NS interface. This effect is fully included here, even though Ψ does not appear explicitly in the expressions
which follow. The reason is that the order parameter quantifies the degree of phase coherence between
electrons and holes, but does not itself affect the dynamics of the quasiparticles.
(The BdG equation
(21.12) contains ∆ not Ψ.)
21.3.
NS INTERFACE
415
(The modeindex
n
has been suppressed for simplicity of notation.)
The reflected and
transmitted wave has vector of coefficients
c
out
N
≡
(
c

e
(N
1
)
, c
+
e
(N
2
)
, c
+
h
(N
1
)
, c

h
(N
2
)
)
.
(21.23)
The ˆ
s
matrix
s
N
of the normal region relates these two vectors,
c
out
N
= ˆ
s
N
c
in
N
.
(21.24)
Because the normal region does not couple electrons and holes, this matrix has the block
diagonal form
ˆ
s
N
(
ε
) =
ˆ
s
0
(
ε
)
0
0
ˆ
s
0
(

ε
)
*
,
ˆ
s
0
≡
r
11
t
12
t
21
r
22
.
(21.25)
Here ˆ
s
0
is the unitary ˆ
s
matrix associated with the singleelectron Hamiltonian
H
0
. The
reflection and transmission matrices
r
(
ε
) and
t
(
ε
) are
N
×
N
matrices,
N
(
ε
) being the
number of propagating modes at energy
ε
. (We assume for simplicity that the number of
modes in leads N
1
and N
2
is the same.)
The matrix
s
0
is unitary (ˆ
s
†
0
ˆ
s
0
= 1) and satisfies the symmetry relation ˆ
s
0
(
ε, B
)
ij
=
ˆ
s
0
(
ε,

B
)
ji
.
Andreev reflection
For energies 0
< ε <
∆
0
there are no propagating modes in the superconductor. We can
then define an ˆ
s
matrix for Andreev reflection at the NS interface which relates the vector
of coefficients
(
c

e
(N
2
)
, c
+
h
(N
2
)
)
to
(
c
+
e
(N
2
)
, c

h
(N
2
)
)
.
The elements of this ˆ
s
matrix can
be obtained by matching the wavefunctions (21.19) at
x
= 0 to the decaying solutions
in S of the BdG equation. If terms of order ∆
0
/
F
are neglected (the socalled Andreev
approximation, the result is simply
c

e
(N
2
) =
α
e
i
φ
c

h
(N
2
)
,
c
+
h
(N
2
) =
α
e

i
φ
c
+
e
(N
2
)
,
(21.26)
where
α
≡
exp[

i
arccos(
ε/
∆
0
)]. Andreev reflection transforms an electron mode into a
hole mode, without change of mode index. The transformation is accompanied by a phase
shift, which consists of two parts:
1. A phase shift

arccos(
ε/
∆
0
) due to the penetration of the wavefunction into the
superconductor.
2. A phase shift equal to plus or minus the phase of the pair potential in the supercon
ductor (
plus
for reflection from hole to electron,
minus
for the reverse process).
416
CHAPTER 21.
MESOSCOPIC SUPERCONDUCTIVITY
General relations
We can combine the 2
N
linear relations (21.26) with the 4
N
relations (21.24) to obtain a
set of 2
N
linear relations between the incident wave in lead N
1
and the reflected wave in
the same lead:
c

e
(N
1
) = ˆ
s
ee
c
+
e
(N
1
) + ˆ
s
eh
c

h
(N
1
)
,
c
+
h
(N
1
) = ˆ
s
he
c
+
e
(N
1
) + ˆ
s
hh
c

h
(N
1
)
.
(21.27)
The four
N
×
N
matrices ˆ
s
ee
, ˆ
s
hh
, ˆ
s
eh
, and ˆ
s
he
form together the scattering matrix ˆ
s
of the
whole system for energies 0
< ε <
∆
0
. An electron incident in lead N
1
is reflected either as
an electron (with scattering amplitudes ˆ
s
ee
) or as a hole (with scattering amplitudes ˆ
s
he
).
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 Physics, Cubic crystal system, periodic structures, Reciprocal lattice, Lattice Vibrations