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 a
1
, a
2
, and a
3
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 R
could be obtained from another point R as
R
= R + m
1
a
1
+ m
2
a
2
+ m
3
a
3
(1.1)
where m
i
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 a
i
. We choose a
1
as shortest period of the lattice,
a
2
as the shortest period not parallel to a
1
, a
3
as the shortest period not coplanar to
a
1
and a
2
.
• Vectors a
i
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 V
0
V
0
= (a
1
[a
2
a
3
]) (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, C
n
: 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, S
n
: Rotation C
n
, 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 |a
1
| = a, V
0
= 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 a
i
and the angles α
i
.
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.
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