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

138 C.W.J. Beenakker
and we have defined σe ≡ 1, σh ≡−1. The labels e and h indicate the electron
or hole character of the wave function. The index n labels the modes, Φn(y, z)
is the transverse wave function of the n-th mode, and En its threshold energy:
� �p2 y + p 2� z /2m + V (y, z)
�
Φn(y, z) =EnΦn(y, z) . (12)
The eigenfunction Φn is normalized to unity, � dy � dz |Φn| 2 =1.
In the superconducting lead S the eigenfunctions are
Ψ ± �
iη
n,e(S) =
e e /2
e−iηe �
1
/2 √2q (E e
n
2 /∆ 2 − 1) −1/4
× Φn(y, z)exp(±iq e nx) , (13a)
Ψ ±
�
iη
n,h (S) =
e h /2
e−iηh �
1
�2q (E /2 h
n
2 /∆ 2 − 1) −1/4
× Φn(y, z)exp(±iq h nx) . (13b)
We have defined
√ 2m
� [EF − En + σ e,h (E 2 − ∆ 2 ) 1/2 ] 1/2 , (14)
q e,h
n =
η e,h = σ e,h arccos(E/∆) . (15)
The wave functions (10) and (13) have been normalized to carry the same
amount of quasiparticle current, because we want to use them as the basis for
a unitary scattering matrix. The direction of the velocity is the same as the
wave vector for the electron and opposite for the hole.
A wave incident on the Andreev billiard is described in the basis (10) by
a vector of coefficients
c in =(c + e ,c −
h ) , (16)
as shown schematically in Fig. 4. (The mode index n has been suppressed for
simplicity of notation.) The reflected wave has vector of coefficients
c out =(c − e ,c +
h ) . (17)
The scattering matrix SN of the normal region relates these two vectors, cout N =
SNcin N . Because the normal region does not couple electrons and holes, this
matrix has the block-diagonal form
SN(E) =
� S(E) 0
0 S(−E) ∗
�
. (18)
Here S(E) is the unitary scattering matrix associated with the single-electron
Hamiltonian H. ItisanN ×N matrix, with N(E) the number of propagating
modes at energy E. The dimension of SN(E) isN(E)+N(−E).