Carsten Timm: Theory of superconductivity

As above, we keep only terms up to second order in the fluctuations (Gaussian approximation), i.e., terms
proportional to δψ 2 , δψ ϕ, or ϕ 2 . We get
∫ [
F [δψ, ϕ] ∼ ✭
= d 3 r ✭✭✭✭✭✭✭
2αψ 0 δψ + 2βψ0δψ 3 + αδψ 2 + 3βψ0δψ 3 2 + 1 ( ) ∗
2m ∗ i ∇δψ ·
i ∇δψ
(
+ 1 ( ) )
]
∗
2m
✭✭✭✭✭✭✭✭✭✭✭✭✭✭✭✭✭✭
∗ i ∇δψ · ψ 0 ∇ϕ + (ψ 0 ∇ϕ) ∗ ·
i ∇δψ + 1
2m ∗ (ψ 0∇ϕ) ∗ · ψ 0 ∇ϕ (+ const). (6.40)
Note that the first-order terms cancel since we are expanding about a minimum. Furthermore, up to second order
there are no terms mixing amplitude fluctuations δψ and phase fluctuations ϕ. We can simplify the expression:
[
−2αδψ 2 + 1
2m ∗ (
i ∇δψ ) ∗
·
i ∇δψ −
∫
F [δψ, ϕ] ∼ = d 3 r
= ∑ [(
−2α + 2 k 2 )
}{{} 2m ∗ δψkδψ ∗ k − α β
k > 0
2 k 2
2m ∗
} {{ }
> 0
]
2 α
2m ∗ β (∇ϕ)∗ ∇ϕ
ϕ ∗ kϕ k
], (6.41)
in analogy to the case T > T c . We see that amplitude fluctuations are gapped (massive) with an energy gap
−2α = α ′ (T c − T ). They are not degenerate. Phase fluctuations on the other hand are ungapped (massless) with
quadratic dispersion
ϵ ϕ k = −α 2 k 2
β 2m ∗ . (6.42)
The appearance of ungapped so-called Goldstone modes is characteristic for systems with spontaneously broken
continuous symmetries. We state without derivation that the heat capacity diverges like C ∼ √ T c − T for
T → Tc − , analogous to the case T > T c .
6.3 Ginzburg-Landau theory for superconductors
To describe superconductors, we have to take the charge q of the particles forming the condensate into account.
We allow for the possibility that q is not the electron charge −e. Then there are two additional terms in the
Landau functional:
• The canonical momentum has to be replaced by the kinetic momentum:
where A is the vector potential.
• The energy density of the magnetic field, B 2 /8π, has to be included.
i ∇ → i ∇ − q A, (6.43)
c
Thus we obtain the functional
∫
F [ψ, A] ∼ =
[
d 3 r α |ψ| 2 + β 2 |ψ|4 + 1 ∣( ∣∣∣
2m ∗ i ∇ − q )
c A ψ
∣
2
+ B2
8π
]
. (6.44)
Minimizing this free-energy functional with respect to ψ, we obtain, in analogy to the previous section,
1
2m ∗ (
i ∇ − q c A ) 2
ψ + α ψ + β |ψ| 2 ψ = 0. (6.45)
37