Carsten Timm: Theory of superconductivity

which is valid within the film and away from vortex cores so that |ψ| 2 ∼ = −α/β. However, the current is confined
to the thin film. We can write
j(r) = K(x, y) δ(z) (7.94)
with a surface current density K. Thus
K(x, y) =
∫ ∞
−∞
∫
dz j(r) =
If the thickness is d and A is approximately constant across the thickness, we have
and finally
⇒
film
K(x, y) = − c
4π
j(r) = − c
4π
(
dz −
c )
4πλ 2 A(r). (7.95)
d
A(x, y, 0)
λ2 (7.96)
d
A(r) δ(z)
λ2 (7.97)
∇ 2 A = d A δ(z). (7.98)
λ2 This result exhibits the new length scale that controls the spatial variation of A and thus of the current j,
λ ⊥ := λ2
d , (7.99)
which is large, λ ⊥ ≫ λ, for a thin film. We see that in thin films, λ ⊥ assumes the role of the penetration depth
λ. Since λ ⊥ is large for thin films, one could say that thin films are always effectively of type II.
We here do not discuss the full derivation of Pearl but only consider the “far field” for large r ≫ λ ⊥ and show
that it does not lead to a diverging free energy for a single vortex. We assume that most of the magnetic flux of
Φ 0 penetrates the film for ϱ = √ x 2 + y 2 λ ⊥ . Then the magnetic field above (and below) the film looks like a
monopole field for r ≫ λ ⊥ .
z
B
This field is easy to obtain from symmetry:
∫
da · B = Φ 0 ⇒ 2πr 2 B = Φ 0 (7.100)
By symmetry,
half space
for z>0
⇒ B(r) = Φ 0
2πr 2 and B(r) = Φ 0 ˆr for z > 0. (7.101)
2πr2 B(r) = sgn z Φ 0 ˆr. (7.102)
2πr2 65