Carsten Timm: Theory of superconductivity

Note that this only works for ∆ 0 ≠ 0 since we have divided by ∆ 0 . If the density of states is approximately
constant close to E F , the equation simplifies to
∫ω D
1 ∼ = V 0 D(E F ) dξ tanh √ β
2 ξ2 + ∆ 2 0
2 √ . (10.42)
ξ 2 + ∆ 2 0
−ω D
The integral is easily evaluated numerically, leading to the temperature dependence of ∆ 0 :
∆ 0
∆ (0)
0
0
T c
T
For weak coupling we have already seen that
(
)
∆ 0 (0) ∼ 1
= 2ω D exp −
. (10.43)
V 0 D(E F )
We can also obtain an analytical expression for T c : If T approaches T c from below, we can take the limit ∆ 0 → 0
in the gap equation,
1 ∼ = V 0 D(E F )
∫ω D
−ω D
by parts
= V 0 D(E F )
dξ tanh β 2 |ξ|
2 |ξ|
{
ln βω D
2
∫
= V 0 D(E F )
tanh βω D
2
−
∫
ω D
0
βω D /2
0
dξ tanh β 2 ξ
ξ
= V 0 D(E F )
∫
βω D /2
0
dx tanh x
x
}
dx
ln x
cosh 2 x
. (10.44)
In the weak-coupling limit we have βω D = ω D /k B T ≫ 1 (this assertion should be checked a-posteriori). Since
the integrand of the last integral decays exponentially for large x, we can send the upper limit to infinity,
{
1 ∼ = V 0 D(E F ) ln βω D
tanh βω ∫ ∞ }
D
− dx
ln x
(
∼=
2
} {{
2
} cosh 2 V 0 D(E F ) ln βω D
+ γ − ln π )
, (10.45)
x
2
4
0
∼ = 1
where γ is again the Euler gamma constant. This implies
( )
1
exp
∼= 2βω D
e γ (10.46)
V 0 D(E F ) π
⇒ k B T c
∼
2e γ (
)
=
π
ω 1
D exp −
. (10.47)
V 0 D(E F )
This is exactly the same expression we have found above for the critical temperature of the Cooper instability.
Since the approximations used are quite different, this agreement is not trivial. The gap at zero temperature and
the critical temperature thus have a universal ratio in BCS theory,
2∆ 0 (0)
k B T c
∼
2π = ≈ 3.528. (10.48)
eγ This ratio is close to the result measured for simple elementary superconductors. For example, for tin one
finds 2∆ 0 (0)/k B T c ≈ 3.46. For superconductors with stonger coupling, such as mercury, and for unconventional
superconductors the agreement is not good, though.
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