Carsten Timm: Theory of superconductivity

10.5 Ultrasonic attenuation and nuclear relaxation
To conclude the brief survey of experimental consequences of BCS theory, we discuss the effects of time-dependent
perturbations. They will be exemplified by ultrasonic attenuation and nuclear relaxation, which represent two
distinct ways in which quasiparticle interference comes into play. Quite generally, we write the perturbation part
of the Hamiltonian as
H 1 = ∑ kk ′ ∑
σσ ′ B k ′ σ ′ kσ c † k ′ σ ′ c kσ , (10.91)
where B k′ σ ′ kσ are matrix elements of the perturbation between single-electron states of the non-interacting system.
In the superconducting state, we have to express c, c † in terms of γ, γ † ,
c k↑ = u k γ k↑ + v k γ † −k,↓ , (10.92)
c k↓ = u k γ k↓ − v k γ † −k,↑ , (10.93)
where we have assumed u −k = u k , v −k = v k . Thus, with σ, σ ′ = ±1,
H 1 = ∑ ∑
) (
)
(u ∗ k ′γ† k ′ σ
+ σ ′ v ∗ ′ k ′γ −k ′ ,−σ ′ u k γ kσ + σv k γ † −k,−σ
kk ′
= ∑ kk ′ ∑
σσ ′ B k′ σ ′ kσ
σσ ′ B k′ σ ′ kσ
(
u ∗ k ′u kγ † k ′ σ ′ γ kσ + σu ∗ k ′v kγ † k ′ σ ′ γ † −k,−σ
+ σ ′ v ∗ k ′u kγ −k′ ,−σ ′γ kσ + σσ ′ v ∗ k ′v kγ −k′ ,−σ ′γ† −k,−σ
)
. (10.94)
It is useful to combine the terms containing B k ′ σ ′ kσ and B −k,−σ,−k ′ ,−σ ′ since both refer to processes that change
momentum by k ′ − k and spin by σ ′ − σ. If the perturbation couples to the electron concentration, which is the
case for ultrasound, one finds simply
B −k,−σ,−k′ ,−σ ′ = B k ′ σ ′ kσ. (10.95)
This is often called case I. Furthermore, spin is conserved by the coupling to ultrasound, thus
Adding the two terms, we obtain
H ultra = 1 2
∑ ∑
kk ′ σ
B k ′ σkσ
B k′ σ ′ kσ = δ σσ ′ B k′ σkσ. (10.96)
(
u ∗ k ′u kγ † k ′ σ γ kσ + σu ∗ k ′v kγ † k ′ σ γ† −k,−σ
+ σvk ∗ ′u kγ −k′ ,−σγ kσ + vk ∗ ′v kγ −k′ ,−σγ † −k,−σ + u∗ ku k ′γ † −k,−σ γ −k ′ ,−σ
)
− σu ∗ kv k ′γ † −k,−σ γ† k ′ σ − σv∗ ku k ′γ kσ γ −k′ ,−σ + vkv ∗ k ′γ kσ γ † k ′ σ
. (10.97)
Assuming u k , v k , ∆ k ∈ R for simplicity, we get, up to a constant,
H ultra = 1 2
∑ ∑
kk ′ σ
B k′ σkσ
+ σ (u k ′v k + v k ′u k )
[
(
)
(u k ′u k − v k ′v k ) γ † k ′ σ γ kσ + γ † −k,−σ γ −k ′ ,−σ
(γ † k ′ σ γ† −k,−σ + γ −k ′ ,−σγ kσ
)]
. (10.98)
We thus find effective matrix elements B k ′ σkσ(u k ′u k − v k ′v k ) for quasiparticle scattering and B k ′ σkσ σ(u k ′v k +
v k ′u k ) for creation and annihilation of two quasiparticles. Transition rates calculated from Fermi’s golden rule
contain the absolute values squared of matrix elements. Thus the following two coherence factors will be impor-
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