Page 404 - Electrical Properties of Materials

386 Superconductivity

Introducing the new parameters,
m
2
λ = (14.60)
2 2
4e ψ μ 0
0
and
2 3/2 eH c μ 0
2
k = λ , (14.61)

and making use of eqns (14.43) and (14.46), we can rewrite eqns (14.56) and
(14.57) in the forms,

2 2 2 3
d ψ k A ψ
= – 1– ψ + (14.62)
2 2 2
dx 2 λ 2 2H λ μ ψ 3
c 0 0
and
2
d A 1 ψ 2
– A = 0. (14.63)
2
dx 2 λ ψ 2
0
In the absence of a magnetic field, A ≡ 0; eqn (14.62) gives ψ = ψ 0 ,asit
should. In the presence of a magnetic field the simplest approximation we can
make is to take κ = 0, which still gives ψ = ψ 0 . From eqn (14.63),
A = A(0)e –x/λ , (14.64)

leading to

1 –x/λ –x/λ
B =– A(0)e = B a e . (14.65)
λ
Thus, we can see that the magnetic flux density inside the super-
conductor decays exponentially, and λ appears as the penetration depth.
Better approximations can be obtained by substituting

ψ = ψ 0 + ϕ (14.66)

into eqns (14.62) and (14.63) and solving them under the assumption that ϕ is
small in comparison with ψ 0 . Then ψ also varies with distance, and B has a
somewhat different decay; but these are only minor modifications and need not
concern us.
The main merit of the Landau–Ginzburg theory is that by including the
kinetic energy of the superconducting electrons in the expression for the Gibbs
free energy, it can show that the condition of minimum Gibbs free energy leads
to the expulsion of the magnetic field. The expulsion is not complete, as we
assumed before in the simple thermodynamic treatment; the magnetic field
can penetrate to a distance, λ, which is typically of the order of 10 nm.
Thus, after all, there can be no such thing as the break-up of the supercon-
ductor into alternating normal and superconducting regions—or can there? We
have solved eqn (14.62) only for the case when κ is very small. There are per-
haps some other regions of interest. It turns out that another solution exists for
the case when