Carsten Timm: Theory of superconductivity

Since Ω is small, we can expand the integral up to linear order in Ω,
α s ∼
1 =
α n Ω
∫ ∞
−∞
⎛
∫
= − ⎝
[ ] 2 Ds (|ω|) ω 2 − ∆ 2 0
dω
D n (E F ) ω 2 (−Ω) ∂n F
∂ω
−∆ 0
−∞
= −n F (∞)
} {{ }
= 0
= 2 n F (∆ 0 ) =
⎞
∫ ∞ [ ] 2
+ ⎠
|ω| ω 2 − ∆ 2 0
dω √
ω2 − ∆ 2 ω 2
0
∆ 0
⎛
−∆ ∫ 0
∂n F
∂ω = − ⎝
−∞
+
⎞
∫ ∞
∆ 0
⎠ dω ∂n F
∂ω
+ n F (∆ 0 ) − n F (−∆ 0 ) + n F (−∞) = n F (∆ 0 ) + 1 − n F (−∆ 0 )
} {{ }
= 1
2
e β∆ 0 + 1
. (10.107)
Inserting the BCS prediction for ∆ 0 (T ), we can plot α s /α n vs. temperature:
α s
α n
1
0 T c
T
We now turn to the relaxation of nuclear spins due to their coupling to the electrons. We note without derivation
that the hyperfine interaction relevant for this preocess is odd in momentum if the electron spin is not changed
but is even if the electron spin is flipped, i.e.,
B −k,−σ,−k ′ ,−σ ′ = −σσ′ B k ′ σ ′ kσ (10.108)
(recall σσ ′ = ±1). This is called case II. The perturbation Hamiltonian now reads
H NMR = 1 2
∑ ∑ (
B k′ σ ′ kσ u ∗ k ′u kγ † k ′ σ
γ ′ kσ + σu ∗ k ′v kγ † k ′ σ
γ † ′ −k,−σ
kk ′ σσ ′
+ σ ′ v ∗ k ′u kγ −k′ ,−σ ′γ kσ + σσ ′ v ∗ k ′v kγ −k′ ,−σ ′γ† −k,−σ − σσ′ u ∗ ku k ′γ † −k,−σ γ −k ′ ,−σ ′
+ σu ∗ kv k ′γ † −k,−σ γ† k ′ σ ′ + σ ′ v ∗ ku k ′γ kσ γ −k′ ,−σ ′ − v∗ kv k ′γ kσ γ † k ′ σ ′ )
. (10.109)
Assuming u k , v k , ∆ k ∈ R, we get, up to a constant,
H NMR = 1 ∑ ∑ [
)
B k
2
′ σ ′ kσ (u k ′u k + v k ′v k )
(γ † k ′ σ
γ ′ kσ − σσ ′ γ † −k,−σ γ −k ′ ,−σ ′
kk ′ σσ ′ (
)]
+ σ (u k ′v k − v k ′u k ) γ † k ′ σ
γ † ′ −k,−σ − σσ′ γ −k ′ ,−σ ′γ kσ . (10.110)
Compared to ultrasonic attenuation (case I) there is thus a change of sign in both coherence factors. An analogous
derivation now gives the interchanged coherence factors
F + (k, k ′ ) = 1 (
1 + ∆ )
k∆ k ′
(10.111)
2 E k E k ′
for quasiparticle scattering and
F − (k, k ′ ) = 1 2
(
1 − ∆ )
k∆ k ′
E k E k ′
(10.112)
104