because of the periodicity of the potential, the Bloch functions only con-tain Fourier components q that are connected to k via reciprocal lattice vectors.For fixed k, the set of equations couple Ck to all the Ck−Q via the Fouriercomponent of the potential Vions(Q). In principle, the set of infinite coupledalgebraic equations (412) could be used to find the coefficients Ck−Q and theeigenvalue En,k. The Bloch function is expressed in terms of the coefficientsCk−Q as

φn,k(r) =∑Q

Ck−Q1√V

exp[i ( k − Q ) . r

]

=1√V

exp[

+ i k . r

] ∑Q

Ck−Q exp[− i Q . r

](413)

Using this, the Bloch function can be expressed in terms of the periodic functionun,k(r) via

un,k(r) =∑Q

Ck−Q exp[− i Q . r

](414)

In order to make this approach tractable, it is necessary to truncate the infiniteset of coupled equations (412) to a finite set. However, if this set of equations aretruncated, it would require approximately 103 to 106 plane wave componentsbefore convergence is attained in three dimensions. Therefore, other methodsare frequently used.

201

7.5 The Bloch Wave Vector

The Bloch wave vector k plays a role similar to that of the momentum of afree electron. In fact, it reduces to the momentum quantum number in thelimit Vions(r) → 0. However, for a non-zero crystal potential, k is not equalto the eigenvalue of the electron momentum p = − i h ∇ since it differsby amounts that are determined by the reciprocal lattice vectors Q and thecoefficients Ck+Q. That is,

p φn,k(r) = h k φn,k(r) − i h√V

exp[i k . r

]∇ un,k(r) (415)

Thus, h k is known as the crystal momentum.

The crystal momentum can always be chosen to be in the first Brillouin zoneby making the transformation

k = k′ + Q (416)

On substituting this relation into the expression for the Bloch function, onefinds that it can be re-written as

φn,k(r) =1√V

exp[i k . r

]un,k(r)

=1√V

exp[i k′ . r

] (exp

[i Q . r

]un,k(r)

)=

1√V

exp[i k′ . r

]un,k(r) (417)

where

un,k(r) = exp[i Q . r

]un,k(r) (418)

is identified as a periodic function of the type that is used in Bloch’s theorem.The new function un,k(r) transforms like un,k(r) as it has the periodicity of theBravais lattice since

exp[i Q . R

]= 1 (419)

Due to the periodic translational symmetry, the eigenvalue problem can bereduced to finding a solution for the periodic function un,k(r) in a single cell ofthe lattice. The total number of energy eigenfunctions must correspond to thesum of the numbers of electron states originating from each atom in the crystal,and there may be many basis atoms in the unit cell. As an isolated atom isexpected to have an infinite number of excited levels, and as the number ofdifferent k points in the Brillouin zone is equal to the number of primitive cellsin the crystal, there must be infinitely many energy eigenfunctions with fixed k.

202

The different one-electron states with fixed k are distinguished by the index n.The energy En,k is a continuous function of k, forming energy bands. This isseen by examination of the eigenvalue equation, when the Bloch functions areexpressed as

φn,k(r) =1√V

exp[i k . r

]un,k(r) (420)

This procedure leads to the energy eigenvalue equation

Hk un,k(r) =[

h2

2 m

(− i ∇ + k

)2

+ Vions(r)]un,k(r)

= En,k un,k(r) (421)

Due to the Born-von Karman boundary conditions, each energy band in theBrillouin zone contains N different states. The different k values are not partof a continuum but form a discrete dense set of points. The energy eigenvaluesEn,k, therefore, although a continuous function of k, only exist at a finite set ofpoints.

7.6 The Density of States

A physical quantity A may be expressed in terms of the quantities An,k associ-ated with the individual electrons in each of the occupied Bloch states (n, k) inthe solid. That is, the quantity A is given by

A = 2∑n,k

An,k (422)

where the sum runs over each level (n, k) that is occupied by an electron. Thefactor 2 originates from the spin degeneracy. Since the different k states aredense and uniformly distributed in the Brillouin zone, the summation may berepresented by an integration. The volume ∆3k of phase space associated witha Bloch state is given by

∆3k =( 2 π )3

V(423)

The quantity A is expressed as the integral

A = 2V

( 2 π )3∑

n

∫En,k <EF

d3k An,k (424)

where the integration over k runs over the volume of occupied states in the firstBrillouin zone. Thus, for the partially filled bands the integration runs overa volume of k space enclosed by a surface of constant energy εF , and for thecompletely filled bands it runs over the entire Brillouin zone.

The integration over k space may be converted into an integral over theenergy ε, by introducing the one-electron density of states ρ(ε). The density

203

of states per spin is defined by the sum over Dirac delta functions for eachone-electron state

ρ(ε) =∑n,k

δ( ε − En,k )

= V∑

n

∫d3k

( 2 π )3δ( ε − En,k ) (425)

If the quantity An,k only depends on (n, k) through En,k, then

An,k = A(En,k) (426)

so the quantity A can be represented as an integral over the density of states

A = 2∫ εF

−∞dε ρ(ε) A(ε) (427)

The density of states ρ(ε) can be calculated by noting that the infinitesimalintegral

∫ ε+∆ε

ερ(ε) dε ≈ ρ(ε) ∆ε is the number of states in the energy range

between ε and ε + ∆ε, or the allowed number of k values between ε and ε + ∆εin each of the energy bands. Thus, on integrating over an energy range ∆ε andusing the definition of the density of states in terms of the Dirac delta function,one finds

ρ(ε) ∆ε ≈ V∑

n

∫ ε+∆ε

ε

dε

∫d3k

( 2 π )3δ( ε − En,k )

=V

( 2 π )3∑

n

∫d3k

(Θ(ε+ ∆ε− En,k) − Θ(ε− En,k)

)(428)

where Θ(x) is the Heaviside step function. Thus, the density of states is ex-pressed by an integral over a volume of k space enclosed by surfaces of constantenergy ε and ε+ ∆ε. Furthermore, since ∆ε is an infinitesimal quantity, ∆ε canbe expressed in terms of the perpendicular distance between the two surfaces ofconstant energy.

Let Sn(ε) be the surface En,k = ε lying within the primitive cell and letδk(k) be the perpendicular distance between the surfaces Sn(ε) and Sn(ε+ ∆ε)at point k. Then, as Sn(ε) is a surface of constant ε and ∇ En,k is perpendicularto that surface

ε + ∆ε ≈ ε + | ∇ En,k | δk(k)

δk(k) ≈ ∆ε| ∇ En,k |

(429)

204

δδδδk

Sn(ε+∆ε)(ε+∆ε)(ε+∆ε)(ε+∆ε)

Sn(ε)(ε)(ε)(ε)

d2Sn

k3

Figure 119: The density of states is given in terms of a weighted integrationover a surface in k-space of constant energy, Sn(ε).

Hence, the density of states can be expressed as an integral over a surface ofconstant energy

ρ(ε) ≈ V∑

n

∫Sn(ε)

d2S

( 2 π )31

| ∇ En,k |(430)

This gives an explicit relation between the density of states and the band struc-ture.

Since En,k is periodic, it is bounded from above and below for each value ofn. This implies that there will be values of k in each Brillouin zone where thegroup velocity vanishes,

∇ En,k = 0 (431)

As the band energy En,k is a periodic function of three variables it must haveat least one maximum and one minimum in the Brillouin zone and six saddlepoints. At each of these k points, the integrand in the expression for ρ(ε) di-verges. Other divergences may be expected which originates from k points nearthe Brillouin zone boundary, where the dispersion relation is expected to havezero slope. These divergences give rise to van Hove singularities in the density ofstates. L. van Hove provided a general discussion of these types of singularitiesusing the Morse index theorem40.

In three dimensions these singularities are integrable. That is, the integra-tion over the surface area yield a finite value for ρ(ε). In the three-dimensionalcase the divergences show up in the slopes of the density of states ∂ρ(ε)

∂ε , andare the van Hove singularities. The van Hove singularities at the density of

40L. van Hove, Phys. Rev. 89, 1189 (1953), also see the discussion by H. P. Rosenstock,Phys. Rev. 97, 290 (1955).

205

ε/εε/εε/εε/ε0000

Den

sity

of

stat

es ρ

(ε)

ρ(ε

) ρ

(ε)

ρ(ε

)

ε/εε/εε/εε/ε0000

Den

sity

of

Stat

es ρ

(ε)

ρ(ε

) ρ

(ε)

ρ(ε

)

ε/εε/εε/εε/ε0000

Den

sity

of

stat

es ρ

(ε)

ρ(ε

) ρ

(ε)

ρ(ε

)

ε/εε/εε/εε/ε0000

Den

sity

of

stat

es ρ

(ε)

ρ(ε

) ρ

(ε)

ρ(ε

)

Figure 120: The types of van Hove singularities in the one-electron density ofstates of a three-dimensional solid.

states occur at the values of E where ∇ En,k vanishes at some points of thesurface Sn(ε). Typical van Hove singularities occur at the band edges where thedensity of states varies as

√| ε | . Although the density of states ρ(ε) at van

Hove singularities does not diverge in three dimensions, the derivatives divergeand can give rise to anomalies in thermodynamics as can be seen by examiningthe Sommerfeld expansion.

In low-dimensional systems, the divergence can show up directly as a diver-gence in the density of states.

——————————————————————————————————

7.6.1 Exercise 35

The energy dispersion relation at a van Hove singularity has a zero gradient. Inthe vicinity of the van Hove singularity, the d-dimensional dispersion relationcan be written as

Ek = E0 + E1

i=d∑i=1

αi k2i a

2i (432)

where the coefficients αi determine whether the extremum is a maximum, min-imum or saddle point. The coefficients are given by

αi = ± 1 (433)

Characterize the different types of van Hove singularities41 in the density ofstates and sketch the energy dependence in the vicinity of the singularity for

41One theorem states that a periodic function of d variables must have a number of at least(dn

)van Hove singularities with n negative coefficients αi in the d-dimensional primitive

unit cell. The expression

(dn

)is the binomial coefficient(dn

)=

d!

( d − n )! n!

where d > n.

206

d = 1, 2 and d = 3.

——————————————————————————————————

7.7 The Fermi Surface

The ground state of the electronic system has the lowest possible energy. Fornon-interacting electrons, the electrons occupy the lowest possible eigenvalues.However, the distribution of electrons must satisfy the restriction imposed bythe Pauli exclusion principle, which states that no uniquely specified electronstate can be occupied by more than one electron. This means that a spin de-generate state cannot be occupied by more than two electrons, one for eachspin value. Thus, the ground state of H0 is represented by a Slater determinantwave function in which two electrons are placed in the lowest energy eigenstate,and, successively, two more are placed in the next lowest states, until all the Ne

electrons have been placed in states. In the following, the convention is adoptedthat the electrons which are associated with the states (n, k) have k restrictedto be within the first Brillouin zone.

On distributing all the electrons into the lowest one-electron energy eigen-states in a manner consistent with the Pauli exclusion principle, one finds twodifferent types of ground states:

(i) Insulators.

In insulators, a number of bands are completely filled and all other bandsare completely empty. No band is partially filled. In this case, there must exist

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

E k [

Ryd

berg

s ]

Γ ΓX W L XK

µ

Figure 121: The calculated electronic dispersion relation for Si. The chemicalpotential µ does not cross the energy bands.

an energy interval which separates the lowest unoccupied band state and the

207

highest occupied band state. The density of states must be zero in this energyinterval. The width of the interval, in which ρ(ε) = 0, is the threshold energyrequired to excite an electron from an occupied to an unoccupied state. Thisenergy interval is defined to be the band gap. In an insulator, the chemical po-

Figure 122: The calculated electronic density of states for Si. The chemicalpotential µ occurs in the gap in the density of states. [After Chelikowski et al.(1973).]

tential µ falls in the band gap. An insulating state can only occur if the numberof electrons Ne is equal to an even number times the number of primitive unitcells N in the direct lattice. This is because each band can be occupied by 2 Nelectrons. For example, C being tetravalent when it crystallizes in the diamondstructure is insulating, and has a band gap of over 5 eV. The elements Si andGe are also insulating, but have smaller band gaps which are 1.1 eV and 0.67eV, respectively.

(ii) Metals.

A number of bands may be partially filled. In this case, the highest occu-pied Bloch states have an energy εF which lies within the range of one or morebands. This case corresponds to a metal, in which the one-electron density ofstates at εF is non-zero, ρ(εF ) 6= 0. Systems with an odd number of electronsper unit cell should be metallic, such as the simple mono-valent metals like Naor K. However, systems with two electrons per unit cell can be metallic. Forexample, divalent Mg is metallic. Mg crystallizes in the hexagonal close-packedsystem and, hence, has four electrons per unit cell. The small distance betweenthe atoms is responsible for the large dispersion of the bands which allows thebands to overlap. The overlapping of the bands leads to divalent Mg beingmetallic.

For each partially filled band, there will be a surface in the three-dimensional

208

Electron Bands for Al

-0.25

0

0.25

0.5

0.75

1

1.25

Ek [

Ryd

berg

s ]

Γ ΓΧ ΧW L K W

f.c.c.

µ

Figure 123: The electronic dispersion relations for metallic Al. The chemicalpotential µ or Fermi energy εF cuts across the bands.

Density of States for b.c.c. Li

0

0.25

0.5

0.75

0 1 2 3 4

ε [ eV ]

ρ(ε)

[ St

ates

/eV

]

εF

[After Ham (1962)]

Figure 124: The electronic density of states ρ(ε) for metallic Li. The chemicalpotential µ or Fermi energy εF occurs in the energy region where the density ofstates is finite.

k space which separates the occupied from the unoccupied states. The set of allsuch surfaces forms the Fermi surface. The Fermi surface is determined by theequation

En,k = εF (434)

Since En,k is periodic in the reciprocal lattice, the Fermi surface may either berepresented within the full periodic reciprocal lattice or in a single unit cell ofthe reciprocal lattice. The Fermi surface is represented in the extended zonescheme, if the full reciprocal lattice is used. If the Fermi surface is representedwithin a single primitive unit cell of the reciprocal lattice, it is represented in areduced zone scheme.

209

210

8 Approximate Models

Some of the earlier approaches to electronic structure of solids will be discussedin this chapter. These methods are not in common use, and are not reliablemethods for calculating electronic structures. These older methods also ne-glect the effect of electron-electron interactions. By contrast, the most commonmethod in use today is based on the Density Functional approach of Kohn andSham, which is quantitatively reliable and includes the effect of electron-electroninteractions. Nevertheless, the older methods were important in the develop-ment of the subject and yield important insights into the results of electronicstructure calculations. We shall focus our attention to two different modelswhich are suitable for different regimes of the energy spectrum. First, we shallexamine the nearly-free electron model that treats continuum states E > 0and in which the potential of the lattice of ions is introduced as a small pertur-bation. Then, we shall examine the tight-binding model, which describes howatomic bound states with E < 0 are perturbed when they are brought togetherto form a crystal.

8.1 The Nearly-Free Electron Model

In the nearly-free electron approach to electronic structure calculations, one as-sumes that the periodic potential due to the lattice is small. This assumption isnot justified, apriori, as the potential is of the order of 10 eV. However, the effectof the potential can be much smaller than this estimate, and for these cases, thenearly-free electron model gives results which can be used to phenomenologicallydescribe metals found in groups I, II, III, and IV of the periodic table. Thesematerials have an atomic structure which consists of s or p electrons outside aclosed shell configuration.

The nearly-free electron model provides a good description of conductionelectron states due to two principal reasons:-

(i) The region in which the electron - ion interaction is strongest is in thevicinity of the ion. However, since this region is occupied by the core electronsand the Pauli principle forbids the conduction electrons to enter this region, theeffective potential is weak.

(ii) In the region of space where the conduction electrons reside, the motionof other conduction electrons effectively screen the potential.

Since in the nearly-free electron approximation the effective potential is as-sumed to be small, perturbation theory may be used.

211

8.1.1 Perturbation Theory

The wave function for an electron in a Bloch state with wave vector k is givenby

φk(r) =∑Q

Ck−Q1√V

exp[i ( k − Q ) . r

](435)

where Q are reciprocal lattice vectors and the coefficients Ck have to be deter-mined. The coefficients satisfy the set of coupled algebraic equations[

h2

2 m( k − Q )2 − E

]Ck−Q +

∑Q′

Vions(Q−Q′) Ck−Q′ = 0 (436)

where the sum runs over all the reciprocal lattice vectors Q′. For fixed k, thereis an equation for each Q value. The different solutions of this set of equationsfor fixed k are labelled by the index n.

If one neglects the potential due to the lattice, one obtains the empty latticeapproximation. This is the result of the zero-th order perturbation theory. Tozero-th order in the perturbing potential Vions, the set of equations reduce to[

E(0)k − Q − E

]Ck−Q = 0 (437)

where the zero-th order energy eigenvalues are given by

E(0)k − Q =

h2

2 m( k − Q )2 (438)

and the zero-th order energy eigenfunctions are

φ(0)k (r) =

1√V

exp[i ( k − Q ) . r

](439)

If, for a given k, the energies associated with the set of reciprocal lattice vectorsQ

1, . . . , Q

mare degenerate,

E(0)k − Q

1= E

(0)k − Q

2= . . . = E

(0)k − Q

m

(440)

then the zero-th order approximation for φk(r) can be made of any linear su-perposition of the functions φ(0)

k (r) = 1√V

exp[ i ( k − Q ) . r ].

The type of perturbation theory that is appropriate depends on whether thezero-th order eigenvalues are degenerate or not.

212

8.1.2 Non-Degenerate Perturbation Theory

Non-degenerate perturbation theory can be used when the energy separationsbetween the level under consideration, E(0)

k − Q1

, and all other zero-th ordereigenvalues are large compared with the magnitude of the potential

| E(0)k − Q

1− E

(0)k − Q | | Vions(Q1

−Q) | (441)

for fixed k and all Q 6= Q1. This corresponds to the non-degenerate case.

We shall evaluate the one-electron energy eigenvalue to second order in Vions

but first, we need to consider the first order correction to the energy and wavefunction of the state under consideration which, to zero-th order, has momentumk − Q

1. The amplitude corresponding to the plane wave component with this

momentum satisfies the secular equation[E

(0)k − Q

1− E

]Ck−Q

1+∑Q′

Vions(Q1−Q′) Ck−Q′ = 0 (442)

This shall be used to obtain the energy E and the coefficient Ck−Q1

to firstorder in Vions. The term involving the summation is explicitly of the order ofVions, so the coefficients Ck−Q in this term only need to be calculated to zero-thorder in the Vions. Only one coefficient is non-zero to zero-th order in Vions,since

C(0)k−Q = 0 ∀ Q 6= Q

1(443)

Thus, to first order in Vions, only one term survives in the summation and thecoefficient Ck−Q

1satisfies the eigenvalue equation[E − E

(0)k − Q

1

]Ck−Q

1= Vions(0) C(0)

k−Q1

(444)

This equation determines the energy eigenvalue E(1) to first order in Vions. Sincethe energy shift is to be calculated to first order in Vions, the coefficient Ck−Q

1

can be substituted by its zero-th order value C(0)k−Q

1. This procedure yields the

first order approximation for the energy eigenvalue

E(1) = E(0)k−Q

1+ Vions(0) (445)

First order perturbation theory only produces a constant shift in the zero-thorder energy eigenvalues which can be absorbed into the definition of the refer-ence energy. It is also seen from eqn(444) that, to first order, the change in thecoefficient Ck−Q

1remains undetermined, so we may set

C(1)k−Q

1= C

(0)k−Q

1(446)

213

This is seen by substituting the first-order expression for E − E(0)k−Q

1into

eqn(444). In the following discussion, we shall neglect the effect of the averagepotential Vions(0).

The coefficients of the other plane wave components of the Bloch function,Ck−Q, satisfy[

E(0)k − Q − E

]Ck−Q +

∑Q′

Vions(Q−Q′) Ck−Q′ = 0 (447)

This is used to obtain the coefficients C(1)k−Q to first order in Vions. Since the

summand is explicitly of first order in Vions, then the coefficients Ck−Q′ need

only be considered to zero-th order. However, only C(0)k−Q

1is non-zero in this

order so,

C(1)k−Q =

Vions(Q − Q1)

E − E(0)k − Q

C(0)k−Q

1(448)

The coefficients C(1)k−Q and C

(0)k−Q

1completely determine the energy eigenfunc-

tion to first order in Vions.

The energy eigenvalue can now be found to second order in Vions usingthe wave function that have just been calculated to first order in Vions. Onsubstituting the expression for C(1)

k−Q, eqn(448), into the secular equation whichdetermines Ck−Q

1, eqn(442), one finds

(E − E

(0)k − Q

1

)Ck−Q

1=∑Q

| Vions(Q1− Q) |2

( E − E(0)k − Q )

C(0)k−Q

1(449)

Since both the energy and wave function are unchanged to first order in Vions,the lowest order non-zero contribution to the term on the left hand side is foundwhen Ck−Q

1is evaluated in zero-th order and E is evaluated to second order.

Thus, to second order in Vions, the energy eigenvalue E is given by the solutionof (

E − E(0)k − Q

1

)=∑Q

| Vions(Q1− Q) |2

( E − E(0)k − Q )

(450)

or, since the eigenvalue E is approximately equal to E(0)k − Q

1, the energy eigen-

value is given by

E = E(0)k − Q

1+∑Q

| Vions(Q1− Q) |2

( E(0)k − Q

1− E

(0)k − Q )

(451)

214

This relation shows that weakly perturbed non-degenerate bands repel eachother. For example, if

E(0)k − Q > E

(0)k − Q

1(452)

then the second order contribution is negative and E is reduced further belowE

(0)k − Q

1. On the other hand, if

E(0)k − Q < E

(0)k − Q

1(453)

then the second-order contribution is positive and E is increased further aboveE

(0)k − Q

1. Hence, the leading-order effect of the perturbation increases the sep-

aration between the energy bands.

8.1.3 Degenerate Perturbation Theory

The most important effect of the potential occurs when a pair of the free electroneigenvalues are within Vions of each other, but are far from all other eigenvalues.Under these conditions, the eigenvalues are almost doubly-degenerate and onecan use degenerate perturbation theory to couple these energy levels.

In this case, the set of equations can be truncated to only two non-zerocomplex coefficients Ck−Q

1and Ck−Q

2. These two coefficients satisfy the pair

of equations

( E − E(0)k − Q

1) Ck−Q

1= Vions(Q2

− Q1) Ck−Q

2(454)

and( E − E

(0)k − Q

2) Ck−Q

2= Vions(Q1

− Q2) Ck−Q

1(455)

which can be combined to yield the quadratic equation for E

( E − E(0)k − Q

1) ( E − E

(0)k − Q

2) = | Vions(Q1

− Q2) |2 (456)

The two energy eigenvalues are given by the solutions of the quadratic equation

E =( E

(0)k − Q

1+ E

(0)k − Q

2

2

)±

√√√√(E(0)k − Q

1− E

(0)k − Q

2

2

)2

+ | Vions(Q1− Q

2) |2

(457)Whenever the Bloch wave vector k takes on special values such that unperturbedbands cross

E(0)k − Q

1= E

(0)k − Q

2(458)

the energy bands simplify to yield the two energies

E = E(0)k − Q

1± | Vions(Q1

− Q2) | (459)

215

Nearly Free Electron Dispersion Relation

-1

0

1

2

3

0 0.2 0.4 0.6 0.8 1

(kxa/π)

Ek/E

0

2 Vions(1,0,0)

Figure 125: The electronic dispersion relations calculated in the nearly-freeelectron approximation. The crystal potential may lift the degeneracy of thefree electron bands at their crossing points.

If the unperturbed bands cross, the non-zero potential produces a splitting of2 | Vions(Q1

− Q2) |. This result is consistent with that previously found by

using non-degenerate perturbation theory.

The avoided crossings of the bands are expected to occur whenever

E(0)k − Q

1∼ E

(0)k − Q

2(460)

This gives rise to a specific condition on the wave vectors. For convenience ofnotation, let q = k − Q

1so that this criterion takes the form

E(0)q = E

(0)q − Q” (461)

for some reciprocal lattice vector Q” 6= 0. This requires that vector q lies onthe Bragg plane bisecting Q”, as this condition reduces to

Q”2 = 2 q . Q” (462)

The vector q − Q” lies on a second Bragg plane. Thus, the geometric signifi-cance of the condition for the degeneracy of the unperturbed bands, is that theelectronic states satisfy the condition for Bragg scattering.

The origin of the gaps can be easily understood from consideration of thewave functions. When q lies on a single Bragg plane, then the energy eigenvaluesare simply given by

E = E(0)q ± | Vions(Q”) | (463)

The coefficients corresponding to these energies are found from the two coupledequations. In this case, where the unperturbed bands cross, the coefficients are

216

O

q

Q

Q-q

Q

Figure 126: If the vector q lies on a Bragg plane, the vector q −Q also lies ona Bragg plane.

related via

Cq = ± sign(Vions(Q”)

)Cq−Q” (464)

which produces two standing wave solutions. If Vions(Q”) > 0, then the pairof standing wave states are the anti-bonding state

| φ+q (r) |2 ∼ 2

Vcos2

(Q” . r

2

)E+ = E(0)

q + | Vions(Q”) | (465)

and the bonding state

| φ−q (r) |2 ∼ 2V

sin2

(Q” . r

2

)E− = E(0)

q − | Vions(Q”) | (466)

On the other hand, if Vions(Q”) < 0, then the situation is reversed, and theanti-bonding state is given by

| φ+q (r) |2 ∼ 2

Vsin2

(Q” . r

2

)E+ = E(0)

q + | Vions(Q”) | (467)

217

while the bonding state is given by the other form

| φ−q (r) |2 ∼ 2V

cos2(Q” . r

2

)E− = E(0)

q − | Vions(Q”) | (468)

In this context, the wave function

φpq(r) ∼

√2V

sin(Q” . r

2

)(469)

is called p-like as it vanishes at the lattice points, whereas

φsq(r) ∼

√2V

cos(Q” . r

2

)(470)

is called s-like as it is non-vanishing at the positions of the ions, r = R. Theorigin of the gap between the two branches is seen through examination of theaverage potential energy of the s and p like wave functions∫

V

d3r Vions(r) | φs,pq (r) |2 (471)

The s-like electrons congregate at the position of the ions where the potential islower, and the p-like electrons congregate between the ions where the potentialis higher. For an attractive interaction Vions(r) < 0, this leads to φs

q(r) havinga lower energy than φp

q(r), (since Vions(Q”) < 0).

The opening of a gap in the electronic dispersion relation at the Bragg planeis manifested by structure in the one-electron density of states which is associ-ated with the van Hove singularities. For example, in the free electron approxi-mation the surfaces of constant energies are spherical. If the surface of constantenergy ε0 corresponding to the momentum k0 first touches the Brillouin zoneboundary at n points, then for energies ε ∼ ε0, the gap in the dispersion re-lation at the Brillouin boundary impedes the progress of the constant energysurface into the next zone. The effect of the gap on the density of states can beestimated by only considering the free electron states within the first Brillouinzone. In this case, the surface of constant energy has an area S(ε) which is givenby

S(ε) = 4 π k2 − n 2 π k ( k − k0 ) (472)

which is the surface area of the sphere of radius k minus the area of the nspherical caps. The density of states is proportional to

ρ(ε) ∝ S(ε)dk

dε(473)

Hence, on using

dε =h2

mk dk (474)

218

-4 -3 -2 -1 0 1 2 3 4

r/a

Vions(r)

cos(πr/a)2|φQ/2(r)|2

-4 -3 -2 -1 0 1 2 3 4

r/a

|φQ/2(r)|2

sin(πr/a)2

Vions(r)

Figure 127: The Bloch functions with k on the Bragg planes have the formof standing waves. The standing waves that have nodes located between theatoms have lower energies, while the standing waves that have nodes at theatomic positions have higher energies.

one obtains the approximate expression for the density of states

ρ(ε)ρ(ε0)

≈ 4 π k − n 2 π ( k − k0 )4 π k0

=n

2+ ( 1 − n

2)k

k0

=n

2+ ( 1− n

2)√

ε

ε0(475)

Thus, for energies which fall within the gaps at the Brillouin zone boundaries,the density of states should be decreased below the free electron density of states.The states pushed out from the gap region are manifested by an increase in thedensity of states just outside the gap. This type of structure in the electronicdensity of states is clearly seen above the Fermi energy of Li, as found in the

219

Figure 128: A constant energy surface of free electrons in reciprocal space for as.c. solid.

Brillouin Zone DecompositionFree Electron Density of States

0

0.05

0.1

0.15

0 0.5 1 1.5 2 2.5 3

ε [ units of (h2π2/2ma2) ]

ρ(ε)

[ St

ates

(2m

a2 /h2 ) ]

First Zone

Second Zone

Third Zone

Figure 129: The Brillouin Zone decomposition of the free electron density ofstates for a s.c. solid.

calculations of F.S. Ham42.

The Bragg planes have other significance, as can be inferred from the gradi-ent of the energy

E± =( E

(0)q + E

(0)q − Q”

2

)±

√√√√(E(0)q − E

(0)q − Q”

2

)2

+ | Vions(Q”) |2

(476)

42F. S. Ham, Phys. Rev. 128, 82 (1962).

220

which is found as

∇q E± =h2

m

[ (q −

Q”2

)±Q”2

(E

(0)q − E

(0)q − Q”

)√(

E(0)q − E

(0)q − Q”

)2

+ 4 | Vions(Q”) |2

]

(477)On the Bragg plane, one has

E(0)q = E

(0)q − Q” (478)

therefore, the second term in the expression for the gradient drops out on theseplanes. Thus, the gradient of the energy of the mixed bands is given by

∇q E± =h2

m

(q −

Q”2

)(479)

and, as q is on the Bragg plane, the vector q − Q”

2 is parallel to the plane andso is the gradient. The gradient of the energy is perpendicular to surfaces ofconstant energy and so, the constant energy surfaces are usually perpendicularto the Bragg planes at their points of intersection.

Generally, the vanishing of the normal component of the gradient at theBrillouin zone boundary is not dependent on the validity of the nearly-freeelectron approximation, but is a consequence of symmetry. Consider the casein which there is a mirror plane symmetry, σ. The mirror plane is assumed torun through the origin of the Brillouin zone and is parallel to the Brillouin zoneboundary under consideration. Then, the normal component of the gradient isdefined as

Q . ∇k Ek

∣∣∣∣ = limδ → 0

[Ek+δQ − Ek−δQ

2 δ

](480)

However, since the point k is equivalent to the point k −Q, one has

Ek−δQ = E−Q+k−δQ (481)

and, as there exists a mirror plane σ through the origin and perpendicular toQ, one also has

E−Q+k−δQ = EQ+σk+δQ (482)

Noting that as k is on the Bragg plane, k ≡ Q + σk, and substituting theabove equality into the definition, one finds that the normal component of thegradient vanishes at the Brillouin zone boundary

Q . ∇k Ek

∣∣∣∣ = 0 (483)

Thus, at the Brillouin zone boundary, either the normal component of the gra-dient vanishes or the gradient does not exist, i.e. there might be a cusp. Thepresence of other types of symmetry can give rise to similar conclusions.

221

Two Bragg Planes and a Mirror Plane

O

k-δδδδQ

Q/2

-Q+k-δδδδQ

σσσσ

Figure 130: The vanishing of the group velocity at the Brillouin zone boundaryis a consequence of a mirror plane σ parallel to the boundary.

8.1.4 Empty Lattice Approximation Band Structure

Since the nearly-free electron approximation deviates only slightly from the freeelectron approximation, the gross features of the band structure can be foundusing the empty lattice approximation.

Since the Brillouin zone is a three-dimensional object and is highly sym-metric, it is only necessary to specify the bands within an irreducible wedge.Once the bands are specified within the wedge, then by use of symmetry, thebands are completely known throughout the Brillouin zone. Since it is difficultto represent the energy dispersion relations in a three-dimensional volume ofreciprocal space, it is customary to specify the dispersion relations on the linesdefining the boundaries of the irreducible wedge. These lines have high symme-tries.

Consider the case of an f.c.c. Bravais lattice, and consider the bands withinthe first Brillouin zone. The high-symmetry points are marked by special letters.

222

The Brillouin zone for a face-centered cubic lattice

L

X

XK

UΓ

W

Figure 131: The high-symmetry points and lines in the Brillouin zone of anf.c.c. structure.

Γ ≡ (0, 0, 0) ≡ (0, 0, 0)

K ≡ 3 π2 a (1, 1, 0) ≡ 2 π

a ( 34 ,

34 , 0)

W ≡ πa (2, 1, 0) ≡ 2 π

a (1, 12 , 0)

X ≡ 2 πa (1, 0, 0) ≡ 2 π

a (1, 0, 0)

L ≡ πa (1, 1, 1) ≡ 2 π

a ( 12 ,

12 ,

12 )

and, in units of 2 πa , these points correspond to

Γ ≡ (0, 0, 0) The zone center

K ≡ ( 34 ,

34 , 0) The midpoint of the hexagonal edge

X ≡ (1, 0, 0) The square face center

W ≡ (1, 12 , 0) The vertex

L ≡ ( 12 ,

12 ,

12 ) The hexagonal face center

The electron bands are usually plotted against k along the directions

223

Γ → X → W → L → Γ → K → X

All of these are high-symmetry lines. The lengths of the linear segments (inunits of 2 π

a ) are given by

1 12

1√2

√3

23√

24

√104

In the empty lattice approximation, the single-electron energies can be plottedalong these lines in units of E0, where

E0 =h2

2 m

(4 π2

a2

)(484)

The components of the reduced wave vectors are defined as the dimensionlessquantities ki, where

ki =ki a

2 π(485)

It should be noted that two or more Bragg planes may intersect at points lo-cated on some of these lines.

The energies of the various bands can be constructed from the various E0k−Q.

The lowest energy band that is considered is simply E(0)k , where Q

Γ= (0, 0, 0).

This band has the lowest energy, since points in the first Brillouin zone arecloser to the origin than any other reciprocal lattice point Q. The band isnon-degenerate, except if k is located on the Brillouin zone boundary. Theone-electron energy is given by

E(0)k

E0=(k2

x + k2y + k2

z

)(486)

which for Γ → X, reduces to

= k2x for 0 ≤ kx ≤ 1 (487)

For the line segment X → W on the Brillouin zone boundary, this banddispersion relation is evaluated as

= 1 + k2y for 0 ≤ ky ≤ 1

2(488)

The band should be doubly-degenerate for most points on the line X W , sincepoints on this line are equidistant to the origin and the reciprocal lattice pointQ

X= 4 π

a (1, 0, 0). For W → L, this band is described by

E(0)k

E0=

14

+ k2x + ( 1 − kx )2 for

12≤ kx ≤ 1 (489)

224

The band should be at least doubly-degenerate for points on the line W L,since the line is on the Q

L= 4 π

a ( 12 ,

12 ,

12 ) Bragg plane. That is, the line is

symmetrically positioned with respect to the reciprocal lattice points QΓ

andQ

L. For L → Γ, the dispersion relation reduces to

E(0)k

E0= 3 k2

x for 0 ≤ kx ≤ 12

(490)

For Γ → K, the band energy simplifies to

= 2 k2x for 0 ≤ kx ≤ 3

4(491)

The last segment is given by the, lower symmetry, interior line K → X, overwhich the band energy takes the form

= k2x + 9 ( 1 − kx )2 for

34≤ kx ≤ 1 (492)

0

1

2

3

Ek

/ E0

ΓΓΓΓ ΓΓΓΓΧΧΧΧ ΧΧΧΧW L K

(1) (1)

(2)

(1)(1)

(2)

(2)

(2)

(2)

(2)

(3)(4)

(1)

(2)(1)

(2)

Figure 132: The electronic dispersion relations for an f.c.c. compound in theempty lattice approximation. The degeneracies of the various branches areenclosed in parentheses.

The next lowest energy bands can be identified by considering their proxim-ity to other reciprocal lattice points Q. At any point, this just reduces to findingthe nearest Bragg planes. For our particular path, these are found amongst theset of reciprocal lattice vectors Q

X= 4 π

a (1, 0, 0), QL

= 4 πa ( 1

2 ,12 ,

12 ) and

225

QL′

= 4 πa ( 1

2 ,12 ,−

12 ).

The next band to be considered is simply E(0)k−Q

X

, where QX

= 4 πa (1, 0, 0).

In the free electron approximation, this band should be degenerate with E(0)k

along the line segment X W . The dispersion relation for this one-electron bandis given by

E(0)k−Q

X

E0=(

( kx − 2 )2 + k2y + k2

z

)(493)

which, for Γ → X, reduces to

= ( kx − 2 )2 for 0 ≤ kx ≤ 1 (494)

For X → W , the band dispersion relation reduces to

= 1 + k2y for 0 ≤ ky ≤ 1

2(495)

and, indeed, is degenerate with the band E(0)k . On the line W → L, the band

E(0)k−Q

X

is described by the dispersion relation

E(0)k−Q

X

E0=

14

+ ( kx − 2 )2 + ( 1 − kx )2 for12≤ kx ≤ 1 (496)

Since the line W L is on the Bragg plane with reciprocal lattice vector QL, the

band energy E(0)k−Q

X

is higher energy than the band energy E(0)k−Q

L

. Along the

line L → Γ, the dispersion relation of E(0)k−Q

X

is expressed as

= ( kx − 2 )2 + 2 k2x for 0 ≤ kx ≤ 1

2(497)

In the empty lattice approximation, this branch of the band could be expectedto be triply-degenerate since the line is symmetrically placed with respect to thethree equivalent reciprocal lattice points, 4π

a (1, 0, 0), 4πa (0, 1, 0) and 4π

a (0, 0, 1).For Γ → K, this band takes the form

= ( kx − 2 )2 + k2x for 0 ≤ kx ≤ 3

4(498)

The last segment is given by K → X, for which the band takes the form

= ( kx − 2 )2 + 9 ( 1 − kx )2 for34≤ kx ≤ 1 (499)

Since, the line is mostly confined within the interior of the Brillouin zone, theband is generally non-degenerate.

226

The next band in the set is E(0)k−Q

L

, where QL

= 4 πa ( 1

2 ,12 ,

12 ). In the empty

lattice approximation, this band becomes degenerate with E(0)k on the Bragg

plane defined by the reciprocal lattice point QL. The one-electron dispersion

relation is given by

E(0)k−Q

L

E0=(

( kx − 1 )2 + ( ky − 1 )2 + ( kz − 1 )2)

(500)

which, for Γ → X, is just

= 2 + ( kx − 1 )2 for 0 ≤ kx ≤ 1 (501)

This branch of the band should be at least four-fold degenerate, since the lineΓ X is symmetrically positioned with respect to the four reciprocal lattice points4 πa ( 1

2 ,±12 ,±

12 ). For X → W , the branch has a dispersion relation given by

= 1 + ( ky − 1 )2 for 0 ≤ ky ≤ 12

(502)

The line W → L is on the Brillouin zone boundary and, therefore, the freeelectron band E(0)

k−QL

is degenerate with E(0)k at these points. On the line W L,

the dispersion relation is described by

E(0)k−Q

L

E0=

14

+ ( kx − 1 )2 + k2x for

12≤ kx ≤ 1 (503)

For L → Γ, this dispersion is given as

= 3 ( kx − 1 )2 for 0 ≤ kx ≤ 12

(504)

and one expects the band to be non-degenerate. For Γ → K, this band takesthe form

= 1 + 2 ( kx − 1 )2 for 0 ≤ kx ≤ 34

(505)

On the line segments Γ K and K X, the branch of this one-electron bandare generally doubly-degenerate, since the points on the line are symmetricallypositioned with respect to the pair of reciprocal lattice points Q

Land Q

L′. The

last path segment is given by K → X, in which the band has the form of

= 1 + ( kx − 1 )2 + ( 2 − 3 kx )2 for34≤ kx ≤ 1 (506)

The last band we consider is E(0)k−Q

L′, where Q

L′= 4 π

a ( 12 ,

12 ,−

12 ). Our

set of lines only meet this Bragg plane of QL′

at the point K. In the empty

227

lattice approximation, the band should be triply-degenerate at the K point.The dispersion relation is given by

E(0)k−Q

L′

E0=(

( kx − 1 )2 + ( ky − 1 )2 + ( kz + 1 )2)

(507)

which, for Γ → X, is just

= 2 + ( kx − 1 )2 for 0 ≤ kx ≤ 1 (508)

For X → W , the dispersion relation for the band is given by

= 1 + ( ky − 1 )2 for 0 ≤ ky ≤ 12

(509)

For W → L, this band is described by

=14

+ ( kx − 1 )2 + ( 2 − kx )2 for12≤ kx ≤ 1 (510)

For L → Γ, this dispersion is given as

= 2 ( kx − 1 )2 + ( kx + 1 )2 for 0 ≤ kx ≤ 12

(511)

This branch may be expected to be triply-degenerate as the line L Γ is symmetri-cally positioned with respect to the three reciprocal lattice points, 4 π

a (− 12 ,

12 ,

12 ),

4 πa ( 1

2 ,−12 ,

12 ) and Q

L′= 4 π

a ( 12 ,

12 ,−

12 ). The line Γ K is equidistant from

the reciprocal lattice points QL

and QL′

and, therefore, this band should be

degenerate with E(0)k−Q

L

on this line segment. On the line Γ K, the band takesthe form

E(0)k−Q

L′

E0= 1 + 2 ( kx − 1 )2 for 0 ≤ kx ≤ 3

4(512)

The last line segment is given by K → X, for which the band has the form

= 1 + ( kx − 1 )2 + ( 2 − 3 kx )2 for34≤ kx ≤ 1 (513)

which is also doubly-degenerate.

It is seen that some branches of these bands are highly degenerate. For areal solid, the symmetry of the bands is not dictated by the symmetry of theBrillouin zone (which is determined by the symmetry of the Bravais lattice),but instead is dictated by the symmetry of the space group of the lattice whichincludes the atomic basis. When Vions 6= 0, the degeneracy of the variousbranches found in the empty lattice approximation may be lifted. Group theorycan be used to determine whether or not the potential lifts the degeneracy of

228

the branches.

Thus, even in the empty lattice approximation, the method of plotting bandsshows a great deal of structure. The real structure is actually inherent in theBragg planes which generally can be associated with an“energy gap” in the dis-persion relations. The “gap” may or may not extend across the entire Brillouinzone. A gap only appears in the density of states if the “gap” extends across theentire Brillouin zone. The nearly-free electron approximation has been workedout in detail for Al by B. Segall43 and has been compared with the results ofnumerical calculations.

Nearly Free Electron Bands for f.c.c. Al

-0.25

0

0.25

0.5

0.75

1

1.25

Ek [

Ryd

berg

s ]

Γ ΓΧ ΧW L K W

f.c.c.

Figure 133: The the nearly-free electron approximation to the dispersion rela-tions of f.c.c. Al.

For the b.c.c. lattice, the reciprocal lattice vectors are

b1 =12

4 πa

( ex + ey )

b2 =12

4 πa

( ex + ez )

b3 =12

4 πa

( ey + ez ) (514)

The Cartesian coordinates of the high-symmetry points are

43B. Segall, Phys. Rev. 124, 1797 (1961).

229

The Brillouin zone of the body-centered cubic lattice

N H

P

Γ

H

N

Figure 134: The high-symmetry points and lines in the Brillouin zone of a b.c.c.solid.

Γ ≡ (0, 0, 0)

H ≡ (1, 0, 0)

N ≡ ( 12 ,

12 , 0)

P ≡ ( 12 ,

12 ,

12 )

in units of 2 πa .

——————————————————————————————————

8.1.5 Exercise 36

Derive the lowest energy bands of a b.c.c. lattice in the empty lattice approx-imation. Plot the dispersion along the high-symmetry directions (Γ → H →N → P → Γ → N).

——————————————————————————————————

230

8.1.6 Degeneracies of the Bloch States

The degeneracies of the bands at various points in the Brillouin zone, found inthe empty lattice approximation, can be raised by the crystalline potential. Thecharacter and degeneracies of the bands at symmetry points can be ascertainedby the use of group theory44.

Given a Bloch function φn,k(r), one can apply a general point group sym-metry operator O(Aj) to the Bloch function, thereby, transforming it into theBloch function corresponding to the wave vector Aj k

O(Aj) φn,k(r) = φn,Ajk(r) (515)

This is proved by considering the combined operation consisting of the pointgroup operation O(Aj) followed by a translation through a Bravais lattice vectorR. The effect of the combined operation is evaluated as

TR O(Aj) φn,k(r) = TR φn,k(A−1j r)

= exp[− i k . A−1

j R

]φn,k(A−1

j r) (516)

where the second line follows from Bloch’s theorem. However, we note thatthe scalar product remains invariant if both vectors are transformed. We shalltransform the vectors k and ( A−1

j R ) by Aj . Hence, as

k . ( A−1j R ) = ( Aj k ) . ( Aj A

−1j R )

= ( Aj k ) . R (517)

we find that

TR O(Aj) φn,k(r) = exp[− i ( Aj k ) . R

]φn,k(A−1

j r)

= exp[− i ( Aj k ) . R

]O(Aj) φn,k(r)

(518)

Since the quantity

exp[− i ( Aj k ) . R

](519)

is the eigenvalue of the translation operator TR, the Bloch wave vector of thefunction O(Aj) φn,k(r) is Aj k. As this is an energy eigenfunction, the trans-formed function is a Bloch function. That is,

φn,Ajk(r) = O(Aj) φn,k(r) (520)

44L. P. Bouckaert, R. Smoluchowski and E. Wigner, Phys. Rev. 50, 58 (1936).

231

Since the point group symmetry operations commute with the Hamiltonian,

[ H , O(Aj) ] = 0 (521)

the Bloch states O(Aj) φn,k(r) all have the same energy En,k.

A set of basis functions for a representation of the space group can be con-structed by repeatedly applying the point group symmetry operators to anyone of the Bloch functions. The same vector k cannot appear in distinct basescreated from a Bloch function in this manner, since the symmetry operationsform a group. This means that two such bases are either identical or have nowave vector k in common. A basis created from the Bloch function φn,k(r) inthis fashion may be either reducible or irreducible.

An irreducible basis can be constructed by selecting an appropriate subsetof Bloch functions from the above basis set. If one considers the set of wavevectors Ai k, then certain of these points may be equivalent in that

Ai k = Aj k + Q (522)

where Q is a reciprocal lattice vector. The star of k is the set of all the inequiv-alent wave vectors Ai k. More precisely, the star of the wave vector k consistsof the set of all mutually inequivalent wave vectors Ai k, where Ai ranges overall the operations of the point group. Since none of the Bloch wave vectors inthe star are equivalent, the corresponding Bloch functions are all linearly inde-pendent. Hence, the Bloch functions of the star may be used to construct anirreducible basis. The star of k contains fewer wave vectors than the order ofthe point group, if either k lies on a symmetry line or is on the Brillouin zoneboundary. As an example, consider a crystal with a simple tetragonal BravaisLattice and a one-atom basis. The crystal has the D4h point group symmetry.The stars of the wave vectors Γ, Z,M and A each only contain one wave vector.However, the star of the wave vector X contains a total of two points, as doesthe star of wave vector R. For the case of tetragonal symmetry, the star ofa general k vector contains a total of 2! × (2)3 wave vectors. Since the set ofenergies of Bloch states with wave vector k are equal to the set of energies ofBloch states at each wave vector in the star of k, it is only necessary to find theelectronic dispersion relations in an irreducible wedge of the Brillouin zone.

The group of the k vector consists of all symmetry operations which, whenacting on k, lead to an equivalent point. That is, the symmetry operations ofthe group of the k vector satisfy

Aj k = k + Q (523)

where Q is a reciprocal lattice vector. As an example, the groups of the k vectorsfor the points Γ, Z,M,A of a crystal with a simple tetragonal lattice coincidewith the D4h point group of the tetragonal lattice itself. The groups of the k

232

Γ X

Z R

A

M

Figure 135: The tetragonal Brillouin zone and the points of high symmetry.

vectors at X and R are D2h, and D2h is a subgroup of D4h. In general, thegroup of the k vector of the Γ point will always coincide with the point groupof the crystal. However, at a general point, the group of the wave vector onlyconsists of the identity. The group of the k vector has irreducible representa-tions, and these are called the small representations.

The Bloch functions corresponding to the wave vectors of the star of k canbe symmetrized with respect to the small representations. The symmetrizationcan be performed by using the projection method. Although the groups of thewave vectors in the star may be different, the small representation of any onecan be chosen for the symmetrization process. After the symmetrization, theresulting basis functions form an irreducible representation of the space group.Each basis function of the small representation only corresponds to exactly onewave vector in the star and its equivalent wave vectors. The basis functionscorresponding to the different irreducible representations are orthogonal.

The irreducible representations of the space group constructed from theBloch functions are fully determined by the star of the k vector and the smallrepresentation. The basis functions forming the irreducible representation ofthe space group constructed from the Bloch state φn,k(r) are eigenstates of H0

with energy En,k. Barring accidental degeneracies, the degeneracy of this eigen-value is equal to the dimension of its irreducible representation. As k varies inthe Brillouin zone, the eigenvalue En,k and the corresponding basis functionsvary continuously. The group of the k vector also varies as k varies. Wheneverthe dimension of the small representation corresponding to the basis functionφn,k(r) changes, the degeneracy of En,k changes. This may signify that at these

233

points different bands cross or merge together.

Alternatively, at k there are a vast number of bands each corresponding toa different small representation. The degeneracy of each band is given by thedimension of the corresponding small representation. If an irreducible represen-tation of the group of the wave vector k can be decomposed into the irreduciblerepresentations of the group of k0, then on varying k to k0, the branch willsplit into sub-levels. The degeneracies of the sub-levels are determined by thedimensions of the irreducible representations contained in the decomposition.

——————————————————————————————————

As an example, consider the nearly-free electron bands of Zinc Blende. Thematerial has tetrahedral point group symmetry, Td. The point group containstwenty four elements in five equivalence classes. One class consists of the iden-tity E. There is a class of eight C3 operations, which contain the rotation C3

and the inverse rotation C−13 about the four axes [1, 1, 1], [1, 1, 1], [1, 1, 1] and

[1, 1, 1]. There is a class consisting of three C2 operations around the [1, 0, 0],[0, 1, 0] and [0, 0, 1]. There is a class consisting of six S4 operations around the[1, 0, 0], [0, 1, 0] and [0, 0, 1] axes. Finally, there is a group consisting of six σ op-erations which are reflections in the six planes (1, 1, 0), (1, 0, 1), (0, 1, 1), (1, 1, 0),(1, 0, 1) and (0, 1, 1). Therefore, the group has five irreducible representations.The character table is given by

Td E C2(3) C3(8) (S4)(6) σ(6)

Γ1 1 1 1 1 1Γ2 1 1 1 - 1 - 1Γ3 2 2 - 1 0 0Γ4 3 - 1 0 1 - 1Γ5 3 - 1 0 - 1 1

Let us consider the band structure along the high-symmetry directions [1, 1, 1]and [1, 0, 0] directions.

At the Γ point the group of the k vector coincides with the point group ofthe crystal. Since the free electron approximation for the Bloch wave functionfor k = 0 is a constant, it is a basis for the Γ1 representation. Thus, the levelis non-degenerate.

234

Figure 136: A unit cell of the Zinc Blende structure.

At a general point along the eight [1, 1, 1] directions (the Λ high-symmetrylines), the group of the k vector is C3v and contains six elements in three classes.These classes are the identity E, a class consisting of the rotation C3 about the[1, 1, 1] axis and its inverse C−1

3 , and the class consisting of three reflections σ inthe three equivalent (1, 1, 0) planes containing the [1, 1, 1] axis. The charactertable is given by

C3v E C3(2) σ(3)

Λ1 1 1 1Λ2 1 1 - 1Λ3 2 - 1 0

Thus, the branches along the Λ high-symmetry lines are either singly or dou-bly degenerate, when the crystalline potential is introduced. The branch whichemanates from k = 0 with the approximate energy E(0)

k = h2

2 m k2 belongs tothe Λ1 representation as this is compatible with the Γ1 representation.

At the end point L where k = 2 πa ( 1

2 ,12 ,

12 ), the symmetry operations

are identical to those of Λ. In the free electron approximation, the state at Lis doubly degenerate (ignoring spin) since the wave vectors 2 π

a ( 12 ,

12 ,

12 ) and

− 2 πa ( 1

2 ,12 ,

12 ) differ by a reciprocal lattice vector Q = 2 π

a (1, 1, 1). Usingthe compatibility relations, one can show that the other state also has Λ1 sym-

235

metry. These two states are accidentally degenerate, since they are not partnerbasis functions of a multi-dimensional irreducible representation. Therefore, thedegeneracy may be lifted by the presence of a crystalline potential Vions(Q).

On continuing along the Λ high-symmetry line, one reaches the point k =2 πa (1, 1, 1). Since the primitive reciprocal lattice vectors of the f.c.c. lattice

are of the form

b1 =2 πa

(−1, 1, 1)

b2 =2 πa

(1,−1, 1)

b3 =2 πa

(1, 1,−1) (524)

thenQ = b1 + b2 + b3 is equal to 2 πa (1, 1, 1). Thus, the point k = 2 π

a (1, 1, 1)is equivalent to the Γ point. The star consists of just one wave vector. At thispoint, the eight nearly-free electron bands corresponding to

φkj(r) =

1√V

exp[i

2 πa

( ± x ± y ± z )]

(525)

are degenerate. These eight functions form the basis of an eight-dimensionalrepresentation of the group Td which is reducible. In this representation, asymmetry transformation A is represented by the 8 × 8 matrices, D(A), whichare constructed according to the prescription

O(A) φki(r) = φki

(A−1r)

=∑

j

φkj(r) D(A)j,i (526)

The characters of this eight-dimensional representation are given by the traceof the 8 × 8 matrices and, therefore, the character of an operation is just thenumber of wave functions that are unchanged by the transformation.

Class Transformation χ

E x, y, z 8C2(3) x, y, z 0C3(8) y, z, x 2S4(6) x, z, y 0σ(6) y, x, z 4

236

This eight-dimensional representation, Γ, is reduced into the irreducible repre-sentations, Γµ, via

Γ =∑

µ

aµ Γµ (527)

The decomposition can be found from considering the characters. The charac-ters of a symmetry operation A, χ(A), is decomposed into the characters of theirreducible representations, χµ(A), via

χ(A) =∑

µ

aµ χµ(A) (528)

The multiplicity aµ can be found from the orthogonality relation∑i

gi χ(Ai) χµ(Ai) = g aµ (529)

where the sum over i runs over all the equivalence classes of the group, and gi

is the number of symmetry elements in the i-th equivalence class, and g is theorder of the group. This procedure leads to the decomposition

χ(Ai) = 2 χΓ1(Ai) + 2 χΓ4(Ai) (530)

Thus, the eight plane wave basis can be symmetrized into two sets of basisfunctions of Γ1 symmetry and two three-dimensional sets of basis functionsof Γ4 symmetry. The symmetrization process is performed by the use of theprojection method. A projector, Pµ which projects the functions on to anirreducible set of basis functions, is constructed from the symmetry operationsO(A) and the characters of the operations via

Pµ =dim(Γµ)

g

∑A

χµ(A) O(A) (531)

In this, dim(Γµ) is the dimension of the µ-th irreducible representation, i.e.,dim(Γµ) = χµ(E). When the projector acts on an arbitrary combination offunctions with equivalent wave vectors k, φk(r), it produces a basis function,φµ

k(r) for the µ-th irreducible representation

Pµ φk(r) = φµk(r) (532)

In this way, one can construct the set of symmetrized basis functions:

237

Representation Basis functions

Γ1

cos 2πxa cos 2πy

a cos 2πza

Γ1

sin 2πxa sin 2πy

a sin 2πza

Γ4

cos 2πxa sin 2πy

a sin 2πza

sin 2πxa cos 2πy

a sin 2πza

sin 2πxa sin 2πy

a cos 2πza

Γ4

sin 2πxa cos 2πy

a cos 2πza

cos 2πxa sin 2πy

a cos 2πza

cos 2πxa cos 2πy

a sin 2πza

In this basis, all the matrices D(A) representing the symmetry operators Ahave the same block diagonal form. The matrices contains two one-dimensionalblocks and two three-dimensional blocks. Thus, these eight levels may be splitby the application of a potential into two non-degenerate levels and two sets oftriply degenerate levels. Therefore, the degeneracies at the eight approximate(free-electron) bands at Γ are not completely lifted.

Along the X direction (the ∆ high-symmetry line), the wave vectors are ofthe form (k, 0, 0) where 0 < k < 2 π

a . The group of k is C2v. It has fourelements in four classes: the identity E, a two-fold rotation about the [1, 0, 0]axis, and the two diagonal mirror planes σd and σ′d. The character table is givenby

238

C2v E C24 σd σ′d

∆1 1 1 1 1∆2 1 1 - 1 - 1∆3 1 - 1 1 - 1∆4 1 - 1 - 1 1

Therefore, along this direction, all the irreducible representations are one-dimensional. The symmetry of the wave function emanating from (0, 0, 0) belongto ∆1 since this is the only irreducible representation compatible with Γ1. Thisbranch continues up to the X point. The point 2 π

a (1, 0, 0) is equivalent to thepoint − 2 π

a (1, 0, 0), as they are related via the Q vector Q = b2 + b3. Atthe X point, the lowest energy level in the nearly-free electron approximationis doubly degenerate.

The group of the k vector at the X point is D2d and consists of eight el-ements arranged in five classes. These are the identity E, a two-fold rotationabout the x axis C2

4 , a class of two elements which are the two-fold rotationsC2 about the y and z axis, and a class containing two S4 operations about thex axis, and a class of two diagonal reflections σd on the (0, 1, 1) and the (0, 1, 1)planes. Thus, there are five irreducible representations. The character table isgiven by

D2d E C24 (1) C2(2) S4(2) σd(2)

X1 1 1 1 1 1X2 1 1 1 - 1 - 1X3 1 1 - 1 - 1 1X4 1 1 - 1 1 - 1X5 2 - 2 0 0 0

At theX point, the wave functions of the two-fold degenerate energy levels, E(0),found in the nearly-free electron approximation belong to the one-dimensionalX1 and X3 irreducible representations. This degeneracy may be raised by the

239

potential.

On continuing along theX direction, one reaches the pointQX

= 2 πa (2, 0, 0).

The six k points (±2, 0, 0), (0,±2, 0) and (0, 0,±2) are all equivalent to the zonecenter. The group of the wave vector is Td. The six wave functions

φQX

(r) =1√V

exp[± i

4 πa

x

]φQ

Y(r) =

1√V

exp[± i

4 πa

y

]φQ

Z(r) =

1√V

exp[± i

4 πa

z

](533)

can be used as a basis for a six-dimensional representation. In this representa-tion, the characters of the symmetry operations are given by:

Class Transformation χ

E x, y, z 6C2(3) x, y, z 2C3(8) y, z, x 0S4(6) x, z, y 0σ(6) y, x, z 2

This representation is degenerate and can be decomposed via

Γ =∑

µ

aµ Γµ (534)

The multiplicities aµ are calculated from the orthogonality relations∑i

gi χ(Ai) χµ(Ai) = g aµ (535)

which leads to the decomposition

Γ = Γ1 + Γ3 + Γ4 (536)

Thus, the six-dimensional representation of the group of the wave vector (200)is reducible and reduces into a one-dimensional, a two-dimensional and a three-dimensional irreducible representation. The basis functions can be symmetrized

240

using the projection method. The basis functions for the small representationsare

Representation Basis functions

Γ1

cos 4πxa + cos 4πy

a + cos 4πza

Γ3

cos 4πya − cos 4πz

a

2 cos 4πxa − cos 4πy

a − cos 4πza

Γ4

sin 4πxa

sin 4πya

sin 4πza

Hence, the six-fold degenerate free-electron energy level E(0)Q

X

= h2

2m ( 4 πa )2

may have its degeneracy lifted by Vions(Q).

——————————————————————————————————

8.1.7 Exercise 37

Using the symmetrized wave functions at k = ( 2 πa ) (1, 1, 1) in the nearly-free

electron model for Zn Blende

φΓ1 =

√8

N a3cos

2 π xa

cos2 π ya

cos2 π za

φΓ4(x) =

√8

N a3sin

2 π xa

cos2 π ya

cos2 π za

φΓ4(y) =

√8

N a3cos

2 π xa

sin2 π ya

cos2 π za

φΓ4(z) =

√8

N a3cos

2 π xa

cos2 π ya

sin2 π za

(537)

241

show that the matrix elements of the momentum operator between the Γ1 andΓ4 basis functions are given by

| < Γ1 | px | Γ4(x) > |2 = | < Γ1 | py | Γ4(y) > |2 = | < Γ1 | pz | Γ4(z) > |2 =(

2 π ha

)2

(538)while all other matrix elements are zero.

——————————————————————————————————

8.1.8 Exercise 38

Consider the nearly-free electrons bands in f.c.c. Al, on the the high-symmetrylines Γ X and X W . Consider the branches of the free electron bands

E(0)k−Q

i

=h2

2 m( k − Q

i)2 (539)

where Qi

runs over the four reciprocal lattice vectors Qi

= 2πa (1,±1,±1).

Show that, in the free electron approximation, there is some degeneracy betweenthe band branches on these line segments. Set up the secular equation for thesefour bands on the two line segments, and solve them using Matlab. Assume thatthe lattice constant of Al is 4.05 A, and that the non-zero values of Vions(Q)are given by

Vions(1, 1, 1) = 0.023 RydbergsVions(2, 0, 0) = 0.043 Rydbergs

while all other matrix elements are zero.

——————————————————————————————————

8.1.9 Brillouin Zone Boundaries and Fermi Surfaces

The Brillouin zone boundaries play an important role in the understanding ofFermi surfaces. In the empty lattice approximation, the Fermi surface is asphere when represented in the extended zone scheme. The nearly-free electronapproximation introduces a distortion to the sphere which is most marked nearthe Brillouin zone boundaries.

In general, if the spherical Fermi surface crosses a Bragg plane, then thesphere may distort. In particular, the constant energy surface generally shouldbe perpendicular to the Bragg plane at the line where they intersect. Due tothe appearance of the potential Vions(Q) in the expression for the Bloch en-ergy near the Bragg plane, and also due to the accompanying band splitting,

242

the circles of intersection of the constant energy surfaces (corresponding to EF )with the Bragg plane do not match up. This is necessary since the distortion ofthe Fermi surface must conserve the volume enclosed. This volume is equal tothe volume enclosed by the spherical Fermi surface found in the empty latticeapproximation.

The Fermi surface in the reduced Brillouin zone scheme can be constructedfrom the Fermi surface in the extended zone scheme. This is done by translatingthe disjoint pieces of the Fermi surface in the higher order zones by reciprocallattice vectors, so that the pieces fit back into the first Brillouin zone.

The first Brillouin zone is the Wigner-Seitz unit cell of the reciprocal lattice.It encloses the set of points that are closer to Q = 0 than they are to any otherreciprocal lattice vector Q 6= 0. This can be restated as, the first Brillouinzone consists of the volume in the reciprocal lattice which can be accessed fromthe origin without crossing a Bragg plane.

The second Brillouin zone is the volume that can be reached from the firstBrillouin zone by crossing only one Bragg plane.

Likewise, the (n + 1)-th Brillouin zone consists of the points, not in the(n − 1)-th zone, that can be reached from the n-th zone by crossing only oneBragg plane. Alternatively, the n-th Brillouin zone is the volume that can onlybe reached from the origin by crossing a minimum of (n− 1) Bragg planes.

The Fermi surface is constructed by:

(i) Drawing the free electron sphere.

(ii) Distorting the sphere at the Bragg planes.

(iii) For each of the n Brillouin zones, take the portions of the surface inthe n-th zone and translate them by reciprocal lattice vectors so that they laywithin the first Brillouin zone. The resulting surface is the branch of the Fermisurface assigned to the n-th band in the extended zone scheme.

The Hume-Rothery rules provide a correlation of crystal structure with thenumber of electrons per unit cell, or band filling45. It is an empirical rulewhich only applies to alloys of noble metals, such as Cu, Ag and Au, with s-pelements such as Zn, Al, Si, and Ge. It is assumed that the noble metals haveone electron outside the closed d shell, that Zn has two conduction electrons,Al has three electrons, and so on. With these assumptions, then the alloys havean f.c.c. phase for an average number of electrons per atom up to 1.38, while

45T. B. Massalaski, Binary Alloy Phase Diagrams, eds. J. L. Murray, L. H. Bennett andH. Baker, Vol. 1, A.S.M., Metals Park, Ohio, (1986).

243

The Fermi surface and Brillouin zone of b.c.c. Na

ez

ex

ey

Figure 137: The Fermi surface of b.c.c. Na is completely enclosed within thefirst Brillouin zone.

Figure 138: Cu forms in an f.c.c. structure. The Fermi surface and the Brillouinzone boundary intersect in circular rings (necks) around the L points. [After D.Shoenberg, Proc. Roy. Soc. 379, 1 (1983).]

the b.c.c. phase is stable for band-fillings between 1.38 and 1.48. In the b.c.c.structure, the smallest vectors from the zone center to each face of the Brillouinzone have the form 1

22 πa (1, 1, 0), whereas for the f.c.c. lattice these vectors

are of the form 12

2 πa (1, 1, 1). Therefore, the radius of the Fermi sphere, kF , at

244

Figure 139: The experimentally observed (T, n) phase diagram of Cu-Zn andCu-Ga intermetallic alloys. After W. Hume-Rothery, J. Inst. Metals 90, 42(1961). The α phase is f.c.c., the β phase is b.c.c., the γ phase has a complexcubic structure, and a hexagonal ε-phase occurs near 1.75.

which it first makes contact with the Brillouin zone boundary is given by

kF =√

3π

afor f.c.c.

kF =√

2π

afor b.c.c. (540)

When the Fermi sphere first makes contact with the zone boundary, the occupiedband is depressed by Vions(Q) resulting in an energy lowering which stabilizesthe structure. In the free electron approximation, the number of electrons per

Proposed Density of States for Cu-Zn Alloys

0

0.1

0.2

0.3

0.4

0 1 2 3 4 5 6 7 8 9

ε [ eV ]

ρ(ε)

[ St

ates

/ eV

]

b.c.c.

f.c.c.

[After H. Jones (1937)]

Figure 140: The proposed density of states for Cu− Zn alloys.

245

primitive unit cell, n, is given by

n = 2V

N ( 2 π )34 π3

k3F (541)

where

V

N=

a3

4for f.c.c.

V

N=

a3

2for b.c.c. (542)

Thus, one finds that the critical number n is given by√

3π4 = 1.36 for the f.c.c.

and√

2 π3 = 1.48 for the b.c.c. lattices.

The above explanation of the Hume-Rothery rules as proposed by Jones46

does require further modification. Heine and Weaire47 noted that it is neces-sary to consider the presence of the Cu d bands. It is also unclear whether theconcept of a sharp Fermi surface remains valid at these high concentration ofimpurities, and furthermore, it is unknown what the electron-electron interac-tion does to the relative stability of these phases48.

Hume-Rothery observed that the number of valence electrons per unit cellplays a critical role in the stability of intermetallic alloys. Jones’s hypothe-sized that the proximity of the Fermi surface to prominent Bragg scatteringplanes does play an important role in stabilizing these alloys. Apparently, theJones mechanism is also active in intermetallic quasi-crystals. Although quasi-crystals do not have conventional Bragg planes, they do have prominent peaksin their diffraction patterns which can be used to construct a pseudo-Brillouinor Jones zone that takes into account the most important components of theelectronic potential. Friedel49 has noted that Jones’s mechanism is optimal inquasi-crystals because of the high-multiplicity of Bragg planes due to the icosa-hedral symmetry, so that the pseudo-zone is almost spherical. Therefore, it isnot surprising that the Fermi energy is found to occur in a dip in the density ofstates of many quasi-crystals.

46H. Jones, Proc. Phys. Soc. A 49, 250 (1937).47V. Heine and D. Weaire, Solid State Physics, Volume 22, Academic Press, N.Y., (1970).48The above argument is equivalent to assuming that the stable phase is found by minimizing

the band energy given by

E = 2

∫ εF

−∞dε ε ρ(ε)

(543)

and, therefore, double counts the effect of electron-electron interactions.49J. Friedel, Phil. Mag. B 65, 1125 (1992).

246

Al–Cu Hume–Rothery alloys 4241

Figure 6. Al p CB distribution curves in pure Al (full line),γ -Al 35Cu65 (starred line) andi-AlCuFe (line with triangles) alloys as adjusted to the Al 3p intensity at the Fermi level (seetext). For clarity only the Al 3p distribution curve of pure Al is shown.

Figure 7. X-ray diffraction pattern of theδ-Al 39Cu61 phase and its Brillouin–Jones zone.

Our alloys are HR compounds and consequently a pseudo-gap is expected atEF .The one we observe is quite small as compared to the pseudo-gap in approximant andquasicrystalline alloys, whereIF goes down to less than 15% in icosahedral phases.Nevertheless, the FS–BZ interaction should be strong, especially in the case of theγ andδ phases, where the BZ is constructed with 36 faces corresponding to Bragg peaks thatconcentrate almost all the intensity of the x-ray diffraction pattern as shown in figure 7.The partial Al p DOS inγ -Al 4Cu9 calculated using the TB–LMTO–ASA method and thecorresponding calculated Al Kβ spectra are presented in figure 8. In that case, a direct

Figure 141: The X-ray diffraction pattern of δ-Al39Cu61. After V. Fournee etal., J. Phys. CM. 10, 4231 (1998).

4242 V Fournee et al

Figure 8. Experimental (top) and calculated (bottom) Al p DOS curves of theγ -Al 4Cu9 phaseand its Brillouin zone.

comparison withγ -Al 35Cu65 is possible, as it has the sameγ -brass structure. The Fermilevel falls in a pronounced pseudo-gap, confirming the strong FS–BZ interaction. Thewidth of this pseudogap is about 1 eV. After broadening, the relative intensity atEF isIF = 40%, just as the experimental value. Taking the number of facesN as a roughparameter for the sphericity of the BZ, the comparison with the case of the approximantAl 13Fe4 or the quasicrystalline i-AlCuFe, withN equal to 30 and 42 respectively, showsthat the HR mechanism alone is not sufficient to explain the formation of the pseudogapobserved by SXS in Al–TM–TM′ quasicrystalline phases (TM, TM′: transition metals). Thewider depletion cannot be the result of a more spherical shape of the PBZ, differences withthe γ phase being too small.

More likely, the effect of the TM should be invoked. Indeed, it has been shown(Friedel 1992, Trambly de Laissardiere et al 1995) that the presence of TM d states ofenergies close to the gap, coupled to the sp states, results in an increase of the magnitudeof the potential acting on the Fermi electrons and consequently an increase of the widthof the resulting pseudo-gap. Previous experiments on Al–TM alloys have revealed indeedthe strong Al p–TM d hybridization atEF , the result of which is to repel the electronicstates on both sides of the Fermi level, giving rise to an enhanced pseudo-gap (Belinet al 1992, 1994a, Belin-Ferre et al 1996). The p–d hybridization is even responsible forthe opening of an almost true gap in Al2Ru and Ga2Ru semiconductors (Nguyen Manhet al 1992, Fournee et al 1997). Note that these two effects, diffraction by Bragg planesand hybridization with TM d states, should not be considered as distinct effects as thesublattice of the TM atoms contributes predominantly to the scattering potentialVK . Notealso that the HR effect has almost no influence on the flattening of the Al p CB withinthe normalization scheme exposed above. This is consistent with the conclusion of Belin-Ferre and Dubois (1996) who noticed the continuous change in the shape of the Al p CB,from a tan−1(E) behaviour to a parabolic-like one when going from pure Al to perfecti-AlCuFe. Such a dependence of the DOS in the CB is in good accordance with the

Figure 142: The experimentally observed and calculated Al p-density of statesfor γ-Al4Cu9. After V. Fournee et al., J. Phys. CM. 18, 4231 (1998).

247

8.1.10 The Geometric Structure Factor

The potential Vions(r) is a periodic function and can be defined in terms of theionic potentials of the basis atoms Vatom, the lattice vectors R, and the basisvectors rj , via

Vions(r) =∑R

∑j

Vatom(r −R− rj) (544)

The evaluation of the Fourier Transform of the potential can be reduced to anevaluation of the Fourier Transform in one unit cell of the lattice as

Vions(Q) =1V

∫V

d3r exp[− i Q . r

] ∑R,j

Vatom(r −R− rj)

=1V

∫V

d3r∑R,j

exp[− i Q . ( r − R )

]Vatom(r −R− rj)

=1V

∫V ′

d3r′∑R,j

exp[− i Q . r′

]Vatom(r′ − rj)

(545)

where we have used the Laue condition

exp[i Q . R

]= 1 (546)

and the transformation r′ = r − R. Furthermore, since the Bravais lattice vec-tors do not explicitly appear in the summand, the sum over R merely producesa factor of N ∑

R

≡ N (547)

one has

Vions(Q) =N

V

∫V

d3r exp[− i Q . r

] ∑j

Vatom(r − rj)

=N

V

∑j

exp[

+ i Q . rj

] ∫V

d3r” exp[− i Q . r”

]Vatom(r”)

=N

VS(Q) Vatom(Q) (548)

where S(Q) is the geometric structure factor associated with the basis and theother factor is the Fourier transform of the ionic potential

Vatom(Q) =∫

V

d3r exp[− i Q . r

]Vatom(r) (549)

Thus, when the geometric structure factor vanishes, the Fourier component ofthe lattice potential also vanishes and then the lowest order splitting at the

248

Bragg plane also vanishes. Examples of this are provided by the diamond struc-ture and also by the hexagonal close-packed lattice.

The vanishing of the form factor of the ionic potential occurs in the diamondstructure phases of Si (and Ge). The nearly-free electron method may be ap-plied, as the Fourier components of the effective potential from the lattice of ions(the pseudo-potential) are reasonably small since they are orthogonalized withthe 2p (and 3p) core states. We shall assume that the pseudo-potential methodcan be applied and that the pseudo-potential can be approximated by a localform. This assumption is highly questionable since (as we shall see later) thecancellation of part of the ionic potential because of orthogonality with the 2pcores requires that the pseudo-potential should be directional dependent. How-ever, since the values of the local approximation to the pseudo-potential are notknown apriori, the pseudo-potential model can be viewed as only producing afit of the bands and the results should be discarded when they conflict with theresults obtained by rigorous means. The semiconducting materials Si (and Ge)

0

1

2

3

4

5

(ka/2π)

E0 k [

arb

itrar

y un

its ]

8 electrons per primitive unit cell

Γ XL

(0,0,0)

(1,0,0)

(1/2,1/2,1/2)

(1/2,1/2,−3/2) (1,1,0)

(1,1,1)

(2,0,0)

∆Λ

Figure 143: The free electron dispersion relation for an f.c.c. solid.

form in the diamond structure, and have eight atoms in the conventional cubicunit cell and have lattice constants of a = 5.43 (and 5.65) A. Due to the basicf.c.c. structure, only the conventional unit cell reciprocal lattice vectors of theform Q = 2π

a (m1,m2,m3) where the mi are either all even or are all odd yieldnon-zero values of the Fourier components of Vions. Furthermore, since thereare two identical atoms in the basis, the structure factor is given by

S(Q) =(

1 + exp[iπ

2( m1 + m2 + m3 )

] )(550)

249

The structure factor causes the potential to vanish for half of the Q vectorswhich correspond to the set of all even mi. That is, the non-zero Fourier com-ponents correspond either to the set of all odd integers mi or the set of all evenvalues of mi for which m1 +m2 +m3 is a doubly even integer.

From considering the free electron model where the eight atoms in the con-ventional f.c.c. unit cell each contribute four electrons to the valence band, theFermi wave vector of the empty lattice is expected to be given by

32 = 24π3

k3Fa

3

( 2π )3(551)

Hence, the chemical potential is approximately estimated to be

µ =h2

2mk2

F

=(

12π

) 23 h2

2 m

(2πa

)2

= 2.4435h2

2 m

(2πa

)2

(552)

Therefore, to obtain a reasonable description of the structure near the gap, it iscrucial to include the free electron states corresponding to the set of the eightreciprocal lattice vectors (±1,±1,±1), the set of the six (±2, 0, 0) and (0, 0, 0).At the Γ point, the approximate free electron states have energies of 0, 3 and 4 inunits of 0.3754 Rydbergs for Si (in units of 0.3455 Rydbergs forGe). The Fouriercomponents of the pseudo-potential corresponding to Q = 2π

a (2, 0, 0) is ineffec-tual, since S(Q) = 0 for this value of Q. The non-zero values for the pseudo-potential are given by V (Q

(1,1,1)) = +0.149 Ry, V (Q

(2,2,0)) = −0.040 Ry and

V (Q(3,1,1)

) = −0.080 Ry for Si (while V (Q(1,1,1)

) = +0.190 Ry, V (Q(2,2,0)

) =−0.038 Ry and V (Q

(3,1,1)) = −0.035 Ry for Ge). Despite the different size of

the Si and Ge atoms, the pseudo-potentials are almost the same. This simi-larity occurs since the differences between the atoms mainly occur close to thecores where the differences in the true potentials are cancelled by the repulsivecontribution from the core wavefunctions. The higher order pseudo-potentialsmay be neglected, since the pseudo-potential (like the full potential Vions is ex-pected to show an overall decrease with Q−2, for large Q.

The electronic bands of semiconductors with diamond and Zinc-Blende struc-tures have been calculated by Chelikowsky et al.50. To lowest order in thepseudo-potential, a large magnitude gap should open up between the lowestpair of bands at the L point. However, the lowest two bands at the X pointshould remain degenerate for the elemental compounds, due to the vanishing ofthe structure factor. For the binary compounds, the Fourier component of the

50J. R. Chelikowsky, D. J. Chadi and M. L. Cohen, Phys. Rev. B, 8, 2786 (1973).

250

potential does not vanish at the X point, and so the degeneracy of the lowestpair of bands is lifted. The direct band gap opens up between the next two high-est sets of bands at the Γ point and these sets of bands do not cross, so the gapbetween the valence and conduction bands persists throughout the entire Bril-louin zone. However, the minimum energy separation between the valence and

-0.2

0

0.2

0.4

0.6

0.8

1

1.2E k

[ R

ydbe

rgs ]

Γ ΓX W L XK

µ

Figure 144: The nearly-free electron dispersion relation for Si.

GaAs Structure

-0.5

-0.25

0

0.25

0.5

0.75

1

1.25

(ka/2π)

Ek [

Ryd

berg

s ]

ΓL X W L Γ

Figure 145: The nearly-free electron dispersion relation for GaAs.

conduction bands occurs between the valence band at the Γ and the conductionband at the X point. Since the minimum energy to excite an electron betweenthe valence and conduction bands requires a non-zero change in k, this thresholdenergy is known as the indirect gap. The band splitting is mainly caused by themixing of the set of the six (2, 0, 0) free electron states with the set of the eight(1, 1, 1) free electron states. Inspection of the secular equation shows that thepseudo-potential causes a propensity for the (1, 1, 1) states directed parallel to

251

any one vector connecting the central atom to a neighboring atom in the localatomic tetrahedra to combine with the linear superposition of (2, 0, 0) stateswhich has the same orientation. These un-normalized combinations are given

(1/4,1/4,1/4)

(-1/4,-1/4,1/4)

(1/4,-1/4,-1/4)

(-1/4,1/4,-1/4)

(0,0,0)

Diamond Structure

Figure 146: The local atomic tetrahedron in a diamond structure.

by

direction

(1, 1, 1) (2, 0, 0) + (0, 2, 0) + (0, 0, 2)(−1,−1, 1) (2, 0, 0) + (0, 2, 0) − (0, 0, 2)(−1, 1,−1) (2, 0, 0) − (0, 2, 0) + (0, 0, 2)(1,−1,−1) (2, 0, 0) − (0, 2, 0) − (0, 0, 2)

The set of states are the symmetrized components the hybrid atomic (sp3) or-bitals which form a local basis in the tight-binding representation. The appro-priate basis states can then be combined to describe the uni-directional bondingand anti-bonding states of the central and each of the four neighboring atomsof the local atomic tetrahedron. The hybrid states localized on one of the two

252

inequivalent atoms51, are denoted by

direction

(1, 1, 1) φs + φpx+ φpy

+ φpz

(−1,−1, 1) φs − φpx− φpy

+ φpz

(−1, 1,−1) φs − φpx+ φpy

− φpz

(1,−1,−1) φs + φpx− φpy

− φpz

and, when properly normalized, the hybrid wave functions on the atom forman orthonormal set. The gap in the one-electron energy spectra of the com-pounds occurs between the bonding and anti-bonding states, when the splittingcaused by the non-zero Fourier components of the pseudo-potential is largerthan the dispersion of the free electron bands. This inequality suggests thatthe nearly-free electron picture is being pushed to the limits of its applicability.The diamond form of C is stabilized and has a large gap since the 2p stateshave quite similar energies to the 2s states and also since the pseudo-potentialis large, both of which favor the formation of the tetrahedrally bonded (sp3)hybrids.

The Brillouin zone boundaries of hexagonal close-packed materials are alsonot fully gapped, due to the symmetry of the structure. The unit cell of thereciprocal lattice of the (direct space) hexagonal closed packed lattice is a hexag-onal prism. There are two hexagonal planes which have normals pointing alongthe positive and negative z axis. These are Bragg planes. The structure factorvanishes for the Q values which define the Bragg planes that are the hexagonaltop and bottom of the prism. The structure factor can be evaluated as

S(Q) = 1 + exp[i π (

43m1 +

23m2 + m3 )

](553)

which vanishes when m1 = m2 = 0 and m3 = ± 1, corresponding to theQ value of these Bragg planes. The vanishing of the structure factor at theseparticular Bragg planes is a consequence of a glide symmetry. In fact, grouptheory shows that the splitting on these planes is rigorously zero in the absenceof spin-orbit coupling52.

Since the gaps vanish on some faces of the Brillouin zones, it is sometimeshelpful to define a set of zones, called the Jones zones53, which are enclosed by

51The two atoms in the primitive unit cell are not equivalent since their local atomic tetra-hedra have different orientations.

52C. Herring, Phys. Rev. 52, 361 (1937).53H. Jones, Theory of Brillouin Zones and Electronic States in Crystals, North Holland

Publishers, Amsterdam (1960).

253

planes in which gaps do occur. For example, the hexagonal Bravais lattice hasa hexagonal Brillouin zone which is bounded by a set of eight Bragg planes54.For the hexagonal close-packed structure, the gaps in the electronic dispersionrelations either vanish or are small on the 001 Bragg planes. One can constructa Jones zone for the h.c.p. structure which is based on the hexagonal Brillouinzone but, instead of being bounded by the 001 planes, it is bounded by the002 Bragg planes. This Jones zone has twice the volume of the Brillouin zoneand, therefore, contains two states for each unit cell in the crystal. However,the sets of 101, 011 and 111 Bragg planes intersect with the zone, sotruncating the above zone with these twelve equivalent planes leads to anotherJones zone with a smaller volume and a more spherical shape. This minimalvolume Jones zone is bounded by a set of twenty Bragg planes55 and containsa total of

2 − 34

(a

c

)2

+316

(a

c

)4

(554)

states per unit cell. The minimal volume Jones zone for the h.c.p. structure isshown in fig(147).

(002)

(4π/c)

4/3(2π/a)

(1-10)(100)

(0-10)

(1-11)(0-11) (101)

(1-1-1)(0-1-1) (10-1)

Figure 147: The minimal volume Jones zone for the hexagonal close-packedlattice.

The Jones zone concept can be applied to explain the stability of the γ-phasesof Hume-Rothery alloys. The γ-phases have complex cubic structures withfiftytwo atoms in the unit cell56. The calculated magnitudes of the structure

54If the reciprocal lattice vectors are represented by

Q =

(2π

a

)((m1 −m2)

√3

e1 + (m1 + m2) e2 + m3a

ce3

)the hexagonal Brillouin zone is bounded by the sets of 100, 010, 110 and 001 Braggplanes.

55The minimal volume h.c.p. Jones zone is bounded by twenty Bragg planes which are thesix 100, 010 110 planes, the twelve 101, 011, 111 planes and the two 002 Braggplanes

56A. J. Bradley and J. Thewlis, Proc. Roy. Soc. 112, 678 (1926).

254

Jones zone for Bi

(1,0,-1) (0,1,-1)(1,-1,0)

(2,2,1)(2,1,2)

(-1,-2,-2)

Figure 148: The Jones Zone for solid Bi.

factor S(Q) at the reciprocal lattice vectors Q are given by:

Q (110) (200) (211) (220) (310) (222) (321) (400) (330) (411)

|S(Q)| 0.32 0 0.32 0 0.32 2.68 1.05 0 8.85 5.63

As seen in fig(149), the structure factors are large at for the (330) and (441)reciprocal lattice vectors. Therefore, one expects large gaps to open up at the

past for approximants. Theg brass is also known as a struc-turally complex alloy phase stabilized at specific electronconcentrations and, as described below, the unit cell is justsmall enough to be handled with the FLAPW band calcula-tions.

The g brass is characterized by the possession of a com-plex cubic structure containing 52 atoms in the unit cell andformed in many systems at the electron per atom ratioe/a of21/13=1.615.13–16 Mott and Jones17 discussed why thegbrass is stabilized at the specific value ofe/a by assumingsimultaneous contacts of the free-electron Fermi sphere withzones marked out by the planesh330j and h411j, inside ofwhich 90 states per unit cell of 52 atoms or 1.731 states peratom are accommodated. The coincidence of the Fermisphere with the particular zone plane has been referred to asthe Hume-Rotheryshereafter abbreviated as H-Rd matchingcondition 2kF=Kp, where 2kF is the Fermi diameter andKpthe magnitude of the relevant reciprocal lattice vector. Thediscovery of many quasicrystals and their approximants inthe last two decades has been successfully made by using theH-R matching condition as a guide. It is noteworthy that thefulfillment of the 2kF=Kp condition has been confirmed notonly in the Hume-Rothery alloy phases likeg brasses andquasicrystals but also in liquid metals, amorphous alloys, andeven in amorphous semiconductors.18

As mentioned above, the 2kF=Kp condition proposed byMott and Jones is based on the free-electron model. This iscertainly too simple to be applied to a realistic system, wherethe electronic structure is strongly perturbed by the presenceof the orbital hybridization among the neighboring atoms,particularly when the transition metal is involved as a con-stituent element. To overcome this difficulty, we consider theab initio FLAPW band calculations to be best applied for asystem involving various types of orbital hybridizations andto be able to extract the Fermi surface-Brillouin zone inter-action in the most rigorous way from the electronic structurethus obtained.

In the present work, we performed theab initio FLAPWband calculations for the Cu5Zn8 and Cu9Al4 g brasses tosingle out the Fermi surface-Brillouin zone interaction re-sponsible for the formation of the pseudogap at the Fermilevel. As described below, we demonstrate why the Cu5Zn8and Cu9Al4 g brasses are commonly stabilized at thee/avalue of 21/13 in spite of the entirely different solute con-centrations. This is, we believe, the first straightforwarddemonstration for the Hume-Rothery electron concentrationrule in a structurally complex alloy phase.

II. ELECTRONIC STRUCTURE CALCULATIONS

A. Atomic structure

Theg brass of both Cu5Zn8 and Cu9Al4 contains 52 atomsin the conventional body-centered-cubic unit cell with latticeconstants of 8.84 and 8.675 Å, respectively. However, thespace group of these two compounds is slightly different

from each other and is identified to beI43m and P43m,thereby resulting in the primitive cells involving 26 and 52atoms, respectively. Detailed structure information is avail-able elsewhere.19,20 Figure 1 shows the diffraction spectrummeasured on the Cu9Al4 g brass by using the synchrotronradiation with wavelength of 0.500 41 Å.20 The diffractionpeaks with asterisk represent those indexed with Miller indi-ces, the sum of which is odd. They certainly vanish in the

Cu5Zn8 g brass with the space groupI43m. Among diffrac-tion lines with asterisk, only liness210d, s221d, s300d, ands511d were observed but turned out to be fairly weak in theCu9Al4 brass, as clearly seen in the inset to Fig. 1. Hence, wewill not further consider this subtle difference in structurebetween the Cu5Zn8 and Cu9Al4 and discuss the Fermisurface-Brillouin zone interaction on the same footing forboth compounds in the rest of the discussion.

B. FLAPW band calculations

The FLAPW method10–12 treats all electrons and has noshape approximations for the potential and charge density.The exchange-correlation energies are treated within the lo-cal density approximation using the Hedin-Lundqvist param-eterization of the exchange-correlation potential.21 The corestates are calculated fully relativistically and updated at each

FIG. 1. Diffraction spectrum taken at Spring-8 synchrotron ra-diation facility and Rietveld fitting for the Cu9Al4 brass. The insetshows its enlargement at low diffraction angles. The Miller indiceswith asterisk indicate diffraction lines observed in the Cu9Al4 with

space groupP43m but not in the Cu5Zn8 brass with space group

I43m.

FIG. 2. Total DOS derived from the FLAPW band calculationsfor the Cu5Zn8 g brass.

ASAHI et al. PHYSICAL REVIEW B 71, 165103s2005d

165103-2

Figure 149: X-ray diffraction pattern from γ-phase Cu9Al4. [After Asahi et al.(2005).]

twelve (330) and the twentyfour (411) Bragg planes. The two sets of equiva-lent Bragg planes encloses a region around the origin which is nearly spherical,

255

since it is has thirtysix faces. This Jones zone contains ninety electronic statesfor each unit cell in the crystal. In an extended zone scheme, the Jones zonewould be fully occupied if an almost spherical Fermi-surface coincided with theboundaries of this region. In this case, the number of electrons per atom shouldbe n = 90

52 and the structure would be stabilized by the relatively large valuesVions(Q) at the boundaries.

Figure 150: The Jones Zone for the γ-phase of Cu− Zn alloys.

The spin-orbit interaction can lead to the re-occurrence of small gaps in thebands57. The spin-orbit interaction is a relativistic effect, which appears as alow order correction to the non-relativistic limit of the Dirac equation. For aparticle of charge q in the presence of a scalar and vector potential (φ,A), thisprocess yields the single-particle Hamiltonian in the form

H = m c2 +1

2 m

(( p − q

cA ) . σ

)2

+ q φ

− 18 m3 c

p4 +q h

4 m2 c2σ .

[∇ φ ∧ ( p − q

cA )

]+

q h3

8 m2 c2∇2φ

57M. H. Cohen and L. Falicov, Phys. Rev. Lett. 5, 544 (1960).

256

(555)

The first line, apart from the rest energy, coincides with the non-relativisticPauli Hamiltonian

HP =1

2 m

(( p − q

cA ) . σ

)2

+ σ0 q φ (556)

which, together with the identity(σ . a

) (σ . b

)= σ0

(a . b

)+ i σ .

(a ∧ b

)(557)

leads to

HP =[

12 m

σ0

(− i h ∇ − q

cA

)2

+ σ0 q φ

− h q

2 m cσ .

(∇ ∧ A + A ∧ ∇

) ](558)

Furthermore, on using

∇ ∧ A Ψ(r) = Ψ(r)(∇ ∧ A

)− A ∧ ∇ Ψ(r) (559)

and on noting that B = ∇ ∧ A, one obtains the non-relativistic PauliHamiltonian including the anomalous Zeeman interaction

HP =[

12 m

σ0

(p − q

cA

)2

− h q

2 m cσ . B + σ0 q φ

](560)

Thus, all the terms in the first line of equation (555) are found in the non-relativistic theory whereas the terms in the second line represent interactions,Hrel, which have a relativistic origin. The relativistic terms are given by

Hrel = − 18 m3 c

p4 +q h

4 m2 c2σ .

[∇ φ ∧ ( p − q

cA )

]+

q h3

8 m2 c2∇2φ

(561)

The first term which is proportional to p4 represents a relativistic correctionto the kinetic energy. The next term is the spin-orbit interaction which can beinterpreted as being caused by the interaction of the spin with the magnetic fieldproduced by the electron’s own orbital motion. The last term is the Darwinterm, which is often discussed as an interaction with a classical electron offinite spatial extent. Thus, the spin-orbit interaction for an electron is trulya relativistic effect and, unlike the other relativistic corrections, is not verysymmetric. It is given by the pseudo-scalar interaction

− q h

4 m c2σ . ( v ∧ E ) (562)

257

For systems with an inversion symmetry, the spin-orbit interaction does notlift the two-fold spin-degeneracy of the Bloch states. This can be seen by con-sidering time reversal symmetry58. The Hermitean nature of the Hamiltonianimplies that if the Bloch state with wave vector −k

φn,−k(r) =1√V

exp[− i k . r

]un,−k(r)

(χ+

χ−

)(563)

is an eigenfunction with eigenvalue En,−k, then so is the time reversed state

φTn,−k(r) =

1√V

exp[i k . r

]u∗n,−k(r)

(−χ∗−χ∗+

)(564)

The time reversed state has a spin state which is orthogonal to the original one.This is a Bloch state with vector k. Since the system has a center of inversion,then

φn,−k(−r) =1√V

exp[i k . r

]un,−k(−r)

(χ+

χ−

)(565)

is also Bloch state φn,k with wave vector k and has the same energy En,−kandspin state as the original state φn,−k. Hence, for a system with a center ofinversion, the Bloch state with energy En,k must be at least doubly-degenerate(spin-degenerate). It also follows that for every Bloch state φn,k with spin σ,there is a state φn,−k with the same spin and the same energy. This is knownas Kramers’ theorem.

Due to its reduced symmetry, the spin-orbit interaction raises the degener-acy of the bands at high-symmetry points in k space59, such as those on thehexagonal faces of the h.c.p. Brillouin zone.

——————————————————————————————————

8.1.11 Exercise 39

The effect of the Bragg planes on the density of states can be calculated fromthe nearly-free electron model. For simplicity, consider the effect of one Braggplane. The Bloch wave vector k is resolved into components parallel, k‖, andperpendicular, k⊥, to the reciprocal lattice vector Q

k = k⊥ + k‖ (566)

58The time reversal operator τ can be represented by τ = i σy C where σy is the Pauli

matrix and C is the complex conjugation operator.59R. J. Elliott, Phys. Rev. 96, 280 (1954).

258

The energy of the two bands can be written as

Ek,± =h2

2 mk2⊥ + ∆E±(k‖) (567)

where

∆E±(k‖) =h2

2 m

[k2‖ +

12

(Q2 − 2 k‖ Q

) ]

±

( [h2

4 m

(Q2 − 2 k‖ Q

) ]2+ | V (Q) |2

) 12

(568)

describes the splitting of the two bands. (Note that the band energies are notperiodic in k‖. This is a consequence of our artificial assumption that there isonly one Bragg plane.) For each band, the density of state per spin is

ρ±(ε) =V

( 2 π )3

∫d3 k δ( ε − Ek,± ) (569)

Show that the contribution to the density of states from each band is of theform (

2 mh2

)V

4 π2

(kmax‖(ε) − kmin‖(ε)

)(570)

where an equation of the form ε = ∆E±(km‖) defines the maximum and min-imum value of k‖.

Show that, if the constant energy surface cuts the Brillouin zone boundary,i.e.,

E(0)Q

2

− | V (Q) | ≤ ε ≤ E(0)Q

2

+ | V (Q) | (571)

then for the lower band, one has

kmax‖(ε) =Q

2(572)

and if the constant energy does not intersect the Brillouin zone boundary then

kmax‖(ε) =√

2 m ε

h2 + O(|V (Q)|2) (573)

where E(0)Q

2

− | V (Q) | ≥ ε ≥ 0.

Show that the density of states for the upper band is given by

ρ+(ε) =V

4 π2

(m

h2

) (Q

2− kmin‖(ε)

)for ε ≥ E

(0)Q

2

+ | V (Q) | (574)

259

Show that the energy derivative of the density of states, ∂ρ∂ε , is singular at

the energiesε = E

(0)Q

2

± | V (Q) | (575)

——————————————————————————————————

8.1.12 Exercise 40

Consider the point W on the Brillouin zone boundary of an f.c.c. crystal. ThreeBragg planes meet at W. The k value at W is

kW =(

2 πa

)(1,

12, 0) (576)

The three Bragg planes correspond to the reciprocal lattice vectors QX

=2 πa (2, 0, 0), Q

L= 2 π

a (1, 1, 1) and QL′

= 2 πa (1, 1, 1). The four free electron

energies are

E(0)Γ,k =

h2

2 mk2

E(0)L,k =

h2

2 m

(k − Q

L

)2

E(0)L′,k =

h2

2 m

(k − Q

L′

)2

E(0)X,k =

h2

2 m

(k − Q

X

)2

(577)

These four energies are degenerate at W and are equal to E(0)W = h2

2 m k2W .

Show that near W , the first order energies are given by the solutions of

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

E(0)Γ,k − E V1 V1 V2

V1 E(0)L,k − E V2 V1

V1 V2 E(0)L′,k − E V1

V2 V1 V1 E(0)X,k − E

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣= 0

260

where V2 = V (Qx) and V1 = V (Q

L) = V (Q

L′), and that at W the roots

are

E = E(0)W − V2 doubly degenerate

E = E(0)W + V2 ± 2 V1 singly degenerate

Two Bragg planes meet on the line W U W , where the point U is themidpoint of the edge of the square face. The point U corresponds to the k value

kU =(

2 πa

)(1,

14,14) (578)

Show that for points k which are on the line W U W and are close to U , theband energies are given by

E = E(0)k − V2

E = E(0)k +

V2

2± 1

2

√V 2

2 + 8 V 21 (579)

where

E(0)k =

h2

2 mk2 (580)

is the free electron energy at k.

——————————————————————————————————

8.1.13 Exercise 41

Consider a nearly-free electron band structure near a Bragg plane. Let

k =Q

2+ q (581)

and resolve q into the components q‖ and q⊥ parallel and perpendicular to the

Bragg planeQ

2 . Then, the energy bands are given by

E = E(0)Q

2

+h2

2 mq2 ±

(4 E(0)

Q

2

h2

2 mq2‖ + | V (Q) |2

) 12

(582)

It is convenient to express the Fermi energy µ in terms of the energy of thelower band at the Bragg plane

µ = E(0)Q

2

− | V (Q) | + ∆ (583)

261

Show that when 2 V (Q) > ∆ > 0, then the Fermi surface is only composedof states in the lower Bloch band. Furthermore, show that the Fermi surfaceintersects the Bragg plane in a circle of radius ρ where

ρ =

√2 m ∆h2 (584)

Show that, if ∆ > 2 | V (Q) |, the Fermi surface cuts the Bragg plane intwo circles of radius ρ1 and ρ2 such that the area between them is

π

(ρ21 − ρ2

2

)=

4 π mh2 | V (Q) | (585)

This area is measurable through de Haas - van Alphen experiments.

——————————————————————————————————

8.1.14 Exercise 42

In a weak periodic potential the Bloch states in the vicinity of a Bragg planecan be approximated in terms of two plane waves.

Let k be a wave vector with polar coordinates (θ, ϕ) in which the z axisis taken to be the direction Q of the reciprocal lattice vector that defines theBragg plane.

(i) If E < h2

2 m

(Q

2

)2

, show that to order V (Q)2 the surface of energy E

is given by

k(θ, ϕ) =

√2 m E

h2

(1 + δ(θ)

)(586)

where

δ(θ) =m

| V (Q) |2

E

h2 Q2 − 2 h Q cos θ√

2 m E(587)

(ii) Show that | V (Q) |2 results in a shift of the Fermi energy given by

∆µ = µ − µ0 (588)

where

∆µ = − 18| V (Q) |2

µ0

(2 kF

Q

)ln∣∣∣∣ Q + 2 kF

Q − 2 kF

∣∣∣∣ (589)

262

——————————————————————————————————

8.1.15 Exercise 43

Consider an energy E which lies within the gap between the upper and lowerbands at point k on the Bragg plane which is defined by the reciprocal latticevector Q. Let

k =Q

2+ q (590)

(i) Find an expression for the imaginary part of k for E within the gap.

(ii) Show that for E at the center of the gap, the imaginary part of k satisfies

(=m k

)2

= − Q2

2±

√ (Q2

2

)2

+∣∣∣∣ 2 mh2 V (Q)

∣∣∣∣2 (591)

Thus, on solving for k given E, there is a range of =m k when <e k = Q2 .

Complex wave vectors are important for the theory of Zener tunnelling be-tween two bands, caused by strong electric fields. Complex wave vectors alsooccur in the description of states that are localized near surfaces.

——————————————————————————————————

263

8.2 The Pseudo-Potential Method

The failure of the nearly-free electron model is primarily due to the large valuesof the potentials, V (Q), calculated from first principles, and the small values ofthe experimentally observed splittings between the bands. Due to the large valueof the lattice potential, if the wave functions are expanded terms of plane wavesthen very many plane waves (of the order of 106) are needed to obtain conver-gence. Furthermore, band structure calculations with the exact lattice potentialare expected to reproduce the entire set of wave functions ranging from the corewave functions located within the ions, up to the valence and/or conductionwave functions. Since the core electrons are very localized and almost atomic, alarge number of plane waves are needed for an accurate calculation of the corewave functions. Large numbers of plane waves are also needed to calculate thevalence band wave functions. The need for a large number of Fourier compo-

1s2s2p3s3p

core

Si: 1s2 2s2 2p6 3s2 3p2Vatom(r) = - 14 e2/r

Figure 151: The spatial separation of the core and the valence electrons in a Siatom (schematic).

nents to calculate the valence band wave functions can be understood by theconsideration of the fact that the conduction or valence band states have to beorthogonal to the wave functions of the core electrons. Thus, the conductionelectrons should have wave functions that exhibit rapid oscillations in the vicin-ity of the ion cores. Historically, there have been many methods which wereused to avoid the need to use many plane waves. The methods used range fromorthogonalized plane waves, augmented plane waves and pseudo-potentials. Allthese methods have some common features, namely the feature of producingwave functions that require fewer plane wave components in the expansion and,thereby, increase the rate of convergence, and concomitantly diminish the effectof the ionic potential. The pseudo-potential method provides a first principlesway of explicitly finding a smaller effective potential.

The electrons in the valence band move in a periodic potential Vions(r) pro-vided by the ions. The ionic potential already includes a partial screening of

264

the nuclear potential by the ion core electrons.

The valence band Bloch functions φvk,n(r) undergo many oscillations in the

region of the core as they must be orthogonal to the core electron wave functionsφc

k,α(r). In the Dirac notation, the orthogonality condition is expressed as

< φvk,n | φc

k,α > = 0 (592)

The valence band Bloch function can be expressed in terms of a smooth function

ψvk,n(r) (593)

that doesn’t contain the oscillations that orthogonalize the Bloch state, | φvk,n >,

with the core wave states. The smooth function is known as the pseudo-wavefunction. The pseudo-wave function is related to the valence band Bloch func-tion by the definition

| φvk,n > = | ψv

k,n > −∑α

| φck,α > < φc

k,α | ψvk,n > (594)

This definition automatically ensures the othornomality of the core states withthe valence band states without placing any restriction on the form of thepseudo-wave function. The basic idea behind pseudo-potential theory is thatthe smooth pseudo-wave function represents the electronic wave function in theregion between the cores, and may be expressed in terms of only a few planewave components60.

Since the Bloch state, | φvk,n >, satisfies the one-particle Schrodinger equa-

tionH | φv

k,n > = Evk,n | φv

k,n > (595)

one finds that the smooth function satisfies

H | ψvk,n > −

∑α

Ecα | φc

k,α > < φck,α | ψv

k,n > =

= Evk,n

(| ψv

k,n > −∑α

| φck,α > < φc

k,α | ψvk,n >

)(596)

This equation can be re-arranged to yield an eigenvalue equation for the (un-known) smooth function, which has the same energy eigenvalues as the exacteigenfunction. The rearranged equation has the form(

H + V (Evk,n)

)| ψv

k,n > = Evk,n | ψv

k,n > (597)

60J. C. Phillips and L. Kleinman, Phys. Rev. 116, 287 (1959).

265

where

V (Evk,n) =

∑α

(Ev

k,n − Eck,α

)| φc

k,α > < φck,α | (598)

is a non-local and energy-dependent contribution to the potential. The impor-tant point is that this potential may be regarded as being positive and, therefore,counteracts the effect of the large negative potential due to the ions. This canbe seen by taking the expectation value of the energy dependent potential inany arbitrary state | Ψ >

< Ψ | V (Evk,n) | Ψ > =

∑α

(Ev

k,n − Eck,α

)| < Ψ | φc

k,α > |2 (599)

and as the valence electrons have a higher energy than the core electrons, Evk,n >

Eck,α, one finds

< Ψ | V (Evk,n) | Ψ > ≥ 0 (600)

Thus, the potential operator is effectively positive as it increases the expectationvalue of the energy for an arbitrary state.

The operator V is non-local. This can be seen by considering the action of Von an arbitrary wave function Ψ(r). The operator has the effect of transformingthe state through

V (Evk,n) Ψ(r) =

∑α

(Ev

k,n − Eck,α

) ∫V

d3r′ φ∗ck,α(r′) Ψ(r′) φck,α(r) (601)

Thus, the operator when acting on the wave function at position r changes theposition to r′.

If the original one-particle Schrodinger equation for φvk,n(r) has the form(

− h2

2 m∇2 + Vions(r)

)φv

k,n(r) = Evk,n φv

k,n(r) (602)

then the Schrodinger equation for the smooth function ψvk,n(r) has the form(

− h2

2 m∇2 + Vions(r) + V (Ev

k,n))ψv

k,n(r) = Evk,n ψv

k,n(r) (603)

The Schrodinger equation for the smooth wave function has exactly the sameenergy eigenvalues as the original potential. The pseudo-potential is defined as

Vpseudo = Vions(r) + V (Evk,n) (604)

and, as has been shown, the effect of the pseudo potential is much weaker thanthat of Vions(r). Also as the eigenstate ψv

k,n(r) is a smooth function it can beexpanded in terms of a few planes waves

ψvk,n(r) =

∑Q

Ck−Q1√V

exp[i ( k − Q ) . r

](605)

266

Thus, the pseudo-potential may be treated as a weak perturbation and givesresults very similar to those of the nearly-free electron model. Since its energy

-0.5

-0.25

0

0.25

0.5

0.75

1

0 0.5 1 1.5 2 2.5 3 3.5 4

r/a

r R

nl(r

)1s

3s

2s Rc RWS

80%

Figure 152: The schematic spatial variation of the ns atomic wave functionsfor Al. X-ray scattering reveals that the core electrons are localized withinRc ≈ 1.1 atomic units and the linear dimension of the unit cell is given by theWigner-Seitz radius RWS ≈ 3 a.u. It is seen that roughly 80 % of the atomic3s electrons would reside outside the unit cell, and hence can be described asnearly-free electrons.

dependence is weak, Ev can be set to zero in the pseudo-potential.

There are many different forms that a pseudo-potential can take61.

8.2.1 The Pseudo-Potential Theorem

If the valence band Bloch functions | φvk,n > satisfy the energy eigenvalue

equationH | φv

k,n > = Evk,n | φv

k,n > (606)

then one can find a pseudo-wave function | ψk > which satisfies the energyeigenvalue equation

Heff | ψk > = Evk,n | ψk > (607)

with the same eigenvalue as the valence states but in which the effective Hamil-tonian Heff has the form

Heff = H +∑α

| φck,α > < Fα | (608)

61B. J. Austin, V. Heine and L. J. Sham, Phys. Rev. 127, 276 (1962).

267

and the pseudo-wave function is given by

| ψk > = | φvk,n > +

∑α

aα | φck,α > (609)

Proof

On assuming that the conduction band states and the valence states form acomplete orthonormal set, one can expand the pseudo-wave function as

| ψk > =∑n′

an′ | φvk,n′ > +

∑α′

aα′ | φck,α′ > (610)

where an′ and aα′ are coefficients that are to be determined. On substitutingthis form in the energy eigenvalue equation

Heff | ψk > = Evk,n | ψk > (611)

one finds∑n′

an′ H | φvk,n′ > +

∑α′

aα′ H | φck,α′ >

+∑α,n′

an′ | φck,α > < Fα | φv

k,n′ > +∑α,α′

aα′ | φck,α > < Fα | φc

k,α′ >

= Evk,n

( ∑n′

an′ | φvk,n′ > +

∑α′

aα′ | φck,α′ >

)(612)

On taking the matrix elements of the above equation with the valence bandstate < φv

k,n′′ | and using the orthogonality relations between the valence andcore Bloch functions, one finds the condition∑

n′

(Ev

k,n′′ − Evk,n

)an′ δn′′,n′ = 0 (613)

where the sum over n′ can be trivially performed. The above condition is auto-matically satisfied independent of an if n′′ = n. On the other hand, for n′′ 6= n,one has an′′ = 0. Hence, the expansion of the pseudo-wave function with theenergy eigenvalue Ev

k,n only contains the valence band Bloch function with bandindex n.

On taking the matrix elements of eqn(612) with the core state < φck,α′′ |,

using the orthogonality relations and the condition an′ = δn′,n, one finds∑α′

(Ec

k,α′′ − Evk,n

)aα′ δα′′,α′ = − < Fα′′ | φv

k,n > − aα′′

∑α

< Fα′′ | φck,α >

(614)

268

or(Ec

k,α′′ − Evk,n +

∑α

< Fα′′ | φck,α >

)aα′′ = − < Fα′′ | φv

k,n > (615)

which determines aα′′ in terms of | Fα′′ > or vice versa. Either the choice ofFα is arbitrary or the choice of aα is arbitrary.

There are many common choices made for Fα. Previously, we made thechoice

| Fα > = ( E − Eck,α ) | φc

k,α > (616)

Austin, Heine and Sham made an alternate choice:

| Fα > = V | φk,α > (617)

which corresponds to the pseudo-potential

Vpseudo = V

(I −

∑α

| φck,α > < φc

k,α |)

(618)

which, if the cores wave functions were complete within the core region of space,would result in the effective potential being zero within this region.

The non-local pseudo-potential can be approximated by a local potential. Inthis approximation, the pseudo-potential is almost zero within the core. This isa result of the so-called cancellation theorem62.

8.2.2 The Cancellation Theorem

The Austin, Heine and Sham potential is optimal in the sense that the cancella-tion is most complete as the kinetic energy is as close as possible to the energyeigenvalue. This can be seen by minimizing the functional

Λ[ψ] =∫

d3r

[h2

2 m| ∇ ψ |2 − Ek,v | ψ |2

](619)

w.r.t to the coefficients aα. For a pseudo-wave function ψ, the above functionalis evaluated as

Λ[ψ] = − < ψ |(V +

∑α

| φck,α > < Fα |

)| ψ >

= − < ψ | Vpseudo | ψ > (620)

Hence, if we minimize Λ[ψ] the cancellation is optimal. We shall make a variationw.r.t. the coefficients aα, so

δψ =∑α

δaα φk,α (621)

62M. H. Cohen and V. Heine, Phys. Rev. 122, 1821 (1961).

269

Hence, the first-order variation in Λ[ψ], δΛ(1) is given by

δΛ(1) = −∑α

δa∗α < φck,α | Vpseudo | ψ > −

∑α

δaα < ψ | Vpseudo | φck,α >

(622)Since the general form of the operator Vpseudo is not a Hermitean, the extremalcondition δΛ(1) = 0 for arbitrary δaα yields

< ψ | V | φck,α > + < ψ | Fα > = 0 (623)

which is satisfied if| Fα > = − V | φc

k,α > (624)

Thus, the requirement that the cancellation is most complete reduces the pseudo-potential to the Austin, Heine and Sham pseudo-potential.

The cancellation theorem can be understood through classical considera-tions. Classically, the gain in kinetic energy of a conduction electron as it entersthe core region is equal to the potential energy. As the oscillations in φc

k,α(r)give rise to the kinetic energy of the electron in the core region, one expects thepseudo-potential to cancel in the core region. Therefore, the pseudo-potentialfollows the ionic core potential for distances larger than the ionic core radiusRc, at which point the attractive potential almost shuts off. The empty core

Empty Core Approximation Pseudopotential

r

Vps

(r)

-Z e2/r

Rc

Figure 153: The empty core approximation for the atomic pseudo-potential.

approximation to the atomic pseudo-potential63 is given by

Vpseudo(r) = − Z e2

rfor r > Rc

Vpseudo(r) = 0 for r < Rc

(625)63N. W. Ashcroft, Phys. Lett. 23, 48 (1966).

270

Basically, this is a reflection of the fact that the valence electrons do not probethe region of the cores as this region is already occupied by the core electronsand the Pauli exclusion principle forbids the overlap of states.

The Fourier transform of the atomic pseudo-potential is a smooth functionof the wave vector q.

Vpseudo(q) = − 4 π Z e2

q2cos q Rc (626)

Only the values of Vpseudo(q) at the reciprocal lattice vectors Q are physicallyimportant, and Vpseudo(Q) are small at most of Q. When one includes the effectof the screening electron clouds, the pseudo-potential is replaced by the screenedpseudo-potential

Vpseudo(r) = − Z e2

rexp

[− kTF r

]for r > Rc

Vpseudo(r) = 0 for r < Rc (627)

The Fourier transform of the screened pseudo-potential is given by

Vpseudo(q) = − 4 π Z e2

q2 + k2TF

cos q Rc (628)

which is weakened with respect to the original potential. The pseudo-potential

Screened Pseudopotential in Momentum Space

-0.8

-0.6

-0.4

-0.2

0

0.2

0 0.5 1 1.5 2q/2kF

Vps

(q)/E

F

q0

Figure 154: The q-dependence of the screened atomic pseudo-potential.

for the crystalline solid is equal to the Fourier transform of the atomic pseudo-potential multiplied by the structure factor times the inverse of the volume of aunit cell. For a lattice with a one atom basis, this yields the empirical potential

Vions(q) = − N

V

4 π Z e2

q2 + k2TF

cos q Rc (629)

271

The weakening of the potential occurs mainly in the q region around the q0 wherethe potential changes sign. That is, the pseudo-potential method is most efficientfor systems where the most important reciprocal lattice vectors have magnitudessimilar to q0. For example, in Al, one finds that the pseudo-potential at the(2,0,0) reciprocal lattice vector is given by Vions(2, 0, 0) ≈ + 0.5 eV, whereas theFourier transform of the bare lattice potential is of opposite sign and is an orderof magnitude larger at the same reciprocal lattice vector, Vions(2, 0, 0) ≈ − 5eV. The periodic oscillations found in the Ashcroft empty core pseudo-potentialare spurious, since they are due to the sharp cut-off. Nevertheless, changes insign are also found in other pseudo-potentials64. In these cases, the oscillationsare rapidly damped so that only a few changes in sign occur. For example, ontreating the atomic 1s wave functions of Li as hydrogenic-like, then one findsan unscreened pseudo-potential of the form

Vions(Q,Q′) = − 4 π Z e2

Vc (Q−Q′)2+ ( E − E1s )

F ∗(k +Q) F (k +Q′)Vc

(630)

where Vc is the volume of a unit cell of the Bravais lattice and

F (q) =

√κ3

π

8 π κ( q2 + κ2 )2

(631)

in which κ represents the inverse radial dimension of the 1s electron core. Thenon-locality of the pseudo-potential shows up as an extra dependence on k.This extra k dependence can be ignored for s states, since the core electronsonly occupy a tiny fraction of the unit cell volume (about 6 % for Al). Thatmeans that the product κ a is sufficiently large (κ a ≈ 14 for Li), so that thereis only a small variation in F (q) when q is varied.

8.2.3 The Scattering Approach

The pseudo-potential is a potential that gives the same eigenvalues as the fullpotential Vions(r), for the valence electron states. The pseudo-potential may beobtained from scattering theory.

Consider a single ionic scattering center with a spherically symmetric poten-tial V (r) which is zero for r > R. Then for r > R, the radial wave functionhas the asymptotic form

Rl(r, E) = Cl

[jl(kr) − tan δl ηl(kr)

](632)

where

E =h2 k2

2 m(633)

64A. E. O. Animalu and V. Heine, Phil. Mag. 12, 1249 (1965).

272

-0.3

-0.2

-0.1

0

0.1

0.2

0 0.5 1 1.5 2 2.5q/2kF

Vio

ns(q

) [ R

ydbe

rgs ]

Li(1,1,0) (2,0,0)

(2,1,1)

(2,2,0)

q0 q1

εF = 0.3492 RydkF a = (6π2)1/3

−2/3 εF

Figure 155: The q dependence of the pseudo-potential for Li. The reciprocallattice vectors Q for the b.c.c. structure are denoted by the red triangles.

and jj(x) and ηl(x) are the spherical Bessel and Neumann functions. The coef-ficients Cl and the phase shifts δl(E) are obtained by matching the asymptoticform to the solution at some large distance r = R. The exact logarithmicderivative of Rl(r, E) at r = R can be defined as

Ll(E) =R′l(R,E)Rl(R,E)

(634)

The matching condition of the logarithmic derivative of the asymptotic formwith the logarithmic derivative of the wave function at r = R leads to theequation

tan δl(E) =jl(kR) Ll(E) − k j′l(kR)ηl(kR) Ll(E) − k η′l(kR)

(635)

The phase shifts δl(E) determine the scattering amplitude f(θ, E) for a particleof energy E to be scattered through an angle θ. Partial wave analysis yields therelation

f(θ, E) =1

2 i k

∑l

( 2 l + 1 )(

exp[

2 i δl

]− 1

)Pl(cos θ) (636)

The scattering amplitude only depends on the phase shift modulo π. The phaseshift can always be restricted to the range − π

2 to + π2 by defining

δl = nl π + ∆l (637)

where nl is an integer chosen such that

| ∆l | <π

2(638)

273

The value of nl denotes the number of the oscillations in the radial wave func-tion Rl(r, E). The (truncated) phase shifts ∆l produce the same scatteringamplitude as the original phase shift δl(E).

The atomic pseudo-potential is defined as any potential in which the com-plete phase shifts are the truncated phase shifts ∆l and, thus, gives rise to thesame scattering amplitude, but does not produce any bound states (accordingto Levinson’s theorem). The pseudo-radial wave functions Rl(r, E) have nonodes and, thus, have no rapid oscillations. Therefore, the pseudo-radial wavefunction can be represented in terms of a finite superposition of plane waves oflong wave length. The pseudo-potential actually only depends on the functionLl(E). From the knowledge of logarithmic derivative, Ll(E), one can constructthe pseudo-potential. One method has been proposed by Ziman and Lloyd.

8.2.4 The Ziman-Lloyd Pseudo-potential

Ziman and Lloyd independently65 proposed a pseudo-potential which is localin r and is zero everywhere except on the surface of a shell of radius R. Thepotential operator, V ZL, is written as

V ZL =∑

l

Bl(E) δ( r − R ) Pl (639)

where Pl projects onto the states with angular momentum l. Inside the spherethe potential is zero and so the radial wave function is just proportional tojl(kr), since the Neumann function is excluded due to the boundary conditionat r = 0. The amplitude Bl(E) is chosen so as to give the proper asymptoticproperties of the wave function of the true potential V , for r > R.

The pseudo-radial wave functions satisfy the radial Schrodinger equation,given by

− h2

2 m1r2

∂

∂r

(r2

∂

∂rRl

)+[h2 l ( l + 1 )

2 m r2+ V ZL(r)

]Rl(r) = E Rl(r)

(640)The derivative of the pseudo-radial wave function is found by integrating theRadial Schrodinger equation over the shell at r = R

− h2

2 m∂

∂rRl(r)

∣∣∣∣R+

R−

+ Bl(E) Rl(R) = 0 (641)

The pseudo-wave function is matched with the true wave function at the radiusr = R+. The matching condition determines the function Bl(E) in the pseudo-potential in terms of the logarithmic derivative of the true wave function, Ll(E).

65J. M. Ziman, Proc. Phys. Soc. (London) 86, 337 (1965), P. Lloyd, Proc. Phys. Soc.(London), 86, 825 (1965).

274

Thus, the coefficient Bl(E) is related to Ll(E) via

Ll(E) − kj′l(kR)jl(kR)

=2 mh2 Bl(E) (642)

Therefore, the Bl(E), for different l, are determined in terms of the exact valueof logarithmic derivatives. The projection operator is simply given as

Pl =∑m

| l,m > < l,m | (643)

which also gives rise to the non-locality of the pseudo-potential operator. Thepseudo-potential for the solid can be constructed as a superposition of thepseudo-potentials of the ions.

It should be noted that the pseudo-potential only cancels for states of an-gular momentum l if there are core states with angular momentum l otherwise,the electrons experience the full potential. Thus, in C the 2s electron ex-

-0.4

-0.2

0

0.2

0.4

0 4 8 12 16 20

r/a0

r R

3,l(r

)

n = 3 l = 0

l = 1

l = 2

Figure 156: The schematic spatial variation of the atomic wave functions andpseudo-wave functions, for various values of the angular momentum. The pseudowave functions are depicted by the dashed lines and are nodeless.

perience the cancelled pseudo-potential but the 2p electrons interact with thefull potential. The 2p electrons are relatively tightly bound compared with the2s. Thus, the s → p promotion energy is lower than in the other group IVelements Si, Ge, Sn and Pb. This allows C to easily form the tetrahedrallydirected sp3 valence bonds and, therefore, is partly responsible for its abilityto form the diamond structure. Similarly, in the 3d transition metals, the 3delectrons are tightly bound compared with the 4d or 5d electrons in the secondand third series. Thus, the 3d electrons form tightly bound narrow bands, andpseudo-potential theory is inappropriate.

275

-6

-3

0

3

0 2 4 6 8 10 12r/a0

Vl(r

) - Z e2/r

l = 0

l = 1

Figure 157: The schematic spatial-dependence of the (norm-conserving) atomicpseudo-potential, for various values of the angular momentum l.

In summary, the pseudo-potentials can be created from first principles andthen, if the pseudo-potential is weak enough, the nearly-free electron model canbe used to obtain the results for the valence bands of real solids.

——————————————————————————————————

8.2.5 Exercise 44

An electron outside a hydrogen atom with a 1s core state is treated by thepseudo-potential method. Calculate the Bloch wave function for an electronwhich has a pseudo-wave function that can be approximated by a single planewave. Discuss whether this function is appropriate to represent a 2s wave func-tion. Evaluate the magnitude of the pseudo-potential, for low-energy electronstates.

——————————————————————————————————

8.2.6 Exercise 45

Show that the non-local pseudo-potential displayed in eqn.(598) is non-unique.In particular, show that the magnitude of contribution from each core statecan be changed by an arbitrary factor and yet the energy eigenvalue remainsunchanged.

——————————————————————————————————

276

8.2.7 Exercise 46

Consider the pseudo-potential for a hydrogen-like atom for the 3s and 3p states.Fourier transform the energy eigenvalue equation. Show that the contributionsto the pseudo-potential from the 1s and 2s core states do not reduce the mag-nitude of the effective potential on the 3p states. Also show that the non-localnature of the pseudo-potential from the 2p core states is important for the re-duction of the magnitude of the effective potential on the 3p states.

——————————————————————————————————

277

8.3 The Tight-Binding Model

The tight-binding method is appropriate to the situation in which the electrondensity in a solid can be considered to be mainly a superposition of the densi-ties of the individual atoms66. However, the tight-binding method does produceslight corrections to the atomic densities. It should be a good approximationfor the inner core orbitals where the ratio of the radius of the atomic orbit tothe inter-atomic separation is small.

Consider a lattice with a mono-atomic basis. The Hamiltonian for a singleion centered at 0 is H0 and has eigenstates | φm > defined by the eigenvalueequation

H0 | φm > = Em | φm > (644)

The periodic potential of the ions can be written as the sum of the potentialfrom the ion at site 0, V0, and the potential due to all other ions in the crystallinelattice ∆V

Vions = V0 + ∆V (645)

Thus, the Hamiltonian is written as the sum of a single ion Hamiltonian and

Periodic Potential due to a lattice of ions

-2

-1

0

1

-3 -2 -1 0 1 2 3r/a

Vio

ns(r

)

Figure 158: The potential periodic potential Vions(r).

the potential due to the rest of the ions

H = H0 + ∆V (646)

In the tight-binding method it is convenient to define Wannier functions, φn,as a transform of the Bloch functions

φk,n(r) =1√N

∑R

exp[i k . R

]φn( r − R ) (647)

66J. C. Slater and G. F. Koster, Phys. Rev. 94, 1498 (1954).

278

-2

-1

0

1

-4 -3 -2 -1 0 1 2 3 4r/a

∆V(r)

Figure 159: The potential at the origin due to all other atoms (excluding theatom at the origin), ∆V (r).

In the above expression, the Wannier functions are centered around the differ-ent lattice points R. The Wannier states are almost localized states and arecomposed of a linear superposition of the atomic bound states

| φn > =∑m

Cn,m | φm > (648)

The band structure is found from the energy eigenvalue equation for the Blochwave functions

H | φk,n > = Ek,n | φk,n > (649)

or (H0 + ∆V

)| φk,n > = Ek,n | φk,n > (650)

This energy eigenvalue equation is projected onto the atomic wave function| φm > bound to the origin, O, which leads to

< φm | H | φk,n > = < φm |(H0 + ∆V

)| φk,n >

= Ek,n < φm | φk,n > (651)

However, the state | φm > is an eigenstate of the atomic Hamiltonian H0 andso the overlap is given by

< φm | H0 | φk,n > = Em < φm | φk,n >

(652)

279

On substituting this relation into the matrix elements of the eigenvalue equation,the equation reduces to

( Ek,n − Em ) < φm | φk,n > = < φm | ∆V | φk,n >

(653)

The Bloch wave function can be expressed in terms of the Wannier functions,and then the Wannier functions are expressed in terms of the atomic wavefunctions via

φk,n(r) =1√N

∑R

exp[i k . R

]φn( r − R )

=1√N

∑R,m′

Cn,m′ exp[i k . R

]φm′( r − R )

(654)

The overlap of the Bloch functions and the atomic wave function is expressedas the sum of the overlap of atomic wave functions at the same site and theoverlaps of atomic wave functions centered at different sites√N < φm | φk,n > =

∑m′

δm,m′ Cn,m′ +

+∑m′

∑R 6=0

Cn,m′ exp[i k . R

] ∫d3r φ∗m(r) φm′(r −R)

(655)

Substituting this into the energy eigenvalue equation, one obtains the equation∑m′

(Ek,n − Em

)δm,m′ Cn,m′ +

+(Ek,n − Em

) ∑m′,R 6=0

Cn,m′ exp[i k . R

] ∫d3r φ∗m(r) φm′(r −R)

=∑m′

Cn,m′

∫d3r φ∗m(r) ∆V (r) φm′(r) +

+∑

m′,R 6=0

Cn,m′ exp[i k . R

] ∫d3r φ∗m(r) ∆V (r) φm′(r −R)

(656)

The first term on the left side involves the overlap of two atomic wave functionboth bound to the site 0. These atomic wave functions are part of an orthonor-mal set of eigenfunctions67. The second term on the left hand side involves the

67The eigenfunctions corresponding to the continuum of scattering states have to be dis-carded for reasons which will become obvious.

280

overlap of bound atomic wave functions centered at site 0 and at site R, andmay be expected to be exponentially smaller than the first term.

1 ∣∣∣∣ ∫ d3r φ∗m(r) φm′(r −R)

∣∣∣∣ (657)

The two terms on the right both involve the potential ∆V and the atomic wavefunction φm(r) located at site 0. The first term on the right hand site involvesthe effect of the potential due to the other ions on the central atom. Thisterm represents the effect of the crystalline electric field on the atomic levels.The remaining term represents the delocalization of the electrons. The magni-tudes of the coefficients Cn,m that appear in the expansion of the Wannier statecrucially depend on the ratios of the overlap integrals to the energy difference(Ek,n − Em). The dependence of Cn,m upon (Ek,n − Em), is similar tothe way that the coefficients Ck+Q that occur in the plane wave expansion of

a Bloch state depend on the energy difference (Ek,n − E(0)k+Q). Generally, the

dependence of Cn,m upon (Ek,n − Em) allows one to approximate the Wannierfunctions by retaining only a finite number of atomic wave functions in theirexpansions. That is, the expansion of the Wannier function is truncated by only

-25

-20

-15

-10

-5

0

1 2 3 4 5 6 7 8

E [

Ryd

berg

s ]

1s

1s

2s

2p

H He

Li

Be

B

C

C N O F Ne

B C N O F Ne

BBeLi

Figure 160: The one-electron energies for the first and second row elements ofthe periodic table, as calculated in the Hartree-Fock approximation.

considering atomic wave functions that have energies close to the energy of theBloch state.

The set of equations can be solved approximately by considering the spatialdependence.

If one assumes that the potential ∆V is non-zero only in the range whereφm(r) is negligibly small, both terms on the right hand side will be approxi-mately zero. Thus, in a first order and very crude approximation, it is foundthat Ek,n = Em.

281

On keeping the two-center and three-center integrals in which R is limitedto a few sites which are close to O, and to atomic states with a few energiesclose to Em, the set of equations truncate into a finite set. This set of equationscan be solved to yield the Bloch state energies and the Bloch wave functions.

In general, the band widths are linearly related to the overlap matrix ele-ments, γi,j , where

γi,j(R) = −∫

d3r φ∗i (r) ∆V (r) φj(r −R) (658)

in which φj are atomic wave functions and R represent atomic positions relativeto the central atom 0. Tight-binding overlap integrals have been tabulated foran extensive number of elemental solids by Papaconstantopoulos68. The bandwidths increase with the increase in the ratio of the spatial extent of φi(r) tothe typical separation R. Thus, bands with large binding energies which tend

0

0.5

1

1.5

2

2.5

3

3.5

1 2 3 4 5 6 7 8

Ele

ctro

nic

radi

us [

units

of B

ohr

radi

us ]

1s

1s

2s

2p

Figure 161: The radii of the one-electron wafefunctions for the first and secondrow elements of the periodic table, as calculated in the Hartree-Fock approxi-mation.

to have wave functions with small spatial extents form narrow bands while thehigher energy bands have broader band widths.

The overlap integrals are conventionally expressed in terms of the angularmomentum quantum numbers (l,m) of the atomic wave functions that are quan-tized along the axis joining the atoms. The matrix elements are non-negligibleonly if the z-component of the angular momentum satisfies a selection rule. Thenon-zero overlap matrix elements are then characterized by m. In analogy tothe atomic wave functions, the type of bonding is labelled by the Greek lettersσ, π and δ respectively, corresponding to m = 0, m = ± 1 and m = ± 2.

68Handbook of the Band Structure of Elemental Solids, D. A. Papaconstantopoulos, PlenumPress, NY (1989).

282

The overlap integrals corresponding to ssσ and ppπ bonds are negative, as thelobes of the wave function with the same sign overlap the negative crystal fieldpotential. The ppσ bonds are positive at large to intermediate separations aslobes of opposite sign overlap the negative potential, but become negative atsmall values of R where the overlap of lobes with the same sign start to domi-nate. The spσ overlap is an odd function of R and vanishes for zero separationR = 0 as the different atomic wave functions are orthogonal. The sign of thespσ overlap depends on the ordering of the s and p orbitals along the axis. Thespσ bond is positive if lobes of different sign overlap and is negative if lobes ofthe same sign overlap.

The Helmholtz-Wolfsberg approximation69 consists of replacing the value ofthe potential ∆V by a constant. The magnitude of the potential is factorized outof the integral. Therefore, the overlap integrals merely depend on the displacedatomic wave functions, i and j. The overlap integrals are then written as

γi,j(R) = ∆V ti,j(R) (659)

whereti,j(R) = −

∫d3r φ∗i (r) φj(r −R) (660)

The overlap between hydrogen-like 1s wave functions

φ1s(r) =

√κ3

πexp

[− κ r

](661)

can be evaluated from the Fourier transformed wave function

φ1s(q) =

√κ3

π

8 π κ( q2 + κ2 )2

(662)

The overlap of two wave functions, with a relative displacement R, can beevaluated via the convolution theorem∫

d3r φ∗1s(r) φ1s(r−R) =∫

d3q

( 2 π )3φ1s(−q) φ1s(q) exp

[i q . R

](663)

with the result that

t1s,1s,σ = −(

1 + κ R +13κ2 R2

)exp

[− κ R

](664)

On using the hydrogenic-like 2s and 2p wave functions,

φ2s(r) =

√κ3

π

(1 − κ r

)exp

[− κ r

]69M. Wolfsberg and L. Helmholtz, J. Chem. Phys. 20, 837, (1952).

283

φ2p,0(r) =

√κ3

πcos θ κ r exp

[− κ r

]φ2p,±1(r) =

√κ3

2 πsin θ exp

[± i ϕ

]κ r exp

[− κ r

](665)

one finds that the Fourier transform of the 2s and 2p wave functions are givenby

φ2s(q) =√κ3

32 π κ ( q2 − κ2 )( κ2 + q2 )3

Y 00 (θq, ϕq)

φ2p,0(q) = i

√κ5

364 π κ q

( κ2 + q2 )3Y 0

1 (θq, ϕq)

φ2p,±1(q) = − i

√κ5

364 π κ q

( κ2 + q2 )3Y ±1

1 (θq, ϕq) (666)

where the dependence on the direction of q is expressed through the factorsY m

l (θq, ϕq). The functions Y ml (θ, ϕ) are the spherical harmonics. On using the

+

+

+

-

-

S p m=0

p m = +1m = -1

θez

Polar plot of Ψ(r,θ,ϕ)dm = 0

+

+

--

dm = + 1m = - 1

dm = + 2m = - 2

+

+

-

-

+ +

Polar plot of Ψ(r,θ,ϕ)

Figure 162: A polar plot exhibiting the angular dependence of the sphericalharmonics with quantum numbers l and m.

convolution theorem, the approximate overlap integrals are evaluated as

t2s,2s,σ = −(

1 + κ R +13κ2 R2 +

115

κ4 R4

)exp

[− κ R

]

t2s,2p,σ =115

κ3 R3

(1 + κ R

)exp

[− κ R

]

t2p,2p,σ = −(

1 + κ R +15κ2 R2 − 2

15κ3 R3 − 1

15κ4 R4

)exp

[− κ R

]

t2p,2p,π = −(

1 + κ R +25κ2 R2 +

115

κ3 R3

)exp

[− κ R

](667)

284

where κ determines the spatial extent of the wave function and R is the inter-atomic separation. Typically for a material such as C, the relative strength of

Tight-binding Overlap Integrals

-1

-0.5

0

0.5

1

0 2 4 6 8 10

κ R

t(R

)

tssσ

tspσ

tppσ

tppπ

Figure 163: The dependence of the tight-binding overlap integrals on the inter-atomic separation R.

the bonds are given by the ratios at the radius R where the bonding saturates.Typical values of the relative strengths are given by

t2s,2s,σ : t2s,2p,σ : t2p,2p,σ : t2p,2p,π = − 1 : 1 : 0.75 : − 0.49 (668)

The structure of tight-binding d bands can be found by expressing the Blochfunctions in terms of five atomic d wave functions that correspond to the differenteigenvalues of the z component of the orbital angular momentum mz = ± 2,mz = ± 1 and mz = 0. If mz is quantized along the axis between twoatoms, the tight-binding overlap integrals between these sets of states are de-noted, respectively, by td,d,δ, td,d,π and td,d,σ. The matrix elements for arbitraryorientations are tabulated in the article of Slater and Koster70. Representativeratios of the strengths of the td,d,δ, td,d,π and td,d,σ bonds are given by

td,d,δ : td,d,π : td,d,σ = − 6 : 4 : − 1 (669)

In general, the tight-binding bands obtained by considering d bands alone arehighly inaccurate. Usually, a broad s band crosses the narrow set of d bands.This degeneracy is lifted as the d and s bands hybridize strongly71.

70J. C. Slater and G. F. Koster, Phys. Rev. 94, 1498 (1954).71V. Heine, Phys. Rev. 153, 673 (1967).

285

The Bloch functions are constructed out of localized atomic levels with equalamplitude, but only involves the phase exp[ i k . R ]. Thus, the electrons areequally likely to be found in any atomic cell of the crystal. Also, <e φk,n

shows that the atomic structure is modulated by the sinusoidal variation ofexp[ i k . R ]. Since the mean velocity is given by

v(k) =1h∇Ek 6= 0 (670)

then the electrons have a non-zero velocity and will be able to move through-out the crystal. The non-zero velocity is due to the coherent tunnelling of theelectron between the atoms.

For a lattice with a basis, the Bloch wave function is given

φk(r) =1√N

∑R

exp[ i k . R ]∑j,m

Cj,m φm(r − rj −R) (671)

where rj are the positions of the basis atoms and Cj,m are the amplitudes ofthe orbitals on the j-th basis atom. The equation for the Bloch function has astructure in which the basis atoms in each unit cell can be viewed as formingmolecules72. The molecular wave functions in each lattice cell are then com-bined via the tight-binding method.

8.3.1 Tight-Binding s Band Metal

For a simple s-band metal the Wannier state | φn > can be approximated bythe atomic s wave function. As this s wave function is non-degenerate, one has

| φ1 > ≈ | φs > (672)

as Cs ≈ 1. All other coefficients are set to zero, corresponding to the assump-tion that the energy of the s band, Es, is well separated from the energies ofthe other bands. This is probably a good assumption for the 1s band which isoften regarded as forming part of the core of the ions. The energy eigenvalueequation truncates to(

Es,k − Es

) (1 +

∑R 6=0

exp[i k . R

] ∫d3r φ∗s(r) φs(r −R)

)

= < φs | ∆V | φs > +∑R 6=0

exp[i k . R

] ∫d3r φ∗s(r) ∆V (r) φs(r −R)

(673)

72By using symmetry adapted wave functions, the tight-binding secular equation may beput in block diagonal form. Hence, an N ×N secular equation might be reducible to a set ofsmaller secular equations.

286

The overlap between the atomic wave functions on different sites is defined tobe a function α(R) through∫

d3r φ∗s(r) φs(r −R) = α(R) (674)

The matrix elements of the atomic functions centered at 0 with the tail of thepotential, ∆V , is defined to be β where

< φs | ∆V | φs > = − β (675)

and the matrix elements of the atomic functions centered at 0 and R with thetail of the potential is defined to be γ(R) through∫

d3r φ∗s(r) ∆V (r) φs(r −R) = − γ(R) (676)

The dispersion relation can be expressed in terms of these three functions via

Es,k = Es −

( β +∑

R 6=0 γ(R) exp[i k . R

]1 +

∑R 6=0 α(R) exp

[i k . R

] ) (677)

Since γ(R) = γ(−R) and α(R) = α(−R) the dispersion relation E1,k is aneven periodic function of k. For bonding only to the nearest neighbors, the sumsover R are truncated to run only over the nearest neighbors.

For the f.c.c. structure the dispersion relation becomes

Es,k = Es −

(β + γ(k)1 + α(k)

)(678)

where

γ(k) = 4 γ(

coskx a

2cos

ky a

2+ cos

kx a

2cos

kz a

2+ cos

ky a

2cos

kz a

2

)(679)

and

α(k) = 4 α(

coskx a

2cos

ky a

2+ cos

kx a

2cos

kz a

2+ cos

ky a

2cos

kz a

2

)(680)

Usually α is neglected as it is small. The tight-binding bands are off-set fromEs by an energy β due to the tail of the potential of all other atoms at O,

β = − < φs | ∆V | φs > (681)

287

The band width is governed by the overlap of the central atom’s wave functionwith the nearest neighbor atomic wave function. This overlap, γ, is evaluatedfrom

γ = −∫

d3r φ∗s(r) ∆V (r) φs(r −Rnn) (682)

The band width for the f.c.c. lattice is 12 γ.

For small | k | a one can expand the dispersion relation in powers of k

E1,k = Es − β − 12 γ + γ k2 a2 (683)

which is independent of the direction of k near k = 0. Thus, the constantenergy surfaces are spherical around k = 0.

The gradient of the energy has a component perpendicular to the squareface of the Brillouin zone (the face containing the X point) that is given by

∂Ek

∂kx= 2 a γ sin

kx a

2

(cos

ky a

2+ cos

kz a

2

)(684)

Thus, if E1,k is plotted along any line in k space which is perpendicular to thesquare face, it crosses with zero slope.

The points on the hexagonal face satisfy the equation

kx + ky + kz =3 πa

=32

(2 πa

)(685)

Since there is no plane of symmetry parallel to the hexagonal face, the energyplotted along any line perpendicular to the hexagonal face is not required tocross with zero slope,

∇ E1,k . e ∝ sinkx a

2

(cos

ky a

2+ cos

kz a

2

)+ sin

ky a

2

(cos

kx a

2+ cos

kz a

2

)+ sin

kz a

2

(cos

kx a

2+ cos

ky a

2

)(686)

This only vanishes along the lines joining L (12 ,

12 ,

12 ) to the vertices W (1, 1

2 , 0).

For degenerate levels such as p or d levels, the tight-binding method leadsto a N × N secular equation where N is the orbital degeneracy.

288

For heavy elements, spin-orbit coupling should be included. In this case, thepotential ∆V should have a spin-dependent contribution. The spin-orbit cou-pling breaks the spin-degeneracy and increases the size of the secular equation73

by a factor of 2.

8.3.2 Tight-Binding Bands of Diamond Structured Semiconductors

Tight-binding calculations for semiconducting materials with the diamond struc-ture, such as C, Si or Ge, require a minimum basis set which consists of ones state and three p states for the atom at each of the two lattice sites. Theseeight states are required in order to provide a reasonable description of thevalence bands. More states need to be included in the basis set, in order toyield a reasonable description of the lowest energy conduction bands. We shalluse the Huckle approximation, in which the non-zero overlap matrix elementsbetween atomic wave functions centered on different atomic sites are neglected.Furthermore, the non-zero tight-binding hopping matrix elements will be evalu-ated in the Helmholtz-Wolfsberg approximation. The Bloch wave functions areexpressed in terms of a sum of primitive Bloch functions based on the atomicorbitals with atomic quantum numbers β (φ0

β) centered on the two lattice sites.Hence,

φk,α(r) =1√N

∑R,j,β

Cβ,jα exp

[i k . ( R + rj )

]φ0

β(r −R− rj) (687)

where the coefficients Cβ,jα are to be determined. On substituting the above

ansatz for the wave function into the energy eigenvalue equation, and on takingthe matrix elements with φ0

β(r), one finds that the dominant overlap integralsare either on-site or occur with the atomic states on the atoms of the surroundingtetrahedron. If the central atom has coordinates (0, 0, 0), then the neighboringatoms which are tetrahedrally coordinated with it are located at the set ofequivalent sites

r1 =a

4(1, 1, 1)

r2 =a

4(1, 1, 1)

r3 =a

4(1, 1, 1)

r4 =a

4(1, 1, 1) (688)

These four sites are equivalent since they are related to the each other by a com-bination of the f.c.c. primitive lattice vectors, a(0, 1

2 ,12 ), a( 1

2 , 0,12 ), a(0, 1

2 ,12 ).

For the other lattice site, say the site at a4 (1, 1, 1), the separations to the tetra-

hedrally coordinated neighboring atoms are given by the four equivalent vectors

r5 =a

4(1, 1, 1)

73J. Friedel, P. Lenghart and G. Leman, J. Phys. Chem. Solids, 25, 781 (1964).

289

r6 =a

4(1, 1, 1)

r7 =a

4(1, 1, 1)

r8 =a

4(1, 1, 1) (689)

Thus, the vectors parallel to the bonds with the atoms of the surrounding tetra-hedron have opposite orientations for the two lattice sites. This is expected sincethe two sites are connected via a glide-like inversion symmetry about the pointa8 (1, 1, 1). Thus, the phase factors in the matrix form of the energy eigenvalueequation involve combinations of the four phase factors

exp[ia

4( + kx + ky + kz )

]exp

[ia

4( − kx − ky + kz )

]exp

[ia

4( + kx − ky − kz )

]exp

[ia

4( − kx + ky − kz )

](690)

or their complex conjugates.

On considering the matrix element of the eigenvalue equation with the swave function located at site (0, 0, 0) and on utilizing the tight-binding approx-imations, one obtains the linear equation

0 = ( Es − Ek,α ) Cs,0α − 4 γss,σ gs(k) Cs,1

α

− 4 γsp,σ√3

(gx(k) Cpx,1

α + gy(k) Cpy,1α + gz(k) Cpz,1

α

)(691)

which for the s− s overlaps, involves the combination of the phase factors

gs(k) =14

(exp

[ia

4( kx + ky + kz )

]+ exp

[ia

4( − kx − ky + kz )

]+ exp

[ia

4( kx − ky − kz )

]+ exp

[ia

4( − kx + ky − kz )

] )= cos

kxa

4cos

kya

4cos

kza

4− i sin

kxa

4sin

kya

4sin

kza

4(692)

The s− p hopping matrix elements are found by resolving the p orbitals in newcoordinate systems where the z′-axes are parallel to the ri. In this case, onlythe σ matrix elements of the s and the p wave functions are non-zero, since thez′ component of the angular momenta is conserved. For example, for the px

orbital, one finds that the s− p overlap is given by

γsp,σ

(1√3

exp[ia

4( kx + ky + kz )

]− 1√

3exp

[ia

4( − kx − ky + kz )

]

290

+1√3

exp[ia

4( kx − ky − kz )

]− 1√

3exp

[ia

4( − kx + ky − kz )

] )(693)

The negative signs in this expression occur since the overlap of the central swave function with the px wave functions located at the positions (xj , yj , zj)and (xj , yj , zj) have opposite signs. The above expression is written as

4 γsp,σ√3

gx(k) (694)

where the relevant combination of phase factors is given by

gx(k) = − coskxa

4sin

kya

4sin

kza

4+ i sin

kxa

4cos

kya

4cos

kza

4(695)

Likewise, the sum of the other combinations of phase factors for the s− py ands− pz hopping are given by the analogous expressions

gy(k) = − sinkxa

4cos

kya

4sin

kza

4+ i cos

kxa

4sin

kya

4cos

kza

4(696)

and

gz(k) = − sinkxa

4sin

kya

4cos

kza

4+ i cos

kxa

4cos

kya

4sin

kza

4(697)

The matrix element of the eigenvalue equation with the other s wave functionφ0

s(r − r1) is given by

0 = ( Es − Ek,α ) Cs,1α − 4 γss,σ g

∗s (k) Cs,0

α

+4 γsp,σ√

3

(g∗x(k) Cpx,0

α + g∗y(k) Cpy,0α + g∗z(k) Cpz,0

α

)(698)

The sign of the term proportional to γsp,σ is opposite to that of eqn.(691), sincethe bonds with the atoms in the surrounding tetrahedra have different orienta-tions.

The overlap integrals between the p wave functions on the two atoms canbe evaluated in much the same way. For the overlap between the two p wavefunctions located on atoms separated by the vector a

4 (1, 1, 1), one must expressthe px wave functions along a new z′ axis (ez′) parallel to the line joining theatoms 1√

3(1, 1, 1) and two other mutually orthogonal directions (say, 1√

6(1, 2, 1)

and 1√2(1, 0, 1)). Thus, since the px wave function has the form

x f(r) = ( r . ex ) f(r)(699)

291

and asex =

1√6ex′ +

1√2ey′ +

1√3ez′ (700)

one can express the px wave function in terms of the p wave functions which arequantized in the new coordinate system

φ0px

(r) =1√6φ0

px′(r) +

1√2φ0

py′(r) +

1√3φ0

pz′(r) (701)

Likewise,

φ0py

(r) = − 2√6φ0

px′(r) +

1√3φ0

pz′(r) (702)

andφ0

pz(r) =

1√6φ0

px′(r) − 1√

2φ0

py′(r) +

1√3φ0

pz′(r) (703)

The px− px overlap integral for the a4 (1, 1, 1) bond is then found to be given by

(16

+12

) γpp,π +13γpp,σ =

13

( 2 γpp,π + γpp,σ ) (704)

while the corresponding px− py and px− pz overlap integrals are both given by

13

( γpp,σ − γpp,π ) (705)

On evaluating the matrix element of the px wave function at the origin with theenergy eigenvalue equation, one obtains the linear equation

+4√3γsp,σ gx(k) Cs,1

α + ( Ep − Ek,α ) Cpx,0α − 4

3( γpp,σ + 2 γpp,π ) gs(k) Cpx,1

α

− 43

( γpp,σ − γpp,π ) gz(k) Cpy,1α − 4

3( γpp,σ − γpp,π ) gx(k) Cpz,1

α = 0 (706)

Similar equations are found for the matrix element of the energy eigenvalueequation with the py and pz wave functions. On repeating this procedure forthe other atom located at a

4 (1, 1, 1), one obtains a closed set of coupled linearequations. The energy eigenvalues Ek,α are then given by the solutions of thesecular equation. The 8 × 8 secular matrix is given by

Es − Ek,α −4γss,σgs 0 0 0 −4γsp,σ√

3gx −

4γsp,σ√3

gy −4γsp,σ√

3gz

−4γss,σg∗s Es − Ek,α4γsp,σ√

3g∗x

4γsp,σ√3

g∗y4γsp,σ√

3g∗z 0 0 0

04γsp,σ√

3gx Ep − Ek,α 0 0 −

4(γpp,σ+2γpp,π)3 gs −

4(γpp,σ−γpp,π)3 gz −

4(γpp,σ−γpp,π)3 gy

04γsp,σ√

3gy 0 Ep − Ek,α 0 −

4(γpp,σ−γpp,π)3 gz −

4(γpp,σ+2γpp,π)3 gs −

4(γpp,σ−γpp,π)3 gx

04γsp,σ√

3gz 0 0 Ep − Ek,α −

4(γpp,σ−γpp,π)3 gy −

4(γpp,σ−γpp,π)3 gx −

4(γpp,σ+2γpp,π)3 gs

−4γsp,σ√

3g∗x 0 −

4(γpp,σ+2γpp,π)3 g∗s −

4(γpp,σ−γpp,π)3 g∗z −

4(γpp,σ−γpp,π)3 g∗y Ep − Ek,α 0 0

−4γsp,σ√

3g∗y 0 −

4(γpp,σ−γpp,π)3 g∗z −

4(γpp,σ+2γpp,π)3 g∗s −

4(γpp,σ−γpp,π)3 g∗x 0 Ep − Ek,α 0

−4γsp,σ√

3g∗z 0 −

4(γpp,σ−γpp,π)3 g∗y −

4(γpp,σ−γpp,π)3 g∗x −

4(γpp,σ+2γpp,π)3 g∗s 0 0 Ep − Ek,α

The tight-binding parameters for C, Si and Ge have been tabulated by Chadiand Cohen74, and are given in Rydbergs in the following table:

74D. J. Chadi and M. L. Cohen, Physica Status Solidi, B 68, 405 (1975).

292

Ep − Es γss,σ γsp,σ γpp,σ γpp,π

C 0.544 0.279 − 0.326 − 0.360 0.097

Si 0.529 0.149 − 0.187 − 0.334 0.080

Ge 0.618 0.125 − 0.169 − 0.299 0.077

-0.6

-0.4

-0.2

0

0.2

Ek

[ R

ydbe

rgs

]

ΓΓΓΓ ΓΓΓΓX XW L K

Figure 164: The tight-binding approximation for the valence bands of crystallineSi.

The lowest four bands are completely occupied. At the Γ point these consist ofthe bonding s band which has the energy Es − 4 | γss,σ |, followed by three-folddegenerate bonding bands with energies Ep − 4

3 | ( γpp,σ + 2 γpp,π ) |. Theanti-bonding bonds are unoccupied. The anti-bonding s band has the energyEs + 4 | γss,σ |, whereas the three-fold degenerate anti-bonding p bands havethe energy eigenvalues Ep − 4

3 | ( γpp,σ + 2 γpp,π ) |.

——————————————————————————————————

8.3.3 Exercise 47

Consider two p orbitals, one located at the origin and another at the pointR (cos Θx, cos Θy, cos Θz), where R is the separation between the two ions and

293

Figure 165: The (occupied) valence bands of diamond as determined by AngleResolved Photoemission Spectroscopy experiments [From Jimenez et al., Phys.Rev. B, 56, 7215 (1997).].

the cos θ are the direction cosines of the displacements. The overlap parametersfor the orbitals φi(r) and φj(r) are defined by

γi,j(R) = −∫

d3r φ∗i (r) ∆V (r) φj(r −R) (707)

Show that the overlap parameters are given by

γx,x = ∆V(tppσ cos2 Θx + tppπ sin2 Θx

)γx,y = ∆V

(tppσ − tppπ

)cos Θx cos Θy (708)

Thus, the tight-binding parameters not only depend on the distance, R, butalso depend on the direction.

——————————————————————————————————

8.3.4 Exercise 48

Consider the p bands in a cubic crystal, which have the p wave functions

φpx(r) = x f(r)φp

y(r) = y f(r)φp

z(r) = z f(r) (709)

294

where f(r) is a spherically symmetric function. The energies of the three pbands are found from the secular equation∣∣∣∣ ( Ek − Ep

)δi,j + βi,j + γi,j(k)

∣∣∣∣ = 0 (710)

and

γi,j(k) =∑R

exp[i k . R

]γi,j(R) (711)

andγi,j(R) = −

∫d3r φ∗i (r) ∆V (r) φj(r −R) (712)

andβi,j = γi,j(0) (713)

Show that, using cubic symmetry,

βx,x = βy,y = βz,z = β (714)

and all other overlap matrix elements are zero

βx,y = βy,z = βx,z = 0 (715)

Assuming that only the nearest neighbor overlaps γi,j(R) are non-zero, showthat for a simple cubic lattice γi,j(k) are diagonal in i and j. Hence, the px, py

and pz wave functions generate three independent bands

Ex,k = Ep − β − 2 γppσ cos kxa − 2 γppπ ( cos kya + cos kza )Ey,k = Ep − β − 2 γppσ cos kya − 2 γppπ ( cos kxa + cos kza )Ez,k = Ep − β − 2 γppσ cos kza − 2 γppπ ( cos kxa + cos kya )

(716)

The relative values of these parameters can be estimated from first princi-ples calculations of bulk silicon, where the ratios were found to be given bytppσ : tppπ = 3.98 : − 1 .

——————————————————————————————————

295

8.3.5 Exercise 49

Consider the p bands in a face-centered cubic lattice with nearest neighborhopping γi,j(R). Show that the system is described by a 3 × 3 secular equationwhich is expressed in terms of four integrals

0 =

∣∣∣∣∣∣E − E0

k + M0x − M1

z − M1y

− M1z E − E0

k + M0y − M1

x

− M1y − M1

x E − E0k + M0

z

∣∣∣∣∣∣ (717)

where the functions M0i and M1

i are given by

M0x = 4 γ0 cos

ky a

2cos

kz a

2

M1x = 4 γ1 sin

ky a

2sin

kz a

2(718)

and cyclic permutations. The energy E0k is given by

E0k = Ep − β − 4 γ2

(cos

ky a

2cos

kz a

2+ cos

kx a

2cos

kz a

2+ cos

kx a

2cos

ky a

2

)(719)

Evaluate the integrals γn in terms of the overlap of atomic wave functions byusing the Helmholtz-Wolfsberg approximation. Also show that the three energybands are degenerate at the Γ point, and that when k is directed along the cubeaxis (Γ X) or the cube diagonal (Γ L), two bands are degenerate.

——————————————————————————————————

8.3.6 Exercise 50

The parent compound of the doped high temperature superconductors is La2CuO4

which has the Perovskite structure. In this structure, the CuO2 atoms formplanes. Each Cu atom is surrounded by an octahedra of O atoms of which fouratoms are in the plane. The in-plane Cu − O bonds can serve to define thex and y axes. The O atoms that have the Cu−O bonds parallel to the x axisare denoted as Ox, whereas the other O atoms are denoted by Oy. In this coor-dinate system, the appropriate basis orbitals are the Cu dx2−y2 orbitals, whilethe only Ox states which mix with the Cu states are the px states and the onlyOy states that mix with the Cu are the py states.

Using the tight-binding form of the Bloch wave function

φk =1√N

∑R

exp[i k . R

] (a φd

x2−y2(r) + bx φpx(r−a

2ex) + by φ

py(r−a

2ey))

(720)

296

find the energy bands for the CuO2 planes.

——————————————————————————————————

8.3.7 Exercise 51

Evaluate the tight-binding density of states for the s states of a simple hyper-cubic lattice in d = 1, d = 2, d = 3, d = 4, in which only the nearestneighbor hopping matrix elements t are retained. Calculate the form of thedensity of states when d → ∞.

——————————————————————————————————

8.3.8 Exercise 52

Consider the tight-binding density of states for s states on a tetragonal latticewhere the overlap in the c direction is t′ and the overlap in either the a or bdirection is t. Assume that t t′. Examine the form of the Fermi surfacewhen the band is nearly half-filled. Evaluate the density of states.

——————————————————————————————————

8.3.9 Wannier Functions

Bloch functions are eigenstates of the set of operators TR which produce transla-tions through the discrete lattice vectorsR. The eigenvalues of TR are exp[− i k . R ].The set of periodic translations can be viewed as being produced by a generator,℘, via

TR = exp[− i R . ℘

](721)

The operator ℘ is the crystal momentum operator and has eigenvalues k, andits eigenvectors are the Bloch functions

℘ φn,k(r) = k φn,k(r) (722)

The eigenvalues of ℘ are only defined in a bounded region of k space, sincek ≡ k + Q. One expects that there is an operator < which is canonically con-jugate to ℘. Like the relation between canonically conjugate operators Lz andϕ in quantum mechanics, the eigenvalues of < may be expected to be discretesince k is only defined in a bounded region of space. The eigenstates of < arethe Wannier functions.

297

Consider the position r to have a fixed value, and n to be fixed. That is,we shall consider the Bloch functions to be functions of k. The Bloch functionscan be written as

φk,n(r) =1√N

∑R

exp[i k . R

]fn(r,R) (723)

The Bloch function φk for fixed r is periodic in k, with periodicity given by theprimitive reciprocal lattice vectors Q. Clearly

φk+Q,n(r) =1√N

∑R

exp[i ( k + Q ) . R

]fn(r,R)

=1√N

∑R

exp[i k . R

]fn(r,R)

= φk,n (724)

since Q and R satisfy the Laue condition. Thus, the Bloch functions are periodicfunctions in k space. The Fourier coefficients, fn(r,R), that appear in the kspace Fourier expansion can be found from the inversion formulae

fn(r,R) =1√N

∑k′

exp[− i k′ . R

]φk′,n(r) (725)

where the sum over k′ is restricted to run over the volume of the first Brillouinzone, Ωc.

The simultaneous transformations r → r − R0 and R → R − R0 leavethe function fn(r,R) unchanged

fn(r,R) = fn(r −R0, R−R0) (726)

This is proved by considering the effect of the transformation r → r − R0 onthe definition of the functions fn(r,R)

φk,n(r) =1√N

∑R′

exp[i k . R′

]fn(r,R′) (727)

Applying the transformation on the Bloch function yields

φk,n(r −R0) =1√N

∑R′

exp[i k . R′

]fn(r −R0, R

′) (728)

and then, on transforming the sum over R′ as R′ = R − R0, one has

φk,n(r −R0) =1√N

∑R

exp[i k . (R−R0)

]fn(r −R0, R−R0) (729)

298

On comparing the above expression with the result of Bloch’s theorem

φk,n(r −R0) = exp[− i k . R0

]1√N

∑R

exp[i k . R

]fn(r,R) (730)

one recovers the symmetry relation

fn(r,R) = fn(r −R0, R−R0) (731)

Using the above symmetry of f(r,R) under a translation R0, and on choosingR0 = R, one finds

fn(r,R) = fn(r −R, 0) = φn(r −R) (732)

which shows that the function only depends on the difference r − R. Hence, ithas been shown that the Bloch function can be expressed as

φk,n(r) =1√N

∑R

exp[i k . R

]φn(r −R) (733)

where φn(r) are the Wannier functions75. The Wannier functions at differentsites are orthogonal. Thus, as they are linearly related to the Bloch wave func-tions φk,n(r), the set of Wannier functions form a complete orthogonal set.

The Wannier functions are given in terms of the Bloch functions via theinverse transform

φn(r −R) =1√N

∑k

exp[− i k . R

]φk,n(r) (734)

where the sum over k is restricted over the volume of the first Brillouin zone,Ωc. The Wannier functions defined in this way are not unique76. The Wannierfunctions are eigenstates of the operator <, with eigenvalues of R 77. The

75G. Wannier, Phys. Rev. 52, 191 (1947).76Since the Bloch functions are only defined up to an arbitrary phase factor, if φn,k(r) is a

Bloch function and ϕ(k) ≡ ϕ(k+Q) + 2π M is an arbitrary phase, then exp[ i ϕ(k) ] φn,k(r)is also a Bloch function. This means that there is an arbitrariness in the definition of theWannier functions. The Wannier functions could equally well be defined as

φn(r −R) =1√

N

∑k

exp

[− i k . R

]exp

[i ϕ(k)

]φk,n(r)

The phase factor is usually chosen such that the Wannier function φn(r − R) is maximallylocalized around site R.

77This is most easily seen if one works in a representation where < = − i ∇k. This

is discussed in detail by G. Weinreich in Solids, Elementary Theory for Advanced Students,John Wiley & Sons Inc. New York, (1965).

299

Wannier functions are localized around the site R, as can be seen by substitutingthe expression for the Bloch functions in the above equation

φn(r −R) =

√V

N

∫Ωc

d3k

( 2 π )3exp

[+ i k . ( r − R )

]un,k(r) (735)

The phase factor in the integral over d3k has the effect of localizing the Wannierfunction around r = R, as at this r value the phase of the integral is stationary.The integral is easy to evaluate for free electrons for which un(r) = 1. TheWannier functions appropriate to free electrons in an orthorhombic lattice aregiven by

φn(r) =1

√ax ay az

( sin [ π xax

]π xax

) ( sin [ π yay

]π yay

) ( sin [ π zaz

]π zaz

)(736)

which have amplitudes that decay algebraically outside the unit cell. This alge-

Free Electron Wannier Functions

-0.4

0

0.4

0.8

1.2

-4 -2 0 2 4

x/a

φ φφφ(x)

/ √ √√√a

Figure 166: The Wannier functions for free electrons.

braic decay is found only for bands with infinite width. Bands that have allowedenergies that are separated by forbidden ranges of E of finite width have Wan-nier functions that decay exponentially. Furthermore, the rate of exponentialdecay is dependent on the band width78.

——————————————————————————————————

78W. Kohn, Phys. Rev. 115, 809 (1959), E. I. Blount, Solid State Physics, Vol 13, Acad.Press, (1962).

300

8.3.10 Exercise 53

Prove that the Wannier functions centered on different lattice sites are orthog-onal ∫

d3r φ∗n′(r −R′) φn(r −R) ∝ δn′,n δR′,R (737)

Also show that the Wannier functions are normalized to unity∫d3r | φn(r) |2 = 1 (738)

——————————————————————————————————

8.3.11 Exercise 54

Given that the Bloch functions form a complete set, show that the Wannierfunctions also form a complete set.

——————————————————————————————————

8.3.12 Example of Tight-Binding: Graphene

Carbon in its stable form of graphite is a semimetal. The structure of graphiteconsists of well separated stacks of two-dimensional layers, as the interatomicspacing in the layers is a = 1.42 A, while the interlayer spacing is given byc2 = 3.35 A. As a consequence of the two-dimensional form of the structure,Two-dimensional Graphite Structure

a2

a1

a2 = 3/2 a ex - √3/2 a ey

a1 = 3/2 a ex + √3/2 a ey

Figure 167: The crystal structure of a sheet of graphene.

301

the electronic states can almost be described by a two-dimensional tight-bindingmodel79. A single layer of graphite is also known as Graphene. In the two-

Brillouin Zone of two-dimensional Graphite

Γ Q

P

kx

ky

2π/3a(1,0)

2π/3a(1,1/√3)

2π/3a(0,2/√3)b1 = (2π/3a) ( ex + √3 ey )

b2 = (2π/3a) ( ex - √3 ey )

Figure 168: The first Brillouin Zone of a graphene sheet.

dimensional model, the pz wave functions are out of plane and decouple fromthe s, px and py orbitals. The s, px and py orbitals mix and form bonding andanti-bonding σ orbitals which are separated by a very large band gap. Sincethere are two carbon atoms in the unit cell of a single sheet of graphene, thethree bonding levels are completely occupied with six electrons. The remainingtwo electrons are to be distributed amongst the pz-states. The pz-orbitals formthe bonding and anti-bonding π bands. Their dispersion relations are given bythe solution of the secular equation

0 =∣∣∣∣ Ep − Ek,α − γpp,π gs(k)− γpp,π g∗s (k) Ep − Ek,α

∣∣∣∣ (739)

where

gs(k) = exp[− i kx a

]+ exp

[+ i

kx a

2

]2 cos

√3 ky a

2(740)

Due to the vanishing of gs(k), the bonding and anti-bonding π bands just touchat the single point P ≡ 2π

3a (1, 1√3, 0). Since the remaining two electrons com-

pletely fill the anti-bonding π band, the Fermi surface consists of just one point,namely P . Hence, the two-dimensional model describes graphite as a zero bandgap semiconductor. However, if the small inter-layer hopping is reintroduced,the bands energies show a small modulation as kz is varied. This slight mod-ulation of the energy on the lines H - P - H modulates both the bondingand anti-bonding π bands. As a result, the density of states from the bond-ing and anti-bonding π bands slightly overlap and the material is semimetallic.The Fermi surface of the three-dimensional material consists of small (equalvolume) needle shaped hole pockets and electron pockets centered on the sixvertical edges of the hexagonal Brillouin zone located at 2π

3a (1, 1√3, κ3). The

79F. Bassani and G. Pastori Parravicini, Il Nuovo Cimento B 50, 95, (1967).

302

Two-dimensional Graphite

-1.1

-0.9

-0.7

-0.5

-0.3

-0.1

0.1

(3ka/2π)

Ek

[ Har

tree

s ]

Γ Q PP

π

π

EF

Figure 169: The dispersion relation for the p-bands of graphene.π Density of States of two-dimensional Graphite

0

2

4

6

8

10

12

-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0

ε [ Hartrees ]

ρ(ε)

[ St

ates

/ H

artr

ee ]

εF

Figure 170: The density of states for the π-bands of graphene.

occupied portions of the electronic bands of graphite and graphitic layers sepa-rated by Li or Rb (intercalated graphite) have been measured in angle resolvedphotoemission experiments80 and are found to be in reasonable agreement withthe two-dimensional model.

Carbon Nanotubes

The semi-metallic nature of the single layer of graphite is an importantingredient in the description of carbon nanotubes. These carbon nanotubeswere an unexpected by-product of studies of C60 molecules81. It was found thatit was possible to grow carbon tubules, which may have walls composed of oneor more layers of graphene. A single-walled nanotube may be viewed a beingconstructed from a single graphene sheet, which is cut by two parallel lines,

80I.T. McGovern, W. Eberhardt, E.W. Plummer and J.E. Fischer, Physica B, 99, 415 (1980).81S. Ijima, Nature, 354, 56, (1991).

303

Figure 171: Typical forms of single-walled Carbon Nanotubes. The indices ofthe chiral vector cp are denoted by (p1, p2)

both normal to the chiral vector

cp = p1 a1 + p2 a2 (741)

for integer values of (p1, p2). The lattice points on the line passing throughthe origin must be identified with lattice points connected by cp. It should benoted that the two sub-lattices must remain distinct after this construction hastaken place. (Both these conditions must be satisfied, for the local geometryto be unique at any site on the line.) The resulting structures consists of agraphite tubules or cylinders, which may or may not have a chiral character82.The fundamental translation vector τ is given by the vector originating from O,normal to cp which ends at the first lattice point it meets. This translationalvector is given by

τ =( p1 + 2 p2 )

da1 − ( 2 p1 + p2 )

da2 (742)

where d is the greatest common divisor of (p1 + 2p2) and (2p1 + p2).

The properties of a single nanotubule are governed by cp, that is by thechiral indices (p1, p2). The unit cell for the graphite tubule consists has primitivelattice vectors cp and τ . The radius of the nanotubule is given by

r =| cp |2 π

=√

3 a2 π

√p21 + p2

2 + p1 p2 (743)

82R. Saito, M. Fujita, G. Dresselhaus and M.S. Dresselhaus, Phys. Rev. B 46, 1804 (1992).

304

θ

τ

Ο

cp

Figure 172: The representation of a single-walled carbon nano-tube in terms ofa sheet of graphene. The chiral vector cp is depicted by a red dashed line. Aline parallel to the tube axis is depicted by the dark blue dashed lines. Thechiral angle is denoted by θ.

The length of the unit cell of the nanotube is given by

| τ | =3 ad

√p21 + p2

2 + p1 p2 (744)

Since the area of a single unit cell of graphene is√

272

a2 (745)

the number of unit cells of the honeycomb lattice in the unit cell of the nan-otubule is given by

N =2d

(p21 + p2

2 + p1 p2

)(746)

The zig-zag axis is defined as the direction (0, 1). The chiral angle θ is definedas the angle between cp and the zig-zag axis. The chiral angle governs the screwsymmetry of the graphene structure along the tubule, and is given by

tan θ =√

3 p1

2 p2 + p1(747)

The nanotubues which have mirror planes are considered to be non-chiral.

The finite radius of the tubule causes the crystalline momentum along thec-direction (perpendicular to the tubule’s axis) to be quantized. Each quantized

305

k value forms its own one-dimensional band, which is separated from the nextone-dimensional band by an energy gap due to the finite size. The allowedmomentum values are found to lie on parallel lines which slice through thetwo-dimensional Brillouin zone. This is seen by applying periodic boundaryconditions across the “circumference” of the tubule

k . cp = 2 π nt (748)

where nt is an integer. Hence, the quantization condition is

2 π nt = ( p1 + p2 )32kx a + ( p1 − p2 )

√3

2ky a (749)

Therefore, the allowed values of k fall onto a set of discrete lines. This quanti-zation condition leads to three types of nanotubes. In one set of states, there isa line of quantized crystal momenta (k1, k2) which passes through the P pointwhere the two bands are degenerate. This occurs when

3 nt = 2 p1 + p2 (750)

which is satisfied ifp1 − p2 = 3 j (751)

for integer j, sincent = p1 − j (752)

Thus, if p1 − p2 = 3 j the system is metallic. Since there is a one-dimensionalband of states running through the P point and since the energy of the P pointis equal to the Fermi energy, the system acts like a one-dimensional metal83.Due to the reduced dimensionality of phase space, the density of states at theFermi energy is finite. There are another two types of states where

p1 − p2 = 3 j + 1 (753)

orp1 − p2 = 3 j + 2 (754)

In these other two types of materials, the crystal momentum correspondingto the P point is absent and since there is always a finite energy gap betweenthe occupied and unoccupied states, these other two types of carbon nanotubesare semiconducting. Hence, nanotubes have a broad range of properties whichsensitively depend on the tube’s structure.

83The curvature of the tube may introduce tiny gaps in the density of states at the Fermienergy.

306

Semiconducting Nanotube Density of States

0

1

2

-3 -2 -1 0 1 2 3

E / γpπ

ρ(E

) [ar

bitr

ary

units

]

Metallic Nanotube Density of States

0

1

2

-3 -2 -1 0 1 2 3E / γpπ

ρ(Ε) [

arbi

trar

y un

its]

Figure 173: Typical forms of the density of states for semiconducting and metal-lic nanotubes.

9 Electron-Electron Interactions

In the last chapter, the effects of interactions between electrons were neglectedwhen calculating the energies of single-electron excitations and the single-electronwave functions. The neglect of electron-electron interactions is certainly not jus-tifiable from considerations of the strength of the effect of the Coulomb interac-tions due to the potential of the lattice of nuclei relative to the electron-electroninteractions. However, due to the Pauli exclusion principle, the lowest energyexcitations of an interacting electron gas can be put into a one to one correspon-dence with the excitations of a non-interacting gas of fermions. The effects ofelectron-electron interactions are weak for low-energy excitations and this leadsto the concept of treating the interacting electron system as a Landau FermiLiquid.

9.1 The Landau Fermi Liquid

The Pauli exclusion principle plays an important role in reducing the effect ofelectron-electron interactions. An important result of this blocking principle isthat the low-energy excitations of an electron gas behave very similarly to thoseof a non-interacting electron gas. This allows one to consider the low-energyexcitations as quasi-particles, which have a one to one correspondence with theexcitations of a non-interacting electron gas. This is the basis of the Landautheory of Fermi liquids.

An important step in deriving the Landau theory was proved by J. M. Lut-tinger84, who showed that electrons have scattering rates that vanish as theirenergies approach the Fermi energy, to all orders in the electron-electron inter-action. This can already be be seen from the lowest order calculation of thelifetime of an electron in a Bloch state due to electron-electron interactions. Al-

84J. M. Luttinger, Phys. Rev. 121, 942 (1961), J. M. Luttinger, Phys. Rev. 119, 1153,(1960).

307

though, a rigorous derivation of Fermi Liquid theory85 must consider processesof all orders in the electron-electron interaction, we shall only consider the low-est order processes. Consider the lowest order process, in which an electron,initially in a state k above the Fermi surface, is scattered to a state k − q.In this scattering process, a second electron is excited from an initial state k′

below the Fermi surface to a state k′ + q above the Fermi surface. This processconserves momentum and will conserve energy if

h2 k2

2 m−

h2 ( k − q )2

2 m=

h2 ( k′ + q )2

2 m− h2 k′2

2 m(755)

or( k − k′ ) . q = q2 (756)

For fixed k and k′, the allowed values of the momentum transfer, q, are on thesurface of a sphere of diameter | k − k′ | centered at the point ( k − k′ ) / 2.Thus, the momentum transfer, q, ranges from 0 to k − k′. Therefore, con-servation of energy and momentum ensures that both of the electrons possiblefinal state momenta k − q and k′ + q are located on the surface of sphere ofradius | ( k − k′ ) | / 2, centered at the point ( k + k′ ) / 2. This sphericalsurface also passes through both the points k and k′, which correspond to theelectrons initial state momenta. It is seen that, if the electrons are initiallytravelling parallel, so k ∼ k′, then the radius of the sphere is quite small.Thus, for glancing collisions, the phase space of final states is quite small. By

k

k'

k'-qk+q

Figure 174: The range of allowed final states for the scattering of an electron instate k with an electron of momentum k′ inside the Fermi sphere. The allowedstates are constrained to lie in a ring on the spherical surface. For glancingcollisions, there is a small number of allowed final states.

comparison, the radius of the sphere can be larger for head-on collisions, where85P. Nozieres and J. M. Luttinger, Phys. Rev. 127, 1423 (1962), J. M. Luttinger and P.

Nozieres, Phys. Rev. 127, 1431 (1962).

308

k ∼ − k′. This result shows that there is a larger range of final states available

k

k'

k'-qk+q

Figure 175: The range of allowed final states for the scattering of an electron instate k with an electron of momentum k′ inside the Fermi sphere. The allowedstates are constrained to lie in a ring on the spherical surface. For head oncollisions, there is a large number of allowed final states.

for head-on collisions.

Since electrons are fermions and the Fermi sphere is filled, the Pauli exclusionprinciple further restricts the phase space of the final states. That is, both k − q

and k′ + q must be above the Fermi surface. Hence, one has the additionalrestrictions that

| k − q | ≥ kF (757)

and| k′ + q | ≥ kF (758)

Thus, only a segment of the surface of the sphere represents allowed final statesof possible electron scattering processes. The allowed segment is in the form ofa circular strip. The thickness of the strip becomes small as k approaches kF . Inthe limit | k | → kF , this segment tends to a circle in the plane of intersectionof the sphere and the Fermi surface, unless of course k = − k′. The net resultis that the phase space available for the scattering process vanishes as k → kF ,and the scattering rate vanishes86.

9.1.1 The Scattering Rate

In considering scattering of an electron above the Fermi surface with electronsbelow the Fermi surface, one must consider spin-dependent effects. The scat-tering rate can be estimated from scattering processes in which the an electron

86J. J. Quinn and R. A. Ferrell, Phys. Rev. 112, 812 (1958).

309

of spin σ is only scatters from electrons which have anti-parallel spins, sincethe scattering rate from pairs of electrons with parallel spins is diminished dueto the exchange process. In fact, for short-ranged interactions, the scatteringrate for electrons with parallel spins vanishes identically. Hence, we shall onlyconsider the scattering of of electrons with anti-parallel spins.

We shall consider the finite temperatures scattering rate of an electron ini-tially in the state (k, σ). At finite temperatures, there is a finite probabilitythat single-electron states with energies below µ are unoccupied, and also thatthe single-electron states with energies above µ are occupied. The rate for scat-tering the electron out of the state can be estimated by using the Fermi Goldenrule expression

1τk

=2 πh

1V 2

∑q

(4 π e2

q2 + k2TF

)2 ∑k′

fk′,−σ (1− fk′−q,−σ) (1− fk+q,σ)

× δ( Ek + Ek′ − Ek+q − Ek′−q ) (759)

where the wave-vector kTF is the inverse Thomas-Fermi screening length, whichis given by (

k2TF

k2F

)=

2π

(2 m e2

h2 kF

)(760)

The expression fk′,−σ in the scattering rate represents the Fermi function

f(Ek′) =1

exp[ β (Ek′ − µ) ] + 1(761)

which represents the probability that the one-electron state (k′,−σ) is occupied.The factors (1− fk′−q,−σ) and (1− fk+q,σ) represent the probabilities that theone-electron states (k + q, σ) and (k′ − q,−σ) are unoccupied. These lattertwo factors ensure that the final states are consistent with the Pauli exclusionprinciple. The final factor in eqn(759) expresses the condition of conservationof energy.

We shall evaluate the sum over the Bloch states k′. On denoting the energyloss of the scattered electron by h ω, via

h ω = Ek − Ek+q (762)

then the electron-electron scattering rate becomes

1τk

=2 πV 2

∑q

(4 π e2

q2 + k2TF

)2

(1− fk+q,σ)

×∫ ∞

−∞dω IT (ω, q) δ( Ek − Ek+q − h ω ) (763)

310

where the energies are independent of σ and the k′ dependent factors have beenexpressed as the integral

IT (ω, q) =V

( 2 π )3

∫d3k′ fk′ (1− fk′−q) δ( Ek′ − Ek′−q + h ω ) (764)

The integral can be evaluated in the limit of zero temperature, first by expressingit in the form

IT=0(ω, q) =V

( 2 π )3m

h2

∫d3k′ fk′ (1−fk′−q) δ( k

′ . q − q2

2+m ω

h) (765)

and then by noting that the Fermi functions reduce to step functions. Thevalues of k′ at which the delta function is non-zero lie on the plane

k′ . q =q2

2− m ω

h(766)

Therefore, the integral over k′ is proportional to the area of the plane which isinterior to the Fermi sphere centered on the origin and is exterior to the sphereof radius kF centered on q.

q

k'

k'-qq > 2 kF

q . k' = q2/2 - mωωωω/!

Figure 176: The range of integration of k′ for IT=0(ω, q) when q > 2kF . Thepoint k′ must lie inside the Fermi sphere and must lie outside the sphere centeredon q, if the process is to obey the Pauli principle. Energy conservation constrainsk′ to the plane surface.

For q > 2 kF , the spheres do not overlap. As a result the integral isevaluated as the area of the plane enclosed by the Fermi sphere

IT=0(ω, q) =V

( 8 π2 )m

h2 q

[k2

F −(q

2− m ω

h q

)2 ]for kF >

q

2− m ω

h q> − kF (767)

311

and is zero otherwise, since in these other cases the surface does not intersectwith the Fermi sphere.

q . k' = q2/2 - mωωωω/!!!!

q < 2 kF

q

k'

k'-q

Figure 177: The range of integration of k′ for IT=0(ω, q) for q < 2kF . En-ergy conservation constrains the allowed values of k′ to the surface. The Pauliexclusion principle further restricts the points k′ to an annulus in the plane.

For q < 2 kF , the two spheres overlap. If q − kF > q2 − m ω

h q > − kF

then, as before, the results is

IT=0(ω, q) =V

( 8 π2 )m

h2 q

[k2

F −(q

2− m ω

h q

)2 ]for q − kF >

q

2− m ω

h q> − kF (768)

since the plane does not intersect the region where the spheres overlap. However,if q

2 > q2 − m ω

h q > q − kF , the integral yields

IT=0(ω, q) =V

( 8 π2 )m

h2 q

[ (q

2+

m ω

h q

)2

−(q

2− m ω

h q

)2 ]for

q

2>

q

2− m ω

h q> q − kF (769)

which is the area of the annulus bounded by the two spheres.

From the analysis of the zero temperature form, we note that for small(positive) values of ω and q, one has

IT=0(ω, q) =V

4 π2

m2 ω

h3 q(770)

Furthermore, the function is zero for negative values of ω. This is due to thePauli exclusion principle and conservation of energy. The electron in state

312

(k′,−σ) involved in the T = 0 scattering process is initially inside the Fermisphere and since the only available final states are outside the Fermi sphere, theelectron must be excited in the scattering process. Hence, the electron in statek must lose energy. We also note that the range of q for which the process takesplace is restricted by the condition√

k2F +

2 m ω

h+ kF > q >

√k2

F +2 m ω

h− kF (771)

At finite temperatures, the integration can be performed by using the prop-erty of the delta function, so

IT (ω, q) =V

( 2 π )3

∫d3k′ f(Ek′) (1− f(Ek′ + hω)) δ( Ek′ − Ek′−q + h ω )

(772)We shall express the product of Fermi functions as

f(Ek′) (1−f(Ek′ + hω)) =[

1 + N(hω)] (

f(Ek′) − f(Ek′ + hω))

(773)

where N(x) is the Bose-Einstein distribution function

N(x) =1

exp[ βx ] − 1(774)

Hence, we have found that

IT (ω, q) =V

( 2 π )3

[1 +N(hω)

] ∫d3k′ δ( Ek′ − Ek′−q + h ω )

(f(Ek′)− f(Ek′+hω)

)(775)

or equivalently

IT (ω, q) =V

( 2 π )3

[1 + N(hω)

] ∫d3k′ δ( Ek′ − Ek′−q + h ω )×

×(fk′ (1− fk′−q) − fk′−q (1− fk′)

)where we have again used the properties of the delta function and have intro-duced a pair of terms that cancel. It should be noted that the first term in thebig round parenthesis has a form similar to the form IT (ω, q) given in eqn(764).The second term is identified as having the form similar to IT (−ω, q). Thisshows that our function satisfies the condition

IT (ω, q) =[

1 + N(hω)] [

IT (ω, q) − IT (−ω, q)]

(776)

required from the consideration of the principle of detailed balance.

313

As we shall see later, in the evaluation of the scattering rate, it is justifiableto make the approximation

IT (ω, q) ≈[

1 + N(hω)] [

IT=0(ω, q) − IT=0(−ω, q)]

(777)

We shall formally extend the definition of IT=0(ω, q) to the negative axis, makingit an antisymmetric function. That is, we shall define

IT=0(ω, q) = IT=0(ω, q) for ω > 0 (778)

andIT=0(ω, q) = − IT=0(−ω, q) for ω < 0 (779)

The scattering rate can then be expressed as

1τk

=2 πV 2

∑q

(4 π e2

q2 + k2TF

)2 ∫ ∞

−∞dω

[1 + N(hω)

]× (1− f(Ek − hω)) δ( Ek − Ek+q − h ω ) IT=0(ω, q)

where

IT=0(ω, q) ∼ V

4 π2

m2 ω

h3 q(780)

The integration over q can be evaluated by choosing the direction of k as thepolar axis. The integration over the direction of q can be performed by notingthat the integrand is independent of the azimuthal angle ϕ, and only dependson cos θ through the argument of the delta function. We shall change variablesfrom cos θ to x, where x is defined by

x =h2

mk q cos θ (781)

The angular integration is evaluated as∫dΩ δ( Ek − Ek+q − h ω ) =

2 π mh2 k q

Θ(k −

∣∣∣∣ q2 +m ω

h q

∣∣∣∣ ) (782)

where Θ(x) is the Heaviside step function. Therefore,

1τk

=1

( 2 π )m

h2 k V

∫ ∞

0

dq q

(4 π e2

q2 + k2TF

)2 ∫ ∞

−∞dω

[1 + N(hω)

]× (1− f(Ek − hω)) Θ

(k −

∣∣∣∣ q2 +m ω

h q

∣∣∣∣ ) IT=0(ω, q)

The integration over hω is cut off at Ek−µ by the Fermi function factor, and at− kBT by the Bose-Einstein factor. Therefore, the scattering rate is dominatedby the low-frequency processes, such that

EkF | h ω | (783)

314

The q integration is also dominated by the region

2 kF > q > 0 (784)

On using the approximate expression for IT=0(ω, q), the scattering rate becomes

1τk

=1

( 2 π )3m3

h5 k

∫ 2kF

0

dq

(4 π e2

q2 + k2TF

)2 ∫ ∞

−∞dω

[ω

1− exp[−βhω]

]×(

11 + exp[β(hω − Ek + µ]

)(785)

The integral over ω can be performed exactly using the identity∫ ∞

−∞dx

x

(1 − exp[ − x ] ) ( 1 + exp[ x − y ] )=

12

y2 + π2

( 1 + exp[ − y ] )(786)

Therefore,

1τk

=1

( 2 π )3m3

2 h7 k

∫ 2kF

0

dq

(4 π e2

q2 + k2TF

)2 ( π kB T )2 + ( Ek − µ )2

( 1 + exp[−β(Ek − µ] )

We note that if we had used the bare Coulomb interaction the integral over qwould diverge, so the scattering rate would have diverged. The divergence hasbeen suppressed by the inclusion of screening. On performing the integral overq, the scattering rate is evaluated as

1τk

=1

( 8 π h )kF

k

( e2 kF )2

( h2 kF kF T

2 m )3F

(2kF

kTF

)( π kB T )2 + ( Ek − µ )2

( 1 + exp[−β(Ek − µ] )

where

F (x) =12

(tan−1 x +

x

1 + x2

)(787)

On introducing the electronic plasma frequency ωp via,

ω2p =

e2 k3F

3 π2 m(788)

(ωp ∼ 1015 sec−1) one finds that, in the limit T → 0, the scattering rate can beexpressed as

1τk

≈ π2

128

(3π

) 12

ωp

(Ek − µ

µ

)2

(789)

Thus, the quasi-particle scattering rate vanishes as Ek → µ at zero temper-ature. At finite temperatures, the quasi-particle scattering rate at the Fermienergy87 varies as ( kB T )2. In conclusion, we have indicated why the quasi-particle concept may remain valid in the limit Ek → µ and T → 0.

87E. Abrahams, Phys. Rev. 95, 834 (1954).

315

9.1.2 The Quasi-Particle Energy

The quasi-particle excitation energy Ek is affected by the interaction with theother electrons in the system. The manner in which this change in energyoccurs can be exhibited using perturbation theory. For convenience, we shallassume that the interaction between the electrons is a highly screened pointcontact interaction. To second order in the perturbation, the approximate en-ergy eigenvalue of the state where an electron is added to the Bloch state k isgiven by

E+k = E0

k +∑

|kn|<kF

E0kn

+ < k∏

|kn|<kF

kn | Hint | k∏

|kn|<kF

kn >

+∑

q

∑|km|<kF

∣∣∣∣ < k∏|k

n|<kF

kn | Hint | k − q km + q∏|k

n|<kF ,n 6=m kn >

∣∣∣∣2E0

k + E0km

− E0k−q − E0

km+q

+∑

q

∑|k

m|,|k

m′ |<kF

∣∣∣∣ < k∏|kn|<kF

kn | Hint | k, km′ − q km + q∏|kn|<kF ,n 6=m,m′ kn >

∣∣∣∣2E0

km′

+ E0k

m− E0

km′−q − E0

km

+q

(790)

To second order in the interaction, the ground state energy is given by

Egs =∑

|kn|<kF

E0kn

+ <∏

|kn|<kF

kn | Hint |∏

|kn|<kF

kn >

+∑

q

∑m,m′

∣∣∣∣ < ∏|k

n|<kF

kn | Hint | km′ − q km + q∏|k

n|<kF ,n 6=m,m′ kn >

∣∣∣∣2E0

km′

+ E0k

m− E0

km′−q − E0

km

+q

(791)

The excitation energy for adding an electron to state k is defined by the energydifference

Eexck = E+

k − Egs (792)

To this order, the excitation energy is expressed in terms of two-particle statesas

Eexck = E0

k +∑

|kn|<kF

< k kn | Hint | k kn >

+∑

|k−q|>kF

∑|k

m|<kF

∣∣∣∣ < k km | Hint | k − q km + q >

∣∣∣∣2E0

k + E0k

m− E0

k−q − E0k

m+q

316

−∑

|k+q|<kF

∑|km|<kF

∣∣∣∣ < k + q km | Hint | k km + q >

∣∣∣∣2E0

k+q + E0k

m− E0

k − E0k

m+q

(793)

The terms first order in the interaction represent the interaction of the particlewith the average density due to the other electrons. The last two terms aresecond order terms. The first of this pair represents the scattering of a pair ofelectrons in the initial states k above the Fermi energy and km below the Fermienergy. This pair of particles is scattered into the final states k − q and km + qwhich are both above the Fermi energy. The last term represents a subtraction,as this represents a scattering process that is allowed in the ground state butwhich is forbidden in the state where an extra electron is added to k. Thisparticular scattering process is forbidden since it violates the Pauli exclusionprinciple. That is, the process whereby an electron is scattered from the groundstate to the state k is forbidden when the state k is already occupied by an elec-tron. The k independent terms in the excitation energies are usually absorbedinto a shift of the Fermi energy.

The above expression represents the excitation energy for a state corre-sponding to the non-interacting state in which one electron is added to thesystem. Since, in the limit of large systems, the energy eigenvalues form aquasi-continuum and since the interactions are turned on adiabatically, thequasi-particle does not correspond to a single exact eigenstate but instead cor-responds to a linear superposition of states with almost degenerate energies.However, the corresponding many-body energy eigenstate consists of a linearsuperposition of the state with an electron added to the ground state and stateswhere the added electron is scattered and dressed by many electron-hole pairs.The quasi-particle weight Z−1(k) is defined as the fraction of the initial bareelectron contained in this state. To lowest order in Hint, the quasi-particleweight or wave function renormalization is calculated as

Z(k) = 1 +∑

|k−q|>kF

∑|k

m|<kF

∣∣∣∣ < k km | Hint | k − q km + q >

∣∣∣∣2( E0

k + E0km

− E0k−q − E0

km

+q )2

+∑

|k+q|<kF

∑|k

m|<kF

∣∣∣∣ < k + q km | Hint | k km + q >

∣∣∣∣2( E0

k+q + E0k

m− E0

k − E0k

m+q )2

(794)

which is greater than unity. Thus, the fraction of the bare electron in the quasi-particle state is always less than unity88. This conclusion remains valid to all

88The quasi-particle is actually described as a wave packet constructed from the above exactenergy eigenstates.

317

orders of perturbation theory, if the Fermi Liquid phase is stable. When |k|crosses kF , the quasi-particle changes from a quasi-particle to a quasi-hole. Atzero temperature due to the vanishing of the quasi-particle scattering rate, thedistribution of the number of bare particles has a discontinuity at the Fermienergy of Z(k)−1. This discontinuity is small compared with the discontinuityfor non-interacting electrons which is completely represented by the Fermi func-tion. Thus, the concept of a Fermi surface remains well defined for interactingelectron systems.

The quasi-particle weight has the effect that the excitation energy for a singlequasi-particle is given by the expression

Eqp(k) =Eexc

k

Z(k)(795)

In addition to the shift in the excitation energy, the quasi-particle excitationenergy is reduced by Z(k) and these two effects combine to yield a flattening ofthe quasi-particle’s dispersion relation. The reduced dispersion is interpreted interms of an increase in the effective mass of the quasi-particle. The density ofsingle-electron excitations has a quasi-particle contribution which is given by

ρ(E) ≈∑

k

Z(k)−1 δ

(E − Eqp(k)

)(796)

where E is the excitation energy relative to the Fermi energy. Due to quasi-particle weight factor, the single-electron density of states is narrowed and peaksup near the Fermi energy. As the quasi-particles obey Fermi-Dirac statistics,the quasi-particles can give rise to an enhancement of the coefficient of the linearT term in the low-temperature electronic specific heat.

Despite the apparent simplicity of the Fermi Liquid picture, it is exceed-ingly difficult to quantitatively derive the Fermi Liquid description appropriateto a specific microscopic Hamiltonian. Since the perturbation due to electron-electron interaction is long-ranged, there are divergent terms in the perturba-tion expansion. The divergent terms first appear in the expansion when takento second order. The divergent terms can actually be re-summed to yield finiteresults. The re-summations are made possible by the fact that the long-rangedCoulomb interaction between a pair of electrons in a metal is screened by theother electrons. The screening processes involves the Coulomb interaction toinfinite order. By taking into account the screening of the long-ranged Coulombinteraction, the divergent terms can be summed to infinite order leading tofinite results. That is, the divergence associated with any term can be elim-inated by combining it with a subset of other divergent terms. However, there-summation of all the terms in the perturbation expansion presents a seriouschallenge and so approximations have been developed. These approximationsinvolve the summation of infinite subsets of the terms that appear in the pertur-bation expansion. One such approximation is the Hartree-Fock approximation.

318

The Hartree-Fock approximation is self-consistent first-order perturbation the-ory in that it just consists of the first order terms in the perturbation expansion.However, in these terms, all the wave functions are calculated self-consistentlyby taking the first-order processes into account.

——————————————————————————————————

9.1.3 Exercise 55

Using a perturbation expansion, find the energy of a free electron gas to first-order in the electron-electron interaction.

——————————————————————————————————

319

9.2 The Hartree-Fock Approximation

The Hartree-Fock approximation consists of approximating the many-electronwave function by a single Slater determinant. The resulting approximate wavefunction has the same form as for independent or non-interacting electrons.This should be contrasted with the exact wave function which is expected to becomposed of a linear superposition of Slater determinants. The Hartree-Fockapproximation, therefore, involves finding the best one-electron basis functionsthat takes the average effect of electron-electron interactions into account89.

The Hartree-Fock approximation can be expressed in terms of the Rayleigh-Ritz variational principle90, in which the many-particle wave function is writtenas a single Slater determinant91. The Hamiltonian operator is expressed as

H =∑

i

[p2

i

2 m+ Vions(ri)

]+

12

∑i 6= j

e2

| ri − rj |(797)

The expectation value of the Hamiltonian in a state described by a single Slaterdeterminant Φ of a complete set of, as yet, unspecified single-electron wavefunctions φα,σ(r) is written as

H =i=Ne∏i=1

[ ∫d3ri

]Φ∗α1, . . . αNe

(r1, . . . rNe) H Φα1, . . . αNe

(r1, . . . rNe)

(798)

The expectation value of the energy is evaluated as

E =∑α

∫d3r φ∗α(r)

[− h2

2 m∇2 + Vions(r)

]φα(r)

+12

∑α,β

∫d3r

∫d3r′ φ∗α(r) φ∗β(r′)

e2

| r − r′ |φβ(r′) φα(r)

− 12

∑α,β

∫d3r

∫d3r′ φ∗α(r) φ∗β(r′)

e2

| r − r′ |φα(r′) φβ(r)

(799)

where the sums over α and β run over all the single-particle quantum numberslabelling the Slater determinant Φ. The first term just represents the sum ofone-particle energies of the electrons. The second term represents the interactionenergy between an electron and the average charge density of all the electrons.The last term is the exchange term; it arises due to the Coulomb interactionand the anti-symmetry of the many-electron wave function. The spin indices

89D. R. Hartree, Proc. Cambridge Philos. Soc. 24, 89 (1928).90V. A. Fock, Z. Physik, 61, 126, (1930).91J. C. Slater, Phys. Rev. 35, 210 (1930).

320

have been suppressed in the expression for the energy. The quantum number αneeds to be supplemented by the spin quantum number σ to uniquely specify thestate which will be written as either φα(r) = ψα(r) χσ or φ∗α(r) = χT

σ ψ∗α(r).Therefore, in the matrix elements there are not only an integrations over r, butalso the matrix elements of the spin states have to be evaluated.

The single-electron wave functions are to be chosen such that they minimizethe energy, subject to the constraint that they remain normalized to unity.Hence, subject to this condition, the single-electron wave functions are chosensuch that the first order variation of the energy is identically equal to zero.The minimization is performed by using the Lagrange method of undeterminedmultipliers. First, one forms the functional Ω which is the average value of theHamiltonian minus the Ne constraints that ensure that the one-electron wavefunctions are normalized to unity. The functional Ω is given by

Ω =i=Ne∏i=1

[ ∫d3ri

]Φ∗α1, . . . αNe

(r1, . . . rNe) H Φα1, . . . αNe

(r1, . . . rNe)

−i=Ne∑i=1

λαi

( ∫d3ri φ

∗αi

(ri) φαi(ri) − 1

)(800)

where the λα are the undetermined multipliers. Since φα is an arbitrary complexfunction, the real and imaginary parts are independent. Instead of working withthe real and imaginary parts, we shall consider the function φα and its complexconjugate φ∗α as being independent. The second step of the Lagrange methodconsists of considering the effect of varying the set of φ∗α. The deviation of thevariational functions φ∗α(r) from the extremal function, φ∗HF,α(r), are denotedby δφ∗α, i.e.,

φ∗α(r) = φ∗HF,α(r) + δφ∗α(r) (801)

To first order in the deviation δφ∗α(r), the expectation value of the functional Ωchanges to first order in δφ∗α by an amount δΩ. The change δΩ is evaluated as

δΩ =∑α

∫d3r δφ∗α(r)

[− h2

2 m∇2 + Vions(r) − λα

]φHF,α(r)

+∑α,β

∫d3r

∫d3r′ δφ∗α(r) φ∗HF,β(r′)

e2

| r − r′ |φHF,β(r′) φHF,α(r)

−∑α,β

∫d3r

∫d3r′ φ∗HF,β(r) δφ∗α(r′)

e2

| r − r′ |φHF,β(r′) φHF,α(r)

(802)

The expression for δΩ must vanish identically for any of the independent and ar-bitrary variations δφ∗α(r), if the Hartree-Fock wave functions φHF,α(r) minimizethe average energy. In order for this to be true, the coefficient of δφ∗α(r) mustvanish identically for each value of α. After interchanging the variables r and

321

r′ in the last term, one finds that the normalized Hartree-Fock wave functionsmust satisfy the set of equations

0 =[− h2

2 m∇2 + Vions(r) − λα

]φHF,α(r)

+∑

β

∫d3r′

(φ∗HF,β(r′)

e2

| r − r′ |φHF,β(r′)

)φHF,α(r)

−∑

β

∫d3r′

(φ∗HF,β(r′)

e2

| r − r′ |φHF,α(r′)

)φHF,β(r)

(803)

in order to minimize the energy. To simplify further analysis, we shall explicitlydisplay the spin-dependence by writing

φHF,α(r) = ψα(r) χσ

φHF,β(r) = ψβ(r) χσ′ (804)

This notation recognizes that the spatial component of the wave function, ψα(r),may depend on all the quantum numbers represented by α, including the spinquantum number, as in the un-restricted Hartree-Fock approximation. TheHartree-Fock equations are re-written as

0 =[− h2

2 m∇2 + Vions(r) − λα

]ψα(r) χσ

+∑

β

∫d3r′

(χT

σ′ ψ∗β(r′)

e2

| r − r′ |ψβ(r′) χσ′

)ψα(r) χσ

−∑

β

∫d3r′

(χT

σ′ ψ∗β(r′)

e2

| r − r′ |ψα(r′) χσ

)ψβ(r) χσ′

(805)

In the inner product, the integrations over the position r′ of the spatial com-ponent of the wave function is combined with the matrix elements of the spinwave functions. The spin matrix elements are given by

χTσ′ χσ = δσ′,σ (806)

Since the Coulomb interaction is spin-independent, that last term contains aKronecker delta function that is non-vanishing only when σ = σ′. The set ofHartree-Fock equations are eigenvalue equations for a non-local linear operator

0 =[− h2

2 m∇2 + Vions(r) − λα

]ψα(r)

+∑

β

∫d3r′

(ψ∗β(r′)

e2

| r − r′ |ψβ(r′)

)ψα(r)

322

−∑

β

δσ′,σ

∫d3r′

(ψ∗β(r′)

e2

| r − r′ |ψβ(r)

)ψα(r′)

(807)

There is one such equation for each value of α. In solving the above equationsfor ψα(r), one should consider the functions ψβ(r) as known quantities. Inthis case, the eigenvalue equations are linear in the eigenfunctions, ψα, and theundetermined multipliers, λα, are the eigenvalues. The term proportional to

Vdirect(r) =∑

β

∫d3r′

e2 | ψβ(r′) |2

| r − r′ |(808)

represents a contribution to the potential from the average electrostatic potentialdue to the electron density ρ(r′) =

∑β |ψβ(r′)|2 from all the electrons in the

system. That is, this potential even includes the contribution from an electronin the state α. This potential is independent of the spin states of the electrons,and is called the direct interaction. The last term in the Hartree-Fock equation(808) is non-local, as it relates the unknown eigenfunction ψα(r) to the weightedaverage of the unknown eigenfunction at other points in space, ψα(r′). The non-local potential represented by

V σexch(r, r′) = −

∑β

δσ,σ′ ψ∗β(r′)

e2

| r − r′ |ψβ(r) (809)

is called the exchange interaction. Since the Coulomb interaction is spin inde-pendent, the matrix elements in the non-local exchange potential are non-zeroonly if the spin of state α is identical to the spin of state β. If the spins are anti-parallel, the exchange term is zero. Thus, the exchange term is spin-dependent.The terms with β = α are spurious, since they represent the interactionof an electron with itself. However, their contributions to the direct and ex-change terms cancel exactly. Therefore, there are no self-interaction terms inthe Hartree-Fock approximation. This cancellation of the self-interaction hasthe effect that the linear potential operator is the same for all the single-electronwave functions.

With this notation, the Hartree-Fock equations can be written as[− h2

2 m∇2 + Vion(r) + Vdirect(r)

]ψα(r) +

∫d3r′ V σ

exch(r, r′) ψα(r′) = λα ψα(r)

(810)This set of equations can be solved iteratively. Using approximations for thedirect and exchange potentials, one can solve the equations to find a set ofwave functions which are approximations for the ψα(r). These approximatewave functions are then used to construct new approximations for the directand exchange potentials. The procedure is repeated until self-consistency is

323

achieved. Roothaan92 has shown how, by introducing an appropriate set ofbasis functions θm(r), one can express the Hartree-Fock wave functions ψα(r)as

ψα(r) =M∑

m=1

Cαm θm(r) (811)

and, hence, the Hartree-Fock equations can be reduced to a set of M coupledsimultaneous equations. The resulting set of non-linear simultaneous equationsare known as the Roothaan equations.

k

k'

k

k'

k-k'Hint

Hint

Figure 178: Contributions to the Hartree-Fock interaction energy for a uniformsystem. The process shown in (a) is the direct interaction or Hartree term. Theprocess in (b) is the exchange or Fock term.

The Hartree-Fock approximation can be solved exactly for the free electrongas in which the potential of the lattice of ions is replaced by a constant value.This (unrealistic) case of a uniform potential is of special importance, since thesolution is often used as a starting point to discussing the electronic structureof a non-uniform electron gas. Specifically, the most common method of de-termining electronic structure, the local density functional method, utilizes theexpression for the ground state energy of the uniform electron gas.

9.2.1 The Free Electron Gas.

The Hamiltonian for the free electron gas is invariant under all translations and,as long as the translational symmetry is not spontaneously broken, the Hartree-Fock eigenstates should be simultaneous eigenstates of the momentum operator.Thus, the Hartree-Fock equations for a uniform potential Vions = V0 shouldhave the eigenfunctions

φα,σ = ψk,σ(r) χσ =1√V

exp[i k . r

]χσ (812)

92C. C. J. Roothaan, Rev. Mod. Phys. 23, 68 (1951).

324

where V is the volume of the crystal. That this is true can be seen by substi-tuting the wave functions into the Hartree-Fock eigenvalue equations.

The charge density due to the electrons is a constant, and this combines withthe uniform charge density from the background gas of ions. Due to chargeneutrality, the resulting net direct Coulomb potential from the total chargedensity vanishes

Vions(r) + Vdirect(r) = 0 (813)

In order to evaluate the exchange potential, one has to perform the sumover values of k′, σ′. The sum over k′, σ′ only runs over the occupied states.We shall assume that the Hartree-Fock state does not spontaneously break thespin rotational symmetry and lead to magnetism. Likewise, we shall also as-sume that the Hartree-Fock solution does not break translational invariance.Magnetic solutions which also break translation invariance have been found byOverhauser93. However, it is unclear whether theses inhomogeneous phases arestable when screening is taken into account. Kohn and Nettel94 have pointed outthat the absolute stability of the broken-symmetry Hartree-Fock states requiresthe presence of significant anisotropy (or low dimensions). In the non-magnetictranslationally invariant case, the Hartree-Fock states are spin degenerate, andthe one-particle states are filled according to the magnitude of the kinetic en-ergy. All the one-particle states labelled by (k, σ), where k is contained insidea sphere of radius kF , are filled with electrons. The spin-dependent exchangeterm is evaluated as

Vexch(r, r′) = − 1V

e2

| r − r′ |∑

|k′|≤kF , σ′

δσ,σ′ exp[i k′ . ( r − r′ )

](814)

The exchange potential also has translational invariance, and so it is possiblethat plane waves are eigenfunctions of the Hartree-Fock equations. The ex-change potential is evaluated as the integral

Vexch(r, r′) = − 1( 2 π )3

∫ 2 π

0

dϕ

∫ π

0

dθ sin θ

×∫ kF

0

dk′ k′2 exp[i k′ . ( r − r′ )

]e2

| r − r′ |

= − 1( 2 π )3

2 π e2

×∫ kF

0

dk′ k′

( exp[

+ i k′ | r − r′ |]− exp

[− i k′ | r − r′ |

]i | r − r′ |2

)93A. W. Overhauser, Phys. Rev. Lett. 4, 462 (1960), Phys. Rev. 128, 1437 (1962).94W. Kohn and S. J. Nettel, Phys. Rev. Lett. 5, 8 (1960).

325

(815)

The integration over k′ can be performed with the aid of an identity obtainedby differentiating the expression∫ 1

0

dx cos α x =sin α

α(816)

with respect to α. That is,∫ 1

0

dx x sin α x =(

sin α

α2− cos α

α

)(817)

The resulting expression for the exchange potential is

Vexch(r, r′) = − e2 k4F

2 π2

(sin kF | r − r′ |( kF | r − r′ | )4

− cos kF | r − r′ |( kF | r − r′ | )3

)(818)

The long-ranged oscillatory behavior of the exchange potential is due to thesharp cut off of the integration at kF . This cut off occurs as the Fermi wavevector kF is the largest wave vector associated with the occupied one-electronstates.

-0.04

-0.03

-0.02

-0.01

0

0.01

0 2 4 6 8 10 12 14 16kF r

Vex

c(r) [

uni

ts o

f e2 k

F4 /(2π2 ) ]

Figure 179: The radial dependence of the exchange potential for the free electrongas. This plot emphasizes the oscillations due to the Fermi cut off and does notshow the 1/r divergence at the origin.

The contribution of the exchange potential to the energy eigenvalue λk canbe found from∫

d3r′ Vexch(r, r′) ψk(r′) =1√V

exp[i k . r

] ∫d3r′ exp

[i k . ( r′ − r )

]Vexch(r, r′)

(819)

326

Thus, the contribution of the eigenvalue stemming from exchange potential isjust the Fourier transform of the exchange term, Vexch(k),

Vexch(k) = − e2 k4F

2 π2

∫d3R exp

[i k . R

] (sin kF R

( kF R )4− cos kF R

( kF R )3

)(820)

which can be evaluated directly. An alternate method involves using the con-volution theorem, in which case the expression

Vexch(k) = − V

( 2 π )3

∫d3r′

e2

| r − r′ |

×∫|k′|≤kF

d3k′ ψ∗k′(r′) ψk′(r) exp

[i k . ( r′ − r )

](821)

can be used. The plane wave nature of the eigenfunctions can be utilized towrite the expression for Vexch as

Vexch(k) = − V

( 2 π )3

∫d3r′

e2

| r − r′ |×

×∫|k′|≤kF

d3k′ | ψk′(r′) |2 exp[i ( k′ − k ) . ( r − r′ )

](822)

The electron density, per spin, arising from state k is just | ψk′(r′) |2 = 1V

for | k′ | ≤ kF . Since this is independent of r′, the exchange contribution tothe eigenvalue involves the Fourier Transform of the Coulomb potential. TheFourier transform of the exchange potential is found as

= − 1( 2 π )3

∫d3r′

∫|k′|≤kF

d3k′e2

| r − r′ |exp

[i ( k′ − k ) . ( r − r′ )

](823)

Hence, the expression for the exchange contribution to the eigenvalue λk is givenby

Vexch(k) = − 1( 2 π )3

∫|k′|≤kF

d3k′4 π e2

| k − k′ |2

= − e2

π k

∫ kF

0

dk′ k′ ln| k + k′ || k − k′ |

(824)

The integral can be evaluated as

Vexch(k) = − 2 e2

πkF F

(k

kF

)(825)

327

where

F (x) =12

+1 − x2

4 xln

| 1 + x || 1 − x |

(826)

At k = 0, the function F (0) is unity. At k = kF , the function falls to thevalue F (1) = 1

2 and has a logarithmic singularity in the slope. This singularityin the slope is due to the long-ranged nature of the Coulomb interaction ( 4 π

k2 ).The function F (x) falls to zero in the limit limx → ∞ F (x) → 0. Thus, theeigenvalue λk is given by

λk =h2 k2

2 m− 2 e2

πkF F

(k

kF

)(827)

-1

-0.5

0

0.5

1

0 0.5 1 1.5k / kF

λ k [

Ryd

berg

s ]

rs=3

rs=4

Figure 180: The Hartree-Fock expression for the dispersion relation λk for freeelectrons. The curves corresponding to the different rs values correspond todifferent values of the electronic density.

The total energy of the electron system is given by the sum of the kineticenergy and the exchange energy of the occupied states

EHF = 2∑

k

h2 k2

2 m

−∑k,k′

∫d3r

∫d3r′ ψ∗k(r) ψ∗k′(r

′)e2

| r − r′ |ψk(r′) ψk′(r)

=∑

k

(h2 k2

2 m+ λk

)(828)

where the summations are restricted to the values of k and k′ which are within

328

the Fermi sphere. The Hartree-Fock energy can be re-expressed as

EHF = 2∑

k ≤ kF

h2 k2

2 m

− 2 e2

πkF

∑k ≤ kF

[12

+(k2

F − k2

4 k kF

)ln

| kF + k || kF − k |

](829)

The summations over k can be evaluated by transforming them into integralsover the volume of the Fermi sphere

EHF = 24 π V

( 2 π )3

∫k ≤ kF

dk k2 h2 k2

2 m

− 2 e2

πkF

4 π V( 2 π )3

∫k ≤ kF

dk k2

[12

+(k2

F − k2

4 k kF

)ln

| kF + k || kF − k |

]=

V

π2

h2 k5F

10 m− V e2

π3k4

F

[13

+16

](830)

The number of electrons, per spin, Ne

2 is given by

Ne

2=

V

8 π3

4 π3

k3F (831)

Using this, the Hartree-Fock approximation for the cohesive energy of the freeelectron gas can be expressed as

EHF = Ne

[35h2 k2

F

2 m− 3

4e2

πkF

](832)

An alternative expression is given by introducing a characteristic dimension, orradius rs, such that there exists one electron in a sphere of radius rs a0, wherea0 is the Bohr radius (a0 = h2

m e2 ). Then, the uniform electron density, ρ, isgiven by the equivalent expressions

1ρ

=4 π3

a30 r

3s

=3 π2

k3F

(833)

Thus, the magnitude of the Fermi wave vector kF is given by

kF =(

9 π4

) 13 1rs a0

(834)

329

and so the electronic energy is expressed as

EHF

Ne=

3 h2

10 m a20

(9 π4

) 23 1r2s

− 3 e2

4 π a0

(9 π4

) 13 1rs

=e2

2 a0

(9 π4

) 13[

35

(9 π4

) 13 1r2s− 3

2 π1rs

]=

2.2099r2s

− 0.9163rs

Rydbergs (835)

where 1 Rydberg = e2

2 a0. The Hartree-Fock energy has a minimum at the rs

value given by rs ∼ 4.8 which corresponds to a value of the cohesive energywhich is about 0.1 Rydbergs. Typical materials have spatially varying densities,hence, the local value of rs also varies. For a hydrogen-like atom, the groundstate density is given by

ρ(r) =Z3

π a30

exp[− 2 Z r

a0

](836)

Therefore, typical values of rs correspond to the density at the nuclear position

rs =( 34 )

13

Z

=0.9086Z

(837)

or to the density at the first Bohr radius r = Z a0

rs =( 34 )

13 e

23

Z

=1.7696Z

(838)

Since for metals the density of electrons corresponds to rs values in the rangeof 2 to 5, the exchange term is of similar magnitude to the kinetic energy term.The Hartree-Fock approximation indicates that the cohesive energy is largestfor low density metals, i.e., those with rs ∼ 5.

In the particular case of the free electron gas where the potential due to theions is uniform, the Hartree-Fock approximation coincides with second orderperturbation theory. If an infinite number of higher order terms are included inthe perturbation series95, one obtains the expression for the energy per electron

E

Ne=[

2.2099r2s

− 0.9163rs

+ 0.06218 ln rs − 0.094 + O(rs)]

(839)

95M. Gell-Mann and K. A. Brueckner, Phys. Rev. 106, 347, (1957), W. J. Carr and A. A.Maradudin, Phys. Rev. A 133, 371 (1964).

330

Hartree-Fock Total Energy

-0.16

-0.08

0

0.08

0 2 4 6 8rs

E(r

s)/N

e [ u

nits

of

e2 /(2a

0) ]

Figure 181: The Hartree-Fock approximation for cohesive energy of the freeelectron gas.

in units of e2

2 a0. The energy is a form of an expansion in rs, valid for rs < 1.

Thus, the Hartree-Fock result can be thought of as an approximation whichreproduces the high density limit (small rs limit) correctly. The other terms inthe expression are due to electron correlations. A completely different behavioris expected to occur in the low density limit. In reducing the density from thehigh density metallic limit to the low density limit, the system is expected toundergo a transition to a Wigner crystal phase96. In a Wigner crystal, theelectrons are expected to localize in a b.c.c. structure. The total energy isexpected to be dominated by the electrostatic interaction and the energies ofthe vibrations of the electronic lattice97. The energy of the Wigner crystallinephase is given by

E

Ne=

e2

2 a0

(− 1.792

rs+

2.65

r32s

− 0.73r2s

+ . . .

)(840)

for rs 1.

The electronic wave functions described by a Slater determinant are notdevoid of correlations. The correlations are a result of the Pauli exclusion prin-ciple. The two-particle density-density correlation function for a single Slaterdeterminant can be written as

ρ2(r, r′) =∑α,β

12| φα(r) φβ(r′) − φβ(r) φα(r′) |2

=∑α

| φα(r) |2∑

β

| φβ(r′) |2 −∑α,β

φ∗α(r) φβ(r) φ∗β(r′) φα(r′)

96E. P. Wigner, Phys. Rev. 46, 1002 (1934).97W. J. Carr, R. A. Coldwell-Horsfall, and A. E. Fein, Phys. Rev. 124, 747 (1961).

331

(841)

On making the spin dependence explicit, by writing

φα(r) = ψα(r) χσ

φβ(r) = ψβ(r) χσ′ (842)

one finds that the two-particle density-density correlation function is given by

ρ2(r, r′) =∑α

| ψα(r) |2∑

β

| ψβ(r′) |2 −∑α,β

δσ,σ′ ψ∗α(r) ψα(r′) ψ∗β(r′) ψβ(r)

= ρ(r) ρ(r′) −∑

σ

Gσ(r′, r) Gσ(r, r′) (843)

where Gσ is given by a sum over the single-particle states labelled by α whichhave the spin quantum number σ

Gσ(r, r′) =∑α

ψ∗α(r′) ψα(r) (844)

The last term in the two-particle density-density correlation function is the ex-change term. The exchange term originates from pairs of electrons with parallelspins. In the Hartree-Fock approximation for the free electron gas, the exchangecontribution to the two-particle density-density correlation function ρ2(r, r′) isexpressed in terms of the factors

Gσ(r, r′) =∑

|k′| < kF

ψ∗k′(r′) ψk′(r)

=1V

∑|k′| < kF

exp[i k′ . ( r − r′ )

]

=k3

F

2 π2

[sin kF | r − r′ |( kF | r − r′ | )3

− cos kF | r − r′ |( kF | r − r′ | )2

](845)

where the summation is over the Fermi sphere. The density-density correlationfunction shows a hole in the density of parallel spin electron around the electronand vanishes as | r − r′ | → 0, as expected from the Pauli exclusion principle.The exchange hole describes the exclusion of just one electron. The exchangepotential has a similar form to the density-density correlation function and canbe thought of arising from a deficiency in the density of parallel spin electronsaround an electron at r. The Hartree-Fock approximation is deficient in that itdoes not include a similar correlation hole between electrons with anti-parallelspins98.

98The effect of the Coulomb interaction between pairs of electrons (with anti-parallel spins)is also expected to produce a correlation hole. Recent experiments in which a single photonproduces the emission of two electrons, via a process which necessarily involves inter-electronicinteractions, allows for a mapping of the exchange-correlation hole. [F. O. Schuhman et al.,Physical Review B 73, 041404 (2006).]

332

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5kF r

ρ ρρρ 2(r

)/ρ ρρρ2

Figure 182: The radial dependence of the two-particle density-density correla-tion function for the free electron gas, as calculated in the Hartree-Fock approx-imation. The Pauli-exclusion principle causes the density-density correlationfunction at r = 0 to fall to half of the asymptotic value.

In the Hartree-Fock approximation, the energies of the excited states aregiven by Koopmans’ theorem99. That is, the energy for adding or removing anelectron from the system is given by the eigenvalue λk, if the other one-electronstates in the many-particle Slater determinant are not changed or that the otherelectrons in the ground state are not re-arranged. Thus, in the Hartree-Fock

0

5

10

15

20

25

30

-0.5 0 0.5 1 1.5E [Rydbergs]

ρ qp(

E) [

Stat

es/R

ydbe

rg]

Fermi-level

Hartree-Fock

non-interacting

Figure 183: The Hartree-Fock approximation for the quasi-particle density ofstates for a free electron gas.

99T. A. Koopmans, Physica 1, 104 (1933).

333

approximation, the quasi-particles energies are given by

Eqp(k) = λk − µ (846)

The zero of the quasi-particle energy is chosen to be the renormalized Fermienergy. The quasi-particle density of states, per spin, is given by

ρqp(E) =∑

k

δ( E − Eqp(k) )

=V

2 π2

∫ kF

0

dk k2 δ( E − Eqp(k) )

=V

2 π2k2

(dEqp(k)dk

)−1∣∣∣∣k(E)

(847)

where k(E) is the value of k that satisfies the equation

Eqp(k) = E (848)

The Fermi energy is defined by

Eqp(kF ) = 0 (849)

which fixes µ in terms of kF . At the Fermi energy, the quasi-particle density ofstates is zero since

dEqp(k)dk

=h2 k

m− e2

π

kF

k

+e2

π

( k2F + k2 )2 k2

ln| kF + k || kF − k |

(850)

which diverges logarithmically at k(E = 0) = kF . Thus, the Hartree-Fock ap-proximation for the free electron gas is of limited utility in discussing propertiesof real metals. This is caused by the divergent slope of the one-electron eigen-values near the Fermi surface. This spurious divergence is due to the neglect ofscreening and results in the one-electron density of states falling to zero just atthe Fermi energy.

——————————————————————————————————

Broken Symmetry: The Spiral Spin Density Wave State

The spiral spin density wave state is an example of a state with combinedbroken spin rotational invariance and translational invariance. In the Hartree-Fock approximation, it is represented by a single Slater determinant Φ composedof one-electron states of the form

φk,+(r) =uk√V

exp[ i ( k − Q/2 ) . r ] χ↑ +vk√V

exp[ i ( k + Q/2 ) . r ] χ↓

φk,−(r) =v∗k√V

exp[ i ( k − Q/2 ) . r ] χ↑ −u∗k√V

exp[ i ( k + Q/2 ) . r ] χ↓

(851)

334

where uk and vk are complex coefficients that have yet to be determined. Itshould be noted that k serves to label the one-electron wave functions but doesnot represent the momentum of the electron. The above one-electron statesform an orthonormal basis set if

| uk |2 + | vk |2 = 1 (852)

For uk ≡ 1 and vk ≡ 0, the broken-symmetry Hartree-Fock state reducesto the Hartree-Fock approximation to the free electron gas. The spin density isdefined as

−→S (r) =

h

2

∑k,±

φ∗k,±(r) −→σ φk,±(r) fk,± (853)

where −→σ is the vector Pauli spin matrix and where fk,± are the occupationnumbers of the broken-symmetry one-electron states. The broken-symmetrystate may have a spiral spin density, since the components are evaluated as

Sz(r) =h

2 V

∑k

(| uk |2 − | vk |2

)( fk,+ − fk,− ) (854)

and

Sx(r) =h

2 V

∑k

(uk v

∗k exp[ − i Q . r ] + vk u

∗k exp[ + i Q . r ]

)( fk,+ − fk,− )

Sy(r) = ih

2 V

∑k

(uk v

∗k exp[ − i Q . r ] − vk u

∗k exp[ + i Q . r ]

)( fk,+ − fk,− )

(855)

The z-component of the spin density, just like the charge density, is uniform forthis state. We shall define a complex number with amplitude MT

Q and phase ϕvia

MQ exp[ i ϕ ] =h

V

∑k

vk u∗k ( fk,+ − fk,− ) (856)

so that the components of the spin density can be simply expressed as

Sx(r) = MQ cos( Q . r + ϕ )

Sy(r) = MQ sin( Q . r + ϕ ) (857)

If the amplitude MQ is non-zero, the static magnetic spin density spirals in thex− y plane with wave vector Q. Hence, the state has a broken spin-rotationalinvariance and translational invariance if the product vk u

∗k 6= 0.

The coefficients uk and vk are to be determined by minimizing the energy.The energy of the broken-symmetry Hartree-Fock state is given by minimizingthe expectation value

< Φ | H | Φ > (858)

335

with respect to variations of the complex coefficients uk and vk subject to thenormalization conditions given by eqn(852). Instead of considering the real andimaginary parts of the coefficients uk and vk as independent variables, we shallconsider the pair of coefficients uk and vk and their complex conjugates u∗k andv∗k as independent. The minimization is performed by using Lagrange’s methodof undetermined multipliers, which minimizes the functional

Ω[uk, u∗k, vk, v

∗k] = < Φ | H | Φ > −

∑k,±

fk,± λk,±

(| uk |2 + | vk |2 − 1

)(859)

where λk,± are the undetermined multipliers. The functional is evaluated as

Ω =∑

k

fk,+

(E0

k−Q/2 | uk |2 + E0k+Q/2 | vk |2

)

+∑

k

fk,−

(E0

k−Q/2 | vk |2 + E0k+Q/2 | uk |2

)

− 12

∑k,k′

( fk,+ fk′,+ + fk,− fk′,− )(

4 π e2

V | k − k′ |2

) ∣∣∣∣ uk u∗k′ + vk v

∗k′

∣∣∣∣2

− 12

∑k,k′

( fk,+ fk′,− + fk,− fk′,+ )(

4 π e2

V | k − k′ |2

) ∣∣∣∣ uk vk′ − vk uk′

∣∣∣∣2−∑k,±

fk,± λk,±

(| uk |2 + | vk |2 − 1

)(860)

The first two terms represent the kinetic energies of the independent electrons.The energy due to the direct Coulomb interaction is assumed to cancel withthe corresponding interaction energy due to the positive uniform charge background. The third and fourth terms represent the exchange part of the pairwiseCoulomb interactions between the electrons. The last term represents originatesfrom the normalization condition and vanishes if the one-electron wave functionsare properly normalized. The Lagrange undetermined multipliers λk,± are tobe identified with the quasi-particle energies. The occupation numbers fk,± areone or zero, depending on whether or not the corresponding one-electron wavefunctions φk,±(r) are included in the Slater determinant. Therefore, the fk,±can be treated as independent quantities.

On requiring that Ω be an extrema with respect to a variation of u∗k, oneobtains

0 =[E0

k−Q/2 − λk,+ −∑k′

(4 π e2

V | k − k′ |2

) (fk′,+ | uk′ |2 + fk′,− | vk′ |2

) ]uk

−∑k′

(4 π e2

V | k − k′ |2

) (( fk′,+ − fk′,− ) uk′ v

∗k′

)vk (861)

336

Similarly, varying Ω with respect to v∗k and demanding that the first-order vari-ation is zero, yields

0 =[E0

k+Q/2 − λk,+ −∑k′

(4 π e2

V | k − k′ |2

) (fk′.+ | vk′ |2 + fk′,− | uk′ |2

) ]vk

−∑k′

(4 π e2

V | k − k′ |2

) (( fk′,+ − fk′,− ) vk′ u

∗k′

)uk (862)

On defining the real k-dependent quantities

∆0(k) =∑k′

(4 π e2

V | k − k′ |2

)( fk′,+ + fk′,− ) (863)

and

∆±(k) =∑k′

(4 π e2

V | k − k′ |2

)( fk′,+ − fk′,− ) ( | uk′ |2 − | vk′ |2 ) (864)

which represent the first-order shifts of the bands and the complex quantity

∆(k) =∑k′

(4 π e2

V | k − k′ |2

)( fk′,+ − fk′,− ) uk′ v

∗k′ (865)

which represents the mixing of the bands, the equations can be re-written as

0 =[E0

k−Q/2 − 12

∆0(k) −12

∆±(k) − λk,+

]uk − ∆(k) vk

0 =[E0

k+Q/2 − 12

∆0(k) +12

∆±(k) − λk,+

]vk − ∆∗(k) uk

(866)

The above set of equations can be solved for the eigenvalues λk,+, yielding

λk,+ =(E0

k−Q/2 + E0k+Q/2 −∆0(k)

2

)±

√(E0k−Q/2 − E0

k+Q/2 −∆±(k)

2

)2

+ | ∆(k) |2

(867)We shall choose the solution with the lower sign, since we have assumed thatthe states with weight fk,+ are occupied in the ground state. The undeter-mined multipliers λk,− can be found by an analogous procedure, however, inthis case, the solution corresponding to the upper sign should be chosen. Thekz-dependence of the eigenvalues λk,± is sketched in fig(184). It is seen thatthe dispersion relation consists of two branches which asymptotically approachE0

k−Q/2 and E0k+Q/2, and are separated by a gap of 2 ∆(0) at k = 0. Since the

interaction is a property of the system as a whole, it seems a reasonable guess

337

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5

kz/Q

λ k

λk+

λk-

Figure 184: The form of the quasi-particle dispersion relations in a spiral spindensity wave state in label space.

that the states (k,±) should be filled in the order of increasing quasi-particleenergies λk,±. The functions | uk |2 and | vk |2 are determined as

| vk |2 =12

[1 +

E0k−Q/2 − E0

k+Q/2 − ∆±(k)√( E0

k−Q/2 − E0k+Q/2 − ∆±(k) )2 + 4 | ∆(k) |2

]

| uk |2 =12

[1 −

E0k−Q/2 − E0

k+Q/2 − ∆±(k)√( E0

k−Q/2 − E0k+Q/2 − ∆±(k) )2 + 4 | ∆(k) |2

](868)

These functions, respectively, represent the probability density for finding spin-down or spin-up electrons in the states of the lower quasi-particle branch. Hence,the states in the lower branch are primarily spin-up when the label k ≈ Q/2 andare primarily spin-down when k ≈ −Q/2. The roles of uk and vk are reversedfor the upper branch of quasi-particles. In momentum-space, the mixed spin-character of the states at the label value kz = 0 corresponds to up-spin electronsat the back of the Fermi-surface being mixed with down-spin electrons at thefront of the Fermi-surface. Therefore, the spiral spin density wave state roughlycorresponds to a state which is described in our k-label space as having an up-spin Fermi-sphere centered on k = Q/2 and a down-spin Fermi sphere centeredon k = −Q/2. However, the “Fermi-surfaces” are ellipsoids of revolution thatare expected to merge and then finally a new sheet will re-emerge as the wavevector Q is reduced from Q > 2kF to 2kF < Q. The formation of the new sheet

338

0

0.2

0.4

0.6

0.8

1

1.2

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1kz/Q

| uk

|2

| vk |2 | uk |2

Figure 185: The probability density for finding spin-up |uk|2 and spin-down|vk|2 electrons in the lower branch of quasi-particles in the spiral spin densitywave state.

corresponds to the filling of the upper band. The product uk v∗k is evaluated as

-1.5

-1

-0.5

0

0.5

1

1.5

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

kz/kF

k x/k

F

Figure 186: The ky = 0 section across the “Fermi surfaces” of spiral density wavestates, for ordering waves vector Q = 2.2kF (blue), Q = 2kF and Q = 1.8kF

(red).

339

uk v∗k =

∆(k)√( E0

k−Q/2 − E0k+Q/2 − ∆±(k) )2 + 4 | ∆(k) |2

(869)

On substituting this product into the definition of the gap ∆(k), we find thegap equation

∆(k) =∑k′

(4 π e2

V | k − k′ |2

)( fk′,+ − fk′,− ) ∆(k′)√

( E0k′−Q/2 − E0

k′+Q/2 − ∆±(k′) )2 + 4 | ∆(k′) |2

(870)The spiral magnetization only occurs if the gap equation has a non-zero solutionfor ∆(k). It should be noted that the absolute value of the phase ϕ of thegaps ∆(k) drop out of the gap equation and so the phase can be consideredas being arbitrary. In fact, the absolute phase is spontaneously chosen by thesystem when the system condenses into the spiral spin density wave state. Theabsolute phase is expected to be determined by the interaction between thespiral magnetization and the impurities and surfaces in the solid. The relativeshift of the quasi-particle bands is found to be given by

∆±(k) = −∑k′

(4 π e2

V | k − k′ |2

) ( fk′,+ − fk′,− ) ( E0k′−Q/2 − E0

k′+Q/2 − ∆±(k′) )√( E0

k′−Q/2 − E0k′+Q/2 − ∆±(k′) )2 + 4 | ∆(k′) |2

(871)The set of equations (870), (871) and (863) are to be solved self-consistently forarbitrary distributions of fk,± which conserve the total number of electrons.

It might be expected that for non-zero values of ∆(k), the solution withlowest energy corresponds to the spiral state with Q = 2kF since, in this case,the electrons only occupy states in the lower sub-band which have quasi-particleenergies lower than the free-electron energies due to the opening of the gap. Thispossible optimal value of Q could be considered quite reasonable since 2kF isthe only momentum scale in the system. On the other hand, the system can alsobe viewed as having a static spiral magnetization which, through the exchangeinteraction, simultaneously scatters and flips the spins of the electrons. Thatis, through the electronic interactions, the electrons produce an effective spin-dependent potential V (r,−→σ ) of the form

V (r,−→σ ) =12

(V ∗(Q) exp[ − i Q . r ] σ+ + V (Q) exp[ + i Q . r ] σ−

)= | V (Q) |

(cos( Q . r + ϕ ) σx + sin( Q . r + ϕ ) σy

)∝ −→

S (r) . −→σ (872)

Since the Hartree-Fock approximation is self-consistent, the spin-dependent po-tential must yield the spiral magnetization. This leads one to expect that thebroken-symmetry state will be more stable with Q ≈ 0, at least for small values

340

of the spiral magnetization, since the appropriate response function is largestwhen Q = 0. The reason why the response function is largest for Q ≈ 0 is sim-ply because the number of electrons available for elastic scattering across theFermi-surface involving small momentum transfers Q = 2 kF cos θ, where thepolar angle is θ ≈ π

2 , is proportional to the area 2πk2F sin θ ∆θ. The available

area for small Q is large compared to the corresponding area for large momen-tum transfers, for which θ ≈ 0. The number of electrons available for scatteringthrough large momenta is proportional to the area πk2

F sin θ ∆θ and, therefore,vanishes as ∆θ2. That is, the response function at large momentum transfersis small since it rapidly cuts off for momentum transfers equal to the diameterof the Fermi-sphere Q = 2kF . This reasoning leads one to expect that, if thesymmetry breaking is sufficiently weak, the ferromagnetic state should be stable.

The effective spin-dependent potential only has components at ±Q, andthis is responsible for the unusual form of the quasi-particle dispersion rela-tion. In momentum-space, one has up-spin electrons being spin-flip scatteredthrough momentum +Q. The effective potential with momentum component

kF

Q

Figure 187: A three-dimensional depiction of the momentum-space Fermi-surface of a spiral density wave state with wave vector Q. The Fermi-surface isgapped in annular regions centered on ±Q/2.

Q results in the dispersion relation for spin-up electrons being flattened at thevalue −Q/2. Likewise, since the spin-down electrons are spin-flip scatteredbackward through −Q, the dispersion relation for spin-down electrons is flat-tened at +Q/2 in momentum-space. Hence, for a spiral spin density wave state,the quasi-particle mass enhancements at the Fermi-surface are anisotropic andspin-dependent. Furthermore, the spiral spin density wave state should havesmall annular regions on the Fermi-surface which are gapped.

——————————————————————————————————

341

9.2.2 Exercise 56

Show, using perturbation theory, that the second-order correction to the energyof a free electron gas from the interaction between electrons with anti-parallelspins is given by

∆E(2) = − m

h2

∑k,k′,q

(4 π e2

q2 V

)2 1q . ( k − k′ + q )

(873)

where k < kF , k′ < kF , | k + q | > kF and | k′ − q | > kF . Since thisintegral is dominated by the region q → 0, the dominant values of k and k′

are given by k ∼ kF and k′ ∼ kF . Show that the contribution to ∆E(2) isproportional to ∫

d3q

q3= 4 π

∫dq

q

= 4 π ln q (874)

and, thus, diverges for q → 0.

——————————————————————————————————

The second-order contribution to the energy of the electron gas can be ex-pressed as

∆E(2) =∑

k,σ;k′,σ′;q

(4 π e2

V q2

)2 fk fk′ ( 1 − fk−q ) ( 1 − fk′+q )

E0k + E0

k′ − E0k=q − E0

k′+q

−∑

k,k′,σ;q

(4 π e2

V q2

) (4 π e2

V |k − k′ − q|2

)fk fk′ ( 1 − fk−q ) ( 1 − fk′+q )

E0k + E0

k′ − E0k−q − E0

k′+q

(875)

where fk is one or zero, depending on whether k < kF or not. The first term isthe direct Coulomb interaction and the second term only acts between electronswith the same spin and represents the exchange interaction. The second orderexchange contribution can be evaluated analytically100 and expressed in unitsof Rydbergs per electron as

e2

2 a0

[13

ln 2 − 32 π2

ζ(3)]∼ 0.0484

e2

2 a0(876)

where ζ(n) is Riemann’s zeta-function. The above result for the second-orderexchange energy is independent of rs.

100L. Onsager, L. Mittag and M.J. Stephen, Ann. Physik 18, 71 (1966).

342

Simple second-order perturbation theory does not work for the free electrongas. In fact, the energy corrections from each higher order in the perturbationseries diverges. None the less, perturbation theory can be applied by using moreelaborate techniques which take into account the screening of the Coulomb in-teraction.

343

9.3 The Density Functional Method

The density functional method provides an exact method for calculating the elec-tron density and ground-state energy for interacting electrons in the presence ofa crystalline potential. As such, it can be used to determine the stability of var-ious lattice structures. It can also be used to determine ground-state propertiesor static properties of the electronic systems, such as those provided by elas-tic scattering experiments. It is based on the Hohenberg and Kohn Theorem101.

The Hohenberg-Kohn theorem considers the form of a many-electron Hamil-tonian in which the form of the Coulomb interaction term between pairs of elec-trons, Vint(r, r′), is known, but in which the one-particle potential due to theion cores is considered to be an external potential. Thus, the external potentialVions(r) varies between one crystal structure and the next. The Hamiltonianis written as the sum of the interaction energies of the pairs of electrons, thekinetic energies of all the electrons and the external potentials acting on eachelectron.

The Hohenberg-Kohn theorem starts by noting that every non-degeneratemany-body ground state wave function is associated with a unique external po-tential. The theorem proves that a second map exists between the ground stateelectron density and the external potential. This establishes a unique mappingbetween the many-body ground-state wave function and the ground-state elec-tron density. Therefore, the expectation value of any ground-state property canbe expressed in terms of a unique functional of the electron density. Havingestablished this, the ground state properties and electron density can then bedetermined by using the Rayleigh-Ritz variational principle for the ground stateenergy in which the electron density is the function to be varied.

This leads to a knowledge that, if one can construct the energy functional asthe sum of a unique energy functional and a simple part due to the external po-tential due to the lattice, then one can find the ground state energy and electrondensity. The unique energy functional is not known, however, it is customaryto make the Local Density Approximation (LDA). In this approximation, anun-testable assumption is made about the electron-electron interactions in anon-uniform electron gas. The method also generates eigenvalues which are in-terpreted in terms of the energies of independent Bloch electrons. The energydispersion relations generated this way do show marked similarities with theexperimentally determined bands of simple metals.

101P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).

344

The basis for density functional theory is provided by a theorem proved byHohenberg and Kohn102.

9.3.1 Hohenberg-Kohn Theorem

The Hohenberg-Kohn theorem first assures us that the electron density in asolid, ρ(r), uniquely specifies the electrostatic interaction potential between theNe electrons and the ionic lattice. Since there is a one to one mapping betweenVions(r) and the ground state many-body wave function Ψ, then the densityρ(r) can be used as the basic variable. As a consequence of this theorem andthe Rayleigh-Ritz principle, the energy can be expressed as the sum of a uniquefunctional of the electron density and a simple functional describing the po-tential energy of the electrons arising from their electrostatic interaction withthe ionic lattice. This establishes a variational principle which can be used tocalculate the electron density, ρ(r), and the total energy of the electronic system.

First, it shall be assumed that Vions(r) is not uniquely specified if ρ(r) isgiven. That is, it is assumed that there exists at least two potentials V and V ′

which give rise to the same ground state electron density. These potentials arerelated to the exact ground state many-particle wave functions via the energyeigenvalue equations,

H Ψ(r1, . . . rNe) = E Ψ(r1, . . . rNe

) (877)

andH ′ Ψ′(r1, . . . rNe

) = E′ Ψ′(r1, . . . rNe) (878)

From the Rayleigh-Ritz variational principle, one finds that the primed wavefunction Ψ′(r1, . . . rNe

) provides an upper bound to the ground state energyof the unprimed Hamiltonian H,

E =i=Ne∏i=1

[ ∫d3ri

]Ψ∗(r1, . . . rNe

) H Ψ(r1, . . . rNe)

E <

i=Ne∏i=1

[ ∫d3ri

]Ψ′∗(r1, . . . rNe

) H Ψ′(r1 . . . rNe) (879)

However, as the primed and unprimed Hamiltonian are related through

H = H ′ + V − V ′ (880)

and as Ψ′ is the ground state of H ′ with energy eigenvalue E′, the energiessatisfy an inequality

E < E′ +∫d3r ρ(r) ( V (r) − V ′(r) ) (881)

102P. Hohenberg and W. Kohn, Phys. Rev. 136, B 864 (1964).

345

The assumption that the ground state densities of the primed and unprimedHamiltonian are equal has been used. By virtue of similar reasoning, it can alsobe shown that the energies also satisfy the inequality

E′ < E +∫d3r ρ(r) ( V ′(r) − V (r) ) (882)

where the prime and unprimed quantities are interchanged. Adding these twoinequalities leads to an inconsistency

E + E′ < E + E′ (883)

Therefore, the assumption that the same ground state density can be found fortwo different potentials is false. Furthermore, the potentials can, at most, onlydiffer by a constant V ′(r) − V (r). Thus, the ground state electron density ρ(r)must correspond to a unique V (r). Since there is a unique one to one mappingbetween Ψ and V (r), there is a unique one to one mapping between Ψ and ρ(r).This means that the electron density, ρ(r), can be taken to be the principalvariable. It also implies that the Rayleigh-Ritz principle can be re-stated as avariational principle for an energy functional E[ρ] which is expressed in termsof the electron density.

9.3.2 Functionals and Functional Derivatives

As a mathematical prelude, we shall define functionals and functional deriva-tives.

A functional is a generalization of a function. A function f(r) can be definedas a mapping which maps each point in space, r, to a number. The value ofthe number depends on the position of the point. The functional is similar inthat it maps a scalar function onto a number. The value of the functional, F [ρ],depends upon the function ρ(r), i.e., the values of the function ρ at each pointin space. Functionals are usually expressed in terms of integrals over space,usually as multiple integrals.

A simple example of a functional is given by the number of electrons Ne[ρ],which is a functional of the density. The number of electrons is given by

Ne[ρ] =∫

d3r ρ(r) (884)

It is a functional as different densities may correspond to different number ofparticles i.e., Na has a different density than Li and they have different numbersof electrons.

The classical Coulomb energy is a more interesting functional. The Coulombenergy is defined as the pair-wise sum of interactions

ECoul[ρ] =e2

2

∫d3r

∫d3r′

ρ(r) ρ(r′)| r − r′ |

(885)

346

This yields a number which is the value of the energy, and this number dependson the electron density at all points of space.

Given a functional F [ρ], one can define a functional derivative. The defini-tion of the functional derivative is similar to the definition of a derivative of afunction. However, instead of defining the derivative in terms of the differenceof the function at two nearby points, one defines the functional derivative interms of the difference of the functional for two functions that are close. Forexample, an arbitrary family of functions, ρ′(r), can be defined in terms of afixed function ρ(r) and an arbitrary deviation δρ(r) via

ρ′(r) = ρ(r) + λ δρ(r) (886)

The scale factor λ varies from unity to zero continuously. When λ = 1, thisrelation defines the shape of the deviation δρ(r). If λ is changed continuouslyto zero, the differences between the function ρ′ and the fixed function ρ van-ish. The shape of the deviation λ δρ(r) is arbitrary and does not change, onlythe magnitude of the deviation is changing. The functional derivative can beexpressed in terms of the limit of the difference of the functional evaluated forthese two functions. If one assumes that one may Taylor expand the functionalin powers of λ, one has

F [ρ′] = F [ρ] + λ δ1F [ρ, δρ] +12λ2 δ2F [ρ, δρ] + . . . (887)

since the differences now depend on two functions ρ and δρ. If one defines theterms of first order in λ to have the form

δ1F [ρ, δρ] =∫

d3r δρ(r)δF [ρ]δρ(r)

(888)

then the quantityδF [ρ]δρ(r)

(889)

is independent of the shape of the deviation, and is defined to be the first orderfunctional derivative. Sometimes a functional may depend on the higher orderderivatives of ρ i.e.,

F [ρ] =∫

d3r f(r, ρ,∇ρ) (890)

In this case, one can define a functional derivative in terms of the partial deriva-tives,

∂f

∂ρ(891)

and the vector quantity∂f

∂∇ρ(892)

347

etc., where the functions ρ and ∇ρ etc. are treated as independent variables.This yields the first order variation as

δ1F [ρ, δρ] =∫

d3r

(δρ

∂f

∂ρ+ ∇δρ . ∂f

∂∇ρ

)(893)

If the functions ρ satisfy appropriate conditions at the boundaries of the regionof integration, the equation can be integrated by parts to eliminate the term∫

d3r ( ∇δρ ) .∂f

∂∇ρ(894)

In this case, the first order functional derivative is evaluated as

δF [ρ]δρ(r)

=∂f

∂ρ− ∇ .

∂f

∂∇ρ(895)

The extension to functionals containing higher order derivatives is quite straight-forward.

An alternative method of evaluating functional derivatives is based on theobservation that the functional derivative is independent of the variation δρ.Since δρ is arbitrary, one may choose δρ to have any particular form. Theparticular variation of the form of a Dirac delta function proves to be a usefulchoice

δρ(r) = δ3(r − r1) (896)

since, for this particular choice, the value of δF 1[ρ, δ3(r − r1)] is given by

δ1F [ρ, δ3(r − r1)] =δF [ρ]δρ(r1)

(897)

An example of the first order functional derivative is given by the functionalderivative of the Coulomb energy

δECoul[ρ]δρ(r1)

=e2

2

∫d3r

ρ(r)| r − r1 |

+e2

2

∫d3r′

ρ(r′)| r′ − r1 |

= e2∫

d3rρ(r)

| r − r1 |(898)

In obtaining the second line, we have relabelled the variable of integration. Thefirst order functional derivative of the mono-nomial functional

Fn[ρ] =∫

d3r ρ(r)n (899)

is simply evaluated asδFn[ρ]δρ(r1)

= n ρ(r1)n−1 (900)

348

The delta function method also proves useful for evaluating functional deriva-tives of higher orders.

The first order functional derivative is often encountered in variational prin-ciples. In a variational principle, there exists a function ρ(r) which yields anextremal value of the functional. That is, if the functional is changed by anarbitrary small variation λδρ away from the extremal function, the functionaldoes not change. On regarding the functional F [ρ+λδρ] as a function of λ, theextremal condition is equivalent to

∂

∂λF [ρ+ λδρ]

∣∣∣∣λ=0

= 0 (901)

since the value of the functional does not change to order λ, as λ approacheszero. This equation is satisfied for an arbitrary shape δρ(r), if the functionalderivative is identically zero

δF [ρ]δρ(r)

= 0 (902)

for all r. The extremal function ρ(r) must satisfy this extremal condition for allr. Often, the extremal condition provides an integro-differential equation thatcan be used to uniquely determine ρ(r). The above condition only guaranteesthat for this particular ρ the functional F [ρ] is extremal. In order that thefunctional F [ρ] is minimized, we require that

δ2F [ρ, δρ, δρ′] > 0 (903)

for every δρ.

The second order functional derivative is defined via

δ2F [ρ, δρ, δρ′] =∫d3r

∫d3r′ δρ(r) δρ′(r′)

δ2F [ρ]δρ(r) δρ(r′)

(904)

On using the choiceδρ(r) = δ3(r − r1) (905)

for the deviation centered at r in the first derivative and the choice

δρ′(r′) = δ3(r′ − r2) (906)

when differentiating the second time, one obtains

δ2F [ρ, δρ, δρ′] =δ2F [ρ]

δρ(r1) δρ(r2)(907)

As an example, the second order functional derivative of the Coulomb energy isfound to be

δ2ECoul[ρ]δρ(r1) δρ(r2)

=e2

| r1 − r2 |(908)

349

A second example is provided by the functional derivative of the mono-nomial

Fn[ρ] =∫

d3r ρ(r)n (909)

for real φ. For this functional, the second order functional derivative has theform

δ2Fn[ρ]δρ(r1) δρ(r2)

= δ3(r1 − r2) n ( n − 1 ) ρ(r1)n−2 (910)

etc.

9.3.3 The Variational Principle

Hohenberg and Kohn defined an energy functional of the electron density

E[ρ] = F [ρ] +∫

d3r Vions(r) ρ(r) (911)

in which the energy functional F [ρ] depends on the kinetic energy T given by

T = − h2

2 m

Ne∑i=1

∇2i (912)

and the electron-electron interaction energy, Vint, given by

Vint =12

∑i 6=j

e2

| ri − rj |(913)

The functional F [ρ] can be evaluated as

F [ρ] =i=Ne∏i=1

[ ∫d3ri

]Ψ∗(r1, . . . rNe

)(T + Vint

)Ψ(r1, . . . rNe

)

(914)

The functional F [ρ] is a universal functional of ρ, as the functional F [Ψ] is auniversal functional of Ψ. Furthermore, as will be shown, the energy of theelectronic system E is given by the minimum value of the functional E[ρ] whereρ(r) is the correct ground state density of an Ne electron system associated withthe lattice potential Vions(r). In fact, E is the minimum value of E[ρ] evaluatedfor the set of functions, ρ(r), which correspond to the Ne-electron ground statedensities of arbitrary potentials. Such densities are known as V -representabledensities. Not all densities are V -representable.

350

Let ρ′(r) 6= ρ(r) be an arbitrary density associated with some many-bodywave function Ψ′ 6= Ψ, that is not the ground state of our system. The groundstate energy is defined by

E =i=Ne∏i=1

[ ∫d3ri

]Ψ∗(r1, . . . rNe

) H Ψ(r1, . . . rNe) = E[ρ]

(915)

The Rayleigh-Ritz variational principle asserts that the expectation values ofthe Hamiltonian satisfies the inequality

i=Ne∏i=1

[ ∫d3ri

]Ψ∗(r1, . . . rNe

) H Ψ(r1, . . . rNe)

≤i=Ne∏i=1

[ ∫d3ri

]Ψ′∗(r1, . . . rNe

) H Ψ′(r1, . . . rNe) (916)

and soE[ρ] ≤ E[ρ′] (917)

This establishes the minimum principle for the energy functional

δE[ρ]δρ(r)

= 0 (918)

subject to the constraint that the total number of electrons are fixed∫d3r ρ(r) = Ne (919)

The condition that ρ is V -representable may be replaced by a less stringentcondition of N representable103, which only requires

ρ(r) > 0∫d3r ρ(r) = Ne∫

d3r | ∇ ρ12 (r) |2 < ∞ (920)

Having established the existence of the variational function, the precise form ofthe functional E[ρ] remains to be determined.

103M. Levy, Proc. Nat. Acad. Sci. 76, 6062 (1979).

351

9.3.4 The Electrostatic Terms

Hohenberg and Kohn suggest that one should separate the long-ranged classicalCoulomb energy of the electrons from the functional F [ρ]. This term representsthe average Coulomb interaction between the electrons in the system and, there-fore, represents the Hartree terms. That is, the energy functional representingthe kinetic and electron-electron interaction energies is written as

F [ρ] =e2

2

∫d3r

∫d3r′

ρ(r) ρ(r′)| r − r′ |

+ G[ρ] (921)

The total energy functional is given by

E[ρ] =∫

d3r Vions(r) ρ(r) +e2

2

∫d3r

∫d3r′

ρ(r) ρ(r′)| r − r′ |

+ G[ρ]

(922)

The electrostatic potential φes(r) is given by the sum of the potential due tothe lattice of ions and the electron-electron interaction

− | e | φes(r) = Vions(r) + e2∫

d3r′ρ(r′)

| r − r′ |(923)

This potential may be obtained directly from Poisson’s equation from the densityof the ions and electrons

− ∇2 φes(r) = 4 π | e |(Z ρions(r) − ρ(r)

)(924)

where | e | is the magnitude of the charge on the electron. The electrostaticpotential determines the chemical potential through the variational procedure.The energy functional is minimized w.r.t variations of ρ(r) subject to the con-straint that the density is normalized to Ne. This is performed by using La-grange’s method of undetermined multipliers. The method consists of construct-ing the functional Ω[ρ] as

Ω[ρ] = E[ρ] − µ

( ∫d3r ρ(r) − Ne

)(925)

Then on writing ρ′(r) = ρ(r) + λ δρ(r) and Taylor expanding in λ, one has

Ω[ρ′] = Ω[ρ] + λ

∫d3r δρ(r)

δΩ[ρ]δρ(r)

+λ2

2

∫d3r

∫d3r′ δρ(r) δρ(r′)

δ2Ω[ρ]δρ(r) δρ(r′)

+ . . . (926)

The extremal conditionδΩ[ρ]δρ(r)

= 0 (927)

352

becomesδE[ρ]δρ(r)

= µ (928)

The first order functional derivative of E is evaluated from the Taylor expansionby retaining the terms of first order in λ. The first order term in E, δE1, isevaluated as

δE1 =∫

d3r δρ(r) Vions(r)

+e2

2

∫d3r

∫d3r′

(δρ(r)

ρ(r′)| r − r′ |

+ δρ(r′)ρ(r)

| r − r′ |

)+∫

d3r δρ(r)δG[ρ]δρ(r)

(929)

On interchanging the variables of integration r and r′ in the second part of theCoulomb term and combining it with the first, one obtains

δE1 =∫

d3r δρ(r)(Vions(r) + e2

∫d3r′

ρ(r′)| r − r′ |

+δG[ρ]δρ(r)

)(930)

Since the first two terms are identified with the electrostatic potential, the func-tional derivative is given by

δE[ρ]δρ(r)

= Vions(r) + e2∫

d3r′ρ(r′)

| r − r′ |+

δG[ρ]δρ(r)

= − | e | φes(r) +δG[ρ]δρ(r)

(931)

Hence, Ω is minimized if ρ satisfies the equation

− | e | φes(r) +δG[ρ]δρ(r)

= µ (932)

For large Ne, µ is equal to the chemical potential given by

µ =∂E

∂Ne(933)

9.3.5 The Kohn-Sham Equations

The Kohn-Sham equations104 provide a formal correspondence between themany-body problem and an effective (non-interacting) one-body problem. This

104W. Kohn, L. J. Sham, Phys. Rev. 140, A 1133 (1965).

353

allows the kinetic energy term in the energy functional to be determined.

The kinetic energy functional T [ρ] can be defined via

T [ρ] =i=Ne∏i=1

[ ∫d3ri

]Ψ∗(r1, . . . rNe

) T Ψ(r1, . . . rNe) (934)

so the non-electrostatic contribution to the energy functional may be written asthe sum

G[ρ] = T [ρ] + Exc[ρ] (935)

which defines the exchange and correlation functional Exc[ρ]. The variationalprinciple for the density functional gives

− | e | φes(r) +δExc[ρ]δρ(r)

+δT [ρ]δρ(r)

= µ (936)

Thus, the quantity

− | e | φes(r) +δExc[ρ]δρ(r)

(937)

plays the role of an effective potential, Veff [ρ, r], which not only depends on r,but is also a functional of ρ. The effective potential is given by

Veff [ρ, r] = − | e | φes(r) +δExc[ρ]δρ(r)

(938)

Thus, minimizing the energy functional entails solving the equation

Veff [ρ, r] +δT [ρ]δρ(r)

= µ (939)

Formally, this is equivalent to solving for the ground state of a (non-interacting)problem with the energy functional given by

Es[ρ] = T [ρ] +∫d3r Vs(r) ρ(r) (940)

in which the electron-electron interaction terms are absent. The variationalprocedure leads to

Vs(r) +δT [ρ]δρ(r)

= µ (941)

Since the particles are non-interacting, this equation is solved by exactly findingthe single-particle wave functions φs,α which make up the single Slater determi-nant that represents the non-interacting ground state. The set of single-particlewave functions are given as the solutions of the eigenvalue equation[

− h2

2 m∇2 + Vs(r)

]φs,α(r) = Es,α φs,α(r) (942)

354

and then the electron density is given by

ρ(r) =i=Ne∑i=1

| φs,αi(r) |2 (943)

By analogy, one can find the solution of the effective one-body eigenvalue equa-tion [

− h2

2 m∇2 + Veff [ρ, r]

]φeff,α(r) = λeff,α φeff,α(r) (944)

and the electron density is given by

ρ(r) =i=Ne∑i=1

| φeff,αi(r) |2 (945)

The value of the kinetic energy functional for this effective one-body problemcan be found from the eigenvalues λeff,αi

by

T [ρ] =Ne∑i=1

λeff,αi −∫d3r Veff [ρ, r] ρ(r) (946)

Thus, one also has to minimize the sum of the effective one-body eigenvaluesNe∑i=1

λeff,αi (947)

This shows that the Kohn-Sham equations provide a method of obtaining thekinetic energy functional and also minimizes the energy functional. AlthoughKohn-Sham eigenvalues λeff,α are often used to describe electron excitation en-ergies, they have no physical meaning. In general, the method only provides theground state energy and ground state electron density. However, there is a den-sity functional analogue of Koopmans’ theorem: the eigenvalue of the highestoccupied effective single-particle level is the Fermi energy. All the non-trivialinformation about the many-body ground state is contained in the exchange andcorrelation function. This is usually approximated in an uncontrolled fashionby using the local density functional approximation.

9.3.6 The Local Density Approximation

In the Kohn-Sham equations, the remaining unknown function is the exchangeand correlation functional Exc[ρ]. This contains the information about themany-body interactions. The local density approximation is motivated by anassumption namely, that this functional can be represented as an integral overall space of a function of ρ. This assumes that the functional has no non-localterms. It may be thought that the first correction terms to the local densityfunction could be expressed by expanding it in powers of the gradient105. This105S. K. Ma and K. A. Brueckner, Phys. Rev. 165, 18 (1968), V. Sahni, J. Greunebaum

and J. P. Perdew, Phys. Rev. B 26, 4371 (1982).

355

type of expansion would be justifiable if the density ρ(r) was slowly varying inspace. The first few terms of the gradient expansion of the exchange-correlationenergy would be

Exc[ρ] =∫d3r

[Exc0(ρ(r)) + Exc2(ρ(r)) | ∇ ρ(r) |2 + . . .

](948)

where the coefficients Exc0(ρ(r)) and Exc2(ρ(r)) are ordinary functions of thedensity. The local density approximation neglects the gradient terms and usesthe same form of the exchange-correlation function Exc0(ρ(r)) as it pertains tothe free electron gas. In the free electron gas, the electron density ρ is inde-pendent of r. However, in the local density approximation, the uniform densityappearing in the expression for Exc for the uniform electron gas is replaced bya similar expression but which depend upon the local electron density.

The exchange and correlation terms from the local density approximationare taken from the free electron gas. The energy of the free electron gas iswritten as

E = Ne

[35h2 k2

F

2 m− 3

4e2

πkF − 1

π2( 1 − ln 2 )

m e4

h2 ln kF + O(1)]

(949)

where the first term is due to the kinetic energy, and the second term is the ex-change energy. The final term is the leading term in the high density expansionof the electron correlation energy, as evaluated by Gell-Mann and Brueckner106.For the free electron gas, the electrostatic interaction energy between the elec-trons and the smeared out lattice of ions cancels identically with the Hartreeterm. To obtain the exchange-correlation energy, the kinetic energy term isomitted to find

Exc = Ne

[− 3

4e2

πkF − 0.0311

m e4

h2 ln kF + O(1)]

(950)

Combining this together with the two relations

kF =(

3 π2 ρ

) 13

(951)

andNe = V ρ (952)

the exchange-correlation term can be expressed as

Exc = V ρ

[− 3

4e2

π

(3 π2 ρ

) 13

− 0.0104m e4

h2 ln ρ + O(1)]

(953)

106M. Gell-Mann and K. A. Brueckner, Phys. Rev. 106, 364 (1957).

356

The exchange-correlation energy in the local density approximation is simplygiven by

Exc[ρ] =∫

d3r ρ(r)[− 3

4e2

π

(3 π2

) 13

ρ(r)13 − 0.0104

m e4

h2 ln ρ(r) +O(1)]

(954)

Since the effective potential is given by the sum

Veff [ρ, r] = − e φ(r) +δG[ρ]δρ(r)

(955)

the local density approximation for the exchange and correlation energy func-tional contributes a term to the potential in the Kohn-Sham equations of

Vxc[ρ] = − e2

π

(3 π2 ρ(r)

) 13

− 0.0104m e4

h2 ln ρ(r) + O(1) (956)

which adds to the electrostatic potential. The first term comes from the ex-change interaction, and has the form that was originally proposed by J. C.Slater but has a different coefficient107. The higher order terms come from thecorrelation energy. In practice, the form of the exchange-correlation energy thatis used as an input to the local density approximation is a form which interpo-lates between the high-density limit and the low-density limit. One interpolationformulae which has been used is

Exc[ρ] =∫

d3r ρ(r)[− 3

4e2

π

(3 π2

) 13

ρ(r)13 − 1

6 π2( 1− ln 2 )

m e4

h2 ln ρ(r) +O(1)]

(957)

which is based on the work of Nozieres and Pines108. As the density is reduced,the electrons are expected to undergo a phase transition and form a Wignercrystal. Since the energy is expected to be a non-analytic function at the phasetransition, the interpolation is of doubtful utility. It seems more appropriate touse the results of Monte Carlo calculations109 for the correlation energy of thehomogenous electron gas.

The local density functional approximation has been used to successfully de-scribe many different materials, and fails miserably for some others. Attemptsto justify this expression based on the gradient expansion have failed. Basi-cally, the electron density varies too rapidly for the gradient expansion to beuseful. However, despite the rapid varying charge densities, a generalized gra-dient expansion110, in which appropriately chosen cut-off’s are introduced, has107J. C. Slater, Phys. Rev. 81, 385 (1951).108P. Nozieres and D. Pines, Phys. Rev. 111, 442 (1958).109D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566 (1980).110D. C. Langreth and M. J. Mehl, Phys. Rev. B 28, 1809 (1983), J. P. Perdew and Y.

Wang, Phys. Rev. B, 33, 8800 (1986).

357

Interpolated Total Energy

-0.3

-0.2

-0.1

0

0.1

0 2 4 6 8

rs

E(r

s)/N

e [ R

ydbe

rgs ]

Figure 188: The density dependence of the total energy for a uniform electrongas, as suggested by Nozieres and Pines (1958).

produced some encouraging results.

358

9.4 Static Screening

The response of an electronic system to a static or time-independent externalpotential is quite remarkable in a metal. In a metal, the long-ranged part of thestatic external potential is completely screened out by the electron response.The screening is characterized by the dielectric constant. Classically, the totalelectrostatic potential φes(r) is related to the charge density through Poisson’sequation. In the absence of the external potential, Poisson’s equation is writtenas

− ∇2 φes(r) = 4 π | e |(Z ρions(r) − ρ(r)

)(958)

For a free electron gas, the charge density for the electrons exactly cancels thecontributions from the smeared out charges of the ions. The correspondingpotential is constant, and the reference value φes(r) may be set to be zero.It is expected that a positive external charge with density ρext(r) will inducea change in the electronic density ρind(r). The external charge produces anexternal potential which is defined by the Poisson equation

− ∇2 φext(r) = 4 π | e | ρext(r) (959)

The total potential φes(r) satisfies Poisson’s equation

− ∇2 φes(r) = 4 π | e |(ρext(r) − ρind(r)

)(960)

where ρext is assumed to be a positive charge, and the induced electron densityρind is associated with the negatively charged electrons. The external potentialis related to the total potential via the dielectric constant through the non-localrelation

φext(r) =∫

V

d3r′ ε(r, r′) φes(r′) (961)

In a spatially homogeneous system, the dielectric constant is translationallyinvariant and, therefore, only depends upon the difference r − r′. In this case,the linear response relation is expressed as a convolution

φext(r) =∫

V

d3r′ ε(r − r′) φes(r′) (962)

This non-local relation, which is valid for homogeneous systems, is simpler afterit has been Fourier transformed. The Fourier transform of φext(r) is defined by

φext(q) =1V

∫V

d3r φext(r) exp[− i q . r

](963)

and the Fourier transform of the dielectric constant is defined by

ε(q) =∫

V

d3r ε(r) exp[− i q . r

](964)

359

Hence, the Fourier transform of the convolution is just the product of the re-spective Fourier transforms. Thus, the relation becomes

φext(q) = ε(q) φes(q) (965)

Hence, the total potential is reduced by the dielectric constant

φes(q) =φext(q)ε(q)

(966)

This relation is sometimes used as the definition of the dielectric constant.

The Fourier Transforms of Poisson’s equations, given in eqn.(959) and eqn.(960),yield

q2 φext(q) = 4 π | e | ρext(q) (967)

and

q2 φes(q) = 4 π | e |(ρext(q) − ρind(q)

)(968)

On using the first equation to eliminate ρext(q) in the second, one obtains

q2 φes(q) = q2 φext(q) − 4 π | e | ρind(q) (969)

Taking the induced charge density term to the other side of the equation pro-duces

q2 φext(q) = q2 φes(q) + 4 π | e | ρind(q) (970)

The definition of the dielectric constant, ε(q), can be used to eliminate φext(q),thereby yielding the relation

ε(q) = 1 +4 π | e |q2

ρind(q)φes(q)

(971)

The total scalar potential can be expressed as a potential energy term δV (q)acting on the electrons

δV (q) = − | e | φes(q) (972)

The response function χ(q) is defined as the ratio of the induced density to thepotential, δV (q), which produces it

χ(q) =ρind(q)δV (q)

(973)

Therefore, one finds that the dielectric constant is related to the response func-tion via

ε(q) = 1 − 4 π | e |2

q2χ(q) (974)

Thus, the dielectric constant is related to the response of the charge densityto the total potential. This response function can be calculated via different

360

techniques. However, in making approximations, it is imperative that only theresponse to the total potential is approximated and not the response to theexternal potential. In a metal, it is all the electrons that take part in screeningan external charge. If each electron were to react independently to screen theexternal charge, the external charge density would be over-screened by a factorof Ne as each electron by itself could neutralize a charge of | e |. The simplestapproximate theory of the system’s response to the total field is given by theThomas-Fermi approximation. The Thomas-Fermi theory pre-dates linear re-sponse theory and density functional theory. A more accurate approximationfor weak potentials is based on linear response theory.

The above derivation has the following drawbacks: First, the use of Poisson’sequation only treats the classical direct Coulomb interactions between aggre-gates of electrons, neglecting the effect of the exchange interactions. Second,the assumption of spatial homogeneity neglects the effect of Umklapp interac-tions in a solid. The neglect of Umklapp interactions produced simple algebraicequations relating φ(q) and ρind(q). The inclusion of Umklapp scattering pro-duces an infinite set of coupled equations which has no known analytic solution.

9.4.1 The Thomas-Fermi Approximation

The Thomas-Fermi approximation111 is based on the assumption that the po-tential is slowly varying. Hence, the energy of a Bloch state is given by

h2 k2

2 m− | e | φes(r) (975)

The momentum of the highest occupied energy is r dependent and is defined tobe kF (r). Then kF (r) is given by

h2 k2F (r)

2 m− | e | φes(r) = µ (976)

The above equation shows that kF (r) is slowly varying if φes(r) is slowly vary-ing. Since by assumption kF (r) is slowly varying, the electron density ρ(r) atposition r can be expressed in terms of the local Fermi wave vector

ρ(r) = 21

8 π3

4 π k3F (r)

3(977)

On expressing the Fermi wave vector in terms of the chemical potential and theelectrostatic potential, the total density becomes

ρ(r) =1

3 π2

(2 mh2

) 32(

µ + | e | φes(r)) 3

2

(978)

111L. H. Thomas, Proc. Cambridge Philos. Soc. 23, 542 (1927), E. Fermi, Zeit. fur Physik,48, 73 (1928).

361

r

E

µ = h2kF(r)2/2m − e φes(r)

E0 - e φes(r)

0

Figure 189: The schematic spatial variation of the one-electron energies, asenvisaged in the Thomas-Fermi approximation.

The induced density is then given in terms of the electrostatic potential via

ρind(r) =1

3 π2

(2 mh2

) 32[ (

µ + | e | φes(r)) 3

2

−(µ

) 32]

(979)

The above equation forms the basis of the Thomas-Fermi Theory. On assumingthat φes(r) is small compared with µ, the equation can be linearized to yield

ρind(r) = | e |(∂ρ0

∂µ

)φes(r) (980)

Thus, the Thomas-Fermi response function is given by

χTF = −(∂ρ0

∂µ

)= − 1

2 π2

(2 mh2

) 32

µ12

= − m kF

π2 h2 (981)

This leads to the Thomas-Fermi approximation for the dielectric constant

ε(q) = 1 +4 π e2

q2

(∂ρ0

∂µ

)= 1 +

k2TF

q2(982)

The Thomas-Fermi wave vector is given in terms of the Fermi wave vector by

k2TF =

4 m e2

π h2 kF

=4π

kF

a0(983)

362

and by the alternate expressions

kTF

kF=

√4

π kF a0

=(

163 π2

) 13

r12s

= 0.8145 r12s (984)

Thus, kTF is of the order of kF in a metal, and depends on the density ofmobile electrons available to perform screening. This means that the externalpotential or charge is screened over distances of the order of k−1

TF ∼ 1 Angstrom.

The spatial dependence of the screened potential can be most clearly seenthrough the application of the Thomas-Fermi approximation to the screeningof a point charge Z e in a metal. The charged particle is located at the origin.From the Fourier transform of Poisson’s equation, the external potential is givenby

φext(q) =4 π Z | e |

q2(985)

The total potential is given by

φes(q) =φext(q)ε(q)

=4 π Z | e |q2 + k2

TF

(986)

which no longer shows the divergence associated with a long-ranged interactionwhen q → 0. On performing the inverse Fourier transform, thereby transform-ing the potential back into direct space, one has

φes(r) =Z | e |r

exp[− kTF r

](987)

Thus, the charged impurity is exponentially screened over a distance k−1TF . The

induced charge density is given by

ρind(r) =Z e2

r

(∂ρ0

∂µ

)exp

[− kTF r

]=

Z

r

(k2

TF

4 π

)exp

[− kTF r

](988)

On integrating this over all space, one finds that the screening in a metal isperfect in that the total number of electrons in the induced density is equal toZ.

363

r

φ es(r

)Z e/r exp[-kTFr]

Z e/r

0kTF

-1

Figure 190: A comparison of the spatial dependence of the Thomas-Fermiscreened and the unscreened electrostatic potential.

The Thomas-Fermi approximation is deficient. It over-screens negative pointcharges in the free electron gas. That is, the Thomas-Fermi approximationproduces an unphysical negative total electron density at the position of thepoint charge. Likewise, for isolated atoms, it can be shown that the Thomas-Fermi approximation breaks down as it predicts that the electron density atthe nuclear position is infinite (L. D. Landau and E. M. Lifshitz, QuantumMechanics), i.e.,

limr → 0

ρ(r) ∼ r−32 (989)

The Thomas-Fermi approximation cannot describe negative ions. That is, inthe Thomas-Fermi approximation, the number of electrons must always be lessthan the nuclear charge. Furthermore, the Thomas-Fermi method also precludesthe binding of neutral atoms into molecules112. The Thomas-Fermi method isdeficient as it assumes that the potential is slowly varying in space comparedto the distance over which the electrons adjust to the potential. Therefore, theThomas-Fermi method assumes that a local approximation for the kinetic en-ergy is valid. This is not the case for most simple metals, where the potentialdue to the ions varies over distances of the order of Angstroms. However, theThomas-Fermi theory is valid113 in the limit Z → ∞, and provides a usefuldescription of the bulk of the atom for finite Z.

9.4.2 Linear Response Theory

Linear response theory describes the response of a system to a weak perturbingpotential. In such cases, the response is assumed to be approximately linearin the perturbation, so first-order perturbation theory may be used. The effect112E. Teller, Rev. Mod. Phys. 34, 627 (1962), see also E. H. Lieb, Rev. Mod. Phys. 48,

553 (1976).113E. H. Lieb and B. Simon, J. Chem. Phys. 61, 735 (1977).

364

-0.5

0

0.5

1

1.5

r/a0

ρ(r)/ ρ

0

Figure 191: The spatial dependence of the total electron density near a negativepoint charge, as calculated in the Thomas-Fermi approximation. The electrondensity is unphysical near the origin, since it becomes negative.

of a perturbing potential δV (r) on the electronic system is considered. Theeffect of this one-body potential on the single-particle Bloch functions φn,k(r),is examined via perturbation theory. To first order in the perturbation, theone-electron eigenfunctions are altered. The one-electron eigenfunctions are nolonger Bloch functions, but are approximated by the expressions

ψn,k(r) = φn,k(r) +∑

n′,k′ 6=n,k

Mn′,k′;n,k

En,k − En′,k′φn′,k′(r) (990)

which include the first-order corrections from the perturbation series expan-sion. In the above equation, Mn′,k′;n,k is the matrix element of the perturbingpotential between two Bloch functions,

Mn′,k′;n,k =∫

d3r′ φ∗n′,k′(r′) δV (r′) φn,k(r′) (991)

The induced change in the electron density, to first order in δV (r), is found as

ρind(r) =∑n,k,σ

[ ∑n′,k′ 6=n,k

φ∗n,k(r)Mn′,k′;n,k

En,k − En′,k′φn′,k′(r) + c.c.

](992)

where the summation over n, k runs over all the occupied states and c.c. denotesthe complex conjugated term. Thus, the induced charge density at position ris not just related locally to the perturbation at the same point, but instead isrelated to δV (r′) applied at all the points r′. The induced charge density ρind(r)is expressed as the response to the perturbation δV (r′) through the non-localrelation

ρind(r) =∫

d3r′ χ(r, r′) δV (r′) (993)

365

where the response function χ(r, r′) is given by the expression

χ(r, r′) =∑n,k,σ

[φ∗n,k(r) φn,k(r′)

∑n′,k′ 6=n,k

φ∗n′,k′(r′) φn′,k′(r)

En,k − En′,k′+ c.c.

](994)

In the above expression for χ(r, r′), the summation over n, k, σ runs over allone-electron states that were occupied before the perturbation was turned on.Due to the Pauli exclusion principle, the summation over n′, k′ is restricted tothe unoccupied states. The expression for the response is expected to be modi-fied by the presence of electron-electron interactions.

The expression for the non-interacting response can easily be evaluated forfree electrons. First, the variables k′ and k are interchanged in the complexconjugate term, and then, due to a cancellation between the two terms, therange of one integration in each term is extended over all momentum space.Once again, the variables k′ and k are interchanged in the second term, to yield

χ(r, r′) =4 mh2

∫|k|≤kF

d3k

( 2 π )3

∫d3k′

( 2 π )3

[exp

[i ( k′ − k ) . ( r − r′ )

]

+ c.c.

] (1

k2 − k′2

)(995)

where the integration over k′ now runs over all space. As the Hamiltonianpossesses translational invariance, the response function only depends on thevector R = r − r′. Thus, for the homogeneous electron gas, the real spacelinear response relation is in the form of a convolution. The integrations over thedirections of k and k′ can be evaluated by standard means. The integration canbe transformed so that the integration over the magnitude of k′ extends overthe range (∞,−∞). The resulting integral is evaluated by means of contourintegration, and yields

χ(r, r′) = − 2 mh2

2( 2 π )3

∫ kF

0

dk ksin 2 k | r − r′ || r − r′ |2

(996)

The resulting expression is

χ(r, r′) =2 mh2

1π3

k4F

[cos 2 kF | r − r′ |( 2 kF | r − r′ | )3

− sin 2 kF | r − r′ |( 2 kF | r − r′ | )4

](997)

This is the response to a delta function perturbation at the origin. This deltafunction perturbation requires the electron gas to adjust at very short wave

366

lengths. Instead of having the exponential decay as predicted by the Thomas-Fermi approximation, the response only decays algebraically, with characteristicoscillations determined by the wave vector 2 kF due to the sharp cut off at theFermi surface. That is, 2 kF is the largest wave vector available for a zero-energydensity fluctuation in which an electron is excited from just below to just abovethe Fermi surface. The oscillations in the density that occur in response to apotential are known as Friedel oscillations114.

It is more convenient to consider the Fourier transform of the response func-tion

χ(q) =∫

V

d3r χ(r) exp[− i q . r

](998)

The response function χ(q) is evaluated from

χ(q) = 22 mh2

1V

∑k<kF

[1

k2 − ( k + q )2+

1k2 − ( k − q )2

](999)

where the summation over k runs over the occupied states within the Fermisphere. The summation can be replaced by an integration

χ(q) = − 22 mh2

14 π2

∫ kF

0

dk k2

∫ +1

−1

d cos θ[

1q2 + 2 k q cos θ

+1

q2 − 2 k q cos θ

]= − 2

2 mh2 q

14 π2

∫ kF

0

dk k ln| q + 2 k || q − 2 k |

(1000)

The response is given explicitly by

χ(q) = −(m kF

h2 π2

) [12

+4 k2

F − q2

8 q kFln| 2 kF + q || 2 kF − q |

](1001)

This is the Lindhard function115 for the free electron gas. The Lindhard functionreduces to the value of the corresponding Thomas-Fermi response function atq = 0, which is

χTF = − k2TF

4 π e2(1002)

Thus, for very slowly varying potentials, the response of the free electron gasis identical to the response function found using the Thomas-Fermi approxima-tion. The magnitude of the Lindhard function drops with increasing q, fallingto half the q = 0 value at q = 2 kF . At this point, the slope has a weak114J. Friedel, Phil. Mag. 43, 153 (1952), Nuovo Cimento, Suppl. 7, 287 (1958).115J. Lindhard, Kgl. Danske Videnskab. Selskab. Mat.-Fys. Medd. 28, 8 (1954).

367

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2

q/2kF

χ χχχ(q)

/ χ χχχT

F

Figure 192: The dependence of the Lindhard function on q/2kF .

logarithmic singularity. The electron gas is ineffective in screening the appliedpotential for q ≥ 2 kF as 2 kF corresponds to the largest wave vector at whichelectrons on the spherical Fermi surface can readjust.

The electron density induced by a point external charge of strength Z canbe written as

ρind(r) = Z

∫d3q

(ε(q) − 1ε(q)

)exp[ i q . r ] (1003)

where we have used the relation

ε(q) = 1 − 4 π e2

q2χ(q) (1004)

to simplify the numerator. For short distances, the induced electron densityshows an exponential decay similar to that found in the Thomas-Fermi approx-imation. For large distances, the electron density exhibits Friedel oscillationsand varies as

ρind(r) ∼ 4 Z k3F

π

k2TF k2

F

( k2TF + 2 ( 2 kF )2 )2

cos 2kF r

k3F r3

(1005)

This expression was first derived by Langer and Vosko116.

9.4.3 Density Functional Response Function

The change in the electron density ρind(r) due to an external potential, φext(r),in which electron-electron interactions are included can be obtained from density116J. S. Langer and S. H. Vosko, J. Phys. Chem. Solids, 12, 196 (1960).

368

0

0.05

0.1

0.15

0.2

0 1 2 3 4

kF r

ρ ind

(r)/(

ZkF3 )

rs=4

rs=3

Figure 193: The short-distance variation of the electron density induced toscreen a point charge of strength Z. The values of rs = 4 and rs = 3 have beenused in these calculations. [After Langer and Vosko (1960).]

-0.002

0

0.002

0.004

2 3 4 5 6 7 8 9 10

kF r

ρ ind

(r)/(

ZkF3 )

rs=3

rs=4

Figure 194: The Friedel oscillations in the induced electron density around apoint charge of strength Z. [After Langer and Vosko (1960).]

369

functional theory. The relation between the induced density and the externalpotential is given in terms of the screened response function by

ρind(r) = −∫

d3r′ χs(r − r′) | e | φext(r′) (1006)

Density functional theory yields an effective potential which contains the effectof the electron-electron interactions

| e | φeff (r) = | e | φext(r) −∫

d3r′ ρind(r′)

[| e |2

| r − r′ |+

δ2Exc

δρ(r) δρ(r′)

](1007)

The relation between the induced electron density and the effective potential isgiven by

ρind(r) = −∫

d3r′ χ0(r − r′) | e | φeff (r′) (1008)

where χ0(r−r′) is the Lindhard response function for non-interacting electrons.

The response function, including the effects of the electron-electron interac-tions, can be found by Fourier transforming the above set of equations. Thus,the full response function is given by

ρind(q) = − χs(q) | e | φext(q) (1009)

and the non-interacting response function is given by

ρind(q) = − χ0(q) | e | φeff (q) (1010)

The relationship between the effective and external potential is given by

φeff (q) = φext(q) +(− 4 π | e |

q2+

π | e |k2

TF

Γxc(q))ρind(q) (1011)

This equation can be solved for χs(q) in terms of the non-interacting responsefunction χ0(q).

χs(q) =χ0(q)

1 − | e |2(

4 πq2 − π

k2T F

Γxc(q))χ0(q)

(1012)

The dielectric constant ε(q) is given by

1ε(q)

=φes(q)φext(q)

1ε(q)

= 1 − 4 π | e |q2

ρind(q)φext(q)

1ε(q)

= 1 +4 π e2

q2χs(q) (1013)

370

The exchange contribution to Γxc(q) is given in the limit q → 0 by

Γxc(q) =[

1 +59

(q

2 kF

)2

+73225

(q

2 kF

)4

+ . . .

](1014)

It is noted that if the effect of the exchange-correlation terms to the screeningcould be dropped, then the dielectric constant is approximated by

1ε(q)

≈ 1 +4 π e2

q2χ0(q)ε(q)

(1015)

which is consistent with the result for free-electrons using the Lindhard approx-imation for the response to the total field, and treating the total scalar potentialclassically via Poisson’s equation. In obtaining this approximate result, it wasnecessary to calculate the response of the system to the external potential byincluding processes, to all orders in e2, in which the electron gas is polarized.That is, the electron gas is polarized by the external potential and then the re-sulting polarization and the external potential are screened by the electron gas,ad infinitum. This infinite regression is necessary for the external charge to becompletely screened at large distances, and is a consequence of the long-rangednature of the Coulomb interaction limq → 0

4 π e2

q2 → ∞. This re-emphasizesthe importance of only making approximations in the response to the total po-tential χ and not in the response to the external potential χs.

The response of the electronic system to an applied potential can be usedto examine the stability of a structure. The electronic energy change due tothe perturbation consists of the potential energy of interaction between theions and the electron gas, as well as the change induced into the energy ofelectron-electron repulsions. All of these energies can be expressed in terms ofthe induced charge density.

——————————————————————————————————

9.4.4 Exercise 57

Calculate the Lindhard function for a free electron gas with the dispersion re-lation E

(0)k = h2 k2

2 m in d = 1, d = 2 and d = 3 dimensions, at zerotemperature.

——————————————————————————————————

371

9.4.5 Exercise 58

Consider the Lindhard function for a tight-binding non-degenerate s band on ahyper-cubic lattice with the dispersion relation

Ek = E0 − 2 ti=d∑i=1

cos ki a (1016)

Show that the response function at the corner of the Brillouin zone q =πa (1, 1, 1, ., ., .) diverges as the number of electrons in the band approaches oneper site.

——————————————————————————————————

Worked Example.

We have seen that, in the Hartree-Fock approximation, the density of statesof the uniform electron gas is pathological. We have asserted that these patholo-gies are related to the neglect of screening. In fact, we have shown that simpleperturbation theory for the unform electron gas is unworkable due to divergencescaused by the long-ranged coulomb interaction. Here we shall show that, if itis assumed that the electrons interact via a Thomas-Fermi screened coulombinteraction, the Hartree-Fock approximation is no longer pathological.

The Hartree-Fock quasi-particle energy for a screened coulomb gas is givenby

λk =h2 k2

2 m+ Vexch(k) (1017)

where the exchange energy is related to the screened coulomb interaction

Vexch(k) = −∑k′

∫d3r′

e2

| r − r′ |exp[− kTF | r − r′ | ] ψ∗k′(r′) ψk′(r) exp[ i k . ( r′ − r ) ]

(1018)where the states k′ are occupied by electrons. Since the single-particle wavefunctions ψk′(r) are of the form of plane waves, the exchange interaction hasthe form of a Fourier transform. The Fourier transform is evaluated as

Vexch(k) = − 1( 2 π )3

∫k′<kF

d3k′∫d3r′

e2

| r − r′ |× exp[ − kTF | r − r′ | ] exp[ i ( k − k′ ) . ( r′ − r ) ]

= − 1( 2 π )3

∫k′<kF

d3k′4 π e2

( k − k′ )2 + k2TF

=e2

2 π k

∫k′<kF

dk′ k′ ln∣∣∣∣ ( k − k′ )2 + k2

TF

( k + k′ )2 + k2TF

∣∣∣∣ (1019)

372

-0.3

-0.2

-0.1

0

0 1 2 3 4

k/kF

Vex

ch(k

)

Figure 195: The exchange energy of the screened electron gas as a function ofk/kF .

The result is

Vexch(k) = − e2 kF

2 π

[( k2

TF + k2F − k2 )

2 k kFln(

(k + kF )2 + k2TF

(k − kF )2 + k2TF

)+ 2

kTF

kF

(tan−1 (k − kF )

kTF− tan−1 (k + kF )

kTF

)+ 2

]This expression is a negative function which monotonically increases towardszero as k increases. Furthermore, the derivative of the Vexch(k) and the deriva-tive of the quasi-particle energy λk do not diverge logarithmically as k → kF .Hence, in the screened Hartree-Fock approximation, the quasi-particle densityof states does not exhibit pathological behavior.

——————————————————————————————————

Worked Example: The Spiral Spin Density Wave State

We shall examine a broken-symmetry Hartree-Fock solution for the screenedfree electron gas. We shall replace the Coulomb interaction by a highly-screenedThomas-Fermi interaction of arbitrary strength(

4 π e2

| k − k′ |2

)→

(4 π e2

k2TF

)(1020)

which is separable. For a separable interaction, the band shifts ∆0(k), ∆±(k)and the gap become independent of k. The relative shift ∆± is found to be zeroand the overall shift ∆0 can be absorbed in the definition of the zero of energy.For a fixed value of the gap, the conservation of electron number is ensured by

373

determining µ from the equation

k3F =

34

∑±

[ ∫ kmaxz

kminz

dkz k±⊥(kz)2

](1021)

where

h2

2mk±⊥(kz)2 = µ − h2

2 mk2

z ±

√(h2

2 m

)2

k2z Q

2 + | ∆ |2 (1022)

and the maximum and minimum values of kz are determined from the solutionsof k±⊥(kz) = 0 as either

h2

2m(kz)2 = µ +

h2Q2

8 m±

√h2 Q2

2 mµ + | ∆ |2 (1023)

or zero, depending on the value of µ [see fig(186)]. The integrals can be easilyperformed and expressed in terms of elementary functions. Likewise, the gapequation either has a trivial solution

| ∆ | = 0 (1024)

or it has a non-trivial solution found from the equation

1 =(

4 π e2

k2TF

) (1

8 π2

) ∑±

[±∫ kmax

z

kminz

dkzk±⊥(kz)2√

( h2

2 m )2 Q2 k2z + | ∆ |2

](1025)

which can also be expressed in terms of the same elementary functions. Theright-hand side of the above equation is seen to be a decreasing function of| ∆ |2, therefore, one expects that the Coulomb interaction 4 π e2

k2T F

must exceeda critical value if there is to be a non-trivial solution (| ∆ | 6= 0) of the gapequation. It is instructive to examine the ∆ → 0 limit of the above equationwhich reduces to

1 =(m e2 kF

π h2 k2TF

) [1 +

(4 k2

F − Q2

4 Q kF

)ln∣∣∣∣Q + 2 kF

Q − 2 kF

∣∣∣∣ ] (1026)

This expression confirms that there exists a minimum value of the Coulombinteraction which has to be exceeded if the system is to have a finite amplitudeof the spiral magnetization. The above expression also shows that the para-magnetic free electron gas has a preference to be unstable with respect to aferromagnetic state since the critical value of the interaction required for thesystem to respond with a magnetic spiral of wave vector Q is minimum whenQ = 0. This conclusion is expected to change when the ionic potential is takeninto account or if the material is highly anisotropic.

374

0

1

2

3

0 1 2 3 4

q/kF

χ(q)

/ χ(0

)

d=1d=2d=3

Figure 196: The normalized response function for a free electron gas as a func-tion of momentum transfer q, for different dimensionalities.

For non-zero values of ∆, the determination of the optimal magnitude of Qis more intricate. One way to determine the relative stability of the symmetry-broken states is to compare the values of their total energies E(Q) (per unitvolume). The total energy can be expressed as

E(Q) =∑k,±

λk,± fk,± +(

k2TF

4 π e2

)| ∆ |2 (1027)

From this expression, it can be shown that the value of | ∆ | which minimizesE(Q) corresponds to the solution of the gap equation. It can also be seen thatwhenever the gap equation has a non-trivial solution, the state with broken sym-metry has lower energy than the paramagnetic metal. Furthermore, it can beseen that the condensation energy is proportional to | ∆ |2. For computationalpurposes, it is more convenient to evaluate the total energy E(Q) from

E(Q) =∑±

∫ µ

−∞dE E ρ±qp(E) +

(k2

TF

4 π e2

)| ∆ |2 (1028)

in which the ρ±qp(E) are the quasi-particle density of states (per unit volume) foreach sub-band. It should be noted that the quasi-particle density of states, justlike the values of ∆ and µ, implicity depend on Q. The quasi-particle densityof states (per unit volume) for each sub-band is evaluated as

ρ±qp(E) =2 m

4 π2 h2

(kmax

z (E) − kminz (E)

)(1029)

where the non-zero maximum and minimum values of kz for fixed E are foundfrom

h2

2 mkz(E)2 = E +

h2 Q2

8 m±

√h2 Q2

2 mE + | ∆ |2 (1030)

375

The form of the total quasi-particle density of states is sketched in fig(197). The

0

0.05

0.1

0.15

0 0.5 1 1.5

8mE/h2Q2

ρ qp(

E) [

arb

itrar

y un

its ]

2∆

Figure 197: The total quasi-particle density of states for a spiral spin densitywave state.

quasi-particle density of states is extremely close to the result for a paramagneticmetal since the transition only alters states in a small energy interval with awidth of the order of ∆.

376

10 Stability of Structures

In this chapter, the structural stability of a metal is discussed. The total energyof the metal will be expressed in terms of the energy for a uniform electron gas,and the interaction with the periodic structure will be treated as a perturbation.

10.1 Momentum Space Representation

In the uniform electron gas, the electro-static energy between pairs of electronsand also between the particles forming the background positive charge exactlycancels with the interaction between the electrons and the positive charges.When the periodic potential is introduced as a perturbation, the change in thetotal energy can be expressed in terms of the change in the one-electron eigen-values. However, the inclusion of the Coulomb interaction between the latticeand the electrons will also require that the contributions from electron-electronand ion-ion interaction be explicitly reconsidered in the calculation of the totalenergy.

The energy of a one-electron Bloch state, calculated to second in the poten-tial due to the ionic lattice, can be expressed in terms of the one-electron energyeigenvalues for a free electron gas as

En,k =h2 k2

2 m+ Vions(k, k) +

2 mh2

∑k′ 6=k

| Vions(k′, k) |2

k2 − k′2(1031)

The zero-th order and first order terms in this energy are independent of thelattice structure of the ionic potential. This can be seen by examining the matrixelements

Vions(k′, k) =1V

∫d3r Vions(r) exp

[i ( k − k′ ) . r

](1032)

which, when k = k′, is just the average potential. The sum of the energiesof all the occupied Bloch states, (k, σ), contributes to the total energy of thesolid. The first order contribution from Vions(k, k), like the kinetic energy ofthe free electron gas, does not depend on the structure. These terms combineto produce a volume-dependent contribution to the solid’s total energy.

The other volume-dependent contribution to the total energy of the solidoriginates from the electron-electron interactions and the ion-ion interactions.It is convenient to combine these terms with the energy of the zero-th orderelectron-ion interaction, due to the exact cancellation for the uniform electrongas. This combination is the total electrostatic interaction energy. It can beevaluated in the approximation that the Coulomb interactions between different

377

Wigner-Seitz cells are totally screened117. This means that the ion-ion inter-actions need not be considered explicitly. The electrostatic contribution to theenergy is then written as

Ees =∫

d3r Vions(r) ρ(r) +e2

2

∫d3r

∫d3r′

ρ(r) ρ(r′)| r − r′ |

(1033)

To lowest order in the structure, the electrostatic contribution to the totalenergy can be evaluated by considering the Wigner-Seitz unit cell to be sphericalwith radius RWS . The electron density is given in terms of the Wigner-Seitzradius118 by

ρ =3 Z

4 π R3WS

(1034)

In the case of a uniform electron density, the electron-electron repulsion term isevaluated as

Ees =∫

d3r Vions(r) ρ(r) +35Z2 e2

RWS(1035)

For the free-electron approximation for the kinetic energy to be reasonable, theelectrostatic contribution from the ions should be calculated using the pseudo-potential. We shall use the Ashcroft empty core approximation for the ionicpseudo-potential. Inside the Wigner-Seitz cell, the pseudo-potential reduces tothat of an isolated atom

Vatom(r) = − Z e2

rfor r ≥ Rc

= 0 for r ≤ Rc (1036)

where Rc is the radius of the ionic core. Hence, for a structureless metal, theelectrostatic terms can be expressed as

Ees = − 32Z2 e2

RWS

[1 −

(Rc

RWS

)2]

+35Z2 e2

RWS(1037)

The potential terms inversely proportional to the Wigner-Seitz radius can becombined as − 9

10Z2 e2

RW S. The coefficient α = 9

10 is the Madelung constant fora solid composed of spherical unit cells. In general, the Madelung constant willdepend slightly on the structure of the lattice.

117E. Wigner and F. Seitz, Phys. Rev. 43, 804 (1933), E. Wigner and F. Seitz, Phys. Rev.45, 509 (1934).118The Wigner-Seitz cell radius is related to the electron density parameter rs via

RWS = Z13 rs a0

.

378

For a solid with structure, the electrostatic energy can be expressed as thesum

E = EM + Ec (1038)

where EM is the Madelung energy and Ec is the core energy. The Madelungenergy is the electrostatic energy due to point charges immersed in a neutralizinguniform distribution of electrons. The Madelung energy is given by

EM = − αZ2 e2

RWS(1039)

where α is the structure-dependent Madelung constant. The Madelung con-stants of various structures are evaluated as

Structure α

b.c.c. 0.89593f.c.c. 0.89587h.c.p. 0.89584

simple hexagonal 0.88732simple cubic 0.88006

The Madelung energy is seen to increase as the symmetry is lowered. Theremaining contribution to the electrostatic energy is defined to be the coreenergy. The core energy is given by

Ec =32Z2 e2

RWS

(Rc

RWS

)2

(1040)

and, as it is the electrostatic energy associated with the spherical pseudo-potential core, it is not dependent on the solid’s structure.

The largest structural-dependent contribution to the energy originates fromthe second order terms of the Bloch energies in the electron-ion interaction

E(2)n,k =

2 mh2

∑k′ 6=k

| Vions(k′, k) |2

k2 − k′2(1041)

On summing over all the occupied Bloch states ( | k | < kF ) and both spinvalues σ, one obtains a contribution E2 to the total energy of

E2 =2 mh2

∑|k|<kF ,σ

∑k′ 6=k

| Vions(k′, k) |2

k2 − k′2(1042)

379

In the free electron basis, the matrix elements of the electron-ion interaction,Vions(k′, k), only depends on the momentum difference q = k′ − k.

Vions(k′, k) =1V

∫d3r Vions(r) exp

[i ( k − k′ ) . r

](1043)

The potential due to the lattice can be written as the sum of the individualpotentials from the atoms. The basis position of the j-th atom in the unit cellis denoted by rj and the Bravais lattice vector is denoted by Ri. Thus, thepotential for the lattice of ions is given by

Vions(r) =∑i,j

Vj(r −Ri − rj) (1044)

The matrix elements are then given by

Vions(q) =1V

∫d3r

∑i,j

Vj(r −Ri − rj) exp[− i q . r

]

=1V

∑i,j

∫d3r exp

[− i q . Ri

]Vj(r −Ri − rj) exp

[− i q . ( r −Ri )

](1045)

This can be expressed as

Vions(q) =1V

∑i

exp[− i q . Ri

] ∑j

exp[− i q . rj

]Vj(q)

(1046)

where Vj(q) is related to the Fourier transform of the potential from the j-thatom of the basis

Vj(q) =∫

d3r Vj(r) exp[− i q . r

](1047)

For simplicity, a crystal with a mono-atomic basis is considered. The matrixelements are only non-zero when q is a reciprocal lattice vector Q. The matrixcan be expressed in terms of the structure factor S(Q), via

Vions(Q) =N

VS(Q) V0(Q) (1048)

The structure dependence of the total electronic energy is contained in thesecond order contribution

E2 =N2

V 2

∑k<kF ,σ

2 mh2

∑Q6=0

| S(Q) |2 | V0(Q) |2

k2 − ( k + Q )2(1049)

380

where the sum over k, σ runs over the occupied states (k < kF ), and the termwith Q = 0 is omitted. On interchanging the order of the summations over kand Q, one finds that the second order term can be expressed in terms of theLindhard function χ(q),

E2 =N2

V 2

∑Q6=0

| S(Q) |2 | V0(Q) |2∑k,σ

fk

Ek − Ek+Q

=12N2

V 2

∑Q6=0

| S(Q) |2 | V0(Q) |2∑k,σ

fk − fk+Q

Ek − Ek+Q

=12N2

V

∑Q6=0

| S(Q) |2 | V0(Q) |2 χ(Q) (1050)

The summation over q is limited to the set of non-zero reciprocal lattice vectorsQ (including their multiplicity). For convenience sake, the first few non-zerovalues of the structure factor and the associated reciprocal lattice vectors for anideal hexagonal close-packed crystal are tabulated below.

Q (100) (002) (101) (102) (110) (103) (200) (112) (004)

(Q a2 π

)1.1545 1.225 1.307 1.683 2 2.170 2.309 2.345 2.449

|S(Q)|2 1 4 3 1 4 3 1 4 4

From this we see that, the second-order energy E2 depends on the lattice struc-ture through the number of equivalent reciprocal lattice vectors, the structurefactors | S(Q) |2, and on the electron density through the factors χ(q), and thenature of the ions through V0(q). The latter is often expressed in terms of theThomas-Fermi screened pseudo-potential

V0(q) = − 4 π Z e2cos q Rc

q2 + k2TF

(1051)

where Rc is the radius of the ionic core. The potential has a node at q0 Rc = π2 .

The structural part of the electronic energy depends sensitively on the positionof the node q0 with respect to the smallest reciprocal lattice vectors Q. Recip-rocal lattice vectors close to a node q0 contribute little to the cohesive energy.The system may lower its structural energy, if Q moves away from q0 withoutcausing a change in the volume-dependent contribution to the energy. Recip-rocal lattice vectors greater than 2 kF contribute little to the cohesive energy,

381

Z Z

Z Z

Z Z

4πe2/q2

4πe2/q2 4πe2/q2

4πe2/q2 4πe2/q2 4πe2/q2

χ(q)

χ(q) χ(q)

k+q

k

Figure 198: A diagramatic depiction of the first few terms in the (R.P.A.) ex-pression for the screened Coulomb interaction between two charges Z emmersedin an electron gas. The (red) dashed lines represent the instantaneous Coulombinteraction between the charges 4π

q2 and the (blue) bubbles represent the polar-ization induced in the electron gas χ(q). After summing the infinite number ofterms in this series, the resulting effective interaction is 4 π Z2 e2

q2 ε(q)

since the response of the electron gas is negligible.

In addition to these terms, there is a structural contribution arising fromthe electron-electron interactions which comes from the induced change in theelectron density

E2 es = − 12

∑q

ρ∗ind(q)4 π e2 V

q2ρind(q) (1052)

This term occurs since the effect of electron-electron interactions have beendouble counted. On noting that the ionic potential only has non-zero Fouriercomponents at q = Q, and that

ρind(Q) = χ(Q) Vions(Q) (1053)

one can combine the energy contribution represented by eqn(1052) with thecontribution from the Bloch energies. The factor 4 π e2

q2 χ(q) is related to thedielectric constant ε(q) through

ε(q) = 1 − 4 π e2

q2χ(q) (1054)

The two second order terms can be combined to yield the dominant contributionto the structural energy

Estructural =12N2

V

∑Q6=0

| S(Q) |2 | V0(Q) |2 χ(Q) ε(Q) (1055)

382

Since both pseudo-potential terms V0(Q) include screening, the explicit factorof ε(q) cancels with one factor of ε(q) in the denominators. Thus, the structuralenergy is only screened by one factor of the dielectric constant. The magni-tude of the structural energy is quite small. The maximum magnitude of thepseudo-potential is Z e2

Rcwhich may be as small as 1

2 eV. The magnitude of χis given by the inverse of the Fermi energy which is typically 5 eV. Thus, thestructural energy is of the order of milli-Rydbergs. Since the structure factorvanishes unless q = Q, the structural energy depends on the screened potentialonly at the reciprocal lattice vectors. Note that the pseudo-potential containsa node at the wave vector q0 = π

2 Rc. The structural energy is composed of

negative contributions, but the contributions from the reciprocal lattice vectorswhich are close to the node, contribute little to the stability of the structure. Infact, reciprocal lattice vectors at the node would correspond to the special casein which the band gaps at the appropriate Brillouin zone boundaries are zero.Usually, the opening of a band gap at a Brillouin zone boundary in a conductionband can result in an increased stability of the structure. The electronic statesbelow the “band gap” are depressed and, if occupied, result in a lowering of thesolid’s energy. However, the states above the “band gap,” if empty, are raisedbut don’t contribute to the solid’s energy.

Al is f.c.c. and the reciprocal lattice vectors (1, 1, 1) and (2, 0, 0) are bothlarger than q0. On moving down the column of the periodic table from Al toGa and then In, the ratios of Q/q0 are reduced.

Al Ga In

Q(1, 1, 1)/q0 1.04 0.94 0.93

Q(2, 0, 0)/q0 1.20 1.09 1.08

As the magnitude Q of the six equivalent (2, 0, 0) reciprocal lattice vectorsapproaches q0 in Ga, there is a loss in structural stability and the series un-dergoes a transition from the f.c.c. to a tetragonal structure119. When thistransition occurs, the set of equivalent f.c.c. reciprocal lattice vectors that haveQq0

∼ 1 are split. In the tetragonal structure, as the structure is sheared, thereciprocal lattice vectors undergo different changes. Some values of Q

q0move to

higher values while others move to lower values. This type of transformation119V. Heine and D. L. Weaire, Solid State Physics, 24, 1 (1970).

383

leaves the atomic volume unchanged, but as all the “band gaps” V (Q) increase,the transition lowers the energy of the structure. This structural transition oc-curs when the lowering of the electronic energy outweighs the increase in theMadelung energy.

On proceeding further down the group III column, from In to T l, the re-ciprocal lattice vectors pass through the nodes of the pseudo-potential. As aconsequence of this progression, In has a less distorted structure and T l has aclose-packed structure.

10.2 Real Space Representation

The dominant electronic structural energy is given by a sum over all Q of theform

Estructural =12N2

V

∑Q6=0

| S(Q) |2 | V0(Q) |2 χ(Q) ε(Q) (1056)

where S(Q) is the structure factor evaluated at a reciprocal lattice vector. Thiscan be written as a sum over all vectors q, by using the Laue identity

N2∑Q

δq,Q =∑

R 6=R′

exp[i q . ( R − R′ )

](1057)

where R and R′ are Bravais lattice vectors. The structural energy then takesthe form

Estructural =1

2 V

∑q 6=0

∑R 6=R′

exp[i q . ( R − R′ )

]| S(q) |2 θ(q) (1058)

where θ(q) is defined to be

θ(q) =1V| V0(q) |2 χ(q) ε(q) (1059)

It should be noted that in this approximation, θ(q) is independent of the direc-tion of q. The product of the structure factors can be written as

| S(q) |2 =∑i 6=j

exp[i q . ( ri − rj )

](1060)

Thus, on denoting the position of the atoms by Rj = R + rj , one has

Estructural =12

∑i 6=j

∑q 6=0

exp[i q . ( Ri − Rj )

]θ(q) (1061)

384

-0.1

-0.08

-0.06

-0.04

-0.02

0

0 0.5 1 1.5 2 2.5q/2kF

θ(q)

[Ryd

berg

s]

(1,1,0)

(2,0,0)(2,1,1)(2,2,0)

(2,0,0)(1,1,1) (2,2,0) (3,1,1)

fcc

bcc

Li

(2,2,2)

Figure 199: The variation of θ(q) with q, as calculated for Li.

The Fourier transform of θ(q) is defined as

θ(Ri,j) =∑

q

θ(q) exp[i q . Ri,j

](1062)

where the vector Ri,j denotes the relative position of the two atoms. On chang-ing the sum over q into an integration, θ(Ri,j) is evaluated as

θ(Ri,j) =V

( 2 π )3

∫d3q θ(q) exp

[i q . Ri,j

]=

2 π V( 2 π )3

∫ ∞

0

dq q2∫ 1

−1

d cos θ θ(q) exp[i q Ri,j cos θ

]

=V

( 2 π )2

∫ ∞

0

dq q2 θ(q)

( exp[i q Ri,j

]− exp

[− i q Ri,j

]i q Ri,j

)

=V

2 π2

∫ ∞

0

dq q2 θ(q)(

sin q Ri,j

q Ri,j

)(1063)

Thus, the electronic contribution to the structural energy has the real spacerepresentation

Estructural =12

∑i 6=j

θ(Ri,j) (1064)

385

The Madelung energy, which is the sum over the interaction energies of the ions

EMadelung =12

∑i 6=j

Z2 e2

| Ri,j |(1065)

should also be added to the structural energy. Thus, the total structural energycan be expressed in terms of the sum of pair-potentials Θ(R), where

Θ(R) =Z2 e2

R+ θ(R) (1066)

The pair-potential represents the interaction between a pair of bare ions in thesolid plus the effect of the screening clouds. The pair-potential does not de-scribe the volume dependence of the energy of the solid, but only the structure-dependent contribution to the energy. The pair-potential can be expressed as

Θ(R) =Z2 e2

R+

V

( 2 π2 )

∫ ∞

0

dq q2(

sin q Rq R

)θ(q) (1067)

The first and second term can be combined to yield the interaction between anion and a screened ion. This can be seen by expressing the potential in termsof a dimensionless function V (q) defined by

V0(q) =4 π Z e2

q2 ε(q)V (q) (1068)

Thus, the interaction can be expressed as

Θ(R) =Z2 e2

R

[1 +

2π

∫ ∞

0

dq

(sin q Rq

)4 π e2

q2χ(q)ε(q)

| V (q) |2]

=Z2 e2

R

[1 +

2π

∫ ∞

0

dq

(sin q Rq

)1 − ε(q)ε(q)

| V (q) |2]

=Z2 e2

R

[1 − 2

π

∫ ∞

0

dq

(sin q Rq

)| V (q) |2

]

+Z2 e2

R

[2π

∫ ∞

0

dq

(sin q Rq

)1ε(q)

| V (q) |2]

(1069)

The integral in the first term of the last line can be evaluated with the calculusof residues, and is evaluated in terms of the pole at q = 0. Since V (0) = 1and as R > Rc, the integral is equal to unity. Therefore, the first term cancelsidentically. Hence, the interaction energy between a bare ion and a screened ionis given by the expression

Θ(R) =Z2 e2

R

[2π

∫ ∞

0

dq

(sin q Rq

)1ε(q)

| V (q) |2]

(1070)

386

The long-ranged nature of the Coulomb interaction between the bare ions hasbeen completely eliminated due to the screening120. The repulsive ion-ion in-teraction is reduced to a repulsive core of linear dimension 2 ( Rc + k−1

TF ).The branch cut at q = 2 kF of the logarithmic term in the dielectric constantleads to Friedel oscillations in the potential at asymptotically large distances R

Θ(R) = Acos 2 kF R

( 2 kF R )3(1071)

whereA ∝ cos2 2 kF Rc (1072)

However, at intermediate distances, the pair-potential can be approximatelyexpressed as the sum of three (damped) oscillatory terms121

Θ(R) =Z2 e2

R

3∑n=1

Bn cos(αn 2 kF R + φn

)exp

[− βn 2 kF R

](1073)

where the phase shift depends on the ionic core radius Rc and the electrondensity (through rs). This form is obtained as a result of approximating theLindhard function χ(q) by a ratio of polynomials (Pade approximation). The in-tegration over q can be performed via contour integration. The pairs of complexpoles in the integrand produces terms which have damped oscillatory dependen-cies on R. The fit parameters for Na are given by:

Na

n 1 2 3

αn 0.291 0.715 0.958βn 0.897 0.641 0.271Bn 1.961 0.806 0.023φn

π 1.706 1.250 1.005

while for Mg the interaction is specified by

120The detailed form of the pair potential is very sensitive to the approximation used in thecalculation of the dielectric function.121D. G. Pettifor and M. A. Ward, Solid. State. Commun. 49, 291 (1984).

387

Mg

n 1 2 3

αn 0.224 0.664 0.958βn 0.834 0.675 0.277Bn 5.204 1.313 0.033φn

π 1.599 0.932 0.499

and for Al, one has

Al

n 1 2 3

αn 0.156 0.644 0.958βn 0.793 0.698 0.279Bn 7.964 1.275 0.030φn

π 1.559 0.832 0.431

The contributions to the pair-potential are arranged in order of increasing rangei.e., they are arranged in order of decreasing βn. The Z dependence of thephase shifts determine the position of the minima of the pair-potential. Thispair-potential, although it only has a magnitude of about 10−2 eV, dominatesthe structural energy.

388

-0.003

-0.002

-0.001

0

0.001

0.002

1 2 3 4 5 6

2 kF R / ππππ

ΘΘΘΘ(R

)/ Z2 e2 k F

Al

Na

Mg

Figure 200: The calculated pair potentials for Na, Mg and Al. [After Pettiforand Ward (1984).]

neighbor shell number 1 2 3 4 5

b.c.c.

number of neighbors 8 6 12 24 8neighbor distance

√3

2 1√

2√

112

√3

f.c.c.

number of neighbors 12 6 24 12 24neighbor distance

√2

2 1√

62

√2

√102

h.c.p.

number of neighbors 12 6 2 18 12neighbor distance

√2

2 1 2√3

√6

2

√11√6

389

1

2

3

4

0 0.5 1 1.5 2φφφφ3333/π/π/π/π

Z

hcpbcc

hcp

bcc

fcc

fcc

hcp

fcc Na

Mg

Al

Figure 201: The schematic (Z, φ3) phase diagram.

The energy difference between the f.c.c. and h.c.p. structures are determinedby the third, fourth and fifth nearest neighbors, as the number and positionsof the nearest and next nearest neighbors are the same. Hence, the relativestability of this pair of structures is determined by the reasonably long distancebehavior of the pair-potential. The form of the pair-potential can be used todescribe the relative stability122 of the h.c.p. and f.c.c structures of Na, Mgand Al. At ambient pressure, Na123 and Mg are predicted to be h.c.p. andAl is predicted to have an f.c.c. structure. The f.c.c. form of Mg is unstabledue to a repulsive contribution from the pair-potentials between the (12) fourthnearest neighbor pairs. The h.c.p. form of Al is unstable due to a repulsivecontribution from the pair-potentials between the (12) fifth nearest neighborpairs. This trend is understood as almost entirely being due to the long-rangedcomponent of the pair-potential. Basically, as the value of Z increases, whengoing across the column from Na to Al, the phase shift of the long-ranged in-teraction decreases. This means that the oscillations in the pair-potential moveout to larger distances. This causes the changes in the pair-potential at thepositions of the fourth or fifth nearest neighbors.

Under pressure, these materials are predicted to transform to a b.c.c. phase.The phase shift of the long-ranged component decreases monotonically with in-creasing Rc

rs, which corresponds to increasing pressure. The change in the phase

shifts moves the oscillations in the pair-potentials to distances larger than theneighbor distances. This shows that as the pressure is increased, one may ex-pect the energy differences between the h.c.p. and f.c.c. phases to oscillate. Theenergy differences between the b.c.c. and close-packed phases originate from thecombined (14) first and second nearest neighbors in b.c.c. and the (12) nearest122A. K. McMahan and J. R. Moriarty, Phys. Rev. B 27, 3235 (1983).123Na has the close-packed Samarium structure below T = 5 K, and above 5 K Na has a

b.c.c. structure.

390

neighbors of the close-packed structures. The separations of the neighbors inthe b.c.c. structure should be scaled by a factor of 2−

13 to yield the same elec-

tron density as the close-packed structures. After this scaling, it is found thatthe nearest neighbor distances in the close-packed structures are intermediatebetween the nearest neighbor and the next nearest neighbor distances of theb.c.c. structure. On decreasing the phase shift, one may expect to see the b.c.c.phase become unstable to a close-packed phase when the (8) nearest neighborsexperience the hard core repulsive potential. On further decreasing the phaseshift, the (12) neighbors of the close-packed phase will experience the same hardcore potential at which point, the b.c.c. becomes stable again. This region ofstability of the b.c.c. structure will remain until the (8) next nearest neighborsare compressed to distances where the pair-potential has the form of a hard corerepulsion.

These and similar considerations illuminate the origins of the stability ofdifferent structures124, which are hard to extract from other methods, as thestructural energy typically amounts to only 1% of the cohesive energy of a solid.In general, the cohesive energy of the solid will also involve three and four-atominteractions etc., in addition to the pair-potential. Even though these multi-atom terms contain additional powers of the small factor Vions/EF , they stillcan produce significant contributions to the nearest-neighbor interactions125.More specifically, the three-body terms do seem to be important in describingthe stability of the more open structures found in materials such as the groupIV elements Si, Ge and Sn. To obtain a more accurate description of structuralstability, it is necessary to utilize density functional calculations.

124J. Hafner, From Hamiltonians to Phase Diagrams, Springer Verlag, Berlin (1987).125E. G. Brovman and Yu. M. Kagan, Sov. Phys. J.E.T.P. 25, 365 (1967).

391

11 Metals

In a metal with Ne electrons, the state with minimum energy has the Ne low-est one-electron energy eigenvalue states filled with one electron per state (perspin) in accordance with the Pauli exclusion principle. In a metal, the highestoccupied and the lowest unoccupied state have energies which only differ by aninfinitesimal amount. This energy is called the Fermi energy, εF . Thus, theone-electron states have occupation numbers distributed according to the law

f(ε) = 1 if ε < εF

f(ε) = 0 if ε > εF (1074)

The number of electrons in a solid Ne is dictated by charge neutrality to be equalto the number of nuclear charges N Z. At finite temperatures, the electronoccupation numbers are statistically distributed according to the Fermi-Diracdistribution function

f(ε) =1

1 + exp[β ( ε − µ )

] (1075)

where β−1 = kB T is the inverse temperature. The Fermi-Dirac distributionrepresents the probability that a state with energy ε is occupied. Due to thePauli exclusion principle, the distribution also represents the average occupationof the level with energy ε. The value of the chemical potential coincides withthe Fermi energy at zero temperature µ(0) = εF . Since the solid remainscharge neutral at finite temperatures, the chemical potential is determined bythe condition that the solid contains Ne electrons. For a solid with a density ofstates given by ρ(ε), per spin, the total number of electron is given by

2∫ +∞

−∞dε ρ(ε) f(ε) = Ne (1076)

which is an implicit equation for µ. The factor of two represents the number ofdifferent spin polarizations of the electron.

11.1 Thermodynamics

Due to the Pauli exclusion principle, the density of states at the Fermi energycan often be inferred from measurements of the thermodynamic properties of ametal. As the characteristic energy scale for the electronic properties is of theorder of eV, and room temperature is of the order of 25 meV, the thermody-namic properties can usually be evaluated in the asymptotic low-temperatureexpansion first investigated by Sommerfeld126. The low-temperature Sommer-feld expansion of the electronic specific heat, for non-interacting electrons, shall126A. Sommerfeld, Zeit. fur Physik, 47, 1 (1928).

392

be examined.

11.1.1 The Sommerfeld Expansion

The total energy of the solid can be expressed as an integral

E = 2∫ +∞

−∞dε ρ(ε) ε f(ε) (1077)

Integrals of this type can be evaluated by expressing them in terms of the zerotemperature limit of the distribution and small deviations about this limit.

E = 2∫ µ

−∞dε ρ(ε) ε

+ 2∫ µ

−∞dε ρ(ε) ε

[f(ε) − 1

]+ 2

∫ +∞

µ

dε ρ(ε) ε f(ε) (1078)

The variable of integration in the terms involving the Fermi function is changedfrom ε to the dimensionless variable x defined by

ε = µ + kB T x (1079)

The Fermi function becomes

f(µ + kB T x) =1

1 + exp x(1080)

Thus, the integral becomes

E = 2∫ µ

−∞dε ρ(ε) ε

+ 2 kB T

∫ 0

−∞dx ρ(µ+ kBTx) ( µ+ kBTx )

[f(µ+ kBTx) − 1

]+ 2 kB T

∫ +∞

0

dx ρ(µ+ kBTx) ( µ+ kBTx ) f(µ+ kBTx)

(1081)

The integral over the negative range of x is re-expressed in terms of the newvariable y where

y = − x (1082)

Thus, the energy is expressed as

E = 2∫ µ

−∞dε ρ(ε) ε

393

+ 2 kB T

∫ ∞

0

dy ρ(µ− kBTy) ( µ− kBTy )[f(µ− kBTy) − 1

]+ 2 kB T

∫ +∞

0

dx ρ(µ+ kBTx) ( µ+ kBTx ) f(µ+ kBTx)

(1083)

However, the Fermi function satisfies the relation

1 − f( µ− kBTy ) = f( µ+ kBTy ) (1084)

or equivalently

1 − 11 + exp[ − y ]

=1

1 + exp[ y ](1085)

On setting y back to x, one finds

E = 2∫ µ

−∞dε ρ(ε) ε

+ 2 kB T

∫ +∞

0

dx1

1 + exp x×

×

[ρ(µ+ kBTx) ( µ+ kBTx ) − ρ(µ− kBTx) ( µ− kBTx )

](1086)

The terms within the square brackets can be Taylor expanded in powers ofkB T x, and the integration over x can be performed. Due to the presence ofthe Fermi function, the integrals converge. One then has an expansion which iseffectively expressed in powers of kB T / µ. Thus, the energy is expressed as

E = 2∫ µ

−∞dε ρ(ε) ε

+ 4 kB T

∫ ∞

0

dx∞∑

n=0

x(2n+1)

1 + expx( kBT )2n+1

(2n+ 1)!

(∂

∂µ

)(2n+1)[µ ρ(µ)

] (1087)

The integrals over x are evaluated as∫ ∞

0

dxxn

1 + expx=

∫ ∞

0

dx xn∞∑

l=1

( − 1 )l+1 exp[− l x

]

= n!∞∑

l=1

( − 1 )l+1

ln+1(1088)

which are finite for n ≥ 1. Furthermore, the summation can be expressed interms of the Riemann ζ functions defined by

ζ(m) =∞∑

l=1

1lm

(1089)

394

Using this, one finds that∞∑

l=1

( − 1 )l+1

l2n+2=(

22n+1 − 122n+1

)ζ(2n+ 2) (1090)

The Riemann zeta functions have special values

ζ(2) =π2

6

ζ(4) =π4

90(1091)

Thus, the Sommerfeld expansion for the total electronic energy only involveseven powers of T 2, that is,

E = 2∫ µ

−∞dε ρ(ε) ε

+ 4 ( kB T )2∞∑

n=0

(22n+1 − 1

22n+1

)ζ(2n+ 2) ( kBT )2n

(∂

∂µ

)(2n+1)[µ ρ(µ)

] (1092)

The coefficients may be evaluated in terms of the Riemann ζ functions.

Although the expansion contains an explicit temperature dependence, thereis an implicit temperature dependence in the chemical potential µ. This tem-perature dependence can be found from the equation

Ne = 2∫ +∞

−∞dε ρ(ε) f(ε) (1093)

which also can be expanded in powers of T 2 as

Ne = 2∫ µ

−∞dε ρ(ε)

+ 4 ( kB T )2∞∑

n=0

(22n+1 − 1

22n+1

)ζ(2n+ 2) ( kBT )2n

(∂

∂µ

)(2n+1)[ρ(µ)

] (1094)

Since Ne is temperature independent, in principle, the series expansion can beinverted to yield µ in powers of T .

11.1.2 The Specific Heat Capacity

The electronic contribution of the heat capacity, for non-interacting electrons,can be expressed as

CNe(T ) = T

(∂S

∂T

)Ne

395

=(∂E

∂T

)Ne

(1095)

as the solid remains electrically neutral. Using the Sommerfeld expansion of theenergy, the specific heat can can be expressed as the sum of the specific heat atconstant µ and a term depending on the temperature derivative of µ at constantNe.

CNe= 4 k2

BT∞∑

n=0

(n+ 1)(

22n+1 − 122n

)ζ(2n+ 2) ( kBT )2n

(∂

∂µ

)(2n+1)[µ ρ(µ)

]

+(∂µ

∂T

)Ne

[2 µ ρ(µ)

+ 4 k2BT

2∞∑

n=0

(22n+1 − 1

22n+1

)ζ(2n+ 2) ( kBT )2n

(∂

∂µ

)(2n+2)[µ ρ(µ)

] ](1096)

In the above expression, µ is to be expanded in powers of T about its zero tem-perature value µ = εF . The temperature derivative of the chemical potentialcan be evaluated from the temperature derivative of the equation for the fixednumber of electrons Ne,

0 = 4 k2BT

∞∑n=0

(n+ 1)(

22n+1 − 122n

)ζ(2n+ 2) ( kBT )2n

(∂

∂µ

)(2n+1)

ρ(µ)

+(∂µ

∂T

)Ne

[2 ρ(µ)

+ 4 k2BT

2∞∑

n=0

(22n+1 − 1

22n+1

)ζ(2n+ 2) ( kBT )2n

(∂

∂µ

)(2n+2)

ρ(µ)

](1097)

This equation yields the temperature dependence of µ which can be substitutedback into the expression for the temperature dependence of CN . This yields theleading term in the low-temperature expansion for the electronic-specific heatof non-interacting electrons as

CN = k2B T 4 ζ(2) ρ(µ) + O(k4

B T 3)

= k2B T

2 π2

3ρ(µ) + O(k4

B T 3) (1098)

The coefficient of the linear term is proportional to the density of states, perspin, at the Fermi energy. The result is understood by noting that the Pauliexclusion principle prevents electrons from being thermally excited, unless theyare within kB T of the Fermi energy. There are ρ(µ) kB T such electrons, andeach electron contributes kB to the specific heat. Thus, the low-temperature

396

specific heat is of the order of k2B T ρ(µ). The inclusion of electron-electron

interaction changes this result, and in a Fermi liquid, may increase the coefficientof T . The low-temperature specific heat is enhanced, due to the enhancement ofthe quasi-particle masses. This can be demonstrated by a simplified calculationin which the quasi-particle weight is assumed to be independent of k. Since thequasi-particle width in the vicinity of the Fermi energy is negligible, one hasthe relationship between the quasi-particle density of states and the density ofstates for non-interacting electrons given by

ρqp(E) =∑

k

δ

(Z(k) E − Ek + µ

)

=∑

k

1Z(k)

δ

(E −

( Ek − µ )Zk

)

=∑

k

1Z(k)

δ

(E − Eqp(k)

)(1099)

Also, the quasi-particle density of states at the Fermi energy is un-renormalizedas

ρqp(0) =∑

k

δ

(µ − Ek

)= ρ(µ) (1100)

The γ term in the low-temperature specific heat is calculated from the quasi-particle entropy S defined in terms of the quasi-particle occupation numbersnqp

k by

S = − kB

∑σ,k

[nqp

k ln nqpk + ( 1 − nqp

k ) ln( 1 − nqpk )

]

= − 2 kB

∫ ∞

−∞dE Z ρqp(E)

[f(E) ln f(E) + ( 1 − f(E) ) ln( 1 − f(E) )

](1101)

Thus, in this approximation, the coefficient of the linear T term is given by

γ = limT → 0

CN

T

=(∂S

∂T

)Ne

= k2B Z

2 π2

3ρ(µ) (1102)

In the more general case, the specific heat coefficient is enhanced through ak weighted average of the quasi-particle mass enhancement Zk. For materials

397

like CeCu6, CeCu2Si2, CeAl3 and UBe13, the value of the γ coefficients areextremely large127, of the order of 1 J / mole of f ion / K2, which is 1000times larger than Cu. The quasi-particle mass enhancements are inferred bycomparison to LDA electronic density of states calculations128 and are about 10to 30. The enhancement is assumed to be due to the strong electron-electroninteractions, which the LDA fails to take into account.

——————————————————————————————————

11.1.3 Exercise 59

Calculate the next to leading-order term in the low-temperature electronic-specific heat.

——————————————————————————————————

11.1.4 Exercise 60

CeNiSn is thought to be a zero-gap semiconductor with a V shaped density ofstates. The density of states near the Fermi level is approximated by

ρ(ε) = α0 ε for ε > 0ρ(ε) = − α1 ε for ε < 0 (1103)

where α0 and α1 are positive numbers. Find the leading temperature depen-dence of the low-temperature specific-heat.

——————————————————————————————————

11.1.5 Pauli Paramagnetism

In the absence of spin-orbit scattering effects, the susceptibility of a metal canbe decomposed into two contributions; the susceptibility due to the spins of theelectrons, and the susceptibility due to the electrons orbital motion. The spinsusceptibility for non-interacting electrons gives rise to the Pauli-paramagneticsusceptibility which is positive, and is temperature independent at sufficientlylow temperatures. The susceptibility due to the orbital motion has a negativesign and, therefore, yields the Landau-diamagnetic susceptibility.

127F. Steglich, J. Aaarts, C. D. Bredl, W. Lieke, D. Meschede, W. Franz and H. Schafer,Phys. Rev. Lett., 43, 1982 (1979), H. R. Ott, H. Rudigier, Z. Fisk and J. L. Smith, Phys.Rev. Lett., 50, 1595 (1983), G. R. Stewart, Z. Fisk, J. O. Willis and J. L. Smith, Phys. Rev.Lett. 52, 679 (1984), G. R. Stewart, Z. Fisk and M. S. Wire, Phys. Rev. B, 30, 482 (1984).128C. S. Wang, M. R. Norman, R. C. Albers, A. M. Boring, W. E. Pickett, H. Krakauer,

N. E. Christiensen, Phys. Rev. B, 35, 7260 (1987).

398

The magnetization due to the electronic spins can be calculated from

Mz = −(∂Ω∂Hz

)(1104)

where the grand canonical potential is given by

Ω = − kB T∑α

ln

(1 + exp

[− β ( Eα − µ )

] )(1105)

and where the sum over α runs over the quantum numbers of the single-particlestates including the spin. The applied magnetic field Hz couples to the quantumnumber corresponding to the z component of the spin of the electron, σ, via theZeeman energy

HZeeman = − g | e |2 me c

Hz Sz

= − µB Hz σ (1106)

where the spin angular momentum is given by S = h2 σ and the gyromagnetic

ratio g = 2 originates from the Dirac or Pauli equation. The quantity µB isthe Bohr magneton and is given in terms of the electron’s charge and mass by

µB =| e | h2 me c

(1107)

The energy of a particle can then be written as

Eσ,k = Ek − µB σ Hz (1108)

where σ is the eigenvalue of the Pauli spin matrix σz. The density of states, perspin, in the absence of the field is defined as

ρ(ε) =∑

k

δ( ε − Ek ) (1109)

Thus, in the presence of a field, one has the spin-dependent density of states

ρσ(ε) = ρ(ε+ µBσHz) (1110)

The grand canonical potential can be expressed as an integral over the densityof states

Ω = − kB T

∫ ∞

−∞dε∑

σ

ρ(ε+ µBσHz) ln

(1 + exp

[− β ( ε− µ )

] )

= − kB T

∫ ∞

−∞dε′

∑σ

ρ(ε′) ln

(1 + exp

[− β ( ε′ − µBσHz − µ )

] )(1111)

399

where the variable of integration has been changed in the last line. The summa-tion over σ runs over the values ± 1. The spin contribution to the magnetizationinduced by the applied field is given by

Mz = µB

∫ ∞

−∞dε′

∑σ

σ ρ(ε′)1

1 + exp[β ( ε′ − µBσHz − µ )

]= µB

∫ ∞

−∞dε′

∑σ

σ ρ(ε′) f( ε′ − µBσHz )

= µB

(Ne(σ = 1) − Ne(σ = − 1)

)(1112)

The magnetization due to the spins is just proportional to the number of up-spinelectrons minus the down-spin electrons. The spin susceptibility is given by

χzzp (T,Hz) =

(∂Mz

∂Hz

)(1113)

and is given by

χp(T,Hz) = − µ2B

∫ ∞

−∞dε∑

σ

σ2 ρ(ε)∂

∂εf( ε− µBσHz )

(1114)

It is usual to measure the susceptibility at zero field. Since the derivative of theFermi function is peaked around the chemical potential, only electrons withinkB T of the Fermi energy contribute to the Pauli-susceptibility. At sufficientlylow temperatures, one may use the approximation

− ∂

∂εf = δ( ε − εF ) (1115)

so that the zero temperature value of the Pauli-susceptibility is evaluated as

χp(0) = 2 µ2B ρ(εF ) (1116)

which is inversely proportional to the free electron mass, and is also proportionalto the density of states at the Fermi energy. The finite temperature susceptibilitycan be evaluated by integration by parts, to obtain

χp(T ) = 2 µ2B

∫ ∞

−∞dε f(ε)

∂

∂ερ(ε) (1117)

The zero field spin susceptibility can then be obtained via the Sommerfeld ex-pansion

χp(T ) = 2 µ2B

[ρ(µ) + 2 k2

BT2

∞∑n=0

(22n+1 − 1

22n+1

)ζ(2n+2) (kBT )2n

(∂

∂µ

)(2n+2)

ρ(µ)

](1118)

400

Thus, the spin susceptibility has the form of a power series in T 2. The temper-ature dependence of the chemical potential can be found from the equation forNe. The leading change in the chemical potential ∆µ due to T is given by

∆µ = − k2BT

2 π2

6

(∂ρ(εF )

∂εF

ρ(εF )

)+ O( k4

BT4 ) (1119)

The temperature dependence of the chemical potential depends on the logarith-mic derivative of the density of states, such that it moves away from the regionof high density of states to keep the number of electrons fixed. This leads tothe leading temperature dependence of the Pauli susceptibility being given by

χp(T ) = 2 µ2B

[ρ(εF ) +

π2

6k2

BT2

(∂2ρ(εF )∂ε2F

−( ∂ρ(εF )

∂εF)2

ρ(εF )

)+ O( k4

BT4 )

](1120)

The temperature dependence gives information about the derivatives of the den-sity of states.

The coefficient γ of the linear T term in the low-temperature specific-heatand the zero temperature susceptibility are proportional to the density of statesat the Fermi energy. The susceptibility and specific heat can be used to definethe dimensionless ratio

limT → 0

T χp(T )C(T )

=χp(0)γ

(1121)

This ratio is known as the Sommerfeld ratio. For free electrons, this ratio hasthe value

limT → 0

T χp(T )C(T )

=3 µ2

B

π2 k2B

(1122)

The effect of electron-electron interactions can change this ratio, as they mayaffect the susceptibility in a different manner than the specific heat. The Stonermodel, discussed in the chapter on magnetism, shows that the effect of electron-electron interactions can produce a large enhancement of the paramagnetic sus-ceptibility for electron systems close to a ferromagnetic instability. Thus, neara ferromagnetic instability, the Sommerfeld ratio is expected to be large. Forexample, in Pd the ratio is found to be almost a factor of ten larger than pre-dicted for non-interacting electrons129. However, for heavy fermion materialswhere both C(T )/T and χp(0) are highly enhanced, the value of the Sommerfeldratio is very close to that of non-interacting electrons130.

——————————————————————————————————

129D .N. Budworth, F. E. Hoare and J. Preston, Proc. Roy. Soc. (London) A 257, 250(1960), W. E. Gardner and J. Penfold, Phil. Mag. 11, 549 (1965).130Z. Fisk, H. R. Ott and J. L. Smith, J. Less-Common Metals, 133, 99 (1987)

401

11.1.6 Exercise 61

Determine the field dependence of the low-temperature Pauli susceptibility.

——————————————————————————————————

11.1.7 Exercise 62

Determine the high temperature form of the Pauli susceptibility.

——————————————————————————————————

11.1.8 Landau Diamagnetism

Free electrons in a magnetic field aligned along the z axis have quantized energiesgiven by

Ekz,n =h2 k2

z

2 m+(n +

12

)h ωc (1123)

where

ωc =| e | Hz

m c(1124)

is the cyclotron frequency and n is a positive integer. For a cubic environmentof linear dimension L, the value of kz is given by

kz =2 πL

nz (1125)

The Landau levels have their orbits in the x − y plane quantized and have alevel spacing of h ωc. Each Landau level is highly degenerate. The degeneracyD, or number of electrons with a given n and kz, can be found as the ratio ofthe area of the sample divided by the area enclosed by the classical orbit

D =L2

2 π r2c(1126)

where rc is the radius of the classical orbit. This radius can be obtained byequating the field energy with the zero point energy of the Landau level

m

2ω2

c r2c =

12h ωc (1127)

Thus, the degeneracy is given by

D =L2

2 π hm ωc

D =| e | L2

h cHz (1128)

402

Since Hz ∼ 1 kG, a typical value of the degeneracy is of the order ofD ∼ 1010. These levels can be treated semi-classically as there are an enormousnumber of Landau levels in an energy interval. The number of occupied Landaulevels is given by the Fermi energy µ divided by h ωc,

µ

h ωc=

µ| e | hm c Hz

(1129)

The numerical constant has the value

| e | hm c

∼ 1.16 × 10−8 eV / G (1130)

so, with µ ∼ 1 eV and Hz ∼ 104 G, one finds that the number of occupiedLandau levels is approximately given by

µ

h ωc∼ 104 (1131)

11.1.9 Landau Level Quantization

The Hamiltonian of a free electron in a magnetic field is given by

H =(p +

| e |c

A

)2

/ ( 2 m ) (1132)

Using the gauge A = (0,Hzx, 0) appropriate for a field along the z axis, thenthe Schrodinger equation takes the form

− h2

2 m∇2φ − i

| e | Hz

m cx

(∂φ

∂y

)+

e2 H2z

2 m c2x2 φ = E φ (1133)

This can be solved by the substitution

φ(r) = f(x) exp[i ( ky y + kz z )

](1134)

so that f(x) satisfies

− h2

2 m

(∂2f

∂x2

)+

[ (h ky +

| e | Hz

cx

)2 12 m

−(E − h2 k2

z

2 m

) ]f(x) = 0

(1135)which is recognized as the equation for the harmonic oscillator with energyeigenvalue

E − h2 k2z

2 m=(n +

12

)h ωc (1136)

403

where

ω2c =

e2 H2z

m2 c2(1137)

That is, the motion in the plane perpendicular to the field, Hz, is quantizedinto Landau levels131. The energy spacing between the levels is given by h ωc,where

ωc =| e | Hz

m c(1138)

and the orbit is centered around the position

x0 = − h ky c

| e | Hz(1139)

The momentum dependence of the position x0 has a classical analogy. Thecenter of the classical orbit is determined by its initial velocity vy via

vy = ωc x0 (1140)

so the center of the quantum orbit is determined by py. The energy of theLandau orbit is given by

Ekz,n =h2 k2

z

2 m+(n +

12

)h ωc (1141)

The degeneracy of the n-th level must correspond to the number of kx, ky valuesfor Hz = 0 that collapse onto the Landau levels as Hz is increased.

The degeneracy can be enumerated in the case of periodic boundary condi-tions,

φ(x, y, z) = φ(x, Ly − y, z) (1142)

The periodic boundary conditions imply that

exp[i ky Ly

]= 1 (1143)

or

ky =2 πLy

ny (1144)

The x dependent factor of the wave function f(x) is centered at x0 where

x0 = − h ky c

| e | Hz(1145)

For a sample of width Lx, one must have Lx > x0 > 0, so one has the equality

| e | Hz Lx

h c> − ky > 0 (1146)

131L. D. Landau, Z. Physik, 64, 629 (1930).

404

The degeneracy, D, is the number of quantized ky values that satisfy this in-equality. The degeneracy is found to be

D =| e | Hz Lx

h c/

2 πLy

=| e | Hz Lx Ly

2 π h c(1147)

independent of n. Thus, the degeneracy D of every Landau is given by

D =| e | Hz Lx Ly

2 π h c(1148)

The degeneracy can be expressed in terms of the amplitude of the oscillationsin the x direction, which is defined as the length scale that determines theexponential fall off of the ground state wave function

rc =√

h

m ωc(1149)

The degeneracy of the Landau levels can also be expressed as

D =Lx Ly

2 π r2c(1150)

as previously found from classical considerations. The quantization of the or-bital motion, in the presence of a periodic potential, has been considered byRauh132 and by Harper133. These authors have shown that the periodic poten-tial causes the Landau levels to be broadened or split.

11.1.10 The Diamagnetic Susceptibility

The diamagnetic susceptibility is determined from the field dependence of thegrand canonical potential, Ω,

Ω = 2D Lz

2 π h

∫ ∞

−∞dkz

∑n

(h ωc ( n+

12

) +h2 k2

z

2 m− µ

)Θ(µ−h ωc ( n+

12

)− h2 k2z

2 m

)(1151)

On integrating over kz, one finds

Ω = −83D Lz

2 π

(2 mh2

) 12

µhωc

− 12∑

n=0

(µ − h ωc ( n +

12

)) 3

2

(1152)

132A. Rauh, Phys. Stat. Solidi, B 65, K131 (1974), A. Rauh, Phys. Stat. Solidi, B 69, K9(1975).133P. G. Harper, Ph.D. Thesis, University of Birmingham (1954), P. G. Harper, Proc. Phys.

Soc. London, A 68, 874 (1955).

405

or

Ω = −23

V

( π2 h )m ωc

(2 mh2

) 12

µhωc

− 12∑

n=0

(µ − h ωc ( n +

12

)) 3

2

(1153)

The summation over n can be performed using the Euler-MacLaurin formula

n=N∑n=0

F (n) =∫ N

0

dx F (x) +12

( F (0) + F (N) ) +112

( F ′(N)− F ′(0) ) + . . .

(1154)

This produces the leading-order field dependence of the grand canonical poten-tial, given by

Ω = − 23

V

( π2 h2 )m

(2 mh2

) 12[

25µ

52 − 1

16h2 ω2

c µ12 + . . .

](1155)

The diamagnetic susceptibility is given by the second derivative with respect tothe applied field

χd = −(∂2Ω∂H2

z

)= − µ2

B

V m

3 π2 h2

(2 mh2

) 12

µ12 (1156)

where we have expressed the orbital magnetic moment in terms of the (orbital)Bohr magneton

µB =| e | h2 m c

(1157)

The diamagnetic susceptibility χd can be compared with the Pauli paramagneticsusceptibility χp. For free electrons, the Pauli susceptibility is given by

χp = 2 µ2B ρ(µ)

= 2 µ2B

V

2 π2

m kF

h2

= µ2B

V m

π2 h2

(2 mh2

) 12

µ12 (1158)

Hence, the spin and orbital susceptibilities are related via

χd = − 13χp (1159)

Thus, the Landau diamagnetic susceptibility is negative and has a magnitudewhich, for free electrons, is just one third of the Pauli paramagnetic susceptibil-ity134. The diamagnetism results from the quantized orbital angular momentum134L. D. Landau, Z. fur Phys. 64, 629 (1930).

406

of the electrons. The value of µB in the diamagnetic susceptibility is given bythe band mass m∗, whereas the factor of µB in the Pauli susceptibility is definedin terms of the mass of the electron in vacuum me. In systems such as Bismuth,in which the band mass is smaller than the free electron mass, the diamagneticsusceptibility is larger by a factor of

χd

χp= − 1

3

(me

m∗

)2

(1160)

and the diamagnetic susceptibility can be larger than the Pauli susceptibility.The susceptibility of Bismuth is negative.

In the presence of spin-orbit coupling, the orbital angular momenta are cou-pled with the spin angular momenta. As a result, the components of the totalsusceptibility are coupled. The manner in which the total angular momentumcouples to the field is described by the g factor135.

135Y. Yafet, Solid State Phys. 14, 1 (1963).

407

11.2 Transport Properties

11.2.1 Electrical Conductivity

The electrical conductivity of a normal metal is considered. The applicationof an electromagnetic field will produce an acceleration of the electrons in themetal. This implies that the distribution of the electrons in phase space willbecome time-dependent, and in particular the Fermi surface will be subject toa time-dependent distortion. However, the phenomenon of electrical transportin metals is usually a steady state process, in that the electric current density jproduced by a static electric field E is time independent and obeys Ohm’s law

j = σ E (1161)

where σ is the electrical conductivity. This steady state is established by scatter-ing processes that dynamically balances the time-dependent changes producedby the electric field. That is, once the steady state has been established, theacceleration of the electrons produced by the electric field is balanced by scat-tering processes that are responsible for equilibration.

Since Ohm’s law holds almost universally, without requiring any noticeablenon-linear terms in E to describe the current density, it is safe to assume thatthe current density can be calculated by only considering the first order terms inthe electro-magnetic field. The validity of this assumption can be related to thesmallness of the ratio of λ | e | E

µ where λ is the mean free path, E the strengthof the applied field and µ the Fermi energy. This has the consequence that theFermi surface in the steady state where the field is present is only weakly per-turbed from the Fermi surface with zero field. A number of different approachesto the calculation of the electrical conductivity will be described. For simplic-ity, only the zero temperature limit of the conductivity shall be calculated. Thedominant scattering process for the conductivity in this temperature range isscattering by static impurities.

11.2.2 Scattering by Static Defects

The electrical conductivity will be calculated in which the scattering is due toa small concentration of randomly distributed impurities. The potential dueto the distribution of impurities located at positions rj , each with a potentialVimp(r) is given by

V (r) =∑

j

Vimp(r − rj) (1162)

This produces elastic scattering of electrons between Bloch states of differentwave vectors. The transition rate in which an electron is scattered from thestate with Bloch wave vector k to a state with Bloch wave vector k′ is denotedby 1

τ(k→k′) . If the strength of the scattering potential is weak enough, the

408

transition rate can be calculated from Fermi’s golden rule as

1τ(k → k′)

=2 πh

∣∣∣∣ < k | V | k′ >∣∣∣∣2 δ( E(k) − E(k′) )

=2 πh

1V 2

∑i,j

exp[i (k − k′) . (ri − rj)

] ∣∣∣∣ Vimp(k − k′)∣∣∣∣2 δ( E(k) − E(k′) )

(1163)

where the delta function expresses the restriction imposed by energy conserva-tion in the elastic impurity scattering processes. As usual, the presence of thedelta function requires that the transition probability is calculated by integrat-ing over the momentum of the final state. As the positions of the impuritiesare distributed randomly, the scattering rate shall be configurational averaged.The configurational average of any function is obtained by integrating over thepositions of the impurities

F =∏j

[1V

∫d3rj F (rj)

](1164)

The configurational average of the scattering rate is evaluated as

1τ(k → k′)

= =2 πh

1V 2

∑j

∣∣∣∣ Vimp(k − k′)∣∣∣∣2 δ( E(k) − E(k′) ) (1165)

where only the term with i = j survives. The conductivity can be calculatedfrom the steady state distribution function of the electrons, in which the scat-tering rate dynamically balances the effects of the electric field. This is found,in the quasi-classical approximation, from the Boltzmann equation.

The Boltzmann Equation.

The distribution of electrons in phase space at time t, f(k, r, t), is deter-mined by the Boltzmann equation. The Boltzmann equation can be found beexamining the increase in an infinitesimally region of phase space that occursduring a time interval dt. The number of electrons in the infinitesimal volumed3k d3r located at the point k, r at time t is

f(k, r, t) d3k d3r (1166)

The increase in the number of electrons in this volume that occurs in timeinterval dt is given by(f(k, r, t+ dt) − f(k, r, t)

)d3k d3r =

∂

∂tf(k, r, t) d3k d3r dt + O(dt2)

(1167)

409

This increase can be attributed to changes caused by the regular or deterministicmotion of the electrons in the applied field, and partly due to the irregularmotion caused by the scattering. The appropriate time scale for the changes inthe distribution function due to the applied fields is assumed to be much longerthan the time interval in which the collisions occur. The deterministic motionof the electrons trajectories in phase space results in a change in the numberof electrons in the volume d3k d3r. The increase due to these slow time scalemotions is equal to the number of electrons entering the six-dimensional volumethrough its surfaces in the time interval dt minus the number of electrons leavingthe volume. This is given by

∆ f(k, r, t) d3k d3r = − dt

[∇ .

(r f(k, r, t)

)+ ∇k .

(k f(k, r, t)

) ](1168)

and the slow rates of change in position and momentum of the electrons isdetermined via

r =h k

m

h k = − | e | Em

(1169)

Hence, the deterministic changes are found as

∆ f(k, r, t) d3k d3r = −[∇ .

(h k

mf(k, r, t)

)− ∇k .

(| e | Em h

f(k, r, t)) ]

d3k d3r dt

(1170)

This involves the sum of two terms, one coming from the change of the electronsmomentum and the other from the change in the electrons position. The twogradients in this expression can be evaluated, each gradient yields two terms.One term of each pair involves a gradient of the distribution function, while theother only involves the distribution function itself. One term, originating fromthe change in the electrons position involves the spatial variation of the velocity.From Hamilton’s equations of motion it can be shown that the coefficient of thisterm is equal to the second derivative of the Hamiltonian,

∇ . r = ∇ . ∇p H (1171)

while the similar term originating from the change in particles momentum isjust equal to the negative of the second derivative

∇p . p = − ∇ . ∇p H (1172)

Since the Hamiltonian is ana analytic function these terms are equal magnitudeand of opposite sign. Thus, these terms cancel yielding only

∆ f(k, r, t) d3k d3r =

410

= −[h k

m. ∇

(f(k, r, t)

)−(| e | Em h

. ∇k f(k, r, t)) ]

d3k d3r dt

(1173)

The remaining contribution to the change in number of electrons per unit timeoccurs from the rapid irregular motion caused by the impurity scattering. Thenet increase is due to the excess in scattering of electrons from occupied statesat (k′, r) into an unoccupied state (k, r) over the rate of scattering out of state(k, r) into the unoccupied states at (k′, r). The restriction imposed by the Pauliexclusion principle, is that the state to which the electron is scattered intoshould be unoccupied in the initial state. This restriction is incorporated byintroducing the probability that a state (k, r) is unoccupied, through the factor( 1 − f(k, r, t) ).

∆ f(k, r, t) d3k d3r =∑k′

[1

τ(k′ → k)f(k′, r, t) ( 1 − f(k, r, t) )

− 1τ(k → k′)

f(k, r, t) ( 1 − f(k′, r, t) )

]d3k d3r dt (1174)

On equating these three terms, cancelling common factors of d3k d3r dt oneobtains the Boltzmann equation

∂

∂tf(k, r, t) = −

[∇ .

(h k

mf(k, r, t)

)− ∇k .

(| e | Em h

f(k, r, t)) ]

+ I

[f(k, r, t)

](1175)

where the functional I[f]

is the collision integral and is given by

I

[f(k, r, t)

]=∑k′

[1

τ(k′ → k)f(k′, r, t) ( 1 − f(k, r, t) )

− 1τ(k → k′)

f(k, r, t) ( 1 − f(k′, r, t) )

](1176)

Thus, the Boltzmann equation can be written as the equality of a total derivativeobtained from the regular motion and the collision integral which represents thescattering processes

d

dtf(k, r, t) = I

[f(k, r, t)

](1177)

Since1

τ(k → k′)=

1τ(k′ → k)

(1178)

411

the collision integral can be simplified to yield

I

[f(k, r, t)

]=∑k′

[1

τ(k → k′)

(f(k′, r, t)− f(k, r, t)

) ](1179)

Due to time reversal invariance of the scattering rates and conservation of en-ergy, the collision integral vanishes in the equilibrium state. In equilibrium,the distribution function is time independent and uniform in space. The dis-tribution function, therefore, only depends on k in a non-trivial manner, andcan be written in terms of the Fermi function f0(k). Thus, in this case, thedistributions are related via

f(k, r, t) =1Vf0(k) (1180)

The equilibrium distribution function f0(k) is only a function of the energyE(k). The condition of conservation of energy which occurs implicity in thescattering rate requires f0(k) = f0(k′). Hence, in equilibrium the collisionintegral vanishes.

In the steady state produced by the application of an electric field, theelectron density will be time independent and uniform throughout the metal,and so the temporal and spatial dependence of f(k, r, t) can still be neglected.In this case, the distribution function in momentum space is still related to thedistribution function in phase space via

f(k, r, t) =1Vf(k) (1181)

where f(k) is the non-equilibrium distribution describing the steady state.

The Solution of the Boltzmann equation.

Since the electron distribution in the steady state conduction of electrons isclose to equilibrium one may look for solutions, for f(k) close to the equilibriumFermi-Dirac distribution function. Thus, solutions of the form can be sought

f(k) = f0(k) + Φ(k)∂f0(k)∂E(k)

(1182)

where Φ is an unknown function, with dimensions of energy. It is to be shownthat Φ(k) is determined by the electric field and small compared with the Fermienergy µ. The above ansatz for the non-equilibrium distribution function ismotivated by the notion that the term proportional to Φ occurs from a Taylorexpansion of the steady state distribution function. In other words, the varia-tion of Φ with k occurs from the distortion of the Fermi surface in the steadystate.

412

If the above ansatz for the steady state distribution is substituted into theBoltzmann equation one obtains

− ∇k .

(| e | Em h

f(k, r, t))

= I

[Φ(k)

∂f0(E(k))∂E(k)

](1183)

This shows that the energy Φ has a leading term which is proportional to thefirst power of the electric field. However, in order to obtain a current thatsatisfies Ohm’s law, only the terms in Φ terms linear in E need to be calculated.Therefore, the Boltzmann equation can be linearized by dropping the term thatinvolves the electric field and Φ, since this is second order in the effect of thefield. The linearized Boltzmann equation can be solved by noticing that thecollision integral is equal to the source term which is proportional to the scalarproduct ( k . E ). Hence, it is reasonable to assume that Φ(k) has a similarform

Φ(k) = A(E(k)) ( k . E ) (1184)

where A(E) is an unknown function of the energy, or other constants of motion.Due to conservation of energy, the unknown coefficient can be factored out ofthe collision integral, as can be the factor of ∂f0

∂E since both are only functionsof the energy. It remains to evaluate an integral of the form∫

d3k′ δ( E(k) − E(k′) ) | Vimp(k − k′) |2(

( k′ − k ) . E)

(1185)

The integration over k′ can be performed by first integrating over the magnitudeof q = k − k′. On using the property of the energy conserving delta function,

δ( E(k) − E(k′) ) =2 mh2 δ( q2 − 2 k q cos θ′ ) (1186)

this sets the magnitude of q = 2 k cos θ′, where the direction of k was chosenas the polar axis. For simplicity it shall be assumed that the impurity potentialis short-ranged, so that the dependence of V (q) on q is relatively unimportant.The integration over the factor of q . E can easily be evaluated, and the resultcan be shown to be proportional to just ( k . E ) . That is, on expressing thescalar product as

q . E = ( q sin θ′ cosφ′ Ex + q sin θ′ sinφ′ Ey + q cos θ′ Ez ) (1187)

on integrating over the azimuthal angle φ′

dΩ′ = dθ′ sin θ′ dφ′ (1188)

the terms proportional to Ex and Ey vanish. The integration over the polarangle θ′ produces a factor

8 π2 mh2 k2

∫ 1

−1

d cos θ′ cos3 θ′ | V ( 2 k cos θ′ ) |2 Ez (1189)

413

This yields the result

= 4 π2 mh2 k ( k . E ) 2

∫ 1

−1

d cos θ′ cos3 θ′ | V ( 2 k cos θ′ ) |2 (1190)

which can be expressed as an integral over the scattering angle θ = π − 2 θ′

= 2 π2 mh2 k ( k . E )

∫ 1

−1

d cos θ ( 1 − cos θ ) | V ( 2 k sinθ

2) |2 (1191)

On identifying the non-equilibrium part of the distribution function with

Φ(k)∂f0(k)∂E(k)

= ( k . E ) A(E)∂f0(k)∂E(k)

(1192)

yields the solution for the non-equilibrium contribution of the distribution func-tion as

Φ(k) = + τtr(k)| e |h

(E . ∇k E(k)

)(1193)

Thus, Φ is proportional to the energy change of the electron produced by theelectric field in the interval between scattering events. In the above expression,the term

1τtr(k)

= c2 πh

∑k′

δ( E(k) − E(k′)) | V (k − k′) |2(

1 − cos θ)

(1194)

is identified as the transport scattering rate, in which c is the concentration ofimpurities. The transport scattering rate has the form of the rate for scatteringout of the state k but has an extra factor of ( 1 − cos θ ). In the quantum for-mulation of transport this factor appears as a vertex correction. Basically, theelectrical current is related to the momentum of the electrons in the directionof the applied field. Forward scattering processes do not result in a reductionof the momentum and, therefore, leave the current unaffected. The transportscattering rate involves a factor of ( 1 − cos θ ) where θ is the scattering angle.This factor represents the relative importance of large angle scattering in thereduction of the total current.

The Current Density.

The current density can be obtained directly from the expression.

j = − 2 | e | 1V

∑k

1h

∂E(k)∂k

f(k)

= − 2 | e | 1V

∑k

1h

∂E(k)∂k

(f0(k) + Φ(k)

∂f0(k)∂E(k)

)(1195)

414

where the factor of 2 represents the sum over the electron spins. On viewingthe electron distribution function as the first two terms in a Taylor expansion,the electron distribution function can be described by an occupied Fermi vol-ume which has been displaced from the equilibrium position in the direction ofthe applied field. The displacement of the Fermi volume produces the averagecurrent in the direction of the field. The first term represents the current thatis expected to flow in the equilibrium state. This term is zero, as can be seenby using the symmetry of the energy E(k) = E(−k) in the Fermi function.Due to the presence of the velocity vector 1

h ∇kE(k), it can be seen that thecurrent produced by an electron of momentum k identically cancels with thecurrent produced by an electron of momentum −k.

Thus, the non-zero component of the current originates from the non-equilibriumpart of the distribution function. This can only be evaluated once the Blochenergies are given. The current is given by

j = − 2e2

h2

∑k

τ(k)tr ∇kE(k)(∇kE(k) . E

) (∂f0(k)∂E(k)

)(1196)

On recognizing the zero temperature property of the Fermi function

−(∂f0(k)∂E(k)

)= δ( E(k) − µ ) (1197)

it is seen that the electrical current is carried by electrons in a narrow energyshell around the Fermi surface. On using the symmetry properties of the inte-gral, one finds that only the diagonal component of the conductivity tensor isnon-zero and is given by

σα,β = − 2 δα,β

3e2

h2

∑k

τ(k)tr | ∇kE(k) |2(∂f0(k)∂E(k)

)(1198)

For free electron bands, the conductivity tensor is evaluated as

σα,β = δα,βρ e2 τtrm

(1199)

where ρ is the density of electrons, m is the mass of the electrons and τtr is theFermi surface average of the transport scattering rate τtr(k).

——————————————————————————————————

11.2.3 Exercise 63

Determine the conductivity tensor σα,β(q, ω) which relates the Fourier compo-nent of a current density jα(q, ω) to a time and spatially varying applied electric

415

field with a Fourier amplitude Eβ(q, ω) via Ohm’s law

jα(q, ω) =∑

β

σα,β(q, ω) Eβ(q, ω) (1200)

Assume that ω is negligibly small compared with the Fermi energy so that thescattering rate can be evaluated on the Fermi surface.

The above result should show that in the zero frequency limit ω → 0the q = 0 conductivity is purely real and given by the standard expressionσα,β(0, 0) = δα,β

ρ e2 τtr

m , and decreases for increasing ω. The frequency widthof the Drude peak is given by the scattering rate 1

τtr.

——————————————————————————————————

11.2.4 The Hall Effect and Magneto-resistance.

The Hall effect occurs when an electrical current is flowing in a sample and amagnetic field is applied in a direction transverse to the direction of the cur-rent density. Consider a sample in the form of a rectangular prism, with axesparallel to the axes of a Cartesian coordinate system. The magnetic field is ap-plied along the z direction and a current flows along the y direction. The Halleffect concerns the appearance of a voltage (the Hall voltage) across a samplein the x direction. The Hall voltage appears in order to balance the Lorentzforce produced by the motion of the charged particles in the magnetic field.The initial current flow in the x direction sets up a net charge imbalance acrossthe sample in accordance with the continuity equation. The build up of staticcharge produces the Hall voltage. In the steady state, the Hall voltage balancesthe Lorentz force opposing the further build up of static charge. The sign of theHall voltage is an indicator of the sign of the current carrying particles.

The Hall Electric field is given by

Ex ex = + vy Bz ey ∧ ez (1201)

The Hall voltage VH is related to the electric field and the width of the sampledx via

VH = − Ex dx = − vy Bz dx

= − jy Bz dx

ρ q

(1202)

Hence, measurement of the Hall voltage VH and jy, together with the magnitudeof the applied field Bz, determines the carrier density ρ and the charge q. This

416

is embodied in the definition of the Hall constant, RH

RH =Ey

jx Hz(1203)

which for semi-classical free carriers of charge q and density ρ is evaluated as

RH =1ρ q

(1204)

In other geometries, one notices that the current will flow in a direction otherthan parallel to the applied field. The conductivity tensor will not be diagonal,as will the resistivity tensor. The dependence of the resistivity on the magneticfield is known as magneto-resistance. The phenomenon of transport in a mag-netic field can be quite generally addressed from knowledge of the conductivitytensor in an applied magnetic field. This can be calculated using the Boltzmannequation approach.

The Boltzmann Equation.

The Boltzmann equation for the steady state distribution f(p), in the pres-ence of static electric and magnetic fields, can be expressed as

− | e |(E + v ∧ B

). ∇p f(p) = I

[f(p)

](1205)

Since only a solution for f(p) is sought which contain terms linear in the electricfield E, the equation can be linearized by making the substitution f(p) → f0(p)but only in the term explicitly proportional to E.

− | e | E . ∇p f0(p) − | e | ( v ∧ B ) . ∇p f(p) = I

[f(p)

](1206)

The substitution of the zero-field equilibrium distribution function f0(p) in thefirst term (without any magnetic field corrections) is consistent with the equi-librium in the presence of a static magnet field. This can be seen by examiningthe limit E = 0, where the Boltzmann equation reduces to

− | e | ( v ∧ B ) . ∇p f(p) = I

[f(p)

](1207)

which has the solution f(p) = f0(p) since in this case the collision integralvanishes and the remaining term is also zero as

( v ∧ B ) . ∇p f0(p) = ( v ∧ B ) . ∇p E(p)∂f0(p)∂E(p)

= ( v ∧ B ) . v∂f0(p)∂E(p)

= 0

(1208)

417

since due to the vector identity

( A ∧ B ) . A = 0 (1209)

the scalar product vanishes. This is just a consequence of the fact that a mag-netic field does not change the particles energy. The observation can be usedto simplify the Boltzmann equation as, has been seen, the magnetic force termonly acts on the deviation from equilibrium.

The ansatz for the steady state distribution function is

f(p) = f0(p) + Φ(p)∂f0(p)∂E(p)

= f0(p) + v . C∂f0(p)∂E(p)

(1210)

where C is an unknown vector function. It shall be shown that the vector Cis independent of p. In this case, the collision integral simplifies to the casethat was previously considered. Namely, the collision integral reduces to thetransport scattering rate times the non-equilibrium part of the steady statedistribution function. On cancelling the common factor involving the derivativeof the Fermi function, and using

∇p . v =1m∗ (1211)

one finds

| e | E . v +| e |m∗ c

( v ∧ B ) . C =1τtr

( v . C ) (1212)

The solution of this equation is independent of v, hence C is a constant vector.This can be seen explicitly by substituting the identity

( v ∧ B ) . C = ( B ∧ C ) . v (1213)

back into the Boltzmann equation. The resulting equation can be solved for allv if C satisfies the algebraic vector equation

| e | E +(| e |m∗ c

)B ∧ C =

1τtr

C (1214)

To solve the above algebraic equation it is convenient to change variables

ωc =(| e |m∗ c

)B (1215)

This shall be solved by finding the components parallel and transverse to B.

418

If the scalar product of the algebraic equation is formed with ωc and onrecognizing that ωc . ( ωc ∧ C ) = 0, one finds that the component of Cparallel to the magnetic field is given by

ωc . C = | e | τtr ωc . E (1216)

The transverse component of C can be obtained by taking the vector productof the algebraic equation with ω. This results in the equation

ωc ∧ ( | e | E ) + ωc ∧ ( ωc ∧ C ) =1τtr

ωc ∧ C (1217)

butωc ∧ ( ωc ∧ C ) = ωc ( ωc . C ) − ω2

c C (1218)

so one recovers the relation between the transverse component and C from

ωc ∧ ( | e | E ) + ωc ( ωc . C ) − ω2c C =

1τtr

ωc ∧ C (1219)

by eliminating the longitudinal component. The resulting relation is found as

ωc ∧ ( | e | E ) + ωc ( | e | τtr ωc . E ) − ω2c C =

1τtr

ωc ∧ C (1220)

The transverse component can be substituted back into the original algebraicequation to find the complete expression for C.

| e |(ωc ∧ E + τ ωc ( ωc . E )

)− ω2

c C =1τ2

C − 1τ| e | E (1221)

Therefore, C is given by the constant vector(1 + ω2

c τ2

)C = τ | e |

(E + τ2 ( ωc . E ) ωc + τ ( ωc ∧ E )

)(1222)

which only depends upon E and B but not on the momentum p. This leads tothe explicit expression for the non-equilibrium distribution function of

f(k) = f0(k)

+τ | e |

1 + ω2c τ

2

(v . E + τ2 ( v . ωc ) ( ωc . E ) + τ ( v ∧ ωc ) . E

)∂f0(k)∂E(k)(1223)

The deviation from equilibrium can be interpreted in terms of an anisotropicdisplacement of the Fermi function involving the work done by the electric field

419

on the electron in the time interval between scattering events.

The Conductivity Tensor.

The average value of the current density is given by

j = − | e | 2V

∑k

v(k) f(k) (1224)

where f(k) is the steady state distribution function, and the factor of 2 rep-resents the summation over the electrons spin. On substituting for the steadystate distribution function, and noting that because of the symmetry k → − kin the equilibrium distribution function, no current flows in the absence of theelectric field. The current density j is linear in the magnitude of the electricfield E, and is given by

j = 2 | e2 | τ

1 + ω2c τ

2

∫d3k

( 2 π )3v

(− ∂f0

∂E

)×

×

[( v . E ) + τ2 ( v . ωc ) ( ωc . E ) + ( v ∧ ωc ) . E

](1225)

Thus, the conductivity tensor is recovered in dyadic form as

σ = 2 | e2 | τ

1 + ω2c τ

2

∫d3k

( 2 π )3

(− ∂f0

∂E

)×

×

[v v + τ2 ( v . ωc ) v ωc + τ v ( v ∧ ωc )

](1226)

Furthermore, if E(k) is assumed to be spherically symmetric, one finds that thecomponents of the tensor can be expressed as

σα,β = e2∫

dE ρ(E)(− ∂f0∂E

)v2

3τ

1 + ω2 τ2

[δα,β + τ2ωα ωβ ± ( 1−δα,β ) τ ωγ

](1227)

where in the off-diagonal term the convention is introduced such that γ is chosensuch that (α, β, γ) corresponds to a permutation of (x, y, z). Since the densityof states per unit volume, ρ(E), is proportional to E

32 , the conductivity tensor

can be evaluated, by integration by parts, to yield

σα,β =ρ e2

m

τ

1 + ω2 τ2

[δα,β + τ2ωα ωβ ± ( 1 − δα,β ) τ ωγ

](1228)

where the ± sign is taken to be positive when (α, β, γ) are an odd permutation of(x, y, z) and is negative when (α, β, γ) are an even permutation of (x, y, z). Thus,

420

if the field is applied along the z direction it is found the diagonal componentsof the conductivity tensor are given by

σx,x = σy,y =ρ e2

m

τ

1 + ω2c τ

2

σz,z =ρ e2

mτ (1229)

The non-zero off-diagonal terms are found as

σy,x = − σx,y =ρ e2

m

ωc τ2

1 + ω2c τ

2(1230)

Thus, for the diagonal component of the conductivity tensor are anisotropic.The component parallel to the field is constant while the other two componentsdecrease like ω−2

c in high fields. The off-diagonal components are zero at zero-field, but increase linearly with the field for small ωc but then decreases like ω−1

c

at high fields.

A useful representation of the conductivity is through the Hall angle. Forexample, if one applies the magnetic field along the z-direction and then anelectric field along the x-direction Ex 6= 0, then the current will have an x andy component that can be characterized by a complex number z

z = Jx + i Jy

= σ0 Ex

(1 − i ωc τ

1 + ω2c τ

2

)= σ0

11 + i ωc τ

(1231)

This complex number z lies on a semi-circle of radius σ02 Ex centered on the

point (σ02 Ex, 0), as

z − σ0

2Ex =

σ0

2Ex

(1 − i ωc τ

1 + i ωc τ

)(1232)

and the modulus is just given by

| z − σ0

2Ex | =

σ0

2Ex (1233)

Thus, the number z lies on a semi-circle of radius σ02 Ex passing through the

origin. The Hall angle ΨH is defined as the angle between the line subtendedfrom the point z to the origin and the Jx axis. Thus

tan ΨH =Jy

Jx(1234)

and from the Boltzmann equation analysis of the magneto-conductivity

ΨH = tan−1 ωc τ (1235)

421

Thus, from knowledge of σ0 and E one can find z and, thence, J .

The resistivity tensor ρi,j is obtained from the conductivity tensor σi,j byinverting the relation

Ji =∑

j

σi,j Ej (1236)

to obtainEi =

∑j

ρi,j Jj (1237)

The resistivity tensor is found as

ρ =

ρ0 ρ0 ωc τ 0− ρ0 ωc τ ρ0 0

0 0 ρ0

Thus, for the free electron model the diagonal part of the resistivity tensor is

completely unaffected by the field. There is neither a longitudinal or transversemagneto-resistance.

However, as Hz increases, the transverse component of the electric field Ey

increases. This is the Hall field. The Hall field is given by

Ey = ωc τ ρ0 Jx =Jx Hz

ρ | e |(1238)

Thus, the Hall resistivity is

ρyx =Ey

Jx

=Hz

ρ | e |(1239)

Thus, the Hall constant RH is given by

RH =EY

Hz Jx=

1ρ | e |

(1240)

The magneto-resistivity is usually classified as being longitudinal or trans-verse. The longitudinal magneto resistance is the change in the resistivity tensorρz,z due to the application of a magnetic field along the z direction. The trans-verse magneto-resistance is given by the change in ρx,x or ρy,y due to a fieldHz. The longitudinal magneto-resistance is usually due to the dependence of thescattering rate on the magnetic field, whereas the transverse magneto-resistanceis due to the action of the Lorentz force.

The general features of the magneto-resistance are:-

422

(i) for low fields, Hz such that ωc τ < 1 then

∆ρx,x = ρx,x(Hz) − ρx,x(0) ∝ H2z (1241)

(ii) There is an electric field Ey transverse to Jx and Hz, which has a mag-nitude proportional to Hz.

(iii) For large fields ρx,x(Hz) may either continue to increase with H2z or

saturate.

(iv) For a set of samples all which have different residual resistivities ρzz(T =0,Hz = 0), then the transverse magneto resistance usually satisfies Koehler’srule

∆ρx,x(Hz)ρx,x(T = 0,Hz = 0)

= F

(Hz

ρx,x(0, 0)

)(1242)

Basically, Koehler’s law expresses the fact that ρ(Hz) only depends on Hz

through the combination ωc τ and that ∆ρx,x and ρzz(T = 0,Hz = 0) areboth proportional to τ−1.

The standard form of the relationship between E and J is expressed as avector equation

E = ρ0 J + a ( J ∧ H ) + b H2 J

+ c ( J . H ) H + d T J (1243)

where T is a tensor which only has diagonal components that, when referred tothe crystalline axes, are (H2

x , H2y , H

2z ). That is T is the matrix

H2x 0 0

0 H2y 0

0 0 H2z

(1244)

The five unknown quantities may be determined by five experiments.

(1) When J andH are parallel to the x axis there is the longitudinal magneto-resistance given by

ρx,x = ρ0 + ( b + c + d ) H2 (1245)

(2) When J ‖ x ,H ‖ y then

ρx,x = ρ0 + b H2 (1246)

423

which is the transverse magneto-resistance.(3) With J ‖ (1, 1, 0) i.e.

J =J√2

(1, 1, 0) (1247)

andH =

H√2

(1, 1, 0) (1248)

there is a different longitudinal magneto-resistance

∆ρ = ( b + c +d

2) H2 (1249)

(4) When H = H (0, 0, 1), then a second transverse magneto-resistance isfound as

∆ ρ = b H2 (1250)

but when H = H√2

(1,−1, 0) then the magneto-resistance is found as

∆ ρ = ( b +d

2) H2 (1251)

(5) The constant a makes no contribution to the magneto-resistance, but isfound from the Hall effect. If H is transverse to J then the Hall effect is onlydetermined by a alone, and is isotropic.

The magneto-resistance is usually negative except for cases where the scat-tering is of magnetic origin, such as disorder with spin - orbit coupling or fromKondo scattering by magnetic impurities in metals136.

11.2.5 Multi-band Models

The transverse magneto-resistance for a multi-band model is non-trivial, unlikethe one band free electron model. The resistance can be obtained from thecurrent field diagram, in which the currents originating from the various sheetsof the Fermi surface are considered separately.

For example, a two band model with positive and negative charge carriersproduces two components of the current J+ and J− by virtue of their responsesσ+, σ− in response to the electric field Ex. On assuming that the carriers havethe same Hall angles ΨH , then the total current is found as

Jx =(σ+ + σ−

2

)Ex ( 1 + cos 2 ΨH ) (1252)

136J. Kondo, Prog. Theor. Phys. 32, 37 (1964).

424

and

Jy =(σ+ − σ−

2

)Ex sin 2 ΨH (1253)

Thus,

σx,x = σ0 cos2 ΨH

σy,x = σ0

(ρ+ − ρ−ρ+ + ρ−

)sin ΨH cos ΨH

(1254)

Since the conductivity tensor is anisotropic and given by

σ =

σx,x σx,y 0− σy,x σx,x 0

0 0 σz,z

then the transverse magneto-resistivity can be found from

ρx,x =σx,x σz,z

σz,z ( σ2x,x + σ2

x,y )

=σx,x

( σ2x,x + σ2

x,y )

=1σ0

cos2 ΨH

( cos4 ΨH +(

ρ+ − ρ−ρ+ + ρ−

)2

sin2 ΨH cos2 ΨH )

=1σ0

1

( cos2 ΨH +(

ρ+ − ρ−ρ+ + ρ−

)2

sin2 ΨH )

=1σ0

sec2 ΨH

1 +(

ρ+ − ρ−ρ+ + ρ−

)2

ω2c τ

2

(1255)

since tanΨh = ωc τ .

Furthermore, as

sec2 ΨH = 1 + tan2 ΨH

= 1 + ω2c τ

2 (1256)

then

ρx,x =1σ0

( 1 + ω2c τ

2 )

1 +(

ρ+ − ρ−ρ+ + ρ−

)2

ω2c τ

2

(1257)

This saturates if | ρ+ − ρ− | > 0 and increases indefinitely for a compensatedmetal ρ+ = ρ−. Basically, the positive magneto-resistance occurs because the

425

Lorentz force produces a transverse component of the current in each sheet ofthe Fermi surface. The Lorentz force, the acting on these transverse currentsthen produces a shift of the Fermi surface opposite to the shift produced by theelectric field.

A similar analysis can be performed on the Hall coefficient

RH =E⊥H J

(1258)

The value of E⊥ is the component of the field perpendicular to the current.This is found from the angle θ between J and E

cos θ =Jx

J

sin θ =Jy

J(1259)

Thus

RH =E sin θH J

=E Jy

H J2

=1

σ0 H

(ρ+ − ρ−ρ+ + ρ−

)tan ΨH

cos2 ΨH

1 +(

ρ+ − ρ−ρ+ + ρ−

)2

tan2 ΨH

=ωc τ

σ0 H

(ρ+ − ρ−ρ+ + ρ−

)( 1 + ω2

c τ2 )

1 +(

ρ+ − ρ−ρ+ + ρ−

)2

tan2 ΨH

=1

| e | ( ρ+ − ρ− )

(ρ+ − ρ−ρ+ + ρ−

)2

( 1 + ω2c τ

2 )

1 +(

ρ+ − ρ−ρ+ + ρ−

)2

tan2 ΨH

(1260)

The Hall coefficient saturates to

RH → 1| e | ( ρ+ − ρ− )

(1261)

for large magnetic fields.

426

11.3 Electromagnetic Properties of Metals

Maxwell’s equations relate the electromagnetic field to charges and currentsources ρ(r; t) and j(r; t). Maxwell’s equations can be formulated as

∇ . E(r; t) = 4 π ρe(r; t)∇ . B(r; t) = 0

∇ ∧ B(r; t) =4 πc

j(r; t) +1c

∂E(r; t)∂t

∇ ∧ E(r; t) = − 1c

∂B(r; t)∂t

(1262)

where E andB represent the microscopic electric and magnetic fields, and ρe andj are the microscopic charge and current densities. There are eight equationsfor the six unknown quantities. The six unknown quantities are the componentsof E and B.

The sourceless equations have a formal solution in terms of a scalar potentialφ and a vector potential A which are related to the electric field E and magneticfield B via

E = − ∇ φ − 1c

∂A

∂tB = ∇ ∧ A (1263)

The solutions for the potentials are not unique, as the gauge transformations

A → A′ = A + ∇ Λ (1264)

andφ → φ′ = φ − 1

c

∂Λ∂t

(1265)

where Λ is an arbitrary scalar function, yield new scalar and vector potentials,(A′, φ′), that produce the same physical E and B fields as the original potentials(A,φ). The four quantities φ and A satisfy the four source equations

∇2 φ +1c

∂

∂t

(∇ . A

)= − 4 π ρe (1266)

and

∇2 A − 1c2

∂2A

∂t2− ∇

(∇ . A +

1c

∂φ

∂t

)= − 4 π

cj (1267)

These equations are usually simplified by choosing a gauge condition. The gaugeconditions which are usually chosen are either the Coulomb Gauge

∇ . A = 0 (1268)

427

or the Lorentz Gauge

∇ . A +1c

∂φ

∂t= 0 (1269)

The Lorentz gauge has the advantage that it is explicitly covariant under Lorentztransformations. The Coulomb gauge, also known as the transverse gauge orradiation gauge, is quite convenient for non-relativistic problems in that it sep-arates the effects of radiation from electrostatics.

The space and time Fourier transform of the charge density is defined as

ρe(q, ω) =1V

∫d3r

∫dt exp

[− i ( q . r − ω t )

]ρe(r; t) (1270)

On Fourier transforming the source equations with respect to space and time,one has

− q2 φ(q, ω) +ω

cq . A(q, ω) = − 4 π ρe(q, ω)[

− q2 +ω2

c2

]A(q, ω) + q

(q . A(q, ω) − ω

cφ(q, ω)

)= − 4 π

cj(q, ω)

(1271)

In the wave-vector and frequency domain, the Coulomb gauge condition is ex-pressed as

q . A(q, ω) = 0 (1272)

which shows that the vector potential is transverse to the direction of q. In thetransverse gauge, the equation for the vector potential reduces to[

− q2 +ω2

c2

]A(q, ω) − q

(ω

cφ(q, ω)

)= − 4 π

cj(q, ω)

(1273)

The first term is transverse and the second term is longitudinal. Thus, thecurrent can also be divided into a longitudinal term

jL(q, ω) = q

(q . j(q, ω)

)(1274)

and a transverse term

jT(q, ω) = j(q, ω) − q

(q . j(q, ω)

)(1275)

Thus, the second non-trivial Maxwell equation separates into the transverseequation [

− q2 +ω2

c2

]A(q, ω) = − 4 π

cj

T(q, ω) (1276)

428

and the longitudinal equation

− qω

cφ(q, ω) = − 4 π

cj

L(q, ω) (1277)

In the Coulomb gauge, the other non-trivial Maxwell equation relates the chargedensity to the scalar potential via

− q2 φ(q, ω) = − 4 π ρe(q, ω) (1278)

This is just Poisson’s equation, and it has the solution

φ(q, ω) =4 πq2

ρe(q, ω) (1279)

which is equivalent to Coulomb’s law. When Fourier transformed with respectto space and time, Poisson’s equation yields an instantaneous relation betweenthe charge density and the scalar potential in the form of Coulomb’s law. Al-though this is an instantaneous relation, the signals transmitted by the electro-magnetic field still travel with speed c and are also causal. This is because, inthe Coulomb gauge, the retardation effects are contained in the vector potential.

Poisson’s equation actually has the same content as the longitudinal equa-tion. This can be seen by examining the continuity equation which expressesconservation of charge

∂ρe

∂t+ ∇ . j = 0 (1280)

The continuity equation can be Fourier transformed to yield

− ω ρe(q, ω) + q . j(q, ω) = 0 (1281)

This shows that the fluctuations in the charge density are related to the longi-tudinal current. On solving the continuity condition, one finds that the longi-tudinal current is given by

jL(q, ω) = q

ω

qρe(q, ω) (1282)

On substituting the above expression for the longitudinal current into the lon-gitudinal equation, one finds

− qω

cφ(q, ω) = − 4 π

cqω

qρe(q, ω) (1283)

On cancelling the factors of ω/c and q, one obtains Poisson’s equation. Thisproves that the longitudinal equation is equivalent to Poisson’s equation. Wehave also found that the longitudinal current can be expressed in the forms

jL(q, ω) = q

ω

qρe(q, ω)

=q ω

4 πφ(q, ω) (1284)

so the longitudinal current can be viewed as being produced either by the chargedensity or by the scalar potential.

429

11.3.1 The Longitudinal Response

The currents and charge densities are usually broken down into the externalcontributions and the induced contributions via

j(q, ω) = jind

(q, ω) + jext

(q, ω)ρe(q, ω) = ρe ind(q, ω) + ρe ext(q, ω) (1285)

The external scalar potential is defined in terms of the external charge densityvia Poisson’s equation

− q2 φext(q, ω) = − 4 π ρe ext(q, ω) (1286)

The frequency and wave vector-dependent dielectric constant for a homogeneousmedium, ε(q, ω), is defined by the ratio

ε(q, ω) =φext(q, ω)φ(q, ω)

(1287)

The dielectric constant describes the screening of the external potential by lon-gitudinal or charge density fluctuations. The dielectric constant is related tothe longitudinal conductivity. This can be seen by reducing the relation

jL(q, ω) =

q ω

4 πφ(q, ω) (1288)

into an expression for the induced component of the longitudinal current

jL(q, ω)ind =

q ω

4 π

(φ(q, ω) − φext(q, ω)

)(1289)

Hence, on using the definition of the frequency-dependent dielectric constant toeliminate φext(q, ω), one obtains

jL(q, ω)ind =

q ω

4 π

(1 − ε(q, ω)

)φ(q, ω) (1290)

The total scalar potential φ(q, ω) can be related to the longitudinal electricfield, EL(q, ω), since the electric field can be written as the sum of the timedependence of the vector potential and the gradient of the scalar potential

E(q, ω) =i ω

cA(q, ω) − i q φ(q, ω) (1291)

If the longitudinal part of the electric field is identified as

EL(q, ω) = − i q φ(q, ω) (1292)

then one obtains the relation between the longitudinal current and the longitu-dinal electric field in teh form

jL(q, ω)ind =

i ω

4 π

(1 − ε(q, ω)

)EL(q, ω) (1293)

430

Hence, as the longitudinal conductivity σL is defined by the relation

jL(q, ω)ind = σL(q, ω) EL(q, ω)

(1294)

one finds that the conductivity and the dielectric constant are related through

σL(q, ω) =i ω

4 π

(1 − ε(q, ω)

)(1295)

The frequency-dependent dielectric constant can be expressed in terms of theresponse of the charge density due to the potential

ε(q, ω) =φext(q, ω)φ(q, ω)

ε(q, ω) =φ(q, ω) − φind(q, ω)

φ(q, ω)

ε(q, ω) = 1 − 4 πq2

ρe ind(q, ω)φ(q, ω)

(1296)

The charge density is related to the electron density via a factor of the electron’scharge

ρe ind(q, ω) = − | e | ρind(q, ω) (1297)

and the scalar potential acting on the electrons produces the potential δV (q, ω)where

δV (q, ω) = − | e | φ(q, ω) (1298)

Thus, the frequency-dependent dielectric constant may be written137 as

ε(q, ω) = 1 − 4 π e2

q2ρind(q, ω)δV (q, ω)

= 1 − 4 π e2

q2χ(q, ω) (1299)

where we have used the definition of the frequency-dependent response functionχ(q, ω). The frequency-dependent response function is defined by

χ(q, ω) =ρind(q, ω)δV (q, ω)

(1300)

The real space and time form of the linear response relation can be found byre-writing this relation as

ρind(q, ω) = χ(q, ω) δV (q, ω) (1301)

137H. Ehrenreich and M. H. Cohen, Phys. Rev. 115, 786 (1959).

431

and then performing the inverse Fourier transform. The real space and timeform of the linear response relation is in the form of a convolution

ρind(r, t) =∫

d3r′∫ ∞

−∞dt′ χ(r − r′; t− t′) δV (r′, t′) (1302)

The dependence of the response function on r − r′ is a direct consequence ofour assumption that space is homogeneous. As the response function relatesthe cause and effect in a linear fashion, the response function can be calculatedperturbatively. The induced electron density is found, in real space and time, bytreating the time-dependent potential as a perturbation. The resulting causal,non-local relation is then Fourier transformed with respect to space and time.This procedure is a generalization of our previous treatment of static screening.

The expectation value of the electron density operator ρ(r) at time t, is cal-culated in a state that has evolved from the ground states due to the interaction.The electron density operator is given by

ρ(r) =∑

i

δ3(r − ri

)(1303)

and the time-dependent perturbation is

Hint(t) =∫

d3r′ ρ(r′, t) δV (r′, t) (1304)

The expectation value of the electron density is to be evaluated in the interactionrepresentation. The expectation value of the density is given by

ρ(r, t) = < Ψint(t) | ρint(r, t) | Ψint(t) > (1305)

where the state and operators are expressed in the interaction representation. Inthe interaction representation, the operators evolve with respect to time underthe influence of the unperturbed Hamiltonian H0, and are given by

ρint(r, t) = exp[

+i t

hH0

]ρ(r) exp

[− i t

hH0

](1306)

In the interaction representation, the state evolves under the influence of theinteraction Hint(t). To first order in the perturbation, the ground state is givenby

| Ψint(t) > =

[1 − i

h

∫ t

−∞dt′ Hint(t′) + . . .

]| Ψ0 > (1307)

where | Ψ0 > is the initial ground state eigenfunction of H0. The inducedelectron density is defined as

ρind(r, t) = < Ψint(t) | ρint(r, t) | Ψint(t) > − < Ψ0 | ρint(r, t) | Ψ0 >

(1308)

432

The second term is time independent, as it is the expectation value in the groundstate of the time independent Hamiltonian H0. On substituting the expressionfor the perturbed wave function, one finds a linear relationship between theinduced density and the perturbing potential

ρind(r, t) = − i

h

∫ t

−∞dt′ < Ψ0 |

[ρint(r, t) , Hint(t′)

]| Ψ0 >

= − i

h

∫ t

−∞dt′

∫d3r′ < Ψ0 |

[ρint(r, t) , ρint(r′, t′)

]| Ψ0 > δV (r′, t′)

=∫ +∞

−∞dt′

∫d3r′ χ(r, r′; t− t′) δV (r′, t′) (1309)

This is a causal relation in which the response function is identified as

χ(r, r′; t− t′) = − i

h< Ψ0 |

[ρint(r, t) , ρint(r′, t′)

]| Ψ0 > Θ( t − t′ )

(1310)where Θ(t) is the Heaviside step function. Thus, the response function is atwo time correlation function which involves the ground state expectation valueof the commutator of the density operators at different positions, and differ-ent times. Due to the time homogeneity of the ground state, the correlationfunction only depends on the difference of the two times. For a spatially ho-mogeneous system, the correlation function only depends on the difference r−r′.

The expression can be evaluated by using the completeness relation∑n

| Ψn > < Ψn | = I (1311)

On inserting a complete set of states between the density operators, one obtains

χ(r, r′; t− t′) = − i

h

∑n

[< Ψ0 | ρint(r, t) | Ψn > < Ψn | ρint(r′, t′) | Ψ0 >

− < Ψ0 | ρint(r′, t′) | Ψn > < Ψn | ρint(r, t) | Ψ0 >

]Θ( t − t′ )

(1312)

On expressing the time dependence of the operators in terms of the eigenvaluesof the unperturbed Hamiltonian, H0, the response function reduces to

= − i

h

∑n

exp[

+i

h(t− t′)(E0 − En)

]< Ψ0 | ρ(r) | Ψn > < Ψn | ρ(r′) | Ψ0 >

+i

h

∑n

exp[− i

h(t− t′)(E0 − En)

]< Ψ0 | ρ(r′) | Ψn > < Ψn | ρ(r) | Ψ0 >

(1313)

433

for t − t′ > 0, and is zero otherwise. In the above expression for the responsefunction, the density operators are no longer time-dependent.

Up to this point, our analysis has been completely general. To illustrate thestructure of the response function, we shall now make the assumption that theelectrons are non-interacting. The ground state | Ψ0 > and the excited states| Ψn > can be represented by single Slater determinants composed of the set ofone-electron energy eigenfunctions φαj

(rj); j ∈ 1, 2, . . . Ne and φβj (rj); j ∈1, 2, . . . Ne, respectively. The matrix elements of the one-electron operatorρ(r) are non-zero only if the set of quantum numbers αj ; j ∈ 1, 2, . . . Ne andβj ; j ∈ 1, 2, . . . Ne only differ by at most one element, say the i-th value. Thus,we may permute the indices in the set βj until one has

αi 6= βi (1314)

andαj = βj ∀ j 6= i (1315)

In this case, the matrix elements < Ψ0 | ρ(r) | Ψn > are trivially evaluatedas

< Ψ0 | ρ(r) | Ψn > =∫

d3ri φ∗αi

(ri) δ3( r − ri ) φβi

(ri)

= φ∗αi(r) φβi(r) (1316)

The matrix element is only non-zero if the spin state of α is identical to the spinstate of β, so the spin quantum number is conserved. In the above expression,the single-electron state αi is occupied in the initial state | Ψ0 > and unoccu-pied in the final state | Ψn > while the single-electron state βi is unoccupied inthe initial state | Ψ0 > and occupied in the final state | Ψn > . All the othersingle-electron quantum numbers in | Ψ0 > and | Ψn > are unchanged, i.e.,αj = βj for ∀ j 6= i. Furthermore, the Pauli exclusion principle requires thatβi 6= βj . This shows that the final states of the non-interacting many-electronsystem are obtained by exciting a single electron from the state αi to the stateβi. For non-interacting electrons, the excitation energy En − E0 is simplygiven by the difference in the single-electron energy eigenvalues

En − E0 = Eβi− Eαi

(1317)

Thus, the response function is simply given by

χ(r, r′; t) = − i

h

∑α,β

exp[

+i

ht ( Eα − Eβ )

]φ∗α(r) φβ(r) φ∗β(r′) φα(r′)

+i

h

∑α,β

exp[− i

ht ( Eα − Eβ )

]φα(r) φ∗β(r) φβ(r′) φ∗α(r′)

(1318)

434

for t > 0. The sum over α is restricted to run over the single-particle quantumnumbers that are occupied in the ground state, and the sum over β runs over thequantum numbers that are unoccupied in the ground state. The spin quantumnumber is conserved, that is, σα = σβ .

On evaluating the response function for free electrons, summing over spinstates and using the Bloch state energy eigenvalues, one finds

= − 2 ih V 2

∑|k|<kF

∑|k′|>kF

exp[

+i t

h(Ek − Ek′)

]exp

[− i

h(k − k′) . (r − r′)

]

+2 ih V 2

∑|k|<kF

∑|k′|>kF

exp[− i t

h(Ek − Ek′)

]exp

[+i

h(k − k′) . (r − r′)

](1319)

for t > 0. Since the free electron gas is homogeneous, the response functiononly depends on the distance between the perturbation and the response r− r′.On Fourier transforming the response function with respect to space and time,one obtains χ(q, ω) as

χ(q, ω) =∫ +∞

−∞dt

∫d3r exp

[− i ( q . r − ω t )

]χ(r; t) (1320)

Since the response function contains the Heaviside step function Θ(t), the inte-gral over t can be evaluated in the interval ∞ > t ≥ 0. The integral over tconverges faster if ω is analytically continued into the upper-half complex planeto ω → z = ω + i η. The factor of exp [ − η t ] damps out the oscillationsin the integrand as t → ∞. Thus, in the (q, ω) domain, one finds that theresponse function is complex and is given by the expression

χ(q, ω + iη) =2V

∑|k|<kF |k+q|>kF

[1

h ω + i η + Ek − Ek+q

]

− 2V

∑|k|>kF |k+q|<kF

[1

h ω + i η + Ek − Ek+q

](1321)

The restrictions on the summation over k can be simplified. To see this, weshall introduce a function fk which behaves like the T → 0 limit of the Fermifunction. The function is defined by

fk = 1 for Ek < EF (1322)

andfk = 0 for Ek > EF (1323)

435

-5

0

5

10

15

0 1 2 3 4

q/kF

2mω

/ kF2 h

2kFq+q2

-2kFq+q2

2kFq-q2

-2kFq-q2

Ek+q-Ek

Ek-Ek-q

Figure 202: The region of (ω, q) space available for single-particle excitations,subject to the constraint that the initial state is inside the Fermi sphere. Theregion where the first delta function of eqn(1325) is non-zero is bounded by thetwo blue lines.

The response function can then be written as the sum over all k as

χ(q, ω + iη) =2V

∑k

[fk ( 1 − fk+q )

h ω + i η + Ek − Ek+q

]

− 2V

∑k

[fk+q ( 1 − fk )

h ω + i η + Ek − Ek+q

]

=2V

∑k

[fk − fk+q

h ω + i η + Ek − Ek+q

](1324)

In the last line, it is seen that the factors which explicitly enforce the Pauli-exclusion principle cancel. For ω just above the real axis, i.e in the limit η → 0,the imaginary part of the response function is found as

=m χ(q, ω + iη) = − 2 πV

∑|k|<kF

δ

(h ω + Ek − Ek+q

)

+2 πV

∑|k+q|<kF

δ

(h ω + Ek − Ek+q

)(1325)

From this analysis, one can see that for positive ω, the imaginary part of χ(q, ω)is non-zero in the region of (ω, q) phase space, where

h

2 m( − 2 kF q + q2 ) < ω <

h

2 m( + 2 kF q + q2 ) (1326)

436

It is only in this region that the argument of the first delta function in =m χ(q, ω)has a solution

h

mk . q = ω − h

2 mq2 (1327)

with k < kF . These conditions divide (q, ω) space into non-overlapping regions.

For completeness, the complete expressions for the real and imaginary partsof the Lindhard dielectric function at finite frequencies are given138. The realpart is given by

<e[ε(q, ω)

]= 1 +

k2TF

2 q2

[1 +

kF

2 q

(1 − (2 m ω − h q2)2

4 h2 q2 k2F

)×

× ln

∣∣∣∣∣2 m ω − 2 h q kF − h q2

2 m ω + 2 h q kF − h q2

∣∣∣∣∣+(

1 − (2 m ω + h q2)2

4 h2 q2 k2F

)ln

∣∣∣∣∣2 m ω + 2 h q kF + h q2

2 m ω − 2 h q kF + h q2

∣∣∣∣∣ ]

(1328)

and the imaginary part is given by

=m[ε(q, ω + iη)

]=

π

2k2

TF

q2m ω

h q kF2 m ω < 2 h q kF − h q2

(1329)

=m[ε(q, ω + iη)

]=

π

4k2

TF

q2kF

q

[1 − (2 m ω − h q2)2

4 h2 q2 k2F

]2 h q kF − h q2 < 2 m ω < 2 h q kF + h q2 (1330)

and

=m[ε(q, ω + iη)

]= 0 2 h q kF + h q2 < 2 m ω (1331)

The real part is an even function of ω and the imaginary part is an odd functionof ω. For ω = 0, the response function reduces to the real static responsefunction calculated previously. For | ω | > h

2 m ( 2 kF q+ q2 ), the imaginarypart of the function vanishes, since the denominator never vanishes for any kvalue in the range of integration. In this region of q and ω, there are no poles,therefore, the real part of the response function χ(q, ω) can be expanded inpowers of q2. To the order of q4, one finds

<e χ(q, ω) = +k3

F

3 π2

q2

m ω2

[1 +

35

(h kF q

m ω

)2

+ . . .

](1332)

138J. Lindhard, Kgl. Danske Videnskab. Selskab. Mat.-Fys. Medd. 28, 8 (1954).

437

-80

-40

0

40

80

-4 -3 -2 -1 0 1 2 3 4

2mω/hkF2

ε(q,ω

)

q/kF = 0.3

Figure 203: The frequency dependence of the real (blue) and imaginary (red)parts of the (R.P.A.) dielectric constant for a q value such that q < 2kF .

-0.5

0

0.5

1

1.5

-20 -10 0 10 20

2mω/hkF2

ε(q,ω

)

q/kF = 2.5

Figure 204: The frequency dependence of the real (blue) and imaginary (red)parts of the (R.P.A.) dielectric constant for a q value such that q > 2kF .

438

Thus, for high frequencies such that ω q h kF

m the dielectric constant canbe approximated by

ε(q, ω) = 1 − 4 π ρ e2

m ω2

[1 +

35

(h kF q

m ω

)2

+ . . .

]

= 1 −ω2

p

ω2

[1 +

35

(h kF q

m ω

)2

+ . . .

](1333)

where the expression for the electron density ρ

ρ = 2k3

F

6 π2(1334)

has been used, and the plasmon frequency ωp has been defined via

ω2p =

4 π ρ e2

m(1335)

Thus, the dielectric constant has zeros at the frequencies ω = ωp(q), where

ω2p(q) = ω2

p

[1 +

35

(h kF q

m ωp(q)

)2

+ . . .

](1336)

The finite value of the frequency ωp(0), is a direct consequence of the long-ranged nature of the Coulomb interaction. If the external potential is zero,φext(q, ωp(q)) = 0, and the total potential is non-zero φ(q, ωp(q)) 6= 0, then thereal and imaginary parts of the dielectric constant must vanish, ε(q, ωp(q)) = 0,as

ε(q, ωp(q)) φ(q, ωp(q)) = φext(q, ωp(q))ε(q, ωp(q)) φ(q, ωp(q)) = 0 (1337)

In this case, when the total potential inside the solid φ(q, ωp(q)) is non-zero,the induced density and current fluctuations must be finite. These longitudinalcollective charge oscillations excitations are plasmons. A typical energy rangefor the plasmon energy, h ωp, in metals, ranges from the low values of 3.72 eVfound in K, 5.71 eV found in Na, to values as high as 15.8 eV found in Al. Thedielectric materials Si, Ge etc. also have plasmon energies of the order 16 eV.

One may inquire as to the nature of the excitations at larger q values, suchthat the phase velocity of the plasmons ωp

q becomes smaller than the Fermivelocity vF = h kF

m . At a critical value of q the denominator of the responsefunction may vanish so the response function acquires a sizeable imaginary part.The plasmon excitations merge with a continuum of particle hole excitationswhich have excitation energies given by

h ω(q, k) = Ek+q − Ek (1338)

439

-5

0

5

10

15

0 1 2 3 4

q/kF

2mω

/ kF2 h

2kFq+q2

-2kFq+q2

2kFq-q2

-2kFq-q2

Single-particle excitations

2mωp(q)/kF2h

Damped Plasmons

Figure 205: The phase space of collective (plasmon) excitations and single-particle excitations. The branch of plasmon excitations becomes damped as itmerges with the continuum of electron-hole excitations.

for k < kF . The edges of the continuum stretch from h2

2 m ( 2 kF q + q2 ) toh2

2 m ( − 2 kF q + q2 ). When the plasmon merges into the continuum, it un-dergoes significant broadening. This sort of damping is called Landau damping.Landau damping can also be viewed classically, in terms of electrons surf-ridingthe waves in the potential field. Imagine that a wave with phase velocity ω

q ispropagating through an electron gas, and consider the electrons with velocityalmost parallel and close to the phase velocity of the wave. In the frame ofreference travelling with the wave, the electron is at rest and experiences an es-sentially time independent electric field. The electric field continuously transfersenergy from the wave to the electrons that have the same velocity. If there isa slight mismatch in the velocities, electrons with lower velocity than the wavedraw energy from the wave and accelerate, whereas electrons that are movingfaster lose energy and slow down. This has the consequence that the rate ofenergy loss of the wave is proportional to the derivative of the electron velocitydistribution, evaluated at the wave’s phase velocity.

11.3.2 Electron Scattering Experiments

The longitudinal excitations of the electrons in a metal can be probed by thescattering of a beam of charged particles or fast electrons. The coupling takesplace via the Coulomb interaction

Hint =∑

i

e2

| r − ri |(1339)

440

0

10

20

30

40

50

3 3.5 4 4.5 5

2mω/hkF2

Im[ ω

/ε(q,ω) ]

q/kF = 1.3

Figure 206: The Landau damped plasmon pole in the inverse of the (R.P.A.)dielectric constant.

θ

e

e

k

k' Detector

Thin Film

Figure 207: A beam of high-energy electrons scatters from a thin metal film.

where ri labels the positions of the electrons in the plasma and r is the posi-tion of the incoming high energy electron. If the incident beam is composedof electrons which have high energies, the beam electrons can be considered tobe as classical and are, therefore, distinguishable. This ignores the possibilityof exchange interactions with the electrons in the metal. Analysis of the Mottscattering formula for electrons also shows that the neglect of the exchangescattering is an excellent approximation for scattering through small angle scat-tering. Therefore, we shall consider the charged particles in the beam as beingdistinguishable from the electrons in the solid.

The rate at which a charged particle is scattered inelastically from state k

441

with energy E(k) to state k′ with energy E(k′) is given by

1τ(k → k′)

=2 πh

∣∣∣∣ < k′ Ψn | Hint | k Ψ0 >

∣∣∣∣2 δ( En + E(k′)− E0 − E(k))

(1340)where | Ψ0 > and E0 are the ground state wave function and ground stateenergy of the solid. The final state wave function and energy is given by | Ψn >and En. The momentum and energy loss of the charged particle are defined tobe

h q = h k − h k′

h ω = E(k) − E(k′) (1341)

On performing the integral over the position of the fast charged particle, onehas

1τ(k → k′)

=2 πh

(4 π e2

q2 V

)2 ∣∣∣∣ < Ψn |∑

i

exp[ i q . ri ] | Ψ0 >

∣∣∣∣2 ×

δ

(h ω + E0 − En

)(1342)

The energy conserving delta function can be replaced by an integral over timeby using the identity

δ

(h ω + E0 − En

)=∫ ∞

−∞

dt

2 π hexp[ i ω t ] exp

[it

h( E0 − En )

](1343)

The energy eigenvalues in the exponential time evolution factors can be replacedby the general time evolution operators involving the unperturbed Hamiltonianoperator H0,

1τ(k → k′)

=2 πh2

(4 π e2

q2 V

)2 ∫ ∞

−∞

dt

2 πexp[ i ω t ] < Ψn |

∑i

exp[ i q . ri ] | Ψ0 > ×

< Ψ0 |∑

i

exp[ iH0 t

h] exp[ − i q . ri ] exp[ − i

H0 t

h] | Ψn >

(1344)

The factor involving ri can be expressed as the Fourier transform of the electrondensity operator

1V

∑i

exp[ − i q . ri ] =1V

∫d3r exp[ − i q . r ]

∑i

δ3(r − ri

)=

1V

∫d3r exp[ − i q . r ] ρ(r)

= ρq (1345)

442

Thus, on combining the above expressions, the inelastic scattering rate is foundas

1τ(k → k′)

=2 πh2

(4 π e2

q2

)2 ∫ ∞

−∞

dt

2 πexp[ i ω t ] < Ψn | ρ−q | Ψ0 > ×

< Ψ0 | exp[ iH0 t

h] ρq exp[ − i

H0 t

h] | Ψn >

=2 πh2

(4 π e2

q2

)2 ∫ ∞

−∞

dt

2 πexp[ i ω t ] < Ψ0 | ρq(t) | Ψn > < Ψn | ρ−q(0) | Ψ0 >

(1346)

where the density operator is evaluated in the interaction representation. If thefinal state of the solid | Ψn > is not measured, there is a distribution of possiblefinal states of the solid. If only the final state of the charged particle is measured,and the final state of the solid is not measured, the index n corresponding tothe different possible final states must be summed over

1τ(k → k′)

=∑

n

2 πh2

(4 π e2

q2

)2 ∫ ∞

−∞

dt

2 πexp[ i ω t ] ×

< Ψ0 | ρq(t) | Ψn > < Ψn | ρ−q(0) | Ψ0 >

(1347)

The sum over the final states can be evaluated using the completeness relationwhich leads to the result

1τ(k → k′)

=2 πh2

(4 π e2

q2

)2 ∫ ∞

−∞

dt

2 πexp[ i ω t ] < Ψ0 | ρq(t) ρ−q(0) | Ψ0 >

(1348)

The factor of q−4 shows the scattering process is dominated by small momentumtransfers. The density-density correlation function, S(q, ω), is defined via

S(q, ω) = V 2

∫ ∞

−∞

dt

2 π hexp[ i ω t ] < Ψ0 | ρq(t) ρ−q(0) | Ψ0 > (1349)

On substituting this relation into the scattering rate, we obtain the result

1τ(k → k′)

=2 πh

(4 π e2

q2 V

)2

S(q, ω) (1350)

Thus, it is seen that the long wavelength electron density fluctuations are mainlyresponsible for scattering the incident charged particle. For non-interactingelectrons, S(q, ω) is evaluated as

S(q, ω) = 2∑

|k|<kF |k+q|>kF

δ

(h ω + Ek − Ek+q

)(1351)

443

where the summation over k is over the filled Fermi sphere, subject to the re-striction that the final state be allowed by the Pauli exclusion principle.

The inelastic scattering cross-section can be evaluated in terms of the scat-tering rate, and is found to be given by the expression

d2σ

dΩ dω=

k′

k

(2 mq e

2

h2 q2

)2

h S(q, ω) (1352)

where mq is the mass of the charged particle. Thus, in the Born approximation,the scattering cross-section is directly related to the density-density correlationfunction. This type of correlation function was first introduced by van Hove inthe context of neutron scattering139. Since the Coulomb interaction is relativelystrong, it is frequently necessary to incorporate the effects of multiple-scatteringin order to obtain agreement with experimental data.

The Fluctuation-Dissipation Theorem140 relates the spectrum of electrondensity fluctuations to the imaginary part of the dielectric constant. At finitetemperatures, this relation has the form

S(q, ω) =q2 V

4 π2 e2

[1

exp[ − β h ω ] − 1

]=m

[1

ε(q, ω + iη)

](1353)

The relation between S(q, ω) and the inverse dielectric constant can be seenthrough the following classical argument. The power, per unit volume, dissi-pated by the electromagnetic field of the charged particle is given by

P (r, t) =1

4 πE .

∂D

∂t(1354)

For a negatively charged particle travelling with velocity v(t), the displacementfield D(r, t) is the experimentally controllable quantity and is given by theexpression

D(r, t) = − ∇

[− | e |

| r − r(t) |

](1355)

On Fourier transforming D(r, t) with respect to space and time, one findsD(q, ω). However, D(q, ω) is related to the Fourier transform of the electricfield E(q, ω) via a factor of the dielectric constant

E(q, ω) =D(q, ω)

ε(q, ω + iη)(1356)

On Fourier transforming the expression for the power density, P (r, t), withrespect to r and t, one finds P (q, ω) to be given by

P (q, ω) = − ω

8 π=m

[1

ε(q, ω + iη)

] ∣∣∣∣ D(q, ω)∣∣∣∣2

139L. van Hove, Phys. Rev. 95, 249 (1954).140H. B. Callen and T. A. Welton, Phys. Rev. 83, 34 (1951).

444

=ω

8 π

[=m ε(q, ω + iη)| ε(q, ω + iη) |2

] ∣∣∣∣ D(q, ω)∣∣∣∣2

=ω

8 π

[=m ε(q, ω + iη)

( <e ε(q, ω) )2 + ( =m ε(q, ω + iη) )2

] ∣∣∣∣ D(q, ω)∣∣∣∣2

(1357)

This result implies that the zero in the real part of ε(q, ω) should show up as adelta function peak in the power loss.

——————————————————————————————————

11.3.3 Exercise 64

Use linear response theory to express the change in the electron density inducedby an external charge. Hence, express the inverse of the dielectric constant interms of the exact eigenstates and energy eigenvalues of the interacting many-electron system. Use the resulting expression to find the T = 0 form of thefluctuation-dissipation theorem141.

——————————————————————————————————

Solution 64

In the Coulomb gauge, the Fourier transform of the external charge densityρext(r, t) is related to the external potential via Poisson’s theorem

− q2 φext(q, ω) = − 4 π | e | ρext(q, ω) (1358)

The total field is related to the external charge density and the induced chargedensity via

− q2 φ(q, ω) = − 4 π | e | ρext(q, ω) − 4 π | e | ρind(q, ω) (1359)

The dielectric constant is defined as

1ε(q, ω)

=φ(q, ω)φext(q, ω)

(1360)

which can be expressed as

1ε(q, ω)

= 1 +ρind(q, ω)ρext(q, ω)

(1361)

141P. Nozieres and D. Pines, Nuovo Cimento, 9, 470 (1958).

445

Hence, the linear response relation can be expressed as

ρind(q, ω) =(

1ε(q, ω)

− 1)ρext(q, ω)

=(

1ε(q, ω)

− 1)

q2

4 π | e |φext(q, ω)

=(

1 − 1ε(q, ω)

)q2

4 π e2δV (q, ω) (1362)

However, the interaction operator is given by

Hint(r, t) =∫

d3r ρ(r) δV (r, t) (1363)

where the electron density is given by

ρ(r) =∑

i

δ3(r − ri) (1364)

The induced electron density is evaluated using linear perturbation theory, inthe interaction representation. In the absence of the perturbation, the groundstate is denoted by | Ψ0 > . The perturbation is turned on adiabatically, andthe ground state evolves to the state | Φ0(t) > which, to first order in theinteraction, is given by

| Φ0(t) > =(

1 − i

h

∫ t

−∞dt′ Hint(t′)

)| Ψ0 > (1365)

The induced electron density ρind(r, t) is then given by the expectation value ofthe commutator

ρind(r, t) = − i

h

∫ t

−∞dt′ < Ψ0 | [ ρ(r, t) , Hint(t′) ] | Ψ0 >

= − i

h

∫ t

−∞dt′

∫d3r′ < Ψ0 | [ ρ(r, t) , ρ(r′, t′) ] | Ψ0 > δV (r′, t′)

=∫ ∞

−∞dt′

∫d3r′ χ(r − r′, t− t′) δV (r′, t′) (1366)

The material is assumed to be homogeneous, therefore, the response functionis only a function of the spatial separation r − r′. Furthermore, since H0 isindependent of time, the response function is only a function of t − t′. OnFourier transforming this equation with respect to space and time, one finds

ρind(q, ω) = χ(q, ω) δV (q, ω) (1367)

where

χ(q, ω) = V∑

n

[ | < Ψ0 | ρq | Ψn > |2

h ω + i η + En − E0

+| < Ψ0 | ρq | Ψn > |2

− h ω − i η + En − E0

](1368)

446

The imaginary part of the response function is found as

=mχ(q, ω) = − π V∑

n

| < Ψ0 | ρq |Ψn > |2[δ( h ω + En − E0 )− δ( h ω − En + E0 )

](1369)

Thus, the zero temperature limit of the fluctuation-dissipation theorem has theform

=m 1ε(q, ω)

=4 π2 e2 V

q2

∑n

| < Ψ0 | ρq | Ψn > |2[δ(hω + En − E0)− δ(hω − En + E0)

]=

4 π2 e2

q2 V

[S(q,−ω) − S(q, ω)

](1370)

The first term is only non-zero if 0 > ω, and the second term is only non-zeroin the range ω > 0.

——————————————————————————————————

11.3.4 Exercise 65

Show, using classical electromagnetic theory, that the power loss spectrum of aparticle with charge e moving with velocity v due to plasmons, can be expressedas

P (ω) = − 2 e2

π v=m

[ω

ε(ω + iη)

]lnq0 v

ω(1371)

Assume that the dielectric constant is independent of q, for q < q0 where h q0is the maximum momentum transfer.

——————————————————————————————————

Solution 65

The average power P dissipated by the charged particle can be expressed asthe limit τ → ∞

P =1τ

∫ ∞

0

dt P (t) exp[− t

τ

]=

1τ

∫ ∞

0

dt

∫d3r P (r, t) exp

[− t

τ

]=

14 π τ

∫ ∞

0

dt

∫d3r E(r, t)

∂

∂tD(r, t) exp

[− t

τ

](1372)

where we have inserted an exponential convergence factor. The convergencefactor will be absorbed in the displacement and electric fields. The Fourier

447

transform is expressed as

D(q, ω) =1V

∫d3r

∫ ∞

−∞dt D(r, t) exp

[− i ( q . r − ω t )

](1373)

and the inverse Fourier transformation is given by

D(r, t) =V

( 2 π )3

∫d3q

∫dω

2 πD(q, ω) exp

[+ i ( q . r − ω t )

](1374)

On inserting the expressions for the inverse Fourier transforms into the expres-sion for the average power loss, one finds

P = − i V 2

2 τ ( 2 π )

∫ ∞

−∞

dω

2 π

∫d3q

( 2 π )3E(−q,−ω) ω D(q, ω)

=i V 2

2 τ ( 2 π )

∫ ∞

−∞

dω

2 π

∫d3q

( 2 π )3D(q, ω)

ε(q, ω + i2τ )

ω D∗(q, ω)

=i V 2

2 τ ( 2 π )

∫ ∞

−∞

dω

2 π

∫d3q

( 2 π )3ω

ε(q, ω + i2τ )

| D(q, ω) |2(1375)

in the limit τ → ∞. The Fourier transform of the displacement field is givenby

D(q, ω) =1V

∫d3r

∫ ∞

0

dt exp[− i ( q . r − ω t ) − t

2 τ

]∇ | e |

| r − v t |

= −4 π i q | e |

V q2

∫ ∞

0

dt exp[i ( ω − q . v ) t − t

2 τ

]= −

4 π i q | e |V q2

2 τ1 − i ( ω − q . v ) 2 τ

(1376)

Hence,

P =4 e2

2 τ ( 2 π )3

∫ ∞

−∞dω

∫d3q

q2i ω

ε(q, ω + i2τ )

11

4 τ2 + ( ω − q . v )2

(1377)which in the limit τ → ∞ reduces to

P =2 e2

( 2 π )2

∫ ∞

−∞dω

∫d3q

q2i ω

ε(q, ω + iη)δ( ω − q . v ) (1378)

This yields the expression

P =2 e2

( 2 π )

∫ ∞

−∞dω

∫dq

q v

i ω

ε(q, ω + iη)

[θ( ω + q v ) − θ( ω − q v )

](1379)

448

Hence, on assuming that the dielectric constant is independent of q, from q = 0to an upper cut off q = q0, one obtains the result

P = − e2

π v

∫ ∞

−∞dω

i ω

ε(ω + iη)lnq0 v

ω(1380)

The power loss spectrum, P (ω), is defined in terms of an integral over positivefrequencies

P =∫ ∞

0

dω P (ω) (1381)

On using the symmetry properties of the dielectric constant under the transfor-mation ω → − ω, one finds that the contribution from the real part of theinverse dielectric constant vanishes. Hence, one obtains the final result for P (ω)

P (ω) = − 2 e2

π v=m

[ω

ε(ω + iη)

]lnq0 v

ω(1382)

——————————————————————————————————

In a typical experiment, monochromatic electron beams with energies E(k)of the order of keV fall incident on thin films, and the energies of the scatteredelectrons, E(k′) are analyzed142. Experimentally, it is found that the fast elec-tron loses energy in almost exact multiples of h ωp. That is, the energy lossspectrum shows peaks separated by energies which are multiples of h ωp. Theabove analysis predicts a single pole near ω = ωp. The discrepancy is causedby the use of the Born Approximation which neglects the effects of multiplescattering. The experiments are usually analyzed by fitting the intensities ofthe peaks to a Poisson distribution

In =1n!

(L

λ

)n

exp[− L

λ

]I0 (1383)

where In is the intensity of the n-th plasmon peak, L is the sample thicknessand λ is the mean free path. The mean free path is then compared with thetheoretically derived inelastic scattering cross-section.

The mean free path can be estimated from the scattering cross-section. Onexpressing the density-density correlation function in terms of the imaginarypart of the inverse of the dielectric constant, one finds

d2σ

dΩ dω= − k′

k

(mq e

π h2 q

)2

h V =m

[1

ε(q, ω + iη)

](1384)

For the frequency range

2 m ω > 2 h q kF + q2 (1385)142L. Marton, J. A. Simpson, H. A. Fowler and N. Swanson, Phys. Rev. 126, 182 (1962).

449

Figure 208: The energy loss spectrum, in units of hωp, for electrons of energy20 keV falling on an aluminum foil of thickness 2500 A. [After Marton et al.(1962).]

the imaginary part of χ(q, ω + iη) vanishes as η → 0. Therefore, in this limit,the imaginary part of the dielectric constant also vanishes

=m ε(q, ω + iη) → κ η (1386)

for some value finite of κ. Hence, in this limit, one has

=m

[1

ε(q, ω + iη)

]= − π δ( <e ε(q, ω) ) (1387)

Furthermore, in this region of (ω, q) space, one has the approximate expression

ε(q, ω) = 1−ω2

p(q)ω2

(1388)

where the plasmon dispersion relation is given by

ω2p(q) =

(ω2

p +35q2 v2

F + . . .

)(1389)

and the plasmon frequency by

ω2p =

4 π ρ e2

m(1390)

Thus, one finds the single (plasmon) pole approximation for the inverse dielectricconstant

=m

[1

ε(q, ω + iη)

]= − π δ

(ω2 − ω2

p(q)ω2

)

450

= − πω2

ω + ωp(q)δ( ω − ωp(q) )

= − π

2ωp(q) δ( ω − ωp(q) ) (1391)

for positive ω. The plasmon contribution to the differential scattering cross-section, is found by integrating over the energy loss ω, and is given by

dσ

dΩ=

k′

k

2 m2q ρ V

m h ωp

(e2

h q

)2

(1392)

Thus, the scattering cross-section is directly proportional to the number of elec-trons in the solid. In deriving the differential scattering cross-section, we haveneglected the q dependence of the plasmon dispersion relation. The total plas-mon scattering cross-section is found by integrating the differential cross-sectionover the scattering angle θ. We note that energy and momentum conservationleads to the two conditions

q2 = k2 + k′2 − 2 k k′ cos θ

q2 = ( k − k′ )2 + 4 k k′ sin2 θ

2(1393)

and

( k − k′ ) ( k + k′ ) =2 mq

hωp

( k − k′ ) =2 mq ωp

h ( k + k′ )(1394)

For small scattering angles, θ 1, these can be combined to yield

q2 ≈ k2

(4 sin2 θ

2+

h2ω2p

4 E(k)2

)≈ k2

(θ2 + θ20

)(1395)

Hence, one has

dσ

dΩ≈ k′

k

2 m2q ρ V

m h ωp

e4

h2 k2 ( θ2 + θ20 )

≈ mq e4 ρ V

m h ωp E(k) ( θ2 + θ20 )(1396)

On integrating over the solid angle dΩ, but restricting the range of θ from zeroto a maximum momentum transfer given by 2 kF ∼ θm k, one finds the totalcross-section, σ, for plasmon scattering, is given by

σ ≈ 2 πe4 ρ V

h ωp E(k)mq

mlnθm

θ0(1397)

451

The mean free path, λ, is then found by noting that a trajectory of cross-sectional area σ covers a volume λ σ between consecutive collisions. This mustequal V , the volume of the solid. This leads to the mean free path being givenby

λ−1 ≈ 2 πe4 ρ

h ωp E(k)mq

mlnθm

θ0(1398)

Thus, the mean free path depends linearly on the kinetic energy of the incidentelectron. This value has been found to track the mean free path obtained byfitting the measured intensities of the multi-plasmon peaks.

11.3.5 The Transverse Response

In the Coulomb or radiation gauge, the vector potential describes the transverseelectromagnetic field. It satisfies the equation(

− q2 +ω2

c2

)A(q, ω) = − 4 π

cj

T(q, ω) (1399)

The situation in which there are no transverse external currents impressed onthe system, j

T ext(q, ω) = 0, is considered. Thus, one obtains the microscopic

equation (− q2 +

ω2

c2

)A(q, ω) = − 4 π

cj

T ind(q, ω) (1400)

Ohm’s law can be expressed in the form

jT ind

(q, ω) = σT (q, ω) ET (q, ω) (1401)

where σT is the transverse conductivity, and the total transverse electric field isgiven by

ET (q, ω) = iω

cA(q, ω) (1402)

This leads to(− q2 +

ω2

c2+ i

4 π ωc2

σT (q, ω))A(q, ω) = 0 (1403)

The transverse dielectric function is identified in terms of the optical conduc-tivity

εT (q, ω) = 1 +4 π iω

σT (q, ω) (1404)

The photon dispersion relation can be re-written as(ω

c

)2

εT (q, ω) = q2 (1405)

If εT (q, ω) > 0, then there are undamped transverse electromagnetic waves.

452

-3 -2 -1 0 1 2z

ω > ωω > ωω > ωω > ωp

εεεε(ω) > 0(ω) > 0(ω) > 0(ω) > 0εεεε0000 = 1 = 1 = 1 = 1

z > 0

Figure 209: The spatial dependence of a transverse electromagnetic field fallingincident on a metal, for ω > ωp. The electromagnetic radiation is transmittedthrough the metal.

-3 -2 -1 0 1 2z

ω < ωω < ωω < ωω < ωp

ε(ω) < 0ε(ω) < 0ε(ω) < 0ε(ω) < 0

εεεε0000 = 1 = 1 = 1 = 1

z > 0

Figure 210: The spatial dependence of a transverse electromagnetic field fallingincident on a metal, for ω < ωp. The electromagnetic radiation is attenuatedby the metal.

453

q

ω ωωωωωωωp

cq

√(ω√(ω√(ω√(ωp2 + c2q2)

Figure 211: The energy dispersion for transverse electromagnetic radiation in ametal. The dispersion relation starts off at the plasmon frequency ωp at q = 0and then merges with the dispersion relation of light.

Otherwise, q would be complex which implies that ET (r; t) is attenuated as itenters into the sample. In other words, if =m ε(q, ω) = 0 and <e ε(q, ω) > 0,the material is transparent to transverse electromagnetic waves. The dispersionrelation is given by

εT (q, ω) =(c q

ω

)2

(1406)

Thus, the transverse excitations have a completely different character to thelongitudinal excitations, specially at large q. As q → 0, one expects thatσL(q, ω) → σT (q, ω) since electrons cannot differentiate between transverseand longitudinal waves in this limit. In this limit, the conductivity may bemodelled by the complex Drude expression

σ(0, ω) =ρ e2 τ

m

11 − i ω τ

(1407)

which leads to the dielectric constant being given by

ε(0, ω) = 1 − 4 π e2 ρm ω2

11 + i

ω τ

(1408)

This approximate equality between the longitudinal and transverse dielectricconstants implies that the plasmon frequency also sets the frequency scale forthe interaction of photons with a metal. The dispersion relation for transverseradiation becomes

ε(0, ω) ω2 = c2 q2 (1409)

454

which is given by

ω2 − ω2p = c2 q2 (1410)

Thus, for ω < ωp, the wave will be reflected from a metal. For a typical metal,where ρ ∼ 1022 electrons/cm3, a typical plasmon frequency is 1015 sec−1. Thistypical frequency corresponds to a typical wave length of light in vacuum ofλp ∼ 10−7m. Incident light with longer wave lengths will be reflected fromthe metal. Hence, as εL(0, ω) = εT (0, ω), optical experiments that measureεT (q, ω) produce similar information to energy loss experiments that determineεL(q, ω).

The transverse conductivity may be evaluated directly by linear responsetheory. The vector potential couples to the electrons via the interaction

Hint =| e |

2 m c

∑i

[ (p

i. A(ri, t) + A(ri, t) . pi

)+

| e |c

A(ri, t)2

](1411)

The interaction contains a paramagnetic contribution that involves a coupling tothe momentum density and a diamagnetic contribution that involves a couplingwith the density of the charged electrons. The transverse current density j(r, t)is the mechanical current density e v, and is given by the sum of a paramagneticcurrent j

pand a diamagnetic current j

d

j(r, t) = jp(r, t) + j

d(r, t) (1412)

where the paramagnetic current is given by the symmetric operator

jp(r) = − | e |

2 m

∑i

[δ3(r − ri) pi

+ piδ3(r − ri)

](1413)

and the diamagnetic current is given by

jd(r) = − | e |2

m c

∑i

δ3(r − ri) A(ri) (1414)

To linear order in the vector potential, the interaction can be written as

H1int = − 1

c

∫d3r j

p(r) . A(r) (1415)

The induced paramagnetic current density is then found from linear responsetheory in which the ground state is evaluated to first order in the perturbinginteraction H1

int. The components of the induced paramagnetic current aregiven as a causal convolution of a paramagnetic current - paramagnetic currenttensor correlation function, and the components of the total vector potential.

jαind p(r, t) =

∑β

∫ +∞

−∞dt′

∫d3r′

1cRα,β

j,j (r, r′, t− t′) Aβ(r′, t′) (1416)

455

where the paramagnetic response function is given by the ground state expec-tation value

Rα,βj,j (r, r′, t− t′) = +

i

hΘ( t − t′ ) < Ψ0 |

[jαp (r, t) , jβ

p (r′, t′)]| Ψ0 >

(1417)

This is known as the Kubo formula for the conductivity143. The structure of theKubo formula for the response R is similar to that of the longitudinal responsefunction χ. They both involve the expectation value of a retarded two timecommutator. However, the Lindhard function involves the commutator of thedensity operator and the Kubo formula involves the commutator of the currentoperator.

On Fourier transforming the non-local relation between jp

and A with re-spect to space and time, one has

jαind p(q, ω) =

∑β

1cRα,β

j,j (q, ω) Aβ(q, ω) (1418)

Hence, to linear order in the vector potential, the total transverse current isgiven by

jαind(q, ω) =

∑β

[1cRα,β

j,j (q, ω) − δα,β| e |2

m cρ0

]Aβ(q, ω) (1419)

where it is assumed that the electron density is uniform and is given by ρ0. Thetransverse conductivity is then found with the aid of the relation between thetransverse electric field and the vector potential

ET (q, ω) = iω

cA(q, ω) (1420)

as

σα,βT (q, ω) =

1i ω

[Rα,β

j,j (q, ω) − δα,β| e |2

mρ0

](1421)

The conductivity should be evaluated using a microscopic theory. The conduc-tivity has a real part and an imaginary part that are connected by causality.The conductivity determines the material’s properties and how transverse elec-tromagnetic radiation or light interacts with the electrons in the metal.

The energy loss due to a longitudinal field is related to the inverse of thedielectric constant, but the energy loss of a transverse field is related to the143R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957).

456

conductivity or the imaginary part of the dielectric constant. This can be seenfrom the expression for the time-averaged dissipated power density

P (q, ω) =ω

4 π=m ε(q, ω + iη) | ET (q, ω) |2

=ω3

4 π c2=m ε(q, ω + iη) | A(q, ω) |2 (1422)

As the imaginary part of the dielectric constant is related to the real part of theconductivity,

=m ε(q, ω + iη) =(

4 πω

)<e σ(q, ω) (1423)

the absorption of light measures the conductivity.

11.3.6 Optical Experiments

The optical conductivity can be measured in optical absorption and reflectionexperiments. The wave vector of light in the medium is given by the complexnumber

q =ω

c

(1 +

4 π i σ(ω)ω

) 12

=ω

c

(n + i κ

)(1424)

and this has the effect that intensity of light is exponentially attenuated as itpasses through the material

E(r, t) = E0 exp[i ω (

n z

c− t )

]exp

[− κ ω z

c

](1425)

Experiments measure the absorption coefficient η which is the fraction of lightabsorbed in passing through unit thickness of the material

η =<e(j . E

)n | E |2

= 2κ ω

c(1426)

Another experimental method measures the reflectance of polarized light. Thisinvolves measuring the ratio of the reflected intensity of light to the incidentintensity, and gives rise to the real reflection coefficient. At oblique incidence,with angle of incidence θ, one distinguishes between s and p polarized light. Thes polarized light has the polarization perpendicular to the plane of incidence andthe p polarized light has polarization parallel to the plane of incidence. Thereflectances of the s and p polarized light are given in terms of the complexrefractive index n = n + i κ, via the Fresnel formulas

Rs(θ) =∣∣∣∣ cos θ − ( n2 − sin2 θ )

12

cos θ + ( n2 − sin2 θ )12

∣∣∣∣ (1427)

457

s-polarized

E1

E0

E2k2

k1

k0

θθθθ2

θθθθθθθθ

ε(ω)ε(ω)ε(ω)ε(ω) > 0

εεεε0000 = 1

z > 0

Figure 212: The geometry for s-polarized light falling incident on a metal.

p-polarized

E1

E0

E2

k2

k1

k0

θθθθ2

θθθθθθθθ

ε(ω)ε(ω)ε(ω)ε(ω) > 0

εεεε0000 = 1

z > 0

Figure 213: The geometry for p-polarized light falling incident on a metal.

458

Re z

Im z

z = ω+iηC

Figure 214: The contour of integration C about the simple pole at z = ω + iη.

and

Rp(θ) =∣∣∣∣ n2 cos θ − ( n2 − sin2 θ )

12

n2 cos θ + ( n2 − sin2 θ )12

∣∣∣∣ (1428)

The complex refractive index can then be inferred from measurements of Rs(θ)and Rp(θ). However, it is usual to infer the real part from the imaginary partvia the Kramers-Kronig relation144.

11.3.7 Kramers-Kronig Relation

Causality requires that the frequency be continued in the upper-half complexplane ω + i η in the response functions. This has the consequence that theresponse function is analytic in the upper-half complex plane. Also, it is re-quired that the integrand vanishes over a semi-circular contour at infinity in theupper-half complex plane. With these restrictions, one can consider the Cauchyintegral

( ε(q, ω + iη) − 1 ) =1

2 π i

∫c

ε(q, z) − 1z − ω − i η

dz (1429)

where the contour is taken around the point z = ω + iη. If ε(q, z) does nothave a pole at z = 0, the contour of integration can be deformed to the realaxis and an infinite semi-circular contour in the upper-half complex plane. Inthis case, one finds

ε(q, ω + iη) − 1 =1π i

Pr

∫ +∞

−∞

ε(q, z) − 1z − ω

dz (1430)

in which the contribution of the small semi-circle around the pole at z = ω + i ηhas cancelled with half of the left-hand side. On writing the dielectric constantas the sum of the real and imaginary parts

ε(q, z) = <e ε(q, z) + i =m ε(q, z) (1431)

144H. A. Kramers, Nature, 117, 775 (1926), R. de L. Kronig, J. Opt. Soc. Am. 12, 547(1926).

459

Re z

Im z

C'

z = R eiθ

z = ω+iη

Figure 215: The contour of integration C is deformed to C ′ which includes thereal axis and the semi-circular contour at infinity.

and on equating the real terms, one finds

<e ε(q, ω + iη) − 1 =1πPr

∫ +∞

−∞

=m ε(q, z)z − ω

dz (1432)

The imaginary terms are related via

=m ε(q, ω + iη) = − 1πPr

∫ +∞

−∞

<e ε(q, z) − 1z − ω

dz (1433)

These relations can be recast in the form

<e ε(q, ω + iη) − 1 =2πPr

∫ ∞

0

z =m ε(q, z)z2 − ω2

dz (1434)

and

=m ε(q, ω + iη) = − 2 ωπ

Pr

∫ ∞

0

<e ε(q, z)z2 − ω2

dz (1435)

These are the Kramers-Kronig relations145. They can be used to analyze exper-imental data or as consistency checks.

——————————————————————————————————

11.3.8 Exercise 66

Derive the form of the Kramers-Kronig relation for the imaginary part of thedielectric constant ε(q, ω)

=m ε(q, ω) =4 π σ(q, 0)

ω− 2 ω

πPr

∫ ∞

0

dz<e ε(q, z)z2 − ω2

(1436)

145H. A. Kramers, Nature, 117, 775 (1926), R. de L. Kronig, J. Opt. Soc. Am. 12, 547(1926).

460

valid for a material which has a finite d.c. conductivity σ(q, 0).

——————————————————————————————————

Another sum rule, the optical sum rule, is stated as∫ ∞

0

dω ω =m[εT (q, ω + iη)

]=

π

2ω2

p (1437)

The optical sum rule can be derived by exact methods. However, it can also beproved by noting that at high frequencies

limω → ∞

εT (q, ω) = 1 −ω2

p

ω2(1438)

On expressing the imaginary part of the dielectric constant in terms of the realpart, one can verify the sum rule using contour integration. A more usual formof the optical sum rule is stated in terms of a sum rule for the conductivity∫ ∞

0

dω σ(0, ω) =π

2ρ e2

m(1439)

where ρ is the electron density. Kramers-Kronig relations and sum rules can beestablished for a variety of response functions146. Since the inverse dielectricconstant is the longitudinal response function, 1/ε(q, ω) − 1 also satisfies aKramers-Kronig relation.

——————————————————————————————————

11.3.9 Exercise 67

The n-th moment of the imaginary part of the dielectric constant is defined byMn

Mn =∫ ∞

0

dω ωn =m ε(q, ω + iη) (1440)

Show that M1 is given by

M1 = 2 π2 e2 ρ

m(1441)

and that M−1 is given by

M−1 =π

2

[ε(q, 0) − 1

](1442)

——————————————————————————————————

146P. C. Martin, Phys. Rev. 161, 143 (1967).

461

11.3.10 The Drude Conductivity

Metals have a large conductivity, and as a result, electromagnetic fields onlypenetrate a small distance into the metal before the energy of the field is ab-sorbed and dissipated as Joule heating. For low frequencies, or slowly spatiallyvarying fields, the penetration depth δ can be calculated from Maxwell’s equa-tions using the frequency-dependent Drude electrical conductivity. The Drudeconductivity is calculated by assuming that the photon has a long wavelength,therefore, q ≈ 0. On assuming that the medium is homogeneous and isotropic,one finds that the conductivity tensor is diagonal

σα,β(ω) = δα,β e2 ρ τ

m

11 − i ω τ

(1443)

The Drude formula for the conductivity can be obtained directly from Kuboformulae, for the case of non-interacting electrons. On expressing the Kuboformulae in terms of the completes set of exact eigenstates of the many-particleHamiltonian H0

H0 | Ψn > = En | Ψn > (1444)

for t > 0, one finds

Rα,β(r, r′, t) =i

h

∑n

< Ψ0 | jαp (r) | Ψn > < Ψn | jβ(r′) | Ψ0 > exp

[+ i

t

h(E0 − En)

]− i

h

∑n

< Ψ0 | jβp (r′) | Ψn > < Ψn | jα(r) | Ψ0 > exp

[− i

t

h(E0 − En)

](1445)

On Fourier transforming the Kubo formula with respect to time, one obtains

Rα,β(r, r′, ω) = −∑

n

< Ψ0 | jαp (r) | Ψn > < Ψn | jβ(r′) | Ψ0 >

h ω + i η + E0 − En

+∑

n

< Ψ0 | jβp (r′) | Ψn > < Ψn | jα(r) | Ψ0 >

h ω + i η + En − E0

(1446)

where the convergence factor η is to be assigned a physical meaning. Thisexpression is to be evaluated for non-interacting electrons in which case, thestates | Ψn > can be taken to be Slater determinants. The matrix elementsof the current density operators can be expressed in terms of the one-electronwave functions φγ(r) and φγ′(r) via

< Ψn | jα(r) | Ψ0 > = − | e | h2 i m

[φ∗γ′(r) ∇αφγ(r) − ∇αφ∗γ′(r) φγ(r)

]= − | e | h

m=m

[φ∗γ′(r) ∇αφγ(r)

](1447)

462

where the electron in the state labelled by the one-electron quantum number γis excited to the state with quantum number γ′ in the final state. The energydifference between the initial and final states is given by the energy differencebetween the initial and final energies of the excited electron

En − E0 = Eγ′ − Eγ (1448)

Thus, one has

Rα,β(r, r′, ω) = − e2 h2

m2

∑γ,γ′

=m[φ∗γ(r) ∇αφγ′(r)

]=m

[φ∗γ′(r

′) ∇βφγ(r′)]

h ω + i η + Eγ − Eγ′

+e2 h2

m2

∑γ,γ′

=m[φ∗γ(r′) ∇βφγ′(r′)

]=m

[φ∗γ′(r) ∇αφγ(r)

]h ω + i η + Eγ′ − Eγ

(1449)

where Eγ < µ and Eγ′ > µ.

The conductivity response function will be evaluated for free-electrons. Oninserting the single-electron wave functions, the response function is found as

Rα,β(r, r′, ω) = − e2 h2

4 m2 V 2

∑k,σ;k′

(kα + k′α) (kβ + k′β)exp

[− i

h (k − k′) . (r − r′)]

h ω + i η + Ek − Ek′

+e2 h2

4 m2 V 2

∑k,σ;k′

(kα + k′α) (kβ + k′β)exp

[− i

h (k − k′) . (r − r′)]

h ω + i η − Ek + Ek′

(1450)

where kα and kβ denote the α and β components of the vector k. The summationover k, σ runs over the occupied states k < kF whereas the sum over k′ runsover the unoccupied states with the spin σ but with k′ > kF . The initialand final state spin quantum numbers are identical. On Fourier transformingwith respect to the space variable, and re-arranging the summation index in thesecond term, one finds

Rα,β(q, ω) = − e2 h2

4 m2 V

∑k,σ

(2kα + qα) (2kβ + qβ)f(Ek) − f(Ek+q)

h ω + i η + Ek − Ek+q

(1451)

In this expression, the effect of the Pauli-exclusion principle is automaticallyaccounted for.

463

Due to the large magnitude of c, for fixed ω, this expression can be evaluatedto leading-order in q. In this case, as space is isotropic, the response functionis also isotropic. That is, the response function is diagonal in the indices α andβ and the diagonal components have equal magnitudes. Hence, the diagonalcomponents can be evaluated from the relation

Rα,α(q, ω) =13

3∑β=1

Rβ,β(q, ω) (1452)

The response function can be expressed as

Rα,β(q, ω) = − δα,β 2 e2

3 m V

∑k,σ

Ek

f(Ek) − f(Ek+q)

h ω + i η + Ek − Ek+q

(1453)

On Taylor expanding the Fermi function f(Ek+q) in powers of (Ek+q − Ek),one has

Rα,β(q, ω) = − δα,β 2 e2

3 m V

∑k,σ

Ek

(∂f

∂Ek

) ( Ek − Ek+q )

h ω + i η + Ek − Ek+q

= − δα,β 2 e2

3 m V

∑k,σ

Ek

(∂f

∂Ek

) [1 − h ω + i η

h ω + i η + Ek − Ek+q

](1454)

The first term can be evaluated through integration by parts

− 23

∑σ,k

Ek

(∂f

∂Ek

)= − 2

3

∑σ

∫ ∞

0

dE E ρ(E)(∂f

∂E

)

=23

∑σ

∫ ∞

0

dE f(E)∂

∂E

(E ρ(E)

)(1455)

since the boundary terms vanish. Furthermore, since for free electrons ρ(E) ∝√E, this term is evaluated as

− 23

∑σ,k

Ek

(∂f

∂Ek

)=

∑σ

∫ ∞

0

dE f(E) ρ(E)

= Ne (1456)

as the factor of 23 cancels with the factor of 3

2 from the derivative. Due to thissimplification, the response function is given by

Rα,α(q, ω) =ρ0 e

2

m+

2 e2

3 m V

∑k,σ

Ek

(∂f

∂Ek

)h ω + i η

h ω + i η + Ek − Ek+q

(1457)

464

On substituting this expression into the conductivity, one finds that the firstterm cancels with the diamagnetic current. This cancellation is responsible forprohibiting current to flow occurring in a metal as a response to an appliedmagnetic field. In other words, a normal metal does not superconduct due tothe cancellation of the diamagnetic current.

The conductivity is simply given by

σα,β(q, ω) = δα,β 2 e2

3 i m V

∑k,σ

Ek

(∂f

∂Ek

)h

h ω + i η + Ek − Ek+q

(1458)

The derivative of the Fermi function is only non-zero in the vicinity of the Fermienergy. In the limit, T → 0, the derivative may be expressed as

−(∂f

∂ε

)= δ( ε − εF ) (1459)

The appearance of the derivative of the Fermi function in the expression for theconductivity is a consequence of the Pauli-exclusion principle. Only electronsclose to the Fermi energy can absorb relatively small amounts of energy andbe excited to unoccupied single-electron states and, hence, carry current. Aphenomenological relaxation time τ can be defined via

h

τ= η (1460)

The relaxation time τ can be thought of as the lifetime of the current-carryinghole or the current-carrying excited electron. This lifetime must be causedby a scattering mechanism. In more rigorous treatments of the conductivity,the scattering rate is calculated using microscopic descriptions of the scatteringinteraction. When expressed in terms of the relaxation rate, the conductivitybecomes

σα,β(q, ω) = − δα,β e2 τ

m V

∑k,σ

23

Ek

(∂f

∂Ek

)1 − i τ ( ω − q . v(k) )

(1461)

In the limit, q → 0, one recovers the Drude approximation for the conductivity

σα,β(ω) = δα,β ρ e2 τ

m

11 − i τ ω

(1462)

where ρ is the electron density. The Drude conductivity is purely real in thed.c. limit, and is given by

σα,β(0) = δα,β ρ e2 τ

m(1463)

465

and at finite frequencies has a real part that decays like ω−2

<e σα,β(ω) = δα,β ρ e2τ

m

11 + ω2 τ2

(1464)

Thus, the Drude conductivity has a peak at zero frequency and the width ofthe peak is determined by the relaxation time. The integrated strength of thelow-energy Drude peak in the conductivity is given by∫ ∞

0

dω <e[σα,β(ω)

]= δα,β π ρ e2

2 m(1465)

Hence, the intensity of the Drude peak provides a measure of the number of con-duction electrons in a system of non-interacting electrons. This sum rule, likethe total number of electrons, is unchanged by electron-electron interactions.Electron-electron interactions do renormalize the mass in the expression for theconductivity. However, the interactions also renormalizes the lifetime, τ , by thesame factor, Z, which is the wave function renormalization. In this way, the d.c.limit of the conductivity essentially remains un-renormalized. However, the fre-quency width of the Drude peak is narrowed by the factor Z. In heavy fermionmaterials, such as CeAl3147, the experimentally width of the determined Drudepeak has been found to decrease dramatically at low temperatures as the heavyquasi-particles form. The mass enhancement determined from the width at lowtemperatures is consistent with the large quasi-particle mass determined fromthe low temperature limit of the electronic specific heat.

——————————————————————————————————

11.3.11 Exercise 68

Show that the E field of microwaves, with low frequency ω, satisfy the equation

− ∇2 E(r, ω) =4 π i σ(ω) ω

c2E(r, ω) (1466)

where σ(ω) is the diagonal component of the conductivity tensor. Solve thisequation for the electric field and, therefore, calculate the classical skin depthδ. The classical skin depth is defined as the distance δ that an electric fieldpenetrates into a metal before being attenuated.

——————————————————————————————————

The analysis of Exercise 68 is only valid if the electric field vary slowly overdistances of the order of a mean free path λ. The analysis is only valid for lowfrequencies and dirty metals. However, for good metals, E(r, ω) varies rapidly147A. M. Awasthi, L. Degiorgi, G. Gruner, Y. Dalichaouch and M. B. Maple, Phys. Rev. B

48, 10692 (1993).

466

Figure 216: The frequency and temperature dependence of the Drude peak inUPt3 as measured by S. Donovan, A. Schwartz, and G. Gruner, Phys. Rev.Lett., 79, 1401 (1997).

467

in space. This regime corresponds to the anomalous skin effect148. Since theelectrons do not respond to the field instantaneously and locally, the retardedand non-local response function ought to be used. In this case, one should solveMaxwell’s equations by solving for the Fourier components of the fields E(q, ω)and B(q, ω), and by using an expression for the conductivity tensor in whichboth the wave vector and frequency dependence are kept149. This procedure iscrucial for the discussion of the anomalous skin effect.

——————————————————————————————————

11.3.12 Exercise 69

The conductivity tensor can be expressed as an integral over the Fermi surface,

σα,β(q, ω) =e2

4 π3

∫d2S

| h v(k) |τ vα(k) vβ(k)

1 − i τ ( ω − q . v(k) )(1467)

Consider a clean material with a sufficiently long mean free path λ, such thatqz λ 1 for fixed qz. Show that the transverse conductivity σx,x(qz ez, ω) isgiven by the approximate expression150

σx,x(qz ez, 0) =3π4

σ0

| qz | λ(1468)

——————————————————————————————————

11.3.13 The Anomalous Skin Effect

For clean materials with large mean free-paths λ, the penetration of an electricfield into a metal is governed by the anomalous skin effect. In the low frequencylimit, the component of the electric field parallel to the surface Ex(z, ω) isgoverned by

∂2Ex

∂z2+

ω2

c2Ex = − 4 π i ω

c2jx (1469)

where the surface of the material is the z = 0 plane. We shall assume thatelectrons are specularly reflected from the surface. This boundary conditioncan be understood by imagining that the surface of the metal demarcates theboundary between two identical solids. One solid represents an extension ofthe actual material generated by mirror symmetry. The condition of specularreflection amounts to assuming that the electrons and fields in the mirror image148A. B. Pippard, Proc. Roy. Soc. A 191, 385 (1947), A. B. Pippard, Proc. Roy. Soc. A,

224, 273 (1954).149D. C. Mattis and G. Dresselhaus, Phys. Rev. 111, 403 (1958).150G. E. H. Reuter and E. H. Sondheimer, Proc. Roy. Soc. A 195, 336 (1948).

468

k

k

k'

σσσσk

k'

σσσσk'

z=0

z=0

e

ee

z > 0

Figure 217: Imposing specular boundary conditions on electrons in a solid (lowerframe) can be re-interpreted in terms of two identical crystals (upper frame)related by mirror symmetry σ.

solid behave in the same way as in the actual solid. This leads to the boundarycondition for the electric field being given by(

∂Ex

∂z

)∣∣∣∣z=−ε

= −(∂Ex

∂z

)∣∣∣∣z=ε

(1470)

Therefore, on subsuming the boundary condition in the equation of motion forthe field, one has

∂2Ex

∂z2+

ω2

c2Ex = − 4 π i ω

c2jx + 2 δ(z)

(∂Ex

∂z

)∣∣∣∣z=0

(1471)

Hence, on Fourier transforming with respect to z and using Ohm’s law

jx(qz, ω) = σx,x(qz, ω) Ex(qz, ω) (1472)

one obtains the solution

Ex(qz, ω) =2(

∂Ex

∂z

)∣∣∣∣z=0

ω2

c2 − q2z + 4 π i ωc2 σx,x(qz, ω)

(1473)

In the limit of extremely long mean free path λ → ∞, the conductivitysimplifies to151

σx,x(qz, 0) =3 π σ(0, 0)4 | qz | λ

(1474)

The spatial dependence of the electric field is given by the inverse Fourier151G. E. H. Reuter and E. H. Sondheimer, Proc. Roy. Soc. A 195, 336 (1948).

469

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8

q λλλλ

σσσσ xx(

q z,0

)/σ σσσ x

x(0,

0)

Figure 218: The q-dependence of the low-frequency transverse conductivityσxx(qz, ω).

transform,

Ex(z, ω) =(∂Ex

∂z

)∣∣∣∣z=0

∫ ∞

−∞

dqzπ

exp[− i qz z

]ω2

c2 − q2z + 3 π2 i ωc2 λ | qz | σ(0, 0)

(1475)

On defining δ via

δ−1 =(

3 π2 ω σ(0, 0)c2 λ

) 13

(1476)

one has

Ex(z, ω) =(∂Ex

∂z

)∣∣∣∣z=0

∫ ∞

−∞

dqzπ

| qz | exp[− i qz z

]ω2

c2 | qz | − | q3z | + i δ−3

= − 2 i δ(∂Ex

∂z

)∣∣∣∣z=0

∫ ∞

0

dx

π

x cos( x zδ )

1 + i x3 − i ω2 δ2

c2 x

(1477)

In the last line, we have introduced a dimensionless variable x = qzδ, multipliedthe top and bottom by a factor of iδ3, and folded the region of integration to thepositive x axis. For low frequencies, the decay of the electric field is governedby δ, the anomalous skin depth.

At the surface, the value of the field is given by

Ex(0, ω) = − 2 i δ(∂Ex

∂z

)∣∣∣∣z=0

∫ ∞

0

dx

π

x

1 + i x3 − i ω2 δ2

c2 x

470

-0.2

0

0.2

0.4

0.6

0.8

-8 -6 -4 -2 0 2 4 6 8z/δδδδ

E(z

, ω ωωω)

Im E(z)

Re E(z)

Figure 219: The spatial-dependence of the transverse electric field as calculatedwith the short wave length approximation for the conductivity σxx(qz, ω), as afunction of z

δ .

≈ − 2 δ3

(∂Ex

∂z

)∣∣∣∣z=0

( 1 +i√3

) (1478)

Far from the surface, the field has an exponential decay

Ex(z, ω) ∼(δ

z

)2

exp[− z

δ

](1479)

which decays over a distance δ.

This result for the anomalous skin depth δ was first obtained by Pippard152,up to a numerical factor. Pippard noted that only the fraction of the electronsδλ close to the surface may participate in the screening process. That is, onlythe electrons moving parallel to the surface are strongly affected by the electricfield. The electrons that remain within the penetration depth δ before beingscattered, subtend an angle of

dθ ≈ δ

λ(1480)

The number of the electrons which are capable of responding to the field isproportional to the solid angle dΩ,

dΩ = 2 π sin θ dθ ∼ 2 π dθ (1481)

since θ ≈ π2 . Hence, the effective electron density, ρeff , is given by

ρeff ≈ ρδ

λ(1482)

152A. B. Pippard, Proc. Roy. Soc. A, 191, 385 (1947), A. B. Pippard, Proc. Roy. Soc. A,224, 273 (1954).

471

z > 0δδδδ

λλλλ

e

δθδθδθδθ

Figure 220: The geometry determining the effective number of electrons whichare able to screen the external field.

where ρ is the uniform electron density. This implies that conductivity parallelto the surface should be reduced by the factor δ

λ . Thus, on applying the analysisof the classical skin effect, one recovers the relation

δ−2 ∼ 4 π ωc2

σ(0)δ

λ(1483)

Hence, one has Pippard’s relation

δ−1 ∼(

4 π ω σ(0)c2 λ

) 13

(1484)

which only differs by a numerical factor from the previously given expressionfor the skin depth.

11.3.14 Inter-Band Transitions

The absorption of a photon of wave vector q, may cause an electron to make atransition between the initially occupied state with Bloch wave vector k to a finalstate with wave vector k+ q. Since the wave vector of light q is small comparedwith kF , for a given ω, the final state must be in a different band and must beempty. These are inter-band transitions. Materials with small inter-band gapscan have large dielectric constants. The inter-band contribution to the dielectricconstant can be obtained by neglecting q, thereby producing vertical transitionbetween the different Bloch bands. The imaginary part of the dielectric constant

472

due to inter-band transitions can be written as

=m[ε(ω+iη)

]= 8 π2 h2

(e

m ω

)2 ∫d3k

( 2 π )3

∣∣∣∣ eα . Mk

∣∣∣∣2 δ( Ec,k − Ev,k − h ω )

(1485)The matrix elements for the inter-band transitions are given by

eα . Mk = eα .

∫d3r exp

[− i k . r

]uv,k(r) ∇ exp

[+ i k . r

]uc,k(r)

(1486)

where un,k are the periodic functions of r in the Bloch functions. The sum overk can be transformed into an integral

=m[ε(ω+iη)

]= 8 π2 h2

(e

m ω

)2 ∫d2S

( 2 π )3

∣∣∣∣ eα . Mk

∣∣∣∣2∣∣∣∣ ∇ ( Ec,k − Ev,k )∣∣∣∣Ec−Ev=hω

(1487)where d2S represents an element of the surface in k space defined by the equationh ω = Ec,k − Ev,k. The quantity

J(ω) =∫

d2S

( 2 π )3

∣∣∣∣ eα . Mk

∣∣∣∣2∣∣∣∣ ∇k ( Ec,k − Ev,k )∣∣∣∣Ec−Ev=hω

(1488)

is known as the joint density of states. The joint density of states varies rapidlywith respect to ω at the critical points, which are defined by

∇k ( Ec,k − Ev,k )∣∣∣∣Ec−Ev=hω

= 0 (1489)

The inter-band transitions produce a broad continuum in the absorption spec-trum, and only the van Hove singularities may be uniquely identified in thespectrum. The analytic behavior of the dielectric constant near a singularitymay be obtained by Taylor expanding about the critical point.

When other processes such as electron-phonon scattering are considered,second order time-dependent perturbation theory describes indirect transitions.In this case, a phonon may be absorbed or emitted by the lattice while thephoton is being absorbed. The emission or absorption of the phonon introducesa change of momentum q. Conservation of momentum leads to the momentaof the initial and final state of the electron being related via q = k′ − k.Since the energy of the phonon is usually negligible compared with the energy

473

of the photon, the energy of the absorbed photon is approximately given by thedifference of the electron’s initial and final state energies

h ω ≈ Ec,k − Ev,k′ (1490)

Thus, since q varies continuously, the indirect inter-band transitions have a con-tinuous spectrum. The threshold energy for the inter-band transition is close tothe minimum value of Ec,k − Ev,k′ , for all possible values of k and k′. If theminimum value of Ec,k − Ev,k′ occurs for k = k′ the band gap is known asa direct band gap, whereas if k 6= k′ the band gap is called an indirect band gap.

11.4 The Fermi Surface

The Fermi surface determines most of the thermodynamic, transport and opti-cal properties of a solid. The geometry of the Fermi surface can be determinedexperimentally, through a variety of techniques. The most powerful of thesetechniques is the measurement of de Haas - van Alphen oscillations. The deHaas - van Alphen effect is manifested as an oscillatory behavior of the magne-tization153. The magnetization is periodic in the inverse of the applied magneticfield 1

H . Onsager154 pointed out that the period in 1H is given by the expression

∆(

1H

)= 2 π

| e |h c

Ae (1491)

where Ae is the extremal cross-sectional area of the Fermi surface in the planeperpendicular to the direction of the applied field H . By observing the period ofoscillations for the different directions of the applied field, one can measure theextremal areas for each direction. This information can then be used to recon-struct the Fermi surface of three-dimensional metals155. First, some propertiesof the electron orbits in an applied field will be examined, then the experimentalmethods used in the determination of the Fermi surface will be described.

11.4.1 Semi-Classical Orbits

In the classical approximation, the Hamilton equations of motion are given bythe pair of equations

r = v(k) =1h∇k Ek (1492)

and

h k = − | e |c

v(k) ∧ H (1493)

153W. J. de Haas and P. M. van Alphen, Proc. Amsterdam, Acad. 33, 1106 (1930).154L. Onsager, Phil. Mag. 45, 1006 (1952).155D. Schoenberg, Proc. Roy. Soc. A 170, 341 (1939).

474

From these, one finds that k changes in a manner such that it remains on theconstant energy surfaces. This is found by observing that, from the above equa-tion, k is perpendicular to ∇kEk. Also since k is perpendicular to H, the kspace orbits are a section of the constant energy surfaces with normal alongthe z-axis. That is the orbits traverse the constant energy surfaces, but kz is aconstant.

The real space orbits are perpendicular to the k space orbits. To show this,we shall first prove that k is perpendicular to r. On taking the vector productof the equation of motion with the vector H, one finds

h H ∧ k = − | e |c

H ∧(v(k) ∧ H

)(1494)

The component of the velocity perpendicular to the applied field is given by

r⊥ = r − H

(r . H

)= H ∧

(r ∧ H

)(1495)

where H is the unit vector in the direction of H. Hence,

r⊥ = − c h

| e | HzH ∧ k (1496)

Thus, on integrating this with respect to time, one finds that the displacement∆r⊥ is given in terms of the displacement in k space through

∆r⊥ = − c h

| e | HzH ∧ ∆k (1497)

Thus, the real space orbit is perpendicular to the k space orbit and is scaled bya factor of c h

e Hz.

The period T at which the orbit is traversed is given by the integral overone orbit

T =∮

dk

k(1498)

The rate of change of k is given by the Lorentz Force

k =| e |h2 c

∣∣∣∣ ∇kE ∧ H

∣∣∣∣=

| e |h2 c

Hz

∣∣∣∣ ∇kE⊥

∣∣∣∣ (1499)

where ∇kE⊥ is the component of the gradient perpendicular to H, i.e., theprojection in the plane of the orbit. Thus,

T =∮

dk∣∣∣∣ ∇kE⊥

∣∣∣∣h2c

| e |1Hz

(1500)

475

If semi-classical quantization considerations are applied, then the energy ofthe orbits become quantized as do the orbits themselves. The area enclosed bythe orbits are related to the energy, and so the areas are also expected to bequantized. This shall be shown by two methods, in the first the quantizationcondition is imposed through the energy - time uncertainty relation, and thesecond method will utilize the Bohr-Sommerfeld quantization condition.

Quantization Using Energy - Time Uncertainty.

The relationship between the energy and the areas enclosed by the k spaceorbits can be found from consideration of two classical orbits, one with energyE and another with energy E + ∆E where both orbits are in the same kz

plane. Then, let ∆k be the minimum distance between these two orbits. Thevalue of ∆k is related to ∆E via

∆E =∣∣∣∣ ∇kE⊥

∣∣∣∣ ∆k (1501)

This relation can be substituted into the expression for the period to yield

T =h2c

| e | Hz

1∆E

∮∆k dk (1502)

However, the area between the two successive orbits in reciprocal space is givenby the integral

∆A =∮

∆k dk (1503)

Thus, the period can be expressed as

T =h2c

| e | Hz

∆A∆E

(1504)

The orbits can be quantized through the energy uncertainty relation

En+1 − En =h

T

=| e | Hz

h c

∣∣∣∣ ∆E∆A

∣∣∣∣ (1505)

Furthermore, as the ratio of the energy difference to the difference in areas ofconsecutive orbits is defined as

∆E∆A

=En+1 − En

An+1 − An(1506)

one can cancel a factor of ∆E to find that the area enclosed between consecutiveLandau orbits is quantized

An+1 − An =2 π | e |h c

Hz (1507)

476

This difference equation can be solved to yield the area (in reciprocal space)enclosed by the n-th Landau orbital as

An =(n + λ

)2 π | e |h c

Hz (1508)

where λ is a constant, independent of n. Thus, the area of a Landau orbit ink space is related to n and the applied fieldHz, through the Onsager equation156.

Bohr-Sommerfeld Quantization.

An alternate derivation of the Onsager equation follows from the Bohr-Sommerfeld quantization condition∮

p . dr = 2 π h(n +

12

)(1509)

The mechanical momentum is given by

p = h k − | e |c

A (1510)

The integral is evaluated over an orbit in the x − y plane perpendicular to H.The orbit is obtained from the equation of motion with the Lorentz Force Law

h k = − | e |c

r ∧ H (1511)

The equation of motion can be integrated with respect to time, to yield

h k = − | e |c

r ∧ H (1512)

Thus, the Bohr-Sommerfeld quantization condition reduces to

− | e |c

∮ (r ∧ H + A

). dr = 2 π h

(n +

12

)=

| e |c

(H .

∮dr ∧ r −

∮dr . A

)(1513)

However, the integral ∮dr ∧ r = 2 Ar ez (1514)

is just twice the area enclosed by the real space orbit, and the integral of thevector potential around the loop is given by∮

dr . A = Φ (1515)

156L. Onsager, Phil. Mag. 43, 1006, (1952).

477

where Φ = Ar Hz is the flux enclosed by the orbit. Hence, the magnitude ofthe area of the orbit, Ar, in real space is quantized and is given by

Ar =(n +

12

)2 π h c| e | Hz

(1516)

Since the real space and momentum space orbitals are related via

∆r =h c

| e | Hz∆k (1517)

one can scale the areas of the real and momentum space orbits. Thus, onerecovers the Onsager formulae for the area of the momentum space orbit

An =(n +

12

)2 π | e |h c

Hz (1518)

This quantization condition does not always hold. Some electron orbits do notform closed curves in the Brillouin zone. These open orbits can be understoodby noting that the Bloch state energy is periodic in k space, and if an orbitcrosses the boundary of the first Brillouin zone, it will continue into the nextzone. Sometimes the sections of the orbits in the Brillouin zones can be com-bined, by translating them through reciprocal lattice vectors, to form closedorbits. Trajectories that extends across an entire Brillouin zone can not be usedto construct a closed orbit, and are called open orbits157 Another cause for thefailure of the quantization condition is caused when the field is larger than theenergy gaps between successive bands. This gives rise to the phenomenon ofmagnetic breakdown. In this case, the levels are broadened and the cyclotronorbits tunnel between Bloch states in the different bands, ignoring the bandgaps158. Magnetic breakdown is likely to occur when h ωc µ E2

g , where Eg

is the gap between the bands.

11.4.2 de Haas - van Alphen Oscillations

Given a solid with Hz = 0, surfaces of constant energy do not intersect whenplotted in k space. The consecutive constant energy surfaces, corresponding tothe different allowed values of energy, completely fill momentum space. Thestates on the surfaces which have energy less than µ will be occupied, andthose with energy greater than µ are empty. On applying a magnetic field, Hz,the momentum perpendicular to the field is no longer a constant of motion,but kz is constant. However, as time evolves, an orbit never leaves its surfaceof constant energy. The magnetic field quantizes the orbits. In momentumspace, the allowed orbits form a nested set of discrete Landau tubes. Orbits in157I. M. Lifshitz and M. I. Kaganov, Sov. Phys. Uspekhi, 2, 831 (1959).158A. B. Pippard, Proc. Roy. Soc. A 270, 1 (1962).

478

the regions between the tubes are forbidden. For a general Fermi surface of athree-dimensional crystal, the intersection of the constant energy surfaces witha plane of fixed kz need not be circular, so that the Landau tubes need not havecylindrical cross-sections. However, for free electrons, the zero field constantenergy surfaces are spherical and the Landau tubes are cylindrical. The radiusof the tubes is determined by the energy of the x-y motion, while the heightis determined by the component of the kinetic energy due to motion in the z-direction. For free electrons, an occupied orbit is specified by kz and n. Theenergy of the free-electron orbit is given by

En,kz= ( n +

12

) h ωc +h2 k2

z

2 m(1519)

where

ωc =| e | Hz

m c(1520)

The orbit maps out a circle of area

An =2 π | e |h c

Hz

(n +

12

)(1521)

so the orbits will consist of concentric circles. On varying kz but holding n fixed,the consecutive orbits will map out a tube in k space. The occupied portionsof k space will lie on portions of a series of tubes. These portions will be con-tained in a volume similar to the volume of the Fermi surface, when Hz = 0.The bounding volume must reduce to the volume enclosed by the Fermi surfacewhen the field is decreased. For fields of the order of H ∼ 1 kG, the Fermisurface cuts about 103 such tubes, so the quasi-classical approximation can beexpected to be valid.

As the field increases, the cross-sectional area enclosed by the tubes alsoincreases, as does the number of electrons held by the tubes. The extremal tubemay cross the zero field Fermi surface (H = 0) at which point the electronsin the tube will be entirely transferred into the tubes with lower n values. Thechanging structure gives rise to a loss of tubes from the occupied Fermi volumewhen the field changes by amounts ∆H. Thus, if at some value of kz, theoccupied Landau tube with the largest area has the largest value n given by theextremal area of the Fermi surface A(kF )

2 π | e |h c

Hz ( n +12

) ∼ π k2F (kz) = A(kF ) (1522)

then, on changing Hz to Hz + ∆H the tube becomes unoccupied so the largestLandau tube changes from n to n− 1. This occurs when

( n − 12

) ( Hz + ∆H ) = ( n +12

) Hz

(1523)

479

Thus, the extremal orbit crosses the Fermi surface when Hz is increased by

n ∆H ≈ Hz (1524)

which can be used to eliminate n and relate ∆HHz

to the momentum space areaof the extremal orbit.

A(kF ) = π k2F (kz) =

Hz

∆H2 π | e |h c

Hz (1525)

Thus, n decreases by unity at fields given by

− ∆HH2

= − 2 π | e |h c

1A(kz)

(1526)

In other words, ∆n changes by − 1 with increasing ∆(

1Hz

). The non-

monotonic variation of the occupancy of the extremal orbits or tubes gives riseto oscillations in the Free energy as Hz is varied. This can also be seen fromexamination of the density of states, per spin polarization, for free electrons

ρ(ε) = D∑

n

Lz

∫ ∞

−∞

dkz

2 πδ( ε − n h ωc − h2 k2

z

2 m)

=Lx Ly Lz

4 π2 hm ωc

∑n

∫ ∞

0

dkz δ( ε − n h ωc − h2 k2z

2 m)

=Lx Ly Lz

4 π2 h2 m2 ωc

∑n

θ( ε − n h ωc )√2 m ( ε − n h ωc )

(1527)

where D is the degeneracy of a Landau orbital. The degeneracy is given by theratio of the cross-section of the crystal to the real space area enclosed betweenthe Landau orbits

D =Lx Ly

∆Ar

=Lx Ly

2 π hm ωc (1528)

which increases with increasing field. The density of states has equally spacedsquare root singularities determined by the energies of the Landau levels, butyet still roughly follows the zero field density of states. On changing the fieldthe spacing between the singularities increases. This means that, as the field isincreased, successive singularities may cross the Fermi energy, and give rise tooscillations in physical properties.

Physical properties are expressible as averages which are weighted by theproduct of the Fermi function and the density of states. For zero spin-orbit

480

coupling, the average of A is given by

A =∑

σ

∫ +∞

−∞dε f(ε) ρ( ε − µB σ Hz ) Aσ(ε) (1529)

in which the electronic density of states is spin split by the Zeeman field. Thissplitting is comparable with the effect of h ωc. Increasing the field will produceregular oscillations in the integrand which will show up in A. Due to the thermalsmearing manifested by the Fermi function, the oscillations of A as a functionof 1

Hzcan only be seen at sufficiently low temperatures such that

kB T h ωc (1530)

If this condition is not satisfied, the Fermi function becomes broad and washesout the peaks in the integrand near µ. As

| e | hm c kB

∼ 1.34 × 10−4 k/G (1531)

it is found that, for a typical field of H = 10 kG, the oscillations will only beappreciable below T ∼ 2 K.

——————————————————————————————————

11.4.3 Exercise 70

A non-uniformity of the magnetic field in a de Haas - van Alphen experimentmay cause the oscillations in Mz to be washed out. Calculate the field derivativeof the electron energy

∂En,k

∂H(1532)

for an extremal orbit. Determine the maximum allowed variation of Hz that isallowable for the oscillations to still be observed. Show that it is given by δH,where

δH

H2z

<2 π | e |h c A

(1533)

and A is the area of the extremal orbit.

——————————————————————————————————

11.4.4 The Lifshitz-Kosevich Formulae

The de Haas - van Alphen Oscillations in the magnetization M can be foundfrom the grand canonical potential Ω

Ω = − kB T∑α

ln(

1 + exp[− β ( Eα − µ )

] )(1534)

481

where the sum over α runs over all the one-electron states.

The Lifshitz-Kosevich formulae159 describes the oscillatory parts of M . Thisshall be examined in the T → 0 limit. In the limit T → 0, one has

limT → 0

Ω =∑α

(Eα − µ

)Θ( µ − Eα ) (1535)

where Θ(x) is the Heaviside step function. Also, the total number of electronsis given by

Ne =∑α

Θ( µ − Eα ) (1536)

The dispersion relation for free electrons in an applied field is given by

Eα =h2 k2

z

2 m+(n +

12

)h ωc − µB Hz σ (1537)

so

Ω =| e | Hz V

4 π2 c h

∑σ

n=∞∑n=0

∫ ∞

−∞dkz

(h2 k2

z

2 m+(n +

12

)h ωc − µB Hz σ − µ

)× Θ

(µ − h2 k2

z

2 m− ( n +

12

) h ωc + µB Hz σ

)(1538)

For fixed n, the step function has the effect that the kz integration is limited tothe range of kz values, kz(σ, n) > kz > − kz(σ, n) , where

h2 kz(σ, n)2

2 m= µ −

(n +

12

)h ωc + µB Hz σ (1539)

The kz integration can be performed yielding

Ω = − 43| e | Hz V

4 π2 c h

(2 mh2

) 12 ∑

σ

n=∞∑n=0

(µ −

(n +

12

)h ωc + µB Hz σ

) 32

× Θ(µ − ( n +

12

) h ωc + µB Hz σ

)(1540)

Thus, the summation over n only runs over a finite range of values, where nruns from 0 to n+, where n+ denotes the integer part of

n+ =µ + µB Hz σ

h ωc− 1

2(1541)

159I. M. Lifshitz and A. M. Kosevich, Sov. Phys. J.E.T.P. 2, 636 (1956).

482

Hence,

Ω = − 43| e | Hz V

4 π2 c h

(2 mh2

) 12 ∑

σ

n=n+∑n=0

(µ −

(n +

12

)h ωc + µB Hz σ

) 32

(1542)

The thermodynamic potential shows oscillatory behavior as H increases, sincewhen µ

h ωcchanges by an integer the upper limit of the summation over n also

changes by an integer.

In order to make the oscillatory nature of the summation more explicit, aperiodic function β(x) is introduced. The periodic function is defined as

β(x) =n=+∞∑n=−∞

δ

(x − ( n +

12

))

(1543)

The summation over n in the thermodynamic potential can be expressed interms of an integral over β(x) via

Ω = − 43| e | Hz V

4 π2 c h

(2 mh2

) 12 ∑

σ

∫ x+

0

dx β(x)

(µ − x h ωc + µB Hz σ

) 32

(1544)

where the upper limit of integration is given by

x+ =µ

h ωc+

m c

| e | hµB σ (1545)

However, as

µB =| e | h2 m c

(1546)

the upper limit of integration becomes

x+ =µ

h ωc+

σ

2(1547)

in which the mass of the electron has cancelled in the second term. In general,the spin splitting term will depend on the ratio of the mass of the electron tothe band mass.

On Fourier analyzing β(x), one has

β(x) = 1 + 2∞∑

p=1

cos 2πp ( x − 12

) (1548)

483

which on substituting into the expression for Ω yields the expression

Ω = − 43| e | Hz V

4 π2 c h

(2 mh2

) 12 ∑

σ

[2

5 h ωc

(µ + µB σ Hz

) 52

+ 2∞∑

p=1

∫ x+

0

dx cos 2πp ( x − 12

)

(µ − x h ωc + µB Hz σ

) 32]

(1549)

The first term is non-oscillatory. The term containing the summation producesthe oscillatory terms. The second term can be evaluated by integration by parts

Ip =∫ x+

0

dx cos 2πp ( x − 12

)(

µ

h ωc− x +

µB Hz σ

h ωc

) 32

=∫ x+

0

dx

2 π pd

dx

(sin 2πp ( x − 1

2)) (

x+ − x

) 32

=32

∫ x+

0

dx

2 π psin 2πp ( x − 1

2)(x+ − x

) 12

(1550)

since the boundary term vanishes. Integrating by parts once again, leads to

Ip = − 38 π2 p2

∫ x+

0

dxd

dx

(cos 2πp ( x − 1

2)) (

x+ − x

) 12

=3

8 π2 p2

[x

12+ cosπp − 1

2

∫ x+

0

dx

(cos 2πp ( x − 1

2)) (

x+ − x

)− 12]

(1551)

The integral is performed by changing variables from x to u, where u is definedas

2 p(x+ − x

)=

u2

2(1552)

so that the infinitesimal quantity dx is given by

dx = − 12 p

u du (1553)

In terms of the new variable, u, the integration becomes

Ip =3

8 π2 p2

[x

12+ cosπp − 1√

4 p

∫ u0

0

du

(cos

π

2( u2

0 − u2 − 2 p )) ]

(1554)

484

The cosine term can be decomposed as

cos (π u2

2+ φ ) = cos

π u2

2cosφ − sin

π u2

2sinφ (1555)

so one has the integrals

C(x) =∫ x

0

du cosπ u2

2

S(x) =∫ x

0

du sinπ u2

2(1556)

which for large x have the limits

C(∞) = S(∞) =12

(1557)

Thus, the integral Ip is evaluated as

Ip =3

8 π2 p2

[x

12+ cosπp +

− 1√4 p

(C(u0) cos

π

2( u2

0 − 2 p ) + S(u0) sinπ

2( u2

0 − 2 p )) ]

∼ 38 π2 p2

[x

12+ cosπp − 1√

8 pcos

π

2( u2

0 − 2 p − 12

)

](1558)

Thus, the oscillatory part of the grand canonical potential ∆Ω is

∆Ω ∼(h ωc

) 32 | e | Hz V

4 π4 c h

(2 mh2

) 12

×

×∑

σ

∞∑p=1

2( 2 p )

52

cos(

2πp(

µ

h ωc+σ − 1

2

)− π

4

)(1559)

This depends on the ratio of the extremal cross-section of the zero field Fermisurface, AF = π k2

F , and the difference in areas of the Landau orbits inmomentum space, ∆A = 2π | e |

h c Hz,

∆Ω ∼(h ωc

) 32 | e | Hz V

4 π4 c h

(2 mh2

) 12

×

×∑

σ

∞∑p=1

2( 2 p )

52

cos(

2πp(

c h π k2F

2 π | e | Hz+σ

2− 1

2

)− π

4

)

485

∆Ω ∼(h ωc

) 32 | e | Hz V

4 π4 c h

(2 mh2

) 12

×

×∑

σ

∞∑p=1

2( 2 p )

52

cos(

2πp(AF

∆A+σ − 1

2

)− π

4

)(1560)

Thus, oscillations in the grand canonical potential occur when the number ofLandau orbits inside the extremal cross-sectional area change. The oscillationsoccur in the magnetization Mz as it is related to the grand canonical potentialΩ via

Mz = −(∂Ω∂Hz

)(1561)

Thus, the magnetization also has oscillations that are periodic in 1Hz

. Fur-thermore, for a free electron gas the extremal area of the Fermi surface is justAF = π k2

F , so the period of oscillations is proportional to the extremal cross-sectional area of the zero field Fermi sphere. In addition to the fundamentaloscillations, there are also higher harmonics which can be observed in experi-ments. For the more general situation, where the Fermi surface is non-sphericaldifferent extremal cross-sections will be observed when the magnetic field isapplied in different directions. Measurements of the de Haas - van Alphen os-cillations can be used to map out the Fermi surface.

The Lifshitz-Kosevich formulae160, valid at finite temperatures, yields thecontribution from one extremal area of the Fermi surface to the oscillatory partof the grand canonical potential as

∆Ω =(| e | Hz

2 π h c

) 32 kB T V

( 2 π )12

∑σ

∑p

1p

32

exp[− π p

ωc τ

]sinh 2π2p kBT

h ωc

× cosπp cos[

2πp(AF

∆A+

σ m∗

2 me

)− π

4

](1562)

For a Fermi surface of general shape, the factor ( 2 π )12 in the denominator is

to be identified as the magnitude of the second derivative of the cross-sectionalarea of the Fermi surface evaluated at the extremal area. In the case pertinentto free electrons, the cross-sectional area is given by A(kz) = π ( k2

F − k2z ),

therefore, ∣∣∣∣∂2A(kz)∂k2

z

∣∣∣∣− 12

= ( 2 π )−12 (1563)

160I. M. Lifshitz and A. M. Kosevich, Sov. Phys.-JETP 2, 636 (1956).

486

On performing the sum over the spin polarizations, one obtains the result

∆Ω =(| e | Hz

2 π h c

) 32 2 kB T V

( 2 π )12

∑p

1p

32

exp[− π p

ωc τ

]sinh 2π2p kBT

h ωc

× cosπp(

cosπpm∗

me

)cos[

2πp(AF

∆A

)− π

4

](1564)

The spin splitting factor depends on the ratio of the band mass of the electron tothe electron mass in vacuum. The splitting between the up-spin and down-spinbands has modified the relative phase of the higher harmonics in the oscillations.The interference between the oscillations from the up-spin and down-spin bandsmay even result in the suppression of the fundamental oscillations. For systemswhich are on the verge of ferromagnetism, the spin splitting factor should be en-hanced by including the effective field on the spins due to the interactions withthe other electrons. This formula also includes the exponential damping of theoscillations due to T through the thermal smearing of the Fermi surface and alsohas an exponential damping term depending on the rate for elastic scatteringby impurities 1

τ . Both these effects reduce the amplitude of the de Haas - vanAlphen oscillations161. The oscillations can only be seen at low temperaturesT < 1 K and for samples of high purity, as indicated by small residual resis-tances. The oscillations are only seen in materials where the zero temperaturelimit of the resistivity ρ(0) is less than 1 µΩ cm. The term involving the lifetimecomes from the width of the quasi-particle spectrum, and should also be accom-panied with the change in quasi-particle energy due to interactions. Therefore,the increase in the quasi-particle mass can also be extracted from the amplitudeof the de Haas - van Alphen oscillations. However, the amplitude of the heaviermass bands are small compared with the light quasi-particle bands. In the heavyfermion materials such as CeCu6 and UPt3 quasi-particle masses of about 200free electron masses have been observed in de Haas - van Alphen experiments162.

11.4.5 Geometric Resonances

There are many other probes of the Fermi surface, these include the attenuationof sound waves163. Consider sound waves in a crystal propagating perpendicularto the direction of the applied magnetic field and having a transverse polariza-tion that is also perpendicular to Hz. The motion of the ions is accompaniedby an electric field of the same frequency, wave vector and polarization. Theelectrons interact with the sound wave through the electric field. If the wave161R. B. Dingle, Proc. Roy. Soc. A 211, 257 (1952).162S. Chapman, M. Hunt, P. Meeson, P. H. P. Reinders, M. Springford and M. Norman, J.

Phys. CM. 2, 8123 (1990), L. Taillefer, R. Newbury, G. G. Lonzarich, Z. Fisk and J. L. Smith,J. Mag. Mag. Mat., 63 & 64, 372 (1987).163H. W. Morse and J. D. Gavenda, Phys. Rev. Lett. 2, 250 (1959).

487

length of the mean free path is sufficiently long, i.e. ωc τ 1, the attenuationof the sound waves can be used to determine the Fermi surface.

The electrons follow real space orbits which have projections in the planeperpendicular toHz, which are just cross-sections of the constant energy surfacesin momentum space, but are scaled by h c

| e | Hzand rotated by π

2 . As velocitiesof the ions are much smaller than the electrons’ velocities, the electric field maybe considered to be static. If the phonon wave vector q is comparable to theradius of the real space orbit or, more precisely, the diameter of the orbit inthe direction of q, then the electric field can significantly perturb the electron’smotion. This strongly depends on the mismatch between q−1 and the diameterof the orbit164. When the radius of rc the orbit is such that

2 rc =λ

2(1565)

then the electron may be accelerated tangentially by the electric field at bothextremities of the orbit. The coupling is coherent over the electron’s orbit andthe coupling is strong. When the condition

2 rc = λ (1566)

is satisfied, the electron is sequentially accelerated and decelerated by the field.The coupling is out of phase on the different segments of the electron’s orbit sothat the resulting coupling is weak. In general the condition for strong couplingis that of constructive interference

2 rc = ( n +12

) λ (1567)

and weak coupling occurs when the interference is destructive

2 rc = n λ (1568)

The period differs slightly from the asymptotic large n variation just described.Assume that the projection of the electron’s trajectory on the plane perpendic-ular to the applied field Hz is circular. The energy transfer between the electronand the electronic wave in one orbit is given by

∆E =∫ 2π

ωc

0

dt E(r, t) . v(t) (1569)

The electric field is assumed to be polarized along the y direction, and q isdirected along the x direction. Since the orbit in momentum space is rotatedby π

2 with respect to the real space orbit, one has

vy(t) = v sin ωc t (1570)

164A. B. Pippard, Phil. Mag. 2, 1147 (1957).

488

andx(t) = rc sin ωc t (1571)

Thus, the energy transferred in one period is evaluated as

∆E = Ey v

∫ 2πωc

0

dt exp[i q rc sin ωc t − ωq t

]sin ωc t

∼ Ey v

(2πωc

)J1( q rc ) (1572)

since, in the limit that the phonon can be considered static, ωq → 0, so only theterm with the n = 1 Bessel function need be considered. Hence, the absorptionof transverse sound waves exhibits geometric resonances. The resonances occurfor phonon wave lengths which match the maxima of the Bessel function J1(x).

Only electrons near the Fermi surface can absorb energy from the soundwave. This is caused by the Pauli exclusion principle. The Pauli exclusionprinciple forbids electrons in states far below the Fermi energy to undergo low-energy excitations, since the states with slightly higher energy are completelyoccupied. The strength of the geometric resonances are determined by the dis-tribution of the z component of the electron velocity on the Fermi surface. Theelectrons with the extremal diameter on the Fermi surface are more numerousand, therefore, play a dominant role in the attenuation process. Thus, the soundwave may display an approximately periodic variation in λ where the asymptoticperiod is determined by

∆(

1λ

)=

12 rc

(1573)

By studying the attenuation of sound with different wave vectors, q, and differ-ent magnetic fields H one can map out the Fermi surface165.

11.4.6 Cyclotron Resonances

This experimental method of probing the Fermi surface requires the applicationof an microwave electric field at the surface of a metal. The field is attenuatedas it penetrates into the metal, and is only appreciable with a skin depth δ fromthe surface. Since the field does not penetrate the bulk, electrons can only pickup energy from the field when they are within the skin depth from the surface.

In one experimental geometry, a static (d.c.) magnetic field is applied par-allel to the surface, say in the x direction, so that the electrons undergo spiralorbits in real space. The x component of the electron’s velocity, vx, remainsconstant, but the electrons undergo circular motion in the y − z plane. It isonly necessary to consider the electrons that travel in spirals that are close andparallel to the surface, as it is only these electrons that couple to the microwave165M. H. Cohen, M. J. Harrison and W. A. Harrison, Phys. Rev. 117, 937 (1960).

489

field. The size of the orbit and the electron’s mean free path λ should be muchlarger than the skin depth δ. This holds true when the cyclotron frequency ωc

is large, and for microwave frequencies ω for which the anomalous skin depthphenomenon occurs. The condition of a long mean free path and large cy-clotron frequency is necessary for the electrons to undergo well defined spirals,so ωc τ 1.

The electrons pick up energy from the field only if they are within δ of thesurface. The electrons in the spiral orbits only experience the electric field eachtime they enter the surface region. They enter the skin depth periodically, withperiod

TH =2 πωc

(1574)

which is the period of the cyclotron motion. In general, the period is given bythe expression

TH =h2 c

| e | Hx

(∂A

∂E

)(1575)

where A(E) is the area of the surface of constant energy in reciprocal space.

The electron will experience an E field with the same phase, if the appliedfield has completed an integral number of oscillations during each cyclotronperiod. If the frequency of the a.c. field is ω, then the period of the a.c. fieldTE is given by

TE =2 πω

(1576)

For the a.c. field to act coherently on the electron, the periods must be relatedvia

TH = n TE = n2 πω

(1577)

for some integer n. Hence, this condition requires that the frequency of thecyclotron orbit match with the frequency of the a.c. electric field

ω = n ωc (1578)

so that the a.c. field resonates with the electronic motion in the uniform field.The resonance condition can be written as

1Hx

= 2 π | E | h2 c ωn(

∂A∂E

) (1579)

The factor

mc =h2

2 π

(∂A

∂E

)(1580)

is known as the cyclotron mass. For free electrons, A(E) = π k2 and E =h2 k2

2 m , so the cyclotron mass coincides with the electron’s band mass

mc = m (1581)

490

If the microwave absorption is plotted versus 1Hx

a series of uniformly spacedresonance peaks should be found.

The calculation of the absorption is the simplest in the case when the wavelength of the electromagnetic field λ is much larger than the cyclotron orbit andωc τ 1.

Consider the geometry in which the surface has its normal in the z directionand the d.c. magnetic field is applied parallel to the surface in the x direction

H = ex Hx (1582)

The a.c. electric field is applied in the y direction, so the oscillatory dependenceis given by

E = ey Ey exp[i ( q z − ω t )

](1583)

for | z | δ. The electron in its orbit experiences a rapidly alternating electricfield. For most values of z the contributions cancel. The cancellation only failsat the extremal values of z where the z component of the velocity falls to zero.In this case, the electron velocity is entirely parallel to the planes of constant z.

To the zero-th order approximation, the z component of the electron’s posi-tion can be expressed as

z(t) = z0 +v

ωcsinωc t (1584)

The total change in momentum of the electron due to the oscillating field, inone period, can be calculated in the semi-classical approximation. The changeof momentum and, hence, the current will be in the direction of the a.c. field.The impulse imparted to the electron is given by the integral of the electric fieldevaluated at the electron’s position

h δky = | e |∫ t+TH

t

dt′ Ey exp[i

(q z0 + q

v

ωcsinωc t

′ − ω t′) ]

(1585)

In general, the integral can be evaluated with the aid of the identity

exp[i x sin θ

]=

n=∞∑n=−∞

Jn(x) exp[i n θ

](1586)

where Jn(x) is the Bessel function of order n. When the frequencies preciselysatisfy the resonance condition, the integral is given by the n-th order Besselfunction. However, for large q v

ωcand with an arbitrary frequency mismatch, the

integral can be performed by the method of steepest descents. This yields tothe asymptotic form of the impulse given by

h δky ∼ − | e | Ey

(2 πq v ωc

) 12

exp[i

(q z0 − ω t − π

4

) ](1587)

491

Due to the variation of the phase of the field experienced by the electron as itperforms its orbit, only a fraction(

2 π ωc

q v

) 12

(1588)

of the orbit contributes to the integral. The energy gain of the electron in onetraversal is given by

v h δky = − | e | Ey

(2 π vq ωc

) 12

exp[i

(q z0 − ω t − π

4

) ](1589)

The previous traversal caused a similar displacement, but with t → t − 2 πωc

.However, only the fraction

exp[− 2 π

ωc τ

](1590)

of electrons survive traversing one cyclotron orbit without scattering. Thus,the contribution to the average increase in the energy of the electron from theprevious orbit is given by

∼ v h δky exp[− 2 π

ωc τ( 1 + i ω τ )

](1591)

As the contributions from all the previous orbits all have similar forms, one canobtain the average energy displacement experienced by an electron between onescattering and the next

v h ∆ky = − | e | Ey

(2 π vq ωc

) 12

exp[i

(q z0 − ω t − π

4

) ]F (1592)

The factor F reflects the sum of the probabilities that the electron survive norbits without scattering. The value of F is given by the sum of the contributionsfrom each orbit

F =∞∑

n=0

exp[− 2 π n

ωc τ( 1 + i ω τ )

]

=1

1 − exp[− 2 π

ωc τ ( 1 + i ω τ )] (1593)

When the contributions from the previous orbits survive and are in phase, thereis resonant absorption of the electromagnetic radiation. The resonance is deter-mined by the denominator of F , and is experimentally manifested by resonancesin the surface impedance166.

166M. Ia. Azbel’ and E. A. Kaner, Sov. Phys. J.E.T.P. 5, 730 (1957).

492

To calculate the current at a depth z0, one must examine the orbits on theFermi surface. The orbits circulate around the Fermi surface in sections thatare perpendicular to the d.c. field. Thus, the orbits are in the y− z plane. Theportions of the Fermi surface orbits which contribute most to the current arethose in which the electrons are moving parallel to the surface. These portionsof the orbits are those where kz = 0, and form the effective zone. An electronon the effective zone has kz = 0 and, thus, is moving at the extreme of its orbit.Due to the effect of the a.c. field, this orbit has been displaced by a distance∆ky from the orbit performed by the electron in the absence of the a.c. field.

The total current from the orbits in a section of width dkx, around kx, canbe calculated by considering the contribution of the electrons around the orbit.Due to the phase differences around the orbits only those within a distance

kx

(2 π ωc

q v

) 12

of the effective zone contribute. These orbits are displaced from

their equilibrium positions by an amount ∆ky. Only the displacements fromequilibrium contribute to the current. The contribution to the current densityis

δJy = − | e | 28 π3

dkx kx

(2 π ωc

q v

) 12

exp[iπ

4

]∆ky v

=e2

2 π2dkx kx

(1h q

)F Ey (1594)

where the phases of π4 cancel. The integration over dkx can be converted into an

integral over the effective zone via the polar angle ϕ. If the effective mass andcyclotron frequency are constant over the Fermi surface, then F is also constant.The conductivity σ is proportional to F . When ωc τ n, the conductivityexhibits resonances at n ωc = ω.

The surface impedance is defined as

Z(ω) =4 π i ω Ex(

∂Ex

∂x

) (1595)

This can be crudely estimated from the theory of the (normal) skin depth, as

Z(ω) ∼ ( 1 − i )(

2 π ωσ

) 12

∼ F−12 (1596)

A more rigorous calculation167 requires the use of the theory of the anomalousskin depth, and yields the result

Z(ω) ∼ ( 1 − i√

3 ) ω23 F−

13 (1597)

167L.E. Hartmann and J.M. Luttinger, Phys. Rev. 151, 430 (1966).

493

However, the conclusion remains that the surface impedance shows oscillationswith varying field strength Hx. It should be obvious from the above discussionthat the oscillations provide information on ω

ωc, and that the dominant contri-

bution occurs from the extremal parts of the line of intersection of the Fermisurface with the plane kz = 0. Thus, the cyclotron resonance can be used tostudy points on the Fermi surface168.

The other experimental geometry involves the static magnetic field beingapplied along the normal to the surface. The cyclotron orbits are in the planeparallel to the surface. Since the a.c. electric field is parallel to the orbit, theresonance condition involves the Doppler shift169.

11.5 The Quantum Hall Effect

The Quantum Hall Effect is found in quasi two-dimensional electron systems.Experimentally, electrons can be confined to a two-dimensional sheet in a metaloxide semiconductor field effect transistor. The application of a strong elec-tric field perpendicular to the surface may pull down the conduction band atthe surface of the semiconductor. The energy eigenvalue equation can be sep-arated into parts describing motion parallel and perpendicular to the surface.The wave function is factorized into two parts. One part describes the motionparallel to the surface which can be approximated by a two-dimensional Blochfunction which corresponds to an eigenvalue in a continuum. The other fac-tor describes one-dimensional motion of electrons which are confined near thesurface by the applied field and corresponds to discrete energy levels. If thestates which correspond to the lowest discrete energy level are occupied, theelectrons are confined in the vicinity of the surface and the system correspondsto a two-dimensional electron gas. If the energy levels are lower than the Fermienergy in the metal, these states are occupied by electrons which tunnel from themetal, across the insulating oxide barrier into the semiconductor. After equi-librium has been established, the electrons at the surface of the semiconductorform a two-dimensional electron gas. The Quantum Hall Effect occurs in thetwo-dimensional electron gas when a magnetic field is applied that is so strongthat the mixing of the Landau levels by disorder or electron-electron interac-tions is negligible. The Integer Quantum Hall Effect occurs when the disorder isstronger than the electron-electron interactions. The Fractional Quantum Halleffect occurs when the effect of electron-electron interactions are greater thanthe effects of disorder.

168T. W. Moore and F. W. Spong, Phys. Rev. 125, 846 (1962).169P. B. Miller and R. R. Haering, Phys. Rev. 128, 126 (1962).

494

11.5.1 The Integer Quantum Hall Effect

The Integer Quantum Hall Effect can be understood entirely within the frame-work of non-interacting electrons. The calculation of the Hall coefficient canbe performed using the Kubo formula. However, in applying the Kubo for-mulae, one must recognize that the vector potential has two components: ana.c. component responsible for producing the applied electric field and a secondcomponent which produces the static magnetic field. The usual derivation onlytakes the a.c. component of the vector potential into account as a perturbation.The d.c. component of the vector potential must be added to the paramagneticcurrent operator, yielding the electron velocity operator appropriate for the sit-uation where the weakly perturbing electric field is zero.

The application of a static magnetic field Bz perpendicular to the surfacewill quantize the motion of the electrons parallel to the surface. The motionparallel to the surface is quantized into Landau orbits, and the energy eigenvalueequation reduces to the energy eigenvalue equation of a one-dimensional simpleharmonic oscillator. Due to the confinement in the direction perpendicular tothe surface, kz will not be a good quantum number, and the perpendicularcomponent of the energy will form highly degenerate discrete levels εn. Forlarge enough fields only the lowest level ε0 will be occupied. On choosing aparticular asymmetric gauge for the vector potential,

A(r) = + ey Bz x (1598)

the Hamiltonian for the two-dimensional motion in the x − y plane is given by

H =p2

x

2 m+

m ω2c

2

(x − X

)2

(1599)

where the cyclotron frequency ωc is given by

ωc =| e | Bz

m c(1600)

The operator X is given in terms of the y-component of the momentum operatorpy. Since the Hamiltonian is independent of y, both py and X can be taken asconstants of motion. The momentum component px is canonically conjugate tothe x component of the particle’s position relative to the center of the orbit

x − X = x +py c

| e | Bz(1601)

The energy eigenvalues of the shifted Harmonic oscillator are given by

Eν,0 = E0 + h ωc ( ν +12

) (1602)

where ν is the quantum number for the Landau levels. The Landau levels areindependent of ky and, therefore, are degenerate. Due to the periodic boundary

495

conditions, the values of ky are quantized by

ky =2 π ny

Ly(1603)

for a surface of length Ly. Since the centroid of the wave function is constrainedby

Lx > X > 0 (1604)

the possible value of ky or ny are restricted by

Lx >2 π h c ky

| e | Bz> 0 (1605)

Hence, the total number of degenerate states is given by

D = Lx LyBz | e |2 π h c

=Φ | e |2 π h c

(1606)

where Φ is the total magnetic flux passing through the sample. The fundamentalflux quantum Φ0 is defined as the quantity

Φ0 =2 π h c| e |

(1607)

Therefore, the degeneracy is equal to the number of flux quanta. The densityof states can be approximately expressed as a discrete set of delta functions

ρ(ε) =| e | Bz Lx Ly

h c

∑ν

δ

(ε − E0 − h ωc ( ν +

12

))

=m ωc Lx Ly

2 π h

∑ν

δ

(ε − E0 − h ωc ( ν +

12

))

(1608)

where each delta function corresponds to a Landau level. The weight associatedwith each delta function corresponds to the degeneracy of each Landau level.On defining the cyclotron radius rc by

rc =

√h c

| e | Bz(1609)

one finds that the relative position operator can be expressed as

x − X =rc√2

(a†ky

+ aky

)(1610)

496

and from the Heisenberg equation of motion for x, one finds that the x-componentof the velocity is given by

vx =1i h

[x , H

]= i

rc ωc√2

(a†ky

− aky

)(1611)

as py has been diagonalized. The y-component of the velocity is found from theHeisenberg equation of motion for y and is given by

vy =1i h

[y , H

]= ωc ( x − X ) (1612)

since the commutator[y , X

]= − i h c

| e | Bz. On substituting for the x-

component of the displacement from the center of the orbit, one finds

vy =rc ωc√

2( a†ky

+ aky) (1613)

Hence, vx and vy do not commute. Furthermore, as the velocity operators arenot diagonal in the quantized Landau level indices, the Landau orbitals do notcarry a net current.

The Kubo formula can be expressed in terms of the electron velocity op-erators, which includes the diamagnetic current contributions from the staticmagnetic field. The Kubo formula for the conductivity tensor, per unit area,appropriate for non-interacting electrons has the form

σα,β(ω) =i e2

ω + i η

1Lx Ly

[ ∑ν,ν′,ky

< ν, ky | vα | ν′, ky > < ν′, ky | vβ | ν, ky > ×

× f(Eν) − f(Eν′)h ω + i η + h ωc ( ν − ν′ )

−∑ν,ky

f(Eν)m

δα,β

](1614)

since ky is a constant of motion. On evaluating the Kubo formula170, one findsthat the diagonal components are zero

<e σx,x(0) = 0 (1615)

but, nevertheless, the off-diagonal components are finite and quantized

<e σx,y(0) = − e2

2 π hn (1616)

170P. Streda and L. Smrcka, Phys. Stat. Sol., B, 70, 537 (1975), D. J. Thouless, J. Phys. C,14, 3475 (1981), Q. Niu and D. J. Thouless, Phys. Rev. B, 35, 2188 (1987).

497

clotron energy, separating the filled from the empty lev-els, is the commensuration energy. I assumed he meant:In the n,nw extreme quantum limit, at commensurationwhen n5n/nw51/i , some interaction energy might be-come dominant to drive the 2D system into some newground state. I was not brave enough to ask him: ‘‘Whatdo you mean?’’ But I felt affirmed that I should continueto concentrate on the extreme quantum limit.

Indeed, with the advent of molecular beam epitaxy(Cho, 1995) and the invention of modulation doping toproduce highly perfect 2D electron systems (Stormeret al., 1979), it soon became quite clear to Art Gossard,Horst Stormer, and me that where we wanted to go tolook for new many-body interaction physics should be ahighest-mobility 2DEG sample placed in a most intensemagnetic field.

TWO-DIMENSIONAL MAGNETO-TRANSPORT

In the presence of a perpendicular magnetic field, theenergy levels of a two-dimensional electron collapse, asa result of Landau quantization of its cyclotron orbits,into discrete Landau levels separated by the cyclotronenergy quantum. Scattering broadens the Landau levelsand gives rise to 2D magneto-transport described by theAndo-Uemura theory (Ando and Uemura, 1979). Fig-ure 1 is an example showing the quantum oscillations inthe diagonal resistivity rxx , reflecting the broadenedLandau-level structure of the 2DEG, and the Hall resis-tance rxy , well known from the Drude model. However,when the 2DEG is taken to the extreme condition ofhigh B and low T, much more striking features appear,showing the interplay of disorder and electron-electroninteraction in the system. More specifically, differentphysics phenomena are observed in three distinctly dif-

ferent physical regimes. The first is the disorder-dominant regime, when the sample is dirty with low 2Delectron mobility (e.g., m,105 cm2/V sec in the case ofGaAs). The striking features in the data constitute theintegral quantum Hall effect (IQHE) (von Klitzing et al.,1980), which is understood in terms of the physics ofindependent electrons and their localization in the pres-ence of random impurities in the semiconductors. Thefractional quantum Hall effect (FQHE) is observed inhigh-mobility samples in the regime where the electron-electron interaction dominates. It manifests the many-body interaction physics of the 2DEG in the intense Bfield. Furthermore, even in the cleanest samples, theFQHE series terminates into an insulator in the high-Blimit. This insulator is believed to be an electron crystalpinned by defects to the semiconductor. The third re-gime is this high-m and high-B-field limit, where disorderand interaction play equally important roles and need tobe treated on an equal footing.

QUANTUM PHASE TRANSITIONS IN IQHE

Quantization of the Hall resistance in the natural con-ductance unit e2/h is currently understood in terms ofthe existence of an energy gap, separating the excitedstates from the ground state, and localized states insidethe gap. In the IQHE case, where the quantum numbersare integers identified with the number of completelyfilled Landau levels, the energy gap is the Landau gap ofa cyclotron energy quantum. The accurate quantizationwas shown by Laughlin, using a gedanken experiment,to be a consequence of charge quantization. He showedthat the experiment in effect measures the charge car-ried by the excited electron. The localized states arisefrom disorder in the 2D system, and the data, as shownin Fig. 2, show the localization-delocalization phase tran-sitions. In other words, for B in the plateau regions, thestates at EF are localized, and in between, delocalized.As T is decreased, the range of B for the existence ofdelocalized states decreases and the transition regions

FIG. 1. Magneto-transport coefficients rxx and rxy of a 2DEGin GaAs/AlxGa12xAs at 0.35 K in moderately low B. The in-sert shows the measurement geometry. The magnetic field B isperpendicular to the plane of the 2DEG and to the current I.The voltages V and VH are measured along and perpendicularto I, respectively, rxx5(V/L)/(I/W) is the resistance across asquare, independent of the square size, and rxy5VH /I is theHall resistance independent of the sample width. Data takenby A. Majumdar.

FIG. 2. rxx and rxy of a relatively low-mobility 2DEG inGaAs/AlxGa12xAs. The plateaus in rxy are quantized in thenatural conductance unit e2/h with integer quantum numbersi51,2, . . . . Data taken by H. P. Wei.

892 Daniel C. Tsui: 2D electron gas in intense magnetic fields

Rev. Mod. Phys., Vol. 71, No. 4, July 1999

Figure 221: The Integer Quantum Hall Effect in a GaAs − GaAlAs hetero-junction at T = 66 mK [data taken by H.P. Wei]. The plateau indices are (fromthe right) 1, 2, 3, 4, 6, 8... The odd-indexed plateaus can only be resolved athigh fields. [After D.C. Tsui, Rev. Mod. Phys. 71, 892 (1999).]

where (n − 1) is the quantum number for the highest occupied Landau orbital.That is, the off-diagonal conductivity is quantized and the quantum numbercorresponds to the total number, n, of occupied Landau levels. Since the diag-onal component of the conductivity vanishes, the current flow is dissipationless.As the field is changed, the peaks in the density of states associated with theLandau levels sweep through the Fermi level. The Hall resistivity should, there-fore, show a set of steps as the applied field is increased. This phenomenon isthe integer quantum Hall effect.

Experimentally, it is found that the steps in σx,y(0) are not discontinuous,but instead show a finite slope in the transition region. Furthermore, the diag-onal component of the resistivity is not zero in the region where the transitionbetween successive plateaus occur, but shows spikes there. This phenomenonis associated with impurity scattering. The effect of impurities is to broadenthe set of delta function peaks in the density of states into a set of Gaussians.This allows the transition between the plateaus to be continuous. In fact, ifall the states contributed to the conductivity, the steps of the staircase wouldbe smeared out into a straight line, just like in the Drude theory for three-dimensional metals. However, the states in the tails of each Gaussian are local-ized171, as the deviation from the ideal Landau level energy indicates that thesestates experience a larger impurity potential than average. The large potentialacts to localize the states with energies in the Gaussian tail and so these statesdo not contribute to the Hall conductivity. In fact, in two-dimensions with zerofield, one can show that all the states are localized in an infinite sample. How-171H. Levine, S. B. Libby and A. M. M. Pruisken, Phys. Rev. Lett. 51, 1915 (1983).

498

0

0.5

1

1.5

-3 -1 1 3ε

ρ(ε)

Figure 222: The schematic density of states for a two-dimensional electron gasin a high-magnetic field. Due to the effects of disorder, the states in the tailsof the Landau levels are localized (shaded). Although the region of occurrenceof localized states appears similar to that found in Anderson Localization intwo-dimensions, the effect of the magnetic field breaks time-reversal invarianceand, therefore, yields important differences.

ever, the samples are finite and have edges. The edges have extended states thatcarry current. The edge states can be understood by analogy with the classi-cal motion where there are skipping orbits. In a classical skipping orbit, thecyclotron orbits are reflected at the edges. The classical skipping orbits wouldproduce oppositely directed currents at pairs of edges. Quantum mechanically,the bulk states do not contribute to the current since the velocity operator fora particle with charge q, is given by

vy =py

me− q

me cAy

= ωc ( x − X ) (1617)

and the probability density for the shifted harmonic oscillator is symmetricallypeaked about X. Since the current carried by individual particles in the bulkstates are given by integrals which are almost anti-symmetric

jky,ν = q ωc

∫ Lx

0

dx | φν( x − X ) |2 ( x − X ) (1618)

the bulk contributions to the current vanish. On the other hand for X closeto the boundary, say at X = 0, then the wave function must vanish at theboundary for a hard core potential and so the current is given by

jky,ν = q ωc

∫ Lx

0

dx | φν( x ) |2 x (1619)

499

V

Iφ

Br

Φ

Figure 223: The geometry of Laughlin’s gedanken experiment which identifiesthe quantum of Hall conductance with the electric charge. A magnetic flux Φthreads through the axis of the cylinder. On increasing the magnetic flux bythe flux quantum Φ0, there is a net transfer of one electron from one end of thecylinder to the other.

Since the wave function is cut off at x = 0, the integral is positive and theedge state carries current. The other edge state carries an oppositely directedcurrent. The presence of the confining potential also lifts the degeneracy of thestates in the Landau levels, by increasing the energy of the states close to theboundary. As the wave function of the odd order excited state Landau levelsof the homogeneous system vanish at x = X, one finds that the energy of theLandau levels with hard wall confining potentials increases from ( ν + 1

2 ) h ωc

to ( 2 ν + 32 ) h ωc as X → Lx. The increase in the Hall resistivity only occurs

when the Fermi level sweeps through the itinerant or delocalized portions of thedensity of states.

As the above calculations completely neglects the effect of impurities andlocalization, Laughlin172 proposed a gauge theoretic argument which overcomesthese shortcomings. Laughlin envisaged an experiment in which the two-dimensionalsample is in the form of the surface of a hollow cylinder of radius R and lengthLz. The axis of the cylinder is taken to be along the z-direction. A uniformmagnetic field Br is arranged to flow through the sample in a radial direction.Locally, this field is perpendicular to the plane in which the electrons are con-fined. A second field is arranged to thread through the cylinder parallel to itsaxis, but is entirely contained inside the hollow and falls to zero at r = R.This field does not affect the motion of the electrons directly since it is zeroinside the sample. However, the associated flux threading through the cylinder172R. B. Laughlin, Phys. Rev. B, 23, 5632 (1981).

500

Φ does lead to a finite vector potential AΦ which satisfies

∇ ∧ AΦ = 0 (1620)

Therefore, the vector potential can be written in the form

AΦ = ∇ Λ (1621)

This vector potential does cause the wave functions of the charged particles toacquire an Aharonov-Bohm phase factor of

exp[− i

q

h cΛ]

(1622)

For the vector potential

AΦ =Φ

2 π reϕ (1623)

one finds thatΛ =

Φ2 π

ϕ (1624)

and, hence, the phase factor is given by

exp[− i

q Φ2 π h c

ϕ

](1625)

Thus, on traversing a singly connected path around the cylinders axis, theAharonov-Bohm flux changes the extended state wave function by a factor of

exp[− i

q Φ2 π h c

2 π]

(1626)

Since the fundamental flux quantum Φ0 is defined by

Φ0 =2 π h c| e |

(1627)

the Aharonov-Bohm factor for electrons can be written as

exp[i

ΦΦ0

2 π]

(1628)

Thus, if Φ is an integer multiple of Φ0, i.e.,

Φ = µ Φ0 (1629)

the extended wave functions are single-valued.

The presence of the perpendicular field Br within the sample quantizes themotion into Landau levels. These states may either be localized or may be

501

extended throughout the sample. The vector potential at position z is purelytangential and is given by

A =(Br z +

Φ2 π R

)eϕ (1630)

If the vector potential is absorbed into the wave function by gauge transforma-tion, involving the factor

exp[i

(z R Br +

Φ2 π

)ϕ

Φ0

](1631)

The flux quantization condition is found by requiring that the wave function tobe single-valued. Hence,(

2 π z R Br + Φ)

= µ Φ0 (1632)

must be an integer multiple of Φ0. If Φ is adiabatically increased by ∆Φ = Φ0,then the extended states in a Landau level must be translated along the z-axisby amounts ∆z given by

∆z = − Φ0

2 π R Br(1633)

as the phase of an extended wave function must be single valued. If the fluxthrough the entire surface is equal to ΦS , then ∆z is given by

∆z = − LzΦ0

ΦS(1634)

The phase of the localized states can shift by arbitrary amounts. The presenceof the gap forbids excitation of electrons to states in the higher Landau levels. Inthe pure system, the maximum number of electronic states, M , in a Landau levelis given by the total number of the integer valued quantum numbers µ = mwhich satisfy eqn(1632) with Lz > z > 0,

2 π R Lz Br = M Φ0 (1635)

Hence, a fully occupied Landau level contains a number

M =ΦS

Φ0

=Lz

| ∆z |(1636)

of electrons. Furthermore, for the pure system, the adiabatic increase of Φ by Φ0

results in the transfer of electrons between neighboring extended states. Hence,in the dirty system, all the delocalized electrons in a Landau level are translatedalong the z-direction by one spacing, skipping over the localized states. The netresult is that one electron, per Landau level, is translated across the entire

502

length of the sample. In the absence of an applied electric field, the initial andfinal states have the same energy. Thus, by gauge invariance, increasing theAharonov-Bohm flux by Φ0 maps the system back on itself. However, if thereis an electric field Ez across the length of the cylinder, this process requires anenergy change of

∆E = − q Ez ( − Lz )= − | e | Ez Lz (1637)

per Landau level. The current wrapping around the cylinder Iϕ, from all theelectrons in a single Landau level, is given by

Iϕ = − c∂E

∂Φ(1638)

which leads to a current density

jϕ =c

Lz

| e | Ez Lz

Φ0

=e2

2 π hEz (1639)

Hence, on summing over all the n occupied Landau levels, one has

jϕ =e2

2 π hn Ez (1640)

In this expression n is the number of completely occupied Landau levels withextended states, and the Fermi energy is assumed to lie in an energy rangewhere the bulk states are localized. The Hall conductivity is given by

σH =jyEz

=e2

2 π hn (1641)

Hence, as long as there are Landau levels with extended states, there is an in-teger quantum Hall effect.

The integer quantum Hall effect was measured experimentally by von Kl-itzing173 in 1980. The steps can only be discerned in very clean samples. Atmuch higher fields, where only the lowest Landau level should be occupied, Gos-sard, Stormer, and Tsui discovered a similar type of effect174 which is known asthe fractional quantum Hall effect. This phenomenon involves the effect of theCoulomb repulsion between electrons in the Landau levels. Laughlin showed175

that the energy of the interacting electron states can be minimized by allow-ing the electrons to form a ground state with a different symmetry from the173K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, 494 (1980).174D. C. Tsui, H. L. Stormer and A. C. Gossard, Phys. Rev. Lett. 48, 1559 (1982).175R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983).

503

being fermions they are prevented from condensing intothe lowest energy state. Instead, they fill up successivelythe sequence of lowest-lying energy states, until a maxi-mum is reached and all CFs have been accommodated.The process is equivalent to the filling of states by elec-trons at B50. Hence, from the point of view of CFs, then51/2 state appears equivalent to the case for electronsat B50. In spite of the huge external magnetic field athalf filling of the Landau level, CFs are moving in asimilar fashion to electrons moving in zero field. Thishas been directly observed in experiment. Flux quantumattachment has transformed these earlier electrons andthey are propagating along straight trajectories in a highmagnetic field, where normal electrons would orbit onvery tight circles. The mass of a CF, usually consideredto be a property of the particle, is unrelated to the massof the underlying electron. Instead, the mass depends onthe magnetic field and only on the magnetic field. Infact, it is a mass of purely many-particle origin, arisingsolely from interactions, rather than being a property ofany individual particle. It is another one of these bafflingimplications of e-e interactions in high magnetic fields.The absence of condensation and the lack of an energygap prevents the n51/2 state from showing a quantizedHall resistance. Instead the Hall line is featureless, justas it is for electrons around B50 (see Fig. 18).

The difference between n51/3 and n51/2 is striking.One is a Bose-condensed many-particle state showing aquantized Hall effect and giving rise to fractionallycharged particles. The other is a Fermi sea, in spite ofthe existence of a huge external field, and its particleshave a mass that arises from interactions. One flux quan-tum per electron makes all the difference.

There are many fascinating open questions associatedwith the n51/2 state, such as: how does the mass varywith energy for CFs? and what is the microscopic struc-ture of the particles? Also, how does the electron spin(which we were neglecting throughout this lecture) af-fect CF formation? A beautiful picture of composite fer-mions being tiny dipoles is emerging. While one of thevortices is placed directly on the electron (Pauli prin-ciple), the position of the second vortex is a bit displacedfrom exact center, rendering the object an electric dipolein the 2D plane. There is great promise for future dis-covery and future theoretical insight.

All those other FQHE states

Bose condensation of CBs consisting of electrons andan odd number of flux quanta rationalizes the appear-ance of the FQHE at the primary fractions aroundLandau-level filling factor n5i61/q with quantized Hallresistances RH5h/(ve2) and deep minima in the con-comitant magnetoresistance R. However, a multitude ofother FQHE states have been discovered over the years.Figure 18 shows one of the best of today’s experimentaltraces on a specimen with a multimillion cm2/V sec mo-bility. What is the origin of these other states? The com-posite fermion model offers an extraordinarily lucid pic-ture. We shall discuss it for the sequence of prominentfractions 2/5, 3/7, 4/9, 5/11, . . . and 2/3, 3/5, 4/7, 5/9, . . .(i.e., n5p/(2p61), p52,3,4 . . . ) around n51/2.

At half filling the electron system has been trans-formed into CFs consisting of electrons which carry twomagnetic flux quanta. All of the external magnetic fieldhas been incorporated into the particles and they residein an apparently field-free 2D plane. Since they are fer-mions, the system of CFs at n51/2 resembles a system ofelectrons of the same density at B50. What happens asthe magnetic field deviates from B50? For electronstheir motion becomes quantized into electron-Landauorbits. They fill up their electron-Landau levels, encoun-ter the energy gaps, and exhibit the well-knownIQHE. CFs around n51/2 follow the same route. Asthe magnetic field deviates from exactly n51/2, the mo-tion of CFs becomes quantized into CF-Landau orbits.They fill up their CF-Landau levels, encounter CF-energy gaps, and exhibit an IQHE. However, this is notan IQHE of electrons, but an IQHE of CFs. This IQHEof CFs arises exactly at n5p/(2p61), which are thepositions of the FQHE features. In fact, the oscillatingfeatures in the magnetoresistance R of the FQHEaround n51/2 closely resemble the oscillating featuresin R around B50 and, once they have been shifted fromB50 to n51/2, they coincide with their position. This isvery remarkable in several ways.

CFs ‘‘survive’’ the additional (effective) magnetic field(away from n51/2), and the orbits of these compositeparticles mimic the orbits of electrons in the equivalentmagnetic field in the vicinity of B50. The CFs remain‘‘good’’ particles. In this way, a complex electron many-particle problem at some rational fractional filling factorhas been reduced to a single-particle problem at integer

FIG. 18. The FQHE as it appears today in ultrahigh-mobilitymodulation-doped GaAs/AlGaAs 2DESs. Many fractions arevisible. The most prominent sequence, n5p/(2p61), con-verges toward n51/2 and is discussed in the text.

886 Horst L. Stormer: The fractional quantum Hall effect

Rev. Mod. Phys., Vol. 71, No. 4, July 1999

Figure 224: The Fractional Quantum Hall Effect in an ultra-high mobilitymodulation-doped GaAs−GaAlAs 2-d electron gas. [After H.L. Stormer, Rev.Mod. Phys. 71, 874 (1999).]

bulk. Since Laughlin’s state has a lower symmetry, it can not be adiabati-cally continued to states of the ideal Landau levels, without states crossing theFermi-energy. Hence, Laughlin’s gauge argument does not forbid the possibilityof non-integer quantization of the Hall conductance.

——————————————————————————————————

11.5.2 Exercise 71

Evaluate the Kubo formula for the real part of the diagonal and off-diagonalquantum Hall conductivities by first taking the limit ω → 0 and then tak-ing the limit ε → 0. Also estimate the effects of introducing scattering dueto random impurities. The effect of the scattering lifetime can be introducedby including imaginary parts of the energies of the occupied and unoccupiedsingle-particle states of the form ± i h

2 τ . Choose the signs to ensure that thewave functions for the excited states (with an electron - hole pair) will decay tothe ground state after a time τ . Compare your result with the conductivitiesobtained for the three-dimensional Drude model.

——————————————————————————————————

504

11.5.3 Exercise 72

Using the definition of the Hamiltonian for non-interacting electrons

H =1

2 m

(p − q

cA

)2

(1642)

in the presence of an Aharonov-Bohm phase Φ, show that the total current Iflowing around the cylinder enclosing the flux is given by the derivative

I = − c∂E

∂Φ(1643)

——————————————————————————————————

Solution

The total charge, per unit time, crossing a line parallel to the cylinder’s axisis given by the expectation value

I =q

me 2 π R L

∑j

< Ψ |(pϕj − q

cAϕ(rj)

)| Ψ > (1644)

where A is the vector potential due to the field in the sample. It has beenassumed that the charges are distributed uniformly on the sheet. Therefore, thenumber of electrons crossing the line, in the time interval dt, is just ρ vϕ dt.The Hamiltonian can be written as

H =∑

j

12 me

[p2

zj +(pϕj − q

cAϕ(rj) −

q Φ2 π R c

)2 ](1645)

where the term involving Φ represents the change in the kinetic energy due tothe Aharonov-Bohm field. On taking the expectation value and then taking thederivative with respect to Φ, one obtains

∂E

∂Φ=

q

2 π R me c

∑j

< Ψ |(pjϕ − q

cAϕ(rj)

)| Ψ >

= − 1cI (1646)

——————————————————————————————————

505

11.5.4 The Fractional Quantum Hall Effect

Consider a particle of mass me, confined to move in the x − y plane with auniform magnetic field in the z direction. Using the circularly symmetric gauge,the energy eigenstates are also eigenstates of angular momentum. The single-particle wave functions describing the states in the lowest Landau level withangular momentum m are all degenerate. The wave functions of these statescan be written as

φm(rj) =1√

2 π m! r2c

(xj + i yj

2 rc

)m

exp[−

x2j + y2

j

4 r2c

](1647)

where the length rc is given by

rc =

√h c

| e | Bz(1648)

The probability density for finding a particle at position r has a peak whichform a circle around the origin. The radius of the circle depends on m, so thatfor large m

r ∼ rc√

2 ( m + 1 ) (1649)

The many-particle ground state wave function corresponding to the com-pletely filled lowest Landau level is constructed as a Slater determinant fromthe states of different m. The spins of the electrons are assumed to be fullypolarized by the applied field. The Ne particle wave function is

Ψ ∼∏i>j

(( xi − xj ) + i ( yi − yj )

) Ne∏k=1

exp[− x2

k + y2k

4 r2c

](1650)

This satisfies the Pauli exclusion principle as the wave function vanishes linearlyas ri → rj . The linear vanishing is a signature that the a pair of particles arein a state of relative angular momentum m = 1, together with contributionsfrom states of higher angular momentum. This can be seen by expressing theprefactor as a van der Monde determinant

∏i>j

(zi − zj

)=

∣∣∣∣∣∣∣∣∣z01 z1

1 z21 . . . zNe−1

1

z02 z1

2 z22 . . . zNe−1

2...

...z0Ne

z1Ne

z2Ne

. . . zNe−1Ne

∣∣∣∣∣∣∣∣∣ (1651)

where zj = xj + i yj . Hence, the Ne electrons occupy the zero-th Landaulevels single-particle states with all the angular momentum quantum numbers

506

in the range between m = 0 and m = Ne − 1. The wave function is aneigenfunction of the total angular momentum. The total angular momentumabout the origin is Mz = Ne ( Ne − 1 )

2 h. Since all the one-electron states inthe lowest Landau level are occupied, the many-particle state corresponds to auniform particle density of

ρ =D

Lx Ly=

12 π r2c

(1652)

particles per unit area. Hence, this many-particle state corresponds to the com-pletely filled lowest Landau level.

For larger Bz the lowest Landau level is only partially filled. The wavefunction which minimizes the interactions between pairs of particles is given bythe Laughlin trial wave function176

Ψp ∼∏i>j

(( xi − xj ) + i ( yi − yj )

)p Ne∏k=1

exp[− x2

k + y2k

4 r2c

](1653)

for odd integers p. The power of p in the prefactor has the effect of minimizingthe interactions between particles, since the square of the wave function vanisheslike a power law with power 2p instead of quadratically. This is a consequenceof the pairs of particles being in states with relative angular momentum p. TheLaughlin state is also an eigenstate of total angular momentum with eigenvalueMz = Ne ( Ne − 1 )

2 p h. Since the trial state contains one particle for every pquantum states in the lowest Landau level, the many-particle state correspondsto a uniform particle density of

ρ =1

2 π r2c p(1654)

particles per unit area. The filling factor, ν, is defined as

ν =Ne

D

= NeΦ0

Φ

=ρ Φ0

Bz(1655)

The Laughlin state corresponds to a state with the fractional filling, ν = 1p , of

the lowest Landau level. The energy of the Laughlin ground state is determinedby the Coulomb interaction energy. Since the Coulomb potential is a centralpotential, it conserves the relative angular momentum. In the Laughlin state,the energy is evaluated as177

Eg

Ne= − 0.78213

√p

(1 − 0.211

p0.74+

0.012p1.7

)e2

ε rc(1656)

176R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983).177D. Levesque, J. J. Weis and A. H. MacDonald, Phys. Rev. B, 30, 1056 (1984).

507

The energy per particle, for small p, is lower than any other candidate state byan amount determined by the Coulomb interaction between states with angularmomentum m < p.

11.5.5 Quasi-Particle Excitations

Quasi-particles can either be thermally excited, or can be excited by coupling thesystem to an external perturbation. One method of introducing quasi-particlesinto the fractional quantum Hall state is by slightly changing the applied mag-netic field. The quasi-particle excitations of the Laughlin state, like the quasi-particle excitations of a completely filled Landau level, can be obtained fromconsidering the effect of adding a number of flux quanta, Φ, passing through thecenter of the system. Although the magnetic field generating the extra flux doesnot act on the electrons in the sample, it does add an Aharonov-Bohm phase tothe system. First, we shall consider the completely filled nr = 0 Landau level.

The single-particle wave function φ(r, ϕ) experiences a vector potential ofthe form

A =[Bz r

2+

Φ2 π r

]eϕ (1657)

where Bz is the uniform field and Φ is the Aharonov-Bohm flux. Since thesingle-particle energy eigenstates satisfy[− h2

2 me

1r

∂

∂r

(r∂

∂r

)+

12 me

(− i h

r

∂

∂ϕ− q

c(Bz r

2+

Φ2 π r

))2

− E

]φ(r, ϕ) = 0

(1658)Then, with the ansatz

φ(r, ϕ) =1√2 π

exp[

+ i µ ϕ

]R(r) (1659)

one finds that the radial wave function is given by the solution of[− h2

2 me

1r

∂

∂r

(r∂

∂r

)+

h2

2 me r2

(µ− q Φ

2 π h c− q Bz r

2

2 h c

)2

− E

]R(r) = 0

(1660)Hence, the solutions for the lowest Landau level are of the form

φ(r, ϕ) ∼ exp[i

q Φ2 π h c

ϕ

]exp

[i ν ϕ

]rν exp

[− r2

4 r2c

](1661)

whereν = µ − q Φ

2 π h c(1662)

Since the wave function is single valued, µ must be an integer, say m. Themaximum in the probability density is determined by ν and rc. On increasing

508

ν the particles move away from the origin. On subtracting one flux quantum,Φ0, through the center of the loop, where

Φ0 =2 π h c| e |

(1663)

then the degenerate eigenfunctions transform into themselves, m → m + 1. Ifthe Landau level had been completely filled, then one particle has been pushedto the edge of the system and a hole has been created in the m = 0 orbit. Thisis the quasi-hole excitation in the filled Landau level.

The wave function of the Laughlin state with an excited quasi-hole is givenby similar considerations. The insertion of a flux quanta produces an extraAharonov-Bohm phase. The requirement that the wave function is single valuedrestricts µ to be integer, m. The Laughlin state in which the flux is decreased byone flux quantum Φ0 has m shifted by m → m + 1 which creates a quasi-holeat the origin. The many-particle wave function with a quasi-hole at the originis given by the expression

Ψ+p ∼

∏i

( xi + i yi )∏i>j

(( xi − xj ) + i ( yi − yj )

)p Ne∏k=1

exp[− x2

k + y2k

4 r2c

](1664)

The wave function of the Laughlin ground state with a quasi-hole present at r0is given by

Ψ+p ∼

Ne∏i=1

(xi − x0 + i ( yi−y0 )

) ∏i>j

(( xi − xj ) + i ( yi − yj )

)p Ne∏k=1

exp[− x2

k + y2k

4 r2c

](1665)

where one flux quanta has also been removed from r0. This state has angularmomentum of

Mz =Ne ( Ne − 1 )

2p h + Ne h (1666)

as there is now an extra zero at point r0. Due to the zero, the charge densityof this state is depleted around r0. The charge deficiency is smaller than thataround the position of any electron by a factor of 1

p . Hence, the quasi-hole hasthe fractional charge − q

p . The above argument demonstrating the existenceof fractionally charged quasi-particles can be re-phrased in a classical language.On inserting a microscopic solenoid through the two-dimensional electron gas,and adiabatically increasing the flux through the metal by an amount Φ0, oneintroduces a small electric field. From Faraday’s law of induction,

∇ ∧ E = − 1c

∂B

∂t(1667)

one finds, after using Stoke’s theorem, that the electric field encircles the solenoidand the field at a radius r is given by

2 π r Eϕ(t) = − 1c

∂Φ(t)∂t

(1668)

509

Φ(t)

Eφ

jr

Figure 225: The geometry of Laughlin’s gedanken experiment which identifiesthe fractional charge on the quasi-hole excitations.

Since the two-dimension electron gas exhibits the fractional quantum Hall effect,this field induces a radial current

jr = σr,ϕ Eϕ (1669)

where the transverse conductivity is given by

σr,ϕ = − νe2

h(1670)

and ν is the filling factor. This equation is integrated with respect to time. Onefinds that, when the flux is increased by Φ0, a charge Q is transferred into theregion

Q = 2 π r∫ 0

−∞dt jr(t)

= νe2

h

1c

∫ 0

−∞dt

∂Φ∂t

= νe2

h

1c

Φ0

= ν | e | (1671)

Hence, the quasi-particle excited by the non-uniform flux has a fractional charge.Alternatively, one may notice that by adding p quasi-holes at the same pointand then on adding an electron there, one just obtains the wave function for theLaughlin state with one more electron. Hence, p quasi-holes are neutralized byan extra electron. The operator creating a quasi-hole can be written just as

Sp ∼Ne∏i=1

(xi − x0 + i ( yi − y0 )

)(1672)

510

since it just adds zeros to the wave function.

Creating a quasi-particle is a little more complicated, as adding a flux quan-tum results in the transformation m → m − 1. Hence, the circles contract tothe origin, but the state with m = 0 is already filled in the initial state. Thismust be lifted to the next Landau level, nr = 1. An operator S†p which addsa flux quantum at r0, creating a quasi-particle, can be written just as

S†p ∼Ne∏i=1

(∂

∂xi− i

∂

∂yi− x0 − i y0

r2c

)(1673)

where this operator only acts on the polynomial part of the wave function andnot the exponential part. It reduces the angular momentum of each single-particle state by one unit of h and sends the particle at r0 into the higherLandau levels. This activation process ensures that the quasi-particle excita-tion spectra has a gap. The threshold energies quasi-hole and quasi-particleexcitations are determined by the Coulomb interaction, and for p = 3 have beenevaluated178 as 0.073 e2

rcand 0.026 e2

rc.

Since each quasi-particle of charge qp is attached to one flux quantum, the

statistics are neither fermionic nor bosonic179. Two quasi-particles can be ex-changed by a process whereby one quasi-particle is rotated by π in a semi-circle centered on the other fixed quasi-particle and then the two particles aretranslated in the same direction along the diameter. This process results inan interchange of the electrons between their initial position eigenstates. Thephase of the wave function changes in this permutation. The rotation of thequasi-particle through π around a flux tube produces an Aharonov-Bohm phaseof

πq

p

Φ0

2 π c h=

π

p(1674)

since the quasi-particle has a fractional charge. Thus, the quasi-particles havefractional statistics.

These types of fractional or anyon statistics are only possible in two or lessdimensions. If the permutation of two particles produces a phase difference of π

p ,then the time reversed permutation process must yield a phase change of − π

p .In two-dimensions, the permutation process and the time reversed process aredistinguishable. However, if the two-dimensional process is embedded in threedimensions, the processes are no longer distinct. On flipping over the plane inwhich the particles are contained in, the interchange becomes equivalent to thetime reversed interchange. Hence,

exp[

+ iπ

p

]= exp

[− i

π

p

](1675)

178F. D. M. Haldane and E.H. Rezayi, Phys. Rev. Lett. 54, 237 (1985).179B.I. Halperin, Phys. Rev. Lett. 52, 1583 (1984).

511

which restricts the value of p so that p = 1. Thus, in three dimensions, thequasi-particles have to obey Fermi-Dirac statistics.

11.5.6 Skyrmions

Although we have considered the effect of extremely high magnetic fields, theelectronic spin system is not completely polarized, as we have been assuming.The reasons for the relatively weak coupling between the electronic spin and themagnetic field, compared with the coupling of the orbital motion to the field ismainly due to the small band mass of the electron. The strength of the orbitalcoupling to the field is determined by the quantity

q

2 m∗ c(1676)

where m∗ is the band mass, which in GaAs has the value of m∗ ≈ 0.067 me

where me is the free electron mass. The strength of the spin coupling to thefield is governed by the electron mass and is given by

g q

2 me c(1677)

where g is the gyro-magnetic ratio, which for free electrons would be g ≈ 2.03.The strength of the anomalous Zeeman interaction inGaAs is further reduced bythe strong spin-orbit coupling by reducing g to the value given by the Lande gL

factor, gL ≈ 0.44. Since the importance of the anomalous Zeeman interactionhas been reduced by a factor of about 60, the spin directions are, therefore,determined via the exchange parts of the Coulomb interaction. The magnitudeof the Coulomb interaction is given by the screened interaction

q2

ε rc(1678)

where ε ∼ 12 and rc is the magnetic length. For fields of order B = 10 Tesla,the Coulomb interaction is the same order of magnitude as h ωc. Therefore,the optimal polarization of the ground state is determined by the ratio of theorbital part of the Zeeman interaction and the Coulomb exchange and partiallyspin-polarized states may occur.

The spin magnetization M contributes to the effective magnetic field accord-ing to

Beff = B + 4 π M (1679)

Hence, if we consider excitations in the spin system, the magnetization will bespatially varying, and so will the effective field. A variation in the effective fieldwill result in a local change of the filling factor. The system will respond tothe change of the filling factor by transferring charge. Thus, spin and chargeexcitations are coupled. The lowest energy coupled spin-charge excitations are

512

skyrmions, not the Laughlin quasi-particles.

Consider an electron with spin S moving in the exchange field of the otherfixed electrons. The spin degree of freedom is governed by the effective ZeemanHamiltonian

HZ = − g µB

hBeff (r) . S (1680)

In the lowest energy state, the electron aligns its spin with the static effectivemagnetic field. If the electron is moved around a closed contour, the spin willremain aligned with the local magnetic field all along the contour. However,the spin wave function does not return to its initial value but instead acquires aphase, the Berry phase. The Berry phase is related to the solid angle enclosedby the spin’s trajectory, as mapped onto the unit sphere in spin space. Thesolid angle Ω traced out by the spin when completing the contour is given by

Ω =∮

dϕ ( 1 − cos θ ) (1681)

where the spin direction is specified by the polar coordinates (θ, ϕ). After thecontour is traversed the spin wave function acquires an extra phase is S Ω

h .

The Berry phase can be illustrated by considering a spin one-half in a mag-netic field of constant magnitude oriented along the direction (θ, ϕ). In thiscase, the Zeeman Hamiltonian is given by

HZ = − µB ( B . σ ) (1682)

which can be expressed as

HZ = − µB B

(cos θ sin θ exp[− i ϕ ]

sin θ exp[ + i ϕ ] − cos θ

)(1683)

For fixed (θ, ϕ), the Zeeman Hamiltonian HZ has an eigenstate χ+ given by

χ+ =(

cos θ2

sin θ2 exp[ + i ϕ ]

)(1684)

which corresponds to the eigenvalue

E0 = − µB B (1685)

Thus, in this state the spin is aligned parallel to the applied field. For a staticfield, one has the time-dependent wave function given by

χ+(t) =(

cos θ2

sin θ2 exp[ + i ϕ ]

)exp

[+ i

µB B

ht

](1686)

where the time dependence is entirely contained in the exponential phase factor.

513

If the direction of the field (θ(t), ϕ(t)) is changed very slowly, one expects thespin will adiabatically follow the field direction. That is, if the field is rotatedsufficiently slowly, one does not expect the spin to make a transition to thestate with energy E = + µB B where the spin is aligned anti-parallel to thefield. However, the wave function may acquire a phase which is different fromthe time and energy dependent phase factor expected for a static field. Thisextra phase is the Berry phase δ, and can be calculated from the Schrodingerequation

i h∂

∂t

(α(t)β(t)

)= − µB B

(cos θ(t) sin θ(t) exp[− i ϕ(t) ]

sin θ(t) exp[ + i ϕ(t) ] − cos θ(t)

) (α(t)β(t)

)(1687)

We shall assume that the wave function has the adiabatic form(α(t)β(t)

)=

(cos θ(t)

2

sin θ(t)2 exp[ + i ϕ(t) ]

)exp

[+ i

(µB B

ht − δ(t)

) ](1688)

which instantaneously follows the direction of the field but is also modified bythe inclusion of the Berry phase δ(t). On substituting this ansatz into theSchrodinger equation, one finds that the non-adiabatic terms satisfy

− ∂δ

∂t

(cos θ(t)

2

sin θ(t)2 exp[ + i ϕ(t) ]

)+

∂ϕ

∂t

(0

sin θ(t)2 exp[ + i ϕ(t) ]

)

=i

2∂θ

∂t

(− sin θ(t)

2

cos θ(t)2 exp[ + i ϕ(t) ]

)(1689)

The above equation is projected onto the adiabatic state by multiplying it bythe row matrix (

cos θ(t)2 sin θ(t)

2 exp[ − i ϕ(t) ])

(1690)

On performing the projection, one finds that the derivative of θ w.r.t. t cancelsand that the equation simplifies to

− ∂δ

∂t+

∂ϕ

∂tsin2 θ

2= 0 (1691)

Hence, the Berry phase is given by integrating w.r.t. to t,

δ(t) =∫ t

0

dt′∂ϕ

∂t′sin2 θ(t

′)2

=∫

dϕ sin2 θ

2

=12

∫dϕ ( 1 − cos θ ) (1692)

514

On completing one orbit in spin space, the extra phase is given by

δ =Ω2

(1693)

as was claimed.

Thus, an inhomogeneous effective field on the spin introduces an extra phaseof Ω

2 in the wave function of the electron which is dragged around a contour.This extra phase has the same effect as if the contour contains an additionalcontribution to the magnetic flux given by

∆Φ =Ω2

Φ0

2 π(1694)

since encircling a flux quanta Φ0 produces a phase change of 2 π. Furthermore,as the effective flux enclosed in the region is increased by ∆Φ, and the fillingfraction ν is constant, the number of electrons in the region must change by anamount ∆Ne. Since the number of electrons is related to the flux by the fillingfactor

ν Φ = Ne Φ0 (1695)

the change in the number of electrons is given by

ν ∆Φ = ∆Ne Φ0 (1696)

Hence, the contour encloses an extra charge

∆Q = q ∆Ne

= ν q∆ΦΦ0

= ν qΩ

4 π(1697)

Thus, the extra charge is determined by the Berry phase and the filling fractionν, and also the spin and charge excitations are coupled.

Due to the coupling of spin and charge, a localized spin-flip excitation of afully polarized ground state introduces a non-uniform charge density. Considera skyrmion excitation in the fully filled lowest Landau level180. The groundstate wave function is written in second quantized form as

| Ψ0 > =∏m

a†m,↑ | 0 > (1698)

The creation of a charged spin-flip excitation at the origin requires adding adown spin electron in the statem = 0. However, to allow for the spin excitationto have a finite spatial extent and the charge density to re-adjust, the wave180D. H. Lee and C. L. Kane, Phys. Rev. Lett. 64, 1313 (1990).

515

function needs to be able to reduce the charge density and net spin at the originby redistributing them on neighboring shells. The excited state must also be aneigenstate of total angular momentum Jz. Thus, to a first approximation theexcited state wave function can be written as

| Ψ+ > ≈(v0 + u0 a

†1,↓ a0,↑

)a†0,↓

Ne∏m=0

a†m,↑ | 0 >

≈(v0 a

†0↑ + u0 a

†1,↓

)a†0,↓

Ne∏m6=0

a†m,↑ | 0 > (1699)

where v0 and u0 are variational parameters which, since the wave function isnormalized, must satisfy

| u0 |2 + | v0 |2 = 1 (1700)

Iterating this process leads to the skyrmion wave function181

| Ψ+ > =Ne∏

m=0

(vm a†m↑ + um a†m+1,↓

)a†0,↓ | 0 > (1701)

One expects that as m → Ne one recovers the fully polarized state, so| vm | → 1 and | um | → 0. This is a variational wave function for theexcited state, and the parameters vm and um are to be determined by minimiz-ing the expectation value of H.

The Hamiltonian can be approximated by

H =∑m,σ

εm,σ a†m,σ am,σ +

12!

∑m,m′

Vm,m′ a†m,↑ a†m′+1,↓ am′,↑ am+1,↓ (1702)

which is a simplified version of the skyrmion Hamiltonian. On using the relations

< Ψ+ | a†m,↑ am,↑ | Ψ+ > = | vm |2

< Ψ+ | a†m,↑ am+1,↓ | Ψ+ > = v∗m um

< Ψ+ | a†m+1,↓ am,↑ | Ψ+ > = u∗m vm

< Ψ+ | a†m+1,↓ am+1,↓ | Ψ+ > = | um |2 (1703)

the expectation value of the Hamiltonian is found as

< Ψ+ | H | Ψ+ > =∑m

(εm,↑ | vm |2 + εm+1,↓ | um |2

)+

12!

∑m,m′

Vm,m′ v∗m um u∗m′ vm′ (1704)

181H. A. Fertig, L. Brey, R. Cote and A. H. MacDonald, Phys. Rev. B, 50, 11018 (1994).

516

The energy of this excited state is to be minimized w.r.t. um and vm subject tothe constraint

| vm |2 + | um |2 = 1 (1705)

The minimization is performed using Lagrange’s method of undetermined mul-tipliers, λm. The minimization results in the set of equations

( εm,↑ − λm ) v∗m +12u∗m

∑m′

Vm,m′ um′ v∗m′ = 0 (1706)

and

( εm+1,↓ − λm ) u∗m +12v∗m

∑m′

Vm,m′ vm′ u∗m′ = 0 (1707)

These sets of equations can be solved to yield the undetermined multipliers

λm =(εm↑ + εm+1,↓

2

)±

√ (εm↑ − εm+1,↓

2

)2

+ | ∆m |2 (1708)

where we have defined the parameter

∆m =12!

∑m′

Vm,m′ v∗m′ um′ (1709)

which we expect will decrease with increasing m. The factors | vm |2 and | um |2are then found as

| vm |2 =12

(1 ∓ εm,↑ − εm+1,↓√

( εm↑ − εm+1,↓)2 + 4 | ∆m |2

)| um |2 =

12

(1 ± εm,↑ − εm+1,↓√

( εm↑ − εm+1,↓)2 + 4 | ∆m |2

)(1710)

Far from the center of the spin-flip excitation one expects that the ground statewill be recovered, so the spins will be polarized parallel to the field. Hence, weshall use the upper signs. The equations (1709) and (1710) have to be solvedself-consistently for ∆m. We shall use a real solution for the gap. These arecombined to yield the “gap” equation

∆m =12

∑m′

Vm,m′∆m′√

( εm′↑ − εm′+1,↓)2 + 4 | ∆m′ |2(1711)

The solution uniquely determines the wave function up to an undeterminedphase. From the gap equation, we note that as the Zeeman splitting increases,the magnitude of ∆m decreases. The energy of the skyrmion is reduced by thenon-zero value for ∆m, since spreading out the charge reduces the Coulombinteraction. However, the cost in Zeeman energy for flipping the spins in a large

517

Page 1 of 1

14.10.2008http://phys.thu.edu.tw/~mfyang/qhe/skyrmion.gif

Figure 226: The spin distribution in a skyrmion excitation. The spin distri-bution is radially symmetric, with an overturned spin at the center. The fullypolarized state is recovered at r →∞.

region of space limits the size of the skyrmions.

Once the wave function has been determined, one can examine the spindistribution. The direction of the spin at the point (r, ϕ) is defined as thedirection along which the spin density operator is maximum. The spin densityoperator, projected along a unit vector η in an arbitrary direction (θ′, ϕ′) insecond quantized form is given by

( η . σ(r) ) = Ψ†(r) ( η . σ ) Ψ(r) (1712)

However, the field creation and annihilation operators are given by

Ψ†(r) =∑m,α

a†m,α φ∗m(r) χ†α

Ψ(r) =∑m′,β

am′,β φm′(r) χβ (1713)

Hence, one finds the component of the spin density operator in the form

( η . σ(r) ) =∑

m,m′

∑α,β

φ∗m(r) ( η . σ )α,β φm′(r) a†m,α am′,β

=∑

m,m′

φ∗m(r) φm′(r) sin θ′(a†m,↑ am′,↓ exp[ − i ϕ′ ] + a†m,↓ am′,↑ exp[ + i ϕ′ ]

)

+∑

m,m′

φ∗m(r) φm′(r) cos θ′(a†m,↑ am′,↑ − a†m,↓ am′,↓

)(1714)

On taking the expectation value of the spin density operator in the skyrmionstate, one finds

< Ψ+ | ( η . σ(r) ) | Ψ+ > =∑m

sin θ′ exp[ − i ϕ′ ] φ∗m(r) φm+1(r) um v∗m

518

+∑m

sin θ′ exp[ + i ϕ′ ] φ∗m+1(r) φm(r) vm u∗m

+∑m

cos θ′(| φm(r) |2 | vm |2 − | φm+1(r) |2 | um |2

)(1715)

The wave functions can be expressed in planar polar coordinates as

φm(r, ϕ) =1√2 π

exp[

+ i m ϕ

]Rm(r) (1716)

where Rm(r) is real. Hence, we have

< Ψ+ | ( η . σ(r) ) | Ψ+ > =∑m

sin θ′ exp[ − i ( ϕ′ − ϕ ) ] Rm(r) Rm+1(r) um v∗m

+∑m

sin θ′ exp[ + i ( ϕ′ − ϕ ) ] Rm+1(r) Rm(r) vm u∗m

+∑m

cos θ′(R2

m(r) | vm |2 − R2m+1(r) | um |2

)(1717)

On maximizing w.r.t ϕ′, one finds

exp[− 2 i ( ϕ′ − ϕ ) ]∑m

Rm(r)Rm+1(r) um v∗m =∑m

Rm+1(r)Rm(r) vm u∗m

(1718)Hence, ϕ′ = ϕ, that is, the in-plane component of the spin is directed radiallyoutwards. On substituting this relation into the wave function, one finds thatthe spin density along this direction simplifies to

< Ψ+ | ( η . σ(r) ) | Ψ+ > = sin θ′∑m

Rm+1(r) Rm(r) ( vm u∗m + v∗m um )

+ cos θ′∑m

(| vm |2 R2

m(r) − | um |2 R2m+1(r)

)(1719)

The out of plane component of the spin is determined by θ′. This is found frommaximizing w.r.t. θ′, and leads to

tan θ′ =∑

m( vm u∗m + v∗m um ) Rm+1(r) Rm(r)∑m′

(| vm′ |2 R2

m′(r) − | um′ |2 R2m′+1(r)

) (1720)

At large distances r from the origin, the wave functions are dominated by arange of m values around the value given by

r2 = 2 m r2c (1721)

519

In this case, one has Rm+1(r) ∼ r2√

m rcRm(r), hence, the out of plane angle

at the distance r from the origin is governed by

tanθ′

2∼ r

2√m rc

∣∣∣∣ um

vm

∣∣∣∣ (1722)

Since, the ratio decreases with increasing m, the spin direction varies fromθ′ = π at the origin to θ′ = 0 as r → ∞. The texture can be expressedempirically as

tanθ′

2=

λ

r(1723)

where λ expresses the size of the skyrmion. In fact the size of the skyrmion isdetermined by the m variation of um, or more explicitly on the ratio(

εm+1,↓ − εm,↑

∆m

)(1724)

The size of the skyrmion decreases as the magnitude of the Zeeman splittingincreases. This reflects the fact that the energy required to flip the spins in aregion of large spatial extent becomes prohibitively costly as the Zeeman inter-action is increased.

Skyrmions can also be created in the Laughlin state. The skyrmions havelower energy than the Laughlin quasi-particles for all values of the gyromagneticratio g. The energy difference is largest for g → 0. However, as g → ∞the region over which the spin is varying is reduced, and the energy approachesthat of the Laughlin quasi-particle. In fact, in this limit, the skyrmion becomesidentical to the Laughlin quasi-particle.

Skyrmion excitations have been found in N.M.R. experiments182. The Knightshift is proportional to the spin polarization of the electron gas. The electron gascan be fully spin polarized if ν < 1 but can only be partially spin polarized forν > 1. In this latter case, the partial polarization is limited by ( 2 − ν )/ν, fornon-interacting electrons. The spin polarization is reduced by having a numberof skyrmions present. Skyrmions can be introduced into the system by chang-ing the filling factor from an integer or fractional value. The dependence of theKnight shift on the filling factor in a range close to ν = 1 follows the theo-retical results and indicates that each skyrmion contains about 3.6 flipped spins.

11.5.7 Composite Fermions

The Laughlin wave function describes states with filling fractions 1p , where p

is odd. However, the sequence of fillings at which the fractional quantum Hall182S. E. Barrett, G. Dabbagh, L. N. Pfieffer, K. N. West, and R. Tycko, Phys. Rev. Lett,

74, 5112 (1995).

520

Figure 227: A cartoon of Composite Fermions in which each electron is boundto an even number of flux quanta.

effect is observed is given by the expressions

ν =n

2 n r ± 1(1725)

andν = 1 − n

2 n r ± 1(1726)

where n and r are integers. These two filling fractions are related by approxi-mate electron-hole symmetry, in which occupations of only the lowest Landaulevel are considered.

These other states can be expressed in terms of composite fermions183. Thegeneral Laughlin state can be written as

Ψ2r+1 ∼∏i>j

(( xi − xj ) + i ( yi − yj )

)2r+1 Ne∏k=1

exp[− x2

k + y2k

4 r2c

]

=∏i>j

(( xi − xj ) + i ( yi − yj )

)2r

×

×∏i>j

(( xi − xj ) + i ( yi − yj )

) Ne∏k=1

exp[− x2

k + y2k

4 r2c

](1727)

This can be thought of as taking the state in which the lowest Landau level hasthe filling ν = 1 and attaching an even number 2 r flux quanta to each particle.Then, the statistics of the composite particle (composed of the 2 r flux quantaand the electron) is manifested by the exchange phase. The exchange phase ofthe composite particle will be the sum of the exchange phase of an electron (π)plus a multiple of π for each flux quantum. Hence, the total exchange phase ofthe composite particle wave function will be

π + 2 r π = ( 2 r + 1 ) π (1728)

183J. K. Jain, Phys. Rev. Lett. 63, 199 (1989).

521

and, thus, the composite particle is a fermion.

Consider the state with the general filling factor

ν =n

2 r n ± 1(1729)

If the electrons are attached to 2 r flux tubes, one has composite fermions. Thishas the effect of reducing the magnetic field from Φ to a value Φ∗ given by

Φ∗ = Φ − 2 r Ne Φ0 (1730)

This effective free flux can be either positive or negative. The fractional quantumHall effect of electrons with filling factor ν can be related to the quantum Halleffect of composite fermions with filling factor ν∗. The relation between ν andν∗ is found by first inverting the definitions

ν =Ne Φ0

Φ

ν∗ =Ne Φ0

Φ∗(1731)

in which Φ∗ and, therefore, ν∗ are assumed to be positive. Then, substitutingΦ∗ and Φ into the relation given by eqn(1730) yields

1ν∗

=1ν− 2 r (1732)

orν =

ν∗

2 ν∗ r + 1(1733)

Hence, the fractional quantum Hall effect at general fillings can be related tothe integer quantum Hall effect, with integer fillings ν∗, for composite fermionswhere 2 r flux tubes attached to each electron. Using r = 1, this processyields the hierarchy of ν = 1

3 , 25 , 3

7 , 49 , . . . , which is the most easily

observable sequence of fractional quantum Hall plateaus. The expression forthe filling fractions with the minus sign in the denominator can be obtained byconsidering negative values of Φ∗, for which a minus sign has to be inserted intothe definition of ν∗ in order to keep the filling fraction positive.

522

12 Insulators and Semiconductors

The existence of band gaps is a natural consequence of Bloch’s theorem for pe-riodic crystals. However, the existence of band gaps is a much more universalphenomenon, for example it also appear in amorphous materials. In these cases,the existence of band gaps can be traced back to the energy gaps separating thediscrete bound state energies of isolated atoms. When the atoms are broughttogether to form a solid, each electron will be shared with all the atoms in acrystal like in a giant molecule of N atoms. The set of discrete energy lev-els from each of the N atoms, are nearly degenerate. The binding of the Natomic degenerate atomic states into a N molecular states will involve bonding/ anti-bonding splittings that raises the degeneracy. As the energy spread ofthe bonding anti-bonding states are fixed, the levels form a dense set of discretelevels, which can be approximated by a continuous energy band. The separationbetween consecutive bands is roughly determined by the energy separation ofthe discrete levels the isolated atom. Generally, the low-energy bands have asmall energy spread, and a clear correspondence with the atomic levels can beestablished. However, the higher energy bands tend to have larger band widths,so the bands overlap and the correspondence with the atomic orbitals becomesmore obtuse.

An example is given by the ionic compound LiF , in which the Li ion losesan electron, and the F ion gains an electron in order that each ion only havecompletely filled atomic shells. The Li 1s and F 1s levels are completely filledand are well separated forming the core levels. The F 2s and 2p levels are alsooccupied, but have broader band widths and form the occupied valence bands.The unoccupied Li 2s and the unoccupied F 3s and 3p levels have large bandwidths which strongly overlap yielding a conduction band which has mixed char-acter. The density of states from the completely filled valence band states areseparated by an energy gap from the completely empty conduction band portionof the density of states. By definition, the Fermi energy or chemical potentiallies somewhere in the energy gap. The existence of a gap in the density ofstates at the Fermi energy is the characteristic feature that defines an insulatoror semiconductor.

Elementary semiconductors or insulators, such as diamond C, Si and Ge,all belong to group IV of the periodic table. Group IV elements are semicon-ducting since the outer shells of the individual atoms are half filled. By formingtetrahedral bonds with neighboring atoms, the electronic states form bondingand anti-bonding levels. The electrons completely fill the bonding levels whichare separated from the empty anti-bonding levels by a gap. These are examplesof the 8N rule, which states stable compounds are formed by sharing electronswith atoms so that the outer shells are complete. Compound semiconductorscan be formed by combining group III and group V, elements or group II withgroup VI elements. The most commonly used compound semiconductor is theIII-IV compound GaAs, whereas CdSe and CdTe are group II-IV compounds.

523

Group I and Group VII elements are usually ionic insulators, which have largegaps.

IV Element Lattice Constant A Energy Gap (eV) Structure

C 3.57 5.48 CubicSi 5.43 1.12 CubicGe 5.66 0.664 Cubic

III-V Compound Lattice Constant Energy Gap (eV) Structure

BN 3.62 6.4 CubicBP 4.54 2.4 CubicAlN 3.11 , 4.98 6.2 HexagonalAlP 5.46 2.5 CubicAlAs 5.66 2.15 CubicGaN 3.18 , 5.17 3.44 HexagonalGaP 5.45 2.27 CubicGaAs 5.65 1.42 CubicInN 3.54 1.89 CubicInP 5.87 1.34 CubicInAs 6.06 0.354 Cubic

524

II-IV Compound Lattice Constant Energy Gap (eV) Structure

ZnS 5.41 3.68 CubicZnO 3.25 , 5.21 3.44 HexagonalZnSe 5.67 2.7 CubicZnTe 6.10 2.28 CubicCdS 5.82 2.55 CubicCdSe 6.05 1.9 CubicCdTe 6.48 1.475 Cubic

α−HgS 4.15 , 9.5 2.1 Trigonal

The distinction between a semiconductor and insulator is only by the mag-nitude of the energy gap between the lowest unoccupied state and the highestoccupied state. In insulators this energy gap is large, perhaps larger than 3eV. In semiconductors this energy gap is small, so the electronic properties aredetermined by the electronic states close to the bottom of the conduction bandand the states close to the top of the valence band. The value of the gap de-creases along the column of the periodic table. This can be considered to be aconsequence of the decrease of the pseudo-potential due to the screening by thecore electrons. Alternatively, from the view-point of the tight-binding model,the decrease in the band gap can be associated with the increase of the latticeconstants. The value of the gap also increases with the degree of ionicity.

Semiconductors can also be formed from transition metals, such as FeSe, orlanthanide materials such as SmB6, Y bB12 or Ce3Bi4Pt3, and actinide materi-als such as UNiSn. In these materials the energy gaps can be as small as a fewmeV, and are strongly temperature dependent. It is believed that the smallnessof the energy gaps in these materials is intimately connected to the strength ofelectron-electron interactions.

The density of states close to the band gap can usually be parameterized by afew quantities, such as the value of the band gap, and the effective massesme andmh for the valence and conduction bands. This is true, since the discontinuitiesat the band edges are van Hove singularities. Due to the symmetry one canrepresent the single-particle Bloch energies of the valence band and conductionband states as

Ec(p) = Eg +d∑

i=1

ai p2i

525

Ev(p) = −d∑

i=1

bi p2i (1734)

where the zero of energy was chosen to be at the top of the valence band. Usingthe definition of the effective mass as

1mα,β

=∂2E(p)∂pα∂pβ

(1735)

one finds an effective mass tensor mα,β . The effective mass tensor is symmetricand can be diagonalized. The diagonal elements are positive for the conductionband and negative for the valence band. Typical values of the effective mass areless than unity ∼ 0.04 me, but increase with increasing energy gaps.

Material Eg (eV) m∗e m∗

h

Diamond 5.48 0.36 - 1.08 , - 0.36 , - 0.154Silicon 1.12 0.191 - 0.537 , - 0.153 , - 0.234

Germanium 0.664 0.081 - 0.284 , - 0.044 , - 0.095

There are two types of semiconductors that are frequently encountered. In-trinsic semiconductors and extrinsic semiconductors.

A Intrinsic Semiconductors.

These are pure semiconductors, where the density of states consists of acompletely filled valence band and a completely empty conduction band at T =0, which are separated by a band gap Eg. At temperatures comparable to theband gap

Eg ∼ kB T (1736)

a finite number of electrons can be excited from valence band states to theconduction band. The thermal excited electrons in the conduction band areassociated with empty states in the valence band. For each conduction electronthere is one empty valence band state. A hole is defined as the absence of anelectron in a valence band state. Thus, at finite temperatures, one has a finitenumber of electron - hole pairs. For materials such as Si or Ge the gap in thedensity of states is of the order of 1.12 to 0.66 eV. Thus, the number of thermallyactivated electron hole pairs is expected to be extremely small under ambientconditions.

526

B Extrinsic Semiconductors.

Extrinsic semiconductors are a type of semiconductor that contain impuri-ties. Semiconductors with impurities can have discrete atomic levels that haveenergies which are lower than the empty conduction band and higher than thefull valence band. That is the impurity level lie within the gap.

There are two types of extrinsic or impurity semiconductors.

N type semiconductors have impurities with levels which at T = 0would be filled with electrons. At higher temperatures the electrons can beexcited from the levels into the conduction band. These types of impurities arecalled donors, since at high temperatures they donate electrons to the conduc-tion band. An example of an N type semiconductor is given by semiconductingSi in which As impurities are substituted for some Si ions or alternatively semi-conducting Ge substitutionally doped with P impurities. These are examplesof elemental semiconductors from the IV column of the periodic table dopedwith impurities taken from the V column. Since the impurity ion has one moreelectron than the host material, the host bands are completely filled by takingfour electrons from each impurity, but the impurity ion can still release oneextra electron into the conduction band.

For Si doped with a low concentration of As impurities, each As impuritycan be considered individually. The As atom contains 5 electrons, while theperfect valence band only contains 4 electrons per site. Thus, the As ion hasone extra electron which, according to the Pauli exclusion principle has to beplaced in states above the valence band. In the absence of the impurity potentialof the ionized As atom, this extra electron would go into the conduction bandand would behave very similarly to a free electron with mass me. However,one must consider the effect of the potential produced by the As+ ion and thespatial correlation that this imposes on the extra electron.

The As+ ions has a positive charge, which affects the free conduction elec-tron much the same way as the positive nuclear charge effects the electron in aH atom. The extra electron becomes bound to the donor atom. In the semi-conductor, the binding energy is extremely low and the radius of the orbit islarge. This can be seen by examining the potential produced by the As+ ion

V (r) = − e2

ε r(1737)

which is screened by the dielectric constant ε. The dielectric constant for Sihas a value of about 70. The mass of the electron is the effective mass me. TheBohr radius of the donor orbital is given by

ad(n) =ε h2

me e2n2 (1738)

527

where n is the principal quantum number. The energy of the donor levels belowthe bottom of the conduction band are given by the expression

Ed(n) = − me e4

2 ε2 h2

1n2

(1739)

The lowest state of the donor level is occupied at T = 0, where the electronis located in an orbit of radius ad(1) ∼ 30 A and the energy of the donorlevel Ed(1) ' − 0.02 eV. Thus, there are shallow impurity levels just belowthe bottom of the conduction band, one of these set are occupied at T = 0K. These discrete levels can be represented by a set of delta functions in thedensity of states. For sufficiently large concentrations these impurities can berepresented in terms of a finite impurity band which appears inside the gap inthe density of states of the pure host material. Since the value of the gap iscomparable to room temperature, one expects that under ambient conditionsthere are a finite number of thermally activated conduction electrons availablefor carrying current.

P type semiconductors have impurities with levels that at T = 0 wouldbe empty of electrons. At higher temperatures, electrons from the filled valenceband will be excited into the empty impurity levels. These thermally excitedlevels will be localized on the impurities (for small concentrations of impurities).However, the holes present in he valence band allows the valence electrons toconduct electricity and contribute to the properties of the semiconductor. Theimpurities in P type semiconductors are called acceptors as they accept electronsat finite temperatures. Examples of P type semiconductors are Ga impuritiesdoped substitutionally on the sites of a Si host, or Al impurities substitutedfor the atoms in a Ge crystal. These are examples of impurities from the IIIcolumn of the periodic table being substituted for the atoms in a semiconductorcomposed of an element from the IV column of the periodic table. In this case,the type III impurity provides only 3 electrons to the host conduction band,which at finite T contains one hole per Ga impurity.

The Ga impurity atom shares the electrons of the surrounding Si atom andbecomes negatively charged. The extra hole orbits around the negative ion pro-ducing acceptor levels that lie just above the top of the valence band. A finiteconcentration of acceptor levels is expected to produce a smeared out acceptorband just above the top of the valence band. Due to the smallness of the gapbetween the valence band and the acceptor levels, at room temperature an ap-preciable number of electrons can be excited from the valence band onto theacceptor levels.

12.1 Thermodynamics

The thermodynamic and transport properties of semiconductors are governedby the excitations in the filled valence band or empty conduction band, as the

528

fully filled or empty bands are essentially inert. The excitations of electronson the localized impurity levels of extrinsic semiconductors, do not directlycontribute to physical properties. However, the electrons in the conductionband and the valence band are itinerant and do contribute. To develop a theoryof the properties of semiconductors it is convenient to focus attention on thefew unoccupied states of the valence band rather than the many filled states.This is achieved by reformulating the properties in terms of holes.

12.1.1 Holes

A hole is an unoccupied state in an otherwise completely occupied valence band.The probability of finding a hole in a state of energy ε, Ph(ε) is given by theprobability that an electron is not occupying that state

Ph(ε) = 1 − f(ε) (1740)

Thus, as

Ph(ε) = 1 − f(ε) =1

1 + exp[ −β (ε− µ) ](1741)

one finds that the probability of finding a hole in a state of energy ε is also givenby the Fermi function except that ε → − ε and µ → − µ. That is the energyof the hole is the energy of a missing electron.

The momentum of a hole can be found as the momentum of a completelyoccupied band is zero, since inversion symmetry dictates that for each state withmomentum ke there is a state with momentum −ke with the same energy, andby assumption both are occupied. Then by definition the momentum of a holein ke is that of the full band with one electron missing

kh = 0 − ke (1742)

Thus, the momentum of a hole is opposite to that of the missing electron

kh = − ke (1743)

The charge on the electron is − | e |. The charge on the hole can be obtainedfrom the quasi-classical form of Newton’s laws applied to an electron in anelectric field E,

hdke

dt= − | e | E (1744)

As the momentum of a hole is given by

kh = − ke (1745)

one finds thathdkh

dt= + | e | E (1746)

529

Thus, the charge of the hole is just | e |. This is consistent with considerationsof electrical neutrality. In a solid in equilibrium, the charge of the nuclei is equalin magnitude to the charge of the electrons. Thus

| e |(Z Nn − Ne

)= 0 (1747)

Thus, in equilibrium the number of electrons is related to the number of nucleivia Ne = Z Nn. Now if one electron is removed from the valence band andremoved from the solid, there is just one hole. The total charge of the one holestate is given by

Qh = | e |(Z Nn − ( Ne − 1 )

)(1748)

where the number of electrons is now Ne − 1 = Z Nn − 1. Hence,

Qh = | e | (1749)

Thus, the charge on the hole is minus the charge of the electron.

The velocity of a hole vh can be found by considering the electrical currentcarried by the electrons. The electrical current carried by a full band is zero.The electrical current carried by a hole is the electrical current carried by a fullband minus one electron.

jh

= | e | vh = 0 − ( − | e | ve ) (1750)

Thus,vh = ve (1751)

Thus, the velocity of the hole is the same as the velocity of the missing electron.

The above results indicate that the velocity of the hole is in the same di-rection as the velocity of the missing electron, but the momenta have oppositedirections. This implies that the sign of the hole mass should be opposite of themass of the missing electron. This can also be seen by considering the alternateform of the quasi-classical equation of motion

medve

dt= − | e | E (1752)

and sinceve = vh (1753)

and the charge of the hole is | e |, one has

− medvh

dt= | e | E (1754)

530

Hence,mh = − me (1755)

the mass of the hole is the negative of the mass of the missing electron.

The most important number for the thermodynamic and transport proper-ties of a semiconductor is the number of charge carriers. This can be obtainedfrom analysis of the appropriate semiconductor.

12.1.2 Intrinsic Semiconductors

The number of electrons thermally excited into the conduction band of an intrin-sic semiconductor can be calculated with knowledge of the chemical potentialµ(T ). Consider an intrinsic semiconductor like pure Si which has an energygap Eg of the order of 1 eV. The energy of the top of the valence band shall beset to zero. The energy of the hole is zero at the top of the valence band andincreases downwards, as the energy is that of the missing electron. This agreeswith the fact that mh < 0, as

∂2Ek

∂2k=

1mh

< 0 (1756)

The energy of the conduction electron is given by

Ek = Eg +h2 k2

2 me(1757)

The chemical potential can be obtained from considerations of charge neutrality,which implies that the number of conduction electrons is equal to the numberof holes in the valence band. The number of electrons can be calculated fromthe electron density of states

ρ(ε) =V

2 π2

(2 me

h2

) 32

( ε − Eg )12 for ε > Eg (1758)

The number of electrons in the conduction band, Nc is given by

Nc =∫ ∞

Eg

dε ρ(ε) f(ε) (1759)

The Fermi function can be expressed in terms of ( ε − Eg ) as

f(ε) =1

exp[ β ( ε − Eg ) ] exp[ − β ( µ − Eg ) ] + 1(1760)

If it is assumed that µ = Eg

2 then, as the lowest conduction band state hasthe energy ε = Eg, the Fermi function for the conduction electrons is almostclassical since both exponents are positive. Thus, with the assumption

f(ε) ∼ exp[− β ( ε − µ )

](1761)

531

As shall be shown later the above assumption is valid. The number of thermallyactivated conduction electrons can now be obtained by evaluating the integralover the classical Boltzmann distribution by changing variables from ε to thedimensionless variable x = β ( ε − Eg ). Then

Nc ∼ V

2 π2

(2 me kB T

h2

) 32

exp[− β ( Eg − µ )

] ∫ ∞

0

dx x12 exp

[− x

]= 2 V

(2 π me kB T

h2

) 32

exp[− β ( Eg − µ )

](1762)

which is usually expressed in terms of the thermally De Broglie wavelength ofthe conduction electrons λe defined by

λe =h√

2 π me kB T(1763)

as

Ne = 2V

λ3e

exp[− β ( Eg − µ )

](1764)

The number of thermally excited conduction electrons depends exponentiallyon the unknown quantity µ.

The number of holes in the valence band can be found from a similar cal-culation. The occupation number for the holes in the valence band is givenby

Ph(ε) = 1 − f(ε)

=1

1 + exp[ −β (ε− µ) ]

which is the Fermi function except that ε → − ε and µ → − µ. The densityof states for the holes in the valence band is given by

ρ(ε) =V

2 π2

(− 2 mh

h2

) 32

( − ε )12 for ε < 0 (1765)

Since the value of µ is assumed to be positive and of the order of Eg

2 the Fermifunction can also be treated classically. Thus, the number of holes Nv in thevalence band is given by the integral

Nv =∫ 0

−∞dε Ph(ε) ρ(ε)

=V

2 π2

(− 2 mh kB T

h2

) 32∫ 0

−∞dε

( − ε )12

exp[ − β ( ε − µ ) ] + 1

∼ 2 V(

− 2 π mh kB T

h2

) 32

exp[− β µ

](1766)

532

which also depends on µ.

Since in an intrinsic semiconductor the electrons and holes are created inpairs, one has

Nc = Nv (1767)

Therefore, the equation for the unknown variable µ is given by

exp[

2 β µ]

=(− mh

me

) 32

exp[β Eg

](1768)

or on taking the logarithm, one has

µ =Eg

2+

34kB T ln

∣∣∣∣ − mh

me

∣∣∣∣ (1769)

Hence, at T = 0, the chemical potential lies half way in the gap and onlyacquires a temperature dependence if there is an asymmetry in the magnitudeof the conduction band density of states and the valence band density in thevicinity of the band edge. This result justifies the previous assumption that thedistribution functions can be treated classically. The result also shows that thenumber of thermally activated electrons or holes crucially depends on the gapvia the exponential factor

exp[− β

Eg

2

](1770)

and is extremely small at room temperature for a semiconductor with a gapthat has a magnitude of the order of electron volts.

12.1.3 Extrinsic Semiconductors

In an intrinsic semiconductor the gap in the host material is usually much largerthan the gap between the impurity levels and the band edges. Therefore, underambient temperatures one can neglect one of the bands, as the number of ther-mally excited electrons or holes in that band is small.

For concreteness, consider an N type semiconductor. The bottom of theconduction band is defined as the reference energy E = 0 and the energy ofthe donor levels is defined as − Ed. Let Nd be the number of donor atoms,and nd be the number of electrons remaining in the donor atoms, and nc be thenumber of electrons thermally excited to the conduction band. The importantphysical properties are determined by the number of conduction electrons. Thiscan be calculated from the free energy F .

The free energy F is given in terms of the energy and entropy via the relation

F = E − T S (1771)

533

where the total energy isE = − nd Ed (1772)

as the energy of the thermally activated conduction electrons E ∼ 0 as theyoccupy states at the bottom of the band. The entropy S is given by

S = kB lnΩ (1773)

where Ω is the number of possible states of the system. This is just the numberof ways of distributing nd electrons in the 2 Nd states of the donor atoms. Itis assumed that the donor atoms only have a two-fold spin degeneracy, andthat there is no interaction energy between two electrons occupying the twospin states on the same donor atom. With these assumptions, the number ofaccessible states is given by

Ω =( 2 Nd )!

nd! ( 2 Nd − nd )!(1774)

Hence, the Free energy can be calculated in the thermodynamic limit as

F = − Ed nd

− kB T

[2Nd ln 2Nd − nd lnnd − ( 2Nd − nd ) ln( 2Nd − nd )

](1775)

The chemical potential is found from minimizing the free energy with respectto nd

µ =∂F

∂nd

= − Ed − kB T

[− lnnd + ln( 2Nd − nd )

](1776)

Thus, on exponentiating

exp[β (µ + Ed )

]=

nd

2 Nd − nd(1777)

which leads to the number of electrons occupying the donor orbitals as

nd =2 Nd

exp[ β ( − Ed − µ ) ] + 1(1778)

which is just governed by the Fermi function f(−Ed) and the number of orbitals2 Nd.

The number of thermally excited conduction electrons is given by

nc = 2 V∫

d3k1

exp[ β ( E(k) − µ ) ] + 1

= 2 V(

2 π me kB T

h2

) 32

exp[β µ

](1779)

534

The unknown chemical potential µ can be eliminated from the above two equa-tions and thereby a relation between nc and nd can be found

nc ( 2 Nd − nd )nd

= exp[− β Ed

]2 V

(2 π me kB T

h2

) 32

(1780)

This is the law of mass action for the dissociation reaction

nd → nc + ( Nd − nd ) (1781)

in which filled donors dissociate into conduction electrons and unoccupied donoratoms. This relation can be written as

nc ( 2 Nd − nd )nd

= exp[− β Ed

]2V

λ3(1782)

where λ is the thermal De-Broglie wavelength,

λ =h√

2 π me kB T(1783)

On using the condition of electrical neutrality

Nd = nd + nc (1784)

one can eliminate nd to find the number of conduction electrons as

nc ( Nd + nc )( Nd − nc )

= exp[− β Ed

]2 Vλ3

(1785)

This is a quadratic equation for nc which can be solved to yield the positiveroot

nc = − 12

(Nd +

2 Vλ3

exp[− β Ed

] )

+12

√(Nd +

2 Vλ3

exp[− β Ed

] )2

+8 Nd V

λ3exp

[− β Ed

](1786)

With this expression for the number of conduction electrons one can solve for µfrom

µ = kB T ln[nc λ

3

2 V

](1787)

At sufficiently low temperatures when

λ3 Nd

2 V exp

[− β Ed

](1788)

535

one finds thatµ = − Ed (1789)

as at T = 0 the donor level is only partially occupied as there are 2 Nd orbitalsincluding spin and Nd electrons. At sufficiently high temperatures, the donorlevels are almost completely ionized and

nc ' Nd (1790)

and the chemical potential is found as

µ = kB T ln[Nd λ

3

2 V

](1791)

——————————————————————————————————

12.1.4 Exercise 73

Assume that if the donor levels are localized to such an extent that the Coulombrepulsion between two opposite spin electrons in the same donor atom is ex-tremely large. This assumption makes the occurrence of doubly occupied donoratoms extremely improbable. Show that the number of accessible states is givenby

Ω = 2NdNd!

nd! ( Nd − nd )!(1792)

Hence, show that

nd =Nd

1 + 12 exp

[− β ( Ed + µ )

] (1793)

Thus, the interaction affects the statistics of the occupation numbers. Also showthat the law of mass action becomes

nc ( Nd − nd )nd

=V

λ3exp

[− β Ed

](1794)

——————————————————————————————————

12.2 Transport Properties

Transport in doped semiconductors is mainly dominated by scattering fromdonor impurities. The potential due to the isolated donor impurities is ascreened Coulomb interaction

V (r) = − Z e2

ε rexp

[− kTF r

](1795)

536

The transport scattering rate is given by

1τ(E(k))

=c

h

∫dk′ k′2 δ( E(k)− E(k′) )

∫d cos θ ( 1− cos θ )

(4 π Z e2

ε ( (k − k′ )2 + k2TF )

)2

(1796)where c is the impurity concentration and θ is the scattering angle. The inte-gration over the magnitude of k′ can be performed over the delta function. Theintegration over the angle can be performed as

1τ(E(k))

=2 m c

h3

1k3

∫d cos θ ( 1 − cos θ )

(4 π Z e2

ε ( 1 − cos θ + ( kT F

k )2 )

)2

(1797)Hence, τ(k) ∼ k3. The conductivity can be evaluated from the formulae

σx,x = − 2 e2

V

∑k

h2

m2k2

x τ(k)∂f(E(k))∂E(k)

=2 e2

V

∑k

h2

m2k2

x τ(k) β exp[− β ( E(k) − µ )

]

∼ β exp[β µ

] ∫dk k7 exp

[− β h2

2 mek2

](1798)

On changing the variable of integration from k to the dimensionless variable

x =h2 k2

2 me kB T(1799)

one finds that the temperature dependence of the conductivity is governed by

σx,x ∼ ( kB T )3 exp[β µ

](1800)

The conductivity is proportional to the electron density and the scattering time.For Coulomb scattering, the average scattering time is proportional to T

32 . On

the other hand, scattering from lattice vibrations gives rise to a conductivitythat is just proportional to exp[ β µ ].

12.3 Optical Properties

537

13 Phonons

14 Harmonic Phonons

The Hamiltonian describing the motion of the ions can be formulated by as-suming that the ions move slowly compared with the electrons. Thus, at anyinstant of time, the electrons have relaxed into equilibrium positions and theions are frozen into their instantaneous positions184.

It is assumed that the ions have definite mean equilibrium positions andthat the displacements of the ions from these equilibrium positions are small.First, the situation in which the crystal lattice can be described by a Bravaislattice with a one atom basis is considered. In a later section, the effects of amulti-atom basis will be described. For the case of a one atom basis, the energyof the solid can be calculated in the which the ionic positions are displaced byamounts ui,

r(Ri) = Ri + ui (1801)

where ui is the deviation from the equilibrium position Ri. It is assumed that

r(Ri)

ui

Ri

Figure 228: The atoms are moved from their equilibrium positions Ri by thedisplacements ui.

the total energy can be formulated as a constant plus pair-wise interactionswhich depends on r(Ri)− r(Rj). The pair-wise interaction is given in terms ofthe pair-potentials Θ(r(Ri)− r(Rj)) via

V =12

∑i,j

Θ(r(Ri)− r(Rj))

=12

∑i,j

Θ(Ri −Rj + ui − uj) (1802)

184M. Born and R. Oppenheimer, Ann. Phys. (Leipzig), 84, 457 (1927).

538

The Hamiltonian governing the motion of the ions is just given by the sum ofthe kinetic energies of the ions of mass M and the pair-wise interactions

H =∑

i

P2

i

2 M+ V (1803)

The harmonic approximation assumes that the displacements ui are sufficientlysmall so that H can be expanded in powers of ui. The terms involving ui

describe the change in energies of the ions due to the lattice vibrations. Theexpansion is

H =∑

i

P2

i

2 M+

12

∑i,j

Θ(Ri −Rj)

+12

∑i,j

(ui − uj) . ∇ Θ(Ri −Rj)

+12

12!

∑i,j

[(ui − uj) . ∇

]2Θ(Ri −Rj) + ....

(1804)

However, since the Ri are equilibrium positions, and the total force on the atomlocated at Ri is given by ∑

j

∇ Θ(Ri −Rj) (1805)

the total force must vanish in equilibrium185. Hence, the potential to secondorder in u is just given by

V = Veq + Vharmonic (1806)

where the harmonic potential is given by

Vharmonic =14

∑i,j

∑µ,ν

(uµi − uµ

j ) Θνµ(Ri −Rj) (uν,i − uν,j) (1807)

In the above equation, the quantities Θνµ are defined in terms of the second

derivatives of the pair-potential

Θνµ(Ri −Rj) = ∇µ ∇ν Θ(Ri −Rj + ui − uj)

∣∣∣∣u≡0

(1808)

185Jahn and Teller have shown, by only considering symmetry, that a crystal structure isgenerally not stable if the electronic ground state is degenerate. The only exceptions occurwhen the degeneracy is a two-fold degeneracy associated with an odd-number of electrons, orwhen the system is linear. (H. A. Jahn and E. Teller, Proc. Roy. Soc. A 161, 220 (1937).)

539

The harmonic potential Vharmonic is usually expressed directly in terms of thedisplacements, and not their differences. The harmonic potential is written as

Vharmonic =12

∑i,j

∑µ,ν

uµi D

νµ(Ri −Rj) uν,j (1809)

where the dynamical matrix is given by

Dνµ(Ri −Rj) = δRi,Rj

∑R′′

Θνµ(Ri −R′′) − Θν

µ(Ri −Rj) (1810)

In general, when the interactions are not just pair-wise, the harmonic potentialcan still be defined. The dynamical matrix is defined as a second derivative ofthe total energy, with respect to the lattice displacements

Dνµ(Ri −Rj) =

∂2E

∂uµi ∂uν,j

(1811)

The dynamical matrix Dνµ(R−R′) is a symmetric matrix

Dνµ(R−R′) = Dµ

ν (R−R′) (1812)

due to the analyticity of the pair-potential. Also, for a crystal with a monoatomicbasis, one has

Dνµ(R−R′) = Dν

µ(R′ −R) (1813)

which follows since every Bravais lattice has inversion symmetry. Due to trans-lational invariance, one has the sum rule∑

R

Dνµ(R) = 0 (1814)

This follows from consideration of the absence of energy change due to a uniformdisplacement of the solid

r(Ri) = Ri + ∆ ∀ i (1815)

Under this displacement, the energy change of the solid is zero, and so

0 =12

∑i,j

∑µ,ν

∆µ Dνµ(Ri −Rj) ∆ν (1816)

for an arbitrarily chosen ∆µ.

Since ui,µ represents the displacement canonically conjugate to Pi,µ, themomentum and displacement operators satisfy the commutation relations

[ ui,µ , Pj,ν ] = i h δi,j δµ,ν (1817)

540

and[ ui,µ , uj,ν ] = [ Pi,µ , Pj,ν ] = 0 (1818)

To diagonalize the harmonic Hamiltonian, first a canonical transformationis performed to a representation in which the periodic translational invarianceof the lattice is explicit. The displacement is expressed as

ui =1√N

∑q

uq exp[i q . Ri

](1819)

and the momentum operator becomes

P i =1√N

∑q

pq

exp[i q . Ri

](1820)

where N = N1 N2 N3 is the number of unit cells in the crystal186 On assumingthat the lattice displacements satisfy Born-von Karman boundary conditions187,the wave vectors are quantized as

q =n1

N1b1 +

n2

N2b2 +

n3

N3b3 (1821)

where ni are integers such that 0 < ni < Ni. In the Fourier transformedbasis, the commutation relations become

[ uq,µ , pq′,ν ] = i h ∆q+q′ δµ,ν (1822)

and[ uq,µ , uq′,ν ] = [ pq,µ , pq′,ν ] = 0 (1823)

The quantity ∆q is the Kronecker delta function which conserves wave vectors,modulo reciprocal lattice vectors. The Kronecker delta function is defined via∑

Ri

exp[i q . R

]= N ∆q (1824)

186It should be noted that the operators uq and pqare no longer Hermitean due to the

appearance of the phase factor. However,

u†q = u−q

and

p†q

= p−q

.187M. Born and Th. von Karman, Zeit. fur Physik, 13, 297 (1912), M. Born and Th. von

Karman, Zeit. fur Physik, 14, 15 (1913).

541

This is non-zero if q is a reciprocal lattice vector Q, and is zero otherwise. Interms of the new coordinates, the Hamiltonian becomes

H =∑

q

[ p†q,µ δµ,ν pq,ν

2 M+

12u†q,µ D

µν (q) uq,ν

](1825)

where D(q) is the Fourier Transform of the dynamical matrix

D(q) =∑R

D(R) exp[− i q . R

](1826)

Thus, the harmonic Hamiltonian is diagonal in the quantum number q.

The symmetries of the dynamical matrix D(R) can be used to show that

D(q) =∑

i

D(Ri −Rj) exp[− i q . ( Ri − Rj )

]

=12

∑R

D(R)

(exp

[− i q . R

]+ exp

[+ i q . R

]− 2

)

= − 2∑R

D(R) sin2

(q . R

2

)(1827)

Using the Laue condition, one finds that D(q) is invariant under translationsthrough reciprocal lattice vectors q → q + Q. Also, since D(R) is a real andsymmetric, then D(q) is a real and symmetric matrix. Every real 3×3 symmetricmatrix has three real eigenvalues, thus, one may find three eigenfunctions

D(q) εα(q) = M ω2α(q) εα(q) (1828)

where εα(q) are the eigenvectors and the eigenvalues have been written asM ω2

α(q). Since the eigenvalues are real, the frequencies ωα(q) are either real orpurely imaginary. It is necessary that the eigenvalues are positive if the latticeis to be stable.

The eigenvectors are usually normalized, such that

εα(q) . εβ(q) = δα,β (1829)

Since the eigenvectors form a complete orthonormal set, one may expand thedisplacements and the momentum operators in terms of the eigenvectors as

uq =∑α

Qαq εα(q) (1830)

andp

q=∑α

Pαq εα(q) (1831)

542

This transformation diagonalizes the Hamiltonian in terms of the polarizationindex α. This can be seen as

u†q D(q) uq =∑α,β

Q† βq εβ(q) D(q) εα(q) Qα

q

=∑α,β

M ω2α(q) Q† β

q εβ(q) . εα(q) Qαq

=∑α

M ω2α(q) Q† α

q Qαq (1832)

and alsop†

q. p

q=∑α

P † αq Pα

q (1833)

Hence, the transformed Hamiltonian is diagonal in the polarization indices α

H =∑q,α

[P † α

q Pαq

2 M+

M ω2α(q)

2Q† α

q Qαq

](1834)

The Hamiltonian has the form of 3 N independent harmonic oscillators. Theeigenvalues of the Hamiltonian can be found by introducing boson annihilationand creation operators. The annihilation operator is defined by

aq,α =

√M ωα(q)

2 hQα

q + i

√1

2 h M ωα(q)Pα

q (1835)

and the creation operator is the Hermitean conjugate of the annihilation oper-ator

a†q,α =

√M ωα(q)

2 hQ† α

q − i

√1

2 h M ωα(q)P † α

q (1836)

The commutation relations for the boson operators can be calculated from thecommutation relations of Pα

q and Qαq , so

[ aq,α , a†q′,β ] = ∆q−q′ δα,β (1837)

These are the usual commutation relations for boson creation and annihilationoperators. In second quantized form, the Hamiltonian is expressed as

H =∑q,α

h ωα(q)2

(a†q,α aq,α + aq,α a†q,α

)(1838)

Due to the commutation relations, one can show that when the operator a†q,α

acts on an energy eigenstate it produces an energy eigenstate in which the energyeigenvalue is increased by an amount h ωα(q). Likewise, when the annihilation

543

operator acts on an energy eigenstate it produces an energy eigenstate with aneigenvalue which is lower by h ωα(q). If one assumes the existence of a groundstate of the oscillator | 0q,α > such that

aq,α | 0q,α > = 0 (1839)

then the energy eigenvalue of the ground state is just 12 h ωα(q). The excited

states can be found by the raising operator to be ( nq,α + 12 ) h ωα(q), where

nq,α is a positive integer. Thus, the Hamiltonian may be expressed as

H =∑q,α

h ωα(q)(nq,α +

12

)(1840)

This implies that each normal mode (q, α) has quantized excitations. Thesequantized lattice vibrations are known as phonons. Ab-initio pseudo-potentialcalculations of the pair-potential for simple metals lead to phonon frequenciesωα(q) which are in good agreement with the results of inelastic neutron scat-tering experiments188. This agreement suggests that the effect of multi-atominteractions on the phonon dispersion relations are relatively small.

The completeness relation for the polarization vectors is just∑α

εµα(q) εαν(q) = δµν (1841)

therefore, an arbitrary displacement can be expanded in terms of the polar-ization vectors. The original displacements and momenta can be expressed interms of the phonon creation and annihilation operators via

ui =1√N

∑q,α

√h

2 M ωα(q)εα(q)

(aq,α + a†−q,α

)exp

[i q . Ri

](1842)

and

P i =i√N

∑q,α

√h M ωα(q)

2εα(q)

(a†−q,α − aq,α

)exp

[i q . Ri

](1843)

In the Heisenberg representation, the displacements and momentum operatorsbecome time-dependent. The time dependence occurs through factors of

exp[± i ωα(q) t

](1844)

188L. J. Sham, Proc. Roy. Soc. A, 283, 33 (1965) and E. G. Brovman and Yu. M. Kagan,Sov. Phys. J.E.T.P. 52, 557 (1967).

544

0

0.005

0.01

0.015

0.02

0.025

0.03

h ωq

Γ ΓPH HN

Figure 229: A model calculation of the phonon dispersion relations for a hypo-thetical metal with an unstable b.c.c. crystal structure. A phonon frequencybecome imaginary close to the N point, signalling that the structure is unstable.

appearing along with the phonon creation and annihilation operators.

Excited lattice vibration normal modes that resemble classical waves, in thatthey have a definite displacement and definite phase, cannot be energy eigen-states of the Hamiltonian as they are not eigenstates of the number operator.The classical lattice vibrations are best described in terms of coherent stateswhich are a superposition of states each containing large numbers of phonons.The time dependence contained in the exponential phase factors has the effectthat the displacements associated with each excited normal mode oscillate peri-odically with time. In general, the time-dependent factors have the effect that,if the eigenvalues of the dynamical matrix are such that ω2

α(q) > 0, the variousexpectation values of the displacements can simply be represented as a sum ofperiodic oscillations. On the other hand, if ω2

α(q) < 0, the displacements haveunbounded exponential growth, the harmonic approximation fails, and the lat-tice becomes unstable.

Sometimes, gradual structural changes are accompanied by softening of thephonon dispersion relation for some values of q as the temperature is decreased.For example, in SrT iO3 a phonon mode softens and at the transition tem-perature, the associated displacement becomes static and is only limited byanharmonic interactions189. However, the phonon frequency does not need tosoften all the way down to zero if a structural changes is to occur. The tempera-ture induced change of stability of the low-temperature close-packed structuresof the light alkaline metals to b.c.c. structures at higher temperatures has been189R. A. Cowley, W. J. L. Buyers and G. Dolling, Sol. State. Commun. 7, 181 (1969).

545

H4M and H8 triple-axis spectrometers at the HFBR atBrookhaven National Laboratory. The use of standard andhigh-temperature Displex refrigerators enabled us to performthe measurements over a wide temperature range 12–400 K.Two sample settings were used. The crystal was aligned tohave either the(100) or the (110) planes coincide with thescattering plane of the spectrometer. The sample mosaic re-veals two monocrystalline grains, misaligned by 0.5°, givinga full width at half maximum~FWHM! of 1°. Neutron scat-tering experiments were performed with fixed final or initial

neutron energies,E514.7 meV andE530.5 meV. Pyroliticgraphite PG~002! reflections were used for monochromatorand analyzer. 208-208-208-408 and 408-208-208-408 collima-tions were used, yielding a typical energy resolution ofdE50.4 meV anddE50.7 meV full width at half maximumat DE50, Ef514.7 meV, and 30.5 meV, respectively. Apyrolitic graphite filter was positioned in front of the detectoror in front of the sample forEf-constant andEi-constantmeasurements, respectively. Both constant-q and constant-E scans were utilized to determine the phonon cross section.

III. RESULTS

A. Precursor phenomena in the parent phase

1. Phonon dispersion

Phonon dispersion curves19 were determined from inelas-tic neutron scattering along the high-symmetry directions(z,0,0), (z,z,0), (z,z,z), and (2z,z,z). The results areshown in Fig. 2. The curves are plotted in an extended Bril-louin zone scheme of the fccL21 structure, that is, in thereduced zone scheme for theB2 structure.

Most modes demonstrate only slight temperature depen-dence easily explained by the overall stiffening of the latticeat low temperatures. The most striking feature, however, isthe strong temperature dependence of the (z,z,0) TA2

branch with polarization along (110), discussed in detail inour previous paper.17 As shown in the blowup in Fig. 3~a!~dashed lines!, the dispersion curve has a wiggle atz0'

13 at

room temperature, which develops into a distinct minimumas the temperature is decreased. In the limitz→0 the(z,z,0) TA2 mode corresponds to the elastic constantc8512(c112c12), which is known to be reduced in bccmaterials,20 and shows an anomalous decrease asT→TM inNi 2MnGa ~Refs. 14 and 13!. For the (110)direction, theslopes in the longitudinal (vL'5.43105 cm/s! and (001)transverse (vT1'3.53105 cm/s! dispersion curves atq→0are in good agreement with experimental ultrasound

FIG. 1. ~a! Elastic intensity measured in the(220) Bragg posi-tion as a function of temperature. The disappearance of the Braggpeak at low temperature indicates the martensitic transformation.TM↑ and TM↓ indicate the temperatures at which the martensiticphase disappears on warming up and first appears on cooling.~b!Temperature dependence of the intensity in the (1

3,13,0) elastic

satellite in cubic Ni2MnGa. The peak is characteristic of the pre-martensitic phase.

FIG. 2. Measured acoustic-phonon dispersion curves for the parentL21 phase of Ni2MnGa; the dash-dotted lines show the fcc zoneboundaries. Solid lines are guides to the eye and arrows indicate the observed phonon anomalies.

15 046 54ZHELUDEV, SHAPIRO, WOCHNER, AND TANNER

Figure 230: The measured acoustic phonon dispersion relations for Ni2MnGawhich undergoes a transition between an f.c.c. crystal structure at room tem-perature to a teragonal low-temperature structure with a modulated distortion.The phonon frequency at the modulation vector softens almost to zero as thetemperature is lowered to T = 240 K. [After Zheludev et al. Phys. Rev. B 54,15045 (1996).]

(1-10)

[111]

[11-1]

(112)

(-1-12)

cos α = 1/3

α α

[111]

[11-1]

√2a

√3a/2

√2a

Figure 231: A prism of the b.c.c. lattice which becomes the parent of an h.c.p.unit cell in the b.c.c. → h.c.p. transition. [After W.G. Burgers. Physica 1, 561(1934).]

546

(1-10)

(112)(-1-12)

[111]

[11-1]

cos α = 1/3α

π/3

c'=√8/3 a'

a'

Figure 232: The b.c.c. → h.c.p. transition involves a combination of a uniformshear and a shuffle from the quasi-static transverse 1

2 (1, 1, 0) phonon modes.[After W.G. Burgers. Physica 1, 561 (1934).]

547

attributed190 to the existence of a low-energy transverse phonon mode in theopen b.c.c. structure. Due to the low-energy of the phonon mode, an increasein temperature produces a rapid increase in entropy which stabilizes the high-temperature b.c.c. phase. Henceforth, the discussion will be restricted to thecase of stable lattice structures.

14.1 Lattice with a Basis

When the lattice has a basis composed of p atoms, the analysis of the phononexcitations is similar to the analysis for a crystal with a mono-atomic basis. Thedisplacements are labelled by the lattice vector index i and also by an index jwhich labels the atoms in the basis. Also, the dynamical matrix becomes a3 p× 3 p matrix and there are 3 p normal modes labelled by α. Thus, one has

uji =

1√N

∑q

∑α

Qαq εjα(q) exp

[i q . Ri

](1845)

The polarization vectors εjα(q) generally are complex for a lattice without inver-sion symmetry. The polarization vectors satisfy the generalized orthonormalitycondition

j=p∑j=1

εjβ(q)∗ . εjα(q) Mj = δα,β (1846)

where Mj is the mass of the j-th atom in the basis.

The dynamical matrix for an lattice with a basis can be expressed in termsof the pair-potential. The Fourier transform of the pair potential Θ(k) is intro-duced via the definition

Θ(R) =∑

k

Θ(k) exp[i k . R

](1847)

so that

Θ(k) =1V

∫d3R Θ(R) exp

[− i k . R

](1848)

For a simple metal, it might be permissable to approximate the pair potentialby the screened Coulomb interaction

Θ(k) =1V

(4 π Z2 e2

k2 ε(k)

)V0(k)2 (1849)

where ε(k) is the static dielectric constant. The real space dynamical matrixcan be expressed in terms of the real space pair-potential, which in turn can be190C. Zener, Phys. Rev. 71, 846 (1947).

548

expressed in terms of the definition of its Fourier transform. On utilizing theLaue identity ∑

R

exp[i ( k − q ) . R

]= N

∑Q

δk−q,Q (1850)

one finds that the dynamical matrix can then be written in terms of a sum overreciprocal lattice vectors as

Dµ,jν,j′(q) = N

∑Q

( Q + q )µ exp[i ( Q + q ) . ( rj − rj′ )

]( Q + q )ν Θ(Q+ q)

− N δjj′

∑Q,′′

( Q )µ exp[i Q . ( rj − rj′′ )

]( Q )ν Θ(Q) (1851)

which now involves the set of p basis vectors rj . The eigenvalues of the 3p× 3pdynamical matrix yields the squares of the frequencies, ωα(q) for the 3p phononmodes.

14.2 A Sum Rule for the Dispersion Relations

The eigenvalue equation for the phonon frequencies

D(q) εα(q) = M ω2α(q) εα(q) (1852)

can be related to the Fourier transform of the pair-potential. The Fourier Trans-form of the dynamical matrix is given by

D(q) =∑R

D(R) exp[− i q . R

](1853)

Hence, one has∑R

D(R) exp[− i q . R

]εα(q) = M ω2

α(q) εα(q) (1854)

The dynamical operator is given in terms of the pair-potential by

Dνµ(Ri) = δR

i,0

∑R′′

∇µ ∇ν Θ(0−R′′) − ∇µ ∇νΘ(Ri)

= δRi,0

∑R′′

∇µ ∇ν Θ(R′′) − ∇µ ∇νΘ(Ri) (1855)

and the Fourier Transform of the pair-potential is given by

Θ(R) =∑

k

exp[i k . R

]Θ(k) (1856)

549

On substituting the expression for the Fourier transform of the pair-potentialinto the expression for the Fourier transform of the dynamical matrix, one ob-tains

Dνµ(q) =

∑R

i

∑k

kµ kν Θ(k) exp

[i ( k − q ) . Ri

]

−∑R′′

∑k

kµ kν Θ(k) exp

[i k . R′′

](1857)

Thus, the phonon frequencies satisfy the eigenvalue equation

M ω2α(q) εα(q) =

∑i

∑k

k Θ(k) exp[i ( k − q ) . Ri

]k . εα(q)

−∑

i

∑k

k Θ(k) exp[i k . Ri

]k . εα(q)

(1858)

On making use of the conservation of momentum, modulo the reciprocal latticevectors Q, one has

M ω2α(q) εα(q) = N

∑Q

( q +Q ) Θ(q +Q) ( q +Q ) . εα(q)

− N∑Q

Q Θ(Q) ( Q . εα(q) ) (1859)

It can be seen that the transverse modes only exist because of the periodicity ofthe lattice. That is, if only the Q = 0 reciprocal lattice vector were includedin the sum, the eigenvectors would be longitudinal as ε(q) would be parallel toq. In this case,

M ω2α(q) εα(q) = N q Θ(q) q . εα(q) (1860)

which are the longitudinal sound modes. Hence, the transverse modes only ex-ist because of the periodicity of the lattice. Furthermore, by letting q → 0,one finds that the phonon frequencies M ω2

α(0) must vanish in this limit, if theinteractions are sufficiently short ranged so |Θ(0)| <∞. The transverse modesare examples of Goldstone modes. The transverse modes occur because thecontinuous translational symmetry of the Hamiltonian is spontaneously brokenat the phase transition when the solid is formed. Goldstone’s theorem191 maybe roughly stated as “When a continuous symmetry of the Hamiltonian is spon-taneously broken in a phase transition, a continuous branch of normal modesappears which extend to zero-energy. These normal modes dynamically restore191J. Goldstone, Nuovo Cimento 19, 154 (1961), Y. Nambu, Phys. Rev. Lett. 4, 380 (1960).

550

the broken symmetry.” Goldstone’s theorem also depends on the condition thatlong-ranged forces are not present. If long-ranged forces are present, the sym-metry restoring mode may acquire a finite frequency at q = 0 through theKibble-Higgs mechanism192. In this case, the resulting modes are called Higgsbosons193.

The Sum Rule.

The Sum Rule is obtained by using the orthogonality relations for the polar-ization vectors. On taking the scalar product of the eigenvalue equation for thedispersion relation with the eigenvector eα(q) and summing over the polarizationindex α, one obtains

M∑α

ω2α(q) = N

∑α

∑Q

εα(q) . ( q +Q ) Θ(q +Q) ( q +Q ) . εα(q)

− N∑α

∑Q

εα(q) . ( Q ) Θ(Q) ( Q ) . εα(q)

(1861)

Utilizing the completeness relation for the components of the polarization vec-tors ∑

α

εµα(q) εα,ν(q) = δµν (1862)

one finds the sum rule

M∑α

ω2α(q) = N

∑Q

Θ(q +Q) ( q +Q )2

− N∑Q

Θ(Q) ( Q )2 (1863)

for the phonon frequencies.

——————————————————————————————————

14.2.1 Exercise 74

Show that if the pair-potential is approximated by the non-screened Coulombpotential between the charged nuclei, one finds∑

α

ω2α(q) =

4 π N Z2 e2

M V(1864)

192T. W. B. Kibble, Proc. Oxford Int. Conf. on Elem. Particles, (1965).193P. W. Higgs, Phys. Rev. Lett. 13, 508, (1964).

551

which defines the plasmon frequency for the ions. In the long wavelength limit,one has the longitudinal plasmon mode and at most, two transverse modes de-pending on the presence of long-ranged order. As the longitudinal plasmonmode saturates the sum rule at q = 0, the transverse modes, if they exist,must be acoustic.

——————————————————————————————————

14.3 The Nature of the Phonon Modes

The long wavelength form of the dynamical matrix can easily be calculated from

D(q) = − 2∑R

D(R) sin2

(q . R

2

)(1865)

as

D(q) = − 12

∑R

D(R)(q . R

)2

(1866)

This implies that ωα(q)2 ∝ q2 as q → 0, which gives rise to acoustic modes.These acoustic modes include the two Goldstone modes as well as the longitu-dinal sound mode, which can be considered as a density fluctuation similar tothe sound waves found in fluids. If the crystal is anisotropic, the frequency orsound velocity may depend on the direction of propagation.

In an isotropic solid, there should be one longitudinal and two transversepolarizations (ε ‖ q) and (ε ⊥ q). In an anisotropic solid, the relationbetween ε and q is not so simple, except at high-symmetry points. However,because the polarization vectors are continuous functions of q, one may still usethe terminology of longitudinal and transverse polarizations in the vicinity ofthe high-symmetry points.

A solid with a p atom basis has 3 N p degrees of freedom, 3 N of whichare tied up in the acoustic phonon branches. The other 3 ( p − 1 ) N modesappear as optic branches.

The phonon density of states is given by an integral over the first Brillouinzone, and a summation over the polarization index α.

ρ(ω) =V

( 2 π )3∑α

∫d3q δ( ω − ωα(q) ) (1867)

This can be written as an integral over a surface of constant ωα(q). This surfaceis denoted by Sα(ω) which consists of the points ω = ωα(q), where q is in thefirst Brillouin zone. This yields

ρ(ω) =V

( 2 π )3∑α

∫Sα(ω)

d2S1

| ∇ωα(q) |(1868)

552

0

0.005

0.01

0.015

hω

q [eV

]

Γ ΓPH N

(1/2,1/2,0)(1/2,1/2,1/2)(1,0,0)

L

LL

T

T

TL

T

Figure 233: A model calculation of the phonon dispersion relations for a metalwith a b.c.c. crystal structure. The screening was treated in the Random PhaseApproximation.

0

0.02

0.04

0.06

h ωq [

eV]

(1,0,0) (1,1/2,0) (3/4,3/4,0) (1/2,1/2,1/2)

Γ ΓX W K L

Figure 234: A model calculation of the phonon dispersion relations for a metalwith an f.c.c. crystal structure. The screening was treated in the Random PhaseApproximation.

553

Figure 235: The phonon dispersion relations for h.c.p. Mg measured along theΓK direction. [After R. Pynn and G. L. Squires, Proc. Roy. Soc. 326, 347(1972).]

The van Hove singularities occur when the group velocity vanishes,

∇ωα(q) = 0 (1869)

The occurrence of these singularities in the phonon density of states has beenstudied via topological arguments194. The van Hove singularities are usuallyintegrable in three dimensions, but still give rise to anomalous slopes or discon-tinuities in the derivatives of ρ(ω). An example of the van Hove singularities inthe phonon modes is given by Cu which has two transverse and one longitudi-nal contribution to the density of states. The phonon density of states has beenconstructed from the results of inelastic neutron scattering measurements195.

——————————————————————————————————

14.3.1 Exercise 75

Consider a one-dimensional linear chain, with a unit cell composed of two atoms,one with mass M1 and the other with mass M2. The atoms interact with theirnearest neighbors via a harmonic force, with force constant γ. Find the phonondispersion relation.

——————————————————————————————————

194L. van Hove, Phys. Rev. 89, 1189 (1953).195E. C. Svensson, B. N. Brockhouse and J. M. Rowe, Phys. Rev. 155, 619 (1967).

554

Figure 236: The phonon density of states ρ(ω) constructed from the data ob-tained from inelastic neutron scatteringe experiments on Cu. [After Svenssonet al., Phys. Rev. 155, 619 (1967).]

14.3.2 Exercise 76

Consider a one-dimensional line of ions, with equal masses but alternatingcharges, such that the charge on the n-th ion is

en = e ( − 1 )n (1870)

Assume that the inter-atomic potential has two contributions:

(A) A short-ranged force between nearest neighbors with a force constantC1 = γ.

(B) A Coulomb interaction between all the ions Cn = 2 ( − 1 )n e2

n3 a3

where a is the atomic spacing.

(i) Show that

ω(q)2

ω20

= sin2 q a

2+ σ

∞∑n=1

( − 1 )n

n3

(1 − cos q n a

)(1871)

where ω20 = 4 γ

M and σ = e2

γ a3 .

(ii) Show that ω2(q) becomes soft ω2(q) = 0 at q = πa if σ > 4

7 ζ(3).

(iii) Show that the speed of sound becomes imaginary if σ > 12 ln 2 .

555

Thus, ω2 goes to zero and the lattice becomes unstable for q in the interval(0, π) if σ lies in the range 0.475 < σ < 0.721.

——————————————————————————————————

14.3.3 Exercise 77

Consider a two-dimensional square lattice with a mono-atomic basis. The atomshave mass M and interact with their nearest neighbors and next nearest neigh-bors through a harmonic force of strength γ1 and γ2, respectively. Calculatethe frequencies of the longitudinal and transverse phonons at q = (π

a ) (1, 0).

——————————————————————————————————

14.3.4 Exercise 78

(i) Show that the linear chain with nearest neighbor (harmonic) interactions hasa dispersion relation

ω(q) = ω0 | sinq a

2| (1872)

and that the density of states is given by

ρ(ω) =2π a

1√ω2

0 − ω2(1873)

which has a van Hove singularity at ω = ω0.

(ii) Show that in three dimensions, the van Hove singularities near a maximumof ωα(q) gives rise to a term in the density of states that varies as

ρ(ω) ∝√

ω20 − ω2 (1874)

and, thus, has a singularity in the first derivative of the density of states withrespect to ω.

——————————————————————————————————

14.3.5 Exercise 79

(i) Show that if the wave vector q lies along a 3 , 4 or 6 fold axis, then onenormal mode is polarized along q and the other two modes are degenerate andpolarized perpendicular to q.

556

(ii) Show that if q lies in a plane of mirror symmetry, then one mode has apolarization perpendicular to q and the plane, and the other two modes havepolarizations within the plane.

(iii) Show that if q lies on a Bragg plane that is parallel to a plane of mirrorsymmetry, then one mode is polarized perpendicular to the Bragg plane, whilethe other two modes have polarizations lying within the plane.

——————————————————————————————————

14.3.6 Exercise 80

Consider an f.c.c. mono-atomic Bravais lattice in which the atoms interact viaa nearest neighbor pair-potential Θ.

(i) Show that the frequencies of the phonon modes are given by the eigen-values of a 3 × 3 matrix given by

D(q) =∑R

D(R) sin2

(q . R

2

) [A I + B R R

](1875)

where the sum over R runs over the 12 lattice sites closest to the site R = 0,and the constants A and B are given in terms of the pair-potential and itsderivatives at the nearest neighbor separation d = a√

2via

A = 2 Θ′(d)/d (1876)

and

B = 2[

Θ”(d) − Θ′(d)/d]

(1877)

(ii) Show that when q = (q, 0, 0), the longitudinal and transverse acousticphonon frequencies are given by

ωl(q) =

√8 A + 4 B

Msin

q a

4(1878)

and

ωt(q) =

√8 A + 2 B

Msin

q a

4(1879)

(iii) Find the frequencies when q = q√3

(1, 1, 1).

557

(iv) Show that when q = q√2

(1, 1, 0) then the degeneracy between thetransverse modes is lifted and the frequencies are given by

ωl(q) =

√8 A + 2 B

Msin2 q a

4√

2+

2 A + 2 BM

sin2 q a

2√

2(1880)

and the two transverse modes are

ω1t (q) =

√8 A + 4 B

Msin2 q a

4√

2+

2 AM

sin2 q a

2√

2(1881)

and

ω2t (q) =

√8 A + 2 B

Msin2 q a

4√

2+

2 AM

sin2 q a

2√

2(1882)

——————————————————————————————————

14.3.7 Exercise 81

Consider a phonon with a wave vector along the axis of a cubic crystal. Thenconsider the sums in

D(q) =∑R

D(R) sin2

(q . R

2

)(1883)

be restricted to the sites in two planes perpendicular to q separated by a distanceq . R. In metals, there exists a long-ranged interaction between the planes

D(q) =∑R

D(R) sin2

(q . R

2

)= A

sin 2kF q . R

2kF q . R(1884)

where A is a constant.

(i) Find an expression for ω2(q) and∂ω2(q)

∂q .

(ii) Show that∂ω2(q)

∂q is infinite at q = 2 kF . The kink in the dispersionrelation at the Fermi wave vector is the Kohn anomaly.

——————————————————————————————————

558

14.4 Thermodynamics

A harmonic lattice has an energy given by

Eharmonic = Ecl +∑q,α

h ωα(q)(nq,α +

12

)(1885)

where Ecl is the ground state energy of the lattice in the classical approximation.At finite temperatures, the energy is to be replaced by a thermal average. Thethermal average of the number of phonons in mode (q, α) is denoted by nq,α

and is calculated as the Boltzmann weighted average. Since the energies of theharmonic phonons are additive, then the Boltzmann factor can be expressed asa product of Boltzamnn factors for each normal mode.

1Z

exp[ − β Eharmonic ] =∏q,α

1Zα(q)

exp[ − β h ωα(q) ( nq,α +12

) ] (1886)

This implies that the thermal average can be performed independently for eachphonon mode. Therefore, we can write

nq,α =1

Zα(q)

∞∑n=0

n exp[ − β ( n +12

) h ωα(q) ]

=1

Zα(q)exp[ − 3

2 β h ωα(q) ](1 − exp[ − β h ωα(q) ]

)2

(1887)

However, the partition function for a single phonon mode Zα(q) is given by thenormalization condition for the probability

Zα(q) =∞∑

n=0

exp[ − β (n +12

) h ωα(q) ]

=exp[ − 1

2 β h ωα(q) ]1 − exp[ − β h ωα(q) ]

(1888)

We note that the contribution from the zero point energy drops out of the ratio.Thus, the thermal average of the number of phonons is given by

nq,α = N(ωα(q)) =1

exp[ β h ωα(q) ] − 1(1889)

where N(ω) is the Bose-Einstein distribution function. Therefore, the thermalaveraged energy of the harmonic lattice is given by

Eharmonic = Ecl +∑q,α

h ωα(q)(N(ωα(q)) +

12

)(1890)

559

The Helmholtz free energy F of the lattice is defined in terms of the partitionfunction, Z, by

Z = exp[− β F

]

= exp[− β Ecl

] ∏q,α

nq,α=∞∑nq,α=0

exp[− β h ωα(q) ( nq,α +

12

)]

= exp[− β Ecl

] ∏q,α

(exp[ − 1

2 β h ωα(q) ]1 − exp[ − β h ωα(q) ]

)(1891)

Thus, the Helmholtz free energy F is given by

F = Ecl +∑q,α

h ωα(q)2

+ kBT∑q,α

ln(

1 − exp[− β h ωα(q)

] )(1892)

The pressure P is found from the infinitesimal thermodynamic relation

dF = dE − T dS − S dT

dF = − S dT − P dV (1893)

Hence, the pressure is determined by

P = −(∂F

∂V

)T,N

= −(∂Ecl

∂V

)T,N

− 12

∑q,α

(∂hωα(q)∂V

)T,N

−∑q,α

(∂hωα(q)∂V

)T,N

1

exp[β h ωα(q)

]− 1

(1894)

The first two terms are temperature independent, and the last term depends ontemperature through the average phonon occupation numbers. The pressure isonly temperature dependent if the phonon frequencies depend on the volume V .

The thermal volume expansion coefficient α is defined by

α =1V

(∂V

∂T

)P

(1895)

As the equation of state is a relation between pressure, temperature and volume

P = P (T, V ) (1896)

then the infinitesimal derivatives are related by

dP =(∂P

∂T

)V

dT +(∂P

∂V

)T

dV (1897)

560

For a process at constant P , dP = 0, so one has the relation

(∂V

∂T

)P

= −

(∂P∂T

)V(

∂P∂V

)T

(1898)

The bulk modulus, B, defined by

B = − V

(∂P

∂V

)T

(1899)

is finite as Ecl is expected to be volume-dependent. Hence, the denominator isfinite. The thermal expansion coefficient is non-zero, only if (∂P

∂T )V 6= 0. Usingthe harmonic approximation, the frequencies must be functions of the volumeV if the solid is to undergo thermal expansion.

The specific heat at constant pressure is different from the specific heat atconstant volume. The difference is found by relating the temperature derivativeof the entropy with respect to temperature, at constant volume, to the tempera-ture derivative of the entropy with respect to temperature, at constant pressure.The relation is found by considering the infinitesimal change in entropy, witheither the change in volume or pressure

dS =(∂S

∂T

)V

dT +(∂S

∂V

)T

dV

=(∂S

∂T

)P

dT +(∂S

∂P

)T

dP (1900)

Using the equation of state relating P and V

V = V (T, P ) (1901)

one finds

dV =(∂V

∂T

)P

dT +(∂V

∂P

)T

dP (1902)

Thus, on combining the expression for dS in terms of dV and dT with the aboveequation for dV , one obtains the expression

dS =[ (

∂S

∂T

)V

+(∂S

∂V

)T

(∂V

∂T

)P

]dT +

(∂S

∂V

)T

(∂V

∂P

)T

dP

(1903)

Thus, one has the relation(∂S

∂T

)P

=(∂S

∂T

)V

+(∂S

∂V

)T

(∂V

∂T

)P

(1904)

561

A Maxwell relation can be used to eliminate(

∂S∂V

)T. The Maxwell relation comes

from the analyticity condition on a thermodynamic function with independentvariables (T, V ), which is F (T, V ). Hence, one has(

∂S

∂V

)T

=(∂P

∂T

)V

(1905)

and, thus,

T

(∂S

∂T

)P

= T

[ (∂S

∂T

)V

+(∂P

∂T

)V

(∂V

∂T

)P

]CP = CV + T

(∂P

∂T

)V

(∂V

∂T

)P

CP = CV − T

(∂P∂T

)2

V(∂P∂V

)T

(1906)

Therefore, if there is a difference between CP and CV , then the phonon frequen-cies must be dependent on V .

14.4.1 The Specific Heat

The specific heat at constant volume can be found from the entropy of thephonon gas

S = kB

∑q,α

[ (N(ωα(q)) + 1

)ln(N(ωα(q)) + 1

)−N(ωα(q)) ln N(ωα(q))

](1907)

In this expression N(ω) is the Bose-Einstein distribution function given by

N(ω) =1

exp[ β h ω ] − 1(1908)

and β is proportional to the inverse temperature, i.e., β−1 = kB T . Thespecific heat is given by

CV = T

(∂S

∂T

)V

= kB T∑q,α

∂N(ωα(q))∂T

ln

(N(ωα(q)) + 1N(ωα(q))

)

=∑q,α

∂N(ωα(q))∂T

h ωα(q)

562

= kB

∑q,α

(β h ωα(q)

)2

N(ωα(q))[N(ωα(q)) + 1

](1909)

This can be expressed as an integral over the phonon density of states ρ(ω) via

CV = kB

∫ ∞

0

dω ρ(ω)(β h ω

)2

N(ω)[N(ω) + 1

](1910)

The experimentally determined temperature dependence of the specific heatmay be compared with the results of the above formula, where the phonon den-sity of states is inferred from experiments196. There is a small discrepancy dueto anharmonic effects which have been neglected in the above analysis. We shallexamine the specific heat using some approximate models of the phonon densityof states.

14.4.2 The Einstein Model of a Solid

The Einstein model of a solid considers the phonons to have a fixed frequencyω0 for all q vectors, and is an approximate representation of the optic phonons.In the Einstein model, the phonon density of states is given by

ρ(ω) = 3 N δ(ω − ω0) (1911)

where there are 3 modes per atom. The specific heat is given by

CV = 3 N kB

(β h ω0

)2

N(ω0)[N(ω0) + 1

](1912)

This vanishes exponentially at low temperatures, kB T h ω0, where

N(ω0) ≈ exp[− β h ω0

](1913)

and at high temperatures kB T h ω0

N(ω0) ≈ kB T

h ω0(1914)

so the specific heat saturates to yield the classical result

limT → ∞

CV → 3 N kB (1915)

The Einstein model of the specific heat fails to describe the lattice contributionto low-temperature specific heat of a solid. This is because it fails to describe thelow-energy acoustic phonon excitations which gives rise to a power law tempera-ture variation. The Debye model of a solid provides an approximate descriptionof the low-temperature specific heat of a solid.

196E. C. Svensson, B. N. Brockhouse and J. M. Rowe, Phys. Rev. 155, 619 (1967).

563

But Let’s Take a Closer Look:

High T behavior: Reasonable agreement with experiment

Low T behavior: CV → 0 too quickly as T → 0 !

Figure 237: The specific heat of diamond compared with the results of theEinstein Model, with ΘE = 1320. [After A. Einstein, Ann. Physik 22, 180(1907).]

14.4.3 The Debye Model of a Solid

The Debye model of a solid approximates the phonon density of states for thetwo transverse acoustic mode and longitudinal acoustic mode in an isotropicsolid. The dispersion relations of the phonon modes are represented by thetwo-fold degenerate transverse mode

ωT (q) = vT q (1916)

and the singly degenerate longitudinal mode

ωL(q) = vL q (1917)

The transverse sound velocity, in general, will be different from the longitudinalsound velocity, vL 6= vT . There are 3N such phonon modes in the first Brillouinzone. The phonon density of states is given by the integral over a surface areain the first Brillouin zone

ρ(ω) =V

( 2 π )3∑α

∫d2Sα(ω)

(dωα

dq

)−1

(1918)

If the Brillouin zone is approximated as a sphere of radius qD, then the densityof states is given by

ρ(ω) =V

( 2 π )3∑α

4 π q2(dωα

dq

)−1

(1919)

for q < qD. Using the form of the dispersion relation, the density of states canbe re-written as

ρ(ω) =V

( 2 π2 )

∑α

ω2

v3α

Θ( vα qD − ω ) (1920)

564

This can be further approximated by requiring that the upper limit on thefrequency of all three phonon modes be set to the Debye frequency ωD. In thiscase, the density of states is simply given by

ρD(ω) =V

( 2 π2 )

∑α

ω2

v3α

Θ( ωD − ω ) (1921)

The value of the Debye frequency is determined by the condition∫ ωD

0

dω ρD(ω) = 3 N

=V

( 6 π2 )

∑α

ω3D

v3α

(1922)

Using this condition, the Debye model for the phonon density of states is writtenas

ρD(ω) = 9 Nω2

ω3D

Θ( ωD − ω ) (1923)

Thus, in the Debye model, the phonon density of states varies as ω2 at lowfrequencies and has a cut-off at the maximum frequency ωD.

The temperature dependence of the specific heat of the Debye model is givenby

CV =9 N kB

ω3D

∫ ωD

0

dω ω2

(β h ω

)2

N(ω)[N(ω) + 1

](1924)

The asymptotic low-temperature variation of the specific heat can be found bychanging variable x = β hω. The specific heat can be written in the form

CV = 9 N kB

(kB T

h ωD

)3 ∫ xD

0

dx x4 exp[ x ](exp[ x ] − 1

)2 (1925)

where the upper limit of integration is given by xD = β hωD. At sufficientlylow temperatures, kB T h ωD, the upper limit may be set to infinity yielding

CV = 9 N kB

(kB T

h ωD

)3 ∫ ∞

0

dx x4 exp[ x ](exp[ x ] − 1

)2

=12 π4

5N kB

(kB T

h ωD

)3

(1926)

where the integral has been evaluated as 4 π4

15 . Thus, the low-temperature spe-cific heat varies as T 3 in agreement with experiment. The asymptotic high

565

Debye Model at low T: Theory

vs. Expt.

Quite impressive agreement with predicted CV ∝ T3

dependence for Ar! (noble gas solid)

(See SSS program debye to make a similar comparison for Al, Cu and Pb)Figure 238: The low temperature specific heat of solid argon plotted against

T 3. [After L. Finegold and N. E. Phillips, Phys. Rev. 177, 1383 (1969).]

temperature specific heat, for kB T h ωD, is found from

CV =9 N kB

ω3D

∫ ωD

0

dω ω2

(β h ω

)2

N(ω)[N(ω) + 1

](1927)

noting that the number of phonons is given by N(ω) = kB Th ω , so

CV =9 N kB

ω3D

∫ ωD

0

dω ω2

= 3 N kB (1928)

which is the classical limit. Thus, the Debye approximation provides an interpo-lation between the low-temperature limit and the high-temperature limit, whichis only governed by one parameter, the Debye temperature kB TD = h ωD.

——————————————————————————————————

14.4.4 Exercise 82

Evaluate the integral ∫ ∞

0

dx x4 exp[ x ](exp[ x ] − 1

)2 (1929)

needed in the low-temperature limit of the specific heat for the Debye model.

566

Debye Model: Theory vs. Expt.

Universal behavior for all solids!

Debye temperature is related to “stiffness” of solid, as expected

Better agreement than Einstein model at low T

Figure 239: The specific heats of Cu, Ag, Pb and C. Diamond has an unusuallyhigh Debye temperature.

——————————————————————————————————

14.4.5 Exercise 83

Generalize the Debye model to a d-dimensional solid. Determine the high-temperature and leading low-temperature variation of the specific heat due tolattice vibrations.

——————————————————————————————————

14.4.6 Exercise 84

Show that the leading high temperature correction to the Dulong and Petitvalue of the specific heat due to lattice vibrations is given by

∆CV

CV= − 1

12

∫dω

(h ω

kB T

)2

ρ(ω)∫dω ρ(ω)

(1930)

Also, evaluate the moment of the phonon density of states∫dω ω2 ρ(ω) (1931)

in terms of the pair-potentials between the ions.

——————————————————————————————————

567

14.4.7 Exercise 85

Numerically calculate the phonon density of states for a single phonon mode fora two-dimensional lattice with a dispersion

ω2(q) = ω20

(2 − cos qxa − cos qya

)(1932)

and hence, obtain the temperature dependence of the specific heat. Comparethis with the numerical evaluation of an appropriate Debye model.

——————————————————————————————————

14.4.8 Lindemann Theory of Melting

Lindemann197 assumed that a lattice melts when the displacements due to lat-tice vibrations become comparable to the lattice constants. Although this the-ory does not address the appropriate mechanism, it does give the right order ofmagnitude for simple metals and transition metals. It is assumed that meltingoccurs at a critical value of the ratio

γ =u2

i

a2i

(1933)

The melting occurs when the temperature dependent function γ is equal to acritical value γc which is expected to be of the order of unity. The function γ isgiven by

γ =h

2 M N a2

∑q,α

2 nq,α + 1

ωα(q)(1934)

in which nq,α is the thermal average of the number of phonons in the mode(q, α). The thermally averaged number of phonons of frequency ωα(q) is givenby the Bose-Einstein distribution function

N(ωα(q)) =1

exp[ β h ωα(q) ] − 1(1935)

so

2 N(ωα(q)) + 1 = coth(β h ωα(q)

2

)(1936)

Hence, the temperature dependent function γ can be expressed as

γ =h

2 M N a2

∑q,α

cothβhωα(q)

2

ωα(q)(1937)

197F. A. Lindemann, Z. Phys. 11, 609 (1910).

568

Using the Debye model, the right hand side can easily be evaluated in twolimits: the zero temperature limit and the high temperature limit. In the limitof zero temperature, the mean squared deviation is only due to the zero pointfluctuations of the harmonic phonons. The low temperature limit of the relativemean squared deviation is given by

γ =h

2 M N a2

∑q,α

1vα q

=3 h

2 M N v a2

V

( 2 π )3

∫d3q

1q

=3 h

2 M N v a2

V

( 2 π )32 π q2D (1938)

Using the relation

N =V q3D6 π2

=3 V

4 π a3(1939)

one obtains

γ ∼ 12

(9

2 π2

) 13 h qDM v

∼ 0.4h qDM v

= 0.4kB TD

M v2(1940)

It can be shown that the right hand side is proportional to M− 12 . Therefore, the

above relation implies that if an element is to solidify, it has to have an atomicmass which is greater than a critical value. At high temperatures, the relativemean squared displacement is dominated by the thermally activated phononsand is given by

γ =h

2 M N a2

V

2 π2

∫ qD

0

dq q26 kB T

h v2 q2

=3

M a2

V

2 π2 N

kB T qDv2

=9

M a2 q2D

kB T

v2=(

36π2

) 13 kB T

M v2

∼ 1.54kB T

M v2= 1.54

h2 q2DM k2

B T 2D

kB T (1941)

The Lindemann criterion provides a relation between the Debye temperatureand the melting temperature. The experimental data for alkaline metals andtransition metals suggest that γc has a value near 1

16 , independent of themetal198. Of course, one expects that anharmonic effects may become impor-tant for large displacements of the ions from their equilibrium positions, and198G. Grimvall and S. Sjodin, Physica Scripta, 10, 340, (1974).

569

0

0.5

1

1.5

0 0.5 1 1.5 2 2.5 3(T/TD)

γ/γ c

Tc

Figure 240: The schematic temperature variation of the function γ used in theLindemann criterion for melting.

Figure 241: The variation of the melting temperature with the square of theDebye temperature. [After Grimvall and Sjodin (1974).]

570

one expects that the phase transition will involve the collective behavior of theatoms. In particular, one might expect that the Debye temperature should de-crease as the phase transition is approached199.

Mermin-Wagner Theorem.

The Lindemann theory of melting may be extended to provide an exampleof the Mermin-Wagner theorem200. The Mermin-Wagner theorem states thatfinite temperature phase transitions, in which a continuous symmetry is sponta-neously broken, cannot occur in lower than three dimensions. Basically, if sucha transition occurs, then there should be a branch of Goldstone modes that dy-namically restores the spontaneously broken symmetry201. These normal modesproduce fluctuations in the order parameter. In a periodic solid where continu-ous translational invariance is broken, the Goldstone modes are the transversesound waves. The transverse sound modes have dispersion relations of the formω(q) = v q. The fluctuations in the order parameter are the fluctuations in thechoice of origin of the lattice and, therefore, are just the fluctuations in positionsof any one ion. In d dimensions, at finite temperatures, the fluctuations havecontributions from the region of small q which are proportional to

u2i ∼

∫ qD

0

ddqh

2 M ω(q)kB T

h ω(q)

∼∫ qD

0

dq qd−3 (1942)

where qD is a cut off due to the lattice. The integral diverges logarithmicallyfor d = 2, indicating that the fluctuations in the equilibrium lattice positionswill be infinitely large, thereby preventing the solid from being formed. Like-wise, for lower dimensions such as one dimension, the integral will also divergeat the lower limit. Therefore, no truly one-dimensional solid is stable againsttemperature-induced fluctuations. An analysis of the zero point fluctuationsalso rules out the possibility of a one-dimensional lattice forming in the limit ofzero temperature.

For a harmonic solid, the phonon frequencies are independent of the volumeV . This can be seen by considering the energy of a solid which has expandedin the linear dimensions by an amount proportional to ε.

199An alternate melting criterion was suggested by Born. Born’s criterion [M. Born, J.Chem. Phys., 7, 591, (1939).] is based on the observation that solids differ from liquidsby their rigidity or resistance to shear stresses. Born postulated that the velocity of shearwaves would go to zero at the melting temperature. However, measurements of the shear forceconstant shows that although it decreases as the temperature is increased towards the meltingtemperature, it does remains finite at the melting temperature.200N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 (1966).201J. Goldstone, Nuovo Cimento 19, 154 (1961).

571

The energy of a harmonic solid with static displacements about the originalequilibrium position is given by the harmonic expression

E = Eeq +12

∑i,j

ui D(Ri −Rj) uj (1943)

Now consider the expanded lattice in which the displacements are given by

ui = ε Ri + ui (1944)

Here, ui are the new displacements from the new lattice sites of the lattice whichhas undergone an increase in volume of ( 1 + ε )3, through the application ofexternal forces. The expanded solid has an energy given by

E = Eeq +ε2

2

∑i,j

Ri D(Ri −Rj) Rj +12

∑i,j

ui D(Ri −Rj) uj

(1945)

The terms linear in ε vanish identically, as the total force on an ion must vanishin equilibrium. The total force is the sum of the internal forces opposing theexpansion and the applied external forces that result in the expansion. Sincethe dynamical matrix that governs the lattice displacements ui is unchanged,its eigenvalues, which are the phonon frequencies, are unchanged by expansionof a harmonic solid.

Thermal expansion only occurs for an anharmonic lattice. Thermal expan-sion provides a measure of the volume dependence of the phonon frequencies∂ h ω∂ V or the anharmonicity.

14.4.9 Thermal Expansion

The coefficient of thermal expansion of an insulator can be evaluated from(∂V

∂T

)P

= −(∂P

∂T

)V

/ (∂P

∂V

)T

(1946)

where the pressure is found from P = −(

∂F∂V

)T

. Using the expression for

the free energy of the lattice, one finds that the coefficient of thermal expansioncan be written as

α = − 1B

∑q,α

(∂ h ωα(q)

∂V

) (∂N(ωα(q))

∂T

)(1947)

where B is the bulk modulus and N(ω) is the Bose-Einstein distribution func-tion.

572

The specific heat can be written as

CV =∑q,α

h ωα(q)(∂N(ωα(q))

∂T

)(1948)

On identifying the contributions from each normal mode, one can define aGruneisen parameter for each normal mode

γα(q) = − V

ωα(q)

(∂ ωα(q)∂V

)(1949)

which is a dimensionless ratio of α B VCV

. Thus,

γα(q) = −

(∂ lnωα(q)∂ lnV

)(1950)

The Gruneisen parameter for the entire solid can be expressed as a weightedaverage of the Gruneisen parameter of each normal mode

γ =

∑q,α γα(q) Cq,α∑

q,α Cq,α(1951)

with weights given by Cq,α. This is consistent with the definition of the Gruneisenparameter in terms of thermodynamic quantities

γ =α B V

CV(1952)

For most models, γα(q) is roughly independent of T and is a constant.

γα(q) ∼ γ = −

(∂ lnωD

∂ lnV

)(1953)

Hence, as B is roughly T independent, the specific heat CV tracks the coefficientof thermal expansion α. A typical Gruneisen parameter has a magnitude of ∼1 or 2, and a slow temperature variation, which changes on the scale of TD.

14.4.10 Thermal Expansion of Metals

For a metal, there is an additional contribution to the pressure from the elec-trons. The electronic contribution to the pressure is calculated as

Pel =23Eel

V

V(1954)

573

and as the electronic energy is temperature dependent, the electronic contribu-tion to the pressure is also temperature dependent. This gives an additionalcontribution to the rate of change of pressure with respect to temperature(

∂Pel

∂T

)=

23Cel

V (1955)

Hence, the coefficient of thermal expansion for a metal is determined from

(∂V

∂T

)P

= −

(∂P∂T

)V(

∂P∂V

)T

(1956)

Hence,

α =1B

(γ Clatt

V +23Cel

V

)(1957)

where 23 is the electronic Gruneisen parameter.

14.5 Anharmonicity

Anharmonic interactions give rise to a coupling of the normal modes. They canbe described a collision process in which the number of phonons may not beconserved. For example, two phonons may interact and combine to produce asingle phonon. The interaction which describes this process is the cubic termin the Taylor expansion of the potential in powers of the lattice displacements.For many properties, it is also necessary to also consider the effect of the quarticterms in the expansion, as these give rise to perturbative corrections which areof the same order as the first non-zero corrections produced by the cubic terms.Furthermore, if the lattice is stable against large amplitude lattice vibrations,the quartic terms are non-zero. The anharmonic interactions give rise to thelifetime of phonons, provide temperature dependent corrections to the phonondispersion relations, and contribute to the specific heat. The theory of anhar-monic interactions has been reviewed by Cowley202 and the experimental man-ifestations have been reviewed by Martin203. The anharmonic interactions mayusually be thought of as producing small corrections to the harmonic phonons,except when the systems is on the verge of a structural instability where theyplay an important role. The phonon modes are not the only excitations of thecrystalline lattice, there are also large amplitude excitations like dislocations.Although these excitations may have a large (macroscopic) spatial extent theydo not extend all through the crystal, like the phonon modes, and the deviationsof the atoms from the ideal equilibrium positions can be large, comparable to202R. A. Cowley, Adv. in Phys. 12, 421 (1963).203D. H. Martin, Adv. in Phys. 14, 39 (1965).

574

the lattice spacing. If the lattice displacements in the dislocations were consid-ered to be made up of a superposition of coherent states for each phonon mode,in the absence of the anharmonic interactions, the distortions would disperseand the dislocations would lose their shape. The anharmonic interactions areresponsible for stabilizing these large amplitude, spatially localized, excitationsby balancing the effects of dispersion of the phonon modes. These excitationdo have macroscopically large excitation energies but they do also have macro-scopically large effects. In essence, these dislocations are non-linear excitations,like solitons, and play an extremely important role in determining the actualmechanical properties of any real solid.

——————————————————————————————————

14.5.1 Exercise 86

The full ionic potential of a mono-atomic Bravais lattice has the form

V = Veq +12!

∑R1,R2

∑µ,ν

uµ(R1) Dµ ν(R1 −R2) uν(R2)

+13!

∑R1,R2,R3

∑µ,ν,λ

uµ(R1) uν(R2) uλ(R3) Dµ ν λ(R1, R2, R3)

+14!

∑R1,R2,R3,R4

∑µ,ν,λ,ρ

uµ(R1) uν(R2) uλ(R3) uρ(R4) Dµ ν λ ρ(R1, R2, R3, R4)

(1958)

where u(R) is the displacement from the equilibrium position R.

(i) The sites of the expanded lattice are defined by

R = ( 1 + ε ) R (1959)

Show that if an expansion is made about the sites of the expanded lattice, thenthe dynamical matrix is changed to

Dµ,ν(R−R′) = Dµ,ν(R−R′) + ε δDµ,ν(R−R′) (1960)

where the change in the dynamical matrix is given by

δDµ,ν(R−R′) =∑λ,R”

Dµ ν λ(R,R′, R”) Rλ” (1961)

(ii) Show that the Gruneisen parameter is given by

γα(q) =εα(q) δD(q) εα(q)

6 M ω2α(q)

(1962)

575

——————————————————————————————————

576

15 Phonon Measurements

The spectrum of phonon excitations in a solid can be measured directly, viainelastic neutron scattering or Raman scattering of light.

15.1 Inelastic Neutron Scattering

The neutrons interact with the atomic nuclei by a very short-ranged contactinteraction

Hint =∑

i

2 π h2

mnb δ3( r − r(Ri) ) (1963)

where r is the position of the neutron, and r(Ri) are the positions of the ions.The inelastic neutron scattering cross-section contains information about theground state and all the excited states of the lattice. The various contributionsto the spectrum are analyzed by use of the conservation laws.

In inelastic neutron scattering experiments, the incident neutron energy isgiven by

E =P 2

2 mn=

h2 k2

2 mn(1964)

and the final energy is given by

E′ =P ′2

2 mn=

h2 k′2

2 mn(1965)

The energy transfer from the neutron to the sample is given by

h ω = E − E′ (1966)

The law of conservation of energy demands that this energy is transferred tothe phonon excitations. The conservation law is written as

h ω =∑q′,α

h ωα(q′) ( n′q′,α − nq′,α ) (1967)

where nq,α is the number of phonons initially in the mode with wave vector andpolarization (q′, α) in the initial state and n′q′,α is the number in the final state.

Due to the periodic translational invariance of the crystal, the neutron trans-fers momentum to the phonon modes and the crystal. The principle of conser-vation of momentum yields

P − P ′ = h k − h k′ =∑q′,α

h q′(n′q′,α − nq′,α

)+ h Q (1968)

577

where Q is a reciprocal lattice vector. Thus, even if the scattering is elastic, theneutron may still be diffracted. The use of the two conservation laws allows thedispersion relation ωα(q′) to be determined.

15.2 The Scattering Cross-Section

The inelastic scattering cross-section d2σdΩdω depends on the scattering geometry

through the scattering angle θ and dΩ. The solid angle dΩ is the angle subtendedby the detector to the target. The inelastic neutron scattering cross-section canbe calculated from the Fermi Golden rule for the neutron scattering rate. TheFermi Golden rule expression for the rate at which the neutron is scattered fromstate k to state k′ is given by

1τk→k′

=2 πh

∣∣∣∣ < k∏q′,α

nq′,α | Hint | k′∏q′,α

n′q′,α >

∣∣∣∣2 δ(hω +∑q′,α

h ωα(q′)(nq′,α−n′q′,α))

(1969)where the phonon occupation numbers in the initial state are given by nq′,α andare n′q′,α in the final state. The energy conserving delta function can be writtenin terms of the integral

δ

(hω +

∑q′,α

h ωα(q′)(nq′,α−n′q′,α))

=∫ ∞

−∞

dt′

( 2 π h )exp

[− i t′

(ω +

∑q′,α

ωα(q′)(nq′,α−n′q′,α)) ]

(1970)The exponential factor involving the energy difference between the initial andfinal state can be associated with one factor of the matrix elements of theinteraction Hamiltonian. Thus, the transition rate can be interpreted as anintegral over time of a two-time correlation function

1τk→k′

=1h2

[ ∫ ∞

−∞dt′ <

∏q′,α

nq′,α | < k | Hint | k′ > |∏q′,α

n′q′,α > ×

<∏q′,α

n′q′,α | < k′ | exp[

+i t′

hH0

]Hint exp

[− i t′

hH0

]| k > |

∏q′,α

nq′,α >

](1971)

The interaction Hamiltonian Hint is expressed in the interaction representationas a time-dependent operator, via

Hint(t′) = exp[

+i H0 t

′

h

]Hint(0) exp

[− i H0 t

′

h

](1972)

Hence, the scattering rate is expressed as

1τk→k′

=1h2

[ ∫ ∞

−∞dt′ <

∏q′,α

nq′,α | < k | Hint(0) | k′ > |∏q′,α

n′q′,α > ×

578

× <∏q′,α

n′q′,α | < k′ | Hint(t′) | k > |∏q′,α

nq′,α >

](1973)

The matrix elements involve the initial and final states each of which are prod-ucts of the neutron states k and the states of the lattice. The initial and finalstates of the lattice are defined by the number of quanta in each normal modeand are written, respectively, as |

∏q′,α nq′,α > and |

∏q′,α n′q′,α >. The

above expression involves the matrix elements of the interaction between theneutron and the nuclei in the solid, but unlike the elastic scattering cross-sectionderived previously, the nuclei may be displaced from their equilibrium positionsby ui according to

r(Ri) = Ri + ui (1974)

Hence, the interaction Hamiltonian is given by

Hint(t′) =∑

i

2 π h2 b

mnδ3( r − Ri − ui(t

′) ) (1975)

The matrix elements of the interaction between the initial and final states ofthe neutron, respectively, labelled by momentum k and k′, are given by

< k′ | Hint(t′) | k > =∑

i

2 π h2 b

mn Vexp

[i (k − k′) . (Ri + ui(t

′)) − i ω t′](1976)

Hence, the scattering rate is given by

1τk→k′

=(

2 π h bmn V

)2 ∑i,j

exp[i (k − k′) . (Ri −Rj)

] [ ∫ ∞

−∞dt′ exp

[− i ω t′

]

× <∏q′,α

nq′,α | exp[− i (k − k′) . uj(0)

]|∏α,q′

n′q′,α >

× <∏q′,α

n′q′,α | exp[i (k − k′) . ui(t

′)]|∏q′,α

nq′,α >

](1977)

Since the final states of the phonon modes are not measured, the final statesmust be summed over. On summing over all the final states of the phononmodes, one obtains the expression for the transition rate

1τk→k′

=(

2 π h bmn V

)2 ∑i,j

exp[i (k − k′) . (Ri −Rj)

] [ ∫ ∞

−∞dt′ exp

[− i ω t′

]

× <∏q′,α

nq′,α | exp[− i (k − k′) . uj(0)

]exp

[i (k − k′) . ui(t

′)]|∏q′,α

nq′,α >

](1978)

579

The initial occupations of the phonon modes (q′, α) are to be thermally averagedover. Since in the harmonic approximation the exponential Boltzmann factorfactorizes, the thermal average for each phonon mode can be performed inde-pendently. The thermal average over the mode (q′, α) corresponds to the sumover the occupation number nq′,α with the Boltzmann probability pq,α where

pq,α(nq′,α) =1

Zα(q′)exp

[− β h ωα(q′) nq′,α

](1979)

The normalization of the probability is given by the partition function for thephonon mode (q′, α)

Zα(q′) =∞∑

nq′,α=0

exp[− β h ωα(q′) nq′,α

]

=1

1 − exp[ − β h ωα(q′) ](1980)

To simplify notation, the thermal average of an arbitrary operator A is denotedby angular brackets

< | A | > =∏

q1,α1

( ∞∑nq

1,α1=0

pq1,α1

)<∏q,α

nq,α | A |∏q′,α′

nq′,α′ > (1981)

The thermal averaged transition rate is given by

1τk→k′

=(

2 π h bmn V

)2 ∑i,j

exp[i (k − k′) . (Ri −Rj)

] ∫ ∞

−∞dt′ exp

[− i ω t′

]

× < | exp[− i (k − k′) . uj(0)

]exp

[i (k − k′) . ui(t

′)]| >

(1982)

This transition rate is to be related to the inelastic scattering cross-section.

The inelastic scattering cross-section is defined in terms of the flux F , thescattering rate and the density of final states

ρdΩ(k′) dk′ dΩ =V

( 2 π )3k′2 dk′ dΩ (1983)

via

Fd2σ

dΩdωdΩ dω =

1τk→k′

ρdΩ(k′) dk′ dΩ (1984)

The flux, for a beam with a density of one neutron per volume V , is expressedby

F =h k

V mn(1985)

580

The density of the neutron’s final states is given by

ρdΩ(k′) dk′ dΩ =mn V

( 2 π )3 hk′ dω dΩ (1986)

Hence, the neutron’s scattering cross-section is given by

d2σ

dΩdω=

k′

k

(mn V

2 π h

)2 12 π τk→k′

=k′

kb2 S(k − k′, ω) (1987)

where S(q, ω) is the structure factor. The structure factor is given by the Fouriertransform of the time-dependent correlation function

S(q, ω) =∫ ∞

−∞

dt

2 πexp

[− i ω t

]S(q, t) (1988)

where the correlation function depends exponentially on the lattice displace-ments

S(q, t) =∑i,j

exp[i q . Ri,j

]< | exp

[− i q . uj(0)

]exp

[+ i q . ui(t)

]| >

(1989)This is recognized as the spatial Fourier transform of the nuclear density-densitycorrelation function. The above expression relating the inelastic scattering cross-section to the structure factor was first derived by van Hove204.

It is to be assumed that the displacements ui are sufficiently smaller thanthe lattice spacing so that the correlation function can be expanded in powersof ui.

= exp[− i q . uj(0)

]exp

[+ i q . ui(t)

]= 1 − i q . ( uj(0) − ui(t) )

− 12!

(( q . uj(0) )2 + ( q . ui(t) )2 − 2 ( q . uj(0) ) ( q . ui(t) )

)+ . . .

(1990)

The lattice displacements are then to be expressed in terms of the phononcreation and annihilation operators, and the thermal average of the terms willbe evaluated. The displacement operators are expressed in terms of the phononcreation and annihilation operators via

ui(t′) =

1√N

∑q′,α

√h

2 M ωα(q′)εα(q′) exp

[− i q′ . Ri

]×

204L. van Hove, Phys. Rev. 95, 249 (1954).

581

×(a−q′,α exp[ − i ωα(q′) t′ ] + a†q′,α exp[ + i ωα(q′) t′ ]

)(1991)

where εα(q′) are the phonon polarization vectors. The expectation values ofthe various powers of u in the expansion can be characterized by the numberof phonons that are excited in the scattering process. Since the displacementoperator either creates or destroys a phonon, the only non-zero terms in theexpansion involve even powers of u.

15.2.1 The Zero-Phonon Scattering Process

The zero-th order term in the expansion of the structure factor corresponds tothe term of unity in eqn(1990). The zero-th order contribution to the structurefactor is given by

S0(q, ω) =∑i,j

exp[i q . Ri,j

] ∫ ∞

−∞

dt

2 πexp

[− i ω t

](1992)

The integration over t can be performed leading to a zero-frequency delta func-tion

S0(q, ω) =∑i,j

exp[i q . Ri,j

]δ( ω ) (1993)

expressing the elastic scattering condition. This yields the dominant contribu-tion to the elastic scattering cross-section

d2σ

dΩdω= b2

∑i,j

exp[i q . Ri,j

]δ( ω ) (1994)

since k = k′ for elastic scattering. On integrating over an infinitesimal fre-quency interval ω, around ω = 0, one finds a contribution to the elasticscattering cross-section given by

dσ

dΩ=

∫ ε

−ε

dωd2σ

dΩdω

= b2∑i,j

exp[i q . Ri,j

](1995)

If q corresponds to a reciprocal lattice vector Q, then the scattering is coherentand the intensity of the Bragg peak is proportional to N2. As we shall see later,the intensity of the elastic scattering peak is reduced by quantum and thermalvibrations of the lattice. The intensity of the elastic peak is reduced by theDebye-Waller factor.

582

15.3 The Debye-Waller Factor

There are other terms in the expansion in eqn(1990) which are also independentof time and, hence, lead to contributions to the inelastic scattering spectrum.For example, the expectation value of the two terms

− 12!

(( q . uj(0) )2 + ( q . ui(t) )2

)(1996)

in eqn(1990) are time independent. On utilizing the expression for the latticedisplacements in terms of the phonon creation and annihilation operators, onefinds that the thermal average involves expectation values such as

< | ( a†q1,α1

+ a−q1,α1 ) ( a†q

2,α2

+ a−q2,α2 ) | > (1997)

The terms involving the product of two phonon creation operators or two phonondestruction operators are identically zero. The remaining terms involve theproduct of a phonon creation and annihilation operator. These terms are non-zero, only if q

1= − q

2and α1 = α2, in which case they can be expressed

directly in terms of the phonon occupation numbers. The non-zero contributionis given by

δq1+q

2δα1,α2

(nq

1,α1 + ( 1 + n−q

1,α1 )

)(1998)

Hence, the expectation value of the term proportional to ( q . uj(0) )2 is evalu-ated as

− 12 N

∑q1,α1

h

2 M ωα1(q1)

(q . εα1(q1)

)2 (1 + 2 N(ωα1(q1))

)(1999)

where the thermal average of the phonon occupation numbers have been re-placed by the Bose-Einstein distribution function N(ω). Due to the time andspatial homogeneity of the system, the expectation value of ( q . ui(t) )2 isidentical to that of ( q . uj(0) )2. The two second-order time independentcontributions can be combined with the zero-th order term 1, to give the firsttwo terms in the expansion of the elastic scattering cross-section. The inelasticscattering cross-section can be written as

d2σ

dΩdω= b2

∑i,j

exp[i q . Ri,j

]δ( ω ) W (q) (2000)

where W (q) is of the form

W (q) = 1 − 1N

∑q1,α1

h

2 M ωα1(q1)

(q . εα1

(q1))2 (

2 N(ωα1(q1)) + 1)

+ . . .

(2001)

583

The series expansion for W (q) can be exponentiated to yield the expression forthe Debye-Waller factor

W (q) = exp[− 1

N

∑q1,α1

h

2 M ωα1(q1)

(q . εα1

(q1))2

cothβ h ωα1(q1)

2

](2002)

The Debye-Waller factor represents the reduction of the intensity of the Braggpeak due to the loss of coherent scattering caused by the displacement of the nu-clei from their equilibrium positions205. The Debye-Waller factor represents thecombined effect of thermal206 and quantum fluctuations207 of the nuclei. Theintensities of the Bragg peaks are usually reduced on increasing temperature.However, if the solid undergoes a structural instability, the phonon dispersionrelation may soften as the temperature is lowered and the instability is ap-proached208. The softening of the phonon modes may lead to a reduction inthe Debye-Waller factor with decreasing temperatures. The form of the expo-nent is similar to the factor appearing in the Lindemann criterion for melting,however, instead of depending on the inverse square of the lattice constant, theDebye-Waller factor is proportional to the square of the momentum transferq = k − k′. Due to the dependence on the momentum transfer, the Debye-Waller factor is expected to preferentially reduce the intensity of the Bragg peakswith large Q, while the intensity of the low Q Bragg peaks are not expected tochange appreciably. The Debye-Waller factor also modifies the intensity of theBragg peak in x-ray scattering.

The Debye-Waller factor multiplies the intensities of the multi-phonon pro-cesses, of all orders. This can be ascertained by examining the correlationfunction in the structure factor

< | exp[− i q . uj(0)

]exp

[+ i q . ui(t)

]| > (2003)

For harmonic phonons, one can express the correlation function as

< | exp[− i q . uj(0)

]exp

[+ i q . ui(t)

]| >

= exp[

+ < |(q . uj(0)

) (q . ui(t)

)| >

]×

× exp[− 1

2< |

(q . uj(0)

)2

| >]

exp[− 1

2< |

(q . ui(t)

)2

| >]

205In the case of anomalous scattering in Ge caused by the accumulation of charge associatedwith bonding, the intensity of the forbidden Bragg reflections at 2π

a(2, 2, 2) is not governed

by the Debye-Waller factor. The intensity of this peak decreases more rapidly with increasingtemperature, than is predicted by the Debye-Waller factor. (J. B. Roberto, B. W. Battermanand D. J. Keating, Phys. Rev. B 9, 2599 (1974).)206P. Debye, Ann. Phys. (Leipzig) 43, 49 (1914) see also I. Waller, Z. Phys. 17, 398 (1923).207H. Ott, Ann. Phys. (Leipzig), 23, 169 (1934).208G. Shirane, Rev. Mod. Phys. 46, 437 (1974).

584

(2004)

The proof of this identity proceeds with the aid of the Baker-Hausdorff theoremand can be found in the review article of A. A. Maradudin, E. W. Montroll,G. H. Weiss and I. P. Ipatova, Solid State Phys. Suppl. 3, 307 (1971). Theproduct of the last two factors is identified with the Debye-Waller factor, whichis given by

W (q) = exp[− < |

(q . ui(0)

)2

| >]

= exp[− 1

N

∑q′,α

h

2 M ωα(q′)

(q . εα(q′)

)2

cothβ h ωα(q′)

2

](2005)

The time-dependent term in eqn(2004) can be expanded in powers of the ex-pectation values of the correlation of the displacements

= W (q) exp[< | ( q . uj(0) ) ( q . ui(t) ) | >

]= W (q)

(1 + < | ( q . uj(0) ) ( q . ui(t) ) | > + . . .

)(2006)

Apart from the first, all the terms in this expansion have a non-trivial expo-nential time dependence and, thus, do not contribute to the elastic scattering.Therefore, all the contributions to the scattering cross-section are reduced inintensity by the Debye-Waller factor. The first term in the expansion is foundto be proportional to δ(ω) and gives rise to the elastic scattering. The secondterm is just the one-phonon contribution to the scattering cross-section.

15.3.1 The One-Phonon Scattering Processes.

The one-phonon creation and annihilation contribution to the scattering cross-section originates from the term

W (q) < | ( q . uj(0) ) ( q . ui(t) ) | > (2007)

in the expansion of eqn(2004). On expressing the displacements uj(0) andui(t) in terms of the phonon creation and annihilation operators, eqn(2007)is evaluated as

= W (q)∑q′,α

h

2 N M ωα(q′)

(q . εα(q′)

)2

×

585

×[

[ nq′,α + 1 ] exp[− i ( q′ . Ri,j − ωα(q′) t )

]+ nq′,α exp

[+ i ( q′ . Ri,j − ωα(q′) t )

] ](2008)

Hence, the one-phonon contribution to the inelastic scattering cross-section isfound to be

d2σ

dΩdω=

k

k′b2 W (q)

∑q′,α

h

2 M N ωα(q′)

(q . εα(q′)

)2

×

×[

[ nq′,α + 1 ] δ( ω − ωα(q′) )∑i,j

exp[i ( q − q′ ) . Ri,j

]

+ n−q′,α δ( ω + ωα(q′) )∑i,j

exp[i ( q − q′ ) . Ri,j

] ](2009)

The displacements have been expressed in terms of the normal modes, andnq′,α is just the number of phonons with wave vector q′ and polarization α inthe initial state. On performing the sum over lattice vectors Ri, one finds thecondition for conservation of crystal momentum,∑

i

exp[i ( q − q′ ) . Ri

]= N ∆q−q′ (2010)

modulo Q. Thus, the summation over q can be trivially performed. Then,the second and third terms involve the absorption or emission of a phonon ofwave vector q′ = (q +Q) where Q is a reciprocal lattice vector. These termsare smaller than the coherent Bragg terms by a factor of 1

N . The inelasticone-phonon contributions are coherent as they involve the conservation of mo-mentum, but have intensities that are only proportional to N .

The two contributions to the one-phonon scattering cross-section correspondto different signs of the energy transfer ω. The contribution which is propor-tional to the Bose-Einstein distribution function corresponds to a negative en-ergy transfer.

N(ωα(q′)) δ( ω + ωα(q′) )∑i,j

exp[i ( q + q′ ) . Ri,j

](2011)

It represents processes in which a phonon, that is thermally excited in the initialstate, scatters with the neutron and is subsequently destroyed. The factorN(ωα(q′)) represents the probability that a phonon is thermally excited in the

586

initial state. The other contribution corresponds to a positive energy transferand has the form

[ N(ωα(q′)) + 1 ] δ( ω − ωα(q′) )∑i,j

exp[i ( q − q′ ) . Ri,j

](2012)

It represents processes whereby the neutron interacts with the solid and emitsa phonon. The term proportional to unity corresponds to the spontaneousemission process, whereas the term proportional to N(ωα(q′)) represents thestimulated emission process. The presence of a phonon in the mode (q′, α) en-hances the probability that subsequent phonons will be created in that mode.

At low temperatures, the number of thermally-activated phonons is small,therefore, the inelastic scattering intensity for processes which lead to an increasein the energy of the neutron due to absorption of phonons is small. On the otherhand, the intensity of processes which involve the energy loss by the neutronbeam due to creation of individual phonons has an intensity governed by the1 + N(ωα(q′)) which is almost unity at low temperatures. The rate for inelastictransitions of the incident neutrons obeys the principle of detailed balance. Thatis, although the neutron beam is not in equilibrium with the solid, the transitionrate is such that it drives the beam towards equilibrium. This can be seen byinspection of the one-phonon contribution to the spectrum. The one-phononabsorption and emission spectrum is proportional to

[ 1 + N(ωα(q′)) ] δ( E − E′ − h ωα(q′) ) + N(ωα(q′)) δ( E − E′ + h ωα(q′) )(2013)

The first term represents processes in which the neutron loses energy due to theemission of a phonon, whereas the second term represents processes in whichthe neutron gains energy due to the absorption of a phonon. The ratio of therate at which the neutron beam gains energy to the rate at which the neutronbeam loses energy is given by

W (E → E + hω)W (E + hω → E)

=N(ω)

[ N(ω) + 1 ]= exp

[− β h ω

](2014)

If equilibrium with the beam were to be established, the kinetic energy of theneutron beam would be distributed according to the Boltzmann formula

P (E) =1Z

exp[− β E

](2015)

such that dynamic equilibrium would be established. In this case, the totalnumber of transitions from E → E + h ω precisely equals the number oftransitions in the reverse direction E + h ω → E

P (E) W (E → E + hω) = P (E + hω) W (E + hω → E) (2016)

However, the beam produced by the neutron source is not in equilibrium withthe sample, and would only equilibrate if the beam traverses an infinite path

587

Figure 242: The phonon dispersion relation inferred from inelastic neutron scat-tering experiments on f.c.c. Cu. [After Svensson et al., Phys. Rev. 155, 619(1967).]

length through the sample.

The phonon dispersion relation can be inferred from a measurement of thesingle-phonon scattering peak209. The scattering cross-section for processes inwhich a single phonon is emitted have to satisfy the energy and momentumconservation laws

h2 k2

2 mn=

h2 k′2

2 mn+ h ωα(q′) (2017)

andk = k′ + q′ + Q (2018)

since ω(q′) is periodic with a periodicity of the reciprocal lattice vectors

ωα(q′) = ωα(q′ +Q) (2019)

One can combine the equations as

h2 k2

2 mn=

h2 k′2

2 mn+ h ωα(k − k′) (2020)

In the scattering experiments, the beam of neutrons is generally collimated tohave a definite direction of the k vector, and also to have a definite initial energy.For a given k, the solution of the above equation for the three components of k′

form a two-dimensional surface. For a detector placed in a particular scatteringdirection, the solution only exists at isolated points. On measuring the scat-tering cross-section at the various magnitudes of the final momentum, k′, onefinds sharp peaks in the spectrum. In general, these peaks occur at differentwave vectors than the Bragg peaks. With knowledge of the magnitude of the209R. Weinstock, Phys. Rev. 65, 1 (1944), A. D. B. Woods, B. N. Brockhouse, R. A. Cowley

and W. Cochran, Phys. Rev. 131, 1025 (1963).

588

Figure 243: The single phonon peak in the frequency distribution of the scatter-ing cross-section of KBr, at constant momentum transfer. The intensity of thesingle phonon peak is temperature dependent, as is the intensity of the smoothmulti-phonon background. [After Woods et al., Phys. Rev. 131, 2025 (1963).]

final momentum k′, one can construct k′ − k, and also E′ −E and, hence, findh ωα(q′) for the normal mode. By varying the direction of k′ and the magnitudeof E, one can map out successive surfaces and, therefore, obtain the dispersionrelation.

h2k'2/2mn

h2k2/2mn - hω(k-k')

k=k'+q+Q

k'=k

k'0

Figure 244: A graphical solution of the conditions of conservation of energy andmomentum for a one-dimensional lattice, in which a neutron interacts with thesolid and emits a phonon.

Information about the polarization of the phonon modes can be obtainedfrom the dependence of the intensity on the scattering wave-vector k−k′ as thescattering cross-section is proportional to∣∣∣∣ ( k − k′ ) . εα(q′)

∣∣∣∣2 (2021)

589

The width of the single-phonon peak obtained in experiments have two ori-gins, one is the experimental resolution and the other component is not reso-lution limited. The second component is due to the lifetime of the phonon, τ .The phonon lifetime, according to the energy-time uncertainty principle, givesrise to an energy width of h

τ . The lifetime occurs because the phonons arescattered either by anharmonic processes or by electrons. The small magnitudeof the width of the phonon peaks attests to the effectiveness of the harmonicapproximation and the Born-Oppenheimer approximation.

15.3.2 Multi-Phonon Scattering

Processes in which two phonons are absorbed or emitted satisfy the two conser-vation laws

h2 k2

2 mn=

h2 k′2

2 mn± h ωα1(q1) ± h ωα2(q2) (2022)

andk = k′ ± q

1± q

2+ Q (2023)

Conservation of momentum can be used to express q2

in terms of q1. Conser-

vation of momentum gives rise to the restriction

h2 k2

2 mn=

h2 k′2

2 mn± h ωα1(q1) ± h ωα2(k − k′ ± q

1) (2024)

Since there are six quantities k′ and q1

and only one remaining constraint, thevalues of k’ and q

1are still undetermined. Even if the direction of k′ is fixed in

an experiment, there still remains three unknown quantities q1. The variation

k'k'=k

h2k'2/2mn

h2k2/2mn + hω(k'-k)

0

k+q'=k'+QPhonon absorption

h2k'2/2mn = h2k2/2mn + hω(k'-k)

Figure 245: A graphical solution of the conditions of conservation of energyand momentum for a one-dimensional lattice, in which a phonon present in theinitial state is absorbed by a neutron.

590

Q Q

k-k'k-k'

q'q'

k+q'=k'+Q

εT

εT

O O

Figure 246: The sensitivity of the inelastic neutron scattering intensity to thepolarization vector εT for a transverse phonon of wave vector q′. In the scatter-ing process a transverse phonon present in the initial state is absorbed by theneutron.

0

0.01

0.02

0.03

0.04

0.05

0.06

Γ ΓX XK W

(1,0,0)(1,1,0) (1,1/2,0)(3/4,3/4,0)

Figure 247: Calculated phonon dispersion relations for an f.c.c. solid, showingthe periodicity of the reciprocal lattice along the XWX high-symmetry direc-tion.

of q1

produces a continuously varying final neutron energy. Hence, one obtainsa continuous spectrum. A similar analysis of the higher order multi-phononprocesses also yields a continuous spectrum. Only the one-phonon spectrumgives rise to a single peak.

Thus, in a general coherent scattering experiment with a specific scatteringdirection, the analysis of the scattered neutrons energy provides a spectrumwhich contains a continuous portion superimposed with sharp peaks. The spec-trum may show an elastic Bragg peak depending on the magnitude of k andθ, or if there is isotopic disorder, one may observe incoherent nuclear scatter-ing at zero-energy transfer. The peaks of the one-phonon scattering can beused to map out the dispersion relations. This has been performed for f.c.c.lead. However, some branches were not observed. The intensity of the one-phonon absorption peak is proportional to the Bose-Einstein distribution func-tion N(ωα1(q1)), whereas the one-phonon emission process has intensity pro-

591

portional to [ N(ωα1(q1)) + 1 ]. Thus, it is usual to measure phonon emissionat low temperatures.

The phonon density of states can be obtained directly, if the incoherentscattering is sufficiently strong. In this case, the isotopic disorder producesscattering from individual ions. The resulting incoherent scattering is, there-fore, proportional to the nuclear density-density correlation function evaluatedat the same lattice site. Hence, the incoherent one-phonon scattering processaverages over all the phonon wave vectors. A measurement of the incoherentinelastic scattering spectrum provides a direct measure of the phonon densityof states210.

——————————————————————————————————

15.3.3 Exercise 87

(i) Find a graphical description of the conservation laws for the phonon emissionprocess.

(ii) Show that there is a minimum or threshold energy required for phonon emis-sion.

——————————————————————————————————

210A. T. Stewart and B. N. Brockhouse, Rev. Mod. Phys. 30, 250 (1958), B. N. Brockhouse,Can. J. Phys. 33, 889 (1953).

ez

ex

ey

1/2(1,1,1)

(1,0,0)(1,-1,0)

Figure 248: The XWX high-symmetry line, in the extended zone scheme. Sincethe point X = (1, 1, 0) is equivalent to X = (1, 0, 0), the dispersion relations foran f.c.c. crystal must be symmetric along the XWX line.

592

15.3.4 Exercise 88

(i) Evaluate the Debye-Waller factor for a one, two or three dimensional systemof acoustic phonons.

(ii) Determine the temperature dependence of the integrated intensity of thescattering cross-section, defined by

I(q) =∫ +∞

−∞dω

k′

k

d2σ

dωdΩ(2025)

——————————————————————————————————

15.3.5 Exercise 89

Consider inelastic neutron scattering from a perfect fluid, described by theHamiltonian

H0 =∑

i

P2

i

2 M(2026)

Show that the inelastic scattering cross-section is proportional to

d2σ

dωdΩ∝(

β M

2 π h2 q2

) 12

exp[− β M

2 h2 q2

(h ω − h2q2

2 M

)2 ](2027)

——————————————————————————————————

15.4 Raman and Brillouin Scattering of Light

Since the energy of visible light is of the order of eV and the energy of a typicalphonon is of the order of meV, ( 10−3 eV), it is not possible to observe phononsby direct absorption or emission of light. However, it is possible to observe thephonons in a solid via light scattering. Even though the scattering processesproceed via the same mechanism, the scattering from optical phonons is calledRaman scattering211 and scattering from acoustic phonons is called Brillouinscattering212.

As in neutron scattering, the basic process may involve emission of phononsor absorption of phonons. The conservation laws for the one-phonon absorptionor emission processes are conservation of energy

h ω′ = h ω ± h ωα(q) (2028)

211C. V. Raman, Nature, 121, 619 (1928).212L. Brillouin, Ann. de Phys., (Paris), 17, 88 (1922).

593

Figure 249: The Stokes and anti-Stokes lines in Si as observed in Raman scat-tering experiments. The measurements were performed at temperatures of 20,460 and 770 K. Note the temperature dependence of the ratio of the intensitiesof the Stokes and anti-Stokes lines. [After T. R. Hart, R. L. Aggarawal and B.Lax, Phys. Rev. B 1, 638 (1970).]

and conservation of momentum

h k′ n = h k n ± h q + h Q (2029)

In these expressions (k, ω) and (k′, ω′) are, respectively, the momentum andenergy of the incident beam of photons and the scattered photons, and n is therefractive index of the media. The refraction index reflects the change in thewavelength of the light as it enters the solid. The phonon absorption process (+)gives rise to the anti-Stoke’s shifted line, which has an intensity proportional tothe number of activated phonons

∝ N(ωα(q)) (2030)

The phonon emission process (−) gives rise to the Stoke’s line which has anintensity proportional to

∝ [ 1 + N(ωα(q)) ] (2031)

as it has contributions from spontaneous and stimulated emission of phonons.

The Raman scattering process allows parts of the phonon dispersion relationsto be mapped out. Since the characteristic phonon frequency is given by theDebye frequency h ωD ∼ 10−2 eV, which is small compared with a typicalphoton energy h c k n ∼ 1 eV, the change in photon wave vector k − k′ is

594

θ

k

k'

q

Figure 250: The relation between the scattering angle θ, the initial and finalwave vector of the light and the phonon wave vector q. The magnitudes of thewavevectors k and k′ of the incident and scattered light are almost equal.

small. Therefore, the triangle formed by the initial and final wave vectors isalmost isosceles. The momentum transfer q is given by

| q | = 2 n k sinθ

2

= 2 nω

csin

θ

2(2032)

Since the direction of k and k′ are known from the experimental geometry, thedirection of q can be inferred. Thus, the phonon momentum momentum andthe phonon energy are known, if the small change in the photon energy, h ∆ω,is measured.

For Brillouin scattering, the phonon energy is given by

ωα(q) = vα q (2033)

where vα is the velocity of sound. The magnitude of the phonon’s momentumis given by

q =ωα(q)vα(q)

= 2ω n

csin

θ

2(2034)

However, the energy of the acoustic phonon is equal to the change in photonenergy, ∆ ω,

ωα(q) = ∆ω (2035)

Thus, the velocity of the acoustic phonon is found as

vα(q) =∆ω2 ω

c

ncsc

θ

2(2036)

The experimentally determined spectra has the form of a strong un-scatteredlaser line, surrounded by a small anti-Stoke’s line at higher frequencies, and aslightly more intense Stoke’s line at lower frequencies. The Stokes and anti-Stoke’s line are both separated from the main line by the same frequency shift∆ω. This technique can be used to determine the phonon frequencies. Thewidths of the phonon peaks provide a measure of the imaginary part of the

595

dielectric constant.

The quantum mechanical theory of the Raman effect was first formulatedby Loudon213. Loudon emphasized the role of electrons in mediating the Ra-man scattering. In particular, Loudon noted that although there are Ramanscattering processes which do not involve electrons, their intensities are negli-gible unless they are nearly resonant in which case the photon frequency mustbe comparable to the phonon frequency. Since typical phonon frequencies areof the order of meV, this process will be limited to the far infrared region.For typical ranges of photon frequencies, Raman scattering is dominated byelectron-assisted processes.

The Raman scattering cross-section is calculated with third-order time-dependent perturbation theory, with two powers of the paramagnetic electron-photon interaction (i.e., only the A . p terms) and one power of the electron-phonon interaction. In the Raman process, a photon is absorbed by the solidcreating a virtual electron-hole pair. The pair either emits or absorbs a phonon.The pair recombines by emitting a photon. The transition amplitude consists ofsix terms corresponding to the various possible time-orderings of the three indi-vidual processes. The transition amplitude is conventionally written in terms ofthe Raman tensor. The scattering cross-section is proportional to the squaredmodulus of the Raman tensor. The Raman tensor is both frequency and wavevector-dependent, it is symmetric if the phonon frequency is negligible comparedwith the frequency of the incident light. The Raman tensor involves the sumsof products of matrix elements. One factor comes from the matrix elements ofthe electron’s momentum along the direction of the initial photon’s polarizationand a second factor involves matrix elements of the electrons momentum alongthe polarization of the final photon. The third and last factor represents thematrix elements of the interaction Hamiltonian between electronic states whena phonon with a specific polarization is present.

If one neglects the difference between the wavelengths of the incident andscattered scattered light, then there are selection rules for the phonon modesthat can be Raman scattered. Loudon has examined the form of the Ramantensor for the different point groups. In crystals which do have a center of in-version symmetry, only even-parity phonons are Raman active. The odd-parityphonons are, however, infra-red active and can be observed in optical absorp-tion measurements. From the symmetry, it is seen that the lowest order Ramantransitions are forbidden if each lattice site is an inversion center. Hence, NaCldoes not exhibit first order Raman scattering214 but diamond does215.

Loudon’s analysis assumes the existence of a virtual electron-hole pair in the213R. Loudon, Proc. Roy. Soc. A 275, 218 (1963), R. Loudon, Adv. in Phys. 13, 423

(1964).214M. Born and M. Bradburn, Proc. Roy. Soc. A 188, 161 (1947).215H. M. J. Smith, Phil. Trans. Roy. Soc. A 241, 105 (1948).

596

time

space

(ω,k)

(ω',k') (ω',k')

(ω,k)

(ω(q),q) (ω(q),q)

(ω,k) (ω,k)

(ω,k)(ω,k)

(ω',k') (ω',k')

(ω',k')(ω',k')

(ω(q),q) (ω(q),q)

(ω(q),q) (ω(q),q)

Figure 251: A diagramatic depiction of the virtual processes involved in one-phonon Raman scattering. The incident photon (ω, k) and the scattered photon(ω′, k′) are depicted by red wavy lines. We have depicted a process in which aphonon (ω(q), q) is emitted by an electron (or a hole). Since the process is avirtual process, it is only necessary for energy to be conserved in the initial andfinal states, and not in the intermediate states.

597

intermediate states, and needs modification when the scattering is in resonancewith the intermediate states216. In such cases, the resonance can enhance theRaman scattering by factors which can be as large as 102, as is found in CdS 217.

216J. L. Birman and A. K. Ganguly, Phys. Rev. Lett. 17, 647 (1966), D. L. Mills and E.Burstein, Phys. Rev. 188, 1465 (1969).217R. C. Leite and S. P. S. Porto, Phys. Rev. Lett. 17, 10 (1966).

598

16 Phonons in Metals

An alternate approach to the phonon dispersion in metals is based on a two-component plasma composed of electrons and ions. The approach starts byconsideration of a plasma composed of the positively charged ions with chargeZ | e | and mass M . The plasma of ions support longitudinal charge densityoscillations which occur in the absence of an external potential. Since the totalpotential is related to the external scalar potential via

φ(q, ω) =φext(q, ω)ε(q, ω)

(2037)

then, if φext(q, ω) = 0, one must have ε(q, ω) = 0 for φ(q, ω) 6= 0. Inthis case, one has spontaneous density fluctuations and an induced longitudinalcurrent. On using Poisson’s equation and the condition of continuity of thecharge density, one finds the induced longitudinal current in the form

jL(q, ω) =q ω

4 π

[φ(q, ω) − φext(q, ω)

]

=q ω

4 π

[1 − ε(q, ω)

]φ(q, ω) (2038)

Using the definition of the longitudinal conductivity, one recovers the expressionfor the dielectric constant

ε(q, ω) = 1 − 4 π σ(ω)i ω

(2039)

The Drude expression for the conductivity of a gas of ions of charge Z | e | andmass M is given by

σ(ω) =Z2 e2 ρions τ

M

11 − i ω τ

(2040)

On substituting the ionic Drude conductivity in the expression for the dielectricconstant of the ions, which on assuming the limit ω 1

τ reduces to

ε(q, ω) = 1 − 4 π Z e2 ρ

M ω2(2041)

where the density of ions is given in terms of the electron density ρ via

ρions =ρ

Z(2042)

The condition for plasmon oscillations is given by

ε(q, ω) = 0 (2043)

599

the solution for ω is defined as the ionic plasmon frequency Ωp. The ionicplasmon frequency may be written in terms of the plasmon frequency

Ω2p =

Z m

Mω2

p (2044)

which is independent of q. The ionic plasmon frequency corresponds to an un-screened phonon frequency. Since the factor Z m

M ∼ 14000 and h ωp ∼ 10 eV,

the unscreened phonon frequency is approximately ∼ 110 eV.

16.1 Screened Ionic Plasmons

The above model is inadequate as it neglects the effects of the conduction elec-trons. This effect of the electrons can be included by screening the Coulombinteractions between the charged nuclei

4 π Z2 e2

q2(2045)

with the dielectric constant of the electron gas. In the Thomas-Fermi approxi-mation, the dielectric constant is given by

εeg(q, ω) = 1 +k2

TF

q2(2046)

Thus, within the Born-Oppenheimer approximation, one obtains the frequency-dependent dielectric constant as

ε(q, ω) = 1 − 4 π Z e2 ρ

M ( 1 + k2T F

q2 ) ω2(2047)

The screened ionic plasmons have frequencies which are given by

ε(q, ω) = 1 − 4 π Z e2 ρ

M ( 1 + k2T F

q2 ) ω2= 0 (2048)

Thus,

ω2 =Z m

Mω2

p

q2

q2 + k2TF

(2049)

This is the Bohm-Staver model of the longitudinal phonons in a metal218. Thismodel results in a linear dispersion relation ω(q) ≈ v q, where the velocity vis defined as

v2 =Z m

M

ω2p

k2TF

(2050)

218D. Bohm and T. Staver, Phys. Rev. 84, 836 (1950).

600

As the Thomas-Fermi wave vector is given in terms of the Fermi wave vector by

4 π e2

k2TF

≈ h2 π

m kF(2051)

and the electron density is expressed as

ρ =k3

F

3 π2(2052)

the velocity of sound is related to the Fermi velocity vF = hm kF via

v2 =13Z m

Mv2

F (2053)

Thus, the velocity of sound v is reduced below the Fermi velocity vF as mM ∼

10−3 − 10−5.

16.1.1 Kohn Anomalies

A more accurate treatment of the phonon frequency replaces the Thomas-Fermidielectric function with the Lindhard expression

εeg(q, ω) = 1 − 4 π e2

q2

∫d3k

4 π3

f(Ek+q) − f(Ek)

Ek+q − Ek + h ω(2054)

where f(x) is the Fermi-Dirac distribution function. For free electrons, thedielectric function has singularities in the derivative at q = 2 kF . Thesesingularities correspond to the extremal diameters of the Fermi surface. WalterKohn showed that these singularities should appear in the phonon spectrum219

by producing kinks or infinities in the derivative(∂ω

∂q

) ∣∣∣∣q=2kF

(2055)

The Kohn anomalies have been observed in some metals by inelastic neutronscattering measurements220. The Fermi surface in Lead has been mapped outby this indirect method221. The results are in fair agreement with the Fermisurface inferred from de Haas - van Alphen measurements222.

219W. Kohn, Phys. Rev. Lett. 2, 393 (1959), E. J. Woll Jr. and W. Kohn, Phys. Rev. 126,1693 (1962).220B. N. Brockhouse, K. R. Rao and A. D. B. Woods, Phys. Rev. Lett. 7, 93 (1961).221R. Stedman, L. Almquist, G. Nilsson and G. Raunio, Phys. Rev. 163, 567 (1967).222J. R. Anderson and A. V. Gold, Phys. Rev. 139, 1459 (1965).

601

16.2 Dielectric Constant of a Metal

The dielectric constant of a metal represents the process in which an externalcharge is screened by the combined effects of the electrons and the ions

φext(q, ω) = φ(q, ω) ε(q, ω) (2056)

A dielectric function can be defined for just the electrons in which the totalpotential φ(q, ω) is produced as the response to a total external potential whichis external to the electron gas. That is, the total external potential is consideredto be the sum of the applied external potential and the total potential due tothe ion charge density

φext(q, ω) + φions(q, ω) = φ(q, ω) εel(q, ω) (2057)

Analogously, a dielectric function can be defined for the ions as the responseof the ions to an external potential composed of the applied potential and theelectrons

φext(q, ω) + φel(q, ω) = φ(q, ω) εions(q, ω) (2058)

This goes beyond the Born-Oppenheimer approximation. The total potential isgiven by the sum of the potentials due to the external, electron and ion charges

φ(q, ω) = φext(q, ω) + φions(q, ω) + φel(q, ω) (2059)

The dielectric constant of the metal is given in terms of the dielectric constant ofthe electrons and the dielectric constant of the ions, by adding the two equationsdefining the electronic and ionic dielectric constants(

εions(q, ω) + εel(q, ω))φ(q, ω) = φ(q, ω) + φext(q, ω) (2060)

Then, with the definition of the total dielectric constant, one has the relation

ε(q, ω) =(εions(q, ω) + εel(q, ω) − 1

)(2061)

The dielectric constant of the ions goes beyond the Born-Oppenheimer approxi-mation. It describes how the ions, alone, screen the potential due to the appliedpotential and the potential due to the electrons. The dielectric constant due tothe ions alone is approximated by

εions(q, ω) = 1 −Ω2

p

ω2(2062)

and the electronic dielectric constant (at low frequencies) is given by the Thomas-Fermi approximation

εel(q, ω) = 1 +k2

TF

q2(2063)

602

Hence, the low-frequency dielectric constant is given by the approximate ex-pression

ε(q, ω) = 1 +k2

TF

q2−

Ω2p

ω2(2064)

for ωp ω.

An alternate definition of the dielectric constant of the ions may be intro-duced in which one considers the external potential to be first screened by theelectron gas. Secondly, the resulting dressed-external potential is screened bythe ions. That is, instead of the electron gas screening the external potential ofthe ions and the applied potential

φ(q, ω) =φext(q, ω)εel(q, ω)

+φions(q, ω)εel(q, ω)

(2065)

one considers only the dressed-external potential

φdressed(q, ω) =φext(q, ω)εel(q, ω)

(2066)

It is this dressed-external potential that is screened by the ions to produce thetotal potential. This relation defines the dressed dielectric constant of the ions

φ(q, ω) =φdressed(q, ω)εdressed

ions (q, ω)

=φext(q, ω)

εel(q, ω) εdressedions (q, ω)

(2067)

Hence, the electronic and dressed-ionic dielectric constants are related to thedielectric constant via

ε(q, ω) = εel(q, ω) εdressedions (q, ω) (2068)

Combining this with the relation of the dielectric constant in terms of dielectricconstants of the electrons and ions

ε(q, ω) =(εions(q, ω) + εel(q, ω) − 1

)(2069)

one finds that the dressed-ionic dielectric constant is defined by

εdressedions (q, ω) =

1εel(q, ω)

(εions(q, ω) + εel(q, ω) − 1

)= 1 +

1εel(q, ω)

(εions(q, ω) − 1

)(2070)

603

The dressed-ionic dielectric constant is calculated as

εdressedions (q, ω) = 1 +

1

1 + k2T F

q2

(εions(q, ω) − 1

)

= 1 − 1

1 + k2T F

q2

(Ω2

p

ω2

)(2071)

This can be written in terms of the phonon dispersion relation ω(q)2

εdressedions (q, ω) = 1 −

ω(q)2

ω2(2072)

since the phonon oscillations occur when the dielectric constant vanishes

εdressedions (q, ω(q)) = 0 (2073)

By inspection of the dressed dielectric constant, the phonon frequency is foundas

ω(q)2 =q2

q2 + k2TF

Ω2p (2074)

The introduction of screening by the electron gas has reduced the frequency ofthe ionic density oscillations from the ionic plasmon frequency to a branch oflongitudinal acoustic phonons. The total dielectric constant, which is a productof the dressed dielectric constant and the Thomas-Fermi dielectric constant ofthe electron gas, can now be written in terms of the phonon frequencies as

1ε(q, ω)

=1

1 + k2T F

q2

1

1 − ω(q)2

ω2

=1

1 + k2T F

q2

ω2

ω2 − ω(q)2

(2075)

This is in agreement with the expression discussed earlier.

16.3 The Retarded Electron-Electron Interaction

Consider the screening of the Coulomb interaction between a pair of electronsvia the dielectric constant

4 πq2

→ 4 πε(q, ω) q2

=4 π

k2TF + q2

(1 +

ω(q)2

ω2 − ω(q)2

)(2076)

604

Thus, there is an additional contribution in the effective interaction due to thescreening by the ions. The ω dependence of the interaction represents the factthat the effective interaction is not instantaneous but instead is a retarded in-teraction223. The retarded nature of the attractive interaction between twoelectrons is caused by the involvement of the polarization of the lattice. Oneelectron produces a polarization of the ions in its vicinity, which evolves ona time scale that is determined by the energy transfer ∼ h ωD. Due to thelarge difference between the Fermi velocity an the speed of sound, the originalelectron will have moved considerable distances before the polarization is fullydeveloped. After the dynamical lattice distortion has been created, a secondelectron is attracted by the remanent of the lattice deformation left behind bythe original electron. The effective interaction between a pair of electrons in-volves a momentum transfer q = k − k′ and energy transfer h ω = Ek − Ek′ .The effective interaction has the following limits:

(i) This interaction reduces to the Thomas-Fermi screened electron-electroninteraction when the electron energy transfer is greater than the typical phononfrequency ωD ∼ Ωp. In this case, when ω > ωD, the phonon correction isunimportant.

(ii) The electron-electron interaction is strongly modified at low frequencies,where ω < ωD. The contribution from the phonons is large and of oppositesign to the direct Coulomb repulsion, and exactly cancels at ω = 0. Theimportant point, however, is that the retarded interaction is attractive at lowfrequencies. It exhibits the phenomenon of over-screening and can give rise tosuperconductivity.

16.4 Phonon Renormalization of Quasi-Particles

The electron-phonon interaction can give rise to a change in the quasi-particledispersion relation. The Hartree-Fock contribution to the quasi-particle energyfrom the un-screened electron-electron interaction is

∆E(k) =∑k′

f(Ek′) < k k′ | e2

| r − r′ || k k′ >

=1V 2

∑k′

f(Ek′)∫

d3r

∫d3r′

e2

| r − r′ |

(1 − exp

[i ( k − k′ ) . ( r − r′ )

] )

= ∆EH − 1V

∑k′

f(Ek′)4 π e2

| k − k′ |2

(2077)

223H. Frohlich, Phys. Rev. 79, 845 (1950), also see J. Bardeen and D. Pines, Phys. Rev.99, 1140 (1955).

605

The first term is the Hartree term which is k independent and can be absorbedinto a shift of the chemical potential. The second term is the exchange termwhich depends on k. The exchange term affects the quasi-particle dispersion re-lation. If the effect of phonon screening is included, the exchange term becomes

− 1V

∑k′

f(Ek′)4 π e2

| k − k′ |2 + k2TF

[1 +

h2 ω(k − k′)2

( Ek − Ek′ )2 − h2 ω(k − k′)2

](2078)

In this expression the electronic screening of the exchange interaction is treatedin the Thomas-Fermi approximation, and the screening due to the phonons hasalso been included.

On utilizing the smallness of the Debye frequency with respect to the Fermienergy, and integrating over the magnitude of k′, one can show that the changein energy due to the electron-phonon interaction is given by

−∫

d2S′

8 π3

1h v(k)

4 π e2

| k − k′ |2 + k2TF

h ω(k − k′)2

ln∣∣∣∣µ− Ek − hω(k − k′)µ− Ek + hω(k − k′)

∣∣∣∣(2079)

where k′ lies on the Fermi surface. Substitution of Ek = µ immediately demon-strates that the value of the Fermi energy µ and the shape of the Fermi surfaceare unaltered by the coupling to the phonons which, in the approximation un-der consideration, is given by the Thomas-Fermi quasi-particle theory. Secondly,when the quasi-particle energy is within h ωD of µ, | Eqp(k) | < h ωD, thelogarithmic term can be expanded in inverse powers of h ω. Then, after invokingself-consistency, it is seen that the phonon contribution to the screening changesthe dispersion relation from that of the Thomas-Fermi screened theory to

Eqp(k) =ETF

k − µ

1 + λ(2080)

where 1 + λ is the wave function renormalization due to the phonons. Thatis the wave function contains a coherent quasi-particle component that has arelative weight of

11 + λ

(2081)

the other component is incoherent as the electron is in a linear combinationof other momentum states since it has been scattered by a phonon. The wavefunction renormalization is given by the expression

λ =∫

d2S′

8π3

1h v(k′)

4 π e2

| k − k′ |2 + k2TF

(2082)

This has the result that the quasi-particle velocity is given by

v(k) =1h∇ Eqp(k)

=1

1 + λ

1h∇ ETF

k (2083)

606

Thus, the quasi-particle contribution to the density of states is enhanced by afactor of 1 + λ

ρ(µ) = ( 1 + λ ) ρTF (µ) (2084)

An upper bound to the coupling constant is provided by the inequality

λ <4 π e2

k2TF

∫d2S′

8 π3

1h v(k′)

(2085)

however, the Thomas-Fermi screening length is defined by

4 π e2

k2TF

=(∂ρ

∂µ

)−1

=1

ρ(µ)

=[ ∫

d2S′

4 π3 h v(k′)

]−1

(2086)

Hence, the phonon renormalization factor is usually less than unity

λ < 1 (2087)

Typical values of λ are in the range of 0.2 to 0.8. Estimates of λ for variousmetals are given below224

Metal λ Metal λ Metal λ

Li 0.41 Be 0.23Na 0.16 Mg 0.36 Al 0.43-0.38

Zn 0.38 Ga 0.40In 0.69-0.8 Sn 0.6

Hg 1.0 T l 0.71 Pb 1.1-1.5Ti 0.38 V 0.60Zr 0.41 Nb 0.82 Mo 0.41

Finally, the phonon corrections are negligible for electron energies far from theFermi energy. For example, when

| Eqp(k) | > h ωD (2088)

then the dispersion relation suffers only small corrections

Eqp(k) = ETFk − µ + O

(h ωD

ETFk − µ

)2

(2089)

224W. L. McMillan, Phys. Rev. 167, 331 (1968).

607

Thus, there has to be a kink in the quasi-particle dispersion relation at energiesclose to the Fermi energy.

16.5 Electron-Phonon Interactions

The effect of coupling with the phonons on the quasi-particle spectrum can beused to deduce the form of the electron-phonon interaction. The change in theground state energy of a metal due to the electron phonon interaction, Hint,can be estimated from second order perturbation theory as

∆2E =∑

i

| < Ψ0 | Hint | Ψm > |2

E0 − Em(2090)

It is assumed that the form of the electron - phonon interaction is dominated bythe first non-trivial term in the expansion of potential acting on the electronsin powers of the ionic displacements

Hint =∑

i

ui . ∇RiVions(r) (2091)

Thus, the most important excitation process comes from excited states | Ψm >in which an electron has been scattered from state k to k− q. Also a phonon ofwave vector q has been excited, hence,

Em − E0 = Ek−q + h ω(q) − Ek (2092)

Thus, one can express the second order correction to the ground state energyin a phenomenological manner as

∆2E = −∑k,q

f(Ek) ( 1 − f(Ek−q) )| λq |2

Ek−q + h ω(q) − Ek(2093)

where f(x) is the Fermi function. One can identify an effective electron-electroninteraction, due to the phonons, from the functional derivative of the energy withrespect to the Fermi functions

Veff (q) =δ2 ∆2E

δf(Ek) δf(Ek−q)(2094)

Hence,

Veff (q) = −| λq |2

Ek − Ek−q − h ω(q)

−| λq |2

Ek−q − Ek − h ω(q)

= | λq |2[

2 h ω(q)

h2 ω(q)2 − ( Ek − Ek−q )2

](2095)

608

On identifying the above effective potential with the phonon contribution tothe screened interaction between the electrons, one obtains an expression forthe effective coupling constant | λq |2 as

| λq |2 =1V

4 π e2

q2 + k2TF

h ω(q)2

(2096)

For small q, the coupling constant vanishes linearly with q, since

4 π e2

k2TF

=23µ

ρ(2097)

for q < kTF , the coupling constant varies as

| λq |2 =µ

ρ V

h ω(q)3

=h ω(q) µ3 N Z

(2098)

16.6 Electrical Resistivity due to Phonon Scattering

The electron-phonon scattering contributes to the electrical resistivity. Thephonon gas acts as a source or sink for the electron momentum, thus, the interac-tions with the electron gas reduces the current flow. Hence, the electron-phononinteraction increases the resistivity. The electron-ion interaction is given

byHions =

∑R

V (r −R) (2099)

and as the position of the i-th ion can be written in terms of the equilibriumposition and a displacement

R = Ri + ui (2100)

The potential of the ions is expanded up to linear order in the lattice displace-ments ui

Hions =∑

i

[V (r −Ri) − ui . ∇R V (r −Ri) + . . .

](2101)

The first term represents the static lattice and the second term is the electronphonon interaction. The electron phonon interaction is given by

Hint = −∑

i

ui . ∇R V (r −Ri) (2102)

609

Thus, the interaction produces scattering of the electrons between Bloch statesand, through ui involves the absorption or emission of phonons. The conditionof conservation of energy yields the selection rule

E(k) = E(k′) ± h ω(k − k′) (2103)

This single restriction leads to a two-dimensional surface of Bloch state wavevectors k′ that are allowed final states for the electron initially in Bloch statek. The momentum transfer for these processes is given by q = k − k′. Thesurface of allowed final states must be close to the surface of initial energy ash ω µ, hence, E(q) ∼ E(k − q). The scattering rate out of the state withmomentum k is given by

1τ(k → k′)

=

2 πh

∑α

| λαq |2 f(E(k))

(1 − f(E(k + q))

)

×

[N(ωα(q)) δ

(E(k) − E(k + q) + h ωα(q)

)

+(

1 + N(ωα(q)))δ

(E(k) − E(k + q) − h ωα(q)

) ](2104)

The rate for scattering into the momentum state k is given by

1τ(k′ → k)

=

2 πh

∑α

| λαq |2 f(E(k + q))

(1 − f(E(k))

)

×

[N(ωα(q)) δ

(E(k + q) − E(k) + h ωα(q)

)

+(

1 + N(ωα(q)))δ

(E(k + q) − E(k) − h ωα(q)

) ](2105)

The transport scattering rate is the rate for momentum change of an electronat the Fermi surface is defined by

( k . E )1τf(E(k))

(1 − f(E(k))

)=∑k′

[( k . E )

1τ(k → k′)

− ( k′ . E )1

τ(k′ → k)

](2106)

610

The rate for scattering out of state k will be transformed into a form comparableto the rate for scattering in. The rate for scattering out of momentum state kis re-written as

=2 πh

∑α

| λαq |2 f(E(k))

(1 − f(E(k + q))

)exp

[β ( E(k) − E(k + q) )

]

×

[ (1 + N(ωα(q))

)δ

(E(k) − E(k + q) + h ωα(q)

)

+ N(ωα(q)) δ(E(k) − E(k + q) − h ωα(q)

) ]

=2 πh

∑α

| λαq |2 f(E(k + q))

(1 − f(E(k))

)

×

[ (1 + N(ωα(q))

)δ

(E(k) − E(k + q) + h ωα(q)

)

+ N(ωα(q)) δ(E(k) − E(k + q) − h ωα(q)

) ](2107)

Thus, the transport scattering rate can be expressed as

( k . E )1τf(E(k))

(1 − f(E(k))

)=

(2108)

=2 πh

∑α, q

( q . E ) | λαq |2 f(E(k + q))

(1 − f(E(k))

)

×

[ (1 + N(ωα(q))

)δ

(E(k) − E(k + q) + h ωα(q)

)

+ N(ωα(q)) δ(E(k) − E(k + q) − h ωα(q)

) ](2109)

Furthermore, as

f(E(k + q))(

1 − f(E(k))) (

1 + N(ωα(q)))δ

(E(k) − E(k + q) + h ωα(q)

)= f(E(k))

(1 − f(E(k + q))

)N(ωα(q)) δ

(E(k) − E(k + q) + h ωα(q)

)(2110)

611

the scattering rate can be expressed as

( k . E )1τf(E(k))

(1 − f(E(k))

)=

2 πh

∑α, q

( q . E ) | λαq |2 N(ωα(q))

×

[f(E(k))

(1 − f(E(k + q))

)δ

(E(k) − E(k + q) + h ωα(q)

)

+ f(E(k + q))(

1 − f(E(k)))δ

(E(k) − E(k + q) − h ωα(q)

) ]

=2 πh

∑α, q

( q . E ) | λαq |2 N(ωα(q)) N( − ωα(q))

×

[ (f(E(k + q)) − f(E(k))

)δ

(E(k) − E(k + q) + h ωα(q)

)

+(f(E(k)) − f(E(k + q))

)δ

(E(k) − E(k + q) − h ωα(q)

) ]

=2 πh

∑α, q

( q . E ) | λαq |2 N(ωα(q)) N( − ωα(q))

×

[ (f(E(k) + hωα(q)) − f(E(k))

)δ

(E(k) − E(k + q) + h ωα(q)

)

+(f(E(k)) − f(E(k)− hωα(q))

)δ

(E(k) − E(k + q) − h ωα(q)

) ](2111)

The summation over q is evaluated by transforming it into an integral

=2 πh

∑α, q

( q . E ) | λαq |2 N(ωα(q)) N( − ωα(q))

×

[ (f(E(k) + hωα(q)) − f(E(k))

)δ

(h2

m( k . q ) +

h2

2 mq2 − h ωα(q)

)

+(f(E(k)) − f(E(k)− hωα(q))

)δ

(h2

m( k . q ) +

h2

2 mq2 + h ωα(q)

) ](2112)

The integration over the direction of q is performed in spherical polar coordi-nates, in which the direction of k is fixed as the polar axis. The integral overthe azimuthal angles result in the factors of sinφ and cosφ in

( q . E ) = q cos θ Ez + q sin θ ( sinφ Ey + cosφ Ex ) (2113)

612

vanishing. The sole surviving term, proportional to Ez, can then be written ina manner independent of the choice of axis as k . E which can be factored outof the integral

=2 πh

(2 m π

h2 k2) ( k . E )

∑α

∫dq q2 | λα

q |2 N(ωα(q)) N( − ωα(q))

×

[ (f(E(k) + hωα(q)) − f(E(k))

) ∫ 1

−1

d cos θ cos θ δ(

cos θ +q

2 k−

m ωα(q)h k q

)

+(f(E(k)) − f(E(k)− hωα(q))

) ∫ 1

−1

d cos θ cos θ δ(

cos θ +q

2 k+

m ωα(q)h k q

) ](2114)

On neglecting the term of order vα

vF, cancelling the factors of ( k . E ), and Taylor

expanding the Fermi function factors in powers of the phonon frequencies, onefinds the transport scattering rate for electrons on the Fermi surface is given by

1τf(E(k))

(1 − f(E(k))

)=

= − 2 πh

(2 m π

h2 k3)∑α

∫dq q3 | λα

q |2 N(ωα(q)) N( − ωα(q)) h ωα(q)(∂f(E(k))∂E(k)

)(2115)

On using (∂f(E(k))∂E(k)

)= − β f(E(k))

(1 − f(E(k))

)(2116)

one finds

1τ

= β (4 m π2

h3 k3)∑α

∫dq q3 | λα

q |2 N(ωα(q)) N( − ωα(q)) h ωα(q)

(2117)

The temperature dependence of the transport scattering rate can be evaluatedusing the Debye model for the phonons, and using a linear q dependence of|λα

q |2. The integral over q is evaluated through the substitution z = β h ωα(q)and ωα(q) = vα q to yield

1τ∝ T 5

∫ TDT

0

dzz5

( exp[z] − 1 ) ( 1 − exp[−z] )(2118)

For this temperature range, the number of thermally exited phonons is propor-tional to T 3. One would expect that the scattering rate would be proportional

1τ∝∫

dq q2 N(ω(q))

∝ T 3 (2119)

613

However, as forward scattering is ineffective in transport properties, the trans-port scattering rate is proportional to the change in momentum along the di-rection of the electric field and therefore, is proportional to

( 1 − cos θ ) = 2 sin2 θ

2

≈ 12q2

k2F

(2120)

which produces an additional T 2 dependence. For low temperatures ( T < TD

) , the upper limit on the integration may be set to infinity yielding

σ(T )−1 ∝(T

TD

)5

(2121)

Thus, the combined effect of the factor ( 1 − cos θ ) and 1τ ∝ T 2 produces a

T 5 temperature dependence in the low-temperature resistivity.

At high temperatures ( T > TD ), the range of integration is less thanunity so the integrand may be expanded in powers of z yielding

σ(T )−1 ∝ T 5

∫ TDT

0

dz z3 = T T 4D

∝(T

TD

)(2122)

which is the result for the classical limit of the scattering. This can be consideredto arise merely as the number of thermally activated phonons is given by theclassical expression

N(ωα(q)) =kB T

h ωα(q)(2123)

The above results were first derived independently by Bloch and Gruneisen andthe resulting formula is known as the Bloch - Gruneisen resistivity due to phononscattering.

16.6.1 Umklapp Scattering

Umklapp processes may change the leading low-temperature variation of theresistivity. Umklapp scattering circumvents the factor of ( 1 − cos θ ) whichproduces the extra T 2 factor. When kF is close to the zone boundary, a small qvalue may couple the sheets of the Fermi surface in neighboring Brillouin zones.These are the umklapp processes. They produce a large change in the electronvelocity ∆v, by a phonon induced Bragg reflection.

614

16.6.2 Phonon Drag

The resistivity could decrease faster than T 5 if the system was relatively freeof defects and umklapp scattering could be neglected. This would occur if thephonons were allowed to equilibrate with the electronic system in its steadystate. The combined system of electrons and phonons should have a total mo-mentum, which is conserved in collisions. As a result, the phonon system wouldnot be able to momentum (or current) from the electron system as they drifttogether.

615

17 Phonons in Semiconductors

17.1 Resistivity due to Phonon Scattering

The transport scattering rate in a semiconductor can be obtained from thecollision integral of the Boltzmann equation

I

[f(k)

]=∑

q

λ2q

[f(k) ( 1 − f(k + q) ) N(ω(q)) δ( E(k) − E(k + q) + h ω(q) )

− ( 1 − f(k) ) f(k + q) ( 1 + N(ω(q)) ) δ( E(k + q) − E(k) − h ω(q) )

](2124)

in which f() is the non-equilibrium distribution function. On linearizing aboutthe equilibrium Fermi distribution function

f(k) = f0(k) + A ( k . E )∂f0(k)∂E

(2125)

yields the linearized collision integral.

Using the identity

( 1− f(E(k)) ) f((E(k)+hω(q)) =(

1− exp[β h ω(q)

] ) (f(E(k))− f(E(k)+hω(q))

)(2126)

one finds the result

I

[f(k)

]=

2 m A V

h2 kB T kexp

[− β ( E(k) − µ )

] ∫ 2k

0

dq

2 πq2 Ezλ

2q N(ω(q)) N(−ω(q))

×

[ (q

2 k+

m ω(q)h k q

) (1 − exp

[− β hω(q)

] )

−(

q

2 k−

m ω(q)h k q

) (1 − exp

[+ β hω(q)

] ) ](2127)

For low frequency acoustic phonons, the Bose-Einstein distribution can be ap-proximated by its high temperature form leading to the collision integral

I

[f(k)

]= A ( k . E )

(− ∂f(k)

∂E

)m V

k3

∫ 2k

0

dq

2 πq2 λ2

q

(− q

2 β h ω(q)

)(2128)

The transport scattering rate is found by factoring out the non-equilibrium partof the distribution function

1τ(E)

=m V

k3kB T

∫ 2k

0

dq

4 πq3

N

2 M ω(q)2q2 | V (q) |2

616

=N V

4 π c2m

Mk | V (0) |2 kB T (2129)

Hence, the conductivity in a semiconductor, in which the scattering rate isdominated by phonon scattering, is given by

σx,x ∼ β2 exp[β µ

] ∫dk k3 exp

[− β E(k)

](2130)

Thus, the conductivity has a temperature dependence given by

σx,x ∼ exp[β µ

](2131)

Thus, as expected, the conductivity is still dominated by the number of carriers,but the conductivity has an additional T dependence of T−

32 above and beyond

the prefactor in the number of carriers.

17.2 Polarons

Electron-phonon coupling in semiconductors can be large. For low density ofcarriers, each carrier can cause a distortion of the lattice. The carrier and thesurrounding distortion forms an excitation which is known as a polaron. At lowtemperatures the polaron appears to have a large effective mass, as the motionof the carrier is hindered by the need to drag the surrounding lattice distortion.Thus, there is a low-temperature regime in which the conductivity is governedby the motion of the heavy quasi-particles with an extremely large and tem-perature dependent effective mass. At high temperatures, the conductivity isdominated by incoherent hopping processes, which are thermally assisted by thepresence of a thermal population of phonons.

17.3 Indirect Transitions

In a semiconductor, light can be absorbed in processes where by an electronis excited from the filled valence band into the empty conduction band. Theminimum energy of the photon must be greater than the band gap between theconduction and valence band density of states. Since the speed of light c isso large, the wave vector of the photon absorbed in a transition between twostates with energy difference of the scale of eV is extremely long. Thus, themomentum of the photon is negligible on the scale of the size of the Brillouinzone. This means that in a semiconductor, if only a photon and an electron areinvolved, momentum conservation only allows transitions in which the initialand final state of the electron have the same k value. This type of transition iscalled a direct transition. In some semiconductors the minimum of the conduc-tion band dispersion relation lies vertically above the maximum of the valenceband, and the band gap is called the direct gap. In this case, the threshold for

617

direct absorption should coincide with the gap observed in the density of states.On the other hand, if the energetic separation between the maximum of the va-lence band and the minimum of the conduction band dispersion relations occurat different k values, then the threshold energy for the absorption of a photonin a direct transition should be greater than the separation inferred from justconsidering the density of states alone. This second type of semiconductor hastwo gaps, the indirect band gap inferred from the density of states and a directgap inferred for q = 0 transitions by consideration of the dispersion relations.

If the ions of the lattice are displaced from their equilibrium positions, simpleconservation of momentum arguments do not apply. In this case, it is possibleto have absorption at the indirect gap. At the threshold for indirect transitions,the absorption process involves the absorption or emission of a phonon withwave vector equal to the wave vector Q separating the valence band maximumto the conduction band minimum. The transition rate has to be calculated viasecond order perturbation theory, one power of the interaction involves the ab-sorption of one photon and the other power of the interaction involves eitherthe absorption or emission of one phonon.

The state of the joint system composed of an electron, phonons of wavevector Q and photons of frequency ω is denoted by | Ψ >. This state satisfiesthe equation of motion

i h∂

∂t| Ψ > =

(H0 + Hint

)| Ψ > (2132)

The state is decomposed in terms of eigenstates of H0, | φn > with energy En

| Ψ > =∑

n

Cn(t) exp[− i t

En

h

]| φn > (2133)

Then, one finds that the expansion coefficients Cn(t) satisfy the equation

i h∂

∂tCn(t) =

∑m

< φn| Hint | φm > exp[i t

Em − En

h

]Cm(t) (2134)

Since the system is initially in the ground state, then the state is subject to theinitial condition given by

Cn(0) = δn,0 (2135)To first order, one has

C1n(t) = − i

h

∫ t

0

dt′ < φn| Hint | φm > exp[i t′

E0 − En

h

](2136)

We assume the perturbation has no diagonal elements, therefore, C10 (t) = 0.

To second order, one has

i h∂

∂tC2

n(t) =∑m6=0

< φn| Hint | φm > exp[i t

Em − En

h

]C1

m(t) (2137)

618

18 Impurities and Disorder

If an isolated impurity is introduced into a solid, and the impurity has no low-energy degrees of freedom which can be excited, then it can be treated as animpurity potential Vimp(r). The total Hamiltonian H is written as

H = H0 + Vimp (2138)

where H0 describes the conduction band states of the pure metal. The eigen-states of H0 are the Bloch states φk with energy eigenvalues Ek.

H0 φk = Ek φk (2139)

Since the impurity breaks the periodic translational invariance of the solid, theimpurity potential will scatter an electron between Bloch states with differentBloch wave vectors. The non-zero matrix elements of the potential can bewritten as ∫

d3r φ∗k′(r) Vimp(r) φk(r) = < k′ | Vimp | k > (2140)

If the wave function, in the presence of an impurity, is written as a superpositionof Bloch states

ψα(r) =∑

k

Cα(k) φk(r) (2141)

then the energy eigenvalue Eα can be expressed as

( Eα − Ek ) Cα(k) =∑k′

Cα(k′) < k | Vimp | k′ > (2142)

If the quantity∑

k′ Cα(k′) < k | Vimp | k′ > is well defined and a non-zero function of k, then there exist eigenvalues Eα between every consecutivepair of values of the Bloch energies Ek. For a large system, where Ek are veryclosely spaced, the eigenvalues form a continuum. These eigenstates correspondto weakly perturbed Bloch states. On the other hand, if the potential is attrac-tive, and there is a minimum value of Ek (say Ek = 0), below which there canbe bound states with energies Eα.

The dependence of the bound state energy on the density of states of theordered material can be easily found, for the case where the potential has theproperty that the matrix elements are independent of k and k′. This correspondsto a short-ranged potential. To simplify notation, we shall introduce the matrixelements

< k | Vimp | k′ > = Vimp (2143)

which are of the order of the inverse of the sample’s volume, V −1. In this case,one can easily solve for the bound states. The above equations can be solvedby introducing the quantity γ, defined by

γ =∑k′

Cα(k′) (2144)

619

1

2

3

4

Im z

-4

-3

-2

-1

0-4 -3 -2 -1 0 1 2 3 4 5 6

Re z

Figure 252: The analytic structure of the Green’s function. The Green’s Func-tion has isolated poles on the negative real axis, corresponding to the boundstate energies Eα and a branch cut across the positive real axis..

The expansion coefficients in the eigenvalue equation can be expressed in termsof γ through

Cα(k) = γVimp

Eα − Ek(2145)

The above two equations leads to a self-consistency condition for the boundstate energy Eα

1 =∑

k

Vimp

Eα − Ek(2146)

which shows that, for an attractive potential, there may be a critical value ofVimp needed for a bound state to form.

A more powerful way of solving the same problem involves use of the one-particle resolvent Green’s function. The resolvent Green’s function is definedby the operator

G(z) = ( z − H )−1 (2147)

where z is a complex number. Since H is a Hermitean operator, the matrixelements of the Green’s function can be expressed in terms of a sum of simplepoles at the energy eigenvalues. Since the eigenvalues of the Hamiltonian arecomposed of discrete bound states at negative energies and a semi-continuousspectrum at positive energies, the Green’s function has discrete poles along thenegative z axis and a branch cut across the positive real axis, x = <e z > 0.The imaginary part of the Green’s function is discontinuous along the real axis.The discontinuity is given by

< Ψ |(G(x−iη)− G(x+iη)

)| Φ >= 2 π i

∑n

< Ψ | En > < En | Φ > δ( x− En )

(2148)where the | En > are the energy eigenstates of H corresponding to the energyeigenvalues En.

620

The density of states ρ(ε) is given by the trace of the Green’s function.

ρ(ε) = − 1π=m

[ ∑n

< Ψn | G(ε+ iη) | Ψn >

](2149)

where | Ψn > form a complete orthonormal basis. The identity can be provedby using a specific basis for evaluating the trace. The trace can be evaluatedin the basis of energy eigenstates, since they form a complete orthonormal set.The right hand side of the identity can be evaluated as

ρ(ε) = − 1π=m

[ ∑n

< En | G(ε+ iη) | En >

](2150)

In the limit η → 0, the matrix elements of the Green’s function are given by

ρ(ε) = − 1π=m

[ ∑n

< En |1

ε + i η − H| En >

]= − 1

π=m

[ ∑n

< En |1

ε + i η − En| En >

]=

1π

∑n

< En |η

( ε − En )2 + η2| En >

=∑

n

δ( ε − En ) (2151)

which is the definition of the density of states. Hence, the density of states isgiven by the trace of the imaginary part of the Green’s function.

The resolvent Green’s function can be obtained by expressing the Hamilto-nian in terms of the unperturbed Hamiltonian H0 and the interaction due tothe impurity potential, Hint,

H = H0 + Hint (2152)

Then, the Green’s function

G(z) =(z − H

)−1

=(z − H0 − Hint

)−1

(2153)

is identically equal to

G(z) =(z − H0

)−1

+(z − H0

)−1

Hint G(z) (2154)

This identity can be expressed in terms of the non-interacting resolvent Green’sfunction, G0(z), as

G(z) = G0(z) + G0(z) Hint G(z) (2155)

621

Similarly, one can also prove the analogous identity

G(z) = G0(z) + G(z) Hint G0(z) (2156)

The matrix elements of the non-interacting resolvent Green’s function are easilyevaluated in terms of the matrix elements between the eigenstates of H0, | E0

n >.

< E0n | G0(z) | E0

m > = < E0n |

1z − H0

| E0m >

= < E0n |

1z − E0

m

| E0m >

=< E0

n | E0m >

z − E0m

=δn,m

z − E0m

(2157)

which is diagonal. Since Bloch states of the host material are eigenstates of theunperturbed Hamiltonian H0, the non-interacting resolvent Green’s functiononly has diagonal matrix elements between Bloch states. The matrix elementsare evaluated as

< k′ | 1z − H0

| k > =< k′ | k >

z − Ek

=δk,k′

z − Ek(2158)

The interacting Green’s function can be expressed in terms of the T (z) matrixas

G(z) = G0(z) + G0(z) T (z) G0(z) (2159)

where the T-matrix is defined as

T (z) = Hint

(1 − G0(z) Hint

)−1

(2160)

Thus, the poles of the T-matrix are related to the poles of the Green’s function.For a sufficiently short-ranged potential, the matrix elements of Hint are inde-pendent of k. In this case, the matrix elements of the T-matrix between anypair of Bloch states can be evaluated as

< k′ | T (z) | k > = Vimp

(1 −

∑k”

Vimp

z − Ek”

)−1

(2161)

Since the density of Bloch states ρ0(ε) is defined as

ρ0(ε) = − 1π

∑k

δ

(ε − Ek

)(2162)

622

-0.50

-0.25

0.00

0.25

0.50

-12 -9 -6 -3 0 3 6 9 12ε/t

πρ0(

ε)

Vimp-1En

Figure 253: A graphical solution of the equation for the bound state energy En.The Hilbert transform of the density of states is shown in blue. The imaginarypart of the Hilbert transform is shown in red.

one can express the function in the denominator of the T-matrix as an integral∑k

Vimp

z − Ek=∫ ∞

0

dεVimp ρ0(ε)z − ε

(2163)

This has a discontinuous imaginary part on the positive real axis, therefore, theT-matrix is non-analytic for z on the positive real axis corresponding to thecontinuous spectra of energy eigenvalues. The T-matrix also has isolated polesat the negative energies z = En which are given by the solutions of

1Vimp

=∫ ∞

0

dερ0(ε)

En − ε(2164)

These energies are the energies of bound states225. The bound states are expo-nentially localized around the impurity site. The minimum value of the attrac-tive potential Vimp that produces a bound state strongly depends on the formof the density of states at the edge of the continuum. The critical value of Vimp

denoted as Vc is given by the condition that the bound state energy is zero, i.e.E0 = 0

1Vc

= −∫ ∞

0

dερ0(ε)ε

(2165)

Since ρ0(ε) ∝ ε(d−22 ) near the band edges, the integral converges for three

dimensions and higher, but diverges for one and two dimensions. Hence, anattractive short-ranged interaction will produce a bound state in one and two-dimensions, irrespective of the strength of Vimp. However, in three dimensions,the attractive potential has to exceed a critical value before a bound state canbe formed.

225G. F. Koster and J. C. Slater, Phys. Rev. 96, 1208 (1954), P. A. Wolff, Phys. Rev. 124,1030, (1961).

623

-0.50

-0.25

0.00

0.25

0.50

-12 -9 -6 -3 0 3 6 9 12ε/t

πρ0(

ε)

Vimp-1

E0

Figure 254: A graphical solution of the equation for the energy of a resonanceE0. The Hilbert transform of the density of states is shown in blue. Theimaginary part of the Hilbert transform is shown in red.

If the potential is not strong enough to produce a bound state, then the realpart of the denominator of the T-matrix, i.e.,

1 − Vimp <e∫ ∞

0

dερ0(ε)z − ε

(2166)

may vanish for some values of z on the positive real axis at which the unper-turbed density of states is finite. The solutions usually occur in pairs. If thedenominator vanishes at z = E0 > 0, then the T-matrix has a resonanceat this energy where it varies rapidly with z. The T-matrix does not divergesince the denominator has an imaginary part proportional to the unperturbeddensity of states

Vimp π ρ0(E0) (2167)

which is finite if E0 > 0. The change in the density of states due to theimpurity potential can be found by taking the trace of the equation

G(ε+ iη) = G0(ε+ iη) + G0(ε+ iη) T (ε+ iη) G0(ε+ iη) (2168)

with the Bloch states of the host material. The first term gives rise to thedensity of states of the host material. Hence, the impurity density of states isgiven by the imaginary part of the trace of the remaining term which involvesthe T-matrix. The impurity density of states can be written in terms of anenergy derivative

ρimp(ε) =1π=m

∑k

< k | T (ε+ iη) | k >∂

∂ε

1ε − Ek + i η

(2169)

On using the explicit form of the T-matrix, the impurity density of states isfound to have the form of a logarithmic derivative

ρimp(ε) = − 1π=m

[∂

∂εln(

1 − Vimp

∫ ∞

−∞dε′

ρ0(ε′)ε + i η − ε′

) ](2170)

624

The resonances in the T-matrix may cause the density of states to build upclose to the resonance near the band edge and be depleted at the higher energy(anti-)resonance.

18.1 Scattering by Impurities

The exact eigenstates of a Hamiltonian containing a scattering potential Vimp

satisfies the equation

H | Ψ+ > = ( H0 + Vimp ) | Ψ+ > = E | Ψ+ > (2171)

This can be re-expressed as an integral equation with an initial state given bythe incident state is an eigenstate of H0 corresponding to the plane wave | k >as

( E − H0 + i η ) | Ψ+ > = Vimp | Ψ+ > (2172)

This equation has general solutions which are the superposition of the solutionsof the homogeneous equation and a particular solution of the inhomogeneousequation

| Ψ+ > = | k > +1

E − H0 + i ηVimp | Ψ+ > (2173)

where E = Ek. Or, more formally,

| Ψ+ > = | k > + G0(E + iη) Vimp | Ψ+ > (2174)

where the unperturbed Green’s function G0(z) is defined as

G0(z) = ( z − H0 )−1 (2175)

For future reference, we shall also give the analogous expression

| Ψ+ > = | k > + G(E + iη) Vimp | k > (2176)

involving the exact Green’s function, G(z) defined by

G(z) = ( z − H )−1 (2177)

and the asymptotic incident state | k > . On inserting a complete set ofeigenstates of H0 in eqn(2174), one finds

| Ψ+ > = | k > +∑k′

| k′ > < k′ | G0(E + iη) Vimp | Ψ+ >

= | k > +∑k′

| k′ > 1E − Ek′ + i η

< k′ | Vimp | Ψ+ >

(2178)

625

k+iδ

-k-iδ-∞ +∞

k'

Figure 255: The contour used in the evaluation of the asymptotic behavior ofthe scattered wave.

To ensure that | Ψ+ > − | k > is an outgoing wave, η must be chosen as apositive infinitesimal constant. The asymptotic behavior of the scattered wavefunction can be expressed as

Ψ+(r) =1√V

exp[i k . r

]

+1V

∑k′

∫d3r′

exp[i k′ . ( r − r′ )

]E − Ek′ + i η

Vimp(r′) Ψ+(r′)

(2179)

The sum over k′ in the second term is evaluated as

1V

∑k′

exp[i k′ . R

]E − Ek′ + i η

=1

( 2 π )3

∫d3k′

exp[ i k′ . R ]E − Ek′ + i η

=2 π

( 2 π )32 mi R h2

∫ ∞

0

dk′ k′exp[ i k′ R ] − exp[ − i k′ R ]

k2 − k′2 + i 2 mh2 η

=m

i 2 π2 h2 R

∫ ∞

−∞dk′ k′

exp[ i k′ R ]k2 − k′2 + i 2 m

h2 η

= − m

2 π h2 Rexp[ i k R ] (2180)

The integral in the third line has been evaluated by Cauchy’s theorem, in whichthe contour along the real k axis is closed by a semicircle at infinity in theupper-half complex plane. This contour encloses the pole at k′ = k + i δ,but excludes the pole at k′ = − k − i δ. Thus, the wave function for the

626

kθ

k'

Figure 256: The asymptotic behavior of the stationary state corresponding toan incoming plane wave and an outgoing spherical wave.

stationary scattering state has the solution

Ψ+(r) =1√V

exp[i k . r

]

− m

2 π h2

∫d3r′

exp[i k | r − r′ |

]| r − r′ |

Vimp(r′) Ψ+(r′)

(2181)

This corresponds to a linear superposition of the unscattered wave and a spher-ical outgoing wave emanating from the impurity. Far from the impurity, thewave may be expressed in terms of the scattering amplitude f(k, θ) via

limr → ∞

Ψ+(r) → 1√V

(exp

[i k . r

]+ f(k, θ)

exp[i k r

]r

)(2182)

where the direction of the incident beam defines the z axis, and θ is the anglebetween the z axis and r. The asymptotic form can also be expressed in termsof the phase-shifts δl(k) via a partial-wave analysis. Far from the impuritypotential, r r′, the particles undergoing the scattering are asymptoticallyfree. In this region of space, the wave function can be expressed as a linear

627

superposition of energy eigenstates of the free particle Hamiltonian, with energyE = h2 k2

2 m . Thus,

limr → ∞

Ψ+(r) ∼ 1√V

∞∑l=0

( 2 l + 1 )(al jl( k r) + bl ηl( k r )

)Pl(cos θ)

(2183)where al and bl are coefficients that are to be determined, and jl(x) and ηl(x) arethe Riccati Bessel functions. The Riccati Bessel functions have the asymptoticforms

jl(kr) ∼sin( k r − l π

2 )k r

ηl(kr) ∼ −cos( k r − l π

2 )k r

(2184)

Hence, if one defines the phase shift, δl(k) via

tan δl(k) = − blal

(2185)

then the asymptotic form of the solution can be simply written as

limr → ∞

Ψ+(r) ∼ 1√V

∞∑l=0

( 2 l + 1 )k r

Al sin( k r − lπ

2+ δl(k) ) Pl(cos θ)

(2186)

which is similar to the general solution for the free particle, except that the short-ranged potential has introduced a phase shift in the argument of the trigono-metric function.

The incident particle’s state can be expanded in terms of the free particlestates with energy E and angular momentum l, as

1√V

exp[i k . r

]=

1√V

∞∑l=0

( 2 l + 1 ) il jl(kr) Pl(cos θ) (2187)

Also, the scattering amplitude f(k, θ) can be expanded in terms of the Legendrepolynomials

f(k, θ) =∞∑

l=0

( 2 l + 1 ) fl(k) Pl(cos θ) (2188)

On combining the above expressions with the two asymptotic expression forΨ+(r) and equating the coefficients of exp[ikr] and exp[−ikr], one finds the twoconditions

i−l Al exp[− i δl(k)

]= 1

i−l Al exp[i δl(k)

]= 1 + 2 i k fl(k) (2189)

628

Hence, the partial wave scattering amplitudes fl(k) are given by

( 1 + 2 i k fl(k) ) = exp[

2 i δl(k)]

(2190)

Therefore, the partial-wave scattering amplitudes fl(k) are given by

fl(k) =exp[ 2 i δl(k) ] − 1

2 i k

= exp[i δl(k)

]sin δl(k)

k(2191)

Thus, the amplitude of the asymptotic scattered wave is also determined by thephase shift. The asymptotic form is given by

limr → ∞

Ψ+(r) ∼ 1√V

∞∑l=0

( 2 l + 1 )k r

il exp[i δl

]Pl(cos θ) sin( k r − lπ

2+ δl )

(2192)

Since the on energy-shell T-matrix has matrix elements which satisfy

< k′ | T | k > = < k′ | Vimp | Ψ > (2193)

one finds that

m V

2 π h2 < k′ | T | k > =∞∑

l=0

Pl(cos θ)exp

[i 2 δl

]− 1

2 i k(2194)

The differential scattering cross-section is given in terms of the on shell T-matrixby

dσ

dΩ=(m V

2 π h2

)2 ∣∣∣∣ < k′ | T | k >

∣∣∣∣2 (2195)

Therefore, in general, the angular dependence is expressible in terms of Legendrepolynomials and the phase shifts. In the limit k → 0, only the s-wave phaseshift δ0 is significant. Therefore, in the limit k → 0, one finds

m V

2 π h2 < k′ | T | k > =exp

[i 2 δ0

]− 1

2 i k(2196)

Thus, the scattering cross-section is given by

dσ

dΩ=

sin2 δ0(k)k2

(2197)

and the total cross-section σ is given by

σ =4 π sin2 δ0(k)

k2(2198)

629

The impurity scattering cross-sections are independent of the volume of thesample, and give the largest contribution to the scattering at energies whereδ0(k) = π

2 .

The density of states due to the impurity can be expressed in terms of thephase shift δ0(k). The impurity is assumed to be contained in the host samplewhich has the form of a sphere of radius R. The wave functions are required tovanish at the surface of the sample, r = R. Hence, the phase shift must satisfythe condition

k R + δ0(k) = n π (2199)

This condition quantizes the allowed values of k. Since successive states satisfythis condition with consecutive integers n and n + 1, then the difference in thek values of any two consecutive states is given by

∆k ( R +∂δ0∂k

) = π (2200)

Thus, the number of states per k interval is determined by

1∆k

=1π

(R +

dδ0(k)dk

)(2201)

On multiplying this by dkdEk

, one obtains the impurity density of states, per spin,

as226

ρimp(ε) =1π

(∂δ0∂ε

)(2202)

Systems which have a rapid variation of the phase shift at the Fermi energy,have a large impurity density of states. On integrating the impurity density ofstates with respect to ε, one finds that the number of states due to the impuritywith energy less than ε, N(ε), is given by

N(ε) =1πδ0(ε) (2203)

The condition for electrical neutrality for a charge Z | e | determines the phaseshift at the Fermi energy, through Friedel’s sum rule227

Z =2πδ0(µ) (2204)

where the factor of two occurs due to the spin degeneracy.

The T-matrix for the short-ranged potential is given by

< k′ | T (z) | k > = Vimp

(1 −

∑k”

Vimp

z − Ek”

)−1

(2205)

226In general, the impurity density of states has contributions from all the partial wave

channels ρimp(ε) = 1π

∑l

( 2 l + 1 ) (∂δl∂ε

).227J. Friedel, Phil. Mag. 43, 153 (1952).

630

-0.5

-0.3

-0.1

0.1

0.3

0.5

-6 -4 -2 0 2 4 6

ε/t

δ0(ε)/π

ρimp(ε) tπρ0(ε) t

F(ε) t

t/V

Figure 257: The energy variation of the s-wave phase shift δ0(ε) and impuritydensity of states ρimp(ε) [in units of t ], for an s-wave resonance.

Hence, on using the partial-wave expansion for the T-matrix

m V

2 π h2 < k′ | T | k > =∞∑

l=0

Pl(cos θ)exp

[i 2 δl

]− 1

2 i k(2206)

one finds that the short-ranged potential only produces an l = 0 phase shift.The phase shift is given by

tan δ0(k) = − π Vimp ρ0(ε)1 − Vimp

∑k′

1ε − Ek′

(2207)

It can be seen that at a resonance, the denominator vanishes and the phase shiftis equal to π

2 , modulo π. Hence, the scattering cross-section is maximized at aresonance. Furthermore, since the change in the density of states is given by

ρimp(ε) =1π

∂δ0(ε)∂ε

(2208)

then one has

ρimp(ε) = − 1π

∂

∂εtan−1

[π Vimp ρ0(ε)

1 − Vimp

∑k′

1ε − Ek′

](2209)

which is consistent with eqn(2170). The impurity density of states can be ex-pressed in terms of the phase shift as

ρimp(ε) = ρ−10 (ε)

(1

2 πsin 2 δ0(ε)

(∂ρ0

∂ε

)− 1π2

sin2 δ0(ε)∂

∂ε

∫ ∞

−∞dε′

ρ0(ε′)ε − ε′

)(2210)

On applying Friedel’s sum rule

631

δ0(µ) =π Z

2(2211)

one finds that the density of states at the Fermi level is given by

ρimp(µ) = ρ−10 (µ)

(1

2 πsin( π Z )

(∂ρ0

∂ε

)− 1π2

sin2

(π Z

2

)∂

∂ε

∫ ∞

−∞dε′

ρ0(ε′)ε − ε′

)∣∣∣∣µ

(2212)The first term corresponds to the change in the density of states found in therigid band approximation. As the impurity adds Z electrons to the alloy, theFermi energy changes by an amount

∆µ =Z

2ρ−10 (µ) (2213)

In the rigid-band approximation, the change in the density of states at the Fermienergy is given by

ρimp(µ) = ρ0(µ+ ∆µ) − ρ0(µ)

≈ ∆µ(∂ρ0

∂ε

)+ . . .

≈ ρ−10 (µ)

Z

2

(∂ρ0

∂ε

)(2214)

Therefore, the change in the density of states at the Fermi energy, when calcu-lated in the rigid-band approximation, is finite for all values of Z. On the otherhand, the exact result vanishes for even integer values of Z. In the cases ofodd integer values of Z, the leading term in the exact impurity density of statescomes from the second term of eqn(2212). The second term is independent ofthe sign of Z, and decreases as the average value of Z is decreased. The finiteimpurity density of states at the Fermi energy gives rise to an impurity contri-bution to the specific heat and the susceptibility. These impurity contributionshave been observed in alloys where the charge difference between the host andimpurity ions have different signs and the results agree with the theoretical pre-dictions228.

——————————————————————————————————

18.1.1 Exercise

Consider two impurities which act as s-wave scattering centers that are sepa-rated by a distance R. Determine a formal expression for the exact Green’sfunction for the conduction electrons.

——————————————————————————————————

228A. M. Clogston, Phys. Rev. 125, 439 (1962), A. M. Clogston and V. Jaccarino, Phys.Rev. 121, 1357 (1961).

632

Figure 258: The susceptibility due to non-magnetic impurities in V3Ga versusimpurity concentration x. The susceptibility decreases for both types of impu-rities, even though Z has opposite signs for Cr and Ti. [After A.M. Clogston(1962).].

18.2 Virtual Bound States

The virtual bound state can be envisaged as an (almost) localized level that hasa finite probability amplitude for transitions into the conduction band states.These virtual bound states are most frequently found for 3d transition metalimpurities in metals or in mixed valent lanthanide element impurities in metals.In both these cases, the potential well has a large centrifugal barrier

Vl(r) =h2 l ( l + 1 )

2 m r2(2215)

The centrifugal potential prevents the 3d states from being filled until after the4s states are filled or, in the case of the lanthanide elements, the 4f states re-main unfilled until after the 6s, 5p and 5d states are all occupied. When thenuclear potential is strong enough, such that the 3d or 4f states can be occupiedin the ground state, the ion localizes an electron within the centrifugal barrierin an inner ionic shell. For example, in the Ce atom the 4f wave function islocalized, in that it has a spatial extent of 0.7 a.u. which lies inside the core-like5s and 5p orbitals. However, its’ ionization energy is small and comparable tothe ionization energy of the band-like 6s and 6p orbitals. As the localized stateis degenerate with the conduction band states, there is a finite probability am-plitude for an electron in the 4f level to tunnel through the barrier. The virtualbound state describes an extended state which, through resonant scattering,builds up a significant local character. The virtual bound state in a metal maybe modelled by a Hamiltonian which is the sum of three terms

H = H0 + HV = Hc + Hd + HV (2216)

where Hc describes the electrons in the conduction band, the Hamiltonian Hd

represents the (isolated) localized d level on the impurity and the term HV

633

0

0.02

0.04

0.06

0.08

0 20 40 60 80

Z r / a0

P(r)

r2

4f

5d

6s

Figure 259: The schematic spatial dependence of 4f electron densities relativeto the 5d and 6s electron densities.

describes the coupling. The conduction band Hamiltonian is expressed in termsof the number of conduction electrons in the Bloch states (k, σ) with dispersionrelation Ek through

Hc =∑k,σ

Ek nk,σ

=∑k,σ

Ek c†k,σ ck,σ (2217)

where c†k,σ and ck,σ, respectively, create and annihilate an electron in the con-duction band state with Bloch wave vector k and spin σ. Likewise, the energyfor an electron in the localized d state is given by the binding energy Ed timesthe number of d electrons of spin σ,

Hc =∑

σ

Ed nd,σ

=∑

σ

Ed d†σ dσ (2218)

where d†σ and dσ respectively create and annihilate an electron of spin σ inthe localized d state. The hybridization or coupling term is given by the spinconserving Hamiltonian

HV =1√N

∑k,σ

[V (k) c†k,σ dσ + V ∗(k) d†σ ck,σ

](2219)

The first term represents a process whereby an electron in the d orbital tun-nels into the conduction band, and the Hermitean conjugate term representsthe reverse process. It is assumed that the conduction band states have beenorthogonalized to the localized states, so that the conduction band fermion op-erators anti-commute with all the local fermion operators.

634

The Resolvent Green’s function can be calculated from the expression

( z − H0 )1

z − H= 1 + HV

1z − H

(2220)

Evaluating the matrix elements of this equation between the one-electron eigen-states of H0 yields the coupled equations

( z − Ed ) < d | 1z − H

| d > = 1 +1√N

∑k

V (k) < d | 1z − H

| k >

(2221)

and

( z − Ek ) < d | 1z − H

| k > =1√N

V ∗(k) < d | 1z − H

| d >

(2222)

These equations can be combined to yield the matrix elements of the resolventGreen’s functions as

Gd,d(z) = < d | 1z − H

| d >

=1

z − Ed − Σ(z)(2223)

where the d-electron self-energy Σ(z) is defined by

Σ(z) =1N

∑k

| V (k) |2

z − Ek(2224)

The real part of the self-energy can be interpreted as producing a renormaliza-tion of the energy of the localized level Ed. The imaginary part of the self-energycan be interpreted as giving rise to an width or lifetime τ such that

h

2 τ= − =m Σ(Ed + iη) (2225)

The conduction band Resolvent Green’s function is evaluated, from a similarset of coupled equations as

Gk,k′(z) = < k | 1z − H

| k′ >

=δk,k′

z − Ek+

1N

V (k)z − Ek

Gdd(z)V ∗(k′)z − Ek′

(2226)

The matrix elements of the T-matrix between different Bloch states is identifiedas

< k | T (z) | k′ > = V (k) Gd,d(z) V ∗(k′) (2227)

635

0.0

1.0

2.0

3.0

-6 -5 -4 -3 -2ε

ρ d( ε)

Ed

2∆

Figure 260: The impurity d-density of states for a virtual bound state.

From these equations, it can be seen that the density of states of the impurityd level is given in terms of the imaginary part of Σ(ε+ iη) via

ρd(ε) = − 1π=m

[Gd,d(ε+ iη)

]= − 1

π

=m Σ(ε+ iη)(ε − Ed − <e Σ(ε+ iη)

)2

+(=m Σ(ε+ iη)

)2

(2228)

The impurity density of states is approximately in the form of a Lorentziancentered on Ed, and has a width given by =m Σ(ε− iη). The width is given by

=m Σ(Ed − iη) =π

N

∑k

| V (k) |2 δ( Ed − Ek )

≈ 1N

π | V |2 ρ0(Ed) (2229)

which is related to the Fermi Golden rule expression for the rate for the local-ized electron to tunnel into the conduction band density of states ρ0(ε). Thus,the virtual bound state can be interpreted in terms of a narrow band density ofstates which is weakly coupled to the extended conduction band states.

——————————————————————————————————

18.2.1 Exercise

Determine a formal expression for the change in the conduction band densityof states due to the presence of a single impurity with a virtual bound state.

636

——————————————————————————————————

18.3 Disorder

Give a distribution of impurities in a solid, the potential in the solid will benon-uniform. The thermodynamic properties of the solid can be expressed interms of the energy eigenvalues, or alternatively the poles of the Green’s func-tion. For a macroscopic sample, the exact distribution of impurities will not bemeasurable and the thermodynamic properties are expected to be representa-tive of all distributions of impurities. Therefore, the average value of a quantitycan be represented by averaging over all configurations of the impurities. It caneasily be shown that the configurational averaged density of states is given bythe discontinuity across the real axis of the configurational averaged resolventGreen’s function.

The Hamiltonian of a binary (A-B) alloy, with site disorder, may be repre-sented by

H = H0 + V (2230)

The Hamiltonian H0 describes the tight-binding bands of a pure metal with adispersion relation

Ek = − t

d∑i=1

cos ki ai (2231)

The randomness appears as a shift of the binding energies of the atomic orbitals,due to the potential operator V . The potential V is written as the sum of localor on-site potentials

V =∑R

ER | φR > < φR | (2232)

where the state | φR > represents the single electron Wannier state at site R.The single site energies ER can take on the values EA or EB depending on thetype of atom present at site R.

The average Green’s function is given by

G(z) =(z − H0 − V

)−1

(2233)

which can be expressed as

G(z) =(z − H0 − Σ(z)

)−1

(2234)

where the operator Σ(z) is complex and is known as the self-energy due todisorder. Since the configurational averaged Green’s function has translational

637

invariance, then so does the self-energy. The disorder self-energy represents theeffect that the randomly distributed impurities have on the eigenvalue spectrum.The spectrum may be composed of extended states and localized states229. Dueto the fluctuations in the random potential, the energy eigenvalues correspond-ing to localized eigenstates may be broadened to form continua.

The averaged Green’s function can be calculated by expanding the Green’sfunction in powers of the potential and then performing the configurational aver-age. For strongly fluctuating potentials, the resulting power series may be slowlyconvergent, or it may not even be convergent at all. Therefore, to increase therate of convergence, it may be preferable to expand the Green’s function aboutthe self-energy. This procedure leads to the coherent potential approximation.

18.4 Coherent Potential Approximation

The potential difference between a specific realization of the potential V due tothe impurities and the self-energy can be expressed as

∆V (z) = V − Σ(z) (2235)

The resolvent Green’s function for this type of disordered impurity problem canbe expressed as

G(z) =(z − H0 − Σ(z) − ∆V (z)

)−1

(2236)

which can be expressed in terms of the T-matrix via

G(z) = G(z) + G(z) T (z) G(z) (2237)

where the T-matrix is given by

T (z) = ∆V (z)(

1 − G(z) ∆V (z))−1

(2238)

On taking the configurational average, one finds that the averaged T-matrixmust be zero

T (z) = ∆V (z)(

1 − G(z) ∆V (z))−1

= 0 (2239)

This equation can be used to obtain the self-energy.

For the A−B alloy the effective potential is

∆V (z) =∑R

( ER − Σ(z) ) | φR > < φR | (2240)

229P. W. Anderson, Phys. Rev. 109, 1492 (1958).

638

The concentration of A atoms is denoted by c, so the concentration of B atomsis ( 1 − c ). It is assumed that the two types of atoms are randomly dis-tributed on the lattice sites, such that there is one atom at each lattice site.It is also assumed that the T-matrix can be represented as a sum of single-siteT-matrices, in which the scattering is referenced to an appropriately chosen av-eraged medium. This is the single-site approximation. The averaged T-matrixcan be written as

T (z) = cEA − Σ(z)

1 −[EA − Σ(z)

]< R0 | G(z) | R0 >

+ ( 1 − c )EB − Σ(z)

1 −[EB − Σ(z)

]< R0 | G(z) | R0 >

(2241)

The Coherent Potential Approximation230 (C.P.A.) sets

T (z) = 0 (2242)

The resulting equations are non-trivial to solve since the Green’s function in thedenominator is formed from a sum over the Bloch states and also involves theself-energy.

< R0 | G(z) | R0 > =1N

∑k

1z − Σ(z) − Ek

= < R0 | G0( z − Σ(z) ) | R0 > (2243)

where G0(z) is the Green’s function for the tight-binding Hamiltonian. Never-theless, this can be solved numerically or alternatively, if the sum over Blochenergies can be evaluated analytically, an analytic solution may be found.

The C.P.A. is expected to be valid in various limits. These include the limitof a dilute concentration of impurities 1 c, weak scattering t | EA − EB |and trivially in the atomic limit, where the single-site approximation is exact.In general, the C.P.A. may be only trusted to yield the density of states andthermodynamic properties 231, and not transport properties. The density ofstates obtained from this method resembles a smeared version of the weightedsums of the density of states of a solid composed of A atoms and the density ofstates composed of B atoms. For small magnitudes in the differences of the siteenergies, the two components overlap, but they separate for large differencesin the site energies. When the bands are split, the widths of the componentbands are drastically modified from the ideal superposition. The change in the230P. Soven, Phys. Rev. 156, 809 (1967), B. Velicky, S. Kirkpatrick and H. Ehrenreich, Phys.

Rev. 175, 747 (1968).231R. J. Elliott, J. A. Krumhansl and P. L. Leath, Rev. Mod. Phys. 46, 465 (1975).

639

0

0.2

0.4

0.6

0.8

-2 -1.5 -1 -0.5 0 0.5 1 1.5

ω/W

ρ(ω)

0.00.51.01.5

c = 0.1 (Ea-Eb)/W

Figure 261: The dependence of the CPA density of states ρ(ε) on the strengthof the disorder (Ea−Eb

W ), where 2W is the band width of the pure material.

widths of the split bands reflects the increasing separation between sites of thesame type which decreases the tendency to form bands. The effect of the impu-rity scattering is to produce a smearing, which washes out any structure suchas van Hove singularities. Transport properties crucially depend on the spa-tial extended nature of the energy eigenstates, which may be destroyed by thefluctuations in the random potentials. This type of phenomenon is completelyabsent in C.P.A., and can lead to the energy eigenstates becoming localized 232.

——————————————————————————————————

18.4.1 Exercise

Assume that after diagonalizing the tight-binding Hamiltonian H0, one obtainsa density of states ρ0(z) of the form

ρ0(z) =2

π W 2

√W 2 − z2 (2244)

Determine the form of associated Green’s function G0(z). Show that the CPAequations can be reduced to a cubic equation for x(ε) = ε − Σ(ε). (Be sure tocheck whether the cubic equation yields spurious solutions not present in Soven’sinitial equation.) Determine the bounds of the CPA spectra and compare themwith the Saxon-Huttner bounds of

| EX − ε | < W (2245)

where X is either A or B. Determine the condition for the solution of the CPAequations to exhibit a pole in z(ε).

232P. W. Anderson, Phys. Rev. 109, 1492 (1958).

640

——————————————————————————————————

18.5 Localization

The phenomenon of disorder induced localization is easiest to understand interms of states at the tail edge of a band. Just as one impurity with a suffi-ciently strong attractive potential may cause a bound state to form around it,a bound state may also be formed for a number of nearby atoms with weakerattractive interactions, in which case the bound state may be of larger spatialextent. In both cases, they will produce localized states with energies belowthe continuum of the density of states. A distribution in the spatial separationof the impurity atoms will smear the spectrum of these discrete bound states.The localized states manifest themselves as low-energy tails to the density ofstates233. As the strength of the disorder is increased, the number of localizedstates in the tails of the density of states will increase. One surprising featureis that a sharp energy, the mobility edge234, separates the states that extendthroughout the crystal from the localized states. The length scale over whichthe states on the localized side of the mobility edge decay are extra-ordinarilylong235. These states and cannot be treated by perturbation methods but re-quire renormalization group types of approach.

On using the electron-hole symmetry for states at the top of the band, onediscovers that the states at the top edge of the band will also become localizeddue to disorder, and also have a mobility edge. On increasing the strength ofthe disorder, the mobility edges will move towards the middle of the bands.A disorder driven metal insulator transition will occur when the mobility edgecrosses the Fermi energy. This type of transition is known as the Andersontransition236. The effect of many-body interactions complicate the physics onthe metallic side of the Anderson transition, where weak localization occurs237.On the insulating side of the transition, conduction will still be possible butonly due to the thermal excitation of electrons to the itinerant states above themobility edge, or by thermally assisted tunnelling processes. For sufficientlystrong disorder all the states in the band will become localized. All states inone-dimensional and two-dimensional systems must become localized, for arbi-trarily small strengths of disorder. However, this localization will only show upin experiments if the length scale over which the states are localized is smallerthan the sample size.

233P. W. Anderson, Phys. Rev. 109, 1492 (1958).234N. F. Mott, Adv. in Phys. 16, 49 (1967).235D. J. Thouless, Phys. Rev. Lett. 39, 1167 (1977).236N. F. Mott, Metal-Insulator Transitions, Taylor and Francis, London (1974).237P. A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985).

641

0

0.1

0.2

0.3

0.4

0.5

-4 -2 0 2 4

ε/W

ρ(ε)

εF

ε+ε−

Figure 262: The density of states of a disordered band ρ(ε), showing the upperand lower mobility edges ε± and the Fermi-energy εF .

18.5.1 Anderson Model of Localization

In a doped semiconductor such as P doped Si, as the impurity concentrationis increased, it is expected that the energy levels of the isolated impurities willbroaden and form bands. For large concentrations, the impurity level wave func-tions are expected to overlap and become extended. Thus, it is expected thata metal-insulator transition will occur as a function of impurity concentration.The metal insulator transition can be described by a tight-binding model of adisordered system

H =∑i,σ

εi c†i,σ ci,σ −

∑i,j,σ

t c†i,σ cj,σ (2246)

where t are the nearest neighbor tight-binding hopping matrix elements and thesum over (i, j) are assumed to run over pairs of nearest neighbors lattice sites.The site energies εi are assumed to be random variables uniformly distributedover an energy width ∆V

P (ε) =1

∆Vfor − ∆V

2< ε <

∆V2

= 0 otherwise (2247)

The degree of disorder is measured by the dimensionless parameter ∆V/t.

For sufficiently large ∆V/t the states are expected to all be localized. Thecritical value of (∆V/t)c is expected to be dependent on the dimensionality ofthe lattice. In three dimensions, the critical value is estimated as (∆V/t)c ∼ 15.For ∆V/t less than the critical value, the states around the center of the tight-binding bands are extended while states near the band edges are localized. There

642

r/aR

e [φ

k(r)

]

+ exp[-|r|/ξ]

- exp[-|r|/ξ]

Figure 263: A schematic depiction of a the Real part of a wave function φk(r)localized due to randomness. The exponential envelope of the wave function isdepicted by the dashed red lines.

are energies Ec, called mobility edges that separate the localized and extendedstates. When the chemical potential µ crosses the mobility edge, the statesat the Fermi energy change their characters and a metal non-metal transitionoccurs. This is known as the Anderson transition.

The wave functions corresponding to extended and localized states havedifferent characters. A wave function for the disordered solid can be expressedas a linear combination of atomic wave functions

ψ(r) =∑R

C(R) φ(r −R) (2248)

A delocalized wave function has an amplitude C(R) which does not decay tozero at large distances. A localized wave function is expected to decay to zerowith an exponential envelope

| C(R) | ∼ exp[− | R | /ξ

](2249)

The spatial extent of the envelope is given by the correlation length ξ. The cor-relation length is expected to depend on the energy E of the energy eigenstate.The correlation length is expected to diverge as E approaches the mobility edgeEc. In the Anderson transition, the spectral density of localized states is ex-pected to be continuous. Numerical studies 238 show that the wave functionexhibits long-ranged fluctuations close to the critical value of ∆V/t, and ap-pears to be self-similar when viewed at all (sufficiently long) length scales.

238J. T. Edwards and D. J. Thouless, 5, 807 (1972).

643

18.5.2 Scaling Theories of Localization

Since numerical studies of Anderson localization are hampered by finite-sizeeffects which tend to obscure the effect of localization, Licciardello and Thou-less239 introduced a number g(L) which describes the sensitivity of energy eigen-values on the boundary conditions, for a system with linear dimension L. TheThouless number is defined as the ratio

g(L) =∆EδE

(2250)

where ∆E is the shift in energy levels that occurs when the boundary conditionson the wave function are changed from periodic to anti-periodic. The quantityδE is the mean spacing of the energy levels of the finite size sample. If the wavefunctions are exponentially localized, it is expected that

g(L) ∝ exp[− 2 L

ξ(E)

](2251)

Hence in localized states, g(L) tends to zero for ξ(E) < L since the wavefunctions of localized state are insensitive to the choice of boundary condition.On the other hand, if the wave functions are extended, the energy shift due tothe different boundary conditions should be proportional to

h

τ(2252)

where τ is the time required for the electron to diffuse to the boundary ofthe sample. This diffusion time is at most algebraically dependent on L. Thedifferent dependencies of g(L) on L provide a simple criterion in numericalstudies as to whether the states are extended or localized. An elegant argumenthas been put forth240 which indicates that the quantity g(L), the Thoulessnumber, is proportional to the conductance, G(L)

g(L) =G(L) h

2 e2(2253)

The conductance G(L) is related to the conductivity σ via a factor of the areadivided by the length

G(L) = Ld−1 σ

L(2254)

Hence, the Thouless number g(L) is related to the conductivity by

g(L) ∝ Ld−2 σ (2255)

239D. C. Licciardello and D. J. Thouless, Phys. Rev. Lett. 35, 1475 (1975), D. C. Licciardelloand D. J. Thouless, J. Phys. C, 8, 4157 (1975).240J. T. Edwards and D. J. Thouless, J. Phys. C 5, 807 (1972).

644

The scaling theory of localization241 is based upon the length dependenceof g(L). A scale change, of a d-dimensional system with linear dimension L, isproduced when the length scale L is changed to b L. It is expected that g(bL)is related to g(L) and the factor b, and nothing else. This is summarized in theformula

g(bL) = f [b, g(L)] (2256)

where f(x) is a universal scaling function, which only depends on the dimen-sionality d of the lattice. An infinitesimal scale change is defined by

b = 1 +dL

L(2257)

so that the scale can be changed continuously. A scaling function β[g(L)] isintroduced via the definition

β[g(L)] =d ln g(L)d lnL

=∂f(b, g)/∂b

g(L)(2258)

The functional β[g] completely specifies the scaling property of the conductivityin disordered systems. It is assumed that β(g) is a smooth continuous functionof g which is independent of L. This implies that the change in the effectivedisorder of the system can be uniquely determined at one length scale fromknowledge of its value at a smaller length scale.

The asymptotic forms of β can be found in the asymptotic limits g → 0and g → ∞. In the strongly localized regime g → 0 where the wave functionis exponentially localized, one finds that since

g(L) ∝ exp[− 2

L

ξ

](2259)

for L ξ, then

β(g) = − 2L

ξ∝ ln g + Const. (2260)

Thus, β(g) tends to − ∞ as g → 0. In the metallic limit g → ∞ and Ohm’slaw applies so σ is finite and independent of L, if the length scale is greater thanthe mean free path λ. Therefore, in this region one has

β(g) → ( d − 2 ) (2261)

The qualitative dependence of β(g) on g can be determined from continuity andthe use of perturbation expansions in g and g−1. The variation of β(g) withln g is shown in Fig(264). From this, one finds that the system is localized forall spatial dimensionalities less than or equal to two, d < 2. For d < 2,241F. J. Wegner, Zeit. fur Physik 25, 327 (1976).

E. Abrahams, P. W. Anderson, D. C. Licciardello and T. V. Ramakrishnan, Phys. Rev. Lett.42, 673 (1979).

645

Figure 264: A sketch of the universal curve showing the dependence of β(g) onln g. The dashed line for d = 2 represents the un-physical variation required forthere to be a jump in the conductivity. [After Abrahams et al. (1979).]

as β(g) is always negative, then g(L) scales to zero on increasing L. In two-dimensions, the conductivity decreases with increasing L. The decrease of σis logarithmic at large values of g and exponential at small values of g. Thus,in two-dimensions and in the limit of large systems, no matter how weak therandomness is, the states are always localized. By contrast, for d > 2, thereis a critical value of gc such that for g > gc the system scales to the metalliclimit where β(g) = ( d − 2 ). As β(g) is positive for g > gc, then when L isincreased above λ, limL → ∞ g(L) → ∞. Since β(g) is negative for g < gc,then g(L) scales to zero on increasing L. At the critical value of gc, the scalingfunction is zero, β(gc) = 0, and varies approximately linearly with ln g/gc so itcan be shown that the slope s determines the exponent of the correlation lengthas s−1. From the scaling theory, one can infer the dependence of conductivityon the concentration of impurities, c. For c > c0 , close to the metal insulatortransition, the conductivity scales as

σ = σ0 ( c − c0 )1 (2262)

where the exponent of unity can be obtained exactly via perturbation theory242.

242L. P. Gor’kov, A. I. Larkin and D. E. Khmel’nitskii, J.E.T.P. Lett. 30, 228 (1979).

646

19 Magnetic Impurities

19.1 Localized Magnetic Impurities in Metals

When transition metal or rare earth impurities are dissolved in simple metals,the electronic states on the impurities hybridize with the conduction band statesand form a Friedel virtual bound state. Since the impurity states are localizedthe Coulomb interaction U between two electrons occupying these states is largeand has to be taken into consideration. The Hamiltonian can be expressed as

H = H0 + Hint (2263)

where the Hamiltonian H0 represents the non-interacting conduction band andthe virtual bound state

H0 =∑k,σ

Ek c†k,σ ck,σ +

∑σ

Ed d†σ dσ

+∑k,σ

[V (k) c†k,σ dσ + V ∗(k) d†σ ck,σ

](2264)

and the Coulomb interaction U between a pair of electrons in the (spin onlydegenerate) impurity state is given by

Hint = U d†↑ d†↓ d↓ d↑ (2265)

This is the Anderson impurity Hamiltonian. The Anderson impurity modelis exactly soluble using numerical renormalization group243, or Bethe-Ansatztechniques244. The mean-field solution will be outlined below.

19.2 Mean-Field Approximation

The interaction term Hint can be expressed in terms of fluctuations of thenumber of d-electrons with spin σ

∆nσ = d†σ dσ − < | d†σ dσ | > (2266)

and the average valuenσ = < | d†σ dσ | > (2267)

When written in terms of the fluctuations of the occupancy of the d-orbitals,the Hamiltonian takes the form

Hint = U ∆n↑ ∆n↓ + U∑

σ

∆nσ n−σ + U n↑ n↓ (2268)

243K. G. Wilson, Rev. Mod. Phys. 47, 773 (1975).244N. Andrei, K. Furuya and J. H. Lowenstein, Rev. Mod. Phys. 5, 331 (1983), A. M.

Tsvelik and P. B. Weigmann, Adv. in Phys. 32, 453 (1983).

647

In the mean-field approximation, the term quadratic in the occupation numberfluctuations is neglected, yielding

HMF = U∑

σ

nσ n−σ − U n↑ n↓ (2269)

The localized electrons experience an effective spin-dependent binding energygiven by

Hd =∑

σ

( Ed + U n−σ ) d†σ dσ (2270)

where n−σ is the average number of electrons in the localized level of spin −σ.The spin-dependent occupation number is found as an integral over the densityof states of the virtual bound state, and is given by

nσ =∫ ∞

−∞dε f(ε) ρσ

d (ε) (2271)

where the spin-dependent impurity density of states is given by

ρσd (ε) = − 1

π=m

[1

ε + i η − Ed − U n−σ − Σ(ε+ iη)

](2272)

The self-energy can be approximated by a constant imaginary part with value ∆and a small energy shift that can be absorbed into the definition of Ed. Hence,the spin-dependent density of states can be approximated by a Lorentzian

ρσd (ε) ≈ 1

π

∆(ε − Ed − U n−σ

)2 + ∆2(2273)

where the width of the Lorentzian is given by

∆ = π∑

k

| V (k) |2 δ(Ed − Ek)

∼ π | V |2 ρ0(Ed) (2274)

Thus, at T = 0, one finds that the average occupation of the localized statewith spin σ is given by the expression

nσ =1π

cot−1

(Ed − µ + U n−σ

∆

)(2275)

Since cot θ is defined for θ on the interval 0 to π and runs between ∞ and − ∞,then cot−1 x has values that run from π to 0. The two coupled equations for nσ

and n−σ have to be solved self-consistently. This can be done by introducingthe parameters

x =µ − Ed

∆

y =U

∆(2276)

648

-1

-0.5

0

0.5

1

E-µ

ρ↑(E)− ρ↓(E)

∆ Ed+Un-µ

m = 0

-1

-0.5

0

0.5

1

E-µ

− ρ↓(E) ρ↑(E)

U(n↑-n↓)

m > 0

Figure 265: The spin-dependent Anderson-impurity d-density of states. Thelocal density of states of the up-spin sub-band is degenerate with the down-spinsub-band, when there is no local-moment is present. In the case where a local-moment has been formed, the up-spin sub-band (blue) is shifted by an energyUm relative to the down-spin sub-band (red). [After P.W. Anderson (1961).].

which are dimensionless measures of the position of the Fermi energy relative tothe d level and the Coulomb interaction. The pair of self-consistency equationsbecome

cotπ n↑ = ( y n↓ − x )cotπ n↓ = ( y n↑ − x )

(2277)

The non-magnetic phase is described by

n↑ = n↓ = n (2278)

This has a unique solution for n, in the range 0 < n < 1 which is given bythe solution of

cotπ n = ( y n − x ) (2279)

The non-magnetic solution corresponds to a partial occupation of the localizedlevels. In this case, the virtual bound state does not posses a magnetic moment.However, if y is large the equations have two degenerate magnetic solutions.The magnetic solutions only occur for sufficiently large values of y, and whenthey occur they are stable because they minimize the energy. The boundarywhich separates the regions were magnetic moments can occur from regions werethe magnetic moment is zero, can be found by expressing the self-consistencyequations in terms of the variable m defined by

nσ = n +12σ m (2280)

(σ = ±1) so that the magnetization is simply given by

m = n↑ − n↓ (2281)

649

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4

∆/U

( µ−E

d)/U

m > 0

m = 0

Figure 266: The mean-field phase diagram of the single-impurity AndersonModel. The phase space is separated into two regions, one where there is afinite local-moment and the other where there is no local-moment. [After P.W.Anderson (1961).].

The condition that a solution with an infinitesimal value of m first occurs can befound by linearizing the self-consistency equations in powers of m. On equatingthe coefficients of the first two terms in the expansion in m to zero, one finds

cotπ n = ( y n − x )π

sin2 π n= y (2282)

The first equation defines the non-magnetic solution, and the vanishing of thecoefficient of the term linear inm allows a solution with an infinitesimal non-zerovalue of m to occur. The above equations can be re-written as

x

y=

12 π

( θ − sin θ )

1y

=1

2 π( 1 − cos θ ) (2283)

where θ = 2 π n. This pair of equations define the phase boundary line sepa-rating the areas of phase space in which the impurity is magnetic from the areain which the impurity is non-magnetic. The tendency for magnetism is strongestwhen the d-d interaction, U , is large and when n is close to 1

2 , i.e., when Ed

and Ed + U are positioned symmetrically about the Fermi level. In this case,the total number of d-electrons of both spins is almost unity. The non-magnetic

650

Figure 267: The magnitudes of the local magnetic moments of Fe (as determinedfrom the high-temperature Curie-Weiss susceptibility) dissolved in various tran-sition metal alloys. [After A.M. Clogston et al. (1962).].

solution occurs when U is small or when the d-level is either almost completelyfilled or almost completely empty.

For large y, the magnetic solutions are described by

nσ ≈ 1 − 1π y

n−σ ≈ 1π ( y − x )

(2284)

These local moment solutions are doubly-degenerate and correspond to the spin-up and spin-down states of the impurity. It is to be expected that the solutionshould have a continuous symmetry with respect to the orientation of the impu-rity spin. However, the spin-rotational invariance has been specifically brokenby the mean-field approximation through the choice of a specific quantizationaxis. The spin-rotational invariance is expected to be restored by interactionswith conduction electrons which results in resonant scattering processes thatsuccessively flip the impurity spin.

Thus, the mean-field solution of the Anderson model contains solutions withand without local magnetic moments245. The appearance of magnetic momentsof transition metal impurities in metals can be interpreted in terms of the changeof position and width of the virtual bound state246.

245P. W. Anderson, Phys. Rev. 124, 41 (1961).246A. M. Clogston, B. T. Matthias, M. Peter, H. J. Williams, E. Corenzwit and R. C.

Sherwood, Phys. Rev. 125, 541 (1962).

651

19.3 The Atomic Limit

In the case when the hybridization is set to zero, the d-orbital is entirely local-ized. The local level is entirely decoupled from the conduction band and themodel is exactly soluble. The local d level can be described in terms of theeigenstates of the d number operator. The four basis states that correspond tothe d level being unoccupied, with energy 0, two states which correspond to thed-state being occupied by one electron, with energy Ed and one state in whichthe d-level is occupied by two electrons. The doubly-occupied state has energy2 Ed + U .

The projection operators for the atomic energy states can be expressed interms of the electron number operators. The projection operator for the unoc-cupied state P0 is given by

P0 = ( 1 − nσ ) ( 1 − n−σ ) (2285)

The projection operator for the singly occupied state with spin σ, Pσ is givenby

Pσ = nσ ( 1 − n−σ ) (2286)

while the projection operator for the doubly-occupied state Pσ,−σ is simply givenby

Pσ,−σ = nσ n−σ (2287)

The projection operators Pα satisfy the definition of a projection operator

Pα Pα = Pα (2288)

as can be explicitly seen by noting that the number operators satisfy nσ nσ =nσ. The projection operators have eigenvalues of either one or zero. Further-more, since the four states are all the energy eigenstates of H, and hence forman orthogonal and complete set, one has

Pα Pβ = 0 (2289)

if α 6= β and ∑α

Pα = I (2290)

The time-ordered single-electron Green’s function Gdd;σ(t) is defined as

Gdd;σ(t, t′) = − i

h< | T dσ(t) d†σ(t′) | > (2291)

where the d-electron creation and annihilation operators are given in the Heisen-berg representation so

A(t) = exp[

+i

hH t

]A exp

[− i

hH t

](2292)

652

and T is Wick’s time-ordering operator which is defined as

T A(t) B(t′) = Θ(t− t′) A(t) B(t′) − ( 1 − Θ(t− t′) ) B(t′) A(t) (2293)

where A and B are both fermion creation and/or annihilation operators. There-fore, the Green’s function can be expressed as

Gdd;σ(t, t′) = − i

hΘ(t− t′) < | dσ(t) d†σ(t′) | >

+i

h( 1 − Θ(t− t′) ) < | d†σ(t′) dσ(t) | > (2294)

The first contribution to the time-ordered Green’s function, describes the prob-ability amplitude that an electron of spin σ added to the d state at time t′ isstill found in the state at a later time t. The second contribution describesan analogous probability amplitude that the whether the state from which anelectron of spin σ is removed at time t is still empty at a later time t′.

In the atomic limit, the Green’s function can be explicitly evaluated fromthe definition by projecting the state | > onto each of the four atomic energyeigenstates | 0 >, | dσ >, | d−σ > and | dσ d−σ >. If the state | > is projectedonto the state | 0 > where the d-level unoccupied, the Green’s function can beevaluated as

Gdd;σ(t, t′) = − i

hΘ(t− t′) < 0 | exp

[− i

hEd ( t − t′ )

]| 0 >

= − i

hΘ(t− t′) exp

[− i

hEd ( t − t′ )

](2295)

whereas if it is projected onto the state which is singly-occupied by an electronof spin σ, | dσ > , one finds

Gdd;σ(t, t′) =i

h( 1 − Θ(t− t′) ) < dσ | exp

[− i

hEd ( t − t′ )

]| dσ >

=i

h( 1 − Θ(t− t′) ) exp

[− i

hEd ( t − t′ )

](2296)

These results are not dependent on the interaction U since the excited states,respectively, consist of a single added electron or the unoccupied state. On theother hand, if one projects onto a state which is initially singly occupied by anelectron of spin −σ, |d−σ > , the Green’s function is evaluated as

Gdd;σ(t, t′) = − i

hΘ(t− t′) < d−σ | exp

[− i

h( Ed + U ) ( t − t′ )

]| d−σ >

= − i

hΘ(t− t′) exp

[− i

h( Ed + U ) ( t − t′ )

](2297)

653

whereas if the state is projected onto the doubly occupied state | dσ d−σ >,the Green’s function is expressed as

Gdd;σ(t, t′) =i

h( 1 − Θ(t− t′) ) < dσ d−σ | exp

[− i

h( Ed + U ) ( t − t′ )

]| dσ d−σ >

=i

h( 1 − Θ(t− t′) ) exp

[− i

h( Ed + U ) ( t − t′ )

](2298)

Hence, on inserting the completeness relation for the projection operators in thedefinition of the Green’s function and then evaluating the expectation value foreach of the terms, one finds that the time-dependent Green’s function can beexpressed as

Gdd;σ(t, t′) = − i

hΘ(t− t′) (1− nd,σ)

((1− nd,−σ) exp

[− i

hEd (t− t′)

]+ nd,−σ exp

[− i

h(Ed + U) (t− t′)

] )+

i

h( 1 −Θ(t− t′) ) nd,σ

((1− nd,−σ) exp

[− i

hEd (t− t′)

]+ nd,−σ exp

[− i

h(Ed + U) (t− t′)

] )(2299)

Due to the time homogeneity of the ground state, the Green’s function onlydepends on the time difference (t − t′). The Fourier transform of the Green’sfunction is calculated from

Gdd;σ(ω) =∫ ∞

−∞dt exp

[i ω ( t − t′ )

]Gdd;σ(t−t′) exp[ − i

η

h| ( t − t′ ) | ]

(2300)where a factor of exp[ − i η

h | ( t − t′ ) | ] has been inserted to make the integralconvergent. Hence, one obtains the frequency-dependent Green’s function as

Gdd;σ(ω) = (1− nd,σ)[

1− nd,−σ

h ω − Ed − i η+

nd,−σ

h ω − Ed − U − i η

]+ nd,σ

[1− nd,−σ

h ω − Ed + i η+

nd,−σ

h ω − Ed − U + i η

](2301)

The Green’s function can also be obtained by using the equations of motion.It can be shown that

i h∂

∂tGdd:σ(t) = δ(t) < | [ dσ(t) d†σ(0) + d†σ(0) dσ(t) ] | >

− i

h< | T [ dσ(t) , H(t) ]− d†σ(0) | > (2302)

654

where the first term on the right-hand side originates from the explicit time-dependence in the definition of Wick’s time-ordering operator. Since the expec-tation value is to be evaluated at equal times, and the equal-time creation andannihilation operators satisfy the anti-commutation relation

[ dσ , d†σ ]+ = 1 (2303)

one finds

i h∂

∂tGdd:σ(t) = δ(t) − i

h< | T [ dσ(t) , H(t) ]− d†σ(0) | > (2304)

The second term comes from the time-dependence of the annihilation operatorin the Heisenberg representation. If the atomic Hamiltonian is decomposed as

H = H0 + Hint (2305)

where H0 described the atomic binding energy

H0 =∑

σ

Ed d†σ dσ (2306)

and Hint describes the Coulomb interaction between a pair of opposite spinelectrons

Hint =U

2

∑σ

d†σ d†−σ d−σ dσ (2307)

Due to the commutation relation

[ dσ(t) , H0(t) ]− = Ed dσ(t) (2308)

and[ dσ(t) , Hint(t) ]− = U dσ(t) d†−σ(t) d−σ(t) (2309)

The equation of motion becomes(i h

∂

∂t− Ed

)Gdd;σ(t) = δ(t) + U Fdd;σ(t) (2310)

where Fdd;σ(t) is defined as

Fdd;σ(t) = − i

h< | T dσ(t) nd,−σ(t) d†σ(0) | > (2311)

The equation motion for the two-particle Green’s function Fdd;σ(t) is evaluatedas(i h

∂

∂t− Ed

)Fdd;σ(t) = δ(t) nd,−σ(t)− i

hU < | T dσ(t) nd,−σ(t) nd,−σ(t) d†σ(0) | >

(2312)On using the identity

nd,−σ nd,−σ = nd,−σ (2313)

655

the equation reduces to(i h

∂

∂t− Ed

)Fdd;σ(t) = δ(t) nd,−σ(t) + U Fdd;σ(t) (2314)

On Fourier transforming the pair of equations of motion for Fdd;σ and Gdd;σ,one finds the pair of coupled algebraic equations

( h ω − Ed ∓ i η ) Gdd;σ(ω) = 1 + U Fdd;σ(ω)( h ω − Ed − U ∓ i η ) Fdd;σ(ω) = nd,−σ (2315)

where the plus or minus signs, respectively, pertain to the cases where nd,σ iseither unity or zero. This set of algebraic equations is closed and can be solvedto yield

Fdd;σ(ω) =nd,−σ

h ω − Ed − U ∓ i η(2316)

and

Gdd;σ(ω) =1

h ω − Ed ∓ i η+

U nd,−σ

(h ω − Ed ∓ i η ) (h ω − Ed − U ∓ i η )

=1 − nd,−σ

h ω − Ed ∓ i η+

nd,−σ

h ω − Ed − U ∓ i η(2317)

Hence, the equations of motion yields recovers the explicit form of the solutionas

Fdd;σ(ω) = nd,−σ

(1 − nd,σ

h ω − Ed − U − i η+

nd,−σ

h ω − Ed − U + i η

)(2318)

and

Gdd;σ(ω) = (1− nd,σ)[

1− nd,−σ

h ω − Ed − i η+

nd,−σ

h ω − Ed − U − i η

]+ nd,σ

[1− nd,−σ

h ω − Ed + i η+

nd,−σ

h ω − Ed − U + i η

](2319)

The poles of the Green’s function represent the energies required to either addor remove an electron of spin σ from the d-state the system. The excitationenergy required to put an additional particle in the d shell is, therefore, eitherEd or Ed + U depending on whether the d-state of the impurity is initiallyunoccupied or singly occupied.

19.4 The Schrieffer-Wolf Transformation

If the local magnetic impurity has a narrow width and is almost completelyoccupied by one electron, then the Anderson Model can be mapped onto a

656

model of a localized magnetic moment by the Schrieffer-Wolf transformation247.The zero-th order Hamiltonian can be considered to be the terms in which thehybridization is set to zero. Thus, for the present purposes one may write

H = H0 + HV (2320)

where H0 describes the ionic d states and the conduction band states.

H0 =∑k,σ

Ek c†k,σ ck,σ +

∑σ

Ed d†σ dσ

+ U d†↑ d†↓ d↓ d↑ (2321)

The Hamiltonian HV is the hybridization which couples the local and conductionband states.

HV =∑k,σ

[V (k) c†k,σ dσ + V (k)∗ d†σ ck,σ

](2322)

The Schrieffer-Wolf transformation is based on a canonical transformation whichacts on the operators A and is of the form

A′ = exp[

+ S

]A exp

[− S

](2323)

where S is an anti-Hermitean operator. That is, the operator S satisfies

S† = − S (2324)

Thus, if the operator A is Hermitean then A′ is also Hermitean. The canonicaltransformation leads to the same expectation values if the states | Ψ > arealso transformed as

| Ψ′ > = exp[

+ S

]| Ψ > (2325)

In particular the eigenvalues of H ′ and H are identical. The Schrieffer-Wolftransformation S is chosen such that terms linear in the hybridization HV vanishin the transformed Hamiltonian H ′. This can only be achieved if S is assumedto be of the same order as HV . In this case, the transformed Hamiltonian canbe expanded in powers of HV and S. On retaining the terms up to second-order,one finds

H ′ = H0 + HV + [ S , H0 ]

+ [ S , HV ] +12!

[ S , [ S , H0 ] ] + . . . (2326)

247J. R. Schrieffer and P. A. Wolff, Phys. Rev. 149, 491 (1966).

657

The operator S is chosen such that the terms linear in V (k) vanish. Hence, itis required that S satisfies the linear equation

[ S , H0 ] = − HV (2327)

This is an operator equation, and S is determined if all its matrix elements areknown. This requires that a complete set of states be used. The simplest set ofcomplete sets correspond to the eigenstates of H0, | φn > with eigenvalues En.In this case, matrix elements are found as

< φm | S | φn > =< φm | HV | φn >

Em − En(2328)

Thus, the operator S connects states which differ through the presence of anadditional conduction electron and a deficiency of an electron in the local orbital,and vice versa. The energy denominators are of the form Ek − Ed or Ek −Ed − U depending on the state of occupation of the local level. Thus, the anti-Hermitean operator S can be expressed in terms of the four creation operatorsfor the localized level. The operator S is found as

S =1√N

∑k,σ

[V (k)

c†k,σ dσ

Ek − Ed

(1 − d†−σ d−σ

)

+ V (k)c†k,σ dσ

Ek − Ed − U

(d†−σ d−σ

)− V (k)∗

d†σ ck,σ

Ek − Ed

(1 − d†−σ d−σ

)− V (k)∗

d†σ ck,σ

Ek − Ed − U

(d†−σ d−σ

) ](2329)

Having determined the operator S, the Hamiltonian to second-order in V isgiven by

H ′ = H0 + [ S , HV ] +12!

[ S , [ S , H0 ] ] + . . .

= H0 +12!

[ S , HV ] + . . . (2330)

The transformed Hamiltonian H ′ contains an interaction term whereby theconduction electrons are scattered from the different singly occupied states ofthe d impurity. On expressing the conduction band factors in terms of thematrix elements of the Pauli-spin matrices

sαk,k′ =

12

∑δ,γ

c†k,δ < δ | σα | γ > ck′,γ (2331)

658

and likewise for the local operators

Sα =∑δ,γ

d†δ < δ | Sα | γ > dγ (2332)

one finds that in addition to a potential scattering term there is also an interac-tion between the components of the spin density operators. Since the directionof the axis of spin quantization can be chosen arbitrarily, the interaction mustbe invariant under global rotations of the spin directions. The spin-flip contri-bution of the interaction is of the form

H ′spin−flip =

12

∑k,k′,σ

[ (1

Ek′ − Ed

)−(

1Ek′ − Ed − U

) ]

×(V (k) V (k′)∗ c†k′,−σ ck,σ d

†σ d−σ + V (k)∗ V (k′) c†k,σ ck′,−σ d

†−σ dσ

)(2333)

To describe scattering of electrons close to the Fermi energy one may set Ek =Ek′ , then the effective exchange interaction has the strength

Jk,k′ = 2 <e(V (k) V (k′)∗

) [1

Ed − Ek− 1

U + Ed − Ek

](2334)

The total spin-dependent part of the interaction is recognized as just involvingthe scalar product of the Fourier components of the two spin densities.

Hint = −∑k,k′

Jk,k′ S . σk,k′(0) (2335)

For a singly occupied level, where Ed − µ is negative, the coefficient Jk,k′ alsohas a negative sign if U is sufficiently large, so that the energy is lowered when-ever the expectation values of both the spin density operators are anti-parallel.Thus, classically, the energy is lowered whenever the polarization produced bythe conduction electron gas is anti-parallel to the spin of the local moment. Thistype of coupling is known as an anti-ferromagnetic interaction. The alternativetype of coupling occurs when the sign of Jk,k′ is positive, and the ferromagneticinteraction attempts to polarize the conduction electron spin density to be par-allel to the local spin density.

19.4.1 The Kondo Hamiltonian

The resulting Hamiltonian is the Kondo Hamiltonian248, it contains an inter-action between the localized magnetic moment and the spins of the conduction248J. Kondo, Prog. Theor. Phys. 32, 37 (1964).

659

electrons. The Hamiltonian can be expressed as

H = H0 + Hint (2336)

where H0 represents the Hamiltonian for the conduction electrons

H0 =∑k,σ

Ek c†k,σ ck,σ (2337)

and the interaction is given by

Hint = − J S . s(0) (2338)

where S is a local moment and s(0) is the spin of the conduction electrons atthe position of the impurity spin. The components of the conduction electronspin is given in terms of matrix elements of the Pauli-spin matrices

sα(0) =1

2 N

∑k,k′;γ,δ

c†k,δ < δ | σα | γ > ck′,γ (2339)

It is convenient to write the spin-dependent interaction in terms of the spinraising and lowering operators for the local spin and the conduction electronspin density

S± = Sx ± i Sy

s± = sx ± i sy (2340)

with the aid of the identity

S . s = Sz sz +12

( S+ s− + S− s+ ) (2341)

Hence, the interaction is written as

Hint = − J

N

12

∑k,k′

[Sz ( c†k,↑ ck′,↑ − c†k,↓ ck′,↓ )

+ S+ c†k,↓ ck′,↑ + S− c†k,↑ ck′,↓

](2342)

where J is expected to be negative.

19.5 The Resistance Minimum

The Kondo effect249 results in a minimum in the resistivity of metals. The mini-mum in the resistivity is due to the increasing T 5 resistivity caused by electron-phonon scattering and a decreasing contribution from the impurity spin-flipscattering, which in an intermediate temperature regime follows a ln T varia-tion

ρ(T ) = ρ(0) + b T 5 + c ρ1 Jρ(µ) S ( S + 1 ) ln kBT ρ(µ) (2343)

660

Figure 268: The minimum in the resistivity of alloys containing Fe impuri-ties. The resistivities are normalized at T = 4.2 K. [After M.P. Sarachik et al.(1964).].

k k

k'

S+ S-

k kk'

S+S-

Figure 269: The second-order contributions to the T-matrix for spin-flip scat-tering of an up-spin electron (solid blue) by a localized spin (red). The dottedblue line indicates an intermediate state with a hole in state k′.

where c is the concentration of impurities. Then, for negative J , the resistivityshows a minimum with depth proportional to c which occurs at a concentration-dependent temperature

Tmin =(ρ1 |J | ρ(µ) S(S + 1)

5 b

) 15

c15 (2344)

in agreement with experimental findings250.

The lnT term in the resistivity comes from scattering process to third-orderin J . This can be seen by considering the T-matrix for non spin-flip scatteringof an up-spin electron in second-order. The T-matrix will be evaluated on theenergy shell Ek = Ek′ , and E will be set to the ground state energy. To lowestorder, the non spin-flip scattering matrix elements are given by

< k′ ↑ | T (1)(E + iη) | k ↑ > = − J

2 NSz (2345)

whereas to second-order, one finds four non-zero contributions, two contribu-tions from the spin-flip part ( S± ) of the interactions and two contributions249J. Kondo, Prog. Theor. Phys. 32, 37 (1964).250A. D. Caplin and C. Rizzuto, Phys. Rev. Lett. 21, 746 (1968).

661

from the non spin-flip part ( Sz ). The non spin-flip part gives rise to a term inT (2)(E + iη) of

< k′ ↑ | T (2)zz (E + iη) |k ↑ > =

(J

N

)2 (Sz

2

)2 ∑k1,k2

< k′ ↑ | ( c†k1,↑ ck′1,↑ − c†k1,↓ ck′1,↓ )

× 1E − H0 + iη

( c†k2,↑ ck′2,↑ − c†k2,↓ ck′2,↓ ) | k ↑ >

(2346)

As only the spin-up terms contribute to the scattering of the spin-up electronthe term simplifies to yield

=(J

N

)2 (Sz

2

)2 ∑k1,k2

< k′ ↑ | c†k1,↑ ck′1,↑1

E − H0 + iηc†k2,↑ ck′2,↑ | k ↑ >

(2347)

This has two contributions, one which corresponds to k′ = k1 and k = k′2 andthe other with k′ = k2 and k = k′1. The sum of these terms are evaluated as

=(J

N

)2 (Sz

2

)2 ∑k2

1 − f(Ek2)

Ek − Ek2+ iη

−(J

N

)2 (Sz

2

)2 ∑k1

f(Ek1)

Ek1− Ek + iη

=(J

N

)2 (Sz

2

)2 ∑k1

1Ek − Ek1

+ iη(2348)

The singularity at Ek1= Ek yields a finite result when integrated over k1.

Thus, there is no non-analytic behavior originating from the Sz terms in theinteraction, which is just of the order J2 ρ(µ) which is just a factor of J ρ(µ)smaller than the leading contribution to the T-matrix.

The two spin-flip contributions to the T-matrix are given by

< k′ ↑ | T (2)+−(E + iη) | k ↑ > =

(J

2 N

)2 ∑k1,k2

< k′ ↑ | S+c†k1,↓ck′1,↑1

E − H0 + iηS−c†k2,↑ck′2,↓| k ↑ >

(2349)

and

< k′ ↑ | T (2)−+(E + iη) | k ↑ > =

(J

2 N

)2 ∑k1,k2

< k′ ↑ | S−c†k1,↑ck′1,↓1

E − H0 + iηS+c†k2,↓ck′2,↑| k ↑ >

(2350)

662

respectively. These terms are calculated to be

< k′ ↑ | T (2)+−(E + iη) | k ↑ > =

(J

2 N

)2 ∑k1,k2

< k′ ↑ | c†k1,↓ck′1,↑S+ S−

E − H0 + iηc†k2,↑ck′2,↓| k ↑ >

=(

J

2 N

)2

S+ S−∑k2

f(Ek2)

Ek − Ek2+ iη

(2351)

and

< k′ ↑ | T (2)−+(E + iη) | k ↑ > =

(J

2 N

)2 ∑k1,k2

< k′ ↑ | c†k1,↑ck′1,↓S− S+

E − H0 + iηc†k2,↓ck′2,↑| k ↑ >

=(

J

2 N

)2

S− S+∑k1

1 − f(Ek1)

Ek − Ek1+ iη

(2352)

In this case, the two terms cannot be combined to give a result independent ofthe Fermi function, as S+ and S− do not commute. In this case, one can usethe identities

S+ S− = S ( S + 1 ) − ( Sz )2 + Sz

S− S+ = S ( S + 1 ) − ( Sz )2 − Sz (2353)

The terms proportional to S ( S + 1 ) − ( Sz )2 combine to yield an analyticcontribution to the T-matrix of

< k′ ↑ | T (2)sf (E + iη) | k ↑ > =

(J

2 N

)2

( S ( S + 1 ) − ( Sz )2 )∑k1

1Ek − Ek1

+ iη

(2354)

whereas the remaining contribution is proportional to Sz and the integration isdivergent at Ek = Ek1

but the integration is cut off by the Fermi function.

< k′ ↑ | T (2)sf (E + iη) | k ↑ > =

(J

2 N

)2

Sz∑k1

2 f(Ek1) − 1

Ek − Ek1+ iη

(2355)

At finite temperatures, either the Fermi function provides a cut-off for the sin-gularity when the scattered particle is on the Fermi surface Ek = µ, or if thescattered particle is off the Fermi surface, the excitation energy acts as a cut off.In the latter case, the second-order contribution to the real part of the T-matrixcan be evaluated as

< k′ ↑ | T (2)sf (E + iη) | k ↑ > ∼ − 2

N

(J

2

)2

Sz ρ(µ) ln∣∣∣∣ ( Ek − µ ) ρ(µ)

∣∣∣∣(2356)

663

Figure 270: The impurity (Fe) contribution to the electrical resistivity of CuFealloys as a function of lnT . [After M.D. Daybell and W.A. Steyert (1967).].

which is divergent when Ek approaches µ. Thus, this second-order term can beas large as the first-order term which is also proportional to Sz. The scatteringrate which enters into the resistivity is proportional to the thermal average ofthe square of the T-matrix, and involves all possible final spin states of thescattered electron (i.e. k′, ↑ and k′, ↓). On noting that for non-polarized spins

< ( Sz )2 > =13S ( S + 1 ) (2357)

one finds that the rate of scattering from magnetic impurities with concentrationc is given by

1τ

= 32πhc ρ(µ)

(J

2

)2S ( S + 1 )

3

(1 + 2 J ρ(µ) ln

∣∣∣∣ kB T ρ(µ)∣∣∣∣ + . . .

)(2358)

The factor of three in front occurs since the spin-flip scattering rate (k, ↑) →(k′, ↓) is twice as large as the non-spin-flip scattering rate (k, ↑) → (k′, ↑). Theabove scattering rate gives rise to a logarithmically increasing resistivity formagnetic impurities in simple metals. Since the logarithmic divergence is causedby spin-flip scattering in the intermediate states, the application of a field shouldsuppress the Kondo effect. The resistivity does not diverge at T = 0 and, hence,the exact T-matrix also does not diverge. The logarithmic dependence found inperturbation theory saturates when all the logarithmically divergent scatteringprocesses are taken into account. The leading-order logarithmic coefficient ofeach term in the perturbation expansion series (in powers of J ρ(µ)) can becalculated by various means251. In the ferromagnetic case, where J > 0, thesaturation occurs at a characteristic Kondo energy or Kondo temperature TK

given by

J ρ(µ) ln∣∣∣∣ kB TK ρ(µ)

∣∣∣∣ = − 1 (2359)

251A. A. Abrikosov, Physics 2, 5 (1965).

664

0

0.25

0.5

0.75

1

-1 -0.5 0 0.5 1

Jz ρ(µ)

J + ρ(µ)

J+2- Jz

2 = Const

FAF

Fixed Point

Figure 271: The renormalization flow of the anisotropic Kondo Hamiltonianas calculated with a cut-off renormalization scheme. The system is shown toeither scale to ferromagnetic Ising fixed points (F) or to the strong-couplingantiferromagnetic fixed point. [After Anderson (1970).]

or

kB TK = ρ(µ)−1 exp[− 1

|J | ρ(µ)

](2360)

and all the results are finite. The above formula also defines the Kondo tem-perature when the local exchange interaction J is antiferromagnetic.

For the case of anti-ferromagnetic coupling, the physics scales to a strongcoupling fixed point252 so the solution must be obtained by other means suchas Bethe-Ansatz253. The properties of the anti-ferromagnetic solution includethe cross-over from a high temperature (T > TK) Curie susceptibility for thefree impurity moments to a Pauli paramagnetic susceptibility for T < TK .Also, the specific heat originating from the impurity changes from a constantvalue at high temperatures to a low-temperature form having a linear T de-pendence. This indicates that the magnetic moments of the impurity are beingremoved and that at low temperatures, the properties are those of a narrowvirtual bound state of width kB TK located near the Fermi energy254. In facta variational analysis shows255 that the magnetic moments are being screenedby a compensating polarization of conduction electrons, and that the cloud andmoment form a singlet bound state of binding energy kB TK . For T < TK

the conduction electrons occupy the bound state and the moment is screened,for T > TK the bound state is thermally depopulated and the system ex-hibits properties of the free moments. From the perspective of the Anderson252P. W. Anderson, J. Phys. C, 3, 2436 (1970).253N. Andrei, K. Furuya and J. H. Lowenstein, Rev. Mod. Phys. 5, 331 (1983), A. M.

Tsvelik and P. B. Weigmann, Adv. in Phys. 32, 453 (1983), P. Schlottmann, Phys. Reps.181, 1 (1989).254M. A. Daybell and W. A. Steyert, Rev. Mod. Phys. 40, 380 (1968), D. K. Wohlleben

and B. R. Coles, in Magnetism, eds. G. T. Rado and H. Suhl, Academic Press, N.Y. 1973.255K. Yosida, Phys. Rev. 147, 223 (1966).

665

0.1

0.15

0.2

T χ/

(gμ B

)2

0

0.05

0 5 10 15 20 25

k B

T/TK

Figure 272: The universal temperature dependence of the effective local mag-netic moment squared (blue) and kB TK times the reduced impurity suscepti-bility (red) for the S = 1

2 Kondo Model. The Kondo effect quenches the localmoments at low temperatures.

T = 0 Density of States

0

5

10

15

20

0 0.5 1 1.5 2 2.5 3E [ Energy ]

ρ(E

) [ S

tate

s / E

nerg

y ]

Ν ∆

Ef Ef + Uµ

kBTK

∆

Figure 273: The schematic (T = 0) electronic density of states for an N -foldspin and orbitally degenerate Anderson impurity model, as appropriate for Ceimpurities. The Coulomb interaction splits the localized density of states ρ(E)into a singly-occupied level at Ef and an unoccupied level at Ef +U . In addition,there is a narrow Abrikosov-Suhl resonance just above the Fermi-energy µ.

666

impurity model, the density of states that is found at high temperatures followsdirectly from Anderson’s picture of a spin-split virtual bound state. However,as T decreases below TK , the density of states shows a sharp peak of widthkB TK growing in the vicinity of µ. In the low-temperature limit, the height ofthe Abrikosov-Suhl256 peak saturates on the order of ( kB TK )−1. Thus, thelow-temperature properties can be directly understood in terms of the virtualbound state with a density of states which is very large ∝ T−1

K . The propertiesof this low-temperature Fermi liquid were established by Nozieres257.

256A. A. Abrikosov, Physics 2, 5 (1965), H. Suhl, Physics 2, 39 (1965).257P. Nozieres, Ann. Phys., 10, 19 (1985).

667

20 Collective Phenomenon

21 Itinerant Magnetism

21.1 Stoner Theory

The Stoner theory of itinerant magnetism258 examines the stability of a bandof electrons to Coulomb interactions. The Hamiltonian is expressed as the sumof two terms, H0 the non-interacting electrons in the Bloch states and Hint

describing the Coulomb repulsion between the electrons

H = H0 + Hint (2361)

The Hamiltonian for the non-interacting electrons in the Bloch states is writtenas

H0 =∑k,σ

Ek nk,σ (2362)

The interaction Hamiltonian is given by

Hint =U

2

∑i,σ

ni,σ ni,−σ (2363)

where U represents the short-ranged Coulomb interaction between a pair ofelectrons occupying the orbitals on the i-th lattice site. The operator ni,σ cor-responds to the number of electrons of spin σ which occupy the i-th lattice site.It is assumed that the band is non-degenerate, therefore, there is only one or-bital per lattice site which due to the limitations imposed by the Pauli exclusionprinciple can only hold a maximum of two electrons.

The interaction is treated in the mean-field approximation. First it shall beassumed that translational invariance holds, so that the orbitals in each unitcell have the same occupation numbers. Also the Hamiltonian is expanded inpowers of the fluctuation operator ∆ni,σ = ni,σ − nσ so that

Hint =U

2

∑i,σ

[∆ni,σ ∆ni,−σ + nσ ∆ni,−σ + n−σ ∆ni,σ + n−σ nσ

](2364)

and then the second-order fluctuations are ignored. This leads to the interactionenergy being approximated in terms of single-particle operators

Hint ≈ U

2

∑i,σ

[ni,σ n−σ + ni,−σ nσ − n−σ nσ

]

=U

2

∑k,σ

[nk,σ n−σ + nk,−σ nσ − n−σ nσ

](2365)

258E. C. Stoner, Rep. Prog. in Phys. 11, 43 (1948).

668

Thus, in the mean-field approximation, the Hamiltonian is given by

HMF =∑k,σ

( Ek + U n−σ ) nk,σ − NU

2

∑σ

n−σ nσ (2366)

The single particles have the spin-dependent energy eigenvalues

Eσ(k) = Ek + U n−σ (2367)

The magnetization is given by

Mz = g µB12

(n↑ − n↓

)= µB

∫ ∞

−∞dε f(ε)

(ρ(ε − U n↓) − ρ(ε − U n↑)

)(2368)

This equation has non-magnetic solutions with n↑ = n↓ and may have fer-romagnetic solutions in which the number of up-spin electrons is greater thanthe number of down spin electrons n↑ 6= n↓. In the ferromagnetic state, theStoner model predicts that the up-spin sub-bands are rigidly shifted relativelyto the down-spin bands by the exchange splitting ∆ which has a magnitude ofU ( n↑ − n↓ ). On increasing U from zero, the ferromagnetic solutions first

-1.5

-1

-0.5

0

0.5

1

1.5

-4 -2 0 2 4 6ε

∆

ρ↑(ε)

−ρ↓(ε)

εF

Figure 274: A schematic depiction of the spin sub-band density of states, ex-pected for a ferromagnetic solution of the Stoner model.

become stable at T = 0 when Mz ∼ 0. For an infinitesimal magnetization, theequation can be linearized to yield

( n↑ − n↓ ) = U

(n↑ − n↓

) ∫ ∞

−∞dε f(ε)

∂

∂ερ(ε) (2369)

669

since the higher-order terms are negligible. On canceling a factor of Mz fromboth sides of the equation, one finds a self-consistency equation which is satisfiedfor infinitesimal magnetizations. The ferromagnetic state has the lowest energywhen the self-consistency equation is satisfied. The integral in the resultingself-consistency equation can be performed via integration by parts yielding

1 = U

∫ ∞

−∞dε f(ε)

∂

∂ερ(ε)

= − U

∫ ∞

−∞dε ρ(ε)

∂

∂εf(ε) (2370)

At low temperatures, the derivative of the Fermi function can be replaced by adelta function at the Fermi energy.

− ∂

∂εf(ε) = δ(ε − µ) (2371)

This yields the Stoner criterion for ferromagnetism as

1 < U ρ(µ) (2372)

where ρ(µ) is the density of states per spin at the Fermi energy.

If the Stoner criterion is satisfied the paramagnetic state is unstable to theferromagnetic state, and a spontaneous magnetic moment Mz occurs in the tem-perature range 0 ≤ T ≤ Tc. The magnetization is given by the solution of thenon-linear equation, eqn(2368). The non-linear equation shows that the magne-tization increases with increasing U , and saturates to a value which is one Bohrmagneton per electron, for low density materials which the bands have a fillingof less than one electron per atom. In systems which have bands that are more

0.6

0.8

1

Uρ(μ) > 1

0

0.2

0.4

0 0.2 0.4 0.6 0.8 1

M/μB

Figure 275: A graphical solution of the non-linear equation for the magnetiza-tion M , when the Stoner criterion U ρ(µ) > 1 is satisfied.

than half-filled, the saturation magnetic moment is equal to a Bohr magneton

670

per unoccupied state. At finite temperatures, the value of the magnetizationis reduced and disappears at a critical temperature Tc. Unfortunately, Stonertheory does not predict reasonable values for the critical temperatures.

In the paramagnetic state, Stoner theory predicts that the (T = 0) suscep-tibility should be exchange enhanced over the non-interacting susceptibility χ0

p

via

χp =χ0

p

1 − U ρ(µ)(2373)

For systems which are close to the ferromagnetic instability, the susceptibilityshould take on large values. This is the case for Pd in which the d band isalmost completely occupied.

——————————————————————————————————

21.1.1 Exercise 90

Determine the critical temperature Tc predicted by Stoner theory.

——————————————————————————————————

21.1.2 Exercise 91

Determine the paramagnetic susceptibility by using Stoner theory.

——————————————————————————————————

21.2 Linear Response Theory

The spatially varying magnetization Mz(r) of a paramagnetic system producedby a spatially varying applied magnetic field Hz(r) can be expressed in termsof the z− z component of the magnetic susceptibility tensor, through the linearrelationship

Mz(r) =∫

d3r′ χz,z(r, r′) Hz(r′) (2374)

This is a special case of the more general relation

Mα(r) =∫

d3r′∑

β

χα,β(r, r′) Hβ(r′) (2375)

For translational invariant systems, the expression for the response function isonly a function of the difference r − r′. Also for non-magnetic systems, that

671

possess spin rotational invariance, the susceptibility tensor is diagonal and thediagonal components are related via

χx,x(r − r′) = χy,y(r − r′) = χz,z(r − r′) (2376)

The relation between the magnetic response and the applied field becomes sim-pler, after Fourier transforming. The Fourier transform of the magnetization isdefined as

M(q) =∫d3r exp

[− i q . r

]M(r) (2377)

The Fourier transform of the magnetization is related to the Fourier transformof the applied field via

Mα(q) =∑

β

χα,β(q) Hβ(q) (2378)

The response function can be evaluated from perturbation theory, in which theZeeman interaction

HZeeman = −∫

d3r′ M(r′) . H(r′)

= −∫

d3q M(q) . H(q) (2379)

is treated as a small perturbation.

For convenience, χz,z(q) shall be calculated by reducing it to a previouslyknown case. The change in density of electrons of spin σ, with Fourier com-ponent q, produced in response to an applied spin-dependent potential. TheFourier component of the potential is given by

Vσ(q) = − gµB

2Hz(q) σ + U ρ−σ(q) (2380)

Thus, the charge density is given by the two coupled equations

ρσ(q) = χ0(q)(− g µB

2Hz σ + U ρ−σ(q)

)(2381)

one for each spin polarization. In the above expression, χ0(q) is the Lind-hard density-density response function, per spin. The z-component of the q-dependent magnetization is defimed as

Mz(q) =(g µB

2

) (ρ↑(q) − ρ↓(q)

)(2382)

On combining these equations, one finds that the z-component of the q-dependentmagnetization produced by a magnetic field applied along the z-direction is givenby

Mz(q) = − g µB

2χ0(q)

(g µB Hz(q) + U

2 Mz(q)g muB

)(2383)

672

Thus, it is found that the Pauli paramagnetic susceptibility is given by

χz,zp (q) =

Mz(q)Hz(q)

= − g2 µ2B

42 χ0(q)

1 + U χ0(q)(2384)

It is usual to use re-write this expression in terms of the reduced non-interactingmagnetic susceptibility defined by

χz,z0 (q) =

χz,z0 (q)g2µ2

B

(2385)

which results inχz,z

0 (q) = − 12χ0(q) (2386)

instead of the density-density response function χ0(q). This yields the result

χz,zp (q) =

(g µB

2

)2 4 χz,z0 (q)

1 − 2 U χz,z0 (q)

(2387)

Since the reduced non-interacting magnetic susceptibility is positive, and U ispositive, the paramagnetic susceptibility is enhanced for sufficiently small valuesof U .

21.3 Magnetic Instabilities

The reduced non-interacting susceptibility, χz,z0 (q), may have maxima at cer-

tain values of q, say Q, which are determined by the band structure and theoccupancy of the non-interacting bands. If the values of the non-interactingsusceptibility at these maxima are finite, then the denominator of the Pauli-paramagnetic susceptibility may become small at these q values, for sufficientlysmall values of U . This has the effect that, for small U , the Pauli-paramagneticsusceptibility is enhanced at these Q values. If U is increased further, therewill be a critical value of U , Uc at which point the denominator will fall tozero and the susceptibility at Q will become infinite. The divergence of thesusceptibility at Q indicates that an infinitesimal applied field can produce afinite staggered magnetization Mz(Q). Although this argument assumed thatthe z-axis was the axis of quantization, the argument could have been made ifthe quantization axis was assumed to be along any arbitrarily chosen direction.The infinitesimal field may be produced by a spontaneously statistical fluctua-tion, and have an arbitrary direction. This field will force the system to ordermagnetically by having a finite M(Q) in the spontaneously chosen direction.The system, by spontaneously choosing a direction for the magnetization, hasspontaneously broken the symmetry of the Hamiltonian.

673

The critical value of U , above which the paramagnetic state becomes unsta-ble to a state with a modulated spin density M(Q), is given by

1 = 2 Uc χz,z0 (Q) (2388)

If a non-interacting system is considered which has a maximum in χz,z0 (Q) at

Q = 0 the above expression reduces to the Stoner criterion for ferromagnetismas

limQ → 0

χz,z0 (Q) → 1

2ρ(µ) (2389)

where ρ(µ) is the density of states, per spin, at the Fermi energy. Thus, whenQ = 0, it is found the critical value of U is given by the criterion

1 = Uc ρ(µ) (2390)

and for values of U larger than the critical value the paramagnetic state is un-stable to the formation of a ferromagnetic state.

For values of U greater than Uc, the mean-field analysis has to be modifiedto include the effect of the spontaneous magnetization. For a ferromagnet, theinteraction produces a rigid splitting between the up-spin bands and down-spinbands by an amount ∆ = U ( n↑ − n↓ ) called the exchange splitting. Forisotropic systems, the magnetic response will crucially depend on the directionof the applied field compared to that of the spontaneous magnetization. Fora ferromagnet, the longitudinal response (produced by a field which is parallelto the spontaneous magnetization M) will be finite, as this corresponds to pro-cesses which excites the system as it stretches the magnitude of M . However,the transverse response will be infinite as this corresponds to applying a fieldthat will rotate the direction of the spontaneous magnetization until it alignswith the applied field. As the system is isotropic, this global spin rotation canbe achieved without requiring any finite energy excitations. The zero-energyexcitations that uniformly rotate the magnetization in a ferromagnet are theq = 0 Goldstone modes associated with the spontaneously broken continuousspin rotational invariance of the Hamiltonian.

In three-dimensional systems with almost spherical Fermi surfaces, the in-stability can only occur at 2 kF . This can lead to a spin density wave whichhas a periodicity which is incommensurate with the underlying lattice. In low-dimensional systems, such as two-dimensional and one-dimensional organic ma-terials, there can be large sheets of the Fermi surface which can produce a largenon-interacting susceptibility at the Q value connecting these sheets. The largeresponse of the system to the interaction can produce a spin density wave inwhich the magnetization is modulated with this wave vector.

For tight-binding bands which satisfy the perfect nesting condition

Ek+Q = − Ek (2391)

674

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1kxa/π

k ya/π

Q

3 2 1 0

-1 -2 -3

Figure 276: The nesting vector Q = 2πa ( 1

2 ,12 .0) for a perfectly nested two-

dimensional tight-binding model. The constant energy surfaces are depicted ascontours.

for some Q, the non-interacting susceptibility can be evaluated as an integralover the density of states

χz,z0 (Q) =

12

∑k

f(−Ek) − f(Ek)2 Ek

=12

∫ ∞

−∞dε ρ(ε)

f(−ε) − f(ε)2 ε

= − 12

∫ ∞

−∞dε ρ(ε)

f(ε)ε

(2392)

sine ρ(−ε) = ρ(ε). From this one can see that if the density of states ρ(0) isnon-zero, the susceptibility χz,z

0 (Q) will diverge logarithmically or faster thanlogarithmically when µ → 0. The divergence occurs when two large portions ofthe Fermi surface are connected by the wave vectorQ, which allows the system torearrange the electrons at the Fermi surface by zero-energy excitations involvinga momentum change Q. Thus, in this case, there is no energy penalty to beincurred in producing a spin density wave M(Q). The perfect nesting conditionoccurs at Q = π

a (1, 1, 1) for non-degenerate tight-binding bands with a simplecubic lattice, where

Ek = − 2 ti=3∑i=1

cos ki a (2393)

Since the bands are symmetric around ε = 0, the non-interacting susceptibil-

675

0.00

0.10

0.20

0.30

0.40

0.50

-6 -3 0 3 6

µ/t

U-1

Figure 277: The condition that a paramagnet becomes unstable relative to aNeel state with ordering vector 2π

a ( 12 ,

12 ,

12 ), for three-dimensional tight-binding

band, is satisfied for µ values close to half-filling.

ity diverges for half-filled bands. In this case, the critical value of U is zero.Hence, the paramagnetic state will become unstable to a state in which themagnetization exhibits spatial oscillations with wave vector Q, even with aninfinitesimally small value of U . In real space, the staggered magnetization ofthis ordered state is given by

M(r) = MQ

∑i

cos πria

(2394)

The magnetic moments on the nearest-neighboring lattice sites are antiparal-lel, which constitutes an anti-ferromagnetic ordering. Since anti-ferromagneticordering was first proposed by Louis Neel to describe classical magnets259, thistype of ordering is known as Neel ordering. Unfortunately, the Neel state is notan exact ground state for a quantum system.

The occurrence of an anti-ferromagnetically ordered state may be accompa-nied by a metal-insulator transition. This process was first discussed by J. C.Slater260. Physically, the appearance of anti-ferromagnetic order could result ina doubling of the size of the real space unit cell. The electrons of spin σ travelingin the solid experience a periodic potential which contains a contribution due tothe interaction with electrons of opposite spin. The doubling of the size of thereal space unit cell, produced by the magnetic order, results in the volume of theBrillouin zone being halved. The new periodicity caused by the sub-lattice mag-netization shows up as a spin-dependent contribution to the potential, and mayproduce gaps (∆ ∝ U MQ) in the electronic dispersion relations at the surface

259L. Neel, Ann. de Physique, 17, 64 (1932), Ann. de Physique 5, 256 (1936).260J. C. Slater, Physical Review, 82, 538, (1951).

676

-2.5

-1.5

-0.5

0.5

1.5

2.5

-1 -0.5 0 0.5 1

ka/π

E kα/t

Ek+

Ek-

2∆

Figure 278: A schematic description of the mean-field electronic bands for a Neelantiferromagnet. The dashed lines indicates the electronic bands of the high-temperature paramagnetic state. In the Neel state, the size of the Brillouin zoneis halved, and an energy gap of 2 ∆ appears at the boundary of the magneticBrillouin zone.

of the new Brillouin zone. If the magnitude of the spin-dependent potential islarge enough, a gap may occur all around the Brillouin zone resulting in a gapin the density of states. If the Fermi energy lies, in the gap, as is expected for ahalf-filled band, the state will be insulating. Such insulating anti-ferromagneticstates occur in undoped La2CuO4, which is the parent material of some high-temperature superconductors. Although the insulating anti-ferromagnetic statedoes not have low-energy electronic excitations, it does have low-energy spinexcitations in the form of Goldstone modes. These are spin waves, which havethe dispersion relation ω = c q.

21.4 Spin Fluctuations near Ferromagnetic Instabilities

The dynamical magnetic response of a paramagnetic system to a time and spa-tially varying applied magnetic field of wave-vector q and frequency ω is givenby the dynamical response χz,z

p (q;ω). The imaginary part of this response func-tion yields the spectrum of magnetic excitations. The imaginary part of thereduced susceptibility can be measured directly by inelastic neutron scatteringexperiments, in which the neutron’s spin interacts with the electronic spin den-sity via a dipole-dipole interaction. A simple extension of our previous analysisshows that, in the mean-field approximation, the response for a paramagneticmaterial is given by

χz,zp (q;ω) =

(g µB

2

)2 4 χz,z0 (q, ω)

1 − 2 U χz,z0 (q;ω)

(2395)

677

Let us examine the imaginary part of the response function for a paramagneticmetal, such as Pd, which is on the verge of an instability to a ferromagneticstate. Then,

=m[χz,z

p (q;ω)]

=(g µB

2

)2 4 =m χz,z0 (q, ω)[

1 − 2 U <e χz,z0 (q;ω)

]2+[

2 U =m χz,z0 (q;ω)

]2(2396)

which on using the approximation to the Lindhard susceptibility,

<e χz,z0 (q;ω) ≈ 1

2ρ(µ)

=m χz,z0 (q;ω) ≈ π

8ω

q vFρ(µ) (2397)

shows that the system exhibits a continuum of quasi-elastic magnetic excita-tions. As the value of U is increased so that a ferromagnetic instability isapproached, the spectrum is enhanced at low frequencies. These magnetic exci-

0

1

2

3

4

5

0 0.5 1 1.5 2ω/εF

Im χ

(q; ω

) [un

its o

f εF]

0.950.90.850.70.0

q/kF=0.7U/Uc

Figure 279: The spectrum of magnetic excitations with momentum transferq/kF = 0.7 for a paramagnet for various values of the Coulomb interaction U .

tations are known as paramagnons. The lifetime of the paramagnon excitationsand the frequency of the excitations soften as the value of U is increased towardsthe critical value Uc. Basically, this represents a slowing down of the rate atwhich a small region of ferromagnetically aligned spins relax back to the equi-librium (paramagnetic) state. The existence of large amplitude paramagnonfluctuations not only manifest themselves in the inelastic neutron scatteringcross-section (which is directly proportional to =m [ χα,β(q;ω) ]), and an en-hanced susceptibility but also leads to a logarithmic enhancement of the linear

678

-1

-0.5

0

0.5

1

1.5

2

0 0.5 1 1.5 2 2.5 3q/kF

ω/εF

2.0

1.5

1.0

0.5 0.25

U=0.8Uc

Figure 280: The phase space of magnetic excitations for a paramagnet withthe Coulomb interaction U = 0.8 Uc. The values of Imχ(q;ω), in units ofεF , are depicted as contours. The paramagnon-excitations are revealed as abroadened low-energy resonance with small momentum transfers. Due to thePauli exclusion principle, only excitations with ω > 0 are allowed.

T term in the electronic specific heat261, and an enhancement in the T 2 term inthe electrical resistivity262. These characteristics have been observed in metallicPd 263.

The above mean-field type of analysis has shown that close to a magneticinstability, there will be large-amplitude Gaussian fluctuations. This continuousspectrum of excitations is expected to soften as the instability is approached,261N. F. Berk and J. R. Schrieffer, Phys. Rev. Lett. 17, 433 (1966), S. Doniach and S.

Engelsberg, Phys. Rev. Lett. 17, 750 (1966).262P. Lederer and D. L. Mills, Phys. Rev. Lett. 20, 1036 (1968).263A. I. Schindler and B. R. Coles, J. Appl. Phys. 39, 956 (1968).

Figure 281: The magnetic inelastic neutron scattering cross-section for the en-hanced paramagnet Ni3Ga. [After B. R. Bernhoeft, S. M. Hayden, G. G. Lon-zarich, D. McK. Paul and E. J. Lindley, Phys. Rev. Lett. 62, 657 (1989).]

679

say by increasing U . The fluctuations are expected to be long-ranged and long-lived. However, the above mean-field analysis is expected to fail close to thetransition, where critical fluctuations should be taken into account. Unlike mostother phase transitions, the phase transition that has just been described firstoccurs at T = 0. The critical fluctuations are not thermally excited and can-not be treated classically but are zero point fluctuations associated with theexistence of a quantum critical point264.

21.4.1 Ferromagnetic Spin Waves

For values of U greater than Uc where the system is ferromagnetic, the q = 0transverse response shows a sharp zero-energy mode that represents the Gold-stone mode of the system. In the ferromagnetically ordered state, these excita-tions form a sharp (delta function-like) branch of spin waves which stretch upfrom ω = 0 at q = 0. The transverse response functions are equal

χx,x(q;ω) = χy,y(q;ω) (2398)

but, in the ferromagnetic state, differ from the longitudinal response

χx,x(q;ω) 6= χz,z(q;ω) (2399)

The transverse response can be expressed in terms of the spin-flip responsefunction involving the spin raising and lowering operators

M±(q) = Mx(q) ± i My(q) (2400)

The spin-flip response functions are

χ+,−(q;ω) = χ−,+(q;ω) = 2 χx,x(q;ω) (2401)

In the random phase approximation265, the reduced transverse dynamic suscep-tibility has the form

χ+−(q;ω) =χ+−

0 (q;ω)

1 − U χ+−0 (q;ω)

(2402)

where χ+−0 (q;ω) is the mean-field transverse susceptibility. In the ferromagnetic

state, the mean-field susceptibility is given by

χ+−0 (q;ω) =

1N

∑k

(fk↑ − fk+q↓

∆ + εk+q − εk − h ω

)(2403)

which involves the spin-split sub-bands. The spectra of spin-flip excitations isproportional to the imaginary part of the transverse susceptibility. The spectra264J. A. Hertz, Phys. Rev. B, 14, 1165 (1976).265T. Izuyama, D.J. Kim and R. Kubo, J. Phys. Soc. Jpn. 18, 1025 (1963).

680

-15

-10

-5

0

5

10

-4 -3 -2 -1 0 1 2 3 4

ka2m

a2 E

σ(k)

/ ћ2

M = 1/4

kF↑

kF↓

∆

E↓(k)

E↑(k)

Figure 282: A schematic description of the mean-field electronic bands fora ferromagnet. The dashed lines indicates the electronic bands of the high-temperature paramagnetic state. In the ferromagnetic state, the up-spin sub-band is lower than the down-spin sub-band by an energy ∆.

for spin-flip processes, which decreases the z-component of the magnetizationby h/N , is given by

Im

[χ+−(q;ω)

]=

Im χ+−0 (q;ω)

[ 1 − U Re χ+−0 (q;ω) ]2 + [ U Im χ+−

0 (q;ω) ]2

(2404)The spectra is composed of a continuum of slightly modified Stoner excitations,which exist throughout the (q, ω) region where

Im χ+−0 (q;ω) 6= 0 (2405)

For small q, this region has the form of a distorted triangle with an apex atν = ∆ at q = 0, as seen in Fig(283). For free electrons, the energy of the upperboundary of the Stoner continuum is given by

hω =h2

2m(k2

F↑ − k2F↓ + 2kF↑q + q2) (2406)

The energy of the lower boundary of the Stoner continuum is given by

hω =h2

2m(k2

F↑ − k2F↓ − 2kF↑q + q2) (2407)

For weak ferromagnets (for which M/µB is small), the Stoner continuum firstreaches ω = 0 at the wave vector qc = kF↑−kF↓ which connects the up-spin anddown-spin Fermi surfaces. Nevertheless, the imaginary part of the susceptibilityis non-zero for smaller q values at energies which are below the energy of theStoner continuum even though Imχ+−

0 (q : ω) = 0. The ω values of these smallq excitations are given by the solutions of the equation

1 − U Re χ+−0 (q;ω) = 0 (2408)

681

-1

0

1

2

0 0.5 1 1.5 2 2.5 3

q/kF

ω/ε

F

0.5 0.25

1

2

48Spin Waves

∆/εF M=0.5

Figure 283: The phase-space for spin-flip excitations. Since the mean-fieldstate is stable against single-particle excitations, the allowed spin-flip (Stoner)excitations must have ω > 0. The upper and lower boundaries of the Stonercontinuum are depicted by the blue lines. The contours of the imaginary partof the transverse susceptibility are depicted as black lines. The branch of spinwave excitations is marked by a red line.

The spectrum of magnetic excitations contains a sharp delta function located atthese points in (q, ω) space. The solutions of eqn.(2408) describes the dispersionrelation of a branch of collective modes. The above equation is guaranteed tohave a solution at ω = 0 and q = 0, since for q = 0 the above equation reducesto

1 = U Re χ+−0 (0, ω)

=U

N

∑k

(fk↑ − fk↓

∆ − hω

)= U

( n↑ − n↓ )∆ − h ω

(2409)

which at ω = 0 reduces to

1 =U ( n↑ − n↓ )

∆(2410)

The above condition is identified with the self-consistency condition for the ex-change splitting in Stoner-Wohlfarth theory. The zero-energy collective magnetization-lowering excitations are Goldstone modes266 since, when they are combined co-herently, they produce a uniform rotation of the magnetization, and, thereby,restore the spontaneously broken symmetry. By expanding eqn.(2408) for small

266J. Goldstone, Nuovo Cimento, 19, 154 (1961).

682

Figure 284: The spin wave dispersion relation inferred from inelastic neutronscattering experiments on an f.c.c. Cobolt alloy. [After R.N. Sinclair and B.N.Brockhouse, Phys. Rev. 120, 1638 (1960).]

(q, ω), the collective spin-wave excitations are found to obey the dispersion re-lation

h ω = D q2 (2411)

where the spin-wave stiffness constant D is given by the expression,

D =U

3N∆

∑k

[ (fk↑ + fk↓

2

)∇2εk −

(fk↑ − fk↓

∆

)|∇εk|2

](2412)

This dispersion relation differs from the dispersion relation of most other Gold-stone modes, which are usually linear in q, since for a ferromagnet the orderparameter is a conserved quantity. At q = 0, it is easy to see that the spectrafrom the Goldstone modes consists of a delta function located at ω = 0 with astrength governed by the value of the magnetization. From the Kramers-Kronigrelations, one finds that the real part of the q = 0 response has a simple pole atω = 0, therefore the static transverse response is divergent as is expected for anisotropic ferromagnet. For qc > q, the Goldstone modes have lower energiesthan the threshold for the Stoner excitations. For large q values, the branch ofspin-wave modes merges with the continua of Stoner excitations as a resonancewhich is then rapidly damped out.

By contrast, the uniform static longitudinal response is not divergent in theferromagnetic state. The reduced longitudinal susceptibility, evaluated within

683

-1

0

1

2

0 0.5 1 1.5 2 2.5 3

q/kF

ω/εF

21

0.5

0.25

0.125

0.125

-2qkF↑/kF2+(q/kF)

2

-2qkF↓/kF2+(q/kF)

2

2qkF↑/kF2+(q/kF)

22qkF↓/kF

2+(q/kF)2

M=0.5

Figure 285: The phase-space for longitudinal magnetic excitations. Since themean-field state is stable against single-particle excitations, the allowed non-spin-flip excitations must have ω > 0. The upper and lower boundaries ofthe continuum excitations are depicted by the blue lines. The contours of theimaginary part of the longitudinal susceptibility are depicted as black lines.

the Random Phase Approximation, is given by the expression

χzz(q;ω) =14

∑σ

(χσ(q;ω) ( 1 + U χ−σ(q;ω) )1 − U2 χσ(q;ω) χ−σ(q;ω)

)(2413)

where χσ(q;ω) represents the mean-field response of the electrons with spin σ

χσ(q;ω) =1N

∑k

(fk,σ − fk+q;σ

εk+q − εk − h ω

)(2414)

The phase space for longitudinal magnetic excitations resembles the phase spacefor paramagnetic fluctuations in that the continuum extends to ω = 0 at q = 0.Although there does exist a low-energy resonance in the continuum, for appre-ciable values of Mz/µB , its intensity is greatly reduced below the intensity ofthe resonance in the transverse fluctuations.

The Hartree-Fock approximation to the strong ferromagnetic stateMz/µB =1 is an exact eigenstate of the Hamiltonian, since when there are no down-spinelectrons the effect of Coulomb interaction is zero. Because kF↓ = 0, one has

∆ = U n↑ ≥h2 k2

F↑

2 m(2415)

and, therefore, the spectrum of Stoner excitations is either gapped for all mo-mentum transfers or at most reaches ω = 0 at the isolated point q = kF↑.

684

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2

q/kF↑

h ω/ εF ↑

Stoner Continuum

Spin Waves?

∆/εF↑

Figure 286: The possible phase-space for the spin-flip excitations of a Strong Fer-romagnet. A Strong Ferromagnet is defined as being fully polarized, Mz/µB =1.

However, even though the Hartree-Fock state is an exact energy eigenstate forMz/µB = 1, it may not be the lowest-energy state or ground state. In par-ticular, the strong ferromagnetic state is unstable for a half-filled tight-bindingband, as can be seen by examining the above expression for the spin-wave stiff-ness constant D. Since fk↑ = 1 and fk↓ = 0, the spin-wave stiffness constantreduces to

D =U

3N∆

∑k

[12∇2εk −

(1∆

)|∇εk|2

](2416)

The first-term vanishes on summing over the entire Brillouin zone and thesecond-term is manifestly negative. Hence, the spin wave stiffness constantis negative. Therefore, the strong ferromagnetic state is unstable, since byemitting spin waves with finite q the system can lower its energy. This resultis consistent with the prediction that the ground state of the half-filled tight-binding band is the Neel insulating state.

An anti-ferromagnet also has Goldstone modes, but unlike the ferromagnetthe order parameter (the sub-lattice magnetization) for the anti-ferromagnet isnot a constant of motion. This results in the dispersion of the Goldstone modesbeing linear in q, ω = c q, similar to the transverse sound waves in a crystallinesolid.

21.5 The Slater-Pauling Curves

The reduced magnetization can be obtained normalized saturated magnetization(T → 0,H → ∞) for AB alloys containing 3d atoms. When the reducedmagnetization µ/µB is plotted against the total number of 3d and 4s electrons

685

n, the curves fall on-top of straight-lines with slopes of unit magnitude267 knownas the Slater-Pauling curves268.

0

0.5

1

1.5

2

2.5

7 8 9 10 11

μ/μ B

n

FeNi

FeCo

FeCr

FeV

FeMn

CoNi

NiCu

NiCr

NiV

NiMn

Mn Fe Co Ni Cu

Figure 287: The reduced magnetization of 3d AB alloys versus the total numbern of 3d and 4s electrons. [After J. Crangle and G. C. Hallam (1963).]

The Slater-Pauling curves indicate whether the magnetic properties of thesealloys can be described by an itinerant electron model in which there is onecommon rigid band and a common exchange splitting269. The line with positiveslope can be considered as the result of the chemical potential being located inthe minimum in the down-spin density of states. A typical example of a densityof states with this minimum270 is given by spin-polarized b.c.c. Fe. The pinning

J B Staunton et al

The theory proves that the relevant one-body effectivepotential can, in principle, contain all the effects of electroncorrelations although, in practice, approximations mustbe made. All the theorems and methods of the densityfunctional (DF) formalism were soon generalized [21, 22]to deal with cases in which spin-dependent properties playan important role, such as in magnetic systems (SDF). Theenergy thus becomes a functional both of the density andof the local magnetic densitym(r). The proofs of thesetheorems are provided in the original details and there havebeen many developments of a formal nature [12, 23].

The many-body effects of the complicated quantum-mechanical problem are buried in the so-called exchange-correlation part of the energy functionalExc[n(r),m(r)].The exact solution is intractable for macroscopic systemsand some approximation must be made. The so-called‘local approximation’ is the most widely used owing toits simplicity and its success in describing the ground stateand equilibrium properties of a great many different typesof materials. The local approximation (LSDA) takes asits starting point the energy of a uniformly spin-polarizedhomogeneous electron gasεxc(n(r), |m(r)|) [21, 24] sothatExc can be written in the form

Exc[n,m] ≈∫

dr n(r)εxc(n(r), |m(r)|). (7)

The functional derivative of this quantity with respect tom(r) provides the effective magnetic fields for the single-electron equations, namely the spin-polarized band structurediscussed in the introduction:

veff [n,m; r] = V ext (r)+ e2∫

dr′n(r)

|r − r′|+ δE

xc

δn(r)[n,m] (8)

Beff [n,m; r] = Bext (r)+ δExc

δm(r)[n,m] (9)

whereV ext describes an external potential such as a latticearray of nuclei andBext an external magnetic field. Theelectron density and magnetization are given by

n(r) =∫ εF

dε∑i

tr(φ∗i (r, ε)φi(r, ε)) (10)

m(r) =∫ εF

dε∑i

tr(φ∗i (r, ε)σφi(r, ε)) (11)

where theφi(r, ε) obey the Schrodinger–Pauli (Kohn–Sham) equation

(−∇2+ veff (r)1− σ ·Beff (r))φi(r, ε) = εφi(r, ε) (12)

whereεF is the system’s Fermi energy.

3.1. Elemental metals

Electronic structure (band theory) calculations using theLSDA for the pure crystalline state are routinely performedthese days. For the elemental magnetic, transition metals,nearly rigidly exchange-split, spin-polarized bands are

0

10

10

20

20

30

30

0-0.2-0.4-0.6 0.2 0.4

Energy (Ry)

Den

sity

of

Stat

es (

stat

es p

er R

y pe

r sp

in)

Majority Spin

Minority Spin

Figure 1. The spin-polarized electronic density of states ofBCC Fe in units of states Ry−1 using the one-electronpotentials from [15].

obtained which are expected from the simpler Hubbard-model treatment described in the last section. Examplesof band theory calculations for the magnetic 3d transitionmetals BCC iron and FCC nickel can be found in the bookby Moruzzi et al [15]. The figure on p 170 of their book(we show a similar figure in figure 1) shows the density ofstates of BCC iron as a function of energy. The densitiesof states for the two spins are almost (but not quite) rigidlyshifted. As is typical for BCC structures, the d band hasthree major peaks. The Fermi energy resides in the topof the d bands for the so-called majority spins betweenthe upper two peaks. The saturation magnetization,Ms ,that results from this ‘self-consistent’ calculation is 2.2µB ,which is in good agreement with experiment.

In nickel and cobalt the majority spin d bands arecompletely occupied and the Fermi energy lies in theprominent peak in the minority spin d density of states.The d band width has been a topic for close scrutiny overthe years owing to the width extracted from photoemissionmeasurements being much smaller than that of bandstructure calculations. This serves to emphasize the factthat the SDF theory is a theory for the ground state whereasexcited states are probed by spectroscopic measurements.In particular the theory does not correctly describe thecorrelated motion of the electrons as they are excited intostates necessary for them to leave the metal. On theother hand all the ground state properties such asMs

and the lattice spacing are in very good agreement withexperimental values.

The nearly rigidly split spin-polarized bands of these3d elemental transition metal magnets are actually specialcases. We now show how this simple picture is lost assoon as the electronic structures of ferromagnetic alloys areconsidered. We focus on compositionally disordered alloysfor this purpose, since these provide the starting point formuch of the discussion in this article.

2358

Figure 288: The spin-split density of states of b.c.c. Fe. [After J. B. Stauntonet sl., (1998).]

of the chemical potential to the minimum constrains the number of down-spin3d electrons to be three. Hence, the reduced magnetization is given by(

µ

µB

)= n − 6 − 2 n4s↓ (2417)

267J. Crangle and G. C. Hallam, Proc. Roy. Soc. 272, 119 (1963).268J. C. Slater, J. Appl. Phys. 8, 385 (1937).269J. Friedel, Nuovo Cimento, 2, 287 (1958).270J. B. Staunton, S. S. A, Razee, M. F. Ling, D. D. Johnson and F. J. Pinski, J. Phys. D,

Appl. Phys. 31, 2355 (1998).

686

The line with negative slope can be described as having a completely occupiedup-spin 3d band, containing five electrons, and the magnetization decreases sincethe number of down-spin holes increases as n increases. Therefore, one expectsthat (

µ

µB

)= 10 − n + 2 n4s↑ (2418)

This describes the dominant linear variations. The assumption of one commonrigid band for the AB alloy can be quantified by using the C.P.A.271. The den-sity of states of the two elements are centered on εA and εB and, if |εA − εB |is smaller than the band width W , one can expect the density of states will beunsplit. On the other hand, if |εA − εB | is greater than the band width W , onecan expect the density of states will be split and one expects deviations fromthe dominant linear variation. If the exchange splittings are markedly differentbetween the A and B atoms, then one expects that the subbands of a specificspin may resemble unsplit bands while the subbands of the other spin may beconsidered as split. Therefore, in this case, the polarized bands are not sub-jected to a rigid splitting.

21.6 The Heisenberg Model

The above model of itinerant magnetism is believed to be appropriate for tran-sition metals only involves one type of electrons. Another model is appropriatefor materials which contain two types of electrons, such as rare earth materials,in which the magnetic moments occur in the f states which are inner orbitalsburied deep inside the f ion and the interaction is mediated by the itinerantconduction electrons.

The spin localized at site Ri is denoted by Si. The spin at site i interactswith the conduction electrons spin σ near site i via a local exchange interaction

Hint = − J∑

i

Si . σi (2419)

which acts like a localized magnetic field of 2g µB

J Si. This localized magneticfield polarizes the conduction electrons, producing a polarization at site j of(

2g µB

)J Si χ

z,z(Ri −Rj) (2420)

This polarization then interacts with the spin at site j via the local exchangeinteraction leading to an oscillatory interaction between pairs of localized spinsof the form

H = −∑i,j

J(Ri −Rj) Si . Sj

271H. Hasegawa and J. Kanamori, J. Phys. Soc. Japan, 31, 383 (1971).

687

-0.01

-0.005

0

0.005

0.01

0 2 4 6 8 10 12J(

r)/J

0

kFr

Figure 289: The form of the oscillatory R.K.K.Y. interaction J(r) between twolocal moments mediated via the conduction band.

= −∑i,j

(2

g µB

)2

J2 χz,z(Ri −Rj) Si . Si (2421)

The oscillations in the interaction are produced by the oscillations of the re-sponse function of the conduction electrons. The Fourier transform of J(R)shows the oscillation frequency is 2 kF . This interaction was discovered inde-pendently, by Ruderman and Kittel, Kasuya and Yosida272.

22 Localized Magnetism

The nearest neighbor Heisenberg exchange interaction couples spins localizedon adjacent lattice sites

H = − J

2

∑R,δ

S(R+ δ) . S(R) (2422)

where δ are the vectors connecting pairs of adjacent sites. This interactionHamiltonian can be derived from the model of itinerant magnetism, for large Uin the case when the bands are half filled. In this case, there is a spin at eachlattice site and the exchange between the spins is the anti-ferromagnetic superexchange interaction found by Anderson273. The exchange constant is given interms of the tight-binding matrix element t and the Coulomb repulsion via

J = − 4t2

U(2423)

272M. Ruderman and C. Kittel, Phys. Rev. 96, 99 (1954), T. Kasuya, Progr. Theor. Phys.16, 45 (1956), K. Yosida, Phys. Rev. 106, 803 (1957).273P.W. Anderson, Phys. Rev. 115, 2 (1959).

688

The Heisenberg Hamiltonian can be expressed as

H = − J

2

∑R, δ

[Sz(R) Sz(R+δ) +

12

(S+(R) S−(R+δ) + S−(R) S+(R+δ)

) ](2424)

For a ferromagnetic exchange, J > 0, the ground state of a three-dimensionallattice of spins of magnitude S consists of parallel aligned spins. The directionof quantization is chosen as the direction of the magnetization. All the spinshave their z-components of the spin maximized

| Ψg > = |∏R

( mR = S ) > (2425)

This state has a total magnetization proportional to N S, and is an eigenstateof the Hamiltonian, with energy Eg = − 3 N J S2 since the z component ofthe Hamiltonian is diagonal and the spin flip terms vanish as the effect of theraising operators acting on the fully-polarized state

S+(R) | mR = S > = 0S+(R+ δ) | mR+δ = S > = 0 (2426)

are both zero. The excitations of this system are the spin waves. The excitedstate wave function corresponding to a single spin wave is given by

| q > =1√N

∑R

exp[− i q . R

]S−(R) | Ψg > (2427)

It corresponds to a state with total spin N S − 1 since one spin is flippedover. This state is a coherent superposition of all states in which only one spinis flipped and has total momentum of q. The excitation energy is found from

( H − Eg ) | q > = 2 J S(

3 − cos qxa − cos qya − cos qza)| q > (2428)

At long wave-lengths, the spin wave excitation energy hω(q) is found as

h ω(q) = 2 J S

(3 − cos qxa − cos qya − cos qza

)∼ J S q2 a2 (2429)

which is the branch of collective Goldstone modes that dynamically restorethe spontaneously broken spin rotational invariance of the ferromagnet. Thequadratic variation of the frequency near q = 0 is a general consequence of the

689

total spin being a constant of motion. In the limit q → 0, the spin wave statejust corresponds to a uniform reduction of the total magnetization since∑

R

exp[− i q . R

]S−(R) → S−Tot (2430)

The above excitations of the spin system are small amplitude excitationsthat have a close resemblance to the harmonic phonons of a crystalline lattice.The effects of the interactions could be expected to produce small anharmoniccorrections to these excitations, providing them with a lifetime and a renor-malized dispersion relation. Not all the excitations can be expressed as smallamplitude excitations, some systems have large amplitude soliton excitationsthat cannot be treated by perturbation theory. However, the small amplitudeexcitations can be adequately treated as harmonic modes, as can be seen froman analysis based on the Holstein-Primakoff transformation.

22.1 Holstein-Primakoff Transformation

The Holstein-Primakoff transformation provides a representation of localizedspins, which enables the low-temperature properties of an ordered spin systemto be analyzed in terms of boson operators. The technique is particularly usefulfor systems where the magnitude of the spin S is large S > 1. The Hamilto-nian can be expanded in terms of boson operators, providing a description of theground state, small amplitude spin fluctuations and the anharmonic interactionbetween them.

The Holstein-Primakoff transformation274 of the spins represents the effectof the spin operators by a function of bosons operators. The components ofthe spin operators at the i-th site Sα

i can be defined by their action on theeigenstates of Sz

i

Szi | mi > = mi h | mi > (2431)

In particular the spin raising and lowering operators

S±i = Sxi ± i Sy

i (2432)

have the commutation relations with Sx[Sz

i , S±i

]= ± h S±i (2433)

This can be used to show that the operators S±i have the effect of raising andlowering the magnitude of the eigenvalue Sz by one unit of h

S±i | mi > =√

S ( S + 1 ) − mi ( mi ± 1 ) h | mi ± 1 > (2434)274T. Holstein and H. Primakoff, Phys. Rev. 58, 1098 (1940).

690

The Holstein-Primakoff transformation represents the local ( 2 S + 1 ) basisstates | mi > by the infinite number of basis states of a local boson numberoperator a†i ai. The local boson basis states, | ni >, are defined through

a†i ai | ni > = ni | ni > (2435)

The relation between the local spin basis states and boson states is provided bythe boson representation of the z-component of the spin operator

Szi = S − a†i ai (2436)

Thus, the state where the spin is aligned completely along the z-axis (mi = S)is the state with boson occupation number of zero, and the state with thelowest eigenvalue of the spins z-component (mi = − S) corresponds to thestate where 2 S bosons are present. The states with higher number of bosonsare un-physical and must be projected out of the Hilbert space. The effectsof the spin raising and lowering operators are similar to the boson annihilationand creation operators, within the space of physical states. The correspondencebetween the spin and raising and lowering operators and the boson operatorscan be made exact, by multiplying the boson creation and annihilation operatorswith a function that ensures that only the physical acceptable boson states formthe Hilbert space of the spin system. This is achieved by representing the spinlowering operator as

S−i = a†i

( √2 S − a†i ai

)(2437)

which decouples states with more than 2 S bosons present. The raising operatoris given by the Hermitean conjugate

S+i =

( √2 S − a†i ai

)ai (2438)

This transformation respects the spin commutation relations.

The nearest neighbor ferromagnetic (J > 0) Heisenberg Hamiltonian

H = − J

2

∑R,δ

S(R) . S(R+ δ) (2439)

can be expressed as

H = − J

2

∑R,δ

[Sz(R) Sz(R+ δ) +

12

(S+(R) S−(R+ δ) + S−(R) S+(R+ δ)

) ](2440)

691

ωS

Figure 290: The coherent spin precession of a classical spin wave, as seen alongits direction of propagation q.

On representing this Hamiltonian in terms of the boson operators, and expand-ing in powers of 1

S , one finds

H ≈ − J

2

∑R,δ

[( S − a†R aR ) ( S − a†R+δ aR+δ ) + S

(a†R aR+δ + a†R+δ aR

) ](2441)

The terms of order S2 just represents the classical ferromagnetic ground state, inwhich all the spins are aligned. The terms of order S represent excitations fromthe ground state and can be put in diagonal form by expressing them in termsof the Fourier transformed boson operators. The spatial Fourier transform ofthe boson operators are defined as

aR =1√N

∑q

exp[i q . R

]aq (2442)

and the creation operator is given the Hermitean conjugate expression

a†R =1√N

∑q

exp[− i q . R

]a†q (2443)

Substitution of the creation and annihilation operators in the Hamiltonian, andperforming the sums over the spatial index, yields the following approximateexpression for the Hamiltonian

H ≈ − J

2N Z S2 + J S

∑q

a†q aq

∑δ

(1 − cos q . δ

)(2444)

The above expression is the sum of the ground state energy and the energies ofharmonic normal modes that represent the excitations of the spin waves fromthe ferromagnetic ground state. The terms of higher-order in 1

S yield quantumcorrections to the ground state energy, the spin wave energies and also produceanharmonic interactions between the spin waves.

692

Figure 291: The approximate T32 decrease of the magnetization Mz of a single

crystal of Ni at low temperatures. [After B. E. Argyle, S. H. Charap and E. W.Pugh, Phys. Rev. 132, 2051 (1963).].

The thermally excited spin waves have the effect of reducing the magnetiza-tion from the fully saturated T = 0 value

Mz(T ) − N S =∑

q

N(ω(q)) (2445)

where N(ω) is the Bose-Einstein distribution function. Since the ferromagneticspin waves have a dispersion relation which is ω(q) ∼ q2 at small q, one findsthat the temperature induced change in the magnetization is given by

M(T ) − M(0) ∼(kB T

J S

) 32

(2446)

for a three-dimensional lattice at low temperatures. Likewise, the thermal av-erage value of the energy can be calculated as

E(T ) + N ZJ

2S2 =

∑q

h ω(q) N(ω(q))

E(T ) − E(0) ∼ N J S

(kB T

J S

) 52

(2447)

This should result in the low-temperature specific heat being proportional to

693

T32 for a long-range ordered insulating ferromagnet.

22.2 Spin Rotational Invariance

In the limit of zero applied magnetic field, the Heisenberg exchange Hamiltonianis invariant under the simultaneous continuous rotation of all the spins. As thereis no preferred choice of z-axis, the energy of the ferromagnetic state of a fullypolarized spin system with a total spin ST = N

2 has the same energy as the statewhere all the spins are oriented along the direction (θ, ϕ). The ferromagneticstate where the spins are fully-polarized along the z-axis is given by

| 0 > =∏j

| mj =12> (2448)

The state where the polarization is rotated through (θ, ϕ) is given by

| θ, ϕ > =∏j

(cos

θ

2ij + exp

[i ϕ

]sin

θ

2S−j

)| 0 > (2449)

This can be proved by representing the spin vector operator for one site σj interms of its component along the unit vector η in the direction (θ, ϕ)

η . σj = sin θ cosϕ σx j + sin θ sinϕ σy j + cos θ σz j (2450)

where σx j , σy j and σz j are the three Pauli spin matrices for the spin at sitej. Thus, the representation of the operator for the η component of the spin atsite j is found as

η . σj =

cos θ sin θ exp

[− i ϕ

]

sin θ exp[

+ i ϕ

]− cos θ

This above component of the spin operator has two eigenstates. The eiegen-state

| θ, ϕ + > = cosθ

2| + > + exp

[+ i ϕ

]sin

θ

2| − > (2451)

corresponds to the eigenvalue of +1 and

| θ, ϕ − > = cosθ

2| − > − exp

[− i ϕ

]sin

θ

2| + > (2452)

694

corresponds to the eigenvalue of −1. The un-rotated ferromagnetic state has allthe spins in the | + > spinor state and after rotation all the spins have maximaleigenvalue along the direction (θ, ϕ). In the rotated state each spin is describedby the | θ, ϕ + > spinor state. Therefore, the rotated ferromagnetic state hasall the spins aligned along the same direction. This classical ferromagnetic statehas an infinitesimal overlap with the un-rotated ferromagnetic state, since

< 0 | θ, ϕ > =(

cosθ

2

)N

= exp[N ln cos

θ

2

](2453)

which vanishes in the limit N → ∞. The rotated state can be consideredto be a Bose-Einstein condensate of the q = 0 spin waves. For example onexpanding the rotated state in powers of exp[ i ϕ ], one finds

| θ, ϕ > =∞∑

n=0

A(n) exp[i n ϕ

]( S−Tot )n

n!| 0 >

=∞∑

n=0

cos(N−n) θ

2sinn θ

2exp

[i n ϕ

]( S−Tot )n

n!| 0 >

(2454)

where the total spin operator is defined by

S−Tot =∑

j

S−j (2455)

since ( S−j )2 ≡ 0. The un-normalized states with n spin flips present aredefined as

| n > =( S−Tot )n

n!| 0 > (2456)

These states have the normalization

< n | n > = C

(Nn

)(2457)

Hence, the rotated spin state can be expressed as

| θ, ϕ > =∞∑

n=0

cos(N−n) θ

2sinn θ

2exp

[i n ϕ

]| n > (2458)

The number of q = 0 spin flips, n, are distributed with the binomialprobability

P (n) =| < n | θ, ϕ > |2

< n | n >(2459)

which yields

P (n) = C

(Nn

)cos2(N−n) θ

2sin2n θ

2(2460)

695

Since N is a macroscopic number, the distribution of spin flips is a sharp Gaus-sian distribution with a peak at nmax = N sin2 θ

2 , and a width given by∆n = N

12 sin θ

2 cos θ2 . The number of bosons in the q = 0 spin wave mode

is macroscopic and of the order N . This is a coherent representation, and it isthe quantum state that is closest to a classical state as possible.

The coherent state corresponds to the classical state in which the total spinis oriented along the direction (θ, ϕ). This can be established by examining thematrix elements of the spin operators. The matrix elements of the total spinlowering operator between states with total numbers of q = 0 spin flips closeto the maximum of the wave packet are found first by noting that

< n + 1 | S−Tot | n > =(N − n

)C

(Nn

)(2461)

since S− can only have a non-zero effect on the N − n up-spins and createsan extra down-spin. The expectation value of the spin lowering operator in therotated state is found to be

< θ, ϕ | S−Tot | θ, ϕ > =∑

n

exp[− i ϕ

]A(n+ 1) A(n) < n + 1 | S−Tot | n >

= tanθ

2exp

[− i ϕ

] ∑n

A(n)2 ( N − n ) C(Nn

)=

N

2sin θ exp

[− i ϕ

](2462)

and likewise, the matrix elements of the spin raising operator are given by thecomplex conjugate. Finally, the matrix elements of the Sz

Tot is given by

< n | SzTot | n > = (

N

2− n ) C

(Nn

)(2463)

Thus, one has

< θ, ϕ | SzTot | θ, ϕ > =

∑n

A(n)2 < n | SzTot | n >

=∑

n

A(n)2( N − 2 n )

2C

(Nn

)=

N

2cos θ (2464)

Therefore, the components of the total spin operator have matrix elements be-tween the coherent state that exactly corresponds to the components of theclassical vector.

Thus, the different classical ferromagnetic states are represented by coher-ent states which are superpositions of states with arbitrary numbers of excited

696

Goldstone modes. The thermal average in a ferromagnetic state should yielda zero magnetization for a system in the thermodynamic limit. The thermalaverage has to be taken, in the thermodynamic limit, in the presence of an ar-bitrary small magnetic field. In this case, the different classical states have zerooverlap and can be considered as being in disjoint portions of Hilbert space. Inthis quasi-static state the field may then be driven to zero leading to a non-vanishing vector order parameter.

——————————————————————————————————

22.2.1 Exercise 92

Determine the spin wave spectrum for an isotropic Heisenberg ferromagnet inthe presence of an applied magnetic field. Do the conditions of Goldstone’s the-orem apply, and what happens to the excitation energy of the q = 0 spin wave?

——————————————————————————————————

22.3 Anti-ferromagnetic Spinwaves

One can obtain an approximate spin wave spectrum for an anti-ferromagnet (J < 0 ) in a Neel state. Neel ordering275 shall be considered on a crystalstructure that can be decomposed into two interpenetrating sub-lattices. Thespins on one sub-lattice (the A sub-lattice sites) shall be oriented parallel tothe z-axis, and the spins on the second sub-lattice (the B sub-lattice) are anti-parallel to the z-axis. In order for the bosons to represent excitations, it isnecessary to switch the directions of the spins on the B sub-lattice Sz

i → − Szi ,

Sxi → Sx

i and Syi → − Sy

i . This is a proper rotation of π about the x-axisso that the commutation relations remain the same. The Holstein-Primakofftransformation for the operator representing the z-component of the B spins isof the form

Szi = b†i bi − S (2465)

and the spin raising operators for the B spins are

S+i = b†i

( √2 S − b†i bi

)(2466)

which decouples states with more than 2 S bosons present. The lowering oper-ator is given by the Hermitean conjugate expression

S−i =( √

2 S − b†i bi

)bi (2467)

275L. Neel, Ann. de Physique, 17, 64 (1932), Ann. de Physique 5, 256 (1936).

697

7350 M Hagiwara et al

Moreover, we have found a significant contribution of the non-linear term in the expressionof the AFMR frequency to the temperature dependence of the resonance point in the high-field low-frequency region.

The format of this paper is as follows. In section 2 we describe the crystal and magneticstructures of MnF2. Experimental details are given in section 3. In section 4 we presentresults of the AFMR experiments and their analysis.

2. Crystal and magnetic structures of MnF2

The crystal structure of MnF2 belongs to the tetragonal space group D144h with two molecules

per unit cell. The lattice constants at room temperature area = 4.8734A andc = 3.3103A(Griffel and Stout 1950).

From the neutron scattering study of Erickson (1953) below the Neel temperature(TN = 67.34 K) (Heller and Benedek 1962), the magnetic structure of MnF2 was determined.In the ordered phase, the spins at body centre sites point antiparallel to those at the cornersites with the spin easy axis parallel to thec axis. The main origin of the magnetic anisotropyis the dipole–dipole interaction.

Figure 1. The temperature dependence of the molar magnetic susceptibilities parallel andperpendicular to thec axis of MnF2 measured at 100 Oe.

3. Experimental details

3.1. Sample preparation and characterization

Single crystals of MnF2 were grown by the Bridgman method. A commercially availablepowder of MnF2 with 4 N purity was placed in a Pt crucible, melted at about 900C andcooled at the rate of 1C h−1 under an atmosphere of mixed N2 and HF gas. A transparentpink coloured single crystal was obtained. A thin disc of MnF2 with thec axis perpendicularto the plane was cut from a larger crystal.

In order to characterize the crystal, we have measured the temperature dependence ofthe magnetization (M) under an external magnetic field (H ) using a SQUID magnetometer(Quantum Design’s MPMS2). The results are shown in figure 1. There is no anisotropyin the susceptibility (M/H ) along thec axis (χ‖) and c plane (χ⊥) above about 67 K.On the other hand,χ‖ and χ⊥ behave quite differently below∼67 K, which is a typical

Figure 292: The temperature-dependence of the anisotropic magnetic suscepti-bility of MnF2 parallel and perpendicular to the c-axis. [After M. Hagiwara,K. Kamatsumata, I. Yamada and H. Suzuki, J. Phys. C.M. 8, 7349 (1996).].

for the B sub-lattice.

An semi-classical spin wave Hamiltonian can be found276 by expanding theHamiltonian in powers of S−1. The expansion yields the expression

H ≈ J∑i,j

[( S − a†i ai ) ( S − b†j bj ) − S ( a†i b

†j + ai bj )

](2468)

On defining the Fourier transformed operators

aq =

√2N

∑i

exp[− i q . Ri

]ai

bq =

√2N

∑j

exp[− i q . Rj

]bj (2469)

then the Hamiltonian can be reduced to

H = N z J S2 − J S∑

q

[z ( a†q aq + b†−q b−q )

+∑

δ

exp[− i q . δ

]( a†q b

†−q + aq b−q )

](2470)

Since this form is not diagonal in the a and b operators, it is necessary to usethe Bogoliubov canonical transformation to diagonalize the Hamiltonian. The276P.W. Anderson, Phys. Rev. 86, 694 (1952).

698

form of the Bogoliubov transformation is given by

αq = exp[

+ S

]aq exp

[− S

]β−q = exp

[+ S

]b−q exp

[− S

](2471)

where the anti-Hermitean operator S is given by

S =∑

q

θq

2

(b†−q a

†q − aq b−q

)(2472)

and where θq still has to be determined. The transformation is evaluated as

αq = coshθ

2aq − sinh

θ

2b†−q

β−q = coshθ

2b−q − sinh

θ

2a†q (2473)

in which θq is to be chosen so that the Hamiltonian is diagonal. The inversetransformation is given by

aq = coshθ

2αq + sinh

θ

2β†−q

b−q = coshθ

2β−q + sinh

θ

2α†q (2474)

After substitution of the inverse Bogoliubov transformation into the Hamilto-nian, the Hamiltonian takes the form

H = N z J S2

− J S∑

q

[z cosh θq −

∑δ

exp[−iq . δ] sinh θq

] (α†q αq + β†q βq

)

− J S∑

q

[− z sinh θq +

∑δ

exp[−iq . δ] cosh θq

] (α†q β

†−q + βq α−q

)(2475)

The value of θq which eliminates the off-diagonal terms in the Hamiltonian isfound as

tanh θq =1z

∑δ

exp[− i q . δ

](2476)

The resulting approximate Hamiltonian can be interpreted in terms of a zero-point energy and a sum of harmonic normal modes, as was done for the ferro-magnet. However, unlike the case of ferromagnetic spin waves, the amplitudeof anti-ferromagnetic spin waves can be anomalously large.

——————————————————————————————————

699

502 C. G. Windsor and R. W. H. Stevenson

ill 2 2 2

000 001

Figure 1. The reciprocal lattice diagram of a simple cubic antlferromagnet m a 110 plane. N and M mark the elastic nuclear and magnetic reflectlon pomts. The thm lines show the corresponding zone boundaries. Heavy lines show a possible neutron-scattering hagram for the experimental settings used. k and k represent the mcident and scattered neutron wave vectors. K represents the wave vector transfer k -k, while q represents the transfer relative to the centre of the

zone.

I

0 0.2 0.4 0.6 Wave vector Iq I (1-1)

Figure 2. Spin waves m RbMnFs at 4 2 OK wlth q vectors dstnbuted over a 1l0 Plane. The smooth curves show the calculated dispersion along Pamcdar directions w t h JI = 3 4 OK, J2 = J 3 = 0.0 OK. These values were found from a lest- Squares analysis, the exact direction of all the q vectors bemg taken into accounl* The fact that the linear part of the curve extends so close to the o r i p reflects the

very small anisotropy field.

Figure 293: The spin wave dispersion relation of the antiferromagnet RbMnF3

in the 110 plane. [After C. G. Windsor, and R. W. H. Stevenson, Proc. Phys.Soc. 87, 501 (1966).].

22.3.1 Exercise 93

Find the approximate dispersion relation for spin waves of a Heisenberg anti-ferromagnet, J < 0, for spins of magnitude S. The Hamiltonian describesinteractions between nearest neighbor spins arranged on a simple cubic lattice,

H = − J∑R, δ

S(R) . S(R+ δ) (2477)

Assume that S is large so that the classical Neel state can be considered asbeing stable. Also calculate the zero point energy.

——————————————————————————————————

Since tanh θ → 1 in the limit q → 0 then θ → ∞ so both sinh θ2 and

cosh θ2 diverge. In one dimension, the change in the sub-lattice magnetization

< ψ | a†i ai | ψ > diverges logarithmically. In two and three dimensions, thedivergence in

∑q < ψ | a†q aq | ψ > is integrable and converges. There is an

energy change relative to the nominal classical energy of the Neel state, givenby the sum of the zero point energies,

Eg = J z N S2

[1 +

1S

( 1 − Id )]

(2478)

where

Id =2N d

∑q

√√√√ d2 −( d∑

i=1

cos qi a)2

(2479)

700

Spin waves in antiferromagnetic FeFz 315

8 0 r I

70- -7 v

h P c

W

\

<I 0 o> <OOl> 5 0 1 - - 1 - - - I 1 - ~~~~~ 1 - - 1 ~ L - - - _L .--d ~ A

0.2 0.4 0.6 0.8 I .o 9, (i-')

Figure 5. The points are the energies of those spin waves observed which lie close to two symmetry directions. The error bars represent estimates of the probable uncertainty in the measurements. The full curves are calculated from equations (6) and (7) using

parameters giving the best fit to the data throughout the zone.

Table 2. Values obtained for parameters (in cm-1 )in the spin Hamiltonian for FeFz by fitting to the neutron data with different approximations

D J l J 2 J 3

A 6.46 + 0.29 -0.048 k 0.060 3.64 k 0.10 0.194 i 0.060 - 0.10

B(i) 6.69 - 0.074 3.57 B(ii) 6.34 0.0 12 3.73 B(iii) 6.8 1 - 3.6 1 C 6.94 3.5 1 -

0.159 0.209

A, best fit to full Hamiltonian equation (5) including dipolar interaction dispersion and the full Oguchi factor. Final result.

B(i), best fit including dipolar interaction and Oguchi factor in full but not fixing E , at 52.7 cm- l, B(ii), fit obtained omitting Oguchi factor (i.e. a4 = 0), and omitting the dispersion of the dipolar

interaction. E , fixed at 52.7 cm-'. B(iii), fit obtained with clq = $: = Jl = J3 = 0. E , is fixed at 52.7 cm-' and D will include the

dipolar interaction. C, values obtained with clq = $: = J, = J, = 0. J, and D are calculated from E , = 52.7 cm-',

and E,, = 77.1 cm- '. D will include the dipolar interaction.

zero or antiferromagnetic value, while J, is larger and antiferromagnetic. These parameters will be discussed in 4 6. Using them we calculate the zone boundary energies to be Eool(Z) = 79.2 cm-', EIoo(X) = 77.7 cm-', E,,,(R) = 77.7 cm-', E,,,(M) = 76.1 cm-' and E,,,(A) = 76.1 cm-', with errors of + 2 cm-'. The labels are the symmetry points designated in figure 6. It is sometimes useful to represent the dispersion by a formula which reproduces the approximate spin-wave energies in a simple manner. The values of J, and D in row C of table 2 are calculated from E , and a weighted average of the above zone boundary energies at the five special points.

Figure 294: The spin wave dispersion relation of the antiferromagnet FeF2 fortwo high-symmetry directions. The system has a large gap due to single siteanisotropy. [After M. T. Hutchings, B. D. Rainford and H. J. Guggenheim, J.Phys. C. 3, 307 (1970).].

The amplitude of the q = 0 spin wave is divergent, but can be neglected inthe limit N → ∞. As the q = 0 spin wave has zero frequency, one cancompose a state which is a superposition of the q = 0 spin wave excitations.The dynamics of the finite frequency spin wave excitations can be examined forfinite time scales before the zero-energy amplitude wave packet of q = 0 spinwaves diverges re-orienting the sub-lattice magnetization.

Just like in the ferromagnetic state, the magnetic response of an antiferro-magnet to an applied magnetic field depends on the direction of the appliedfield relative to the direction of the order parameter. In the absence of anyanisotropy, it is expected that the application of a magnetic field will causethe Neel state to become unstable to a spin-flop state. In the spin-flop state,the sublattice magnetizations will rotate to be perpendicular to the directionof the applied field, and will also develop small components parallel to the fielddirection that are proportional to the field. The presence of uniaxial anisotropyterm in the Hamiltonian (such as −

∑i D ( Sz

i )2 for S > 12 ) will stabilize

the ground state against the spin-flop transition, for sufficiently small magneticfields277. However, in this case the spin-wave spectrum will develop a gap asGoldstone’s theorem does not apply278. For fields applied that are perpendic-ular to the sublattice magnetization, the transverse susceptibility is constantbelow the Neel temperature. On the other hand, for the field direction parallelto the sublattice magnetization, the longitudinal susceptibility falls to zero asT 2 below TN .277F. Burr Anderson and H. B. Callen, Phys. Rev. 136, A 1068 (1964).278R. Kubo, Phys. Rev. 87, 568 (1952).

701

Figure 295: The concentration-temperature phase diagram AuFe alloys, show-ing the paramagnetic, super-paramagnetic, ferromagnetic, cluster glass and spinglass phases. [After B. R. Coles, B. V. B. Sarkissian and R. H. Taylor, Phil.Mag. B 37, 498 (1978).]

23 Spin Glasses

Spin glasses are found when magnetic impurities are randomly distributed ina metal, such as Fe impurities in gold Au or Mn in Cu. Due to the randomseparations between the moment carrying impurities, the R.K.K.Y. interactionbetween the magnetic moments are also randomly distributed and can take onboth ferromagnetic and anti-ferromagnetic signs. The distribution of interac-tions prevents the local magnetic moments from forming a long-range orderedphase at low temperatures. Nevertheless, the random spin system may freezeinto a spin glass state below a critical temperature. At high temperatures thespins are disordered, and as the temperature is reduced, the spins which aremost strongly interacting progressively build up their correlations and freezeinto clusters. The dynamics of the spin clusters slow down as they grow, and ata critical temperature Tf they lock into the spin glass phase. However, this maynot be the lowest energy state, as in order to reach the ground state there may

702

Journal of Magnetism and Magnetm Materials 31-34 (1983) 1331-1333 1331

T H E E F F E C T O F S P I N - G L A S S O R D E R I N G O N T H E S P E C I F I C H E A T O F C u M n

G.E . B R O D A L E , R .A. F I S H E R , W.E . F O G L E , N .E . P H I L L I P S a n d J. V A N C U R E N

Department of Chemistry and Matertals and Moleeular Re~eareh Dtwston, Lawren(e Berkeley Laboratory. Cahforma, Berkele), CA 94720. USA

Unmerstt of

Hlgh-premsmn measurements have shown the nature of the specific heat anomaly associated w, th the transmon to the spin-glass phase m CuMn, and have been used to determine the entropy as a function of the field and temperature

The theoretical demons t ra t ion [1] that order in spin 200 glasses could appear at a well defined crmcal tempera- ture, T~g, explained the observed d iscont lnumes m mag- nehc propert ies [2] at least qualitatively, but left the

[50 failure to observe comparable features in the specific ~, heat more puzzling than ever More recently, the prop- ertles of spin glasses have usually been interpreted m _~ terms of the S h e r r m g t o n - K l r k p a m c k mfmlte-range , 100 model t reated in various extensions and approximat ions x: [3] In particular, the Par ls l -Tolouse hypothesis [4] has -~ been widely mvoked According to this hypothesis, the v co entropy, S, is independent of applied field, H, below the 5 0 instabil i ty line or phase boundary , T~g(H), recognized by deAlmeIda and Thouless [5]. This paper is a review of the results of a cont inuing series of calorimetric measurements designed to determane the thermody- 0 namlc propert ies jus t above and below T,g(H), and to search for effects in the specific heat associated with T,g

I I I

2790 ppm C.~uuMn

t 50

H ( k O e )

o 1 io 2o 5o 45 6o 75

t i Ioo 15 0 200

T (K) Fig 1 Typical specific heat data for CuMn (T,g = 3 89 K)

O

(D

-6 E

t%l (D

o e d

",r"

G0

'O

I I E I I I i '

0 • •

- 0 5

m

, - I 0

800

I i I I k I H ( O e ) I

2 5 5 0 5 5 4 0 4 5 5 0 5 5 6 0 6 5

T (K)

F~g 2 The field dependence of C / T =- A + B H 2 The error bars in the reset correspond to +0 01% of the total measured specffm heat

0 3 0 4 - 8 8 5 3 / 8 3 / 0 0 0 0 0 0 0 0 / $ 0 3 00 © 1983 N o r t h - H o l l a n d

Figure 296: The temperature dependence of the magnetic heat capacity of aCuMn alloy. [After G. E. Brodale, R. A. Fisher, W. E. Fogle, N.E. Phillips andJ. Van Curen, J. Mag. Mag. Mat. 31-34, 1331 (1983).]

have to be large scale reorientations of the spin clusters. Thus, the spin glassstate is not unique but instead is highly degenerate. This occurs as a resultof frustration. The concept of frustration is illuminated by imagining that allthe spins on the magnetic sites are frozen in fixed directions, except one, thenthere is a high probability that the long-ranged interactions between the spinunder consideration and the fixed spins almost average out to zero. At finitetemperatures, the spin under consideration is almost degenerate with respect tothe orientation of the spin as it leads to an insignificant lowering of the energyof the spin glass state.

The experimental signatures of spin glass freezing are a plateau in the staticsusceptibility and a rounded peak in the specific heat. The susceptibility followsa Curie-Weiss law at high temperatures

χ(T ) = cµ2

B S ( S + 1 )3 kB ( T − Θ )

(2480)

where Θ is the strength of the resultant interaction on an individual spin. Foran R.K.K.Y. interaction, the Curie-Weiss temperature Θ should be proportionalto c. At lower temperatures where the spin freeze into clusters, the effectivemoment increases, reflecting the growth of the clusters. The peak in the specificheat, encloses an entropy which is a considerable fraction of

∆S = c kB ln (2S + 1) (2481)

703

Figure 297: The temperature dependence of the a.c. susceptibility of severalAuFe alloys. [After V. Canella and J. A. Mydosh (1972).]

where c is the concentration of magnetic impurities of spin S. This entropyrepresents the entropy of the spins gradually freezing into clusters, but does notcontain the entropy of the frustrated spins. This maximum disappears and isbroadened and shifted to higher temperatures as a magnetic field is applied. Theeffect of the applied field is to order the spins at higher temperatures. Crudeestimates indicate that about 70 % of the spins are already ordered above Tf .The temperature dependence of the resistivity shows a sharp drop or knee at thespin glass freezing temperature. At this temperature, the majority of spin arefrozen in specific directions preventing the logarithmic increase with decreasingtemperature associated with spin flip scattering.

As the spin glass phase is not a ground state but is instead a highly de-generate meta-stable state, the most unusual properties occur in the dynamicalproperties. The low-field a.c. susceptibility shows a very sharp cusp at the spinglass freezing temperature279. The cusp becomes rounded and the temperatureof the peak diminishes as the a.c. frequency is lowered. The susceptibility satu-279V. Canella and J. A. Mydosh, Phys. Rev. B, 6, 420 (1972).

704

Figure 298: The temperature dependence of the a.c. susceptibility of two AuFealloys for zero and various applied fields. [After V. Canella and J. A. Mydosh(1972).]

rates to a finite value at T = 0 which is roughly half the value of the cusp andhas a T 2 variation on the low-temperature side. The d.c. susceptibility showsa memory effect, in that, in field cooled samples (H 6= 0) the susceptibilitysaturates at Tf , and the curve χ(T ) is reversible as it is also followed for in-creasing temperature at fixed field. By contrast, the zero field cooled samples(H = 0) shows a cusp280 at Tf . The susceptibility is zero, by definition, untilthe field is applied. When the field is applied, the susceptibility jumps to a valuesimilar to that found via the a.c. susceptibility measurements and shows thecusp. However, below Tf , the value of the susceptibility also increases with in-creasing measurement time. The magnetization is slowly increasing as the spinsslowly adjust to lower energy state in the presence of the applied field. This iscontrasted to the field cooled state in which the spins have already minimizedthe field energy before the temperature is lowered and they are frozen into thespin glass state.

The spin glass freezing resembles a phase transition, but the nature of theorder is unclear as the spin glass state involves disorder and is a highly degener-ate meta-stable state. Likewise, the description of the low frequency dynamicsof the magnetization is complicated by the existence of long-ranged correlationsbetween large groups of spins. Since there is no well defined order parame-ter, there is no well defined low frequency Goldstone mode. Several importantsteps in the solution of the thermodynamics of the spin glass problem have beenundertaken; this includes the discovery of the nature of the order parameter,280S. Nagata, P. H. Keesom and H. R. Harrison, Phys. Rev. B 19, 1633 (1979).

705

Figure 299: The temperature dependence of the susceptibility of two samplesof CuMn. Curves (b) and (d) were obtained when the samples were cooled inzero field, before the field was applied. Curves (a) and (c) were obtained whenthe samples were cooled in a finite field. [After S. Nagata, P.H. Keesom andH.R. Harrison (1979).]

by Edwards and Anderson281 , the formulation of a model which is exactlysoluble mean-field theory by Sherrington and Kirkpatrick 282. The Sherrington-Kirkpatrick model consists of an Ising interaction

H = −∑i,j

Ji−j Szi . S

zj (2482)

where Ji−j is a randomly distributed long-ranged interaction between the spins.The average value of Ji−j is zero

< Ji−j > = 0 (2483)

and the average value of the square is given by

< J2i−j > =

J2

N(2484)

The averaging over the randomly distributed interactions is not commutative.

23.1 Mean-Field Theory

The simplest mean-field approximation is based on a representation of the freeenergy, for a spin glass with long-ranged interactions between the Ising spinS = 1

2 , in which the exact value of the spin on a site i is replaced by the

281S. F. Edwards and P. W. Anderson, J. Phys. F, 5, 965 (1975).282D. Sherrington and S. Kirkpatrick, Phys. Rev. Lett. 35, 1792 (1975).

706

thermal averaged value mi. The mean-field free energy F [mi] is given by

F [mi] = −∑i,j

Ji,j mi mj + kB T∑

i

[1 +mi

2ln

1 +mi

2+

1−mi

2ln

1−mi

2

](2485)

where the exchange interactions Ji,j are randomly distributed. On minimizingthe Free energy, one finds, the mean-field magnetization at every site. Above thespin glass freezing temperature, the average magnetization at each site is zero.Below the freezing temperature the spin on each site has a non-zero averagevalue, the direction and magnitude varies from site to site and is determined bythe non-trivial solution of

0 = − 2∑

j

Ji,j mj +kB T

2ln∣∣∣∣1 +mi

1−mi

∣∣∣∣ (2486)

On linearizing in mi (only valid for T ≥ Tf ), one obtains the eigenvalueequation which determines the spin glass freezing temperature

kB Tf mi = 2∑

j

Ji,j mj (2487)

in terms of the largest eigenvalue of the random matrix Ji,j . This is solved byfinding a basis λ that diagonalizes the matrix

Ji,j =∑

λ

Jλ < i | λ > < λ | j > (2488)

The basis gives the set of the spin configurations that the spins will be frozeninto below the spin glass freezing temperature. In the limit N → ∞, theeigenvalues of the random exchange matrix are distributed according to a semi-circular law283

ρ(Jλ) =1

2 π J2

√4 J2 − J2

λ (2489)

where, obviously, 2 J is the largest eigenvalue. The spin glass freezing temper-ature is determined as

kB Tf = 4 J (2490)

This mean-field theory predicts a transition temperature which is a factor of2 too large. This is because the mean-field theory needs to incorporate a selfreaction term. Namely, the reaction term includes the effect of the central spinon the neighbors back on itself, before the thermal averaging is performed284.

283S. F. Edwards and R. C. Jones, J. Phys. A 9, 1591 (1978).284D. J. Thouless, P. W. Anderson and R. G. Palmer, Phil. Mag. 35, 593 (1977).

707

23.2 The Sherrington-Kirkpatrick Solution.

The correct mean-field solution for the Sherrington-Kirkpatrick model can beobtained in a systematic manner, starting from the partition function. Althoughthe average value of the partition function Z is easily evaluated, the averagevalue of the Free energy is difficult to evaluate. However, the logarithm of thefree energy can be evaluated with the aid of the mathematical identity

− β F = limn→0

(Zn − 1

n

)(2491)

For finite integer n the configurational average over Ji−j can be evaluated lead-ing to an expression for the partition function for n replicas of the spin systemin which the replicas are interacting. The Gaussian averaged value of Zn isgiven by

Zn =∏i−j

( √N

2 π J2

∫dJi−j exp

[−

N J2i−j

2 J2

] )×

×(Trace exp

[− β

∑i−j

Ji−j Si Sj

] )n

= Trace∏i−j

( √N

2 π J2

∫dJi−j exp

[−

N J2i−j

2 J2−∑α

β Ji−j Sαi Sα

j

] )

Zn = Trace exp[

( β J )2

2 N

∑i,j

∑α β

Sαi Sα

j Sβi Sβ

j

](2492)

where α and β are the indices labelling members of the n different replicas. Thetrace can be evaluated for integer n and then the result can be extrapolated ton → 0. The spin glass order parameter is given by the correlation between thespins of different replicas

qα,β = < | Sαi Sβ

i | > (2493)

which becomes non-zero below the freezing temperature. The Free energy isevaluated by re-writing the trace in terms of a Gaussian integral

Trace exp[

( β J )2

2 N

∑i,j

∑α β

Sαi Sα

j Sβi Sβ

j

]

= Trace∏α,β

( ∫dyα,β β J

√N√

2 πexp

[− ( β J )2

2

(N y2

α,β − 2 yα,β

∑i

Sαi Sβ

i

) ] )

=∏α,β

( ∫dyα,β β J

√N√

2 πexp

[− N

( β J )2

2y2

α,β

] )×

× exp

[N ln Trace exp

[ ∑α,β

( β J )2 yα,β Sα Sβ

] ](2494)

708

In thie above expression, the thermodynamic limit N → ∞ and the limit n →0 have been interchanged. Due to the long-ranged nature of the interaction,the trace is over a single spin replicated n times. In the thermodynamic limitN → ∞, this integral can be evaluated by steepest descents. The saddlepoint value of yα,β is denoted by qα,β . For temperatures above Tf , it is easyto show that the interaction part of the Free energy originates from the termswith α = β as the off-diagonal terms of qα,β are all equal and zero. Therefore,above the freezing temperature, the Free energy is found as

− β F = N ln 2 +N

2( β J )2 (2495)

Just below the spin glass freezing transition, the off-diagonal terms qα,β are allequal and finite. Separating out the terms where α = β and replacing theintegrals by their saddle point values, the n-th power of the partition functionbecomes

= 2 exp[n N

( β J )2

2

(1 − q2 (n− 1)

) ]×

× exp

[N ln Trace exp

[ ∑α6=β

( β J )2 q Sα Sβ

] ]

= exp[n N

( β J )2

2

(1 − 2 q − q2 (n− 1)

) ]×

× exp

[N ln Trace exp

[ ∑α,β

( β J )2 q Sα Sβ

] ](2496)

where in the last line, the sum over pairs of replicas has been extended to includethe term α = β. By introducing a Gaussian integration, the trace over spins indifferent replicas can be evaluated in terms of a trace of the spin in one replica

= exp[n N

( β J )2

2

(1 − 2 q − q2 (n− 1)

) ]×

× exp

[N ln Trace

∫dz√2 π

exp[− z2

2

]exp

[ ∑α

( β J ) z√

2 q Sα

] ]

= exp[n N

( β J )2

2

(1 − 2 q − q2 (n− 1)

) ]×

× exp

[N ln

∫dz√2 π

exp[− z2

2

]2n coshn

[( β J ) z

√2 q

] ](2497)

The saddle point value of q is found by differentiating Zn with respect to q.After an integration by parts and then clearing away fractions, one obtains(

1 + q (n− 1)) ∫

dz√2 π

exp[− z2

2

]coshn

[( β J ) z

√2 q

]

709

=∫

dz√2 π

exp[− z2

2

]coshn

[( β J ) z

√2 q

] (1 + (n− 1) tanh2

[( β J ) z

√2 q

] )(2498)

In the limit n → 0, the order parameter is given by the solution of the equation

q(T ) =∫ ∞

−∞

dz√2 π

exp[− z2

2

]tanh2 β J

√2 q(T ) z (2499)

The temperature variation of the order parameter is given by

q(T ) =12

[1 −

(T

Tf

)2 ]for T < Tf

limT → 0

q(T ) = 1 −(

23 π

) 12 T

TF(2500)

The finite value of the order parameter produces the cusp in the susceptibilityand the low-temperature saturation, since one can show that

χ(T ) =g2 µ2

B

3 kB T

(1 − q(T )

)(2501)

Although the long-ranged model is exactly soluble in the mean-field approxi-mation, the replica symmetric solution does not have the minimum value of theFree energy. The model is only soluble for all temperatures below the freezingtemperature if the symmetry between the different replicas is broken. Replicasymmetry breaking is specific to interacting random systems285, and the exactsolution of the mean-field model involves repeated replica symmetry breaking286.This repeated replica symmetry breaking has the consequence that the dynam-ics of the low-temperature system are frozen and no longer consistent with theergodic hypothesis.

285J. R. L. de Almeida and D. J. Thouless, J. Phys. A, 11, 983, (1978).286G. Parisi, Phys. Rev. Lett. 43, 1754 (1979), G. Parisi, J. Phys. A, 13, L-115, 1101 and

1887 (1980).

710

24 Magnetic Neutron Scattering

The excitations of the electronic system can be probed by inelastic neutron scat-tering experiments. These experiments provide information about the magneticcharacter of the excitations, due to the nature of the interaction.

24.1 The Inelastic Scattering Cross-Section

The neutron scattering occurs through the interaction with the magnetic mo-ments of the electronic system.

24.1.1 The Dipole-Dipole Interaction

A neutron has a magnetic moment given by

µn

= gn µn σn (2502)

where the neutrons gyromagnetic ratio is given by gn = 1.91 and interacts withthe magnetic moments of electrons via dipole-dipole interactions. The magneticfield produced by a single electron moving with velocity v is a dipole field givenby

H = ∇ ∧[ge µB σe ∧ r

| r |3

]− | e |

c

v ∧ r

| r |3(2503)

where r is the position of the field relative to the electron. The interactionbetween the neutron and the magnetic field is given by the Zeeman interaction

Hint = − gn µn σn .

[∇ ∧

(ge µB

σe ∧ r

| r |3

)− | e |

c

v ∧ r

| r |3

]

= gn µn

[σn . ∇ ∧

(ge µB

σe ∧ r

| r |3

)

− | e |2 me c

(p .

σn ∧ r

| r |3+

σn ∧ r

| r |3. p

) ](2504)

The first term is a classical dipole - dipole interaction and the second term is aspin - orbit interaction.

24.1.2 The Inelastic Scattering Cross-Section

The scattering cross-section of a neutron, from an initial state (k, σn) to a finalstate (k′, σ′n), in which the electron makes a transition from the initial state

711

| φn > to the final state | φn′ > is given by

d2σ

dω dΩ=

(gegnµnµB

)2k′

k

(V mn

2 π h2

)2 ∑n,n′

P (n)∣∣∣∣ < φn′ ; k′, σ′N | σN . ∇ ∧

(σe ∧ r

| r |3

)

− 12 h

(p .

σn ∧ r

| r |3+

σn ∧ r

| r |3. p

)| φn; k, σn >

∣∣∣∣2 δ( h ω + En − En′ )

(2505)

Here, the probability that the electronic system is in the initial state is rep-resented by P (n). The neutron’s energy loss h ω and the momentum loss orscattering vector hq are defined via

h ω = E(k) − E(k′)

h q = h k − h k′ (2506)

As the neutron states are momentum eigenstates, the matrix elements of theinteraction can be easily evaluated. The spin component of the magnetic inter-action is evaluated by considering the neutron component of the matrix elements

< k′ | ∇ ∧(σe ∧ r

| r |3

)| k >

= − < k′ | ∇ ∧(σe ∧ ∇ 1

| r |

)| k >

=1V

∫d3 rn exp

[+ i q . rn

]∇ ∧

(σe ∧ ∇ 1

| r |

)=

4 πV q2

(q ∧ ( σe ∧ q )

)exp

[+ i q . re

](2507)

This shows that the neutron only interacts with the component of the electron’sspin σ perpendicular to the scattering vector. Likewise, the orbital componentcan be evaluated as

< k′ | pe∧(σe ∧ r

| r |3

)| k > = − 4 π i

V q2

(σe ∧ ( q ∧ p

e))

exp[

+ i q . re

](2508)

Furthermore, the operator ( q ∧ pe

) commutes with exp[

+ i q . re

]as

q ∧ q ≡ 0. Hence, the neutron scattering cross-section from a multi-electronsystem can be written as

d2σ

dω dΩ=(gegnµnµB

)2k′

k

(2 mn

h2

)2 ∑n,n′

P (n) δ( h ω + En − En′ )

×∣∣∣∣ < φn′ ;σ′n | σn .

∑e

(q ∧

(σe ∧ q

| q |2

)− i

h

q ∧ pe

| q |2

)exp

[+ i q . re

]| φn;σn >

∣∣∣∣2(2509)

712

Since the nuclear Bohr magneton has the value

µn =| e | h2 mp c

(2510)

the coupling constant can be simplified

2 mn

h2

(gngeµnµB

)=

gn e2

me c2= re (2511)

to yield re, the classical radius of the electron. Thus, the scattering cross-sectioncan be written as

d2σ

dω dΩ= r2e

k′

kS(q;ω) (2512)

where the response function is given by

S(q;ω) =∑n,n′

P (n) δ( h ω + En − En′ )

×∣∣∣∣ < φn′ ;σ′n | σn .

∑e

(q ∧

(σe ∧ q

| q |2

)− i

h

q ∧ pe

| q |2

)exp

[+ i q . re

]| φn;σn >

∣∣∣∣2(2513)

This expression still depends on the polarization of the neutrons in the incidentbeam, and also on the polarization of the detector. Polarized neutron scatteringmeasurements reveal more information about the nature of the excitations ofa system. However, due to the reduction of the intensity of the incident beamcaused by the polarization process, and the concomitant need to compensatethe loss of intensity by increase the measurements time, it is more convenientto perform measurements with unpolarized beams. For an un-polarized beamof neutrons, the initial polarization must be averaged over. The averaging canbe performed with the aid of the identity∑

σn

12< σn | σα

n σβn | σn > = δα,β (2514)

which follows from the anti-symmetric nature of the Pauli spin matrices. Foran un-polarized beam of neutrons the response function reduces to

S(q;ω) =∑n,n′

P (n) δ( h ω + En − En′ )∑α,β

(δα,β − qα qβ

)

× < φn |∑

e

(σe +

i

h

q ∧ pe

| q |2

)α

exp[− i q . re

]| φn′ >

× < φn′ |∑

e

(σe − i

h

q ∧ pe

| q |2

)β

exp[

+ i q . re

]| φn >

(2515)

713

where q is the unit vector in the direction of q. On defining the spin densityoperator Sα(q) via

Sα(q) =∑

e

(σe +

i

h

q ∧ pe

| q |2

)α

exp[− i q . re

](2516)

then the response function can be expressed as a spin - spin correlation function

S(q;ω) =∑n,n′

P (n) δ( h ω + En − En′ ) ×

×∑α,β

(δα,β − qβ qβ

)< φn | Sα(q) | φn′ > < φn′ | S†β(q) | φn >

(2517)

Thus, the inelastic neutron scattering measures the excitation energies of thesystem, with intensity governed by the matrix elements < φn | Sα(q) | φn′ >which filters out the excitations of a non-magnetic nature. Furthermore, thescattering only provides information about the magnetic excitations which havea component of the fluctuation perpendicular to the momentum transfer.

In the case where the spin density can be expressed in terms of the atomicspin density due to the unpaired spins in the partially filled shells, such asin transition metals or rare earths, it is convenient to introduce the magneticatomic (ionic) form factor F (q). For a mono-atomic Bravais lattice, this isachieved by decomposing the spin density in terms of the spin density fromeach unit cell

Sα(q) =∑

e

(σe +

i

h

q ∧ pe

| q |2

)α

exp[− i q . re

]=

∑R

exp[− i q . R

] ∑j

(σj +

i

h

q ∧ pj

| q |2

)α

exp[− i q . rj

](2518)

Since the unpaired electrons couple together to give the ionic spin SR, theWigner - Eckert theorem can be used to express the spin density operator as

Sα(q) =∑R

exp[− i q . R

]F (q) SR (2519)

The form factor F (q) is defined as the Fourier transform of the normalized spindensity for the ion. By definition, the form factor is normalized by

F (0) = 1 (2520)

In this case, the inelastic neutron scattering spectrum can be expressed as

d2σ

dω dΩ= r2e

k′

k| F (q) |2 S(q;ω) (2521)

714

where the spin - spin correlation function is expressed in terms of the local ionicspins SR. Of course, it is being implicity assumed that the magnetic scatteringcan be completely separated from the phonon scattering. Thus, the analysis hasignored the existence of phonon excitations, in the case of zero phonon excita-tions, the intensity of the magnetic scattering is expected to be reduced by theDebye-Waller factor of the phonons.

24.2 Time-Dependent Spin Correlation Functions

The spin-dependent correlation function measured in scattering experiments isdenoted by Sα,β(q;ω) and is defined as

Sα,β(q;ω) =∑n,n′

P (n) δ( h ω + En − En′ ) ×

×

[< φn | Sα(q) | φn′ > < φn′ | S†β(q) | φn >

](2522)

and where P (n) is the probability that the system is found in the initial state| φn >. Using the expression for the energy conserving delta function as anintegral over a time variable

δ( h ω + En − En′ ) =∫ ∞

−∞

dt

2 π hexp

[i

h( h ω + En − En′ ) t

](2523)

the spin - spin correlation function can be written as a Fourier transform of atime-dependent correlation function.

Sα,β(q;ω) =1h

∑n,n′

P (n)∫ ∞

−∞

dt

2 πexp

[i ω t

]exp

[i

h( En − En′ ) t

]

×

[< φn | Sα(q) | φn′ > < φn′ | S†β(q) | φn >

]

=∫ ∞

−∞

dt

2 πexp

[i ω t

]1h

∑n,n′

P (n) exp[i

h( En − En′ ) t

]

×

[< φn | Sα(q) | φn′ > < φn′ | S†β(q) | φn >

](2524)

The product of the phase factor and the matrix elements of the S(q) can beexpressed in terms of an operator in the interaction representation

exp[i

h( En − En′ ) t

]< φn | Sα(q) | φn′ >

715

In the experiments on KDSOG-M, approximately 200 g,of CeAl3 and LaAl3 were sealed in aluminum containers andplaced in a helium cryostat for measurements at 8 K. A py-rolytic graphite monochromator and beryllium filter in frontof the detector fixes the final energy at 4.8 meV. The INSwas measured in neutron energy loss up to 80 meV energytransfer. The resolution for the elastic scattering was 0.6meV and the spectra were summed over three scatteringangles: 30°, 50°, and 70°.

Comparison of the data for CeAl3 and LaAl3 measuredon KDSOG-M and HET, as well as inspection of the HETdata taken at low~^f&519°! and high~^f&5136°! scatteringangles, shows that there is one well-defined inelastic mag-netic peak. Its energy is weakly temperature dependent:e;6.4 meV (T.20 K) ande;7.4 meV (T58 K). In accor-dance with the previous KDSOG-M measurements,7 there isalso evidence of broad structureless magnetic scattering ex-tending up to;80 meV energy transfer. This feature of themagnetic response of CeAl3 , and its variation as a functionof temperature and La/Y dilution will be presented in a laterpublication.13

The Ce31 ions in CeAl3 occupy positions with hexago-nal point symmetry and the corresponding CF Hamiltonian is

H5B20O2

01B40O4

0, ~1!

whereOnm are the Steven’s operator equivalents andBn

m arephenomenological CF parameters. The neutron scatteringlaw for unpolarized neutrons is given by:14

S~Q,e!5e

12exp~2e/kBT!f 2~Q!x0F~e!, ~2!

where f 2(Q) is the Ce13 form factor,F(e) is a normalizedLorentzian function characterizing the line shape of the tran-sition, andx0 is the static bulk susceptibility. The solid linesin Fig. 1 are the calculatedS(Q,e), assuming three spectralcomponents of the magnetic response as suggested in Ref. 7.

In the case of interaction of the 4f electrons with the CFthe static susceptibilityx0 is given by the sum of CuriexC

n

and Van VleckxVVmn contributions:

x05(n

xCn 1 (

mÞnxVV

mn , ~3!

xCn 5

gJ2mB

2

kBTrn (

a5x,y,zu^nuJaun&u2, ~4!

xVVmn52gJ

2mB2~rn2rm!

(a5x,y,zu^muJaun&u2

Dmn. ~5!

The observation of just one inelastic peak in the INSspectra is not sufficient for unambiguous determination ofthe two parameters in the CF Hamiltonian@Eq. ~1!#. For this,we need to make use of the observation by Jaccardet al.15 ofa crossing atT541 K of the single-crystal susceptibilitymeasured parallel and perpendicular to the hexagonalc axis.The Hamiltonian@Eq. ~1!# can be rewritten in the parameter-ized form:16

H5WS ~12x!O2

0

21x

O40

60D , ~6!

where21<x<11, so restricting the allowed range of pa-rameter values. If either the full splitting or the position ofthe first excited level is known,W is fixed for each value of

FIG. 2. ~a! Energies of crystal field levels as a function ofx @Eq. ~6!# withW fixed to 1. ~b! Anisotropy of the magnetic susceptibility,Dx5x'2x i ,vx at T541 K.

FIG. 1. Inelastic neutron scattering data from CeAl3 measured on the HETspectrometer with an incident energy of 35 meV. The lines are the results ofa fit to a three-component model as described in Ref. 7.

6047J. Appl. Phys., Vol. 85, No. 8, 15 April 1999 Goremychkin, Osborn, and Sashin

Downloaded 09 Sep 2008 to 129.241.49.120. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp

Figure 300: The spin correlation function S(Q, ε) inferred from inelastic neutronscattering experiments on the paramagnetic compound CeAl3. The spectrumshows a quasi-elastic peak and an inelastic peak (near 7 meV) due to crystalfield excitations. [After E. A. Goremychkin, R. Osborn and I. L. Sashin, J.Appl. Phys. 85, 6046 (1999).].

716

= < φn | exp[

+i

hH0 t

]Sα(q) exp

[− i

hH0 t

]| φn′ >

= < φn | Sα(q; t) | φn′ > (2525)

Hence, the spin - spin correlation function is given by

Sα,β(q;ω) =∫ ∞

−∞

dt

2 πexp

[i ω t

]×

×

[1h

∑n,n′

P (n) < φn | Sα(q; t) | φn′ > < φn′ | S†β(q; 0) | φn >

](2526)

The final states are a complete set of states, therefore, on using the completenessrelation, one finds

Sα,β(q;ω) =∫ ∞

−∞

dt

2 πexp

[i ω t

] [1h

∑n

P (n) < φn | Sα(q; t) S†β(q; 0) | φn >

]

=∫ ∞

−∞

dt

2 πexp

[i ω t

]1h

< | Sα(q; t) S†β(q; 0) | >

(2527)

The correlation function Sα,β(q;ω) is the Fourier Transform with respect totime of the thermal averaged spin - spin correlation function. The inverse spatialFourier transform of the spin density operator and its Hermitean conjugate aregiven by

Sα(q) =1V

∫d3r exp

[− i q . r

]Sα(r)

S†α(q) =1V

∫d3r′ exp

[+ i q . r′

]Sα(r′) (2528)

On inserting the above expressions into Sα,β(q;ω) and using the spatial homo-geneity of the system, one finds that the inelastic neutron scattering spectrumis related to the spatial and temporal Fourier transform of the spin - spin cor-relation function

Sα,β(q;ω) =1V

∫ ∞

−∞

dt

2 π

∫d3r exp

[i ( ω t − q . r )

]1h

< | Sα(r; t) Sβ(0; 0) | >

(2529)

where the brackets < | . . . | > represents the quantum mechanical thermalaverage. Thus, the inelastic neutron scattering probes the Fourier transform ofthe equilibrium spin correlation functions.

717

24.3 The Fluctuation - Dissipation Theorem

The spin - spin correlation function

Sα,β(q;ω) =∑n,n′

P (n) δ( h ω + En − En′ ) < φn | Sα(q) | φn′ > < φn′ | S†β(q) | φn >

(2530)

is consistent with the principle of detailed balance. If the equilibrium probabilityP (n) is given by the Boltzmann expression

P (n) =1Z

exp[− β En

](2531)

then spin-spin correlation function can be re-written as

Sα,β(q;ω) =∑n,n′

P (n) δ( h ω + En − En′ ) < φn′ | S†β(q) | φn > < φn | Sα(q) | φn′ >

= exp[β h ω

] ∑n,n′

P (n′) δ( En′ − En − h ω ) < φn′ | S†β(q) | φn > < φn | Sα(q) | φn′ >

(2532)

where we have used the identity exp[−βEn] δ(hω+En−En′) = exp[βhω] exp[−βEn′ ] δ(hω+En − En′). On interchanging the summation indices n and n′, one finds

= exp[β h ω

] ∑n,n′

P (n) δ( En − En′ − h ω ) < φn | S†β(q) | φn′ > < φn′ | Sα(q) | φn >

= exp[β h ω

]Sβ,α(−q;−ω)

(2533)

which is consistent with the principle of detailed balance for equilibrium pro-cesses.

The correlation function Sα,β(q;ω) is also related to the imaginary part ofthe magnetic susceptibility χα,β(q;ω) via the fluctuation dissipation theorem.

The reduced dynamical magnetic susceptibility is given by the expression

χα,β(r, r′; t− t′) = − i

h< |

[Sα(r, t) , Sβ(r′, t′)

]| > Θ( t − t′ ) (2534)

which can be expressed as

χα,β(r; t) = − i

h

∑n,n′

P (n) exp[i

h( En − En′ ) t

]< φn | Sα(r) | φn′ > < φn′ | Sβ(0) | φn > Θ( t )

+i

h

∑n,n′

P (n) exp[i

h( En′ − En ) t

]< φn | Sβ(0) | φn′ > < φn′ | Sα(r) | φn > Θ( t )

(2535)

718

The Fourier transform is defined as

χα,β(q;ω) =1V

∫ ∞

−∞

dt

2 π

∫d3r exp

[i ( ω t − q . r )

]χα,β(r; t)

(2536)

and is evaluated as

χα,β(q;ω) =1

2 π

∑n,n′

P (n)

[< φn | Sα(q) | φn′ > < φn′ | S†β(q) | φn >

h ω + En − En′ + i δ

−< φn | S†β(q) | φn′ > < φn′ | Sα(q) | φn >

h ω + En′ − En + i δ

](2537)

The imaginary part of the dynamic susceptibility is given by

=m[χα,β(q;ω)

]= − 1

2

∑n,n′

P (n) ×

×

[δ( h ω + En − En′ ) < φn | Sα(q) | φn′ > < φn′ | S†β(q) | φn >

− δ( h ω + En′ − En ) < φn | S†β(q) | φn′ > < φn′ | Sα(q) | φn >

](2538)

The first term is recognized as being Sα,β(q;ω). On replacing P (n) by P (n′) exp[−βhω]in the second term and interchanging the summation indices n and n′, one recog-nizes that the second term is just proportional to exp[−βhω] Sα,β(q;ω). Hence,we have found that

=m[χα,β(q;ω + iδ)

]= − 1

2Sα,β(q;ω)

[1 − exp

[− β h ω

] ](2539)

or

Sα,β(q;ω) = 2(

1 + N(ω))=m

[χα,β(q;ω − iδ)

](2540)

This is the fluctuation - dissipation theorem287. The fluctuation - dissipationtheorem relates the dynamical response of the system to an external perturba-tion to the naturally occurring excitations in the system, such as those measuredin neutron scattering experiments.

287H. B. Callen and T. A. Welton, Phys. Rev. 83, 34 (1951) and R. Kubo, J. Phys. Soc.Japan 12, 570 (1957).

719

24.4 Magnetic Scattering

The neutron scattering cross-section is given in terms of the components of thespin spin correlation function.

As can be seen by inspection from the Holstein-Primakoff representation ofthe spins and the spin waves, the spin correlation function is a non-linear func-tion of the spin wave creation operators. The inelastic scattering cross-sectioncan be expanded in powers of the number of spin waves. The lowest-order termis time-independent and corresponds to Bragg scattering.

24.4.1 Neutron Diffraction

The ω = 0 component of the inelastic scattering cross-section given by theω → 0 limit of

Sα,β(q;ω) =∫ ∞

−∞

dt

2 πexp

[i ω t

]1h

< | Sα(q; t) S†β(q; 0) | > (2541)

diverges if the integrand does not decay rapidly as t → ∞. In this case, thetime-independent component of the spin - spin correlation function given by

limt → ∞

1h

< | Sα(q; t) S†β(q; 0) | > (2542)

produces a Bragg peak with finite intensity since

δ(ω) =∫ ∞

−∞

dt

2 πexp

[i ω t

](2543)

Thus, the intensity of the Bragg peak represents the static correlations. If theergodic hypothesis holds, then in the long time limit the correlation functiondecouples into the product of two expectation values

limt → ∞

1h

< | Sα(q; t) S†β(q; 0) | >

= limt → ∞

1h

< | Sα(q; t) | > < | S†β(q; 0) | > (2544)

and for a static system for which

< | Sα(q; t) | > = < | Sα(q; 0) | > (2545)

then the Bragg peak has an intensity given by

= limt → ∞

1h

< | Sα(q; 0) | > < | S†β(q; 0) | > (2546)

For a paramagnetic system the (quasi-stationary) average value of the spin

720

Figure 301: Left: The neutron diffraction pattern of MnO at temperaturesbelow and above the magnetic ordering temperature. Right: The antiferromag-netic structure of MnO below the Neel temperature. [After C. G. Shull, W. A.Strauser and E. O. Wollan, Phys. Rev. 83, 333 (1951).].

vector is zero< | Sα(q; 0) | > = 0 (2547)

and, thus, there is no magnetic Bragg peak for a paramagnetic system. On theother hand, if there is long-ranged magnetic order with wave vectors Q and withspin oriented along certain directions (say α), then

< | Sα(Q; 0) | > 6= 0 (2548)

thus, the magnetic Bragg peaks are non-zero. The temperature dependence ofthe intensity of the Bragg peaks provides a direct measure of the temperaturedependence of the magnetic order parameter. For a ferromagnet, the magneticBragg peaks coincide with the Bragg peaks due to the crystalline order, so nonew peaks emerge. The Bragg scattering cross-section is given by(dσ

dΩ

)Bragg

= r2e( 2 π N )2

V

∑Q

(1 − q2z

)δ( q − Q )

∣∣∣∣ < | Sz | >∣∣∣∣2

(2549)For a small single domain single crystal, the magnetic elastic scattering is ex-tremely anisotropic as the scattering should be zero for momentum transfersalong the direction of the magnetization.

For anti-ferromagnetic or spin density wave order new Bragg peaks mayemerge at the vectors of the anti-ferromagnetic reciprocal lattice. Analysis ofthe anisotropy of the neutron scattering intensity for anisotropic single crystals

721

Figure 302: The temperature-dependence of the antiferromagnetic order param-eter for MnO. [After C. G. Shull, W. A. Strauser and E. O. Wollan, Phys. Rev.83, 333 (1951).].

leads to the determination of the preferred directions of the magnetic moments.

——————————————————————————————————

24.4.2 Exercise 94

Evaluate the elastic scattering cross-section for a anti-ferromagnetic insulator,using the Holstein-Primakoff representation of the low-energy spin wave exci-tations. Discuss the anisotropy and also the temperature dependence of theintensity of the Bragg peaks.

——————————————————————————————————

24.4.3 Exercise 95

Design a neutron diffraction experiment that will determine if a system has aspiral spin density wave order, as opposed to a magnetic moment that is mod-ulated in intensity. How can the direction of spiral be determined?

722

——————————————————————————————————

24.4.4 Spin Wave Scattering

The spin wave excitations of an ordered magnet show up in the inelastic neutronscattering spectra. In a process whereby a single spin wave is emitted in thescattering process, conservation of energy leads to the energy difference betweenthe initial state En and the final state En′ being given by

En′ − En = h ωq (2550)

The matrix elements in the spin - spin correlation function can be evaluated as

< φn | Sα(q) | φn′ > = <∏q′

nq′ | Sα(q) |∏q”

n′q” > (2551)

where the number of spin waves in the initial state are related to the numberin the final state via

n′q′ = nq′ for q′ 6= q (2552)

and

n′q = nq + 1 (2553)

For a ferromagnet, the matrix elements are evaluated as

< nq | Sz(q) | nq + 1 > = 0 (2554)

while the transverse matrix elements are

< nq | Sx(q) | nq + 1 > = < nq |12

( S+(q) + S−(q) ) | nq + 1 >

=12

√nq + 1

√2 S (2555)

and

< nq | Sy(q) | nq + 1 > = < nq |1

2 i( S+(q) − S−(q) ) | nq + 1 >

= +1

2 i

√nq + 1

√2 S (2556)

Thus, the inelastic neutron scattering from the single spin wave excitations ofa ferromagnet is given purely by the diagonal components of the transversespin-spin correlation function, as the longitudinal components are zero. Theoff-diagonal terms cancel. The cross-section for the spin wave emission processis given by(d2 σ

dω dΩ

)emit

= r2e( 2 π)2

V

N S

2

(1 + q2z

) ∑q′,Q

δ( h ω − h ω(q′) ) δ( q − q′ −Q )[

1 +N(ω(q′))]

(2557)

723

Likewise, the absorption process has a scattering cross-section given by(d2 σ

dω dΩ

)abs

= r2e( 2 π)2

V

N S

2

(1 + q2z

) ∑q′,Q

δ( h ω + h ω(q′) ) δ( q − q′ −Q )N(ω(q′))

(2558)The intensities of the emission and absorption processes are consistent with theprinciple of detailed balance (only valid for total equilibrium), and give rise toa Stokes and anti-Stokes line in the spectrum of the scattered neutrons.

——————————————————————————————————

24.4.5 Exercise 96

Evaluate the two lowest-order terms in the inelastic scattering cross-section foran anti-ferromagnetic insulator, using the Holstein-Primakoff representation ofthe low-energy spin wave excitations. Discuss the differences between the spec-trum obtained from magnetic scattering and that found in measurements of thephonon excitations.

——————————————————————————————————

24.4.6 Critical Scattering

Just above the temperature where magnetic ordering occurs, the inelastic neu-tron scattering cross-section in the paramagnetic phase shows a softening orbuild up at low frequencies and becomes sharply peaked at q values close tothe magnetic Bragg vectors Q. Below the ordering temperature, the intensitytransforms into the Bragg peak. This phenomenon of the build up of inten-sity close to the Bragg peak is known as critical scattering. The Bragg peakis extracted from the inelastic scattering spectrum by extracting a delta func-tion δ(ω), i.e., the inelastic scattering cross-section is integrated over a smallwindow dω. On invoking the fluctuation dissipation theorem and then notingthat if, in the paramagnetic phase, the main portion of the scattering occurswith frequencies such that β ω 1, then one finds that by using the Kramers- Kronig relation, the intensity of the critical scattering is given by the staticsusceptibility. For q values close to the Bragg peak, the susceptibility varies as

∝ 1( q − Q )2 + ξ−2

(2559)

where ξ the correlation length, in the mean-field approximation, is given by

ξ−1 = a

√Tc

6 ( T − Tc )(2560)

724

Thus, the critical scattering diverges as ( T − Tc )−1 as the transition temper-ature is approached.

725

Figure 303: The electrical resistance of Hg as a function of temperature. [AfterH. Kammerlingh Onnes (1911).].

25 Superconductivity

The electrical resistivity ρ(T ) of metals at low temperatures is expected to bedescribed by the Drude model

ρ(T ) ∝ m

e2 τ(2561)

The resistivity should vary with temperature according to

ρ(T ) ∼ ρ(0) + A T 2 + B T 5 (2562)

since the scattering rates for scattering from static impurities, electron-electronscattering and phonon scattering are expected to be additive. The resistivityof a perfect metal, without impurities, may be expected to vanish at T = 0.However, it was discovered by Kammerlingh Onnes288 that the resistivity ofa metal may become so small as to effectively vanish for all temperatures be-low a critical temperature Tc. This indicates that the scattering mechanismssuddenly becomes ineffective for temperatures slightly below the critical tem-perature, where the metal seems to act like a perfect conductor. The resistivityis so small that persistent electrical currents have been observed to flow withoutattenuation for very long time periods. The decay time of a super-current infavorable materials is apparently not less than 10,000 years.

288H. Kammerlingh Onnes, Comm. Phys. Lab. Univ. Leiden, Nos. 119, 120, 122 (1911).

726

B B

SN

T > TcT < Tc

Figure 304: A schematic depiction of the Meissner effect. The magnetic induc-tion field B is excluded from the bulk of a superconducting sample, in contrastwith the normal state.

25.1 Experimental Manifestation

The first manifestation of superconductivity is zero resistance, below Tc. An-other manifestation of superconductivity was found by Meissner and Ochsen-feld289, which is flux exclusion. A superconductor excludes the magnetic induc-tion field B from its interior, irrespective of whether it was cooled from aboveTc to below Tc in the presence of an applied field, or whether the field is onlyapplied when the temperature is smaller than Tc. In other words, the Meissnereffect excludes time independent magnetic field solutions from inside the su-perconductor. The Meissner effect distinguishes superconductivity from perfectconductivity, as a static magnetic field can exist in perfect conductor.

The perfect conductor has the property that the current produced by anapplied electric field increases linearly with time. Therefore, a perfect conductorexcludes electric fields from within its bulk. Maxwell’s equations reduce to

− 1c

∂B

∂t= 0

∇ ∧ B =4 πc

j

∇ . B = 0 (2563)

Thus, a perfect conductor only excludes a time varying magnetic field, but nota static magnetic field.

The Meissner effect shows hat the magnetic induction inside a superconduc-tor is zero. However, the magnetic induction B can be expressed in terms ofthe applied field H and the magnetization M via

B = H + 4 π M (2564)

289W. Meissner and R. Ochsenfeld, Naturwiss. 21, 787 (1933).

727

H- 4

π M

Type I

Type II

Hc1 Hc2Hc

Figure 305: A schematic depiction of the dependence of the (diamagnetic) mag-netization on applied magnetic field for a type I and a type II superconductor.

so B = 0 implies that

M = − 14 π

H (2565)

so that perfect diamagnetism implies that the susceptibility is given by

χ = − 14 π

(2566)

The perfect diamagnetism does not hold for arbitrarily large applied magneticfields. For fields larger than a critical magnetic field, the induction inside thesuperconductor becomes non-zero. For a type I superconductor, the appliedfield fully penetrates into the bulk of the superconductor above the critical fieldHc. The magnetization drops discontinuously to zero at Hc. The value of Hc

depends on temperature according to

Hc(T ) = Hc(0)(

1 − T 2

T 2c

)(2567)

For a type II superconductor, the induction first starts penetrating into thebulk at the lower critical field Hc1, For fields larger than the lower critical field,the magnetization deviates from linear relation associated with perfect diamag-netism. The magnitude of the magnetization is reduced as the applied field isincreased above Hc1. The magnetization falls to zero at the upper critical fieldHc2, at which point the applied field fully penetrates into the bulk.

The experimental observations of a drop in the resistivity and the Meissnereffect demonstrate that the transition to the superconducting state is a phasetransition as the properties are independent of the history of the sample. Fora type I superconductor, the bulk superconductivity is completely destroyed atHc(T ).

728

25.1.1 The London Equations

A phenomenological description of superconductivity was developed by the Lon-don brothers290. Basically, this description is based on two phenomenologicalconstitutive equations for the electromagnetic field and its relation to currentand density. The first London equation is of the form

j(r, t) = − ns e2

m cA(r, t) (2568)

which expresses the extreme diamagnetic response of a superconductor. Here,ns is density of superfluid electrons. It should be noted that London’s firstequation is not gauge invariant. The vector potential in London’s first equationhas to be calculated with a specific choice of gauge condition. If the supercon-ducting current is to be conserved, one must require that ∇ . A = 0. TheLondon equation describes the microscopic current in the superconductor thatscreens the applied magnetic field. This is slightly different from the conditionof perfect conductivity in a metal. In a perfect conductor, the time derivativeof the current is related to the electric field. In deriving the London equationfrom the condition of perfect conductivity, it has been assumed that the electricfield is transverse. The condition for perfect conductivity has been integratedwith respect to time, and the constant of integration has been chosen to be zero.The choice of the constant of integration allows a constant current to screen thestatic applied magnetic field. In order that the continuity equation be satisfiedin a steady state, a gauge condition must be imposed such that ∇ . A(r, t) = 0and one also requires that the perpendicular component of A vanish at the sur-face. This gauge condition defines the London gauge.

The second London equation comes from Maxwell’s equations

∇ ∧ B(r, t) =4 π j(r, t)

c+

1c

∂

∂tE(r, t) (2569)

and with the definitions

B(r, t) = ∇ ∧ A(r, t)

E(r, t) = − 1c

∂

∂tA(r, t) (2570)

one finds(∇ ∧ ∇ ∧ +

1c2

∂2

∂t2

)A(r, t) = − 4 π ns e

2

m c2A(r, t) (2571)

This is referred to as the second London equation. The quantity ns e2

m c2 has unitsof inverse length squared and is used to define the London penetration depth290F. London and H. London, Proc. Roy. Soc. (London), A 149, 71 (1935), F. London,

Phys. Rev. 74, 562 (1948).

729

λL, via

4 π ns e2

m c2=

1λ2

L

(2572)

The second London equation expresses the Meissner effect. Namely, thata superconductor excludes the magnetic induction field B from the bulk of itsvolume. However, the field does penetrate the region at the surface and extendsover a distance λL into the superconductor. This can be seen by examiningvarious cases in which a static applied magnetic field is produced near a super-conductor. The geometry is considered in which the applied field is parallel tothe surface.

Let the surface be the plane z = 0, which separates the superconductorz > 0 from the vacuum z < 0. The external field is applied in the x direction,so B = B0 x for z < 0. The vector potential inside the superconductor mustsatisfy the boundary condition Az(z = 0) = 0 as any current should be per-pendicular to the surface. The London gauge requires the non-zero componentsof the vector potential to be Ax and Ay. Thus, the vector potential must beparallel to the surface. The static solution for the vector potential that satisfiesthe boundary conditions on the current for the semi-infinite solid is

A(z) = A0 exp[− z

λL

](2573)

An additional boundary condition at z = 0 is that Bx should be continuous.Hence, as the equation for the magnetic induction simplifies to

Bx(z) = − ∂Ay(z)∂z

(2574)

one finds that the vector potential is directed parallel to the surface, but is alsoperpendicular to the applied field. The only non-zero component of the vectorpotential in the superconductor is found as

Ay(z) = + λL Bx(0) exp[− z

λL

]for z > 0 (2575)

London’s first equation then implies that a supercurrent, jy(z), flows in a regionnear the surface of the superconductor which, through Ampere’s law, producesmagnetic field that screens or cancels the applied field. The magnetic inductionand the supercurrent are non-zero in the superconductor only within a distanceof λL from the surface. Hence, λL is called the penetration depth.

25.1.2 Thermodynamics of the Superconducting State

The phase transition to a superconducting state, in zero field, is a second or-der phase transition. This can be seen by examining the specific heat which

730

0

0.2

0.4

0.6

0.8

1

1.2

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1

z / λ

Ay(

z)

Vacuum Superconductor

Ay(z) = A0 exp [ -z / λ ]Ay(z) = A0

Figure 306: The spatial dependence of the vector potential A showing that themagnetic field only penetrates the superconductor over a distance of the orderof λ.

exhibits a discontinuous jump at Tc. The absence of any latent heat impliesthat the entropy is continuous, and since the entropy is obtained as a first orderderivative of the Free energy the transition is not first order. The non-analyticbehavior of the specific heat, which is obtained from a second derivative of thefree energy defines the transition to be second order.

In the presence of a field the transition is first order. The thermodynamicrelations are derived from the Gibbs free energy G in which M plays the role ofthe volume V and the externally applied field H plays the role of the appliedpressure P . Then, G(T,H) has the infinitesimal change

dG = − S dT − M . dH (2576)

where S is the entropy and T is the temperature. Since G is continuous acrossthe phase boundary at (Hc(T ), T )

Gn(T,Hc(T )) = Gs(T,Hc(T )) (2577)

which on taking an infinitesimal change in both T and H = Hc(T ) so as to stayon the phase boundary, one finds

( Ss − Sn ) dT = ( Mn − Ms ) dHc(T ) (2578)

The magnetization in the normal state is negligibly small, but the supercon-ducting state is perfectly diamagnetic, so

Mn − Ms = +1

4 πHc(T ) (2579)

This shows that in the presence of an applied field the superconducting transi-tion involves a latent heat, L, given by

L = T ( Sn − Ss )

= − T

4 πHc

∂Hc

∂T(2580)

731

Thus, the transition is first order in the presence of an applied field.

In the limit that the critical field is reduced to zero, the transition tempera-ture T is reduced to the zero field value Tc, and the entropy becomes continuousat the transition. However, there is a change in slope at Tc, which can be foundby taking the derivative of Sn − Ss with respect to T , and then letting Hc → 0

∂

∂T( Sn − Ss ) = − 1

4 π

[Hc

(∂2Hc

∂T 2

)+(∂Hc

∂T

)2]

= − 14 π

(∂Hc

∂T

)2

(2581)

The specific heat may show a discontinuity or jump at Tc that is a measure ofthe initial slope of the critical field

Cs − Cn =T

4 π

(∂Hc

∂T

)2

(2582)

The discontinuous jump in the (zero field) specific heat is a characteristic of amean-field transition. For temperatures below Tc the specific heat is exponen-tially activated

Cv ∼ γ Tc exp[− ∆

kB T

](2583)

The activated exponential behavior of the specific heat suggests that there is anenergy gap in the excitation spectrum. The existence of a gap is confirmed bya threshold frequency for photon absorption by a superconductor. Above Tc,the absorption spectrum is continuous and photons of arbitrarily low frequencycan be absorbed by the metal. However, for temperatures below Tc, there isa minimum frequency above which photons can be absorbed. The thresholdfrequency is related to ∆.

In most superconductors, the interaction mechanism that is responsible forpairing is mediated by the electron-phonon coupling. This was first identi-fied through the insight of Frohlich291, who predicted that the superconductingtransition temperature Tc should be proportional to the phonon frequency. Fur-thermore, as the square of the phonon frequency is inversely proportional to themass of the ions M , the superconducting transition temperature should dependupon the isotopic mass through

Tc ∝ M− 12 (2584)

This isotope effect was confirmed in later experiments by Maxwell 292 andReynolds et al.293 on simple metals. However, in transition metals the ex-ponent of the isotope effect is reduced and may become zero, and in α− U the291H. Frohlich, Phys. Rev. 79, 845 (1950).292E. Maxwell, Phys. Rev. 78, 447 (1950), Phys. Rev. 79, 173 (1950).293C. A. Reynolds, B. Serin, W. H. Wright and L. B. Nesbitt, Phys. Rev. 78, 487 (1950).

732

Figure 307: The dependence of the superconducting transition temperature Tc

of Sn on the isotopic mass M on a log-log plot. The critical temperature isassumed to vary as Tc ∝ M−α. The two straight lines yield α ≈ 0.487 andα ≈ 0.505. [After E. Maxwell, Phys. Rev. 86, 25 (1952).].

exponent is positive. The occurrence of a positive isotope effect does not nec-essarily signify the existence of alternate pairing mechanisms, but can indicatethe effect of strong electron-electron interactions.

25.2 The Cooper Problem

The electron-electron interaction in a metal, is attractive at low frequencies.The attractive interaction originates from the screening of the electrons by theions, but only occurs for energy transfers less than h ωD . The effective attrac-tion is retarded, and occurs due to the attraction of a second electron with theslowly evolving polarization of the lattice produced by the first electron. Coopershowed that two electrons, which are close to the Fermi energy, will bind intopairs whenever they experience an attractive interaction, no matter how weakthe interactions is294.

Consider a pair of electrons of spin σ and σ′, excited above the Fermi en-ergy. Due to the interaction between the pair of particles, the center of massmomentum q will be a constant of motion, but not the relative motion. Thus,the wave function of the Cooper pair with total momentum q can be written as

| Ψq > =∑

k

C(k) | σ, k + q σ′,−k > (2585)

Due to the Pauli exclusion principle the single-particle energies E(−k) and E(k+q) must both be above the Fermi energy µ. The wave function is normalized

294L. Cooper, Phys. Rev. 104, 1189 (1956).

733

k

-k -k'

k'

kF

µ+hωDµ

Figure 308: The scattering process involved in the pairing problem consideredby Cooper. Two electrons in states (σ, k) and (−σ,−k) above the Fermi-surfaceare scattered into states (σ, k′) and (−σ,−k′), if all the states are within anenergy of hωD of the Fermi energy.

such that ∑k

| C(k) |2 = 1 (2586)

The wave function must be an energy eigenstate of the Hamiltonian H and,thus, satisfies

H | Ψq > = E(q) | Ψq > (2587)

where E(q) is the total energy of the pair of electrons. On projecting out C(k)using the orthogonality of the different momentum states | σ, k + q σ′,−k > ,one finds the secular equation(

E(q) − Ek − Ek+q

)C(k) = − 1

N

∑k′

V (k, k′) C(k′) (2588)

The attractive pairing potential V , ( − V < 0 ), scatters the pairs of electronsbetween states of different relative momentum. The summation over k′ is re-stricted to unoccupied Bloch states within h ωD the Fermi surface, where theinteraction is attractive. The above equation has a solution for the amplitudeC(k) which is given by

C(k) =α(k)

E(q) − Ek − Ek+q(2589)

where α is given by

α(k) = − 1N

∑k′

V (k, k′) C(k′) (2590)

This equation can be solved analytically in the case where the potential is sepa-rable, such as the case where V is just a constant. In such cases, the summation

734

over k′ can be performed to yield a result which is independent of k. For sim-plicity, the separable potential shall be assumed to be attractive and have amagnitude of V when both k and k′ are within h ωD of the Fermi surface.Then, α is independent of k and

α = − α

N

∑k′

V (k, k′)Ek′ + Ek′+q − E(q)

(2591)

Thus, the energy eigenvalue is determined from the equation

1 = − 1N

∑k′

V (k, k′)Ek′ + Ek′+q − E(q)

(2592)

For Cooper pairs with zero total momentum q = 0, this equation reduces to

1 = V

∫ µ+hωD

µ

dερ(ε)

2 ε − E(0)(2593)

The density of states ρ(ε) can be approximated by a constant ρ(µ), and theintegral can be performed, yielding

1 =V ρ(µ)

2ln(

2 h ωD + 2 µ − E(0)2 µ − E(0)

)(2594)

This can be inverted to give the energy eigenvalue as

E(0) = 2 µ − 2 h ωD

exp[

2V ρ(µ)

]− 1

(2595)

This eigenvalue is less than the minimum energy of the two independent elec-trons, thus, the electrons are bound together. It is concluded that, due to thesharp cut off of the integral at the Fermi energy, the electrons bind to formCooper pairs no matter how small the attractive interaction is. The bindingenergy is small and is a non-analytic function of the pairing potential V , thatis, the binding energy cannot be expanded as a power series in V .

In the case that the pairing potential is spin rotationally invariant, the totalspin of the pair S is a good quantum number. The pairing states can be catego-rized by the value of their spin quantum number and the projection of the totalspin along the z-axis. On pairing two spin one-half electrons, there are fourpossible state, a spin singlet state S = 0 and a spin triplet state S = 1 whichis three-fold degenerate. The four Cooper pair wave functions corresponding tothese states have to obey the Paul-exclusion principle and are written as

ψS=0(r1, r2) =∑

k

CS=0(k)12

(φk(r1) φ−k(r2) − φ−k(r1) φk(r2)

)×

×(χ+ 1 χ− 2 + χ− 1 χ+ 2

)(2596)

735

for the spin singlet pairing. The three spin triplet pair wave functions are

ψS=1,m=1(r1, r2) =∑

k

CS=1(k) φk(r1) φ−k(r2) χ+ 1 χ+ 2

ψS=1,m=0(r1, r2) =∑

k

CS=1(k)12

(φk(r1) φ−k(r2) + φ−k(r1) φk(r2)

)×

×(χ+ 1 χ− 2 − χ− 1 χ+ 2

)ψS=1,m=−1(r1, r2) =

∑k

CS=1(k) φk(r1) φ−k(r2) χ− 1 χ− 2

(2597)

Thus, for singlet pairing one must have

CS=0(k) = CS=0(−k) (2598)

which requires that, when expanded in spherical harmonics, the expansion onlycontains even components of orbital angular momentum. For triplet pairing,one has

CS=1(k) = − CS=1(−k) (2599)

thus, the triplet pair can only be composed of odd values of orbital angularmomentum.

Most superconductors that have been found have singlet spin pairing and arein a state which is predominantly in a state of orbital angular momentum l = 0.The high Tc superconductor such as Sr doped La2CuO4 found by Bednorz andMuller295 in 1986 (Tc = 35 K) or Y Ba2Cu3O7 (Tc = 90 K) form exceptionsto this rule. These materials evolve from an anti-ferromagnetic insulator phaseat zero doping, but as the doping increases they lose the antiferromagnetismand become metallic paramagnets. A superconducting phase appears for dop-ing concentrations above a small critical concentration. The superconductivityis exceptional, not just in the magnitude of the transition temperature Tc butalso in that the pairing is singlet, but with an appreciable admixture of a com-ponent with l = 2 in the pair. Due to this admixture, the pairing in high Tc

superconductors is sometimes referred to as d wave pairing. In heavy fermionsuperconductors, such as CeCu2Si2, UBe13, UPt3 and URu2Si2, experimentalevidence exists that these materials do not show exponentially activated behav-ior characteristic of a gap. Instead, the specific heat and susceptibility showpower law variations296. This, and the multiple superconducting transitionsfound in UPt3 and Th doped UBe13, indicate that the order parameter is dom-inated by components with non-zero angular momentum297. If the symmetry295J. G. Bednorz and K. A. Muller, Zeit. fur Physik, B, 64, 189 (1986).296H. R. Ott, H. Rudigier, T. M. Rice, K. Ueda, Z. Fisk and J. L. Smith, Phys. Rev. Lett.

52, 1915 (1984).297H. R. Ott, H. Rudigier, Z. Fisk and J. L. Smith, Phys. Rev. B, 31, 1615 (1985).

736

-r/2

r/2

ξ

Figure 309: A schematic depiction of the overlap of Cooper pairs. The coherencelength is denoted by ξ.

of the order parameter is that of a state with definite angular momentum, thesuperconducting gap may vanishes at points or lines on the normal state Fermisurface.

It is customary to represent the wave function of the Cooper pair in terms ofrelative coordinates r = r1 − r2 and center of mass coordinates R = r1 + r2

2 .Thus, the Cooper pair wave function is written as

ψ(r1, r2) → ψ(r,R) (2600)

and as the pair usually is in a state with zero total momentum, q = 0, thecenter of mass dependence can be ignored.

The mean square radius of the Cooper pair wave function is given by

ξ2 =∫

d3r r2 | ψ(r) |2 (2601)

but

ψ(r) =∑

k

C(k) exp[i k . r

](2602)

Thus,

ξ2 =∫

d3r r2∑k,k′

C(k) C∗(k′) exp[i ( k − k′ ) . r

]=

∑k

| ∇k C(k) |2

=43

(h vF

2 µ − E

)2

=43

(vF

2 ωD

)2

exp[

4V ρ(µ)

](2603)

737

For a binding energy of order 10 K and a Fermi velocity vF of the order of 106

m/sec, one obtains a pair size ξ of order 104 Angstroms. The coherence lengthξ is much greater than the average spacing between the electrons.

The standard weak-coupling theory of superconductivity due to Bardeen,Cooper and Schrieffer, (B.C.S.)298, treats the Cooper pairing of all electronsclose to the Fermi surface in a self-consistent manner.

25.3 Pairing Theory

25.3.1 The Pairing Interaction

The attractive pairing interaction can be obtained from the electron-phononinteraction, via an appropriately chosen canonical transform. The energy of thecombined electron phonon system can be expressed as the sum

H = H0 + Hint (2604)

where the non-interacting Hamiltonian is given by

H0 =∑k,σ

Ek c†k,σ ck,σ +

∑q,α

h ωα(q) a†q,α aq,α (2605)

and the interaction term is given by

Hint =∑k,σ

∑q,α

λq c†k+q,σ ck,σ

(aq,α + a†−q,α

)(2606)

The Hamiltonian will be transformed via

H ′ = exp[

+ S

]H exp

[− S

](2607)

where S is chosen in a way that will eliminate the interaction term (at least toin first order). The operator S can be thought of as being of the same order asHint. The transformation proceeds by expanding the transformed Hamiltonianin powers of S

H ′ = H0 + Hint +[S , H0

]+

12

[S ,

[S , H0

] ]+[S , Hint

]+ O

(H3

int

)(2608)

298J. Bardeen, L. N. Cooper and J. R. Schrieffer, Phys. Rev. 108, 1175 (1957).

738

and then choosing S such that

Hint =[H0 , S

](2609)

Since this is an operator equation, this is solved for S by taking matrix elementsbetween a complete set. A convenient complete set is provided by the eigenstates| φm > of H0, which have energy eigenvalues Em. Thus, the matrix elementsof S are found from the algebraic equation

< φm | Hint | φn > = ( Em − En ) < φm | S | φn > (2610)

The complete set of energy eigenstates are energy eigenstates of the combinednon-interacting electron and phonon Hamiltonians. The non-zero matrix el-ements only occur between states which involve a difference of unity in theoccupation number of one phonon mode ( either q or − q ) , and also a changeof state of one electron ( k to k + q ). The anti-Hermitean operator S can berepresented in second quantized form as

S = −∑k,σ

∑q,α

c†k+q,σ ck,σ

[λq aq,α

Ek+q − Ek − h ω(q)+

λq a†−q,α

Ek+q − Ek + h ω(q)

](2611)

The transformed Hamiltonian contains the effects of the interaction only throughthe higher order terms

H ′ = H0 +12

[S , Hint

]+ O

(H3

int

)(2612)

On evaluating the commutation relation, one finds a renormalization of theelectron dispersion relation of order | λq |2, and electron-electron interactionterms. The electron-electron interaction terms combine and can be written as

H ′int =

∑k,σ;k′,σ′

∑q,α

| λq |2 h ω(q)

( Ek+q − Ek )2 − h2 ω(q)2c†k+q,σ ck,σ c

†k′−q,σ′ ck′,σ′

(2613)Thus, for electrons within h ω(q) of the Fermi energy there is an attractive inter-action between the electrons. This interaction depends on the energy transferbetween the electrons, and the energy transfer corresponds to a frequency. Asthe interaction is frequency-dependent, it corresponds to a retarded interaction.Since the interaction is only attractive at sufficiently low frequencies, the in-teraction is only attractive after long time delays. In the B.C.S. theory thisinteraction is simplified. The simplification consists of only retaining scatteringbetween electrons of opposite spin polarization and momentum, as this maxi-mizes the phase space of the allowed final states. That is, the momenta andspin are restricted such that

k = − k′ (2614)

739

and alsoσ = − σ′ (2615)

This procedure produces a pairing between electrons of opposite spins.

The B.C.S. Hamiltonian is composed of the energy of electrons in Blochstates and an attractive interaction between the electrons mediated by thephonons. The B.C.S. Hamiltonian is written as

H =∑k,σ

Ek c†k,σ ck,σ −

∑k,k′

V (k, k′) c†−k′,↓ c†k′,↑ ck,↑ c−k,↓ (2616)

25.3.2 The B.C.S. Variational State

The pairing theory of superconductivity considers the ground state to be a statewithin the grand canonical ensemble. That is, the ground state is composed ofa linear superposition of states with different numbers of particles. If required,a ground state in the canonical ensemble can be found by projecting the B.C.S.ground state onto one with a fixed number of particles. The B.C.S. state ischosen variationally, by minimizing the energy.

The B.C.S. ground state is found from anti-symmetrizing the many-particlestate which is composed of products of pairs of wave functions. Each paircorresponds to the two Bloch states (k, ↑) and (−k, ↓). For each wave vector k,the pair state ((k, ↑), (−k, ↓)) is occupied with probability amplitude u(k) andunoccupied with probability amplitude v(k). The probability amplitudes areoften referred to as coherence factors.

| ΨBCS > =∏k

(v(k) + u(k) c†k,↑ c

†−k,↓

)| 0 > (2617)

The amplitudes satisfy the constraint

| u(k) |2 + | v(k) |2 = 1 (2618)

The normal state for non-interacting electrons just corresponds to the specialcase,

| u(k) |2 = Θ( µ − Ek ) (2619)

and the phase of the u(k) for each occupied wave function are completely unde-termined. The functions u(k) and v(k) in the variational ansatz are variationalparameters that are to be found by minimizing the expectation value of theHamiltonian, which includes the pairing interaction.

740

The expectation value for the appropriate energy, in the B.C.S. state, isgiven by

E − µ N = < ΨBCS | ( H − µ N ) | ΨBCS >

=∑

k

2 ( Ek − µ ) | u(k) |2 −∑k,k′

V (k, k′) v∗(k) u(k) u∗(k′) v(k′)

(2620)

The term involving the double sum is eliminated by introducing a quantity

∆(k) =∑k′

V (k, k′) u∗(k′) v(k′) (2621)

On minimizing the energy with respect to u(k) and v∗(k), subject to the con-straint of conservation of probability, one finds

0 =[

2 ( Ek − µ ) + λ

]u∗(k) − ∆(k) v∗(k)

0 = λ v(k) − ∆(k) u(k) (2622)

where λ is the Lagrange undetermined parameter. These equations can besolved to yield

| u(k) |2 =12

(1 −

Ek − µ

Eqp(k)

)(2623)

and

| v(k) |2 =12

(1 +

Ek − µ

Eqp(k)

)(2624)

where we have defined

u(k) v∗(k) =∆(k)

2 Eqp(k)(2625)

The first two equations can be multiplied and equated to the modulus squaredof the third equation according to the identity

| u(k) |2 | v(k) |2 =(u(k) v∗(k)

)∗ (u(k) v∗(k)

)(2626)

The resulting expression can be solved for the quasi-particle energy, Eqp(k),resulting in

Eqp(k) = +√

( Ek − µ )2 + | ∆(k) |2 (2627)

The factor | u(k) |2, is the probability of finding an electron of momentum kand spin σ in the B.C.S. ground state and, therefore, is just n(k). Unlike aFermi liquid, where n(k) is discontinuous at the Fermi surface with magnitudeZ(k)−1, in the superconductor the distribution drops smoothly to zero as k in-creases above kF . Thus, the concept of Fermi surface is not well defined in the

741

0

0.2

0.4

0.6

0.8

1

1.2

-10 -8 -6 -4 -2 0 2 4 6 8 10

(Ek−µ)/∆

|u(k)|2 |v(k)|2

Figure 310: The energy dependence of the probability for finding a pair |u(k)|2,and the probability |v(k)|2 that the pair is absent.

superconducting state. The energy Eqp(k), relative to µ, turns out to be theenergy required to create a quasi-particle of momentum k from the ground state.The quasi-particle is either of the form of an added electron or a hole. Withthe B.C.S. ground state both of these quasi-particle excitations leave a singleunpaired electron in an otherwise perfectly paired B.C.S. state. The minimumenergy required to create two quasi-particles, that is two individual electrons,is just 2 ∆(kF ) . That is, there is a gap of 2 ∆(kF ) in the excitation spectra ofthe superconductor.

25.3.3 The Gap Equation

The “energy gap” parameter satisfies the non-linear integral equation

∆(k) =∑k′

V (k, k′)∆∗(k′)

2 Eqp(k′)(2628)

where V (k, k′) is the attractive pairing interaction mediated by the phonons. Itshould be noted that the phase of u(k′) v∗(k′) is the phase of a pair k′ relativeto the rest of the ground state. Furthermore, if the phase of the pairs for eachk′ value were randomly distributed, then ∆(k) would most probably be zero.Hence, the gap equation implies that there is a relation between the phases ofall the wave functions of the pairs of electrons299. The pairing interaction canbe approximated by the attractive s-wave potential

V (k, k′) = V for | Ek − µ | < h ωD

299To be more precise, the spatial extent of a Cooper pair is given by the coherence lengthξ which is extremely long compared to the average separations between adjacent electrons.Thus, as many as 106 Cooper pairs have their centers of mass located in a volume ξ3 containedbetween the two electrons which form a Cooper pair. However, all the Cooper pairs with theircenters of mass in this spatial region are phase coherent.

742

V (k, k′) = 0 for | Ek − µ | > h ωD (2629)

In this case, one finds that the gap has s-wave symmetry, and is give by

∆(k) = ∆(0) for | Ek − µ | < h ωD

∆(k) = 0 for | Ek − µ | > h ωD (2630)

For other types of potentials, the gap may have p or d wave symmetry and,therefore, vanish at lines or points on the Fermi surface. For the s-wave case,the gap in the quasi-particle dispersion relation at the Fermi energy is given bythe solution of

1 = V ρ(µ)∫ hωD

−hωD

dε1

2√

ε2 + | ∆(0) |2

= V ρ(µ)∫ hωD

0

dε1√

ε2 + | ∆(0) |2

= V ρ(µ) sinh−1 h ωD

| ∆(0) |(2631)

which is solved as| ∆(0) | =

h ωD

sinh 1V ρ(µ)

(2632)

This gap 2 ∆(0) just corresponds to the minimum energy required to break thes-wave Cooper pair. At finite temperatures, the superconducting gap satisfiesthe equation

∆(k) =∑k′

V (k, k′)∆(k′)

2 Eqp(k′)( 1 − 2 f(Eqp(k′)) )

=∑k′

V (k, k′)∆(k′)

2 Eqp(k′)tanh

βEqp(k′)2

(2633)

The tanh factor is a decreasing function for increasing temperature, therefore,for the equation to have a non-trivial solution the denominator has to decreasewith increasing temperature. This can only happen if | ∆(T ) | decreases withincreasing temperature. For sufficiently high temperatures, the equation can bereduced to

∆(T ) = ∆(T ) V ρ(µ)β hωD

2(2634)

which only has the trivial solution ∆(T ) = 0, when 2 kB T h ωD V ρ(µ).The critical temperature below which the gap is non-zero, ∆(Tc) = 0, is given

743

by the linearized equation300

1 = 2 ρ(µ) V∫ hωD

0

dεtanh βc ε

2

2 ε

= ρ(µ) V∫ hωD

0

dεtanh βc ε

2

ε

= ρ(µ) V∫ βhωD

2

0

dztanh zz

= ρ(µ) V(

lnβhωD

2−∫ ∞

0

dz ln z sech2z

)= ρ(µ) V

(lnβhωD

2− ln

π

4 exp γ

)(2635)

The critical temperature Tc is given by

kB Tc = 1.14 hωD exp[− 1

V ρ(µ)

](2636)

The critical temperature is a non analytic function of the coupling constant.The critical temperature is proportional to hωD and, therefore, proportionalto M− 1

2 , as is expected from the isotope effect. Deviations from the classicalisotope effect are found in transition metals such as Ru and Os, and in theactinides where the Coulomb interaction strength is large. The strength of thecoupling constant can be estimated from knowledge of the Debye frequency andthe critical temperature. Typical values of ρ(µ) V are quite small, of the orderof 0.2.

300The analysis described here assumes that the density of states at the Fermi energy isroughly constant. This is not the case for MgB2 [Nagamatsu et al., Nature, 410, 63, (2001)].Since in MgB2 Boron is isoelectronic with graphite and forms layers which have the samestructure as the layers of Graphite, the electronic density of states is expected to show anenergy dependence similar to that of graphite.

744

Metal ΘD(K) Tc(K) ρ(µ) V

Zn 235 0.9 0.18Cd 164 0.56 0.18Hg 70 4.16 0.35Al 375 1.2 0.18Ga 320 1.1 0.18In 109 3.4 0.29T l 100 2.4 0.27Sn 195 3.75 0.25Pb 96 7.22 0.39

The ratio of the T = 0 energy gap 2 ∆(0) to the superconducting transitiontemperature has the theoretical value301

2 ∆(0)kB Tc

= 3.53 (2637)

whereas the experimentally determined values in Al, Ga, In, T l, Sn and Pb are3.4, 3.5, 3.6, 3.6, 3.5 and 4.4.

Below the critical temperature, the superconducting order parameter ∆(T )is finite. Above the critical temperature ∆(T ) = 0 and the B.C.S. state reducesto the normal state. Just below the critical temperature, one has

∆(T )2 =8 π2

7 ζ(3)k2

B Tc ( Tc − T ) T → Tc (2638)

Thus, the order parameter has a typical mean-field variation with an exponentof β = 1

2 close to Tc.

25.3.4 The Ground State Energy

The normal state is unstable to the B.C.S. state only if the B.C.S. state has alower energy. At T = 0 the stability can be found by examining the energy

E − µ N = < ΨBCS | ( H − µ N ) | ΨBCS >

= 2∑

k

| u(k) |2 ( Ek − µ ) −∑

k

| ∆(k) |2

2 Eqp(k)

301J. Bardeen and J. R. Schrieffer, Prog. in Low Temp. Phys. 3, 170, (1961).

745

Fig. 25: Temperature dependence of the energy gap according to the BCS theory and comparison with experimental data.

One of the fundamental formulae of the BCS theory is the relation between the energy gap ∆(0)at T = 0, the Debye frequency ωD and the electron-lattice interaction potential V0 :

∆(0) = 2~ωD exp

(

− 1

V0N (EF )

)

. (27)

HereN (EF ) is the density of single-electron states of a given spin orientation atE = EF (the other spinorientation is not counted because a Cooper pair consists of two electrons with opposite spin). Althoughthe interaction potential V0 is assumed to be weak, one of the most striking observations is that theexponential function cannot be expanded in a Taylor series around V0 = 0 because all coefficients vanishidentically. This implies that Eq. (27) is a truely non-perturbative result. The fact that superconductivitycannot be derived from normal conductivity by introducing a ‘small’ interaction potential and applyingperturbation theory (which is the usual method for treating problems of atomic, nuclear and solid statephysics that have no analytical solution) explains why it took so many decades to find the correct theory.The critical temperature is given by a similar expression

kBTc = 1.14 ~ωD exp

(

− 1

V0N (EF )

)

. (28)

Combining the two equations we arrive at a relation between the energy gap and the critical temperaturewhich does not contain the unknown interaction potential

∆(0) = 1.76 kBTc . (29)

The following table shows that this remarkable prediction is fulfilled rather well.

element Sn In Tl Ta Nb Hg Pb∆(0)/kBTc 1.75 1.8 1.8 1.75 1.75 2.3 2.15

In the BCS theory the underlying mechanism of superconductivity is the attractive force betweenpairs of electrons that is provided by lattice vibrations. It is of course highly desirable to find experimentalsupport of this basic hypothesis. According to Eq. (28) the critical temperature is proportional to theDebye frequency which in turn is inversely proportional to the square root of the atomic mass M :

Tc ∝ ωD ∝ 1/√M .

If one produces samples from different isotopes of a superconducting element one can check this relation.Figure 26 shows Tc measurements on tin isotopes. The predicted 1/

√M law is very well obeyed.

Figure 311: The reduced superconducting gap ∆(T )∆(0) for In, Sn and Pb as a

function of the reduced temperature TTc

. The experimental data are comparedwith the prediction of the B.C.S. theory. [After Ivar Giaever and Karl Megerle,Phys. Rev. 122, 101 (1961).].

= 2∑

k

| u(k) |2 ( Ek − µ ) − ρ(µ)∫ hωD

0

dε| ∆(0) |2√

ε2 + | ∆(0) |2

=∑

k

[( Ek − µ ) −

( Ek − µ )2√( Ek − µ )2 + | ∆(k) |2

]− | ∆(0) |2

V

(2639)

The first term represents the kinetic energy. The last term represents the at-tractive interaction. The integral in the last term is evaluated with the aid ofthe gap equation. The condensation energy, ∆E, is defined as the differencebetween the energy of the superconducting state and the normal state

∆E = < ΨBCS | ( H − µ N ) | ΨBCS > − 2∫ 0

−∞dε ε ρ(µ+ ε)

(2640)

The condensation energy is evaluated by writing the sum over k as an integralover the density of states.

∆E =∫ hωD

0

dε ρ(µ+ ε)

[ε − ε2√

ε2 + | ∆(0) |2

]

746

+∫ 0

−hωD

dε ρ(µ+ ε)

[ε − 2 ε − ε2√

ε2 + | ∆(0) |2

]

− | ∆(0) |2

V

=∫ hωD

0

dε ρ(µ+ ε)

[ε − ε2√

ε2 + | ∆(0) |2

]

+∫ 0

−hωD

dε ρ(µ+ ε)

[− ε − ε2√

ε2 + | ∆(0) |2

]

− | ∆(0) |2

V(2641)

The integral over states below the Fermi energy, ε < 0, can be transformed toan integral over positive ε. This leads to the condensation energy being givenby

∆E = 2 ρ(µ)∫ hωD

0

dε

[ε − ε2√

ε2 + | ∆(0) |2

]− | ∆(0) |2

V

= 2 ρ(µ)∫ hωD

0

dε

[ε −

√ε2 + | ∆(0) |2

]

+ 2 ρ(µ)∫ hωD

0

dε| ∆(0) |2√

ε2 + | ∆(0) |2− | ∆(0) |2

V

(2642)

The integrals are evaluated with the aid of the substitution

ε = | ∆(0) | sinh θ (2643)

which gives the result

∆E = h2ω2D ρ(µ)

[1 −

√1 +

(| ∆(0) |h ωD

)2 ]+

| ∆(0) |2

V− | ∆(0) |2

V

≈ − 12ρ(µ) | ∆(0) |2 + . . . (2644)

The condensation energy comes from the attractive potential which is largerthan the increase in the kinetic energy caused by the confinement of the pairwithin the coherence length ξ. The net lowering can be understood in terms ofthe quasi-particle dispersion relation. The electrons with energy within | ∆(0) |of µ have their energy lowered by an amount | ∆(0) |. The net lowering ofenergy is just the number of electrons, ρ(µ) | ∆(0) |, times the energy lowering| ∆(0) |. Therefore, the B.C.S. state has lower energy than the normal statewhenever the gap is non-zero.

747

25.4 Quasi-Particles

The B.C.S. Hamiltonian can be solved for the quasi-particle excitations, in themean-field approximation, by linearizing the pairing interaction terms. In anormal metal, the only allowed matrix elements are between initial and finalstates which have the same number of electrons. However, since for a super-conductor the average is to be evaluated in the B.C.S. ground state, matrixelements between operators with different numbers of pairs are non-zero. Thesegive rise to the anomalous expectation values. For example, the anomalous ex-pectation value associated with adding a pair of electrons ((k′, ↑), (−k′, ↓)) tothe superconducting condensate is given by the probability amplitude

< ΨBCS | c†k′,↑ c†−k′,↓ | ΨBCS > = u∗(k′) v(k′) (2645)

The linearized mean-field Hamiltonian is given by

HMF − µ N =∑

k

(( Ek − µ ) c†k,↑ ck,↑ + ( E−k − µ ) c†−k,↓ c−k,↓

)−∑k,k′

V (k, k′) < ΨBCS | c†−k′,↓ c†k′,↑ | ΨBCS > ck,↑ c−k,↓

−∑k,k′

V (k, k′) c†−k′,↓ c†k′,↑ < ΨBCS | ck,↑ c−k,↓ | ΨBCS >

+∑k,k′

V (k, k′) < ΨBCS | c†−k′,↓ c†k′,↑ | ΨBCS > < ΨBCS | ck,↑ c−k,↓ | ΨBCS >

(2646)

The anomalous expectation value leads to a term in the Hamiltonian withstrength

∆(k) =∑k′

V (k, k′) u∗(k′) v(k′) (2647)

which corresponds to a process in which two electrons ((k, ↑), (−k, ↓)) are ab-sorbed into the condensate. The mean-field Hamiltonian also contains the Her-mitean conjugate which represents the reverse process in which two electronsare emitted from the condensate.

HMF − µ N =∑

k

(( Ek − µ ) c†k,↑ ck,↑ + ( E−k − µ ) c†−k,↓ c−k,↓

)

−∑

k

[∆(k) ck,↑ c−k,↓ + c†−k,↓ c

†k,↑ ∆∗(k)

]+

| ∆(0) |2

V

(2648)

In the absence of an electromagnetic field, the order parameter ∆(k) can bechosen to be real. The mean-field Hamiltonian involves terms in which the con-densate emits or absorbs two electrons. This is reminiscent of the treatment of

748

anti-ferromagnetic spin waves, using the method of Holstein and Primakoff, ex-cept here the Hamiltonian involves fermions rather than bosons. The quadraticHamiltonian can be diagonalized by means of a canonical transformation.

We shall define two new fermion operators via the transformation

αk = exp[

+ S

]ck,↑ exp

[− S

](2649)

and

β†k = exp[

+ S

]c†−k,↓ exp

[− S

](2650)

where S is an anti-Hermitean operator, S† = − S. The energy eigenvalues ofthe Hamiltonian can be found directly from the transformed Hamiltonian

H ′MF = exp

[+ S

]HMF exp

[− S

](2651)

as they have the same eigenvalues and the eigenstates are related via

| φ′n > = exp[

+ S

]| φn > (2652)

The operator S is chosen to be of the form

S =∑

k

θk

(c†k,↑ c

†−k,↓ − c−k,↓ ck,↑

)(2653)

Explicitly, the transformation yields

αk = ck,↑ cos θk − c†−k,↓ sin θk

β†k = c†−k,↓ cos θk + ck,↑ sin θk (2654)

Rather than working with the transformed Hamiltonian, we shall express theoriginal Hamiltonian in terms of the transformed operators. Hence, we shallrequire the inverse transformation which expresses the original electron andholes operators in terms of the new quasi-particles. The inverse transformationis expressed in terms of the transformation matrix but with θk → − θk so onehas

ck,↑ = αk cos θk + β†k sin θk

c†−k,↓ = β†k cos θk − αk sin θk (2655)

The mean-field Hamiltonian is expressed in terms of the new operators and θk ischosen so that the terms that are not represented in terms of the quasi-particle

749

number operators vanish. The normal terms in the Hamiltonian are found as∑k

(( Ek − µ ) c†k,↑ ck,↑ + ( E−k − µ ) c†−k,↓ c−k,↓

)

=∑

k

( ε(k) − µ )

[sin2 θk

(αk α

†k + βk β

†k

)+ cos2 θk

(α†k αk + β†k βk

) ]

+∑

k

( ε(k) − µ ) sin 2θk

(α†k β

†k + βk αk

)(2656)

The anomalous terms are evaluated as

−∑

k

(∆(k) ck,↑ c−k,↓ + c†−k,↓ c

†k,↑ ∆(k)

)

= −∑

k

<e[

∆(k)]

sin 2θk

(β†k βk − αk α

†k

)

+∑

k

<e[

∆(k)]

cos 2θk

(α†k β

†k + βk αk

)(2657)

The off-diagonal terms can be made to vanish by choosing

tan 2θk = −<e[

∆(k)](

Ek − µ) (2658)

Thus, θk decreases from a value less than π4 to less than − π

4 as ε(k) variesfrom hωD below µ to hωD above µ. The factors cos θk and sin θk are foundto be related to the factors u(k) and v(k) in the B.C.S. ground state wavefunction. After this value has been chosen, the Hamiltonian is expressed as thesum of a constant and terms involving the number operators of the α and βquasi-particles

HMF = E0 +∑

k

Eqp(k)(α†k αk + β†k βk

)(2659)

This procedure shows that the excitations are quasi-particles as they are stillfermions. Furthermore, these quasi-particles have excitation energies which havethe dispersion relation

Eqp(k) = +√

( Ek − µ )2 + | ∆(k) |2 (2660)

The canonical transformation shows that the quasi-particles are part electronand part hole like. Basically, this is a consequence that the quasi-particleexcitation consists of a single unpaired electron (k, σ), in the presence of the

750

Excitation Energies

-2

-1

0

1

2

-2 -1 0 1 2

k/kF

E(k

)/ µ

electron addition

electron removal kF-kF

Forbidden excitations E<0

Allowed excitations E>0

Figure 312: The excitation energies for adding an electron or removing an elec-tron (creating a hole) from the normal state of a metal. Electrons can only beadded to states with k > kF , and holes can only be created for k < kF . Theallowed excitation energies are positive.

Quasiparticle Excitation Energies

-2

-1

0

1

2

-2 -1 0 1 2k/kF

E(k

)/ µ

-kF kF

µ

∆∆

Figure 313: The excitation energies for creating a quasi-particle in the super-conucting state of a metal. The minimum energy required to create a quasi-particle is ∆.

751

condensate. This specific state can be produced from the ground state, eitherby adding the electron (k, σ) to the system or by breaking a Cooper pair byremoving the partner electron (−k,−σ). We note that the quasi-particles areeigenstates of the spin operator. The α quasi-particle is a spin-up excitationas it is composed of an up-spin electron and down-spin hole, whereas the βquasi-particle is a spin-down excitation. From the dispersion relation, one findsthat the B.C.S. superconductor is actually characterized by the presence of agap in the excitation spectrum. That is, there is a minimum excitation energy2 | ∆(kF ) | corresponding to breaking a Cooper pair and producing two inde-pendent quasi-particles.

——————————————————————————————————

25.4.1 Exercise 97

Evaluate the constant term in the mean-field B.C.S. Hamiltonian. Show thatthe variational B.C.S. ground state is the lowest energy state of the mean-fieldHamiltonian by showing that the quasi-particle destruction operators annihilatethe B.C.S. state

αk | ΨBCS > = 0

βk | ΨBCS > = 0 (2661)

——————————————————————————————————

25.5 Thermodynamics

Since the quasi-particles are fermions, the entropy S due to the gas of quasi-particles is given by the formulae

S = − 2 kB

∑k

[( 1− f(Eqp(k)) ) ln[ 1− f(Eqp(k)) ] + f(Eqp(k)) ln[ f(Eqp(k)) ]

](2662)

By the usual procedure of minimizing the grand canonical potential Ω withrespect to the distribution f(Eqp(k)) , one can show that the non-interactingquasi-particles are distributed according to the Fermi-Dirac distribution func-tion. Therefore, the quasi-particle contribution to the specific heat is just givenby

Cqp(T ) = 2∑

k

Eqp(k)(∂f(Eqp(k))

∂T

)

752

0

1

2

3

4

5

6

0 1 2 3 4E/∆

ρ qp(E) /ρ

(µ)

Figure 314: The normalized quasi-particle density of states for an s-wave super-conductor. An s-wave superconductor has a gap all around the Fermi-surface,and hence the quasi-particle density of states exhibits a gap.

= 2∑

k

Eqp(k)[− Eqp(k)

T+

∂Eqp(k)∂T

] (∂f(Eqp(k))

∂E

)

= − 2T

∫ +∞

−∞dE ρqp(E)

[E2 − T

2∂∆(T )2

∂T

] (∂f

∂E

)(2663)

The average of the temperature derivative of the square of the quasi-particle en-ergy is given by the temperature derivative of the gap. In the above expression,we have introduced the quasi-particle density of states

ρqp(E) =∑

k

δ

(E − Eqp(k)

)(2664)

Since, in the mean-field approximation, the square of the gap has a finite slopefor T just below Tc and is zero above,

∆(T )2 ∼ ∆(0)2(

1 − T

Tc

)Θ( Tc − T ) (2665)

the specific heat has a discontinuity at Tc. In B.C.S. theory, the magnitudeof the specific heat jump has the value given by, 3.03 ∆2(0) ρ(µ) / Tc. Thus,the value of the specific heat jump found in weak coupling B.C.S. theory, whennormalized to the normal state specific heat, is given by

∆C(Tc)C(Tc)

=Cs − Cn

Cn

=12

7 ζ(3)= 1.43 (2666)

753

Material ∆CC

∣∣∣∣Tc

Zn 1.3Cd 1.4Hg 1.4Al 1.4Ga 1.4In 1.7T l 1.5Sn 1.6Pb 2.7

This ratio is a measure of the quantity

12 k2

B Tc

(∂∆2(T )∂T

) ∣∣∣∣Tc

∼(

∆(0)kB Tc

)2

(2667)

The values of the specific heat jumps for strong coupling materials tend to behigher than the B.C.S. value, for example the normalized jump for Pb is as largeas 2.71. This trend is understood as being due to inelastic scattering processeswhich tend to suppress Tc more than ∆(0). The heavy fermion superconductorsshow that the normalized specific heat discontinuities are significantly smallerthan the B.C.S. ratio.

Low Temperatures.

The gap in the quasi-particle density of states could be expected to show upin an activated exponential dependence of the low-temperature electronic spe-cific heat, for T Tc. For these temperatures the order parameter is expectedto have saturated, and so if one considers the Fermi liquid as being well formedthen the quasi-particle contribution is given by

Cqp(T ) = − 2T

∫ +∞

−∞dE ρqp(E) E2

(∂f

∂E

)(2668)

The B.C.S. quasi-particle density of states is evaluated as

ρqp(E) =∑

k

δ

(E − Eqp(k)

)

=∫ ∞

−∞dε ρ(ε) δ

(E −

√( ε − µ )2 + ∆(T )2

)

754

Figure 315: The specific heat for normal and superconducting states of Al. Thespecific heat shows a jump at the transition temperature Tc = 1.163 K. [AfterN. E. Phillips, Phys. Rev. 114, 676 (1959).]

∼ ρ(µ)| E |

| ε − µ |

= ρ(µ)| E |√

E2 − ∆(T )2for | E | > ∆(T )

(2669)

In evaluating the B.C.S. density of states, the conduction band electron densityof states has been approximated by a constant value. The resulting B.C.S.quasi-particle density of states has a gap of magnitude 2 ∆(T ) around the Fermienergy. This yield an exponentially activated behavior of the specific heat,

Cqp(T ) ∼ 9.17 γ Tc exp[− ∆(0)

kB T

](2670)

found in B.C.S. theory. For superconductors where order parameter has p ord wave symmetry, the gap may vanish either on lines or points of the Fermisurface. The vanishing of the gap can give rise to a power law behavior of thelow temperature specific heat of the superconductor302.

302H. R. Ott, H. Rudigier, Z. Fisk and J. L. Smith, Phys. Rev. B, 31, 1615 (1985).

755

25.6 Perfect Conductivity

The current is composed of the sum of a paramagnetic current and a diamagneticcurrent. The paramagnetic current can be evaluated from the Kubo formula.There are two contributions to the paramagnetic current: one contribution isfrom the particles in the condensate, the other contribution is from the excitedquasi-particles. The condensate contribution is proportional to

jp(q;ω) =

e2 h2

8 m2 c

1V

∑k

( 2 k − q )(

( 2 k − q ) . A) ∣∣∣∣ u(k) v(k + q)− u(k + q) v(k)

∣∣∣∣2 ×

×(

1 − f(E(k)) − f(E(k − q))E(k − q) + E(k) + h ω

+1 − f(E(k − q)) − f(E(k))E(k) + E(k − q) − h ω

)(2671)

The coherence factor,(u(k) v(k + q) − u(k + q) v(k)

)(2672)

occurs, since, in the B.C.S. ground state, adding an electron with specific spinand momentum produces the same final state as adding a hole of opposite spinand momentum. The process of exciting an electron from the Bloch state (k, σ)to the Bloch state (k+q, σ) occurs with the probability amplitude v(k+q) u(k),whereas the process of exciting an electron from the Bloch state (−k − q,−σ)to the Bloch state (−k,−σ) occurs with probability amplitude − u(k+ q) v(k).Since these two processes start from the unique B.C.S. ground state and leadto exactly the same final state, their probability amplitudes should be added.The need to include the coherence factors becomes immediately transparent, ifone expresses the electron creation and annihilation operators in terms of thequasi-particle operators

ck,↑ = αk cos θk + β†k sin θk (2673)

andc−k,↓ = βk cos θk − α†k sin θk (2674)

From this, one sees that the up-spin electron and down-spin hole componentsof the current operator

( k +12q )(a†k+q,↑ ak,↑ − a†−k,↓ a−k−q,↓

)(2675)

combine to produce the usual quasi-particle contributions to the current, like

( k +12q ) ( cos θk+q cos θk + sin θk+q sin θk )

(α†k+q αk − β†k βk+q

)(2676)

756

and the anomalous contributions, such as

( k +12q ) ( cos θk+q sin θk − sin θk+q cos θk )

(α†k+q β

†k − βk+q αk

)(2677)

It is the anomalous contributions which give rise to the condensate componentof the paramagnetic current. Due to the appearance of the coherence factorin the expression for the condensate component of the paramagnetic current,and the finite value of the denominator when ω < 2 | ∆(0) |, the condensatecontribution vanishes in the limit q → 0. The condensate contribution alsovanishes identically in the normal state. The quasi-particle contribution to theparamagnetic current has a coherence factor given by(

u(k) u(k + q) + v(k) v(k + q))

(2678)

This coherence factor represents the contribution from thermally activated quasi-particles in the initial state. One contribution represents a process, in which thethermally excited quasi-particle is viewed as an unpaired electron in the Blochstate (k, σ) which is subsequently excited to the Bloch state (k + q, σ). Thisprocess occurs with probability amplitude u(k) u(k+q). Alternatively, the samequasi-particle may be viewed as a thermally exited single hole in the Bloch state(−k,−σ) which is subsequently excited to the final Bloch state (−k − q,−σ).This process is associated with the probability amplitude v(k) v(k + q). Thecoherence factor is given by the sum of the two probability amplitudes. Thiscoherence factor tends to unity as q → 0, since it reduces to the normalizationcondition. Hence, the quasi-particle contribution to the paramagnetic currentis given by

jp(q;ω) =

e2 h2

4 m2 c

1V

∑k

( 2 k − q )(

( 2 k − q ) . A) ∣∣∣∣ u(k) u(k + q) + v(k) v(k + q)

∣∣∣∣2 ×

×(

f(E(k)) − f(E(k − q))E(k − q) − E(k) + h ω

+f(E(k − q)) − f(E(k))E(k) − E(k − q) + h ω

)(2679)

In this expression E(k) is the quasi-particle energy in the superconductor. Inthe static limit with uniform fields, ( ω → 0 , q → 0 ), the paramagneticcurrent reduces to

jp(0; 0) = 2

e2 h2

m2 c

1V

∑k

k ( k . A )(− ∂f(E(k))

∂E

)(2680)

The total current is found by combining the paramagnetic current with thediamagnetic current

j(0; 0) = − 2e2 h2

m2 c

1V

∑k

k ( k . A )(∂f(E(k))

∂E

)− ρ e2

m cA

757

= − 2e2 h2

m2 c

16 π2

∫dk k4 A

(∂f(E(k))

∂E

)− ρ e2

m cA

= −[e2 h2

m2 c

13 π2

∫dk k4

(∂f(E(k))

∂E

)+

ρ e2

m c

]A

(2681)

In the normal state, where the gap in E(k) vanishes, the derivative of the Fermifunction can be approximated as(

∂f(E(k))∂E

)= − δ( E(k) − µ )

= − 2 mh2 δ( k2 − k2

F ) (2682)

which leads to the vanishing response as

ρ =k3

F

3 π2(2683)

Thus, in the normal state current does not flow in response to a static vectorpotential. However, in the superconducting state the total current is given by

j = − ρ e2

m cA

[1 +

2 µk5

F

∫dk k4

(∂f(E(k))

∂E

) ](2684)

and as there is a gap on the Fermi surface, the derivative of the Fermi functionis always exponentially small. Because of the finite superconducting gap, thesecond term is small and the cancellation does not occur. In the superconductingstate, this reduces to the London equation

j = − ρ e2

m cA (2685)

This shows that a current will flow in a superconductor in response to a uniformstatic vector potential, that is the current will screen an applied magnetic field.This leads to the Meissner effect.

25.7 The Meissner Effect

In the superconducting state, the susceptibility is expected to be dominatedby the diamagnetic susceptibility produced by the supercurrent shielding theexternal field. The Pauli spin susceptibility will also be modified by the su-perconductivity, and provide information about the pairing. The zero fieldsusceptibility is defined as a derivative of the magnetization, χs(T ) = (∂M

∂H ).The magnetization, produced by the electronic spins aligning with a magneticfield applied along the z-axis, is given by

Mz =(g µB

2

) ∑k

[f(E↑(k)) − f(E↓(k))

](2686)

758

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2

T / Tc

χ(T

) / χ

n

Figure 316: The spin susceptibility as a function of the reduced temperature TTc

for a BCS s-wave superconductor.

which is given in terms of the Fermi distribution for quasi-particles with spin σand quasi-particle energy Eσ(k).

For singlet pairing, the magnetic field couples to the spins of the quasi-particles via the Zeeman energies and, as can be seen from inspection of thematrix, only the time reversal partners pair. The quasi-particles consist ofbroken pairs, i.e. electrons of spin σ and holes of spin − σ. Since a down-spin hole has the same Zeeman energy as an up-spin electron, the quasi-particleenergies depend on field through

Eσ(k) = EH=0(k) −(g µB σ H

2

)(2687)

and so the spin susceptibility takes the usual form

χs(T ) = − 2(g µB

2

)2 ∫ +∞

−∞dE ρqp(E)

(∂f

∂E

)(2688)

which involves the B.C.S. quasi-particle density of states. The Pauli suscepti-bility tends to zero as T → 0 in an exponentially activated way

χp(T ) ∼ exp[− ∆(0)

kB T

](2689)

The exponential vanishing of the spin susceptibility occurs as the electrons formsinglet pairs in the ground state, and the finite spin moment is caused by ther-mal population of quasi-particles.

Thus, in the spin-singlet phases, the spin susceptibility could be expectedto vanish as T → 0. However, spin-orbit coupling will produce a residualsusceptibility that depends on the ratio of the superconducting coherence length,ξ0 to the mean free path due to spin-orbit scattering, lso. In the presenceof spin-orbit coupling, the spin is no longer a good quantum number for the

759

single-particle eigenstates, and the spin-up and spin-down states are mixed. Inthe limit that the strength of the spin-orbit coupling λ

−→L .−→S is so large that

λ ∆0, the average value of σz for a single-particle state tends to zero.The spin susceptibility is, therefore, reduced. The scattering has the effect thata significant contribution to the normal state χ(T ) comes from single-particlestates separated by an energy of the order of the spin-orbit scattering rate,which is by our assumption greater than ∆. As an opening up of a gap at theFermi energy is not expected to change the contribution of these higher energystates, one finds that the susceptibility in the superconducting state can remaincomparable in magnitude to the normal state value. According to Anderson,the normalized susceptibility should have the two limits,

χs(0)χn

= 1 − 2 lso

π ξ0(2690)

for strong spin-orbit scattering and for weak spin-orbit scattering, one has

χs(0)χn

=π ξ06 lso

(2691)

Hence, a partial Meissner effect at T = 0 can be found in a conventionalsuperconductor.

25.8 Landau-Ginzburg Theory

Superconductors can be divided into two categories, which depend on theirmacroscopic characteristics when an applied magnetic field is present. The clas-sification is based on the length scale over which the magnetic field is screened,λL, relative to the length scale over which the superconducting order parameterchanges, ξ. The latter length is given by the spatial extent of the Cooper pairwave function or coherence length ξ0

ξ0 =h vF

π ∆(0)(2692)

Type I Superconductors.

Type I superconductors are materials with short penetration depths λ andlong coherence lengths ξ. In particular, in type I superconductors the ratio mustsatisfy the inequality

κ =λ

ξ<

1√2

(2693)

Type I superconductors include simple (non-transition) metals, such as Al,where the penetration depths are of the order of λ ∼ 300 A and the co-herence lengths are ξ ∼ 1 × 104 A as vF is large.

760

Figure 317: The magnetization curves of annealed polycrystaline Pb (A) andPbIn alloys (B-D). [After J. D. Livingston, Phys. Rev. 129, 1943 (1963).]

Type II Superconductors.

Type II superconductors include materials with long penetration depths andshort coherence lengths. Type II superconductors can be satisfy the inequality

κ =λ

ξ>

1√2

(2694)

Type II materials include the transition metals, rare earths and intermetalliccompounds. For example, in V3Ga, the band mass is very large, so λL is verylarge of the order of 2000 A. Since these have high effective masses, the Fermivelocities are small ( vF ∼ 104 m /s ) and as Tc and ∆(0) are high, then ξ issmall ( ∼ 50 A).

Since λL and κ diverge the same way at Tc the dimensionless ratio κ is ap-proximately temperature independent.

Metal ξ (A) λ (A)

Al 16,000 490-515Sn 2,300 510Pb 830 390Cd 7,600 1,100Nb 380 390

If a magnetic field H < Hc is applied to a small superconductor, the fieldis excluded from the superconductor, but if H > Hc the field will penetrate

761

and the superconductor will undergo a transition to the normal state. If a fieldis applied normal to the surface of a large superconducting slab then, because∇ . B = 0, the field has to penetrate the slab.

In a type I superconductor the magnetic field will concentrate into regionswhere

| B | = Hc (2695)

which are normal, and regions where

B = 0 (2696)

which are superconducting. The condensation energy density in the supercon-ducting state is H2

c

8 π , and the diamagnetic energy of the normal state is also H2c

8 π .These regions are separated by a domain wall which has positive energy. Theenergetic cost of forming a domain wall of area A can be estimated as

E

A∼ ξ

H2c

8 π− λ

H2c

8 π(2697)

The term ξH2

c

8 π is the energetic cost of setting the order parameter to zero. The

diamagnetic energy is reduced by λH2

c

8 π . Because of the positive domain wallenergy in a type I superconductor, the number of domains and domain wallswill be minimized. The domain pattern will have a scale of subdivision whichis intermediate between ξ and the sample size.

In type II superconductors, a similar separation occurs, but as the domainwall energy is negative, the superconductor will break up into as many normalregions as possible. These normal regions have the form of magnetic flux car-rying tubes that thread the sample, which are known as vortices. Each vortexcarries a minimum amount of flux Φs, the superconducting flux quantum. Theflux quanta in a superconductor differs from the flux quanta in a normal metal,as the current in a superconductor is carried by Cooper pairs with charge 2 e,whereas in a normal metal current is carried by electrons with charge e. Theflux in a superconductor is quantized in units of Φs.

Φs =h c

2 e= 2.07 × 10−7 Gauss cm2 (2698)

The vortices first enter the superconductor at a critical field Hc1. The vorticesare arranged on the sites of a triangular lattice. The superconductor becomessaturated with vortices when it becomes completely normal at an upper criticalfield Hc2.

The magnetization M is linear in field up to Hc1 with susceptibility − 14 π .

At Hc1 the magnitude of the magnetization has a cusp and the magnitude fallsto zero at Hc2.

762

In order to discuss the spatial variation of superconductivity due to the mag-netic field, it is necessary to extend the microscopic B.C.S. theory to inhomo-geneous systems. The microscopic generalization was performed by Gor’kov303.This resulted in a set of equations which had previously been proposed by Lan-dau and Ginzburg304 as a phenomenological description of superconductivity.

The Landau-Ginzburg equations are based on the phenomenological form ofthe Helmholtz free-energy functional expressed in terms of a superconductingorder parameter

F [∆] = F0 +∫

d3r

[α(T ) | ∆(r) |2 +

β

2| ∆(r) |4 + γ | ∇ ∆(r) |2 + . . .

](2699)

The gradient term allows a slow variation of the order parameter, and is writ-ten in a form appropriate for cubic crystals. For a homogeneous system, fortemperatures close to Tc, the order parameter is homogeneous and small. Onlythe leading terms in the expansion in powers of ∆(T ) need be retained. Onminimizing the Helmholtz free-energy with respect to ∆∗, one finds

| ∆(T ) |2 = − α(T )β

(2700)

and the free-energy simply becomes

F [∆] = F0 − Vα(T )2

2 β(2701)

One can identify the bulk superconducting order parameter with the B.C.S.result

| ∆(T ) |2 = 10.2 k2B Tc ( Tc − T ) (2702)

close to the transition temperature. Furthermore, since the difference in theFree-energy between the normal and superconducting state is given by the crit-ical field energy density, one has

α(T )2

2 β=

Hc(T )2

8 π(2703)

Hence, one can identify the coefficients in the Landau free-energy functional as

V α(T ) = ρ(µ)(T − Tc

Tc

)(2704)

andV β = 0.098 ρ(µ) ( kB Tc )−2 (2705)

For a pure metal, the coefficient of the gradient term has a magnitude given by

V γ ∼ ρ(µ) ξ20 (2706)303L. P. Gor’kov, J.E.T.P. 9, 1364 (1960), L. P. Gor’kov, J.E.T.P. 10, 593 (1960).304V. I. Ginzburg and L. D. Landau, Zh. Eksp. i. Teor. Fiz. 20, 1064 (1950).

763

In the presence of a static magnetic field, the vector potential is given by thesolution of

B = ∇ ∧ A (2707)

However, the solution is invariant under the gauge transformation

A → A′ = A + ∇Λ (2708)

Under this transformation, the one-electron wave functions are also transformedaccording to

φ(r) → φ′(r) = exp[− i

| e |2 π h c

Λ]φ(r) (2709)

Hence, the gap also acquires a phase through the gauge transformation

∆(r) → ∆′(r) = exp[− 2 i

| e |2 π h c

Λ]

∆(r) (2710)

Since the free energy must be gauge invariant, in the presence of fields, thegradient terms must be replaced by the gauge invariant terms

− i h ∇ → − i h ∇ +2 | e |c

A (2711)

Thus, on adding the field energy, one has the Ginzburg-Landau Free energyfunctional

F [∆] = F0 +∫

d3r

[α(T ) | ∆(r) |2 +

β

2| ∆(r) |4

+γ

h2

∣∣∣∣ ( − i h ∇ +2 | e |c

A

)∆(r)

∣∣∣∣2 +B(r)2

8 π

](2712)

In this expression B is the internal field. The original formulation of Ginzburgand Landau only contained a factor of e rather than 2 e in the gradient term.The factor of two occurs since the charge on a Cooper pair is 2 e.

25.8.1 Extremal Configurations

In equilibrium, the free energy functional is to be minimized. The minimizationof the free energy requires that the variation of both the order parameter andthe field distributions are extrema of the functional. On making the variations,of the fields from their extremal values

∆(r) = ∆ext(r) + δ∆(r)A(r) = Aext(r) + δA(r) (2713)

764

one finds the first order functional derivatives are given by

δF

δ∆∗(r)= α ∆ext(r) + β | ∆ext(r) |2 ∆ext(r)

+γ

h2

(− i h ∇ +

2 | e |c

Aext

)2

∆ext(r)

δF

δA(r)=

14 π

∇ ∧ Bext(r) +2 | e | γh2 c

∆∗ext(r)

(− i h ∇ +

2 | e |c

Aext

)∆ext(r)

+2 | e | γh2 c

∆ext(r)(

+ i h ∇ +2 | e |c

Aext

)∆∗

ext(r) (2714)

if the fields satisfy appropriate boundary conditions at infinity. On utilizing thestatic form of Maxwell’s equation

∇ ∧ B =4 πc

j (2715)

the Euler-Lagrange equations become

0 =γ

h2

(− i h ∇ +

2 | e |c

Aext

)2

∆ext(r) + α ∆ext(r) + β | ∆ext(r) |2 ∆ext(r)

j = i2 | e | γ

h

(∆∗

ext(r) ∇ ∆ext(r)− ∆ext(r) ∇ ∆∗ext(r)

)− 8 e2 γ

h2 c| ∆ext(r) |2 Aext

(2716)

The extremal fields are the measurable physical fields. Hence, the subscript cannow be dropped without any ensuing ambiguity.

The above equations show that, close so the critical temperature, the gapparameter satisfies a non-linear Schrodinger equation. That is, the gap satis-fies a one-particle Schrodinger equation which includes a Hartree-like term dueto point contact interactions. The relation between the gap and the electricalcurrent is also very similar to the relation between a one-particle wave functionand the current. It is customary to write the gap, ∆, as being proportional tothe Cooper pair wave function Ψ.

25.8.2 Characteristic Length Scales

The Landau-Ginzburg equations contain two characteristic length scales. Theseare the coherence length ξ which governs the relaxation of ∆ around an inhomo-geneity. The coherence length can be found by examining the one-dimensionalLandau-Ginzburg theory. For a homogeneous system in equilibrium, the orderparameter has the value

| ∆0 |2 = − α

β> 0 (2717)

765

for T < Tc. On scaling the order parameter to the bulk value

∆(z) = Ψ(z) ∆0 (2718)

the Landau-Ginzburg equation

− γ∂2 ∆∂z2

+ α ∆(z) + β ∆(z) | ∆(z) |2 = 0 (2719)

reduces toγ

α

∂2 Ψ∂z2

− Ψ(z) + Ψ(z) | Ψ(z) |2 = 0 (2720)

The coherence length ξ(T ) is defined as

ξ2(T ) = − γ

α> 0 (2721)

The coherence length is given in terms of the B.C.S. coherence length ξ0 via

ξ(T ) = 0.74 ξ0

(Tc

Tc − T

) 12

(2722)

Thus, the coherence length diverges at the superconducting transition temper-ature.

The spatial variation of Ψ can be found by imposing a boundary conditionat an inhomogeneity, say Ψ(0) = 0. The equation can be integrated using theintegrating factor (

∂Ψ∗

∂z

)(2723)

leading to

− 12ξ2(T )

∣∣∣∣∂Ψ∂z

∣∣∣∣2 − 12| Ψ(z) |2 +

14| Ψ(z) |4 = C (2724)

where C is a constant of integration. It is assumed that as z → ∞, one recoversthe bulk superconductivity. Hence, one has limz → ∞ Ψ(z) → 1. This fixesthe constant of integration C = − 1

4 . Hence, the equation can be written as acomplete square

ξ2(T )∣∣∣∣∂Ψ∂z

∣∣∣∣2 =12

(1 − |Ψ(z) |2

)2

(2725)

The pair of first order differential equation have the solutions

Ψ(z) = ± tanh(

z√2 ξ(T )

)(2726)

Thus, the near a superconducting - normal metal interface, the order parametervaries over the length scale provided by the temperature dependent coherence

766

length, ξ(T ).

The penetration depth can be obtained by assuming a constant value for theorder parameter. In these circumstances, the Landau-Ginzburg equations yielda diamagnetic current given by

j(r, t) = − 8 e2 γh c

∆20 A(r, t) (2727)

This is the London equation, if one identifies the phenomenological constants

ns e2

m c=

8 e2 ∆20 γ

h c(2728)

This provides a local relation between the current density and the electromag-netic field. On taking the curl of the equation, one obtains

∇ ∧ j = − 8 e2 γh c

∆20 B(r, t) (2729)

Hence, on using Maxwell’s equations, one finds that the magnetic field will onlypenetrate a distance λL inside the superconductor

Bx(z) = Bx(0) exp[− z

λL

](2730)

where λl(T ) is given by

λL(T )−2 =32 π e2 ∆2

0 γ

h c2(2731)

The B.C.S. theory yields, the temperature dependence of the penetration lengthas

λL(T ) =1√2λL(0)

(Tc

Tc − T

)(2732)

which also diverges at Tc.

From the above analysis, once concludes that the ratio of λL(T ) to ξ(T ) isalmost temperature independent close to Tc. The ratio

κ =λL(T )ξ(T )

(2733)

is the Landau - Ginzburg parameter. For a pure material, the value of κ isevaluated in terms of the zero temperature values

κ = 0.96λL(0)ξ0

(2734)

Hence, there is at most only a slight temperature variation.

767

Type I superconductors are classified as those where κ > 1√2, and type II

superconductors are those for which κ < 1√2. The value of κ can be obtained

directly from the experimentally measured values of Hc(T ) and λL(T ) from

κ = 2√

2| e |h c

Hc(T ) λ2L(T ) (2735)

25.8.3 The Surface Energy

The Landau-Ginzburg equation can be used to calculate the energy of a sur-face separating a normal and superconducting region. The normal material isassumed to be located at z < 0 and the superconducting region is located atz > 0. The boundary conditions are

for z → − ∞∆ = 0B = Hc (2736)

for the normal metal and

for z → + ∞∆ = ∆0

B = 0 (2737)

in the superconductor.

Inside the bulk superconductor, the free energy is given by

F [∆] = F0 +∫

d3r

[α(T ) | ∆(r) |2 +

β

2| ∆(r) |4

+γ

h2

∣∣∣∣ ( − i h ∇ +2 | e |c

A

)∆(r)

∣∣∣∣2 +B(r)2

8 π

](2738)

However, since the field penetrates into the superconductor, there should also bea magnetic contribution due to the magnetization produced by Hc, This energyis

− M Hc = − B − Hc

4 πHc (2739)

The bulk normal and superconducting states have the same Gibbs free-energy,since B = Hc, therefore they can coexist. At the boundary, the energy willvary as both ∆(z) and B(z) vary. Hence, the total energy is just

G[∆] = F0 +∫

d3r

[α(T ) | ∆(r) |2 +

β

2| ∆(r) |4

768

+γ

h2

∣∣∣∣ ( − i h ∇ +2 | e |c

A

)∆(r)

∣∣∣∣2 +B(r)2

8 π− B Hc

4 π+

H2c

4 π

](2740)

The surface energy, per unit area, is found by subtracting the bulk energydensity which is

H2c

8 π(2741)

for both phases. After dividing the free energy by the area, the surface energyσ is found as

σ =∫ ∞

−∞dz

[α(T ) | ∆(z) |2 +

β

2| ∆(z) |4

+ γ

∣∣∣∣∂∆∂z

∣∣∣∣2 +4 e2 γh2 c2

| ∆(z) |2 A2 +( B(z) − Hc )2

8 π

](2742)

This energy has different signs depending on whether κ is greater or smallerthan 1√

2. For a type I superconductor, the surface energy is positive. For a type

II superconductor the surface energy is negative.

The surface energy in the extreme type I limit can be easily calculated. Since,the penetration depth is vanishingly small, it provides a negligible contributionto the surface energy. Hence, for κ 1, the surface energy is simply given by

σ ∼∫ ∞

0

dz

[α(T ) | ∆(z) |2 +

β

2| ∆(z) |4 + γ

∣∣∣∣∂∆∂z

∣∣∣∣2 +H2

c

8 π

](2743)

Since the condensation energy density is such that

α(T )2

∆20(T ) = − Hc(T )2

8 π(2744)

the surface energy can be scaled to

σ = − α(T ) ∆20(T )

∫ ∞

0

dz

[ξ2(T )

(∂Ψ∂z

)2

+12

(1 − Ψ2(z)

)2 ](2745)

The spatial variation of the order parameter is given by

Ψ(z) = tanh(

z√2 ξ(T )

)(2746)

After evaluating the integrals, one finds the surface energy is positive

σ =4√

23

ξ(T )Hc(T )2

8 π(2747)

769

Hence, a type I superconductor minimizes the number of domain walls.

On the other hand, in the extreme type II limit, the surface energy can beestimated by using the London equation. In this limit, the superconductingorder parameter can be set to the bulk value. The surface energy reduces to

σ =∫ ∞

−∞dz

[− H2

c (T )8 π

+4 e2 γh2 c2

| ∆0 |2 A2 +( B(z) − Hc )2

8 π

](2748)

Using the definition of the London penetration depth, the surface free energycan be re-written as

σ =∫ ∞

−∞dz

[− H2

c (T )8 π

+H2

c (T )8 π λ2

L

A2(z) +( B(z) − Hc )2

8 π

](2749)

From the London equations, one finds that the spatial variation of the magneticfield and vector potential is given by

Ay(z) = λL Hc(T ) exp[− z

λL

]Bx(z) = Hc(T ) exp

[− z

λL

](2750)

The surface energy is found, from an elementary integration, to be given by

σ = − λL(T )H2

c (T )8 π

(2751)

Thus, for κ(T ) 1 the surface energy is negative. Hence, in the presence of afield, a type II superconductor will allow the creation of vortices or form manymicroscopic domains.

The macroscopic properties of the two classes of superconductors are dra-matically different.

25.8.4 The Little-Parks Experiment

The Little-Parks experiment demonstrates flux quantization in a superconduc-tor305, and thereby provides a direct experimental confirmation of the chargeon a Cooper pair. Consider a superconductor in the shape of a thin cylindricalshell of thickness d and radius R. A magnetic field, parallel to the cylindricalaxis, threads through the hollow center of the cylinder. The superconducting305W. Little and R. D. Parks, Phys. Rev. 133 A, 97 (1964).

770

gap is assumed to have a constant magnitude, ∆, but has a spatially varyingphase

∆(r) = | ∆ | exp[i θ(r)

](2752)

Inside the superconductor, the free energy is given by

F [∆] = F0 +∫

d3r

[α(T ) | ∆ |2 +

β

2| ∆ |4

+γ

h2

(h ∇ θ +

2 | e |c

A

)2

| ∆ |2 +B(r)2

8 π

](2753)

The magnetic field induces a current jϕ which wraps around the cylinder. Thecurrent is given by

j = i2 | e | γ

h

(∆∗(r) ∇ ∆(r)− ∆(r) ∇ ∆∗(r)

)− 8 e2 γ

h2 c| ∆(r) |2 A

= − 4 | e | γh

| ∆ |2 ∇ θ(r) − 8 e2 γh2 c

| ∆ |2 A

(2754)

On integrating the current around a loop encircling the cylinder, one finds

jϕ 2 π R = − 4 | e | γh

| ∆ |2 δθ − 8 e2 γh2 c

| ∆ |2 Φ (2755)

where δθ is the change of the phase on going around the loop. In deriving thisexpression, we have used Stokes’s theorem∮

A . dr =∫

d2S . B

= π R2 B

= Φ (2756)

where Φ is the total flux enclosed by the cylinder. Since the order parametermust be single valued, then on performing one loop around the cylinder’s axisthe phase must change by an integer multiple of 2 π,

δθ = 2 π n (2757)

for some integer value of n. Hence, one has

jϕ 2 π R = − 8 π | e | γh

| ∆ |2(n +

2 | e |2 π h c

Φ)

(2758)

Hence, on defining the superconducting flux quantum as

Φs =2 π h c2 | e |

(2759)

771

one has

jϕ 2 π R = − 8 π | e | γh

| ∆ |2(n +

ΦΦs

)(2760)

The expression for the current is equivalent to the gradient terms in the freeenergy, since

∇ θ =2 πR

n eϕ

A =R B

2eϕ (2761)

so

F [∆] = F0 +∫

d3r

[α(T ) | ∆ |2 +

β

2| ∆ |4

+4 π2 γ

R2

(n +

2 | e |2 π h c

π R2 B

)2

| ∆ |2 +B(r)2

8 π

](2762)

Thus, one finds that the phase variation of the order parameter is such as tominimize | n + Φ

Φs|. That is, n will jump discontinuously once, every time Φ is

increased by Φs. Hence, the current is a periodic function of the flux Φ. Since nis now fixed, the free energy can be minimized with respect to the magnitude ∆.The non-zero kinetic energy term increases the coefficient of the term quadraticin the order parameter, whereas the quartic term is unaffected.(

α(T ) +4 π2γ

R2( n +

ΦΦs

)2)

∆2 +β

2∆4 (2763)

Hence, the gap parameter is given by

∆2 = −(α(T ) +

4 π2γ

R2( n +

ΦΦs

)2)β−1 (2764)

and is a periodic function of the flux. The variation of the gap is measurable. Infact, the critical temperature Tc(H) at which the gap vanishes is also a periodicfunction of Φ. The periodic variation of resistance as a function of the field wasobserved by Little and Parks306.

25.8.5 The Critical Current

If a current is run through a superconductor, there is a maximum value of thecurrent that can be drawn while the material remains superconducting. Forcurrents larger than the critical current, the material transforms to the normalstate.

306W. Little and R. D. Parks, Phys. Rev. 133 A, 97 (1964).

772

We shall assume the sample has the geometry of a thin film. The supercon-ducting gap is assumed to have a constant magnitude, ∆, but has a spatiallyvarying phase

∆(r) = | ∆ | exp[i θ(r)

](2765)

Inside the superconductor, the free energy is given by

F [∆] = F0 +∫

d3r

[α(T ) | ∆ |2 +

β

2| ∆ |4 + γ ( ∇ θ )2 | ∆ |2

](2766)

The current is related to the spatially varying phase

j = − 4 | e | γh

| ∆ |2 ∇ θ(r) (2767)

The relation between the magnitude of the order parameter and the gradientof the phase can be found by minimizing the free-energy, with respect to theamplitude at constant pase gradient. This leads to the equation

γ ( ∇ θ )2 + α + β | ∆ |2 = 0 (2768)

which can be solved for ∇ θ as

∇ θ =1√γ

(− α − β | ∆ |2

) 12

(2769)

Hence, the current is given in terms of the amplitude by

j = −4 | e | √γ

h| ∆ |2

(− α − β | ∆ |2

) 12

(2770)

This expression shows that there is a maximum value of the current, as a func-tion of | ∆ |. The maximum value of the current is found for

| ∆ |2 = − 2 α3 β

(2771)

and is given by

jc =8 | e | √γ α

3 h β

(− α

3

) 12

(2772)

Thus, if | ∆ | is finite, there is a maximum value of the current. For zero current,the magnitude of the order parameter is given by the value

| ∆ |2 = − α

β(2773)

and decreases with increasing j, until j reaches jc, at which point

| ∆ |2 = − 2 α3 β

(2774)

For larger values of j, the system becomes unstable to the normal state.

773