Heiss W.D. (ed.) Quantum dots.. a doorway to - tiera.ru

4 Scattering Formulation
Andreev Billiards 137
In the step-function model (6) the excitation spectrum of the coupled electronhole
quasiparticles can be expressed entirely in terms of the scattering matrix
of normal electrons [24].
The scattering geometry is illustrated in Fig. 4. It consists of a finite
normal-metal region N adjacent to a semi-infinite superconducting region S.
The metal region represents the Andreev billiard. To obtain a well-defined
scattering problem we insert an ideal (impurity-free) normal lead between N
and S. We assume that the only scattering in the superconductor consists of
Andreev reflection at the NS interface (no disorder in S). The superconductor
may then also be represented by an ideal lead. We choose a coordinate system
so that the normal and superconducting leads lie along the x-axis, with the
interface at x =0.
S N
0
x
c- e
ch
+
c e + ch -
Fig. 4. Normal metal (N) containing an Andreev billiard, coupled to a superconductor
(S) by an ideal lead. The dashed line represents the NS interface. Scattering
) are indicated schematically
states c in =(c + e ,c −
h )andcout =(c − e ,c +
h
We first construct a basis for the scattering matrix. In the normal lead N
the eigenfunctions of the BdG equation (1) can be written in the form
Ψ ± � �
n,e(N) =
10 1
�k Φn(y, z)exp(±ik
e
n
e nx) , (10a)
Ψ ±
� �
n,h (N) =
01 1
�k Φn(y, z)exp(±ik
h
n
h nx) , (10b)
where the wavenumbers k e n and k h n are given by
k e,h
n =
√ 2m
� (EF − En + σ e,h E) 1/2 , (11)