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Superconductivity and Superfluidity
where a and b are model parameters. Using the principle of minimum free energy of the
system in a steady state, we can find the relation between these parameters:
d(∆W )
dn s
= −a + b · n s = 0. (4.13)
Whence
b = a n s
(4.14)
and the energy gain in the transition to an ordered state:
∆W = − a 2 n s. (4.15)
The reverse transition from the superconducting state to a normal state occurs at the
critical magnetic field strength, H c . This is required to create the density of the magnetic
energy H2 c
8π
. According to the above description, this equation is therefore obtained:
H 2 c
8π = a 2 n s. (4.16)
In order to express the parameter a of GL-theory in terms of physical characteristics of
a sample, the density of “superconducting” carriers generally charge from the London’s
equation (4.8). 2
The important step in the Ginzburg-Landau theory is the changeover of the
concentration of superconducting carriers, n s , to the order parameter Ψ
|Ψ(x)| 2 = n s . (4.17)
At this the standard Schrodinger equation (in case of one dimension) takes the form:
−
2m [∇Ψ(x)]2 − aΨ 2 (x) + b 2 Ψ4 (x) = E. (4.18)
Again using the condition of minimum energy
dE
dΨ = 0 (4.19)
2 It should be noted that due to the fact that the London equation does not correctly describes the ratio of the
penetration depth with a density of carriers, one should used the revised equation (9.13) in order to find the
a.
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