Introduction to modern solid state physics

Introduction to Modern

Solid State Physics

Yuri M. Galperin

FYS 448

Department of Physics, P.O. Box 1048 Blindern, 0316 Oslo, Room 427A

Phone: +47 22 85 64 95, E-mail: iouri.galperinefys.uio.no

Contents

I Basic concepts 1

1 Geometry of Lattices ... 3

1.1 Periodicity: Crystal Structures . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 The Reciprocal Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 X-Ray Diffraction in Periodic Structures . . . . . . . . . . . . . . . . . . . 10

1.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2 Lattice Vibrations: Phonons 21

2.1 Interactions Between Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2 Lattice Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3 Quantum Mechanics of Atomic Vibrations . . . . . . . . . . . . . . . . . . 38

2.4 Phonon Dispersion Measurement . . . . . . . . . . . . . . . . . . . . . . . 43

2.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3 Electrons in a Lattice. 45

3.1 Electron in a Periodic Field . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.1.1 Electron in a Periodic Potential . . . . . . . . . . . . . . . . . . . . 46

3.2 Tight Binding Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.3 The Model of Near Free Electrons . . . . . . . . . . . . . . . . . . . . . . . 50

3.4 Main Properties of Bloch Electrons . . . . . . . . . . . . . . . . . . . . . . 52

3.4.1 Effective Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.4.2 Wannier Theorem → Effective Mass Approach . . . . . . . . . . . . 53

3.5 Electron Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.5.1 Electric current in a Bloch State. Concept of Holes. . . . . . . . . . 54

3.6 Classification of Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.7 Dynamics of Bloch Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.7.1 Classical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.7.2 Quantum Mechanics of Bloch Electron . . . . . . . . . . . . . . . . 63

3.8 Second Quantization of Bosons and Electrons . . . . . . . . . . . . . . . . 65

3.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

i

ii CONTENTS

II Normal metals and semiconductors 69

4 Statistics and Thermodynamics ... 71

4.1 Specific Heat of Crystal Lattice . . . . . . . . . . . . . . . . . . . . . . . . 71

4.2 Statistics of Electrons in Solids . . . . . . . . . . . . . . . . . . . . . . . . 75

4.3 Specific Heat of the Electron System . . . . . . . . . . . . . . . . . . . . . 80

4.4 Magnetic Properties of Electron Gas. . . . . . . . . . . . . . . . . . . . . . 81

4.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5 Summary of basic concepts 93

6 Classical dc Transport ... 97

6.1 The Boltzmann Equation for Electrons . . . . . . . . . . . . . . . . . . . . 97

6.2 Conductivity and Thermoelectric Phenomena. . . . . . . . . . . . . . . . . 101

6.3 Energy Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.4 Neutral and Ionized Impurities . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.5 Electron-Electron Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.6 Scattering by Lattice Vibrations . . . . . . . . . . . . . . . . . . . . . . . . 114

6.7 Electron-Phonon Interaction in Semiconductors . . . . . . . . . . . . . . . 125

6.8 Galvano- and Thermomagnetic .. . . . . . . . . . . . . . . . . . . . . . . . 130

6.9 Shubnikov-de Haas effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.10 Response to “slow” perturbations . . . . . . . . . . . . . . . . . . . . . . . 142

6.11 “Hot” electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6.12 Impact ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6.13 Few Words About Phonon Kinetics. . . . . . . . . . . . . . . . . . . . . . . 150

6.14 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

7 Electrodynamics of Metals 155

7.1 Skin Effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

7.2 Cyclotron Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

7.3 Time and Spatial Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . 165

7.4 ... Waves in a Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . 168

7.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

8 Acoustical Properties... 171

8.1 Landau Attenuation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

8.2 Geometric Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

8.3 Giant Quantum Oscillations. . . . . . . . . . . . . . . . . . . . . . . . . . . 174

8.4 Acoustical properties of semicondictors . . . . . . . . . . . . . . . . . . . . 175

8.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

CONTENTS iii

9 Optical Properties of Semiconductors 181

9.1 Preliminary discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

9.2 Photon-Material Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . 182

9.3 Microscopic single-electron theory . . . . . . . . . . . . . . . . . . . . . . . 189

9.4 Selection rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

9.5 Intraband Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

9.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

9.7 Excitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

9.7.1 Excitonic states in semiconductors . . . . . . . . . . . . . . . . . . 203

9.7.2 Excitonic effects in optical properties . . . . . . . . . . . . . . . . . 205

9.7.3 Excitonic states in quantum wells . . . . . . . . . . . . . . . . . . . 206

10 Doped semiconductors 211

10.1 Impurity states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

10.2 Localization of electronic states . . . . . . . . . . . . . . . . . . . . . . . . 215

10.3 Impurity band for lightly doped semiconductors. . . . . . . . . . . . . . . . 219

10.4 AC conductance due to localized states . . . . . . . . . . . . . . . . . . . . 225

10.5 Interband light absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

III Basics of quantum transport 237

11 Preliminary Concepts 239

11.1 Two-Dimensional Electron Gas . . . . . . . . . . . . . . . . . . . . . . . . 239

11.2 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

11.3 Degenerate and non-degenerate electron gas . . . . . . . . . . . . . . . . . 250

11.4 Relevant length scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

12 Ballistic Transport 255

12.1 Landauer formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

12.2 Application of Landauer formula . . . . . . . . . . . . . . . . . . . . . . . 260

12.3 Additional aspects of ballistic transport . . . . . . . . . . . . . . . . . . . . 265

12.4 e − e interaction in ballistic systems . . . . . . . . . . . . . . . . . . . . . . 266

13 Tunneling and Coulomb blockage 273

13.1 Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

13.2 Coulomb blockade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

14 Quantum Hall Effect 285

14.1 Ordinary Hall effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

14.2 Integer Quantum Hall effect - General Picture . . . . . . . . . . . . . . . . 285

14.3 Edge Channels and Adiabatic Transport . . . . . . . . . . . . . . . . . . . 289

14.4 Fractional Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . 294

iv CONTENTS

IV Superconductivity 307

15 Fundamental Properties 309

15.1 General properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

16 Properties of Type I .. 313

16.1 Thermodynamics in a Magnetic Field. . . . . . . . . . . . . . . . . . . . . 313

16.2 Penetration Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

16.3 ...Arbitrary Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318

16.4 The Nature of the Surface Energy. . . . . . . . . . . . . . . . . . . . . . . . 328

16.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

17 Magnetic Properties -Type II 331

17.1 Magnetization Curve for a Long Cylinder . . . . . . . . . . . . . . . . . . . 331

17.2 Microscopic Structure of the Mixed State . . . . . . . . . . . . . . . . . . . 335

17.3 Magnetization curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

17.4 Non-Equilibrium Properties. Pinning. . . . . . . . . . . . . . . . . . . . . . 347

17.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352

18 Microscopic Theory 353

18.1 Phonon-Mediated Attraction . . . . . . . . . . . . . . . . . . . . . . . . . . 353

18.2 Cooper Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

18.3 Energy Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

18.4 Temperature Dependence ... . . . . . . . . . . . . . . . . . . . . . . . . . . 360

18.5 Thermodynamics of a Superconductor . . . . . . . . . . . . . . . . . . . . 362

18.6 Electromagnetic Response ... . . . . . . . . . . . . . . . . . . . . . . . . . . 364

18.7 Kinetics of Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . 369

18.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376

19 Ginzburg-Landau Theory 377

19.1 Ginzburg-Landau Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 377

19.2 Applications of the GL Theory . . . . . . . . . . . . . . . . . . . . . . . . 382

19.3 N-S Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388

20 Tunnel Junction. Josephson Effect. 391

20.1 One-Particle Tunnel Current . . . . . . . . . . . . . . . . . . . . . . . . . . 391

20.2 Josephson Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395

20.3 Josephson Effect in a Magnetic Field . . . . . . . . . . . . . . . . . . . . . 397

20.4 Non-Stationary Josephson Effect . . . . . . . . . . . . . . . . . . . . . . . 402

20.5 Wave in Josephson Junctions . . . . . . . . . . . . . . . . . . . . . . . . . 405

20.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

CONTENTS v

21 Mesoscopic Superconductivity 409

21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409

21.2 Bogoliubov-de Gennes equation . . . . . . . . . . . . . . . . . . . . . . . . 410

21.3 N-S interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412

21.4 Andreev levels and Josephson effect . . . . . . . . . . . . . . . . . . . . . . 421

21.5 Superconducting nanoparticles . . . . . . . . . . . . . . . . . . . . . . . . . 425

V Appendices 431

22 Solutions of the Problems 433

A Band structure of semiconductors 451

A.1 Symmetry of the band edge states . . . . . . . . . . . . . . . . . . . . . . . 456

A.2 Modifications in heterostructures. . . . . . . . . . . . . . . . . . . . . . . . 457

A.3 Impurity states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458

B Useful Relations 465

B.1 Trigonometry Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465

B.2 Application of the Poisson summation formula . . . . . . . . . . . . . . . . 465

C Vector and Matrix Relations 467

vi CONTENTS

Part I

Basic concepts

1

Chapter 1

Geometry of Lattices and X-Ray

Diffraction

In this Chapter the general static properties of crystals, as well as possibilities to observe

crystal structures, are reviewed. We emphasize basic principles of the crystal structure

description. More detailed information can be obtained, e.g., from the books [1, 4, 5].

1.1 Periodicity: Crystal Structures

Most of solid materials possess crystalline structure that means spatial periodicity or trans￾lation symmetry. All the lattice can be obtained by repetition of a building block called

basis. We assume that there are 3 non-coplanar vectors a1, a2, and a3 that leave all the

properties of the crystal unchanged after the shift as a whole by any of those vectors. As

a result, any lattice point R0

could be obtained from another point R as

R0 = R + m1a1 + m2a2 + m3a3 (1.1)

where mi are integers. Such a lattice of building blocks is called the Bravais lattice. The

crystal structure could be understood by the combination of the propertied of the building

block (basis) and of the Bravais lattice. Note that

• There is no unique way to choose ai

. We choose a1 as shortest period of the lattice,

a2 as the shortest period not parallel to a1, a3 as the shortest period not coplanar to

a1 and a2.

• Vectors ai chosen in such a way are called primitive.

• The volume cell enclosed by the primitive vectors is called the primitive unit cell.

• The volume of the primitive cell is V0

V0 = (a1[a2a3]) (1.2)

3

4 CHAPTER 1. GEOMETRY OF LATTICES ...

The natural way to describe a crystal structure is a set of point group operations which

involve operations applied around a point of the lattice. We shall see that symmetry pro￾vide important restrictions upon vibration and electron properties (in particular, spectrum

degeneracy). Usually are discussed:

Rotation, Cn: Rotation by an angle 2π/n about the specified axis. There are restrictions

for n. Indeed, if a is the lattice constant, the quantity b = a + 2a cos φ (see Fig. 1.1)

Consequently, cos φ = i/2 where i is integer.

Figure 1.1: On the determination of rotation symmetry

Inversion, I: Transformation r → −r, fixed point is selected as origin (lack of inversion

symmetry may lead to piezoelectricity);

Reflection, σ: Reflection across a plane;

Improper Rotation, Sn: Rotation Cn, followed by reflection in the plane normal to the

rotation axis.

Examples

Now we discuss few examples of the lattices.

One-Dimensional Lattices - Chains

Figure 1.2: One dimensional lattices

1D chains are shown in Fig. 1.2. We have only 1 translation vector |a1| = a, V0 = a.

1.1. PERIODICITY: CRYSTAL STRUCTURES 5

White and black circles are the atoms of different kind. a is a primitive lattice with one

atom in a primitive cell; b and c are composite lattice with two atoms in a cell.

Two-Dimensional Lattices

The are 5 basic classes of 2D lattices (see Fig. 1.3)

Figure 1.3: The five classes of 2D lattices (from the book [4]).

6 CHAPTER 1. GEOMETRY OF LATTICES ...

Three-Dimensional Lattices

There are 14 types of lattices in 3 dimensions. Several primitive cells is shown in Fig. 1.4.

The types of lattices differ by the relations between the lengths ai and the angles αi

.

Figure 1.4: Types of 3D lattices

We will concentrate on cubic lattices which are very important for many materials.

Cubic and Hexagonal Lattices. Some primitive lattices are shown in Fig. 1.5. a,

b, end c show cubic lattices. a is the simple cubic lattice (1 atom per primitive cell),

b is the body centered cubic lattice (1/8 × 8 + 1 = 2 atoms), c is face-centered lattice

(1/8 × 8 + 1/2 × 6 = 4 atoms). The part c of the Fig. 1.5 shows hexagonal cell.

1.1. PERIODICITY: CRYSTAL STRUCTURES 7

Figure 1.5: Primitive lattices

We shall see that discrimination between simple and complex lattices is important, say,

in analysis of lattice vibrations.

The Wigner-Zeitz cell

As we have mentioned, the procedure of choose of the elementary cell is not unique and

sometimes an arbitrary cell does not reflect the symmetry of the lattice (see, e. g., Fig. 1.6,

and 1.7 where specific choices for cubic lattices are shown). There is a very convenient

Figure 1.6: Primitive vectors for bcc (left panel) and (right panel) lattices.

procedure to choose the cell which reflects the symmetry of the lattice. The procedure is

as follows:

1. Draw lines connecting a given lattice point to all neighboring points.

2. Draw bisecting lines (or planes) to the previous lines.