Carsten Timm: Theory of superconductivity

vortex core. For a vortex along the z-axis we choose cylindrical coordinates ϱ, φ, z. We then have to solve the
Ginzburg-Landau equations with the boundary conditions
From symmetry, we have
and we can choose
where
ψ(ϱ = 0) = 0, (6.95)
|ψ(ϱ → ∞)| = ψ 0 , (6.96)
B(ϱ → ∞) = 0. (6.97)
ψ(r) = |ψ| (ϱ) e iϕ = |ψ| (ϱ) e −iφ , (6.98)
v s (r) = ˆφ v s (ϱ), (6.99)
B(r) = ẑB(ϱ) (6.100)
A(r) = ˆφA(ϱ) (6.101)
B(ϱ) = 1 ϱ
∂
ϱA(ϱ). (6.102)
∂ϱ
Choosing a circular integration path of radius r centered at the vortex core, the enclosed fluxoid is
Φ ′ (r) = Φ(r) − mc ∮
ds · v s = 2πrA(r) − mc
e
e 2πr v s(r) = ! Φ 0
⇒
∂S
A(r) − mc
e v s(r) = Φ 0
2πr . (6.103)
This relation follows from fluxoid quantization and thus ultimately from the second Ginzburg-Landau equation.
To obtain j(r), ψ(r), and A(r), one also has to solve the first Ginzburg-Landau equation
and Ampère’s Law
(
1
4m i ∇ + 2e ) 2
c A ψ + α ψ + β |ψ| 2 ψ = 0 (6.104)
j = c ∇ × B. (6.105)
4π
This cannot be done analytically because of the non-linear term β |ψ| 2 ψ. For small distances ϱ from the core,
one can drop this term, thereby linearizing the Ginzburg-Landau equation. The solution is still complicated, the
result is that |ψ| increases linearly in ϱ. Numerical integration of the full equations gives the results sketched here
for a cut through the vortex core:
~ λ
ψ(
r)
B( r)
0
~ ξ
x
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