Carsten Timm: Theory of superconductivity

V
V int
r barrier
x
There is a finite barrier for vortex-antivortex unbinding at a separation r barrier determined from ∂V/∂r = 0. This
gives
where K barrier := K(r barrier ). The barrier height is
For small currents we have r barrier ≫ r 0 and thus
The rate at which free vortices are generated is
r barrier = 2π k BT K barrier
2F Magnus
, (7.117)
∆E := V (r barrier ) − V (r 0 ) ∼ = V (r barrier )
The recombination rate of two vortices to form a pair is
since two vortices must meet. In the stationary state we have
and thus a resistance of
Since the Magnus force is F Magnus ∝ I we have
= 2π k B T K barrier ln r barrier
− 2F Magnus r barrier
r
( 0
= 2π k B T K barrier ln r )
barrier
− 1 . (7.118)
r 0
∆E ∼ = 2π k B T K(l → ∞) ln r barrier
. (7.119)
} {{ } r 0
≡ K
R gen ∝ e −β∆E ∼ =
(
rbarrier
r 0
) −2πK
. (7.120)
R gen = R rec ⇒ n v ∝ √ R gen ∝
R ∝ n v ∝
R rec ∝ n 2 v, (7.121)
(
rbarrier
r 0
) −πK
(7.122)
(
rbarrier
r 0
) −πK
. (7.123)
r barrier ∝
1
F Magnus
∝ 1 I
(7.124)
so that
( ) −πK 1
R ∝ = I πK . (7.125)
I
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