Page 405 - Electrical Properties of Materials

The Landau–Ginzburg theory 387

ψ ψ 0 and B = B a . (14.67)

So we claim now that there is a solution where the magnetic field can penet-
rate the whole superconducting material, and this happens when the density
of superconducting electrons is small. Then (choosing for this case the vector
potential zero at x =0),

A(x)= B a x, (14.68)
n is an integer; otherwise ψ di-
and neglecting the last term in eqn (14.62) we get verges as x →∞.

2 2
2
2
d ψ κ B x
a
=– 1– ψ. (14.69)
2 2 2
dx 2 λ 2 2H λ μ 0
c
Now this happens to be a differential equation that has been thoroughly
investigated by mathematicians. They maintain that a solution exists only when
√
B a = μ 0 H c κ 2/(2n + 1). (14.70)
The maximum value of B a occurs at n =0, giving

√
B a = μ 0 H c κ 2. (14.71)
√
When κ> 1/ 2, the magnetic field inside the superconductor may exceed
the critical field. You may say this is impossible. Have we not defined the crit-
ical field as the field that destroys superconductivity? We have, but that was
done on the basis of diamagnetic properties. We defined the critical field only
for the case when the magnetic field is expelled. Abrikosov (Nobel Prize, 2003)
argued, still within the Landau–Ginzburg theory, that in certain materials for
√
which κ> 1/ 2, superconductivity may exist up to a magnetic field, H c2 .The
new critical magnetic field is related to the old one by the relationship

√
H c2 = κ 2H c . (14.72)
–M
Up to H c1 the superconductor is diamagnetic, as shown in Fig. 14.11,
where –M is plotted against the applied magnetic field. Above H c1 the mag-
netic field begins to penetrate (beyond the ‘diamagnetic’ penetration depth)
and there is complete penetration at H c2 , where the material becomes nor-
mal. Materials displaying such a magnetization curve are referred to as type
II superconductors, while those expelling the magnetic field until they become
normal (dotted lines in Fig. 14.11) are called type I superconductors. H c1 H c H c2 H
A two-dimensional analysis of a type II superconductor shows that the in-
tensity of the magnetic field varies in a periodic manner with well-defined Fig. 14.11
maxima as shown in Fig. 14.12(a). Since the current and the magnetic field Magnetization curves for type I and
are uniquely related by Maxwell’s equations, the current is also determined. It type II superconductors. The area
is quite clear physically that the role of the current is either ‘not to let in’ or under both magnetization curves is
‘not to let out’ the magnetic field. the same.