Quinn J.J., Yi K.-S. Solid State Physics: Principles and Modern Applications
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Uncorrected Proof
BookID 160928 ChapID 12 Proof# 1 - 29/07/09
384 12 Many Body Interactions: Green’s Function Method
Fig. 12.8. Diagrammatic representation of quasiparticle scattering of (a)zero
momentum transfer and (b) of finite momentum transfer
not in its ground state, there are quasielectrons and quasiholes present that 511
change the energy of the quasiparticle of momentum p. We can get a simple 512
picture of the Fermi liquid interaction between quasiparticles by consider- 513
ing the Feynman diagrams that describe the scattering processes that take 514
a pair of quasiparticles in states (p,σ)and(p
,σ
) from this initial state to 515
an equivalent final state. These processes are represented in diagrammatic 516
terms in Fig. 12.8a and b. Here, the interaction W (denoted by wavy lines) 517
will be taken as the RPA screened interaction. In the first term (a) W (q,ω) 518
corresponds to zero momentum transfer since p → p and p
→ p
.Thisterm 519
is exactly zero since the Coulomb interaction is cancelled by the interaction 520
with the uniform background of positive charge at q = 0. The second term 521
(b) gives the same final state as the initial state only if σ = σ
.ThenW (q,ω) 522
is W(p −p
, 0) since the momentum transfer is p −p
and there is no change 523
in energy. 524
Of course, higher order processes in the effective interaction could be 525
important, but we will ignore them to get the simplified picture. We take 526
Landau’s f
σσ
(p, p
)tobeequalto 527
f
σσ
(p, p
)=
(
4πe
2
(p−p
)
2
ε(p−p
,0)
if σ = σ
0ifσ = σ
.
(12.110)
With this simple approximation a rather good estimate of m
∗
(and hence of 528
the electronic specific heat) can be obtained. Results for the spin susceptibility 529
are not quite as good, and the estimate of the interaction contribution to the 530
compressibility is poor. One important effect that is omitted is the effect 531
of spin fluctuations (in addition to charge density fluctuations) and another 532
is the local field corrections to the RPA. In order to get a more thorough 533
understanding of these ideas, one needs to read advanced texts on many body 534
theory. 535
Uncorrected Proof
BookID 160928 ChapID 12 Proof# 1 - 29/07/09
12.7 Quasiparticle Interactions and Fermi Liquid Theory 385
Problems 536
12.1. Show explicitly that 537
"
t
t
0
dt
1
"
t
1
t
0
dt
2
"
t
3
t
0
dt
3
H
I
(t
1
)H
I
(t
2
)H
I
(t
3
)
=
1
3!
"
t
t
0
dt
1
"
t
t
0
dt
2
"
t
t
0
dt
3
T{H
I
(t
1
)H
I
(t
2
)H
I
(t
3
)}.
12.2. The complete first-order contributions to G(x, x
)areshowninthe538
figure.
539
(a) Write each term δG
(1)
(x, y) out in terms of noninteraction two particle 540
Green’s function G
(0)
(x, y) and the interaction V (x
1
− x
2
) ≡ U (r
1
− 541
r
2
)δ(t
1
− t
2
). Here, x =(r,t) and one may omit the spin to simplify 542
the notation. 543
(b) Let us introduce the Fourier transform G
αβ
(k)oftheG
αβ
(x, y)as 544
follows: 545
G
αβ
(x, y)=
1
(2π)
4
d
4
k e
ik·(x−y)
G
αβ
(k)
546
G
(0)
αβ
(x, y)=
1
(2π)
4
d
4
k e
ik·(x−y)
G
(0)
αβ
(k),
where k =(k,ω), d
4
k ≡ d
3
k dω,andk ·x ≡ k ·x −ωt.Inaddition,for 547
the interaction given by V (x
1
−x
2
) ≡ U(r
1
−r
2
)δ(t
1
−t
2
)wecanwrite 548
V
(
x, x
)
αα
,ββ
=
1
(2π)
4
"
d
4
k e
ik·(x−x
)
V (k)
αα
,ββ
=
1
(2π)
3
"
d
3
x e
i·(x−x
)
U(k)
αα
,ββ
δ(t − t
),
where V (k)
αα
,ββ
= U (k)
αα
,ββ
=
1
(2π)
3
"
d
3
x e
−ik·x
U(k)
αα
,ββ
is the 549
spatial Fourier transform of the interparticle potential. Express each 550
Uncorrected Proof
BookID 160928 ChapID 12 Proof# 1 - 29/07/09
386 12 Many Body Interactions: Green’s Function Method
term obtained in part (a) in terms of G
(0)
αβ
(k)andV (k) in the momen- 551
tum space. 552
12.3. By definition the noninteracting fermion Green’s function is given by 553
G
(0)
αβ
(xt, x
t
)=−iΦ|T{ψ
Iα
(xt)ψ
†
Iβ
(x
t
)}Φ,
the noninteracting ground state vector is taken to be normalized. Show that
554
G
(0)
αβ
(k,ω)=δ
αβ
θ(k − k
F
)
ω − ¯h
−1
ε
k
+iη
+
θ(k
F
− k)
ω −¯h
−1
ε
k
− iη
.
12.4. Let us define the phonon field operator Φ(x)by
555
Φ(x)=
q
γ(q)e
iq·x
(b
†
q
− b
−q
),
where γ(q)=i
4πZe
2
q
¯hN
2Mω
q
1/2
. Then we can define the phonon propaga- 556
tor by P
0
(2, 1) = iGS | T {Φ(x
2
,t
2
)Φ
†
(x
1
,t
1
)}|GS where Φ(x
2
,t
2
)=557
e
−iHt
2
Φ(x
2
)e
iHt
2
. 558
(a) Take the Fourier transform of P
0
(2, 1) to obtain P
0
(q, ω), the wavevec- 559
tor frequency space representation of P
0
(2, 1). 560
(b) Show that P
0
(q, ω) can be written as 561
P
0
(q, ω)=
2iΩ
p
γ
2
(q)
ω
2
− Ω
2
p
,
where Ω
p
=
4πZ
2
e
2
N
M
1/2
is the plasma frequency of the ions. 562
12.5. Let us consider a Dyson equation given by 563
W (q, ω)=V (q, ω) − V (q, ω)Q
0
(q, ω)W (q, ω),
where
564
Q
0
(q, ω)=2(−1)
"
d
3
k
1
dω
1
d
3
k
2
dω
2
(2π)
8
G
0
(k
1
,ω
1
)G
0
(k
2
,ω
2
)δ(ω
1
− ω
2
+ ω)δ(k
1
−k
2
+q)
is the propagation function for the electron–hole pair.
565
(a) Show that Q
0
(q, ω) can be written as 566
Q
0
(q, ω)=
−2i
(2π)
3
"
k<k
F
d
3
k
1
E(k + q) − E(k)+ω
+
1
E(k + q) − E(k) − ω
.
Note that the single particle propagator G
0
(q, ω)isgivenby 567
G
0
(q, ω)=
i
ω −E(q)(1 − iδ)
.
Uncorrected Proof
BookID 160928 ChapID 12 Proof# 1 - 29/07/09
12.7 Quasiparticle Interactions and Fermi Liquid Theory 387
(b) Show that the solution of the Dyson equation given above is simply 568
W =
V
1+VQ
0
. 569
(c) Show that 1 + VQ
0
is the same as the Lindhard dielectric function 570
ε(q, ω). 571
Uncorrected Proof
BookID 160928 ChapID 12 Proof# 1 - 29/07/09
388 12 Many Body Interactions: Green’s Function Method
Summary 572
In this chapter, we study Green’s function method – a formal theory of many 573
body interactions. Green’s function is defined in terms of a matrix element 574
of time ordered Heisenberg operators in the exact interacting ground state. 575
We then introduce the interaction representation of the state functions of 576
many particle states and write the Green’s function in terms of time ordered 577
products of interaction operators. Wick’s theorem is introduced to write the 578
exact Green’s function as a perturbation expansion involving only pairings of 579
field operators in the interaction representation. Dyson equations for Green’s 580
function and the screened interaction are illustrated and Fermi liquid picture 581
of quasiparticle interactions is also discussed. 582
The Hamiltonian H of a many particle system can be divided into two 583
parts H
0
and H
,whereH
represents the interparticle interactions given, in 584
second quantized form, by 585
H
=
1
2
d
3
r
1
d
3
r
2
ψ
†
(r
1
)ψ
†
(r
2
)U(r
1
− r
2
)ψ(r
2
)ψ(r
1
).
Particle density at a position r
0
and the total particle number N are written, 586
respectively, as n(r
0
)=ψ
†
(r
0
)ψ(r
0
); N =
"
d
3
rψ
†
(r)ψ(r). 587
The Schr¨odinger equation of the many particle wave function Ψ(1, 2,...,N) 588
is (¯h ≡ 1) i¯h
∂
∂t
Ψ = HΨ, where Ψ(t)=e
−iHt
Ψ
H
. Here, Ψ
H
is time independent. 589
The state vector Ψ
I
(t) and operator F
I
(t)intheinteraction representation are 590
written as 591
Ψ
I
(t)=e
iH
0
t
Ψ
S
(t); F
I
(t)=e
iH
0
t
F
S
e
−iH
0
t
.
The equation of motion for F
I
(t)is
∂F
I
∂t
=i[H
0
,F
I
(t)] and the solution for 592
F
I
(t) can be expressed as Ψ
I
(t)=S(t, t
0
)Ψ
I
(t
0
), where S(t, t
0
)istheS matrix 593
given by 594
S(t, t
0
)=T
e
−i
"
t
t
0
H
I
(t
)dt
.
The eigenstates of the interacting system in the Heisenberg, Schr¨odinger, and
595
interaction representation are related by 596
Ψ
H
(t)=e
iHt
Ψ
S
(t)and Ψ
I
(t)=e
iH
0
t
Ψ
S
(t).
At time t=0, Ψ
I
(t =0)=Ψ
H
(t =0)=Ψ
H
.Ψ
H
is the state vector of the fully 597
interacting system in the Heisenberg representation: Ψ
H
= S(0, −∞)Φ
H
. 598
The Green’s function G
αβ
(x, x
) is defined, in terms of ψ
H
α
and ψ
H†
β
,by 599
G
αβ
(x, x
)=−i
Ψ
H
|T{ψ
H
α
(x)ψ
H†
β
(x
)}|Ψ
H
Ψ
H
|Ψ
H
,
where x = {r,t} and α, β are spin indices.
600
In normal product of operators, all annihilation operators appear to the 601
right of all creation operators: for example, 602
Uncorrected Proof
BookID 160928 ChapID 12 Proof# 1 - 29/07/09
12.7 Quasiparticle Interactions and Fermi Liquid Theory 389
N{ψ
†
(1)ψ(2)} = ψ
†
(1)ψ(2) while N{ψ(1)ψ
†
(2)} = −ψ
†
(2)ψ(1).
Pairing or a contraction is the difference between a T product and an
603
N product: T(AB) − N(AB)=A
c
B
c
. The Wick’s theorem states that T 604
product of operators ABC ··· can be expressed as the sum of all possible N 605
products with all possible pairings. 606
Dyson’s equations for the interacting Green’s function G and the screened 607
interaction W are written as G = G
(0)
+ G
(0)
ΣG; W = V + V ΠW. Here, Σ 608
and Π denote the self energy and polarization part, and the simplest of which 609
are given, respectively, by Σ
0
= G
(0)
W ;Π
0
= G
(0)
G
(0)
. In the RPA, the G 610
is replaced by G
(0)
and W is exactly equivalent to
V (q)
ε(q,ω)
,whereε(q, ω)isthe 611
Lindhard dielectric function. 612
The Hamiltonian H of a system with the electron–phonon interaction is 613
divided into three parts: H = H
e
+ H
N
+ H
I
, where 614
H
e
=
k
¯h
2
k
2
2m
∗
c
†
k
c
k
,H
N
=
α
¯hω
α
(b
†
α
b
α
+
1
2
),
and H
I
=
k,k
,q
4πe
2
Ωq
2
c
†
k+q
c
†
k
−q
c
k
c
k
+
k,α,G
γ(α, G)(b
α
− b
†
−α
)c
†
k+q+G
c
k
. 615
Once we know Ψ(t
1
) of the Schr¨odinger equation, i¯h
∂Ψ
∂t
= HΨ, we have 616
Ψ(x
2
,t
2
)=
d
3
x
1
G
0
(x
2
,t
2
; x
1
,t
1
)Ψ(x
1
,t
1
).
For free electrons, G
0
(q, ω) is the Fourier transform of G
0
(2, 1): 617
G
0
(q, ω)=
i
ω −E(q)(1 − iδ)
.
For a “model solid” containing longitudinal phonons as well as electrons, two
618
electrons can scatter via the virtual exchange of phonons and the total inter- 619
action, i.e. the sum of the Coulomb interaction and the interaction due to 620
virtual exchange of phonons, is given, in terms of bare interaction D
0
and 621
polarization Q
0
,byD(q,ω)=
D
0
(q,ω)
1+D
0
(q,ω)Q
0
(q,ω)
. 622
The Dyson equation for the Green’s function can be written 623
G(k, ω)=G
(0)
(k, ω)+G
(0)
(k, ω)Σ(k, ω)G(k, ω).
The electron self energy is Σ(k, ω)=[G
(0)
(k, ω)]
−1
− [G(k, ω)]
−1
and the 624
energy of a quasiparticle is written as E
p
= ε
p
+Σ(p, ω) |
ω=E
p
. Σ(p,E
p
) 625
represents the interaction of a quasiparticle of momentum p with the ground 626
state of the interacting electron gas. The energy of the state is written as 627
E = E
0
+
pσ
δn
pσ
E
pσ
+
1
2
p, p
σ, σ
f
σσ
(p, p
)δn
nσ
δn
p
σ
.
The first term on the right is the ground-state energy, the second is the quasi-
628
particle energy E
pσ
multiplied by the quasiparticle distribution function, and 629
the third represents the interactions of the quasiparticles with one another. 630
Uncorrected Proof
BookID 160928 ChapID 13 Proof# 1 - 29/07/09
13 1
Semiclassical Theory of Electrons 2
13.1 Bloch Electrons in a dc Magnetic Field 3
In the presence of an electric field E and a dc magnetic field B, the equation 4
of motion of a Bloch electron in k-space takes the form 5
¯h
˙
k = −eE −
e
c
v × B. (13.1)
Here, v =
1
¯h
∇
k
ε(k) is the velocity of the Bloch electron whose energy ε(k) 6
is an arbitrary function of wave vector k. In deriving (13.1), we noted that 7
no interband transitions were allowed, and that when k became equal to a 8
value on the boundary of the Brillouin zone this value of k was identical to 9
the value on the opposite side of the Brillouin zone separated from it by a 10
reciprocal lattice vector K. 11
Equation (13.1) can be obtained from an effective Hamiltonian 12
H = ε(
p
¯h
+
e
¯hc
A) − eφ, (13.2)
where ε(k) is the energy as a function of k in the absence of the magnetic
13
field and B = ∇×A. Hamilton’s equations give, since p =¯hk −
e
c
A, 14
v
x
=
∂H
∂p
x
=
1
¯h
∂ε
∂k
x
, (13.3)
15
− ˙p
x
=
∂H
∂x
= ∇
k
ε ·
∂
∂x
e
¯hc
A
− e
∂φ
∂x
=
e
c
v ·
∂A
∂x
− e
∂φ
∂x
. (13.4)
But we also know that ˙p
x
=¯h
˙
k
x
−
e
c
˙
A
x
,or 16
˙p
x
=¯h
˙
k
x
−
e
c
∂A
x
∂t
−
e
c
v
x
∂A
x
∂x
+ v
y
∂A
x
∂y
+ v
z
∂A
x
∂z
. (13.5)
Uncorrected Proof
BookID 160928 ChapID 13 Proof# 1 - 29/07/09
392 13 Semiclassical Theory of Electrons
Equating the ˙p
x
from (13.4) with that from (13.5) gives 17
¯h
˙
k
x
= −eE
x
−
e
c
(v × B)
x
. (13.6)
Since the equation of motion, (13.1) or (13.6), is derived from Hamilton’s
18
equations, (13.3) and (13.4), using the effective Hamiltonian (13.2), p and r 19
must be canonically conjugate coordinates. 20
13.1.1 Energy Levels of Bloch Electrons in a Magnetic Field 21
Onsager determined the energy levels of electrons in a dc magnetic field by 22
noting that 23
¯h
˙
k = −
e
c
v × B (13.7)
could be written as
24
˙
k
⊥
= −
e
¯hc
|v
⊥
|B. (13.8)
Here, v
⊥
is the component of v perpendicular to B and k
⊥
is perpendicular 25
to both B and v. Integrating (13.7) gives 26
k
⊥
=
eB
¯hc
× r
⊥
+ constant. (13.9)
27
We can choose the origin of k and r space such that the constant vanishes 28
for the electron of interest. Thus, the orbit in real space (by this we mean 29
the periodic part of the motion in r-space that is perpendicular to B) will be 30
exactly the same shape as the orbit in k-space except that it is rotated by 31
90
◦
and scaled by a factor
eB
¯hc
.Thisfactor
eB
¯hc
is called l
−2
0
,wherel
0
is the 32
magnetic length. 33
Let us choose B to define the z-direction. Then
˙
k
z
=0andk
z
is a constant 34
of the motion. Now, look at the time rate of change of the energy 35
dε
dt
= ∇
k
ε ·
dk
dt
=¯hv ·
−
e
¯hc
v × B
. (13.10)
This is clearly zero since v is perpendicular to v × B, meaning that ε is a
36
constant of the motion also. Thus, the orbit of a particle in k-space is the 37
intersection of a constant energy surface ε(k)=ε and a plane of constant k
z
38
(see Fig. 13.1). 39
Let us look at the different kinds of orbits that are possible by consid- 40
ering the intersections of a plane k
z
= 0 with a number of different energy 41
surfaces for a simple model ε(k) (see, for example, Fig. 13.2). The orbits can 42
be classified as 43
• Closed electron orbits like A and B 44
• Closed hole orbits like C 45
• Open orbits like D 46
Often one simply repeats the Brillouin zone a number of times to show how 47
the pieces of hole orbits or the open orbits look as illustrated in Fig. 13.3. 48
Uncorrected Proof
BookID 160928 ChapID 13 Proof# 1 - 29/07/09
13.1 Bloch Electrons in a dc Magnetic Field 393
Fig. 13.1. A constant energy surface ε(k)=ε and the orbit of a particle in k-space
Fig. 13.2. Different kinds of orbits of a particle in k-space
Fig. 13.3. Repeated zone scheme of the orbits in k-space
Uncorrected Proof
BookID 160928 ChapID 13 Proof# 1 - 29/07/09
394 13 Semiclassical Theory of Electrons
13.1.2 Quantization of Energy 49
For closed orbits in k-space, the motion is periodic. The real space orbits in 50
the direction perpendicular to B will also be periodic. Because p and r are 51
canonically conjugate coordinates we can apply the Bohr–Sommerfeld quan- 52
tization condition 53
3
p
⊥
·dr
⊥
=2π¯h(n + γ), (13.11)
54
where γ is a constant satisfying 0 ≤ γ ≤ 1, and n =0, 1, 2,....Wecanuse 55
p
⊥
=¯hk
⊥
−
e
c
A
⊥
and the fact that k
⊥
(t)=
eB
¯hc
× r
⊥
(t). Then 56
3
p
⊥
·dr
⊥
=
e
c
B ·
3
r
⊥
× dr
⊥
−
e
c
3
A · dr
⊥
.
But
<
r
⊥
×dr
⊥
is just twice the area of the orbit as is illustrated in Fig. 13.4. 57
Furthermore,
<
A ·dr
⊥
=
"
Surface
∇×A ·dS = B × (area of orbit). Therefore, 58
we obtain 59
3
p
⊥
·dr
⊥
=
e
c
B A =2π¯h(n + γ), (13.12)
where A is the area of the orbit. A depends on energy ε and on k
z
.Weknow 60
that A(ε, k
z
) is proportional to the area S(ε, k
z
) of the orbit in k-space, from 61
(13.9), with S(ε, k
z
)=
eB
¯hc
2
A(ε, k
z
). Thus, the quantization condition can 62
be written 63
S(ε, k
z
)=
2πeB
¯hc
(n + γ). (13.13)
64
Example For free electrons ε =
¯h
2
k
2
2m
.TheareaS(ε, k
z
)isequaltoπk
2
⊥
,where 65
k
2
⊥
+ k
2
z
= k
2
. Therefore, 66
S(ε, k
z
)=π
2mε
¯h
2
− k
2
z
. (13.14)
Fig. 13.4. r
⊥
× dr
⊥
is twice the area of the triangle within the orbit