Advanced Solid State Physics

16.5.2 TheJosephsonEffect

We regard two superconductors separated by a thin insulating layer. In the followingψ 1 denotes
the probability amplitude on one side andψ 2 on the other side of the insulator. We write the time
dependentSchrödingerequations for this problem

i~

∂

∂t

ψ 1 =~Tψ 2 , i~

∂

∂t

ψ 2 =~Tψ 1 , (336)

where~T is the interaction rate between the electron pairs, referred to as electron pair coupling.
Similar to Sec. 16.4.4 we writeψiasψi=

√

niexp(iθi). We now insert these expressions in the above
equations, (336) and multiply the resulting equations withψ∗ 1 andψ∗ 2 , respectively. We obtain

1

2

∂

∂t

n 1 +in 1

∂

∂t

θ 1 =−iT

√

n 1 n 2 exp(iδ) (337)

and

1

2

∂

∂t

n 2 +in 2

∂

∂t

θ 2 =−iT

√

n 1 n 2 exp(iδ) (338)

whereδ=θ 2 −θ 1. We now equate the real and imaginary parts in order to obtain

∂

∂t

n 1 = 2T

√

n 1 n 2 sin(δ)

∂

∂t

n 2 =− 2 T

√

n 1 n 2 sin(δ), (339)

and

∂

∂t

θ 1 =−T

√n

1

n 2

cos(δ),

∂

∂t

θ 2 =−T

√n

2

n 1

cos(δ). (340)

In particular forn 1 ≈n 2 , we have

∂

∂t

(θ 2 −θ 1 ) = 0, (341)

and

∂

∂t

(n 2 +n 1 ) = 0. (342)

Since the current is proportional to∂t∂n 2 we obtain

J=J 0 sin(θ 2 −θ 1 ) (343)

whereJ 0 serves as a proportionality constant. Note thatJ 0 is also proportional toT. Hence, we
obtained that there is a current flow according to the phase differenceδeven if no external field is
applied. This is the dcJosephsoneffect.

We now regard the case of an applied potentialV. The equations (336) apply:

i~

∂

∂t

ψ 1 =~Tψ 2 −qV ψ 1 , i~

∂

∂t

ψ 2 =~Tψ 1 +qV ψ 2. (344)