Carsten Timm: Theory of superconductivity

10.6 Ginzburg-Landau-Gor’kov theory
We conclude this chapter by remarking that Lev Gor’kov managed, two years after the publication of BCS theory,
to derive Ginzburg-Landau theory from BCS theory. The correspondence is perfect if the gap is sufficiantly small,
i.e., T is close to T c , and the electromagnetic field varies slowly on the scale of the Pippard coherence length ξ 0
(see Sec. 5.4). These are indeed the conditions under which Ginzburg and Landau expected their theory to be
valid.
Gor’kov used equations of motion for electronic Green functions, which he decoupled with a mean-field-like
approximation, which allowed for spatial variations of the decoupling term ∆(r). The derivation is given in
Schrieffer’s book and we omit it here. Gor’kov found that in order to obtain the Ginzburg-Landau equations, he
had to take
as anticipated, and (using our conventions)
ψ(r) =
q = −2e, (10.119)
m ∗ = 2m, (10.120)
√
7ζ(3)
4π
√
n
0 s
∆(r)
k B T c
, (10.121)
where
n 0 s :=
n s
1 − T T c
∣ ∣∣∣∣T
→T
−
c
. (10.122)
Recall that n s ∼ 1 − T/T c close to T c in Ginzburg-Landau theory. The spatially dependent gap is thus locally
proportional to the Ginzburg-Landau “condensate wavefunction” or order parameter ψ(r).
Since we have already found that London theory is a limiting case of Ginzburg-Landau theory, it is also a
limiting case of BCS theory. But London theory predicts the two central properties of superconductors: Ideal
conduction and flux expulsion. Thus Gor’kov’s derivation also shows that BCS theory indeed describes a superconducting
state. (Historically, this has been shown by BCS before Gor’kov established the formal relationship
between the various theories.)
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