Carsten Timm: Theory of superconductivity

with ˜k 1 := (−k 1x , k 1y , k 1z ) and k 2 = (k 2x , k 1y , k 1z ) with k 2x > 0, where k1x 2 + k1∥ 2 = k2 F + 2mE and k2 2x + k1∥ 2 =
kF 2 − 2mE, and
( )
Ψ(r) = e −κ 1x α+ e i(q 1x+k 1y y+k 1z z) + α − e i(−q 1x+k 1y y+k 1z z)
β + e i(q1x+k1yy+k1zz) + β − e i(−q1x+k1yy+k1zz) for x ≥ 0. (11.95)
Note that ˜k 1 is the wave vector of a specularly reflected electron. From Eq. (11.93) we get
From the continuity of Ψ 1 , Ψ 2 , and their x-derivatives we obtain
β ± = E ∓ i√ ∆ 2 0 − E2
∆ 0
α ± . (11.96)
1 + r = α + + α − , (11.97)
a = β + + β − , (11.98)
ik 1x − r ik 1x = α + (−κ 1 + iq 1 ) + α − (−κ 1 − iq 1 ), (11.99)
a ik 2x = β + (−κ 1 + iq 1 ) + β − (−κ 1 − iq 1 ). (11.100)
We thus have six coupled linear equations for the six unknown coefficients r, a, α + , α − , β + , β − . The equations
are linearly independent so that they have a unique solution, which we can obtain by standard methods. The
six coefficients are generally non-zero and complex. We do not give the lengthy expressions here but discuss the
results physically.
• The solution in the superconductors decays exponentially, which is reasonable since the energy lies in the
superconducting gap.
• In the normal region there is a secularly reflected electron wave (coefficient r), which is also expected. So
far, the same results would be obtained for a simple potential step. However, explicit evaluation shows that
in general |r| 2 < 1, i.e., not all electrons are reflected.
• There is also a term
Recall that the second spinor component was defined by
Ψ 2 (r) = a e ik2·r for x ≤ 0. (11.101)
|Ψ k2 ⟩ = c −k,↓ |ψ BCS ⟩ . (11.102)
Hence, the above term represents a spin-down hole with wave vector −k 2 . Now k 1∥ = k 2∥ and k1x 2 =
kF 2 − k2 1∥ + 2mE and k2 2x = kF 2 − k2 1∥
− 2mE. But the last terms ±2mE are small since
|E| < ∆ 0 ≪ µ = k2 F
2m
(11.103)
in conventional superconductors. Thus |k 2x − k 1x | is small and the hole is traveling nearly in the opposite
direction compared to the incoming electron wave. This phenomenon is called Andreev reflection.
N
electron
~
k 1
electron
k 1
− k 2
hole
y,z
S
electron−like
quasiparticle
(evanescent)
hole−like
quasiparticle
(evanescent)
x
117