Carsten Timm: Theory of superconductivity

Equation (11.25) can be used to study a Josephson junction in various regimes. First, note that a stationary
solution exists as long as |I| ≤ I c . Then
sin ∆ϕ = const = − I I c
and V ≡ 0. (11.32)
This solution does not exist for |I| > I c . What happens if we impose a time-independent current that is larger
than the critical current? We first consider a strongly damped junction, Q ≪ 1. Then we can neglect the
acceleration term and write
⇒
⇒
⇒
1 d
Q dτ ∆ϕ + sin ∆ϕ = − I I c
(11.33)
1 d
Q dτ ∆ϕ = − I − sin ∆ϕ
I c
(11.34)
d ∆ϕ
−
= Q dτ
I
I c
+ sin ∆ϕ
(11.35)
∫
Q (τ − τ 0 ) = −
∆ϕ
0
∣ ∣∣∣∣∣
d ∆ϕ ′ I > I c 2
I
= − √
I c
+ sin ∆ϕ ′ ( I
) arctan 1 + I ∆ϕ
I c
tan ∆ϕ′
2
√
2 (
I c
− 1
I
) . (11.36)
2
I c
− 1
We are interested in periodic solutions for e i∆ϕ or ∆ϕ mod 2π. One period T is the time it takes for ∆ϕ to change
from 0 to −2π (note that d ∆ϕ/dτ < 0). Thus
Qω p T = −
−2π ∫
0
d ∆ϕ ′ I > I c
I
I c
+ sin ∆ϕ ′
⇒ T = 2π
Qω p
1
√ ( I
I c
) 2
− 1
= 2π
2π = √ ( I
) (11.37)
2
I c
− 1
1
√
2eI c R ( I
) = π 1
√ . (11.38)
2
I c
− 1
eR I2 − Ic
2
0
The voltage V ∝ d ∆ϕ/dt is of course time-dependent but the time-averaged voltage is simply
¯V = 1 T
∫ T
0
dt V (t) = − 1
2e T
∫ T
0
dt d dt ∆ϕ = − 1
2e T
[∆ϕ(T ) − ∆ϕ(0)]
} {{ }
= −2π
= π
e
1
T = R √ I 2 − Ic 2 (11.39)
√
for I > I c . By symmetry, ¯V = −R I2 − Ic
2
current thus look like this:
for I < −I c . The current-voltage characteristics for given direct
I
I c
V/R
0
V
−I c
For |I| ≤ I c , the current flows without resistance. At I c , non-zero DC and AC voltages set in gradually. For
|I| ≫ I c , the DC voltage approaches the ohmic result for a normal contact.
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