Carsten Timm: Theory of superconductivity

This is called the BCS gap equation for reasons to be discussed below. When we have solved it, it is easy to
obtain the original variational parameters in terms of ∆ k ,
(
)
u 2 k = 1 ξ k
1 + √ , (9.29)
2 ξ
2
k
+ ∆ 2 k
(
)
vk 2 = 1 ξ k
1 − √ , (9.30)
2 ξ
2
k
+ ∆ 2 k
∆ k
u k v k =
2 √ ξk 2 + . (9.31)
∆2 k
The relative sign of u k and v k is thus the sign of ∆ k . The absolute sign of, say, u k is irrelevant because of the
invariance of E BCS under simultaneous phase rotations of u k , v k (consider a phase factor of e iπ = −1).
For our special interaction
{
−V 0 for |ξ k | , |ξ k ′| < ω D ,
V kk ′ =
(9.32)
0 otherwise,
the BCS gap equation becomes
⎧
⎪⎨ − 1 ∑
∆ k ′
(−V 0 )
∆ k = N
2 √ for |ξ
ξ 2 k
⎪⎩
′ , |ξ k ′ | < ω D k
+ ∆ 2 k | < ω D ,
′ k ′
0 otherwise,
(9.33)
which can be solved with the ansatz
∆ k =
{
∆ 0 > 0 for |ξ k | < ω D ,
0 otherwise.
(9.34)
We obtain
∆ 0 = V 0
N
⇒ 1 = V 0
N
∑
∆ 0
2 √ ξ
k ′
k 2 + ∆ 2 ′ 0
|ξ k ′ | < ω D
∑ 1
2 √ ξ
k ′
k 2 + ∆ 2 ′ 0
|ξ k ′ | < ω D
∫ω D
= V 0
(9.35)
1
dξ D(µ + ξ)
2 √ , (9.36)
ξ 2 + ∆ 2 0
−ω D
where D(ϵ) is the density of states per spin direction and per unit cell. If the density of states is approximately
constant within ±ω D of the Fermi energy, we obtain
⇒
1 = V 0 D(E F ) 1 2
sinh
∫ω D
−ω D
dξ
√ = V
ξ2 + ∆ 2 0 D(E F ) Arsinh ω D
(9.37)
∆
0
0
1
V 0 D(E F ) = ω D
∆ 0
(9.38)
1
⇒ ∆ 0 = ω D 1
. (9.39)
sinh
V 0 D(E F )
In the so-called weak-coupling limit of small V 0 D(E F ), this result simplifies to
(
)
∆ 0
∼
1
= 2ωD exp −
. (9.40)
V 0 D(E F )
87