Quinn J.J., Yi K.-S. Solid State Physics: Principles and Modern Applications
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Uncorrected Proof
BookID 160928 ChapID FM Proof# 1 - 29/07/09
XII Contents
6.4 SemiclassicalApproximationforBlochElectrons...........168 167
6.4.1 EffectiveMass.................................170 168
6.4.2 ConceptofaHole..............................171 169
6.4.3 EffectiveHamiltonian ofBlochElectron...........172 170
Problems ...................................................174 171
7 Semiconductors ............................................179 172
7.1 General Properties of Semiconducting Material . . . . . . . . . . . . 179 173
7.2 Typical Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 174
7.3 Temperature Dependence of the Carrier Concentration . . . . . 182 175
7.3.1 CarrierConcentration:Intrinsic Case .............184 176
7.4 Donor and Acceptor Impurities . . . . . . . . . . . . . . . . . . . . . . . . . . 185 177
7.4.1 Populationof Donor Levels......................186 178
7.4.2 Thermal Equilibrium in a Doped Semiconductor . . . 187 179
7.4.3 High-ImpurityConcentration....................189 180
7.5 p–n Junction ..........................................189 181
7.5.1 SemiclassicalModel ............................190 182
7.5.2 Rectificationofap–n Junction ..................193 183
7.5.3 Tunnel Diode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 184
7.6 SurfaceSpaceChargeLayers ............................195 185
7.6.1 Superlattices ..................................200 186
7.6.2 Quantum Wells................................200 187
7.6.3 Modulation Doping ............................201 188
7.6.4 Minibands ....................................201 189
7.7 ElectronsinaMagneticField............................203 190
7.7.1 Quantum HallEffect ...........................205 191
7.8 Amorphous Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 192
7.8.1 TypesofDisorder..............................207 193
7.8.2 AndersonModel ...............................207 194
7.8.3 ImpurityBands................................208 195
7.8.4 Density ofStates...............................208 196
Problems ...................................................210 197
8 Dielectric Properties of Solids .............................215 198
8.1 Review of Some Ideas of Electricity and Magnetism . . . . . . . . 215 199
8.2 Dipole MomentPerUnitVolume ........................216 200
8.3 Atomic Polarizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 201
8.4 LocalFieldinaSolid...................................217 202
8.5 MacroscopicField......................................218 203
8.5.1 DepolarizationFactor ..........................218 204
8.6 LorentzField..........................................219 205
8.7 Clausius–MossottiRelation..............................221 206
8.8 Polarizability and Dielectric Functions 207
ofSomeSimpleSystems ................................222 208
8.8.1 Evaluation of the Dipole Polarizability . . . . . . . . . . . . 222 209
8.8.2 Polarizability of Bound Electrons . . . . . . . . . . . . . . . . 224 210
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Contents XIII
8.8.3 Dielectric Functionof aMetal ...................224 211
8.8.4 Dielectric Function of a Polar Crystal . . . . . . . . . . . . 225 212
8.9 OpticalProperties .....................................229 213
8.9.1 WaveEquation ................................229 214
8.10 Bulk Modes ...........................................230 215
8.10.1 LongitudinalModes ............................231 216
8.10.2 TransverseModes..............................232 217
8.11 Reflectivity ofaSolid...................................235 218
8.11.1 OpticalConstants..............................236 219
8.11.2 SkinEffect....................................236 220
8.12 SurfaceWaves.........................................237 221
8.12.1 Plasmon......................................239 222
8.12.2 SurfacePhonon–Polariton.......................240 223
Problems ...................................................243 224
9 Magnetism in Solids .......................................247 225
9.1 ReviewofSomeElectromagnetism .......................247 226
9.1.1 MagneticMomentandTorque...................247 227
9.1.2 Vector PotentialofaMagneticDipole ............248 228
9.2 MagneticMomentofanAtom...........................250 229
9.2.1 OrbitalMagneticMoment.......................250 230
9.2.2 SpinMagneticMoment .........................250 231
9.2.3 Total Angular Momentum and Total Magnetic 232
Moment ......................................251 233
9.2.4 Hund’s Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 234
9.3 Paramagnetism and Diamagnetism of an Atom . . . . . . . . . . . . 252 235
9.4 ParamagnetismofAtoms ...............................255 236
9.5 Pauli SpinParamagnetismofMetals .....................257 237
9.6 Diamagnetismof Metals ................................259 238
9.7 deHaas–vanAlphenEffect..............................262 239
9.8 Cooling by Adiabatic Demagnetization of a Paramagnetic 240
Salt ..................................................265 241
9.9 Ferromagnetism .......................................266 242
Problems ...................................................268 243
Part II Advanced Topics in Solid-State Physics 244
10 Magnetic Ordering and Spin Waves ........................275 245
10.1 Ferromagnetism .......................................275 246
10.1.1 Heisenberg ExchangeInteraction.................275 247
10.1.2 Spontaneous Magnetization .....................277 248
10.1.3 DomainStructure..............................279 249
10.1.4 DomainWall..................................280 250
10.1.5 AnisotropyEnergy.............................281 251
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XIV Contents
10.2 Antiferromagnetism ....................................282 252
10.3 Ferrimagnetism........................................283 253
10.4 Zero-TemperatureHeisenbergFerromagnet................283 254
10.5 Zero-TemperatureHeisenbergAntiferromagnet ............286 255
10.6 SpinWavesinFerromagnet .............................287 256
10.6.1 Holstein–Primakoff Transformation . . . . . . . . . . . . . . . 287 257
10.6.2 Dispersion Relation forMagnons.................291 258
10.6.3 Magnon–MagnonInteractions ...................291 259
10.6.4 MagnonHeatCapacity .........................292 260
10.6.5 Magnetization .................................293 261
10.6.6 Experiments RevealingMagnons.................295 262
10.6.7 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 263
10.7 SpinWavesinAntiferromagnets .........................296 264
10.7.1 GroundStateEnergy...........................299 265
10.7.2 Zero Point Sublattice Magnetization . . . . . . . . . . . . . . 300 266
10.7.3 Finite Temperature Sublattice Magnetization . . . . . . 301 267
10.7.4 Heat Capacity due to Antiferromagnetic Magnons . . 303 268
10.8 ExchangeInteractions ..................................303 269
10.9 ItinerantFerromagnetism ...............................304 270
10.9.1 StonerModel..................................305 271
10.9.2 StonerExcitations .............................305 272
10.10 PhaseTransition.......................................306 273
Problems ...................................................308 274
11 Many Body Interactions – Introduction ....................311 275
11.1 Second Quantization ...................................311 276
11.2 Hartree–FockApproximation............................314 277
11.2.1 Ferromagnetism of a degenerate electron gas 278
inHartree–FockApproximation..................316 279
11.3 SpinDensityWaves ....................................318 280
11.3.1 ComparisonwithReality........................326 281
11.4 Correlation Effects–Divergence of Perturbation Theory . . . . . 326 282
11.5 LinearResponseTheory ................................328 283
11.5.1 DensityMatrix ................................328 284
11.5.2 Propertiesof DensityMatrix ....................329 285
11.5.3 ChangeofRepresentation.......................329 286
11.5.4 EquationofMotion ofDensityMatrix ............331 287
11.5.5 Single Particle Density Matrix of a Fermi Gas . . . . . 332 288
11.5.6 LinearResponseTheory ........................332 289
11.5.7 GaugeInvariance ..............................335 290
11.6 Lindhard Dielectric Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 291
11.6.1 LongitudinalDielectric Constant.................340 292
11.6.2 Kramers–KronigRelation .......................343 293
11.7 Effect of Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 294
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Contents XV
11.8 Screening .............................................349 295
11.8.1 Friedel Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 296
11.8.2 KohnEffect ...................................353 297
Problems ...................................................355 298
12 Many Body Interactions: Green’s Function Method ........361 299
12.1 Formulation...........................................361 300
12.1.1 Schr¨odingerEquation...........................362 301
12.1.2 InteractionRepresentation ......................363 302
12.2 Adiabatic Approximation ...............................366 303
12.3 Green’sFunction.......................................368 304
12.3.1 Averages of Time-Ordered Products 305
ofOperators ..................................368 306
12.3.2 Wick’sTheorem ...............................369 307
12.3.3 LinkedClusters................................372 308
12.4 Dyson’sEquations .....................................372 309
12.5 Green’s Function Approach to the Electron–Phonon 310
Interaction............................................374 311
12.6 ElectronSelfEnergy ...................................382 312
12.7 Quasiparticle Interactions and Fermi Liquid Theory . . . . . . . . 383 313
Problems ...................................................385 314
13 Semiclassical Theory of Electrons ..........................391 315
13.1 Bloch Electronsina dc MagneticField ...................391 316
13.1.1 Energy Levels of Bloch Electrons in a Magnetic 317
Field .........................................392 318
13.1.2 QuantizationofEnergy.........................394 319
13.1.3 CyclotronEffectiveMass........................395 320
13.1.4 Velocity Parallel to B ..........................395 321
13.2 Magnetoresistance .....................................396 322
13.3 Two-BandModelandMagnetoresistance..................397 323
13.4 Magnetoconductivity of Metals . . . . . . . . . . . . . . . . . . . . . . . . . . 401 324
13.4.1 FreeElectronModel............................407 325
13.4.2 Propagation Parallel to B
0
......................410 326
13.4.3 Propagation Perpendicular to B
0
................411 327
13.4.4 Local vs. Nonlocal Conduction . . . . . . . . . . . . . . . . . . . 411 328
13.5 Quantum Theory of Magnetoconductivity of an Electron Gas 413 329
13.5.1 Propagation Perpendicular to B
0
................415 330
Problems ...................................................418 331
14 Electrodynamics of Metals .................................423 332
14.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 333
14.2 Skin Effect in the Absence of a dc Magnetic Field . . . . . . . . . . 424 334
14.3 Azbel–KanerCyclotronResonance .......................427 335
14.4 Azbel–KanerEffect ....................................430 336
14.5 MagnetoplasmaWaves..................................431 337
14.6 DiscussionoftheNonlocalTheory .......................434 338
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XVI Contents
14.7 CyclotronWaves.......................................435 339
14.8 SurfaceWaves.........................................437 340
14.9 MagnetoplasmaSurfaceWaves...........................440 341
14.10 Propagation of Acoustic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 441 342
14.10.1 Propagation Parallel to B
0
......................445 343
14.10.2 Helicon–PhononInteraction .....................446 344
14.10.3 Propagation Perpendicular to B
0
................447 345
Problems ...................................................450 346
15 Superconductivity .........................................455 347
15.1 Some Phenomenological Observations of Superconductors . . . 455 348
15.2 LondonTheory........................................459 349
15.3 MicroscopicTheory–AnIntroduction .....................462 350
15.3.1 Electron–PhononInteraction ....................462 351
15.3.2 CooperPair...................................464 352
15.4 TheBCSGroundState.................................467 353
15.4.1 Bogoliubov–ValatinTransformation ..............469 354
15.4.2 CondensationEnergy...........................472 355
15.5 ExcitedStates.........................................472 356
15.6 Type I and Type II Superconductors . . . . . . . . . . . . . . . . . . . . . 475 357
Problems ...................................................479 358
16 The Fractional Quantum Hall Effect: The Paradigm 359
for Strongly Interacting Systems ...........................483 360
16.1 Electrons Confined to a Two-Dimensional Surface in a 361
PerpendicularMagneticField............................483 362
16.2 IntegralQuantum HallEffect............................484 363
16.3 FractionalQuantumHallEffect..........................485 364
16.4 NumericalStudies......................................486 365
16.5 Statistics of Identical Particles in Two Dimension . . . . . . . . . . 490 366
16.6 Chern–SimonsGaugeField..............................492 367
16.7 CompositeFermionPicture .............................494 368
16.8 Fermi LiquidPicture ...................................499 369
16.9 Pseudopotentials.......................................500 370
16.10 AngularMomentumEigenstates .........................504 371
16.11 CorrelationsinQuantumHallStates .....................506 372
Problems ...................................................510 373
A Operator Method for the Harmonic Oscillator Problem ....515 374
Problems ...................................................518 375
B Neutron Scattering ........................................519 376
References .....................................................523 377
Index ..........................................................529 378
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BookID 160928 ChapID 01 Proof# 1 - 29/07/09
1 3
Crystal Structures 4
1.1 Crystal Structure and Symmetry Groups 5
Although everyone has an intuitive idea of what a solid is, we will consider 6
(in this book) only materials with a well-defined crystal structure. What we 7
mean by a well-defined crystal structure is an arrangement of atoms in a lattice 8
such that the atomic arrangement looks absolutely identical when viewed from 9
two different points that are separated by a lattice translation vector.Afew 10
definitions are useful. 11
Lattice 12
A lattice is an infinite array of points obtained from three primitive transla- 13
tion vectors a
1
, a
2
, a
3
. Any point on the lattice is given by 14
n = n
1
a
1
+ n
2
a
2
+ n
3
a
3
. (1.1)
15
Translation Vector 16
Any pair of lattice points can be connected by a vector of the form 17
T
n
1
n
2
n
3
= n
1
a
1
+ n
2
a
2
+ n
3
a
3
. (1.2)
18
The set of translation vectors form a group called the translation group of the 19
lattice. 20
Group 21
A set of elements of any kind with a set of operations, by which any two 22
elements may be combined into a third, satisfying the following requirements 23
is called a group: 24
• The product (under group multiplication) of two elements of the group 25
belongs to the group. 26
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BookID 160928 ChapID 01 Proof# 1 - 29/07/09
4 1 Crystal Structures
Fig. 1.1. Translation operations in a two-dimensional lattice
• The associative law holds for group multiplication. 27
• The identity element belongs to the group. 28
• Every element in the group has an inverse which belongs to the group. 29
Translation Group 30
The set of translations through any translation vector T
n
1
n
2
n
3
forms a group. 31
Group multiplication consists in simply performing the translation operations 32
consecutively. For example, as is shown in Fig. 1.1, we have T
13
= T
03
+ 33
T
10
. For the simple translation group the operations commute, i.e., T
ij
T
kl
= 34
T
kl
T
ij
for, every pair of translation vectors. This property makes the group 35
an Abelian group. 36
Point Group 37
There are other symmetry operations which leave the lattice unchanged. These 38
are rotations, reflections,andtheinversion operations. These operations form 39
the point group of the lattice. As an example, consider the two-dimensional 40
square lattice (Fig. 1.2). The following operations (performed about any lattice 41
point) leave the lattice unchanged. 42
• E:identity 43
• R
1
,R
3
: rotations by ±90
◦
44
• R
2
: rotation by 180
◦
45
• m
x
,m
y
: reflections about x-axis and y-axis, respectively 46
• m
+
,m
−
: reflections about the lines x = ±y 47
The multiplication table for this point group is given in Table 1.1. The oper- 48
ations in the first column are the first (right) operations, such as m
+
in 49
R
1
m
+
= m
y
, and the operations listed in the first row are the second (left) 50
operations, such as R
1
in R
1
m
+
= m
y
. 51
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1.1 Crystal Structure and Symmetry Groups 5
Fig. 1.2. The two-dimensional square lattice
t1.1 Tabl e 1.1. Multiplication table for the group 4 mm. The first (right) operations,
such as m
+
in R
1
m
+
= m
y
, are listed in the first column, and the second (left)
operations, such as R
1
in R
1
m
+
= m
y
, are listed in the first row
t1.2 Operation ER
1
R
2
R
3
m
x
m
y
m
+
m
−
t1.3 E
−1
= EER
1
R
2
R
3
m
x
m
y
m
+
m
−
t1.4 R
−1
1
= R
3
R
3
ER
1
R
2
m
+
m
−
m
y
m
x
t1.5 R
−1
2
= R
2
R
2
R
3
ER
1
m
y
m
x
m
−
m
+
t1.6 R
−1
3
= R
1
R
1
R
2
R
3
Em
−
m
+
m
x
m
y
t1.7 m
−1
x
= m
x
m
x
m
+
m
y
m
−
ER
2
R
1
R
3
t1.8 m
−1
y
= m
y
m
y
m
−
m
x
m
+
R
2
ER
3
R
1
t1.9 m
−1
+
= m
+
m
+
m
y
m
−
m
x
R
3
R
1
ER
2
t1.10 m
−1
−
= m
−
m
−
m
x
m
+
m
y
R
1
R
3
R
2
E
12
34
Fig. 1.3. Identity operation on a two-dimensional square
The multiplication table can be obtained as follows: 52
• Label the corners of the square (Fig. 1.3). 53
• Operating with a symmetry operation simply reorders the labeling. For 54
example, see Fig. 1.4 for symmetry operations of m
+
,R
1
, and m
x
. 55
Therefore, R
1
m
+
= m
y
. One can do exactly the same for all other prod- 56
ucts, for example, such as m
y
R
1
= m
+
.Itisalsoveryusefultonotewhathap- 57
pens to a point (x, y) under the operations of the point group (see Table 1.2). 58
Note that under every group operation x →±x or ±y and y →±y or ±x. 59
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BookID 160928 ChapID 01 Proof# 1 - 29/07/09
6 1 Crystal Structures
+
1
y
π
2
Fig. 1.4. Point symmetry operations on a two-dimensional square
t2.1 Tabl e 1.2. Point group operations on a point (x, y)
t2.2 Operation ER
1
R
2
R
3
m
x
m
y
m
+
m
−
t2.3 xxy−x −yx−xy −y
t2.4 yy−x −yx−yy x−x
Fig. 1.5. The two-dimensional rectangular lattice
The point group of the two-dimensional square lattice is called 4 mm. The 60
notation denotes the fact that it contains a fourfold axis of rotation and two 61
mirror planes (m
x
and m
y
); the m
+
and m
−
planes are required by the exis- 62
tence of the other operations. Another simple example is the symmerty group 63
of a two-dimensional rectangular lattice (Fig. 1.5). The symmetry operations 64
are E,R
2
,m
x
,m
y
, and the multiplication table is easily obtained from that 65
of 4 mm. This point group is called 2 mm, and it is a subgroup of 4 mm. 66
Allowed Rotations 67
Because of the requirement of translational invariance under operations of the 68
translation group, the allowed rotations of the point group are restricted to 69
certain angles. Consider a rotation through an angle φ about an axis through 70
some lattice point (Fig. 1.6). If A and B are lattice points separated by a 71
primitive translation a
1
,thenA
(and B
) must be a lattice point obtained 72
by a rotation through angle φ about B (or −φ about A). Since A
and B
are 73
lattice points, the vector B
A
must be a translation vector. Therefore, 74
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1.1 Crystal Structure and Symmetry Groups 7
φ−φ
Fig. 1.6. Allowed rotations of angle φ about an axis passing through some lattice
points A and B consistent with translational symmetry
t3.1 Tabl e 1. 3. Allowed rotations of the point group
t3.2 p cos φφn(= |2π/φ|)
t3.3 −110or2π 1
t3.4 0
1
2
±
2π
6
6
t3.5 1 0 ±
2π
4
4
t3.6 2 −
1
2
±
2π
3
3
t3.7 3 −1 ±
2π
2
2
|B
A
| = pa
1
, (1.3)
where p is an integer. But |B
A
| = a
1
+2a
1
sin
φ −
π
2
= a
1
− 2a
1
cos φ. 75
Solving for cos φ gives 76
cos φ =
1 − p
2
. (1.4)
Because −1 ≤ cos φ ≤ 1, we see that p must have only the integral values -1,
77
0, 1, 2, 3. This gives for the possible values of φ listed in Table 1.3. 78
Although only rotations of 60
◦
, 90
◦
, 120
◦
, 180
◦
, and 360
◦
are consistent 79
with translational symmetry, rotations through other angles are obtained 80
in quasicrystals (e.g., fivefold rotations). The subject of quasicrystals, which 81
do not have translational symmetry under the operations of the translation 82
group, is an interesting modern topic in solid state physics which we will not 83
discuss in this book. 84
Bravais Lattice 85
If there is only one atom associated with each lattice point, the lattice is called 86
Bravais lattice. If there is more than one atom associated with each lattice 87
point, the lattice is called alatticewithabasis. One atom can be considered 88
to be located at the lattice point. For a lattice with a basis it is necessary to 89
give the locations (or basis vectors) for the additional atoms associated with 90
the lattice point. 91