4.2 Ginzburg–Landau Theory
57
The maximum difference in the energy density (1/2)μ 0 H c
2 is called the condensation
energy density.
(4) Meissner state
Here, the Meissner state is treated. The magnetic field penetrates only in the vicinity
of the surface, while the magnetic flux density is zero and the order parameter has
the equilibrium value ∞ in most regions of the superconductor. Thus, the spatial
variation in the order parameter can be safely neglected, and (4.4) leads to
i = −
4e
2
m ∗ | ∞ |
2 A.
(4.15)
The rotation of this equation leads to
∇ × ∇ × B = −
1
λ 2 B,
(4.16)
where λ is the penetration depth given by
λ =
m
∗
4μ 0 e 2 | ∞ |
2
1/2
.
(4.17)
Equation (4.16) is called the London equation. Here, it is shown that this equation
describes the Meissner state. Assume that we apply external magnetic field H 0 along
the z–axis to a semi-infinite superconductor that occupies x ≥ 0. The inner magnetic
flux density also has the z–component and depends only on x, the distance from the
surface. Then, we have ∇ × B = −(∂B z /∂x)i y , and (4.16) leads to
∂
2 B z
∂x 2 =
1
λ 2 B z .
(4.18)
This equation can be easily solved. Under the conditions of B z (0) = μ 0 H 0 and a
finite value at x → ∞, we have
B z (x) = μ 0 H 0 exp
−
x
λ
.
(4.19)
Thus, the magnetic flux penetrates only up to the distance λ from the surface, as
shown in Fig. 4.5. This is the reason why λ is called the penetration depth. Since
λ is usually smaller than 100 nm, the inner magnetic flux can be neglected for a
superconductor of usual size, and hence, the Meissner state can be explained.
(5) Quantization of magnetic flux
It is shown here that the magnetic flux inside the superconductor is quantized. It is
expected that the magnetic flux has a high density at the center and is extended up
57
The maximum difference in the energy density (1/2)μ 0 H c
2 is called the condensation
energy density.
(4) Meissner state
Here, the Meissner state is treated. The magnetic field penetrates only in the vicinity
of the surface, while the magnetic flux density is zero and the order parameter has
the equilibrium value ∞ in most regions of the superconductor. Thus, the spatial
variation in the order parameter can be safely neglected, and (4.4) leads to
i = −
4e
2
m ∗ | ∞ |
2 A.
(4.15)
The rotation of this equation leads to
∇ × ∇ × B = −
1
λ 2 B,
(4.16)
where λ is the penetration depth given by
λ =
m
∗
4μ 0 e 2 | ∞ |
2
1/2
.
(4.17)
Equation (4.16) is called the London equation. Here, it is shown that this equation
describes the Meissner state. Assume that we apply external magnetic field H 0 along
the z–axis to a semi-infinite superconductor that occupies x ≥ 0. The inner magnetic
flux density also has the z–component and depends only on x, the distance from the
surface. Then, we have ∇ × B = −(∂B z /∂x)i y , and (4.16) leads to
∂
2 B z
∂x 2 =
1
λ 2 B z .
(4.18)
This equation can be easily solved. Under the conditions of B z (0) = μ 0 H 0 and a
finite value at x → ∞, we have
B z (x) = μ 0 H 0 exp
−
x
λ
.
(4.19)
Thus, the magnetic flux penetrates only up to the distance λ from the surface, as
shown in Fig. 4.5. This is the reason why λ is called the penetration depth. Since
λ is usually smaller than 100 nm, the inner magnetic flux can be neglected for a
superconductor of usual size, and hence, the Meissner state can be explained.
(5) Quantization of magnetic flux
It is shown here that the magnetic flux inside the superconductor is quantized. It is
expected that the magnetic flux has a high density at the center and is extended up