Brewster H.D. Solid State Physics
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282
Electronic Bands
by impurities, we know how they are affected by the ux: each ux quantum
removes an electron from the inner edge and adds an electron to the outer
edge. Then,
II
dt
=e
and
Iv
dt =
S:;
=
hi
e,
so:
2
I=~V
h '
Clearly, the key assumption is that there are no extended states at the
chemical potential in the impurity region.
If
there were - and there probably
are in samples that are too dirty to exhibit the quantum - then the above
arguments break down. Numerical studies indicate that, so long as the strength
ofthe
impurity potential is small compared to
nffi
c
'
extended states exist only
at the centre
of
the Landau band. Hence,
if
the chemical potential
is
above the
centre
of
the band, the conditions
of
our discussion are satisfied.
The other crucial assumption, emphasized by Halperin,
is
that there are
gapless states at the edges
of
the system.
In
the special setup which we assumed,
this was guaranteed because there were
no
impurities at the edges. In the integer
quantum,
these gapless states are a one-dimensional chiral Fermi liquid.
Impurities are not expected to affect this because there can
be
no backscattering
in
a totally chiral system. More general arguments, which we will mention in
the context
of
the fractional
quantum,
relate the existence
of
gapless edge
excitations to gauge invariance.
One might, at first, be left with the uneasy feeling that these gauge
invariance arguments are someho\Y too 'slick.'
To allay these worries, consider the annulus with a wedge cut out, which
is topologically equivalent to a rectangle.
In
such a case, some
of
the Hall
current will be carried by the edge states at the two cuts (i.e. the edges which
run radially at fixed azimuthal angle). However, probes which measure the
Hall voltage between the two cuts will effectively couple these two edges
leading, once again, to annular topology.
Laughlin's argument for exact quantization will apply to the fractional
quantum
if
we can show that the clean system has a gap. Then, we can argue
that for an annular setup similar to the above there are no extended states at
the chemical potential except at the edge. Then,
if
threading q ux quanta
removes p electrons from the inner edge and adds p to the outer edge, as we
p e
2
would expect at v =
pi
q , we would have
0"
xy
=
qh
.
Index
A
Acceptor
Doping
66
Alkali metals
21,22,96
Artificial atoms 178, 180, 181, 184, 188,
190, 191, 192
Atoms 164, 165, 167, 168,
170, 173, 174,
175, 176, 178, 185, 187, 188, 189,
191, 193, 194, 195, 197, 198, 199,
203,204,205,207,208,276
B
Ballistic Conductance 138
Ballistic
transport
134, 138, 141, 142
Boltzmann
equation
53, 54, 55, 56,
75,
76,
77,
78, 84, 119, 124, 125, 129,
133
Bose-Einstein 234, 246, 247
Bosons 231, 234, 247
Bragg scattering 261, 262
Bravais lattices 256, 257, 258, 259
Brillouin zones 23, 32, 114,248,273
Broken Symmetries 36, 37
c
Canonical 205, 210, 211, 232, 233
Carrier Model 121, 125, 127
Carrier
Pockets
25, 27,
129
Charging
Devices 146
Classical Regime 119
Condensed
Matter
35,
260
Conductance line 190
Coupling
170, 171, 172, 191, 195, 198,
202,203,205,207,213,220
Critical
Current
200
Critical Points 35
Crystal
momentum
22,29,50,53,54,
55, 113
Crystals 242, 248,
250, 256, 259
Current
37, 50, 51, 55, 57,
62,
75,
76,
78,79,83,84,85,91,93,94,95,97,
121, 122, 123, 124, 129, 131, 132,
133, 134, 136, 137, 138, 139, 141,
142, 143, 144, 145, 146, 148, 151,
152
Cyclotron 119,
120, 125, 127, 128, 129,
130
D
Debye theory 252, 255
Defect-Phonon 115, 116
Degenerate
165,169,170,171,172,183,
185,188,189,271,278,279
Dependence 22, 44, 54, 55, 56, 57, 60,
61,68,69,70,71,72,73,76,77,78,
79,80,81,82,83,85,87,88,89,93,
97, 105, 106, 112, 115, 116, 117,
118, 122, 132, 157, 158,
160
Derivation
77,
80, 84,
97,
120, 122, 138,
141
Dimensional systems 38,
96,
119, 133
Dimensional Transport 138
Dipole
moment
171,172,215,217,219,
221
Domains 216,217,220,221
Donor
impurity
46, 47,
67,
68
Dynamics 198, 232, 236, 245, 248, 263,
274
284
Dynamics 49,129
E
Effective mass
8,
21,
26, 27,
29, 30, 31,
33, 42, 43,
44,
46, 47,
48,
52,
61, 64,
72,73,
77,
103, 119, 120, 125, 127,
128, 129,
130
Einstein Relation 136, 137
Elastic Theory
250
Electrical 178, 276
Electrical 34, 37, 41,
50, 52, 53, 55, 56,
58,61,63,71,72,75,79,80,81,82,
83,84,87,91,96,116,117,119,137,
150, 158
Electrical conductivity 34,
41,
50, 52,
53,55,56,58,61,63,71,72,75,79,
80,81,82,87,96,116,117,119,158
Electron Dynamics
49,
50
Electron scattering 32,80,98,100,107,
110,
112, 113, 115, 117, 120
Electron waveguides 141,142
Electronic 16,
19,24,30,33,
34, 35, 39,
41,46,51,53,75,76,77,78,79,80,
81, 82, 90, 91, 92, 96, 98, 101, 112,
115, 117, 118,
120, 155, 156, 157
Electron-Phonon
101, 110, 116, 176,
195,204,205,207,264
Ellipsoidal 27,
61,
62, 63,
128, 129
Energy
bands
6,7,8,17,20,23,24,26,
30, 31, 33, 34,
35
Energy quantization 184
F
Fermi surface 21, 22, 23, 24,
25,
27, 28,
32,34,35,50,51,58,61,63,77,79,
96,
100, 112, 121, 122, 127, 129,
131,132
Fermi-Dirac distribution
53,57,59
Fermions 207, 209, 210, 213, 231, 236,
239
Flux quantization
201
Free electron 164, 188, 270, 271, 272,
273,277
Solid
State
Physics
I
Insulators 20, 33, 34, 37,
53,
75,
82,
lIS,
116,117
Integer
quantum
280, 282
Interaction 175, 176, 177, 181, 182, 183,
185, 186, 187, 188, 189, 191, 192,
203, 204, 205, 207, 208, 209, 212,
214, 216,
220, 221, 240, 241, 261,
263,264
Intrinsic semiconductors 25,32,59,63,
64,81,91
Invariance 209,245,250,251,255,279,
281,282
Ion backscattering 153
Ion
i'mplantation 151, 152, 153, 157,
158, 159, 161
Ionized
impurity
59,106,107,109,110
Ising Model 240
K
Kelvin Relations 95
L
Landau
Bands 277
Lattice structures
250, 256
Length 185,
201,
216, 217,
227, 250,
260,
265,272,279
Linear chain 242,245,247,249,255,265
Liquid crystals 38
M
Magnetic energy 217,218, 219, 222, 223
Magnetic field 164, 187, 188, 189, 192,
194, 195, 198, 199,
200, 201, 202,
214, 218, 219, 221, 230, 240, 276,
277,278,280
Magneto-transport 119,120, 124
Magnets 37,
220
Many-Particle 36, 230
Mass
theorem
42,
43,
46,
48
Mean Field Theory 240,
241
Metals
1,
8,
20,
21, 22, 32, 33, 34, 37,
52, 57,
72,
75, 76,
77,
80, 82, 83, 87,
Index
88,89,92,93,94,96,100,110,111,
112, 113, 114, 116, 117
Microcanonical 231, 232
Molecular
32,
242
N
Noble metals 22, 96
Non-degenerate 24, 26, 41,
80,
81
o
Optical
phonons
103, 105
Order
perturbation
166, 167, 168, 171,
270
p
Peltier effect 85, 93,
94
Penetration
depth
152, 156, 200
Perturbation theory 164, 165, 166, 167,
168, 169,
170, 171, 172, 213, 270,
271
Phase transitions 193, 195, 198
Phenomena
31,
34,
75,
84, 87,
133, 139,
146, 151, 152, 157
Phonon
scattering 73,
80, 82, 83, 88,
100, 101, 103, 105, 106, 110, 112,
113, 114, 115, 116, 117, 118
Phonon-boundary 116, 115
Phonon-phonon
82,97, 116, 118
Phonons:
Continuum
250
Physics
164, 169, 189, 190, 191, 207,
255,256,264
Planck Distribution 234, 235
Polymers 38, 152, 158, 159
Polyvalent Metals
24
Q
Quantization 180, 181, 184, 201, 205,
278,281,282
Quantum
dots 133, 134, 136
Quantum
effects 119, 133, 134
Quantum
mechanics 35,164,173,191,
263,278
Quasi-Classical 49,
50
R
Radiation damage 153, 157
Realistic
Phonon 255
285
Reciprocal lattice 249,
250, 258, 259,
260,261,262,264,270,271
Reduced Dimensions 133
s
Scattering Equations 153
Scattering mechanisms
59,73,82,99,
106, 107, 110, 113, 114, 115
Scattering processes
73,
80,
82,
98,
100,
113, 114, 117, 118
Screening effects 107,
109, 110, 114
Seebeck effect
85, 86,
93,
94
Semiconductor
7,
21,
25,
26, 27, 28, 30,
31,32,33,34,44,45,46,47,48,52,
53,58,59,61,63,64,65,66,67,68,
69,71,72,73,75,76,77,80,81,82,
83,86,87,89,90,91,94,95,97,100,
101,
102, 103, 106, 107, 110, 114,
115, 117, 119, 121, 122, 124, 129,
134,
150, 151, 152, 153, 157, 158,
159
Semimetals
21,32,33
Silicon 16, 26, 30,
34,
45,
48, 102, 105,
156,157
Simple Energetics 242
Single electron 146, 148, 149, 178, 181,
187, 188, 263, 278
Solid state
I,
37,
151, 164, 169, 207
Spin 189, 195,208, 230,237,240,241,
265,275,278
States of Matter 35,38
Statistical Mechanics 224,
231
Statistics 193, 234
Superconductivity 195,
202, 203, 207,
209,212,268
Superfluid Density
200
T
Temperature 181, 182, 183, 184, 193,
196,203, 212, 214, 220, 232, 233,
286
240, 242, 247, 248, 254, 255, 260,
265,273,274,275,276
Theory of Electrical
50
Thermal conductivity 53,75,76,
77, 78,
79,
80,81,
82,
83,
85,
87,
91, 92, 96,
115, 116, 117, 118
Thermal
rr-
:sport
75, 79,
80,
lIS,
117
Thermodynamic 221, 235, 279
Thermoelectric
75,
77,
84, 85,
86, 87,
89,91,92,94,95,96
Thermopower
79,
85,86,87,90,91,92,
93, 95, 96,
97
Thomson effect
85,
94,
95
Tight
binding
I,
8,
9,
10,
12,
13, IS, 16,
17,
18,23,33
Transport
I,
8,23,25,27,32,34,35,36,
41,52,53,66,75,77,79,80,84,87,
96,98,100,113,114,115,117,118,
119, 120, 122, 124, 125, 133, 134,
Solid
State
Physics
136, 137, 138, 140,
141
U
Unit cell 258, 259, 261, 272, 276
V
Velocity
8,
40, 49, 55,
62,
76,
82,
100,
106, 112, 116, 118, 122, 123, 137,
139, 141, 143
w
Wavepackets 39,
41
X
X-ray scattering 36
z
Zero Gap
31