Carsten Timm: Theory of superconductivity

Thus the current flowing from left to right is
∫ ∞
I L→R ∝ 2e dω D
h
L(ω) s DR(µ n + ω + eV ) |t| 2 n F (ω) [1 − n F (ω + eV )]
0
{ ( √ ) (
1 ω2 − ∆ 2 0
× 1 +
+ 1 √ )}
ω2 − ∆ 2 0
1 −
2 ω 2 ω
} {{ }
= 1
∫ 0
+ 2e dω D
h
L(|ω|) s DR(µ n + ω + eV ) |t| 2 n F (ω) [1 − n F (ω + eV )]
−∞
{ ( √ ) (
1 ω2 − ∆ 2 0
× 1 −
+ 1 √ )}
ω2 − ∆ 2 0
1 +
2 |ω| 2 |ω|
} {{ }
= 1
= 2e
h
∫ ∞
−∞
dω D s L(|ω|) D n R(µ + ω + eV ) |t| 2 n F (ω) [1 − n F (ω + eV )], (10.83)
where DL s (ω) now is the superconducting density of states neither containing a spin factor of 2 nor the factor
of 2 due describing both electrons and holes as excitations with positive energy—positive and negative energies
are here treated explicitly and separately. We see that the electron-hole mixing does not lead to any additional
factors beyond the changed density of states. With the analogous expression
we obtain
where
Thus
I R→L ∝ 2e
h
I sn ∝ 2e
h
∫ ∞
−∞
∫ ∞
−∞
dω D s L(|ω|) D n R(µ + ω + eV ) |t| 2 n F (ω + eV ) [1 − n F (ω)] (10.84)
dω D s L(|ω|) D n R(µ + ω + eV ) |t| 2 [n F (ω) − n F (ω + eV )]
∼= 2e
h Dn L(E F ) D n R(E F ) |t| 2
∫∞
−∞
dω Ds L (|ω|)
D n L (E F ) [n F (ω) − n F (ω + eV )], (10.85)
⎧
⎪⎨
DL s (|ω|)
|ω|
DL n(E F ) = √ for |ω| > ∆
ω2 − ∆ 2 0 ,
0
⎪ ⎩
0 for |ω| < ∆ 0 .
I sn = G nn
e
∫ ∞
−∞
It is useful to consider the differential conductance
G sn := dI sn
dV
= G nn
∫ ∞
−∞
∫∞
= G nn
−∞
(10.86)
dω Ds L (|ω|)
D n L (E F ) [n F (ω) − n F (ω + eV )]. (10.87)
dω Ds L (|ω|)
D n L (E F )
(
− ∂ n )
F (ω + eV )
∂ω
dω Ds L (|ω|)
D n L (E F ) β n F (ω + eV )[1 − n F (ω + eV )]. (10.88)
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