Page 400 - Electrical Properties of Materials

382 Superconductivity

superconductors should break up into normal and superconducting domains;
experiments show that they do not break up. Consequently, the theory is wrong.
The theory cannot be completely wrong, however, for it predicted the correct
relationship for specific heat. So instead of dismissing the theory altogether,
we modify it by introducing the concept of surface energy. This would suggest
that the material does not break up because maintaining boundaries between
the normal and superconducting domains is a costly business. It costs energy. ∗
∗ This is really the same argument that Hence, the simple explanation for the absence of domains is that the reduction
we used for domains in ferromagnetic in energy resulting from the configuration shown in Fig. 14.10 is smaller than
materials. On the one hand, the more do-
mains we have the smaller is the external the energy needed to maintain the surfaces.
magnetic energy. On the other hand, the The introduction of surface energy is certainly a way out of the dilemma,
more domains we have, the larger is the but it is of limited value unless we can give some quantitative relationships
energy needed to maintain the domain for the maintenance of a wall. The answer was given at about the same time
walls. So, the second consideration will by Pippard and (independently) by Landau and Ginzburg. We shall discuss the
limit the number of domains.
latter theory because it is a little easier to follow.

14.6 The Landau–Ginzburg theory

With remarkable intuition Landau and Ginzburg suggested (in 1950) a
Landau received the Nobel Prize formulation that was later (1958) confirmed by the microscopic theory. We
in 1962, and Ginzburg in 2003. shall give here the essence of their arguments, though in a somewhat modified
form to fit into the previous discussion.
1. In the absence of a magnetic field, below the critical temperature, the Gibbs
∗
∗ From now on, for simplicity, all our free energy is G s (0).
quantities will be given per unit volume. 2. If a magnetic field H a is applied and is expelled from the interior of the su-
1
2
perconductor, the energy is increased by μ 0 H per unit volume. This may
2 a
1
2
be rewritten with the aid of flux density as (1/μ 0 )B . If we now abandon
2 a
the idea of a perfect diamagnet, the magnetic field can penetrate the super-
conductor, and the flux density at a certain point is B instead of zero. Hence
the flux density expelled is not B a but only B a – B, and the corresponding
increase in the Gibbs free energy is
1 1 2
(B a – B) . (14.36)
2 μ 0
3. All superconducting electrons are apparently doing the same thing. We,
therefore, describe them by the same wave function, ψ, where
2
|ψ(x, y, z)| = N s , (14.37)
the density of superconducting electrons. In the absence of an applied
magnetic field the density of superconducting electrons is everywhere the
same.
4. In the presence of a magnetic field the density of superconducting elec-
trons may vary in space, that is ∇ψ = 0. But, you may remember, –i ∇ψ
gives the momentum of the particle. Hence, the kinetic energy of our
superconducting electrons,
1 2
KE = |–i ∇ψ| , (14.38)
2m