Carsten Timm: Theory of superconductivity

Ginzburg-Landau theory also gives us the free energy of the wire. Since we have neglected the α and β terms
when solving the Ginzburg-Landau equation, we must for consistency do the same here,
F = A
∫ L
0
dx 2
4m |ψ′ (x)| 2 = A
∫ L
0
(
dx 2
− α ) ∣ ∣∣∣
− 1 4m β L + 1 ∣ ∣∣∣
2
L ei∆ϕ = A 2 2n s
4m L 2
∫L
0
dx 2 (1 − cos ∆ϕ)
= A L
2 n s
m
(1 − cos ∆ϕ) . (11.14)
F
−π 0
π ∆φ
The free energy is minimal when the phases of the two superconductors coincide. Thus if there existed any
mechanism by which the phases could relax, they would approach a state with uniform phase across the junction,
a highly plausible result.
We can now also derive the AC Josephson effect. Assuming that the free energy of the junction is only changed
by the supercurrent, we have
d
dt F = I sV, (11.15)
i.e., the electrical power. This relation implies that
⇒
⇒
∂F
∂∆ϕ
A
L
d dt
2 n s
m
d
dt ∆ϕ = I sV (11.16)
∆ϕ = −2e
sin ∆ϕ d dt ∆ϕ = −2en s
m
A
L
sin ∆ϕ V (11.17)
V, (11.18)
as stated above. Physically, if a supercurrent is flowing in the presence of a bias voltage, it generates power. Since
energy is conserved, this power must equal the change of (free) energy per unit time of the junction.
11.2 Dynamics of Josephson junctions
For a discussion of the dynamical current-voltage characteristics of a Josephson junction, it is crucial to realize
that a real junction also
1. permits single-particle tunneling (see Sec. 10.4), which we model by an ohmic resistivity R in parallel to
the junction,
2. has a non-zero capacitance C.
This leads to the resistively and capacitively shunted junction (RCSJ) model represented by the following circuit
diagram:
R
C
junction
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