Carsten Timm: Theory of superconductivity

The Gibbs free energy of the domain wall, per unit area, will be denoted by γ. To derive it, first note that for the
given boundary conditions, g s (x → ∞) = g n (x → −∞), i.e., the superconducting free-energy density deep inside
the superconductor equals the normal free-energy density deep inside the normal conductor. This bulk Gibbs
free-energy density is, as derived above,
g s (x → ∞) = g n (x → −∞) = f n (x → −∞) − H2 c
8π = −H2 c
8π . (6.84)
The additional free-energy density due to the domain wall is
( )
g s (x) − − H2 c
= g s (x) + H2 c
8π
8π . (6.85)
The corresponding free energy per area is
γ =
=
∫ ∞
−∞
∫ ∞
−∞
[
]
dx g s (x) + H2 c
=
8π
∫ ∞
−∞
[
dx α |ψ| 2 + β 2 |ψ|4 + 1 ∣( ∣∣∣
2m ∗ i
[
dx f s (x) − B(x)H ]
c
+ H2 c
4π 8π
∂
∂x − q )
c A(x)
ψ
∣
8π − BH ]
c
4π
+ H2 c
. (6.86)
} {{
8π
}
= (B−H c) 2 /8π
2
+ B2
We can simplify this by multiplying the first Ginzburg-Landau equation (6.45) by ψ ∗ and integrating over x:
∫ ∞
−∞
dx
by parts
=
=
∫ ∞
−∞
[
α |ψ| 2 + β |ψ| 4 + 1 ( ∂
2m ∗ ψ∗ i ∂x − q ) ] 2
c A(x) ψ
∫ ∞
−∞
[
dx α |ψ| 2 + β |ψ| 4 + 1 (− 2m ∗ i
∂
∂x − q ) (
c A(x) ψ ∗ (x)
i
∂
∂x − q ) ]
c A(x) ψ
[
dx α |ψ| 2 + β |ψ| 4 + 1 ∣( ∣∣∣ ∂
2m ∗ i ∂x − q )
c A(x) ]
ψ∣
= 0. (6.87)
∣2
Thus
∫ ∞
γ = dx
[− β 2 |ψ|4 + (B − H c) 2 ]
8π
−∞
= H2 c
8π
∫ ∞
−∞
= H2 c
8π
∫ ∞
−∞
[
dx −β β α 2 |ψ|4 +
(1 − B ) ] 2
H c
[ (
dx 1 − B ) ]
2
− |ψ|4 , (6.88)
H c
ψ 4 ∞
where we have drawn out the characteristic energy density H 2 c /8π. The domain wall energy is given by the difference
of the energy cost of expelling the magnetic field and the energy gain due to superconducting condensation.
For strong type-I superconductors, ξ ≫ λ; there is a region of thickness ξ − λ > 0 in which the first term is
already large, while the second only slowly approaches its bulk value (see sketch above). Thus γ > 0 for type-I
superconductors. They therefore tend to minimize the total area of domain walls.
Note that even in samples of relatively simple shape, magnetic flux will penetrate for non-zero applied field. It
will do so in a way that minimizes the total Gibbs free energy, only one contribution to which is the domain-wall
energy. For example in a very large slab perpendicular to the applied field, the flux has to penetrate in some
manner, since going around the large sample would cost much energy. A careful analysis shows that this will
usually happen in the form of normal stripes separated by superconducting stripes, see Tinkham’s book. Such a
state is called an intermediate state. It should not be confused with the vortex state to be discussed later.
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