Eleftherios N. Economou

The Physics of SolidsEssentials and Beyond

With 261 Figures

4) Springer

Contents

Part I An Overview

1 Basic Principles Summarized ............................................................. 31.1 The Atomic Idea: From Elementary Particles to Solids .......... 41.2 Permanent (i.e., Equilibrium) Structures of Matter

Correspond to the Minimum of Their (Free) Energy ............... 61.3 Condensed Matter Tends to Collapse Under the Influence

of Coulomb Potential Energy ......................................................... 91.4 Quantum Kinetic Energy Counterbalances Coulomb

Potential Energy Leading to Stable Equilibrium Structures 101.4.1 Heisenberg's Uncertainty Principle and the

Minimum Kinetic Energy ................................................ 101.4.2 Pauli's Exclusion Principle and the Enhancement

of the Minimum Kinetic Energy .................................... 111.4.3 Schrödinger's Spectral Discreteness and the Rigidity

of the Ground State ........................................................... 141.5 Dimensional Analysis ........................................................................ 151.6 Key Points .......................................................................................... 181.7 Questions and Problems ................................................................... 19

2 Basic Principles in Action ................................................................... 212.1 Size and Energy Scale of Atoms .................................................... 212.2 Why do Atoms Come Together to Form Molecules

and Solids? .......................................................................................... 232.3 Ionic Motion: Small Oscillations .................................................... 272.4 Why do the Specific Heats of Solids go to Zero as T —> 0 K ? . 292.5 When is Classical Mechanics Adequate? ...................................... 31

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2.6 Estimating Magnitudes Through Dimensional Analysis ...........2.6.1 Atomic Radius, R, .....................................................................

2.6.2 Volume per Atom, va a V/Na , in Solids .....................2.6.3 Mass Density, pm ................................................................2.6.4 Cohesive Energy, U -, ............................................................2.6.5 Bulk Modulus, B, and Shear Modulus, fis .................................................

2.6.6 Sound Velocities in Solids, co, cE, ct ............................................................................

2.6.7 Maximum Angular Frequency of AtomicVibrations in Solids, wmax .................................................

2.6.8 Melting Temperature, T,, .......................................................

2.6.9 DC Electrical Resistivity, pe .................................................................................................................

2.7 Key Points ............................................................................................2.8 Questions and Problems ....................................................................

A First Acquaintance with Condensed Matter ..........................3.1 Various Kinds of Condensed Matter .............................................

3.1.1 Monocrystalline and Polycrystalline Atomic Solids3.1.2 Atomic or lonic Compounds and Alloys .......................3.1.3 Molecular Solids ..................................................................3.1.4 Glasses ...................................................................................3.1.5 Polymers ...............................................................................3.1.6 Colloids .................................................................................3.1.7 Gels3.1.8 Liquid Crystals ....................................................................3.1.9 Self-Assembled Soft Matter ...............................................3.1.10 Artificial Structures ............................................................3.1.11 Clusters and Other Finite Systems ................................

3.2 Bonding Types and Resulting Properties ....................................3.2.1 Simple Metals .......................................................................3.2.2 Transition Metals and Rare Earths ................................3.2.3 Covalent Solids .....................................................................3.2.4 Ionic Solids ...........................................................................3.2.5 Van der Waals Bonded Solids ...........................................3.2.6 Hydrogen Bonded Solids ...................................................

3.3 A Short Introduction to Crystal Structures ................................3.3.1 Some Basic Definitions ......................................................3.3.2 Unit and Primitive Cells of Some Commonly

Occurring 3-D Crystal Structures ..................................3.3.3 Systems and Types of 3D Bravais Lattices .................3.3.4 Crystal Planes and Miller Indices ..................................

3.4 Bloch Theorem, Reciprocal Lattice, Bragg Planes,and Brillouin Zones .............................................................................3.4.1 Bloch Theorem .....................................................................3.4.2 Reciprocal Lattice ..............................................................

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Contents

3.4.3 Bragg Planes ........................................................................3.4.4 Brillouin Zones ...................................................................Key Points ...........................................................................................Questions and Problems ...................................................................

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Part II Two Simple Models for Solids

4 The Jellium Model and Metals I: Equilibrium Properties .. 834.1 Introduction ........................................................................................ 844.2 Electronic Eigenfunctions, Eigenenergies, Number of States 864.3 Kinetic and Potential Energy, Pressures,

and Elastic Moduli ............................................................................ 904.4 Acoustic Waves are the Ionic Eigenoscillations in the JM ...... 974.5 Thermodynamic Quantities ............................................................. 101

4.5.1 General Formulas ............................................................... 1014.5.2 Specific Heat, Cv ............................................................... 1044.5.3 Bulk Thermal Expansion Coefficient ............................. 107

4.6 Key Points ........................................................................................... 1074.7 Problems ............................................................................................... 108

5 The Jellium Model and Metals II:Response to External Perturbations .............................................. 1135.1 Response to Electric Field ............................................................... 1135.2 The Dielectric Function ................................................................... 1145.3 Static Electrical Conductivity ......................................................... 1205.4 Phonon Contribution to Resistivity .............................................. 1235.5 Response in the Presence of a Static Uniform

Magnetic Field .................................................................................... 1275.5.1 Magnetic Resonances ......................................................... 1285.5.2 Hall Effect and Magnetoresistance ................................ 1315.5.3 Magnetic Susceptibility, x in ................................................................................................................ 133

5.6 Thermoelectric Response ................................................................. 1405.7 Key Points ........................................................................................... 1435.8 Problems ............................................................................................... 145

6 Solids as Supergiant Molecules: LCAO .......................................... 1496.1 Diversion: The Coupled Pendulums Model .................................. 1496.2 Introductory Remarks Regarding the LCAO Method ............... 1526.3 A Single Band One-Dimensional Elemental "Metal" ............... 1536.4 One-Dimensional Tonic "Solid" ....................................................... 1576.5 One-Dimensional Molecular "Solid" .............................................. 1606.6 Diversion: Eigenoscillations in One-Dimensional "solid"

with two Atoms Per Primitive Cell ............................................... 1636.7 One-Dimensional Elemental sp l "Semiconductor" ................... 164

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6.8 One-Dimensional Compound sp l "Semiconductor" ...................6.9 Key Points ............................................................................................6.10 Problems ................................................................................................

Semiconductors and Other Tetravalent Solids ............................

171174174

1777.1 Lattice Structures: A Reminder ..................................................... 1777.2 Band Edges and Gap ......................................................................... 1787.3 Differentes Between the 1-D and the 3-D Case and Energy

Diagrams .............................................................................................. 1817.4 Metals, Semiconductors, and Ionic Insulators ............................ 1837.5 Holes ...................................................................................................... 1847.6 Effective Masses and DOS ................................................................ 1867.7 Dielectric Function and Optical Absorption ................................ 1887.8 Effective Hamiltonian ......................................................................... 1897.9 Impurity Levels ................................................................................... 191

7.9.1 Impurity Levels: The General Picture ............................ 1917.9.2 Impurity Levels: Doping ................................................... 192

7.10 Concentration of Electrons and Holes at Temperature T ......... 1957.10.1 Intrinsic case ......................................................................... 1977.10.2 Extrinsic case ...................................................................... 197

7.11 Band Structure and Electronic DOS ............................................. 1987.12 Eigenfrequencies, Phononic DOS, and Dielectric Function 2007.13 Key Points ............................................................................................ 2077.14 Problems ................................................................................................ 208

8 Beyond the Jellium and the LCAO: An Outline ..................... 2118.1 Introductory Remarks ...................................................................... 2118.2 The Four Basic Approximations ..................................................... 2128.3 Density Functional Theory .............................................................. 2158.4 Outline of an Advanced Scheme for Calculating

the Properties of Solids .................................................................... 2198.5 Beyond the Four Basic Approximations ...................................... 221

8.5.1 Periodicity Broken or Absent ........................................... 2238.5.2 Electron—Electron Correlations, Quasi-Part icles,

Magnetic Phases, and Superconductivity ..................... 2358.5.3 Electron—Phonon Interactions, Transport

Properties, Superconductivity, and Polarons ............... 2378.5.4 Phonon—Phonon Interactions, Thermal Expansion,

Melting, Structural Phase Transitions, Solitons,Breathers ............................................................................... 238

8.5.5 Disorder and Many Body Effects in Coexistence ....... 2398.5.6 Quantum Informatics and Solid State Systems ........... 240

8.6 Key Points ............................................................................................ 2408.7 Problems ................................................................................................ 241

Contents XIII

Part III More About Periodicity & its Consequences

9 Crystal Structure and Ionic Vibrations ........................................ 2459.1 Experimental Determination of Crystal Structures ................... 2459.2 Determination of the Frequency vs. Wavevector ....................... 2519.3 Theoretical Calculation of the Phonon Dispersion Relation .. 2569.4 The Debye—Waller Factor and the Inelastic Cross-Section ...... 2639.5 Key Points .......................................................................................... 2689.6 Problems ............................................................................................... 269

10 Electrons in Periodic Media. The Roleof Magnetic Field ...................................................................................... 27310.1 Introduction ........................................................................................ 27310.2 Dispersion Relations, Surfaces of Constant Energy, and

DOS: A Reminder .............................................................................. 27410.3 Effective Hamiltonian and Semiclassical Approximation ........ 27610.4 Semiclassical Trajectories in the Presence

of a Magnetic Field ............................................................................ 28010.5 Two Simple but Elucidating TB Models ...................................... 28110.6 Cyclotron Resonance and the de Haas—van Alphen Effect 28710.7 Hall Effect and Magnetoresistance ................................................ 29010.8 Key Points .......................................................................................... 29810.9 Problems ............................................................................................... 299

11 Methods for Calculating the Band Structure ............................ 30111.1 Introductory Remarks ..................................................................... 30111.2 Ionic and Total Pseudopotentials ................................................... 30311.3 Schrödinger Equation, Plane Wave Expansion,

and Bloch's Theorem ........................................................................ 30911.4 Plane Waves and Perturbation Theory ........................................ 31011.5 Muffin—Tin Potential ........................................................................ 31311.6 Schrödinger Equation and the Augmented Plane Wave

(APW) Method .................................................................................. 31311.7 Schrödinger Equation and the Korringa—Kohn—Rostoker

(KKR) Method .................................................................................. 31511.8 The k • p Method of Band Structure Calculations ..................... 31711.9 Key Points .......................................................................................... 32111.10 Problems .............................................................................................. 322

12 Pseudopotentials in Action ................................................................. 32512.1 The One-Dimensional Case ............................................................. 32512.2 The Two-Dimensional Square Lattice .......................................... 327

12.2.1 Spaghetti Diagrams ........................................................... 32712.2.2 Fermi Lines .......................................................................... 330

12.3 Harrison's Construction ................................................................... 336

XIV Contents

12.4 Second-Order Correction to the Total JM Energy ................... 33712.5 Ionic Interactions in Real Space ..................................................... 33812.6 Phononic Dispersions in Metals ..................................................... 34012.7 Scattering by Phonons, Mean Free Path, and the

Dimensionless Constant A in Metals ............................................. 34212.8 Key Points ............................................................................................ 34512.9 Problems ................................................................................................ 346

Part IV Materials

13 Simple Metals and Semiconductors Revisited ............................ 35113.1 Band Structure and Fermi Surfaces of Simple Metals ............. 351

13.1.1 Alkali Metals ......................................................................... 35113.1.2 Alkaline Earths: Be, Mg, Ca, Sr, Ba, and Ra ............. 35413.1.3 Trivalent Metals .................................................................. 35413.1.4 Tetravalent Metals .............................................................. 358

13.2 Band Structure of Semiconductors ................................................. 36013.3 The Jones Zone and the Disappearance

of the Fermi Surface ........................................................................... 36313.4 Mechanical Properties of Semiconductors .................................... 36513.5 Magnetic Susceptibility of Semiconductors .................................. 36813.6 Optical and Transport Properties

of Semiconductors ............................................................................... 37113.6.1 Excitons ................................................................................. 37113.6.2 Conductivity and Mobility in Semiconductors ........... 374

13.7 Silicon Dioxide (SiO2) ...................................................................... 37813.8 Graphite and Graphene .................................................................... 38013.9 Organic semiconductors .................................................................... 38613.10 Key Points ............................................................................................ 38813.11 Questions and Problems .................................................................... 389

14 Closed-Shell Solids ................................................................................... 39314.1 Van Der Waals Solids ......................................................................... 39314.2 Ionic Compounds I: Types and Crystal Structures ................... 39714.3 Ionic Compounds II: Mechanical Properties ................................ 39914.4 Ionic Compounds III: Optical Properties .................................... 40114.5 Key Points ............................................................................................ 40614.6 Problems ................................................................................................ 407

15 Transition Metals and Compounds ................................................... 40915.1 Experimental Data for the Transition Metals ............................ 40915.2 Calculations I: APW or KKR .......................................................... 41215.3 Calculations II: LCAO ....................................................................... 41715.4 Calculations III: The Simple Friedel Model ................................ 421

Contents XV

15.5 Compounds of Transition Elements, I: Perovskites ................... 42315.6 Compounds of Transition Elements, II:

High Tc Superconducting Materials .............................................. 42615.7 Compounds of Transition Metals, III: Oxides, etc . ................... 43015.8 Key Points .......................................................................................... 43415.9 Problems ............................................................................................... 435

16 Artificial Periodic Structures ............................................................. 43716.1 Semiconductor Superlattices ........................................................... 43716.2 Photonic Crystals: An Overview ................................................... 43916.3 Photonic Crystals: Theoretical Considerations ......................... 44316.4 Phononic Crystals .............................................................................. 45016.5 Left-Handed Metamaterials (LHMs) ............................................ 45616.6 Designing, Fabricating, and Measuring LHMs ........................... 46116.7 Key Points ........................................................................................... 46616.8 Problems ............................................................................................... 468

Part V Deviations from Periodicity

17 Surfaces and Interfaces .......................................................................... 47117.1 Surface Preparation .......................................................................... 47117.2 Relaxation and Reconstruction ....................................................... 47217.3 Surface States .................................................................................... 47417.4 Work Function .................................................................................... 47917.5 Measuring the Work Function ......................................................... 48117.6 The p—n Homojunction in Equilibrium ........................................ 48317.7 The p—n Homojunction Under an External Voltage V ............. 48717.8 Some Applications of Interfaces ..................................................... 49117.9 Key Points .......................................................................................... 49417.10 Problems 497

18 Disordered and Other Nonperiodic Solids 49918.1 Introductory Remarks ..................................................................... 49918.2 Alloys and the Hume-Rothery Rule .............................................. 50018.3 Glasses and other Amorphous Systems ..........................................50218.4 Distribution and Correlation Functions ........................................ 50418.5 Quasi-Crystals .................................................................................... 50618.6 Electron Transport and Quantum Interference ......................... 51018.7 Band Structure, Static Disorder, and Localization .....................513

18.7.1 3D Case ................................................................................ 51318.7.2 2D Case ................................................................................ 51718.7.3 1D and quasi 1D Systems ................................................ 518

XVI Contents

18.8 Calculation Techniques ...................................................................... 52218.8.1 Coherent Potential Approximation ................................ 52218.8.2 Weak Localization due to Quantum Interference 52618.8.3 Scaling Approach ................................................................ 52918.8.4 Quasi-One-Dimensional Systems and Scaling ............. 53218.8.5 Potential Well Analogy ..................................................... 533

18.9 Quantum Hall Effect ......................................................................... 53418.10 Key Points ............................................................................................ 53818.11 Problems ................................................................................................ 540

19 Finite Systems .............................................................................................. 54319.1 Introduction .......................................................................................... 54319.2 Metallic Clusters ................................................................................. 54419.3 Fullerenes .............................................................................................. 54519.4 C60-Based Solids ................................................................................. 54919.5 Carbon Nanotubes ............................................................................. 55119.6 Other Clusters ..................................................................................... 55619.7 Quantum Dots ..................................................................................... 557

19.7.1 An Overview ......................................................................... 55719.7.2 Optical Transitions .............................................................. 55819.7.3 QDs and Coulomb Blockade ............................................. 561

19.8 Key Points ............................................................................................ 56419.9 Problems ................................................................................................ 565

Part VI Correlated Systems

20 Magnetic Materials, I: Phenomenology ........................................ 56920.1 Which Property Characterizes These Materials? ......................... 56920.2 Experimental Data for Ferromagnets ............................................. 573

20.2.1 Saturation Magnetization vs Temperaturefor Simple Ferromagnets ................................................... 573

20.2.2 Magnetic Susceptibility of Simple Ferromagnet forT > ............................................................................ 573

20.2.3 Saturation Magnetization vs Temperaturefor Ferrimagnets .................................................................. 574

20.2.4 Magnetic Susceptibility of Ferrimagnets vsTemperature (T > .................................................. 575

20.3 Experimental Data for Antiferromagnets .................................... 57620.3.1 Determination of the Antiferromagnetic

Ordered Structure .............................................................. 57620.3.2 Magnetic Susceptibility vs Temperature ........................ 577

20.4

20.5

Contents

Materials ..............................................................................................20.4.1 Simple Ferromagnetic Materials ......................................20.4.2 Ferrimagnetic Materials .....................................................20.4.3 Antiferromagnetic Materials ............................................Thermodynamic Relations ...............................................................

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20.5.1 Thermodynamic Potentials .............................................. 58020.5.2 Mean Field Approximation (Landau's Approach) 58320.5.3 Why are Magnetic Domains Formed? ........................... 58420.5.4 How Thick is the Bloch Wall? ........................................ 58620.5.5 Examples of Magnetic Domains ...................................... 58620.5.6 Thermodynamics of Antiferromagnets ......................... 587

20.6 Spintronics .......................................................................................... 58820.7 Key Points .......................................................................................... 59220.8 Problems ............................................................................................... 593

21 Magnetic Materials II: Microscopic View .................................... 59521.1 Introduction ........................................................................................ 59521.2 Jellium model and el—el Coulomb Repulsion .............................. 599

21.2.1 Is There Ferromagnetic Order in the JM? ................... 59921.2.2 Magnetic Susceptibility Within the JM in the

Presence of Electron—Electron Interactions ................. 60121.2.3 Is There Antiferromagnetic Order in the JM? 603

21.3 The Hubbard Model .......................................................................... 60721.4 The Heisenberg Model ...................................................................... 613

21.4.1 The Hamiltonian ................................................................. 61321.4.2 Mean Field Approximation .............................................. 61521.4.3 The Ferromagnetic Case, (J3 > 0)

and its spin waves ............................................................... 61721.4.4 The AF Case ........................................................................ 619

21.5 Key Points .......................................................................................... 62221.6 Problems .............................................................................................. 624

22 Superconductivity, I: Phenomenology ............................................ 62522.1 Materials .............................................................................................. 62522.2 Properties of Superconductors ....................................................... 627

22.2.1 Zero DC Resistivity ........................................................... 62722.2.2 Expulsion of the Magnetic Field B

from the Interior of a Superconductor ......................... 62722.2.3 Critical Value of the Magnetic Field Beyond Which

Superconductivity Disappears ........................................ 62922.2.4 Specific Heat and Other Thermodynamic

Quantities ............................................................................ 63222.2.5 Response to Microwave or Far Infrared

EM Radiation ..................................................................... 63422.2.6 Ultrasound Attenuation ..................................................... 635

XVIII Contents

22.2.7 Tunneling Current in Met al/Insulator/Superconductor Junctions ................................................. 635

22.2.8 Temperature Dependence of the SuperconductingGap .......................................................................................... 635

22.2.9 Isotope Effect ...................................................................... 63722.2.10 Relaxation Times for Nuclear Spin ................................ 63822.2.11 Thermoelectric Coefficients ............................................... 638

22.3 Thermodynamic Relations ................................................................ 63922.4 London Equation ................................................................................. 64122.5 Pippard's Generalization .................................................................. 64422.6 Ginzburg-Landau Theory ................................................................ 64522.7 Quantization of the Magnetic Flux ............................................... 65122.8 Key Points ............................................................................................ 65222.9 Problems ................................................................................................ 654

23 Superconductivity, II: Microscopic Theory .................................. 65523.1 Electron-Electron Indirect Attraction ........................................... 65523.2 Cooper Pairs ....................................................................................... 65723.3 Comments .............................................................................................. 65923.4 Corrected Binding Energy and the Critical Temperature 66123.5 Further Corrections to the Formula for 7', .................................. 66323.6 The Bardeen-Cooper-Schrieffer (BCS) Theory .......................... 66423.7 Thermodynamic Quantities .............................................................. 66923.8 Response to Electromagnetic Fields ............................................... 67223.9 Towards Material-Specific Calculations of Superconducting

Quantities .............................................................................................. 67423.10 Josephson Effects and SQUID ....................................................... 67723.11 Key Points ............................................................................................ 68023.12 Problems ................................................................................................ 682

Part VII Appendices

A Elements of Electrodynamics of Continuous Media ............... 685A.1 Field Vectors, Potentials, and Maxwell's Equations ................. 685A.2 Relations Among the Fields ............................................................ 688

B Elements of Quantum Mechanics ..................................................... 697B.1 General Formalism ............................................................................. 697B.2 Bra and Ket Notation ...................................................................... 700B.3 Spherically Symmetrie Potentials ................................................... 702B.4 Perturbation Results ........................................................................... 708B.5 Interaction of Matter with an External

Electromagnetic Field ......................................................................... 711

Contents XIX

C Elements of Thermodynamicsand Statistical Mechanics ..................................................................... 713C.1 Thermodynamic Relations ............................................................... 713C.2 Basic Relations of Statistical Mechanics ...................................... 716C.3 Non-Interacting Particles ................................................................. 718

C.3.1 Non-Int eract ing Electrons ................................................ 718C.3.2 Phonons ................................................................................ 721

D Dielectric Function, E(k, w): Formulasand Uses ....................................................................................................... 723D.1 Uses ....................................................................................................... 724D.2 Expressions for E (k, w) within the JM .......................................... 728D.3 Phenomenological Expressions for the Dielectric Function 730

E Waves in Continuous Elastic Media ................................................ 733E.1 Strains ................................................................................................... 733E.2 Equations of Motion ......................................................................... 733E.3 Connecting Stress and Strain ......................................................... 734E.4 The Elastic Wave Equation ............................................................. 735

F The Method LCAO Applied to Molecules .................................... 737F.1 Formulation of the LCAO Method ................................................ 737F.2 Some Important Examples ............................................................. 740

F.2.1 Covalent Diatomic Molecule ............................................ 740F.2.2 Ionic Diatomic Molecule .................................................. 742

F.3 Hybridization of Atomic Orbitals .................................................. 743F.3.1 sp l Hybrid Atomic Orbitals ............................................ 744F.3.2 spe Hybrid Atomic Orbitals ............................................ 748F.3.3 sp3 Hybrid Atomic Orbitals ............................................ 749

G Boltzmann's Equation ............................................................................ 755

H Tables ............................................................................................................. 759

Solutions of Selected Problems and Answers ...................................... 779

General Reading ................................................................................................ 826

References ............................................................................................................. 837

Index ........................................................................................................................ 849