Advanced Solid State Physics

We perform the same operations as above and we obtain

∂

∂t

n 1 = 2T

√

n 1 n 2 sin(δ)

∂

∂t

n 2 =− 2 T

√

n 1 n 2 sin(δ), (345)

and

∂

∂t

θ 1 =

qV

~

−T

√

n 1

n 2

cos(δ),

∂

∂t

θ 2 =−

qV

~

−T

√

n 2

n 1

cos(δ). (346)

Again, we assumen 1 ≈n 2 and obtain

∂

∂t

(θ 2 −θ 1 ) =

∂

∂t

δ=−

2 qV

~

, (347)

or equivalently

δ(t) =δ(0)−

2 qV t

~

. (348)

The superconducting current in this case is given by

J=J 0 sin

[

δ(0)−

2 qV t

~

]

. (349)

This is the acJosephsoneffect. The current oscillates with a frequency given byω=^2 qV~. This is a
very convenient method to determine ratioe~by measuring the frequency and the voltage.

16.4.7 SQUID

A SQUID (superconducting quantum interference device) is a very precise magnetometer based on
theJosephsoneffect. It consists of twoJosephsonjunctions in parallel, see Fig. 168. We do not
apply a voltage and we assume that a magnetic fluxΦpasses through the interior of the circuit. From
Sec. 16.4.4 we know that

δa−δb=

2 q

~

Φ, (350)

so that we can write

δa=δ 0 +

q

~

Φ, (351)

and

δb=δ 0 −

q

~

Φ. (352)

The total current in this case is given by

J=J 0

[

sin

(

δ 0 +

q

~

Φ

)

+ sin

(

δ 0 −

q

~

Φ

)]

= 2J 0 sin(δ 0 ) cos

(q

~

Φ

)

. (353)

Since the current varies with the magnetic fluxΦ, this is a very elegant method to measure the
magnetic field.