Page 401 - Electrical Properties of Materials

The Landau–Ginzburg theory 383

will add to the total energy. It follows then that the appearance of alternating
layers of normal and superconducting domains is energetically unfavour-
able because it leads to a rapid variation of ψ, giving a large kinetic energy
contribution to the total energy.
Equation (14.38) is not quite correct. It follows from classical
∗
electrodynamics that in the presence of a magnetic field the momentum ∗ For a discussion, see The Feynman lec-
is given by p – eA, where A is the magnetic vector potential. Hence, the tures on Physics, vol. 3, pp. 21–5.
correct formula for the kinetic energy is

1 2 2e is the charge on a superconduct-
KE = |–i ∇ψ –2eAψ| . (14.39)
2m ing electron.

We may now write the Gibbs free energy in the form,

1 2 1 2
G s (B)= G s (0) + (B a – B) + |–i ∇ψ –2eAψ| . (14.40)
2μ 0 2m

5. The value of the Gibbs function at zero magnetic field should depend on
the density of superconducting electrons, among other things. The simplest
choice is a polynomial of the form,
2
4
G s (0) = G n (0) + a 1 |ψ| + a 2 |ψ| , (14.41)
where the coefficients may be determined from empirical considerations. At
a given temperature the density of superconducting electrons will be such
as to minimize G s (0), that is,

∂G s (0)
= 0, (14.42)
∂|ψ| 2

leading to

2
2
|ψ| ≡ |ψ 0 | =– a 1 . (14.43)
2a 2
2
Substituting this value of |ψ| back into eqn (14.41) we get

a 2 1
G s (0) = G n – . (14.44)
4a 2
Let us go back now to eqn (14.22) (rewritten for unit volume),

2
1
G s (0) = G n – μ 0 H . (14.45)
2 c
Comparing the last two equations, we get
It is assumed that a 2 > 0 and
H c =–a 1 /(2a 2 μ 0 ) 1/2 . (14.46) a 1 < 0.