180
Appendix
i =
2e
m ∗ ||
2
(∇ϕ + 2eA).
(A.2.2)
Hence, the kinetic energy density is
2
2m ∗ (∇||)
2
+
m
∗
8e 2 ||
2
i
2
.
(A.2.3)
Following (1.105) in [1], the current density is given by
i = −
H c2
2κ 2 | ∞ |
2
∇ ×
||
2 i z
= −
H c2
2κ 2 | ∞ |
2
∂||
2
∂y
i x −
∂||
2
∂x
i y
. (A.2.4)
Hence, the second term in (A.2.3) leads to
m ∗ H 2
c2
32κ 4 e 2 | ∞ | 4 || 2
⎡
⎣
∂|| 2
∂x
2
+
∂|| 2
∂y
2
⎤
⎦ =
m ∗ H 2
c2
8κ 4 e 2 | ∞ | 4 (∇||) 2 =
2
2m ∗ (∇||) 2 ,
(A.2.5)
where we have used the relationships of | ∞ |
2
= μ 0 H
2
c /|α| and |α| =
(2eλμ 0 H c )
2
/m
∗ from (4.37) and (4.38). Thus, the sum of the condensation energy
density and the kinetic energy density at high magnetic fields is written as
α||
2
+
2
m ∗ (∇||)
2
= μ 0 H
2
c
−|ψ|
2
+ 2ξ
2
(∇|ψ|)
2
,
(A.2.6)
where we have used ψ = /| ∞ | and (4.6). Equation (4.32) is obtained by spatially
averaging this equation.
A.3 Derivation of (4.42)
The current flows azimuthally around a stationary quantized flux line along the z–axis
and its density at radius r from the center is given by (see (1.64) in [1])
j =
φ 0
2πμ 0 λ 2 r
.
(A.3.1)
Hence, the azimuthal velocity of superconducting electrons is
v sθ = −
j
2e| ∞ |
2
= −
m ∗ r
,
(A.3.2)