Download SOLID-STATE PHYSICS II 2007 O. Entin-Wohlman

SOLID-STATE PHYSICS II 2007
O. Entin-Wohlman
1.
SUPERCONDUCTIVITY
The Meissner effect. An ideal superconductor permits no magnetic field in its interior. In
other words, a superconductor is a perfect diamagnet. Not only a magnetic field is excluded
from entering a superconductor, (which might have been explained by perfect conductivity),
but a field in an originally normal (i.e., not superconductive) sample is expelled as the
sample is cooled below the superconductive transition temperature, Tc . This fact cannot be
explained by perfect conductivity. This is the Meissner effect.
On the other hand, the Meissner effect implies that superconductivity can be destroyed once
the magnetic field exceeds a certain value, the critical magnetic field, Hc . Let us denote the
free energy density of a system when it is above the superconducting transition temperature
by fn . As the system is cooled below Tc (in the absence of a magnetic field) it goes into the
superconductive state, since in that state its free energy is lower. Let us denote that free
energy density by fs . Then the critical magnetic field is given by
Hc2 (T )
= fn (T ) − fs (T ) .
8π
(1.1)
(Remember that the free energy of a magnetic field is given by the volume integral of H 2 /8π.)
Equation (1.1) implies that the critical magnetic field depends on the temperature, and it
vanishes at T = Tc . In fact,
Hc (T ) ' Hc (0)(1 − (T /Tc )2 ) .
(1.2)
The Meissner effect is described by the (phenomenological) London equations. Let us consider the motion of the electrons in a perfect conductor, i.e., when they are accelerated by
an electric field without any dissipation. Let us also assume that the number density of such
‘dissipation-less’ electrons is ns , and their common velocity is vs . We can then write
m
dvs
= −eE ,
dt
(1.3)
where E is the electric field, and
dJs
n e2
= s E,
dt
m
1
Js = −evs ns .
(1.4)
On the other hand, the Faraday’s law,
∇×E=−
1 ∂H
,
c ∂t
(1.5)
gives
∂
n e2 ∇ × Js + s H = 0 .
∂t
mc
(1.6)
The London equation states that in a superconducting system, not only the time derivative
above vanishes, but that
∇ × Js +
ns e2
H = 0 , London equation .
mc
(1.7)
Since the Maxwell equation gives us that
∇×H=
4π
J ,
c s
(1.8)
we arrive at the result that in a superconducting system
∇2 H =
1
1
2
H
,
or
∇
J
=
J ,
s
λ2
λ2 s
(1.9)
where the penetration depth λ is given by
λ2 =
mc2
.
4πns e2
(1.10)
We note that the penetration depth diverges as the number density of the dissipation-less
electrons (i.e., superconducting electrons), ns , tends to zero. In other words, the penetration
depth diverges as T → Tc .
Let us consider a superconducting slab of finite thickness d, placed in a parallel magnetic
field, Ha . The slab is perpendicular to the x direction. According to the first equation of
(1.9), the magnetic field within the slab is
H(x) = Aex/λ + Be−x/λ ,
(1.11)
where A and B are constants. They are determined from the two requirements that at
x = d/2 and at x = −d/2, the magnetic field equals the applied field, Ha . Hence,
H(x) = Ha
cosh(x/λ)
.
cosh(d/2λ)
2
(1.12)
The average value of the field within the slab is
Z
2λ
1 d/2
cosh(x/λ)
= Ha tanh(d/2λ) .
hHi =
dxHa
d −d/2
cosh(d/2λ)
d
(1.13)
This average value consists of the (external) applied field, Ha plus the magnetization induced
in the slab, i.e.,
hHi = Ha + 4πM .
(1.14)
When d λ, hHi tends to zero, and therefore
M →−
Ha
, dλ.
4π
(1.15)
The susceptibility is −1/4π, which means that a bulk superconductor is a perfect diamagnet.
On the other hand, when d λ, hHi tends to Ha (1 − (d2 /12λ2 )), and therefore
H d2 a
M →−
.
4π
12λ2
(1.16)
We can estimate from this relation the critical magnetic field in the special case where the
field is parallel to the slab. The critical field in this case is not the critical field of the material
from which the slab is made, but it is the critical field of the material times the ratio λ/d,
Hck =
√
λ
12 Hc .
d
(1.17)
∗ ∗ ∗ exercise: Explain intuitively the result (1.17) (by considering the current needed
to screen the field). What will be the critical magnetic field perpendicular to the slab?
The Ginzburg-Landau (GL) theory. Ginzburg and Landau constructed a theory of superconductivity, by introducing a complex pseudo wave function ψ as an order parameter.
The local density of the superconducting electrons is given by
ns = |ψ(r)|2 .
The order parameter obeys the Ginzburg-Landau equations,
e 2
1 −i∇ − A ψ + β|ψ|2 ψ = −αψ ,
2m
c
and the equation for the supercurrent density Js ,
e2
e ∗
Js = −i
ψ ∇ψ − ψ∇ψ ∗ ) −
|ψ|2 A .
2m
mc
3
(1.18)
(1.19)
(1.20)
Here, α and β are (temperature-dependent) parameters, and A is the vector potential which
represents the action of a constant magnetic field, H = ∇ × A. Note that if we write the
complex order parameter in the form
ψ = |ψ|eiφ ,
(1.21)
then the supercurrent is related to thegradient of the phase,
e e
2
Js = |ψ| ∇φ − A .
m
c
(1.22)
The GL theory introduces a length,
ξ=
1
,
|2mα|
(1.23)
which characterizes the distance over which ψ(r) can vary. Near Tc ξ diverges as (Tc −T )−1/2 ,
since α vanishes as (T − Tc ). Thus, superconductivity is described by two lengths, the
coherence length ξ and the penetration length λ. The ratio of these two lengths,
κ=
λ
,
ξ
(1.24)
is approximately temperature-independent. Type II superconductors are those for which
√
κ > 1/ 2. The GL equations can be derived from the free energy density, f , which takes
the form
β 4
e 2 H 2
1 f = fn0 + α|ψ| + |ψ| +
,
−i∇ − A ψ +
2
2m
c
8π
2
(1.25)
by minimizing the free energy with respect to the complex order parameter.
In a bulk superconductor, and in the absence of the magnetic field (A = 0) we can take the
order parameterψ to be real, since in this case all the coefficient of the differential equation
are real. In a homogenous bulk superconductor we do not expect any spatial variation, and
therefore Eq. (1.19) takes the form
αψ + βψ 3 = 0 .
(1.26)
This has two solutions: either the bulk is simply not a superconductor, i.e., ψ = 0, or
2
ψ 2 ≡ ψ∞
=−
4
α
.
β
(1.27)
This result leads to the identification of the temperature dependence of the coefficient α,
α ∝ −Tc + T .
(1.28)
Namely, for temperatures below the transition temperature Tc where α is negative, the
superconducting order parameter is non zero. It vanishes (continuously) at T = Tc , and
then system phase transforms into the normal state. Since this happens continuously (by
construction) the GL equation describes properly the second order phase transition. Note
that this argument ignores the temperature dependence of the other coefficient β. It assumes
that β depends only weakly on the temperature.
Note that the superconducting free energy density, α|ψ|2 + β2 |ψ|4 is zero when ψ is zero, and
is −|α|2 /(2β when |ψ|2 = −α/β, namely, it is lower in the superconducting state.
Let us now assume that the superconductor occupies only half of the space, x > 0 (no fields
are applied). Then the GL equation becomes
αψ + βψ 3 −
1 ∂ 2ψ
=0.
2m ∂x2
(1.29)
Denoting
p
ψ = ψ∞ f = ( |α|/β)f ,
(1.30)
(ψ∞ is the full value of the order parameter in the bulk superconductor), this equation
becomes
2
1 d2 f
3
2d f
+
f
−
f
≡
ξ
+ f − f3 = 0 ,
2m|α| dx2
dx2
(1.31)
where we have used the definition (1.23) for the superconducting coherence length ξ. It is
quite straightforward to solve Eq. (1.31). Denoting f 0 ≡ df /dx, and f 00 ≡ d2 f /dx2 , we
multiply Eq. (1.31) by f 0 . Then, f f 0 = (1/2)df 2 /dx, f 3 f 0 = (1/4)df 4 /dx, and f 0 f 00 =
(1/2)df 02 /dx, and hence
d 2 02
1 ξ f + f2 − f4 = 0 .
dx
2
(1.32)
This implies that the combination of terms within the brackets do not depend on x, and
consequently their value is the same as for x → ∞, i.e., f = 1 [see Eq. (1.30)]. Namely,
1
1
1
ξ 2 f 02 + f 2 − f 4 = V (1 − f 2 (x))2 = ξ 2 f 02 (x)
2
2
2
df
dx
x
V
= √ V f (x) = tanh( √ ) .
2
1−f
2ξ
2ξ
5
(1.33)
We see that near the boundary, the order parameter decays to zero over a scale length of
order ξ.
∗ ∗ ∗ exercise: Find and discuss the order parameter of a superconducting slab of width
d, placed normal to the x axis (the slab is infinite along the y and the z directions). Discuss
in particular the cases d ξ and d ξ.
The critical current. There are certain cases in which one can assume that the absolute
value of the order parameter, |ψ|, does not vary spatially, however, its phase φ, does [see Eq.
(1.21)]. This occurs when the spatial change in |ψ| has to occur over distances far smaller
than ξ, and hence will cost too much kinetic energy. For example, if |ψ| has to change over
the width d of a thin film, its change will be of order x/ξ ' d/ξ 1. In such cases the GL
equations, [see Eqs. (1.22)) and (1.25)] take the form
e
e
|ψ|2 (∇φ − A) ≡ e|ψ|2 vs ,
m
c
mv 2 H 2
β
.
f = fn0 + α|ψ|2 + |ψ|4 + |ψ|2 s +
2
2
8π
Js =
(1.34)
In a very thin film or wire, of thickness d λ, one may neglect the magnetic energy density
H 2 /(8π) as compared to the kinetic energy (the latter is of order λ2 , and the former of order
d2 ). The super conducting free energy density is then
fs = α|ψ|2 +
β 4
mv 2
|ψ| + |ψ|2 s .
2
2
(1.35)
minimizing it with respect to |ψ|2 , we find
α + mvs2 /2
|α| mvs2 2
2
|ψ| = −
=
1−
= ψ∞ 1 − (ξmvs ) .
β
β
2|α|
2
(1.36)
The supercurrent is then
Js =
2
eψ∞
vs
2
1 − (ξmvs )
.
(1.37)
We see that the supercurrent vanishes when vs = 0 and vs = 1/(mξ). Its maximal value
(found below) is the maximal supercurrent that the system can carry. The value of vs at
the maximum is obtained by minimizing Js with respect to vs , yielding mvs2 /2 = |α|/3.
Inserting this value into Eq. (1.36), we find
2 2
|ψ|2 = ψ∞
,
3
6
(1.38)
namely, in the presence of current, the superconducting order parameter is reduced as compared to its value in the bulk (and in the absence of a current). The critical current, Jc , is
given by inserting the results of the minimization into the first of Eqs. (1.34),
Jc =
2
eΨ2∞
2 |α| 1/2
3 3m
.
(1.39)
Note that the critical current vanishes at the transition temperature Tc , where α = 0.
Flux quantization.
Consider the first of Eqs. (1.34), applied to a superconductor in the shape of a ring.
Integrating it over a path located well inside the superconductor, where there are no currents,
that encloses the hole of the ring gives
I
e
e
|ψ|2 (∇φ − A) · d` .
0=
m
c
(1.40)
Stokes theorem gives
I
Z
A · d` =
Z
∇ × A · dS =
B · dS = Φ ,
(1.41)
where Φ is the flux enclosed by the ring (since the magnetic field does not penetrate deep
into the superconductor). On the other hand, the phase must change by 2πn, since the
order parameter is single valued. Therefore
Φ=
2πe
n,
c
(1.42)
Putting back Planck constant , we find that the flux is quantized in units of the flux quantum
(in fact, half of it).
The upper critical field, Hc2 . Let now consider how superconductivity is nucleated in the
bulk, in the presence of a magnetic field H (along the z direction). We use the gauge
ay = Hx ,
(1.43)
and ignore in the GL differential equation the cubic term, since the system is only barely
superconducting and the order parameter is therefore small. Then
2πH 2 4πi
∂
1
−∇ +
Hx
+
x2 ψ = 2 ψ ,
Φ0
∂y
Φ0
ξ
2
7
(1.44)
where the superconducting flux quantum is
Φ0 =
πc
.
e
(1.45)
By substituting
ψ(r) = eiky y+ikz z f (x) ,
(1.46)
we find
−
1
∂ 2 f 2πH 2
2
2
(x
−
x
)
f
=
+
−
k
0
z f ,
∂x2
Φ0
ξ2
x0 =
ky Φ0
.
2πH
(1.47)
The problem now is the same as the Schrödinger equation for an harmonic oscillator, with
the eigenvalues
1
1 eH n = (n + )ωc = (n + )
.
2
2 mc
(1.48)
In our case, however, the energy is given by the term on the right hand side of Eq. (1.47).
The maximal magnetic field (corresponding to the lowest value of n, i.e., n = 0) which the
system can support is therefore
Φ0 1
2
H=
− kz .
2π ξ 2
(1.49)
The truly maximal field is obtained for kz = 0 (i.e., no variation of the order parameter
along the axis parallel to the field). Hence
Hc2 =
Φ0
.
2πξ 2
(1.50)
Note that this field vanishes at the superconducting transition temperature.
Cooper Pairs. Consider the Schrödinger for two fermions in the Fermi sea interacting
through a potential λV (r1 , r2 ),
T1 + T2 + λV (1, 2) Ψ(1, 2) = EΨ(1, 2) ,
(1.51)
where T denotes the kinetic energy Hamiltonian. Let us define the eigenfunctions of the
Hamiltonian H0 ,
H0 φn + (T1 + T2 )φn = En φn .
8
(1.52)
Then
Ψ(1, 2) = φ0 (1, 2) +
X
φn (1, 2)
n6=0
1
hφ |λV |Ψ(1, 2)i ,
E − En n
(1.53)
and
E − E0 = hφ0 |λV |Ψ(1, 2)i .
(1.54)
(This can be shown by applying the operator T1 + T2 − E to Eq. (1.53), and using for the
normalization hφ0 |Ψi).
When the system is confined to a large box of unit volume, then the unperturbed wave
functions are
φk1 k2 (1, 2) = eik1 ·r1 eik2 ·r2 .
(1.55)
Let us assume that the interaction does not depend on the spin, and that the two electrons
have opposite spins so that they are distinguishable. Let us also restrict the sum over n in
Eq. (1.53) such that
X
X
→
n
.
(1.56)
k1 k2 >kF
Namely, the excited states of the two electrons are above the filled Fermi sea. In a homogeneous medium the total momentum is conserved, and therefore we write
1
P =k1 + k2 , k = (k1 − k2 ) ,
2
1
r =r1 − r2 , R = (r1 + r2 ) ,
2
(1.57)
and hence
Ψ(1, 2) = eiP·R ΨPk (r) .
(1.58)
The first exponential in Eq. (1.58) describes the motion of the center-of-mass, the second is
the internal wave function of the interacting pair.
We next use Eq. (1.58) in Eq. (1.53), with
1
| P ± k| < kF ,
2
1
Γ ≡ | P ± t| > kF .
2
9
(1.59)
This yields
ΨPk (r) = eik·r + λ
X
eit·r
t∈Γ
κ2
m
ht|V |ΨPk i ,
− t2
(1.60)
while from Eq. (1.54)
κ2 − k 2 = λmhk|V |ΨPk i .
(1.61)
The last term in Eq. (1.60) describes virtual transitions by which the two electrons which
are initially within the Fermi sea can change their energy.
In order to solve Eq. (1.60) we modify it a bit. We assume that the interaction is nonlocal,
V (r) → V (r, r0 ) ,
Z
Z
−ik·r
dre
V (r)Ψ(r) → drdr0 e−ik·r V (r, r0 )Ψ(r0 ) .
(1.62)
The local potential is obtained as the limit
V (r, r0 ) → V (r)δ(r − r0 ) .
In addition we assume that the nonlocal potential is separable,
Z
0
0
V (r, r ) = u(r)u(r ) ,
dre−ik·r u(r) = u(k) .
With these assumptions, Eq. (1.61) becomes [see Eq. (1.62)]
Z
2
2
κ − k = mλu(k) dru(r)ΨPk (r)
Z
X
m
eit·r 2
ht|V
|Ψ
i
= mλu(k) dru(r) eik·r + λ
Pk
κ − t2
t∈Γ
X
m
κ2 − k 2
= mu2 (k) + mλ
u(t) 2
u(t)
.
κ − t2
m
t∈Γ
(1.63)
(1.64)
(1.65)
Rearranging this equation, we finally obtain
X u2 (t)
1
u2 (k)
= 2
+
≡ F (κ) ,
mλ
κ − k 2 t∈Γ κ2 − t2
(1.66)
κ2 − k 2
∆E =
.
m
(1.67)
for the allowed energies,
10
Equation (1.66) has only one solution for a repulsive interaction, λ > 0,
κ2 − k 2 ' λu2 (k) .
(1.68)
This solution exists also when the interaction is attractive; however, then there is another
solution, Since the first term of Eq. (1.66) is negligible in the macroscopic limit, we have
Z ∞ 2
1
t dt u2 (t)
=
.
(1.69)
2 2
2
m|λ|
kF 2π t − κ
Namely, assembling all constants into an effective interaction λe ,
1
k2
' ln 2 F 2 .
|λe |
kF − κ
(1.70)
κ2 = kF2 − kF2 e−1/|λe | .
(1.71)
We therefore find that
Hence, 1. the energy shift is negative (independent of the volume); 2. the energy shift
has an essential singularity as a function of the coupling, and cannot be obtained within
perturbation theory; 3. the energy shift is greatest when the total momentum P vanishes.
There is a bound pair for an arbitrary small attractive interaction.
∗ ∗ ∗ exercise: Repeat the analysis for two fermions in free space (namely, there are no
other fermions in the system).
The origin of the attractive interaction. Consider first screening of an external source
by all charges in the metal, ions and electrons. The total dielectric function is defined by
relating the (Fourier component of) the total potential in the metal to the potential exerted
by an external charge,
Φtot = Φext .
(1.72)
We may consider separately the response of the electrons and the response of the ions. For
the electrons we have
el Φtot = Φext + Φion .
(1.73)
tot
ion
= Φext + Φel .
b Φ
(1.74)
The response of the ions is
11
We can therefore write
−
Φtot = Φext + Φel + Φion ≡ Φtot ,
el + ion
b
(1.75)
= el + ion
b −1 .
(1.76)
so that
But the response of the metal to an external charge can yet be viewed in a different for, by
writing
Φtot =
1 Φext
.
ion
el
d
(1.77)
is the dressed one, i.e., the
is the ‘bare’ dielectric response of the ions, while ion
Here, ion
d
b
response of the ions to the an external charge dressed by the effect of the electrons. But
since Eq. (1.76) yields
ion − 1 = el 1 + b el
,
(1.78)
we see that
ion
d = 1+
ion
b −1
.
el
(1.79)
The bare dielectric function of the ions may be estimated as follows. An ac field, D(ω)
exerts a force on the ion charge, such that
M ẍ = −Ω2P x + qD(ω) ,
(1.80)
where ΩP is the frequency of the oscillator (representing the ion), given by the plasma
frequency,
Ω2P =
4πq 2 n
,
M
(1.81)
where n is the ion concentration and q is the charge of each ion. Hence,
x=
q
1
D(ω) ,
2
M ΩP − ω 2
(1.82)
and consequently the polarization induced in the ion system, P , is
P = nqx =
nq 2
1
D(ω) ,
2
M ΩP − ω 2
12
(1.83)
where n is the ion concentration. The field D is related to the external applied field E by
D = E + 4πP = E +
4πnq 2
1
D 2
.
M
ΩP − ω 2
(1.84)
We therefore find
Ω2P D = 1− 2 E ,
ω
(1.85)
leading to
ion
=1−
b
Ω2P
.
ω2
(1.86)
The total dielectric function of the metal is therefore [see Eq. (1.76)]
Ω2P
.
ω2
(1.87)
kT2 F
=1+ 2 ,
q
(1.88)
= el −
Using now for el the Thomas Fermi expression,
el
where q is the wave vector of the applied field, we find
k 2 Ω2 /(1 + kT2 F /q 2 ) = 1 + T2F 1 − P
.
q
ω2
(1.89)
However,
Ω2P
1+
2
kT
F
q2
≡ ω 2 (q) ,
(1.90)
the dispersion relation of the phonons. Hence
1 1 ω2
=
.
k2
ω 2 − ω 2 (q)
1 + qT2F
(1.91)
This tells us that for ac frequencies exceeding the phonon frequency, the (retarded) Coulomb
interaction of two electrons can become attractive.
The BCS Hamiltonian. The ‘reduced’ Hamiltonian, which is assumed to include all interactions important for superconductivity, reads
H=
X
kσ
ζk nkσ +
X
kk0
13
Vkk0 c†k↑ c†−k↓ c−k0 ↓ ck0 ↑ .
(1.92)
Here, c†kσ is the operator that creates an electron at state of wave vector k and spin σ,
ckσ is the operator that destroys such a state (these operators obey the anti commutation
fermionic relations), nkσ is the number operator of electrons in the state k with spin σ
nkσ = c†kσ ckσ ,
(1.93)
and Vkk0 is the pairing interaction: it destroys a pair of electrons of opposite spins and
momenta, and creates another pair of opposite spins and momenta. Finally, ζk is the single
electron energy, measured from the Fermi energy.
The BCS Hamiltonian may be solved using the mean-field approximation, which we have
already encountered in the discussion of the Heisenberg exchange interaction. We replace
c†k↑ c†−k↓ c−k0 ↓ ck0 ↑ =⇒ hc−k0 ↓ ck0 ↑ ic†k↑ c†−k↓ + hc†k↑ c†−k↓ ic−k0 ↓ ck0 ↑ − hc−k0 ↓ ck0 ↑ ihc†k↑ c†−k↓ i ,
(1.94)
and denote
bk = hc−k↓ ck↑ i , b∗k = hc†k↑ c†−k↓ i .
(1.95)
The model Hamiltonian, HM , that results reads
HM =
X
ζk nkσ −
X
(∆k c†k↑ c†−k↓ + ∆∗k c−k↓ ck↑ − ∆k b∗k ) ,
(1.96)
k
kσ
where we have defined
∆k = −
X
Vkk0 bk0 ≡
k0
X
Vkk0 hc−k↓ ck0 ↑ i .
(1.97)
k0
This model Hamiltonian is diagonalized by the Bogoliubov transformation:
†
†
ck↑ = u∗k γk0 + vk γk1
, c†−k↓ = −vk∗ γk0 + uk γk1
.
(1.98)
∗ ∗ ∗ exercise: Check that in order for the new operators γk0 and γk1 to be fermions
(that is, to obey the anti commutation relations), it suffices that |uk |2 + |vk |2 = 1.
∗ ∗ ∗ exercise: Show that
†
γk0 = uk ck↑ − vk c†−k↓ γk1
= u∗k c†−k↓ + vk∗ ck↑ .
Explain the meaning of these operators.
14
(1.99)
The next step is to insert Eqs. (1.98) into the model Hamiltonian (1.96). This procedure
gives
HM =
X
i
h
†
†
†
†
2
∗ ∗
2
2
ζk (|uk | − |vk | )(γk0 γk0 + γk1 γk1 ) + 2|vk | + 2uk vk γk1 γk0 + 2uk vk γk0 γk1
k
+
Xh
†
†
(∆k uk vk∗ + ∆∗k u∗k vk )(γk0
γk0 + γk1
γk1 − 1) + (∆k vk∗ 2 − ∆∗k u∗k 2 )γk1 γk0
k
+
(∆∗k vk2
−
†
†
∆k u2k )γk0
γk1
+
∆k b∗k
i
.
(1.100)
2ζk uk vk + ∆∗k vk2 − ∆k u2k = 0
(1.101)
Now we see that when the condition
is satisfied, the Hamiltonian becomes diagonal. Moreover, upon multiplying this condition
by ∆∗k /u2k we obtain
∆∗k vk
= (ζk2 + |∆k |2 )1/2 − ζk ≡ Ek − ζk = real .
uk
(1.102)
v E − ζ
1
ζk k
k
k
2
2
=⇒ |vk | = 1 − |uk | =
1−
.
=
uk
|∆k |
2
Ek
(1.103)
This means that
Inserting all these results into the model Hamiltonian (1.100), we finally obtain
HM =
X
X
†
†
Ek (γk0
(ζk − Ek + ∆k b∗k ) +
γk0 + γk1
γk1 ) .
(1.104)
k
k
The first term here is a constant, and the second is just the Hamiltonian of free fermions,
with excitation energy Ek . We see from Eq. (1.102) that the excitation spectrum has a
gap, of magnitude |∆k |. Another important point is that ∆k itself, as given in Eq. (1.97),
becomes
∆k = −
X
Vkk0 hc−k0 ↓ ck↑ i = −
k0
X
Vkk0 u∗k0 vk0 h1 − γk∗0 0 γk0 0 − γk† 0 1 γk0 1 i ,
(1.105)
k0
where, using Eq. (1.102)
u∗k vk =
∆k
.
2Ek
(1.106)
Since the model Hamiltonian is diagonal, we know that
†
†
hγk0
γk0 i = hγk1
γk1 i =
15
1
e
βEk
+1
≡ f (Ek ) .
(1.107)
Using this in Eqs. (1.105) and (1.106) yields the gap equation,
∆k = −
X
Vkk0
k0
∆k0
βE 0
tanh k .
2Ek0
2
(1.108)
In particular, with the simplifying BCS assumption, Vkk0 = −V , we find
1
1 X tanh(βEk /2)
=
.
V
2 k
Ek
(1.109)
One should note that the BCS assumption about the pairing potential V cannot hold over
the entire Brillouin zone; it is expected to be valid in a narrow range of energies about
the Fermi energy. That narrow energy range is limited by a certain energy, ωc . When
the pairing potential is due to the electron-phonon interaction, then ωc is a typical phonon
energy. Taking all this into account, we may convert Eq. (1.109) into the famous BCS form,
p
Z ωc
tanh 12 (β ζ 2 + ∆2 )
1
p
=
,
(1.110)
dζ
N (0)V
ζ 2 + ∆2
0
where N (0) is the density of states at the Fermi level. This equation determines the temperature dependence of the gap ∆, and in particular, it yields the transition temperature
into the superconducting state. At the transition temperature Tc , ∆ vanishes, and therefore
we have
1
=
N (0)V
Z
ωc
dζ
0
tanh 12 (βc ζ)
ζ
16
=⇒ kB Tc = 1.13ωc e−1/N (0)V .
(1.111)