Quinn J.J., Yi K.-S. Solid State Physics: Principles and Modern Applications
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Uncorrected Proof
BookID 160928 ChapID 13 Proof# 1 - 29/07/09
13.5 Quantum Theory of Magnetoconductivity of an Electron Gas 415
where ξ =
¯hq
2
2mω
c
and L
α
n
(ξ) is the associated Laguerre polynomial of order n. 373
For n
<nwe have f
n
n
(q)=(−1)
n−n
f
nn
(q). Some useful facts about the 374
functions f
n
n
are 375
f
n
n
(q)=i
n−n
∞
−∞
dx e
iqx
u
n
(x)u
n
(x),
f
n
n
(−q)=f
nn
(q)=(−1)
n
−n
f
n
n
(q),
∞
n=0
f
2
n
n
(q)=1,
∞
n
=0
(n
− n)f
2
n
n
(q)=ξ, (13.146)
∂f
n
n
∂q
=
¯h
2mω
c
1/2
X
(−)
n
n
(q),
(n
− n −ξ)f
n
n
(q)=ξ
1/2
X
(+)
n
n
(q).
13.5.1 Propagation Perpendicular to B
0
376
For the case of q =(0,q,0) we find that 377
j
x
(q, ω)=σ
xx
E
x
(q, ω)+σ
xy
E
y
(q, ω),
j
y
(q, ω)=σ
yx
E
x
(q, ω)+σ
yy
E
y
(q, ω),
j
z
(q, ω)=σ
zz
E
z
(q, ω).
(13.147)
The nonvanishing components of σ
can be evaluated and they are written as 378
σ
xx
=
ω
2
p
4πiω
⎡
⎣
1 −
2mω
c
¯h
1
N
nk
y
k
z
α
f
0
(ε
nk
z
)
∂f
n+α,n
∂q
2
α
α
2
− (ω/ω
c
)
2
⎤
⎦
,
σ
yy
=
imω
2
p
ω
2π¯hω
c
1
N
nk
y
k
z
α
f
0
(ε
nk
z
)f
2
n+α,n
α
α
2
− (ω/ω
c
)
2
, (13.148)
σ
xy
= −σ
yx
=
iω
c
2ωq
∂(q
2
σ
yy
)
∂q
,
σ
zz
=
ω
2
p
4πiω
⎡
⎣
1 −
2¯h
mω
c
1
N
nk
y
k
z
α
f
0
(ε
nk
z
)k
2
z
f
2
n+α,n
α
α
2
− (ω/ω
c
)
2
⎤
⎦
.
In these equations, the sum on α is to be performed from −n to ∞ (because
379
0 ≤ n
= n + α ≤∞). The summations over n, k
y
,k
z
extend over all values of 380
the quantum numbers for which ε
nk
z
≤ ζ,whereζ is the chemical potential 381
of the electron gas in the field B
0
. This restriction is indicated by a prime 382
following the summation sign. 383
Uncorrected Proof
BookID 160928 ChapID 13 Proof# 1 - 29/07/09
416 13 Semiclassical Theory of Electrons
The semiclassical limit can be obtained by replacing the sum over n by an 384
integral. Remember that in general we can write 385
nk
y
k
z
=⇒ 2
L
2π
2
n
mω
c
L
¯h
0
dk
y
dk
z
=
mω
c
¯h
Ω
2π
2
dk
z
n
, (13.149)
whereΩisthevolumeofthesampleΩ=L
3
. We define n
0
=
ρ
¯hω
c
−
1
2
and 386
let n = n
0
sin
2
θ. For zero temperature, we can integrate over θ from θ =0to 387
θ =
π
2
instead of summing over n.Forn
0
1 it is not hard to see that the 388
main contribution to the integrals comes from rather large values of n.For 389
large n, we can approximate 390
f
n+α,n
J
α
(4n +2α +2)
1/2
ξ
1/2
J
α
(w sin θ), (13.150)
where w =
qv
F
ω
c
. By substituting into the expressions for the components of σ 391
we obtain 392
σ
xx
=
3α
4πiω
∞
α=−∞
s
α
(w)
1+αω
c
/ω
, (13.151)
where
393
s
α
(w)=
π/2
0
dθ sin
3
θ [J
α
(w sin θ)]
2
. (13.152)
Equation (13.151) is the semiclassical expression we already obtained in
394
(13.122) in the collisionless limit. 395
The quantum mechanical conductivity tensor can be written as the sum 396
of a semiclassical term and a quantum oscillatory part 397
σ(q, ω)=σ
SC
(q, ω)+σ
QO
(q, ω), (13.153)
398
where the semiclassical part σ
SC
has been given earlier. As an example of the 399
quantum oscillatory part we give, without derivation, one example 400
σ
QO
zz
=
3
2
δ
2
ω
2
p
4πiω
1+3
ω
2
ω
2
c
∞
α=−∞
1
α
2
− (ω/ω
c
)
2
1+w
∂
∂w
c
α
(w)
.
(13.154)
Here, δ
2
is a quantum oscillatory function of the de Haas–van Alphen type 401
and is given by 402
δ
2
= π
k
B
T
ζ
0
1
¯hω
c
2ζ
0
∞
ν=1
(−1)
ν
ν
−1/2
sin(
2πνζ
0
¯hω
c
−
π
4
)
sinh(
2π
2
νk
B
T
¯hω
c
)
. (13.155)
The function c
α
(w), with w =
qv
F
ω
c
, was defined by (13.122) in the discussion 403
of the semiclassical conductivity. If k
B
T becomes large compared to ¯hω
c
,the 404
amplitude of the quantum oscillations becomes negligibly small and σ reduces 405
Uncorrected Proof
BookID 160928 ChapID 13 Proof# 1 - 29/07/09
13.5 Quantum Theory of Magnetoconductivity of an Electron Gas 417
to the semiclassical result σ
SC
given previously. What the quantum mechanical 406
conductivity tensor contains but what the semiclassical one does not is the 407
quantum structure of the energy levels. This, of course, determines all the 408
quantum effects like 409
1. de Haas–van Alphen oscillations in the magnetism 410
2. Shubnikov–de Haas oscillations in the resistivity 411
3. Quantum oscillations in acoustic attenuation, etc. 412
Uncorrected Proof
BookID 160928 ChapID 13 Proof# 1 - 29/07/09
418 13 Semiclassical Theory of Electrons
Problems 413
13.1. Take direction of current flow to make an angle θ with x axis as is shown 414
in Fig. 13.8. First, transform to x
− y
frame. Then, put j
y
=0,andcheck 415
for what angles θ the magnetoresistance fails to saturate. 416
13.2. Consider a band for a simple cubic structure given by E(k)=E
0
[cos 417
(k
x
a)+cos(k
y
a)+cos(k
z
a)], where a is the lattice constant. Let an electron at 418
rest (k =0)att = 0 feel a uniform external electric field E which is constant 419
in time. 420
(a) Find the real space trajectory [x(t),y(t),z(t)]. 421
(b) Sketch the trajectory for E in a [120] direction. 422
13.3. Consider an electron in a state with a linear energy dispersion given 423
by E = ±¯hv
F
|k|,wherek is a two-dimensional wave vector. (It occurs in the 424
low-energy states in a graphene – a single layer of graphite.) 425
(a) When a dc magnetic field B is applied perpendicular to the graphene 426
layer, write down the area S(ε)andsketchS(ε
n
) for various values 427
of n. 428
(b) Solve for the quantized energies ε
n
and plot the resulting ε
n
for 429
−5¯hω
c
≤ ε
n
≤ 5¯hω
c
. 430
(c) What can you say about the effective mass of the particle in a graphene 431
subject to the magnetic field B? 432
13.4. The energy of an electron in a particular band of a solid is given by 433
ε(k
x
,k
y
,k
z
)=
¯h
2
k
2
x
2m
x
+
¯h
2
k
2
y
2m
y
,
where −
π
a
<k
i
<
π
a
is the first Brillouin zone of a simple cubic lattice. 434
(a) Determine v
i
(k)fori = x, y,andz. 435
(b) Show that ¯hk
i
(t)=
√
2m
i
ε cos ω
c
t where i = x and y foradcmagnetic 436
field B
0
in the z-direction. 437
(c) Determine ω
c
in terms of m
i
, B
0
,etc. 438
Fig. 13.8. A simple geometry of current flow
Uncorrected Proof
BookID 160928 ChapID 13 Proof# 1 - 29/07/09
13.5 Quantum Theory of Magnetoconductivity of an Electron Gas 419
Fig. 13.9. A constant energy surface ε(k) in a two-dimensional system
13.5. Consider two-dimensional electrons with a linear dispersion given by 439
E =¯hv
F
|k|,wherek is a two-dimensional wave vector. Now apply a dc 440
magnetic field B perpendicular to the system. We shall assume that τ is 441
constant. 442
1. Write down the v(ε
F
,s) and the periodic part of the position vector 443
R
p
(ε, s). 444
2. Evaluate the Fourier coefficients v
n
(ε), and discuss the conductivity tensor 445
σ defined by j = σ · E. 446
13.6. Consider an electron in a two-dimensional system subject to a dc mag- 447
netic field B perpendicular to the system. The constant energy surface of the 448
particle is shown in in Fig. 13.9. 449
(a) Sketch the orbit of the particle in real space. 450
(b) Sketch the velocity v
y
(t) as a function of t. 451
Uncorrected Proof
BookID 160928 ChapID 13 Proof# 1 - 29/07/09
420 13 Semiclassical Theory of Electrons
Summary 452
In this chapter, we study behaviors of Bloch electrons in the presence of 453
a dc magnetic field. Energy levels and possible trajectories of electrons are 454
discussed, and simple two band model of magnetoresistance is illustrated 455
including the effect of collisions. General expression of semiclassical mag- 456
netoconductivity tensor is derived by solving the Boltzmann equation of 457
the distribution function, and the results are applied to the case of free 458
electrons. The relationship between the local and nonlocal descriptions are 459
discussed. Finally, quantum mechanical theory of magnetoconductivity ten- 460
sor is described and quantum oscillatory behavior in magnetoconductivity of 461
Bloch electrons is compared with its semiclassical counterpart. 462
In the presence of an electric field E and a dc magnetic field B(= ∇×A), 463
an effective Hamiltonian is given by H = ε(
p
¯h
+
e
¯hc
A) − eφ, where ε(k)isthe 464
energy as a function of k in the absence of B. The equation of motion of a 465
Blochelectronink-space takes the form 466
¯h
˙
k = −eE −
e
c
v × B.
Here, v =
1
¯h
∇
k
ε(k) is the velocity of the Bloch electron whose energy ε(k)is 467
an arbitrary function of wave vector k. The orbit in real space will be exactly 468
the same shape as the orbit in k-space except that it is rotated by 90
◦
and 469
scaled by a factor
eB
¯hc
: k
⊥
=
eB
¯hc
× r
⊥
. The factor
eB
¯hc
is l
−2
0
,wherel
0
is the 470
magnetic length. The orbit of a particle in k-space is the intersection of a 471
constant energy surface ε(k)=ε and a plane of constant k
z
: 472
dε
dt
= ∇
k
ε ·
dk
dt
=¯hv ·
−
e
¯hc
v × B
=0.
The area of the orbit A(ε, k
z
) in real space is proportional to the area S(ε, k
z
) 473
of the orbit in k-space: S(ε, k
z
)=
eB
¯hc
2
A(ε, k
z
). The area S(ε, k
z
) is quan- 474
tized by S(ε, k
z
)=
2πeB
¯hc
(n + γ) and the cyclotron effective mass is given by 475
m
∗
(ε, k
z
)=
¯h
2
2π
∂S(ε,k
z
)
∂ε
. The Bloch electron velocity parallel to the magnetic 476
field becomes 477
v
z
(ε, k
z
)=−
¯h
2πm
∗
(ε, k
z
)
∂S(ε, k
z
)
∂k
z
.
The transverse magnetoresistance is defined by
R(B
z
)−R(0)
R(0)
=ΔR(B
z
). The 478
simple free electron model gives ΔR(B
z
) = 0, which is different from the 479
experimental results. 480
The current density is given by j(r,t)=
2
(2π)
3
"
(−e)vf
1
d
3
k. In the pres- 481
ence of a uniform dc magnetic field B
0
, the semiclassical magnetoconductivity 482
of an electron gas is written as 483
σ =
e
2
2π
2
¯h
2
τ(ε
F
)
F.S.
dk
z
m
∗
(k
z
)
∞
n=−∞
v
n
(ε
F
,k
z
)v
∗
n
(ε
F
,k
z
)
1+iτ(ε
F
)[ω − q · v
s
− nω
c
(ε
F
,k
z
)]
,
Uncorrected Proof
BookID 160928 ChapID 13 Proof# 1 - 29/07/09
13.5 Quantum Theory of Magnetoconductivity of an Electron Gas 421
where v
n
(ε, k
z
) is defined by 484
v
n
(ε, k
z
)=
ω
c
(ε, k
z
)
2π
2π/ω
c
0
ds v(ε, k
z
,s)e
iq·R
p
(ε,k
z
,s)−inω
c
s
.
Here, R
p
(ε, k
z
,s) denotes the periodic part of the position vector in real space. 485
For the free electron model m
∗
(k
z
)=m is a constant independent of 486
k
z
and the periodic part of the position vector is given by R
p
(ε, k
z
,s)=487
v
⊥
ω
c
(sin ω
c
s, −cos ω
c
s, 0) . For the propagation q ⊥ B
0
, i.e. q =(0,q,0), the 488
nonvanishing components of semiclassical conductivity σ are 489
σ
xx
=3σ
0
∞
n=−∞
s
n
(w)
1 − iτ (nω
c
− ω)
; σ
yy
=
3σ
0
w
2
∞
n=−∞
n
2
g
n
(w)
1 − iτ (nω
c
− ω)
;
490
σ
xy
= −σ
yx
=
3σ
0
i
2w
∞
n=−∞
ng
n
(w)
1 − iτ (nω
c
− ω)
; σ
zz
=3σ
0
∞
n=−∞
c
n
(w)
1 − iτ (nω
c
− ω)
.
Here, c
n
(w)=
1
2
"
1
−1
d(cos θ)cos
2
θJ
2
n
(w sin θ), s
n
(w)=
1
2
"
1
−1
d(cos θ)sin
2
θ 491
[J
n
(w sin θ)]
2
,andg
n
(w)=
1
2
"
1
−1
d(cos θ) J
2
n
(w sin θ). 492
In the presence of a vector potential A
0
=(0,xB
0
, 0), the electronic states 493
are described by H
0
=
1
2m
p
2
x
+(p
y
+
e
c
B
0
x)
2
+ p
2
z
. The eigenfunctions and
494
eigenvalues of H
0
can be written as 495
|ν = |nk
y
k
z
=
1
L
e
ik
y
y+ik
z
z
u
n
x +
¯hk
y
mω
c
,
ε
ν
= ε
n,k
y
,k
z
=
¯h
2
k
2
z
2m
+¯hω
c
(n +
1
2
).
The quantum mechanical version of the nonvanishing components of σ
are 496
given, for the case of q =(0,q,0), by 497
σ
xx
=
ω
2
p
4πiω
⎡
⎣
1 −
2mω
c
¯h
1
N
nk
y
k
z
α
f
0
(ε
nk
z
)
∂f
n+α,n
∂q
2
α
α
2
− (ω/ω
c
)
2
⎤
⎦
,
σ
yy
=
imω
2
p
ω
2π¯hω
c
1
N
nk
y
k
z
α
f
0
(ε
nk
z
)f
2
n+α,n
α
α
2
− (ω/ω
c
)
2
,
σ
xy
= −σ
yx
=
iω
c
2ωq
∂(q
2
σ
yy
)
∂q
,
σ
zz
=
ω
2
p
4πiω
⎡
⎣
1 −
2¯h
mω
c
1
N
nk
y
k
z
α
f
0
(ε
nk
z
)k
2
z
f
2
n+α,n
α
α
2
− (ω/ω
c
)
2
⎤
⎦
,
Uncorrected Proof
BookID 160928 ChapID 13 Proof# 1 - 29/07/09
422 13 Semiclassical Theory of Electrons
where f
n
n
(q
y
) is the two-center harmonic oscillator integral: 498
f
n
n
(q
y
)=
∞
−∞
u
n
(x +
¯hq
y
mω
c
)u
n
(x)dx
and
499
X
(±)
n
n
(q
y
)=(n +1)
1/2
f
n
,n+1
(q
y
) ± n
1/2
f
n
,n−1
(q
y
).
The quantum mechanical conductivity tensor is the sum of a semiclassical
500
term and a quantum oscillatory part: 501
σ(q, ω)=σ
SC
(q, ω)+σ
QO
(q, ω).
Uncorrected Proof
BookID 160928 ChapID 14 Proof# 1 - 29/07/09
14 1
Electrodynamics of Metals 2
14.1 Maxwell’s Equations 3
There are two aspects of the electrodynamics of metals. The first is linear 4
response theory and the second is the problem of boundary conditions. We 5
have already discussed linear response theory in some detail. Its application to 6
waves in an infinite medium is fairly straightforward. The problem of boundary 7
conditions is usually much more involved. We shall cover some examples of 8
each type in the rest of this chapter. 9
We consider an electromagnetic disturbance with space–time dependence 10
of the form e
iωt−iq·r
. Maxwell’s equations can be written 11
∇×E = −
1
c
∂B
∂t
, (14.1)
and
12
∇×B =
1
c
∂E
∂t
+
4π
c
j
T
+4π∇×M
s
. (14.2)
In (14.2) j
T
is the total current in the system; it includes any external currents 13
and the diamagnetic response currents in the medium. The term M
s
is the 14
spin magnetization in the case of a system containing spins. Equation (14.1) 15
can be written 16
B = ξ × E, (14.3)
where ξ =
cq
ω
. Therefore, the magnetic induction B can be eliminated from 17
(14.2): 18
ξ ×(ξ × E)+E =
4πi
ω
j
T
+
4π
c
ξ ×M
s
,
or
19
ξ(ξ · E) − ξ
2
E + E =
4πi
ω
j
T
+
4π
c
ξ × M
s
. (14.4)
Normally, the total current j
T
can be written as 20
j
T
= j
0
+ j
ind
, (14.5)
Uncorrected Proof
BookID 160928 ChapID 14 Proof# 1 - 29/07/09
424 14 Electrodynamics of Metals
where j
0
is some external current and j
ind
is the current induced (in the 21
electron gas) by the self-consistent field. The spin magnetization M
s
and the 22
induced current j
ind
are found in terms of the self-consistent fields E and B 23
from linear response theory. For example j
ind
might simply be the electron 24
current density 25
j
e
= σ · E, (14.6)
and the spin magnetization M
s
will be some similar function of B 26
M
s
= α · B. (14.7)
For the moment, let us ignore the effect of spin to drop the term M
s
. Then, 27
(14.4) can be solved for j
T
. 28
j
T
=Γ· E, (14.8)
29
where 30
Γ =
iω
4π
-
(ξ
2
− 1)1 − ξξ
.
. (14.9)
31
If we choose q
x
= 0 (as we did in linear response theory) Γ can be written 32
Γ =
ic
2
4πω
⎛
⎜
⎝
q
2
y
+ q
2
z
−
ω
2
c
2
00
0 q
2
z
−
ω
2
c
2
−q
y
q
z
0 −q
y
q
z
q
2
y
−
ω
2
c
2
⎞
⎟
⎠
. (14.10)
Notice that Γ
is diagonal for propagation parallel or perpendicular to the dc 33
magnetic field (which we take to be in the z-direction). 34
14.2 Skin Effect in the Absence of a dc Magnetic Field 35
Consider a semi-infinite metal to fill the space z>0 and vacuum the space 36
z<0. Let us consider the propagation of an electromagnetic wave parallel 37
to the z-axis. Electromagnetic radiation is a self-sustaining oscillation of any 38
medium in which it propagates. Therefore, we need no external “driving” 39
current j
0
, and the total current is simply the electronic current 40
j
e
(q,ω)=σ(q,ω) ·E(q,ω). (14.11)
But Maxwell’s equations require that j
T
=Γ· E, and we have just seen that 41
j
T
= j
e
. Therefore, the electromagnetic waves must be solutions of the secular 42
equation 43
| Γ − σ |=0, (14.12)