Carsten Timm: Theory of superconductivity

tant. The first one is
(u k ′u k − v k ′v k ) 2 = 1 4
{(
1 +
)
ξ k ′
ξ
√
(1 + √ k
ξ
2
k ′ + ∆ 2 k ξ
2 ′ k
+ ∆ 2 k
ξ k ′ ξ
− 2√ √ k
+
ξ
2
k ′ + ∆ 2 k ξ
2 ′ k
+ ∆ 2 k
= 1 (
1 + ξ kξ k ′
− ∆ )
k∆ k ′
2 E k E k ′ E k E k ′
(
1 −
)
)
ξ k ′
ξ
√
(1 − √ k
ξ
2
k ′ + ∆ 2 k ξ
2 ′ k
+ ∆ 2 k
)}
(10.99)
with E k = √ ξk 2 + ∆2 k
. If the matrix elements B and the normal-state density of states are even functions of the
normal-state energy relative to the Fermi energy, ξ k , the second term, which is odd in ξ k and ξ k ′, will drop out
under the sum ∑ kk
. Hence, this term is usually omitted, giving the coherence factor
′
F − (k, k ′ ) := 1 (
1 − ∆ )
k∆ k ′
(10.100)
2 E k E k ′
relevant for quasiparticle scattering in ultrasound experiments.
factor
(
σ 2 (u k ′v k + v k ′u k ) 2 = 1 + ∆ k∆ k ′
E k E k ′
}{{}
= 1
Analogously, we obtain the second coherence
)
=: F + (k, k ′ ) (10.101)
for quasiparticle creation and annihilation.
We can gain insight into the temperature dependence of ultrasound attenuation by making the rather crude
approximation that the matrix element B is independent of k, k ′ and thus of energy. Furthermore, typical
ultrasound frequencies Ω satisfy Ω ≪ ∆ 0 and Ω ≪ k B T . Then only scattering of quasiparticles by phonons
but not their creation is important since the phonon energy is not sufficient for quasiparticle creation.
The rate of ultrasound absorption (attenuation) can be written in a plausible form analogous to the current
in the previous section:
where now
α s ∝
∫ ∞
−∞
dω D s (|ω|) D s (|ω + Ω|) |B| 2 F − (ω, ω + Ω) [n F (ω) − n F (ω + Ω)] , (10.102)
F − (ω, ω ′ ) = 1 2
With the approximations introduced above we get
α s ∝ D 2 n(E F ) |B| 2
∫∞
−∞
( )
1 − ∆2 0
|ω| |ω ′ . (10.103)
|
dω D (
s(|ω|) D s (|ω + Ω|) 1 ∆ 2 )
0
1 −
[n F (ω) − n F (ω + Ω)] . (10.104)
D n (E F ) D n (E F ) 2 |ω| |ω + Ω|
The normal-state attentuation rate is found by letting ∆ 0 → 0:
⇒
α n ∝ D 2 n(E F ) |B| 2
α s
α n
= 1 Ω
∫ ∞
−∞
∫∞
−∞
dω n F (ω) − n F (ω + Ω)
2
dω D s(|ω|) D s (|ω + Ω|) |ω| |ω + Ω| − ∆ 2 0
D n (E F ) D n (E F ) |ω| |ω + Ω|
= 1 2 D2 n(E F ) |B| 2 Ω (10.105)
[n F (ω) − n F (ω + Ω)] . (10.106)
103