4.2 Ginzburg–Landau Theory
55
fourth and fifth terms are the magnetic and kinetic energies, respectively. The order
parameter, , and the vector potential, A, are determined in such conditions that the
energy density is minimized with respect to these quantities. These conditions are
respectively given by
1
2m ∗ (−i∇ + 2eA)
2
+ αα + β||
2
= 0,
(4.3)
i =
1
μ 0
∇ × ∇ × A =
ie
m ∗
∗
∇ − ∇
∗
−
4e
2
m ∗ ||
2 A,
(4.4)
where i is the superconducting current density (see Appendix A.1). These equations
are called the Ginzburg-Landau (GL) equations.
(2) Transition at T = T c
Here, we treat the transition at T = T c in the absence of magnetic field. We can
assume that A = 0. Spatial variation in can be disregarded, and (4.2) leads to
F s (0) = F n (0) + α||
2
+
1
2
β||
4
.
(4.5)
When this is minimized with respect to , the condition ∂F s /∂||
2
= 0 leads to
||
2
= −
α
β
≡ | ∞ |
2
,
(4.6)
where ∞ is the equilibrium value of . The parameter α should depend on the
temperature as
α ∝ T c − T ,
(4.7)
and the condition ∂
2
F s (0)/∂
||
2
2 = β > 0 should be satisfied, so that the superconducting state is stable at temperatures below T c . The free energy density in this
case is:
F s (0) = F n (0) −
α
2
2β
,
(4.8)
which indicates that the free energy is lower in the superconducting state. At temperatures above T c , the state of (4.6) can no longer be realized, but the normal state of
= 0 with a lower energy is the result. Thus, the transition to the normal state at
T = T c is explained, as shown in Fig. 4.4.
55
fourth and fifth terms are the magnetic and kinetic energies, respectively. The order
parameter, , and the vector potential, A, are determined in such conditions that the
energy density is minimized with respect to these quantities. These conditions are
respectively given by
1
2m ∗ (−i∇ + 2eA)
2
+ αα + β||
2
= 0,
(4.3)
i =
1
μ 0
∇ × ∇ × A =
ie
m ∗
∗
∇ − ∇
∗
−
4e
2
m ∗ ||
2 A,
(4.4)
where i is the superconducting current density (see Appendix A.1). These equations
are called the Ginzburg-Landau (GL) equations.
(2) Transition at T = T c
Here, we treat the transition at T = T c in the absence of magnetic field. We can
assume that A = 0. Spatial variation in can be disregarded, and (4.2) leads to
F s (0) = F n (0) + α||
2
+
1
2
β||
4
.
(4.5)
When this is minimized with respect to , the condition ∂F s /∂||
2
= 0 leads to
||
2
= −
α
β
≡ | ∞ |
2
,
(4.6)
where ∞ is the equilibrium value of . The parameter α should depend on the
temperature as
α ∝ T c − T ,
(4.7)
and the condition ∂
2
F s (0)/∂
||
2
2 = β > 0 should be satisfied, so that the superconducting state is stable at temperatures below T c . The free energy density in this
case is:
F s (0) = F n (0) −
α
2
2β
,
(4.8)
which indicates that the free energy is lower in the superconducting state. At temperatures above T c , the state of (4.6) can no longer be realized, but the normal state of
= 0 with a lower energy is the result. Thus, the transition to the normal state at
T = T c is explained, as shown in Fig. 4.4.