Carsten Timm: Theory of superconductivity

This gives a contribution to the field energy of
∫
∫
d 3 r B2
8π = Φ2 0
32π 3
d 3 r 1 ∫∞
r 4 = Φ2 0
32π 3 4π dr
r 2 = Φ2 0 1
8π 2 , (7.103)
λ ⊥
λ ⊥
which is finite for an infinity film. The value of the lower cutoff does not matter for this. The cutoff has to be
present, since the monopole field is not a valid approximation for small r.
We next obtain the current from Ampère’s law in integral form:
d
∆r
B
∮
dr · B = 4π c
∫
da · j (7.104)
and with the vector character restored
Note that the sheet current is thus
⇒
r+∆r ∫
2 dr ′ Φ 0
2π(r ′ ) 2 = 4π ∆r d j(r)
c
(7.105)
⇒
r
c ∆r d j(r) = Φ ( )
0 1
π r − 1
∼= Φ 0 ∆r
r + ∆r π r 2 (7.106)
⇒ j(r) = Φ 0 1
4π 2 r 2 d
(7.107)
j(r) = Φ 0
4π 2
K(r) = Φ 0
4π 2
ˆφ
r 2 d . (7.108)
ˆφ
r 2 . (7.109)
For large r we have |ψ| = ψ 0 = √ −α/β (note that typically λ ⊥ ≫ ξ). Then the second Ginzburg-Landau
equation gives
j = i q
2m ∗ (−i∇ϕ − i∇ϕ)ψ2 0 −
q2
m ∗ c ψ2 0A = ψ0
2 q
(
m ∗ ∇ϕ − q )
c A . (7.110)
Thus we can rewrite the gradient term in the free energy as
∫
d
d 2 r
The contribution from large r is
1
2m ∗ ψ2 0
(∇ϕ − q ) 2
∫
c A = d
d 2 r ψ2 0(m ∗ ) 2
2m ∗ ψ 4 0 q2 j · j = d ∫
m ∗
d 2 r
2q 2 ψ0
2 j · j. (7.111)
dm ∗ ∫
2q 2 ψ0
2
d 2 r Φ2 0 1
16π 4 r 4 d 2 = m∗ Φ 2 ∫∞
0
32π 4 q 2 ψ0 2d 2π
λ ⊥
dr
r 3 = m∗ Φ 2 0 1
8π 3 q 2 ψ0 2d , (7.112)
λ 2 ⊥
which is also finite for an infinite film.
We conclude that the free energy of an isolated vortex is finite in a superconducting film. It is then plausible
and indeed true that the interaction energy of a vortex-antivortex pair does not diverge for large separations r but
saturates for r ≫ λ ⊥ . Since for the far field of a single vortex the magnetic-field energy, Eq. (7.103), dominates
over the energy due to the gradient term, we expect the large-r interaction to be dominated by the Coulomb-type
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