62
4 Fundamental Electromagnetic Properties of Superconductors
is called the Ginzburg-Landau parameter. κ is almost independent of temperature.
We have | ∞ |
2
= μ 0 H c
2
/|α| using (4.6) and (4.12). Then, (4.17) is rewritten as
λ = (m
∗
|α|)
1/2
/(2eμ 0 H c ). Eliminating (m
∗
|α|)
1/2 in the expression of ξ in terms of
λ, we have
κ =
2
√
2eμ 0 H c λ
2
.
(4.37)
Then, (4.35) is reduced to
H c2 =
√
2κH c .
(4.38)
A type II superconductor is a superconductor with H c2 higher than H c , i.e., with
κ higher than 1/
√
2. It is required that superconducting current can flow around
the center in a region of about λ from the center so as to keep the structure of
the quantized magnetic flux stable, as shown in Fig. 4.5. That is, superconducting
electrons of sufficient density must exist in this region. This also shows that the
condition that λ is larger than ξ is needed for type II superconductors.
Eliminating (m
∗
|α|)
1/2 in λ by using ξ , (4.35) is written as
H c2 =
2eμ 0 ξ 2 =
φ 0
2πμ 0 ξ 2 .
(4.39)
This formula is used to estimate the coherence length using the observed upper
critical field.
The lower critical field of a type II superconductor is also expressed using H c and
κ as
H c1 =
H c
√
2κ
(logκ + 0.081).
(4.40)
4.3 Flux Flow State
In this section the case is treated where the flux lines with density B flow with velocity
v under the Lorentz force. In this situation a macroscopic electric field of strength
E = B × v
(4.41)
is induced. This equation is called Josephson’s formula. This can be expected from
(2.52) for the general case. That is, when a substance moves with velocity V in a
space of uniform magnetic flux density, it is equivalent to the situation where flux
4 Fundamental Electromagnetic Properties of Superconductors
is called the Ginzburg-Landau parameter. κ is almost independent of temperature.
We have | ∞ |
2
= μ 0 H c
2
/|α| using (4.6) and (4.12). Then, (4.17) is rewritten as
λ = (m
∗
|α|)
1/2
/(2eμ 0 H c ). Eliminating (m
∗
|α|)
1/2 in the expression of ξ in terms of
λ, we have
κ =
2
√
2eμ 0 H c λ
2
.
(4.37)
Then, (4.35) is reduced to
H c2 =
√
2κH c .
(4.38)
A type II superconductor is a superconductor with H c2 higher than H c , i.e., with
κ higher than 1/
√
2. It is required that superconducting current can flow around
the center in a region of about λ from the center so as to keep the structure of
the quantized magnetic flux stable, as shown in Fig. 4.5. That is, superconducting
electrons of sufficient density must exist in this region. This also shows that the
condition that λ is larger than ξ is needed for type II superconductors.
Eliminating (m
∗
|α|)
1/2 in λ by using ξ , (4.35) is written as
H c2 =
2eμ 0 ξ 2 =
φ 0
2πμ 0 ξ 2 .
(4.39)
This formula is used to estimate the coherence length using the observed upper
critical field.
The lower critical field of a type II superconductor is also expressed using H c and
κ as
H c1 =
H c
√
2κ
(logκ + 0.081).
(4.40)
4.3 Flux Flow State
In this section the case is treated where the flux lines with density B flow with velocity
v under the Lorentz force. In this situation a macroscopic electric field of strength
E = B × v
(4.41)
is induced. This equation is called Josephson’s formula. This can be expected from
(2.52) for the general case. That is, when a substance moves with velocity V in a
space of uniform magnetic flux density, it is equivalent to the situation where flux