4.2 Ginzburg–Landau Theory
61
is independent of the temperature. For clean superconductors, however, ξ and ξ 0 are
roughly the same at low temperatures. If the number density of scattering centers
for electrons such as impurities increases, the electron mean free path, l, decreases,
which results in a decrease in the coherence length as (ξ 0 l)
1/2 . ξ 0 does not change
with l in this case either.
(7) Upper critical field and type II superconductor
In the vicinity of the upper critical field the distance between flux lines decreases,
and hence, the magnetic flux overlaps, resulting in an almost uniform magnetic flux
density. Hence, we can assume that B ∼ = μ 0 H 0 around the transition to the normal
state. The order parameter has a small value and the term proportional to |
4 can be
safely neglected in the free energy density given by (4.2). Rewriting the kinetic energy
density, the Ginzburg–Landau free energy density is expressed as (see Appendix A.2)
s (B) = F n (0) + μ 0 H c
2
2
+ 2ξ
2
(∇|ψ|)
2
+
1
2
μ 0 H 0
2
.
(4.32)
Here, represents a spatial average. The Gibbs free energy density in the
superconducting state is given by
(4.33)
On the other hand, the Gibbs free energy density in the normal state is
(4.34)
Hence, the transition to the normal state occurs at the magnetic field at which the
spatial variation in the second term in (4.33) is zero. That is, the upper critical
field is the magnetic field at which the increase in the kinetic energy consumes the
condensation energy. Abrikosov determined the upper critical field as the maximum
value of the magnetic field at which (4.3), with neglect of the higher order term of
, has a solution. This theoretical treatment leads to the same result as the above
treatment using the Gibbs free energy density, although the latter process is essential
from the viewpoint of thermodynamics. According to the result Abrikosov obtained,
the upper critical field is given by
H c2 =
4eμ 0 H c
2
λ
2
.
(4.35)
The ratio of the two characteristic lengths
κ =
λ
ξ
(4.36)
61
is independent of the temperature. For clean superconductors, however, ξ and ξ 0 are
roughly the same at low temperatures. If the number density of scattering centers
for electrons such as impurities increases, the electron mean free path, l, decreases,
which results in a decrease in the coherence length as (ξ 0 l)
1/2 . ξ 0 does not change
with l in this case either.
(7) Upper critical field and type II superconductor
In the vicinity of the upper critical field the distance between flux lines decreases,
and hence, the magnetic flux overlaps, resulting in an almost uniform magnetic flux
density. Hence, we can assume that B ∼ = μ 0 H 0 around the transition to the normal
state. The order parameter has a small value and the term proportional to |
4 can be
safely neglected in the free energy density given by (4.2). Rewriting the kinetic energy
density, the Ginzburg–Landau free energy density is expressed as (see Appendix A.2)
s (B) = F n (0) + μ 0 H c
2
2
+ 2ξ
2
(∇|ψ|)
2
+
1
2
μ 0 H 0
2
.
(4.32)
Here, represents a spatial average. The Gibbs free energy density in the
superconducting state is given by
(4.33)
On the other hand, the Gibbs free energy density in the normal state is
(4.34)
Hence, the transition to the normal state occurs at the magnetic field at which the
spatial variation in the second term in (4.33) is zero. That is, the upper critical
field is the magnetic field at which the increase in the kinetic energy consumes the
condensation energy. Abrikosov determined the upper critical field as the maximum
value of the magnetic field at which (4.3), with neglect of the higher order term of
, has a solution. This theoretical treatment leads to the same result as the above
treatment using the Gibbs free energy density, although the latter process is essential
from the viewpoint of thermodynamics. According to the result Abrikosov obtained,
the upper critical field is given by
H c2 =
4eμ 0 H c
2
λ
2
.
(4.35)
The ratio of the two characteristic lengths
κ =
λ
ξ
(4.36)