19.4. Flux quantization in a superconductor
269
FIGURE 19.2
Magnetic field B and modulus of the macroscopic (pair) wavefunction |ψ| in
the neighbourhood of a flux filament.
type II case |ψ| has the variation shown in this figure. Solutions of the coupled
GL equations for A and ψ can be obtained which exhibit this behaviour.
An important result is that the flux through a vortex line is quantized. To
see this, we write
ψ = e
iφ
|ψ|
(19.57)
as before. The expression for the electromagnetic current is
q
2
(
)
φ
j em = −
A
∇
m
− q
|ψ|
2
(19.58)
as in (19.26), but in (19.58) we are leaving the charge parameter q undetermined for the moment; the mass parameter m will be unimportant. Rearranging, we have
m
φ
A = −
j em +
∇ .
(19.59)
q 2 |ψ| 2
q
Let us integrate equation (19.59) around any closed loop C in the type II
superconductor, which encloses a flux (or vortex) line. Far enough away from
the vortex, the screening currents j em will have dropped to zero, and hence
∮
∮
1
1
A · ds = q
∇φ · ds = [φ]
(19.60)
q
C
C
C
where [φ] is the change in phase around . If the wavefunction ψ is singleC
C
valued, the change in phase [φ] for any closed path can only be zero or an
C
integer multiple of 2π. Transforming the left-hand side of (19.60) by Stokes’
Theorem, we obtain the result that the flux Φ through any surface spanning
269
FIGURE 19.2
Magnetic field B and modulus of the macroscopic (pair) wavefunction |ψ| in
the neighbourhood of a flux filament.
type II case |ψ| has the variation shown in this figure. Solutions of the coupled
GL equations for A and ψ can be obtained which exhibit this behaviour.
An important result is that the flux through a vortex line is quantized. To
see this, we write
ψ = e
iφ
|ψ|
(19.57)
as before. The expression for the electromagnetic current is
q
2
(
)
φ
j em = −
A
∇
m
− q
|ψ|
2
(19.58)
as in (19.26), but in (19.58) we are leaving the charge parameter q undetermined for the moment; the mass parameter m will be unimportant. Rearranging, we have
m
φ
A = −
j em +
∇ .
(19.59)
q 2 |ψ| 2
q
Let us integrate equation (19.59) around any closed loop C in the type II
superconductor, which encloses a flux (or vortex) line. Far enough away from
the vortex, the screening currents j em will have dropped to zero, and hence
∮
∮
1
1
A · ds = q
∇φ · ds = [φ]
(19.60)
q
C
C
C
where [φ] is the change in phase around . If the wavefunction ψ is singleC
C
valued, the change in phase [φ] for any closed path can only be zero or an
C
integer multiple of 2π. Transforming the left-hand side of (19.60) by Stokes’
Theorem, we obtain the result that the flux Φ through any surface spanning