Solid State Physics-1 second edition by Philip Hofmann complete book

COURSE CONTENTS:

1 Crystal Binding and Structure.......................................... 1

1.1 Classification of Solids by Binding Forces (B) ............................. 2

1.1.1 Molecular Crystals and the van der Waals Forces (B).... 2

1.1.2 Ionic Crystals and Born–Mayer Theory (B) ................... 6

1.1.3 Metals and Wigner–Seitz Theory (B) ............................. 9

1.1.4 Valence Crystals and Heitler–London Theory (B) ......... 11

1.1.5 Comment on Hydrogen-Bonded Crystals (B)................. 12

1.2 Group Theory and Crystallography ............................................... 13

1.2.1 Definition and Simple Properties of Groups (AB).......... 14

1.2.2 Examples of Solid-State Symmetry Properties (B)......... 17

1.2.3 Theorem: No Five-fold Symmetry (B) ........................... 20

1.2.4 Some Crystal Structure Terms

and Nonderived Facts (B) ............................................... 22

1.2.5 List of Crystal Systems and Bravais Lattices (B) ........... 23

1.2.6 Schoenflies and International Notation

for Point Groups (A) ....................................................... 26

1.2.7 Some Typical Crystal Structures (B) .............................. 28

1.2.8 Miller Indices (B)............................................................ 31

1.2.9 Bragg and von Laue Diffraction (AB) ............................ 31

Problems .................................................................................................. 38

2 Lattice Vibrations and Thermal Properties..................... 41

2.1 The Born–Oppenheimer Approximation (A) ................................ 42

2.2 One-Dimensional Lattices (B)....................................................... 47

2.2.1 Classical Two-Atom Lattice with Periodic Boundary

Conditions (B)................................................................. 48

2.2.2 Classical, Large, Perfect Monatomic Lattice,

and Introduction to Brillouin Zones (B).......................... 51

2.2.3 Specific Heat of Linear Lattice (B)................................. 61

2.2.4 Classical Diatomic Lattices: Optic and Acoustic

Modes (B) ....................................................................... 64

2.2.5 Classical Lattice with Defects (B) .................................. 70

2.2.6 Quantum-Mechanical Linear Lattice (B) ........................ 76

2.3 Three-Dimensional Lattices .......................................................... 85

2.3.1 Direct and Reciprocal Lattices and Pertinent

Relations (B)................................................................... 85

2.3.2 Quantum-Mechanical Treatment and Classical

Calculation of the Dispersion Relation (B)..................... 87

2.3.3 The Debye Theory of Specific Heat (B)......................... 92

2.3.4 Anharmonic Terms in The Potential /

The Gruneisen Parameter (A)......................................... 99

2.3.5 Wave Propagation in an Elastic

Crystalline Continuum (MET, MS) ................................ 103

Problems.................................................................................................. 108

3 Electrons in Periodic Potentials ......................................... 113

3.1 Reduction to One-Electron Problem ............................................. 114

3.1.1 The Variational Principle (B) ......................................... 114

3.1.2 The Hartree Approximation (B) ..................................... 115

3.1.3 The Hartree–Fock Approximation (A) ........................... 119

3.1.4 Coulomb Correlations and the Many-Electron

Problem (A) .................................................................... 135

3.1.5 Density Functional Approximation (A).......................... 137

3.2 One-Electron Models .................................................................... 148

3.2.1 The Kronig–Penney Model (B) ...................................... 148

3.2.2 The Free-Electron or Quasifree-Electron

Approximation (B) ......................................................... 158

3.2.3 The Problem of One-Electron in a Three-Dimensional

Periodic Potential............................................................ 173

3.2.4 Effect of Lattice Defects on Electronic States

in Crystals (A) ................................................................ 205

Problems.................................................................................................. 209

4 The Interaction of Electrons and Lattice Vibrations....... 213

4.1 Particles and Interactions of Solid-state Physics (B)..................... 213

4.2 The Phonon–Phonon Interaction (B)............................................. 219

4.2.1 Anharmonic Terms in the Hamiltonian (B).................... 219

4.2.2 Normal and Umklapp Processes (B)............................... 220

4.2.3 Comment on Thermal Conductivity (B)......................... 223

4.2.4 Phononics (EE)............................................................... 225

4.3 The Electron-Phonon Interaction.................................................. 225

4.3.1 Form of the Hamiltonian (B) .......................................... 226

4.3.2 Rigid-Ion Approximation (B) ......................................... 229

Also read :Solid State Physics-1 (reciprocal lattice ) chapter no.2 by handwritten notes

4.3.3 The Polaron as a Prototype Quasiparticle (A) ................ 232

4.4 Brief Comments on Electron–Electron Interactions (B)................ 242

4.5 The Boltzmann Equation and Electrical Conductivity .................. 244

4.5.1 Derivation of the Boltzmann Differential

Equation (B).................................................................... 244

4.5.2 Motivation for Solving the Boltzmann Differential

Equation (B).................................................................... 246

4.5.3 Scattering Processes and Q Details (B)........................... 247

4.5.4 The Relaxation-Time Approximate Solution

of the Boltzmann Equation for Metals (B)...................... 251

4.6 Transport Coefficients ................................................................... 254

4.6.1 The Electrical Conductivity (B)...................................... 254

4.6.2 The Peltier Coefficient (B).............................................. 254

4.6.3 The Thermal Conductivity (B)........................................ 255

4.6.4 The Thermoelectric Power (B) ....................................... 255

4.6.5 Kelvin’s Theorem (B) ..................................................... 256

4.6.6 Transport and Material Properties in Composites

(MET, MS) ...................................................................... 257

Problems .................................................................................................. 263

5 Metals, Alloys, and the Fermi Surface ............................. 267

5.1 Fermi Surface (B) .......................................................................... 267

5.1.1 Empty Lattice (B) ........................................................... 268

5.1.2 Exercises (B)................................................................... 269

5.2 The Fermi Surface in Real Metals (B)........................................... 273

5.2.1 The Alkali Metals (B) ..................................................... 273

5.2.2 Hydrogen Metal (B)........................................................ 273

5.2.3 The Alkaline Earth Metals (B)........................................ 273

5.2.4 The Noble Metals (B) ..................................................... 273

5.3 Experiments Related to the Fermi Surface (B) .............................. 275

5.4 The de Haas–van Alphen effect (B) .............................................. 276

5.5 Eutectics (MS, ME) ....................................................................... 280

5.6 Peiperl’s Instability of Linear Metals (B) ......................................... 281

5.6.1 Relation to Charge Density Waves (A)........................... 284

5.6.2 Spin Density Waves (A) ................................................. 285

5.7 Heavy Fermion Systems (A) ......................................................... 285

5.8 Electromigration (EE, MS) ............................................................ 286

5.9 White Dwarfs and Chandrasekhar’s Limit (A).............................. 288

5.9.1 Gravitational Self-Energy (A)......................................... 289

5.9.2 Idealized Model of a White Dwarf (A) ........................... 289

5.10 Some Famous Metals and Alloys (B, MET) .................................. 292

Problems .................................................................................................. 293

6 Semiconductors ................................................................... 295

6.1 Electron Motion ............................................................................ 298

6.1.1 Calculation of Electron and Hole Concentration (B)...... 298

6.1.2 Equation of Motion of Electrons in Energy Bands (B)... 304

6.1.3 Concept of Hole Conduction (B).................................... 307

6.1.4 Conductivity and Mobility in Semiconductors (B)......... 309

6.1.5 Drift of Carriers in Electric and Magnetic Fields:

The Hall Effect (B) ......................................................... 311

6.1.6 Cyclotron Resonance (A) ............................................... 313

6.2 Examples of Semiconductors ........................................................ 320

6.2.1 Models of Band Structure for Si, Ge and II-VI

and III-V Materials (A)................................................... 320

6.2.2 Comments about GaN (A).............................................. 326

6.3 Semiconductor Device Physics ..................................................... 327

6.3.1 Crystal Growth of Semiconductors (EE, MET, MS) ...... 327

6.3.2 Gunn Effect (EE)............................................................ 328

6.3.3 pn-Junctions (EE) ........................................................... 330

6.3.4 Depletion Width, Varactors,

and Graded Junctions (EE)............................................. 333

6.3.5 Metal Semiconductor Junctions —

the Schottky Barrier (EE) ............................................... 336

6.3.6 Semiconductor Surface States and Passivation (EE) ...... 337

6.3.7 Surfaces Under Bias Voltage (EE) ................................. 339

6.3.8 Inhomogeneous Semiconductors

Not in Equilibrium (EE) ................................................. 340

6.3.9 Solar Cells (EE).............................................................. 346

6.3.10 Transistors (EE).............................................................. 352

6.3.11 Charge-Coupled Devices (CCD) (EE)............................ 353

Problems.................................................................................................. 353

7 Magnetism, Magnons, and Magnetic Resonance .............. 355

7.1 Types of Magnetism...................................................................... 356

7.1.1 Diamagnetism of the Core Electrons (B)........................ 356

7.1.2 Paramagnetism of Valence Electrons (B)....................... 357

7.1.3 Ordered Magnetic Systems (B) ...................................... 360

7.2 Origin and Consequences of Magnetic Order ............................... 373

Also read :Solid State Physics-1 (phonons and lattice vibrations ) chapter no.3 handwritten notes by Miss Najma

7.2.1 Heisenberg Hamiltonian ................................................. 373

7.2.2 Magnetic Anisotropy and Magnetostatic

Interactions (A)............................................................... 385

7.2.3 Spin Waves and Magnons (B) ........................................ 390

7.2.4 Band Ferromagnetism (B) .............................................. 407

7.2.5 Magnetic Phase Transitions (A) ..................................... 416

7.3 Magnetic Domains and Magnetic Materials (B)............................ 422

7.3.1 Origin of Domains and General Comments (B).............. 422

7.3.2 Magnetic Materials (EE, MS) ......................................... 432

7.3.3 Nanomagnetism (EE, MS) .............................................. 434

7.4 Magnetic Resonance and Crystal Field Theory............................. 435

7.4.1 Simple Ideas About Magnetic Resonance (B) ................ 435

7.4.2 A Classical Picture of Resonance (B) ............................. 436

7.4.3 The Bloch Equations and Magnetic Resonance (B)........ 439

7.4.4 Crystal Field Theory and Related Topics (B) ................. 445

7.5 Brief Mention of Other Topics ...................................................... 453

7.5.1 Spintronics or Magneto electronics (EE) ......................... 453

7.5.2 The Kondo Effect (A) ..................................................... 457

7.5.3 Spin Glass (A)................................................................. 458

7.5.4 Solitons (A, EE) .............................................................. 460

Problems .................................................................................................. 461

8 Superconductivity............................................................... 463

8.1 Introduction and Some Experiments (B) ....................................... 463

8.1.1 Ultrasonic Attenuation (B).............................................. 467

8.1.2 Electron Tunneling (B) ................................................... 467

8.1.3 Infrared Absorption (B) .................................................. 467

8.1.4 Flux Quantization (B) ..................................................... 467

8.1.5 Nuclear Spin Relaxation (B) ........................................... 468

8.1.6 Thermal Conductivity (B)............................................... 468

8.2 The London and Ginzburg–Landau Equations (B)........................ 469

8.2.1 The Coherence Length (B).............................................. 471

8.2.2 Flux Quantization and Fluxoid (B)................................ 475

8.2.3 Order of Magnitude for Coherence Length (B) .............. 476

8.3 Tunneling (B, EE) ......................................................................... 478

8.3.1 Single-Particle or Giaever Tunneling.............................. 478

8.3.2 Josephson Junction Tunneling ........................................ 479

8.4 SQUID: Superconducting Quantum Interference (EE) ................. 483

8.4.1 Questions and Answers (B) ............................................ 485

8.5 The Theory of Superconductivity (A) .......................................... 486

8.5.1 Assumed Second Quantized Hamiltonian

for Electrons and Phonons in Interaction (A).................. 486

8.5.2 Elimination of Phonon Variables and Separation

of Electron–Electron Attraction Term Due

to Virtual Exchange of Phonons (A)............................... 490

8.5.3 Cooper Pairs and the BCS Hamiltonian (A) ................... 494

8.5.4 Remarks on the Nambu Formalism and Strong

Coupling Superconductivity (A)..................................... 504

8.6 Magnesium Diboride (EE, MS, MET) .......................................... 505

8.7 Heavy-Electron Superconductors (EE, MS, MET) ....................... 506

8.8 High-Temperature Superconductors (EE, MS, MET) ................... 506

8.9 Summary Comments on Superconductivity (B)............................ 509

Problems.................................................................................................. 512

9 Dielectrics and Ferroelectrics............................................. 513

9.1 The Four Types of Dielectric Behavior (B) .................................. 513

9.2 Electronic Polarization and the Dielectric Constant (B) ............... 514

9.3 Ferroelectric Crystals (B) .............................................................. 520

9.3.1 Thermodynamics of Ferroelectricity

by Landau Theory (B) .................................................... 521

9.3.2 Further Comment on the Ferroelectric Transition

(B, ME) ........................................................................... 524

9.3.3 One-Dimensional Model of the Soft Mode

of Ferroelectric Transitions (A) ...................................... 525

9.4 Dielectric Screening and Plasma Oscillations (B)......................... 529

9.4.1 Helicons (EE) ................................................................. 531

9.4.2 Alfvén Waves (EE)......................................................... 533

9.4.3 Plasmonics (EE) ............................................................. 534

9.5 Free-Electron Screening................................................................ 534

9.5.1 Introduction (B) .............................................................. 534

9.5.2 The Thomas–Fermi, and Debye–Huckel Methods

(A, EE) ............................................................................ 535

9.5.3 The Lind hard Theory of Screening (A) .......................... 538

Problems.................................................................................................. 543

10 Optical Properties of Solids................................................ 545

10.1 Introduction (B)............................................................................. 545

10.2 Macroscopic Properties (B)........................................................... 546

10.2.1 Kronig–Kramer’s Relations (A)....................................... 550

Also read :solid state physics-II online Exam answer key for GCUF 2020

10.3 Absorption of Electromagnetic Radiation–General (B) ................. 552

10.4 Direct and Indirect Absorption Coefficients (B) ........................... 553

10.5 Oscillator Strengths and Sum Rules (A) ....................................... 560

10.6 Critical Points and Joint Density of States (A).............................. 561

10.7 Exciton Absorption (A)................................................................. 562

10.8 Imperfections (B, MS, MET) ........................................................ 563

10.9 Optical Properties of Metals (B, EE, MS) ..................................... 565

10.10 Lattice Absorption, Restrahlen, and Polaritons (B)....................... 571

10.10.1 General Results (A) ........................................................ 571

10.10.2 Summary of the Properties of ε(q, ω) (B)....................... 578

10.10.3 Summary of Absorption Processes:

General Equations (B) .................................................... 579

10.11 Optical Emission, Optical Scattering and Photoemission (B) ....... 580

10.11.1 Emission (B) ................................................................... 580

10.11.2 Einstein A and B Coefficients (B, EE, MS) .................... 581

10.11.3 Raman and Brillouin Scattering (B, MS) ........................ 582

10.11.4 Optical Lattices (A, B) .................................................... 584

10.11.5 Photonics (EE) ................................................................ 584

10.11.6 Negative Index of Refraction (EE) ................................. 585

10.12 Magneto-Optic Effects: The Faraday Effect (B, EE, MS) ............. 587

Problems .................................................................................................. 589

11 Defects in Solids .................................................................. 591

11.1 Summary About Important Defects (B)......................................... 591

11.2 Shallow and Deep Impurity Levels in Semiconductors (EE) ........ 594

11.3 Effective Mass Theory, Shallow Defects, and Superlattices (A)... 595

11.3.1 Envelope Functions (A) .................................................. 595

11.3.2 First Approximation (A) ................................................. 596

11.3.3 Second Approximation (A)............................................. 597

11.4 Color Centers (B) .......................................................................... 600

11.5 Diffusion (MET, MS) .................................................................... 602

11.6 Edge and Screw Dislocation (MET, MS) ...................................... 603

11.7 Thermionic Emission (B) .............................................................. 605

11.8 Cold-Field Emission (B)................................................................ 608

11.9 Microgravity (MS)......................................................................... 610

Problems .................................................................................................. 611

12 Current Topics in Solid Condensed–Matter Physics ...... 613

12.1 Surface Reconstruction (MET, MS) .............................................. 614

12.2 Some Surface Characterization Techniques (MET, MS, EE) ........ 615

12.3 Molecular Beam Epitaxy (MET, MS) ........................................... 617

12.4 Heterostructures and Quantum Wells............................................ 618

12.5 Quantum Structures and Single-Electron Devices (EE) ................ 619

12.5.1 Coulomb Blockade (EE) ................................................. 620

12.5.2 Tunneling and the Lindauer Equation (EE).................... 623

12.6 Superlattices, Bloch Oscillators, Stark–Wannier Ladders ............. 626

12.6.1 Applications of Superlattices and Related

Nanostructures (EE)........................................................ 629

12.7 Classical and Quantum Hall Effect (A) ......................................... 631

12.7.1 Classical Hall Effect – CHE (A) ..................................... 631

12.7.2 The Quantum Mechanics of Electrons

in a Magnetic Field: The Landau Gauge (A) ................. 634

12.7.3 Quantum Hall Effect: General Comments (A)................ 636

12.8 Carbon – Nanotubes and Fullerene Nanotechnology (EE)............ 640

12.9 Amorphous Semiconductors and the Mobility Edge (EE) ............ 642

12.9.1 Hopping Conductivity (EE)............................................ 642

12.10 Amorphous Magnets (MET, MS) ................................................. 644

12.11 Soft Condensed Matter (MET, MS) .............................................. 644

12.11.1 General Comments ......................................................... 644

12.11.2 Liquid Crystals (MET, MS) ............................................ 645

12.11.3 Polymers and Rubbers (MET, MS) ................................ 645

Problems.................................................................................................. 648

Appendices ................................................................................... 651

A Units.............................................................................................. 651

B Normal Coordinates ...................................................................... 653

C Derivations of Bloch’s Theorem................................................... 657

C.1 Simple One-Dimensional Derivation– ............................ 657

C.2 Simple Derivation in Three Dimensions ........................ 660

C.3 Derivation of Bloch’s Theorem by Group Theory ......... 661

D Density Matrices and Thermodynamics........................................ 662

E Time-Dependent Perturbation Theory........................................... 663

F Derivation of The Spin-Orbit Term from Dirac’s Equation............ 664

G The Second Quantization Notation for Fermions and Bosons ...... 666

G.1 Bose Particles ................................................................. 666

G.2 Fermi Particles................................................................ 668

H The Many-Body Problem.............................................................. 669

H.1 Propagators..................................................................... 670

H.2 Green Functions.............................................................. 671

Also read :Solid state physics MCQs with solutions 2021

H.3 Feynman Diagrams......................................................... 671

H.4 Definitions ...................................................................... 671

H.5 Diagrams and the Hartree and Hartree–Fock

Approximations .............................................................. 672

H.6 The Dyson Equation ....................................................... 675

I Brief Summary of Solid-State Physics.......................................... 676

J Folk Theorems .............................................................................. 690

K Handy Mathematical Results ........................................................ 694

L Condensed Matter Nobel Prize Winners

(in Physics or Chemistry) .............................................................. 696

M Problem Solutions ......................................................................... 700

M.1 Chapter 1 Solutions ........................................................ 700

M.2 Chapter 2 Solutions ........................................................ 708

M.3 Chapter 3 Solutions ........................................................ 727

M.4 Chapter 4 Solutions ........................................................ 733

M.5 Chapter 5 Solutions ........................................................ 739

M.6 Chapter 6 Solutions ........................................................ 744

M.7 Chapter 7 Solutions ........................................................ 750

Crystal Binding and Structure:

It has been argued that solid-state physics was born, as a separate field, with the publication, in 1940, of Fredrick Seitz’s book, Modern Theory of Solids [82]. In that book parts of many fields such as metallurgy, crystallography, magnetism, and electronic conduction in solids were in a sense coalesced into the new field of solid-state physics. About twenty years later, the term condensed-matter physics, which included the solid-state but also discussed liquids and related topics, gained prominently usage (see, e.g., Chaykin and Lubensky [26]). In this book, we will focus on the traditional topics of solid-state physics, but particularly in the last chapter.

Elements form solids because, for some range of temperature and pressure, a solid has less free energy than other states of matter. It is generally supposed that at low enough temperature and with suitable external pressure (helium requires external pressure to solidify) everything becomes solid. No one has ever proved that this must happen. We cannot, in general, prove from first principles that the crystalline state is the lowest free-energy state.

Crystal Structure Determination

After having described different crystal structures, the question is of course how to determine these structures in the first place. By far, the most important technique for doing this is X-ray diffraction. In fact, the importance of this technology goes far beyond solid-state physics, as it has become an essential tool for fields such as structural biology as well. There the idea is that, if you want to know the structure of a given protein, you can try to crystallize it and use the powerful methodology for structural determination by-ray diffraction. We will also use X-ray diffraction as a motivation to extend our formal description of structures a bit.

X-Ray Diffraction

X-rays interact rather weakly with matter. A description of X-ray diffraction can, therefore, be restricted to single scattering, that is, incoming X-rays get scattered not more than once (most are not scattered at all). This is called the kinematic approximation; it greatly simplifies matters and is used throughout the treatment here. In addition to this, we will assume that the X-ray source and detector are very far away from the sample so that the incoming and outgoing waves can be treated as plane waves. X-ray diffraction of crystals was discovered and described by M. von Laue in 1912. Also, in 1912, W. L. Bragg came up with an alternative description that is considerably simpler and serves as a starting point here.

Bragg Theory

Bragg treated the problem as the reflection of the incoming X-rays at a flat crystal plane. These planes could, for example, be the close-packed planes making up the fcc and hcp crystals or they could be alternating Cs and Cl planes making up the CsCl structure. At first glance, this has very little physical justification because the crystal planes are certainly not “flat” for X-rays that have a wavelength similar to the atomic spacing. Nevertheless, the description is highly successful, and we shall later see that it is actually a special case of the more complex Laue description of X-ray diffraction.

Figure 1.8 shows the geometrical considerations behind the Bragg description. A collimated beam of monochromatic X-rays hits the crystal. The intensity of diffracted X-rays are measured in the specular direction. The angle of incidence and emission is 90∘ −Θ. The condition for constructive interference is that the path length difference between the X-rays reflected from one layer and the next layer is an integer multiple of the wavelength 𝜆. In the figure, this means that 2AB = n𝜆, where AB is the distance between points A and B and n is a natural number. On the other hand, we have sin 𝜃 = AB∕d such that we arrive at the Bragg condition

n𝜆 = 2d sin 𝜃. (1.3)

It is obvious that if this condition is fulfilled for one layer and the layer below, it will also be fulfilled for any number of layers with identical spacing. In fact, the X-rays penetrate very deeply into the crystal so that thousands of layers contribute to the reflection. This results in very sharp maxima in the diffracted intensity, similar to the situation for an optical grating with many lines. The Bragg condition can obviously only be fulfilled for𝜆 < 2d, putting an upper limit on the wavelength of the X-rays that can be used for crystal structure determination.

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