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

Andreev Billiards 143
the Thouless energy ET � �/τdwell. (We assume that ET is less than the gap
∆ in the bulk superconductor.) In this context the dwell time τdwell is the
mean time between Andreev reflections (being the life time of an electron or
hole quasiparticle). The condition τerg ≪ τdwell of weak coupling is therefore
sufficient to be able to apply RMT to the entire relevant energy range.
6.1 Effective Hamiltonian
The excitation energies Ep of the Andreev billiard in the discrete part of the
spectrum are the solutions of the determinantal (22), given in terms of the
scattering matrix S(E) in the normal state (i.e. when the superconductor is
replaced by a normal metal). This equation can alternatively be written in
terms of the Hamiltonian H of the isolated billiard and the M × N coupling
matrix W that describes the N-mode point contact. The relation between S
and H, W is [3, 10]
�
T
S(E) =1− 2πiW E − H + iπW W T �−1 W. (30)
The N × N matrix W T W has eigenvalues wn given by
wn = Mδ
π2 �
2 − Γn − 2
Γn
� �
1 − Γn , (31)
where δ is the mean level spacing in the isolated billiard and Γn ∈ [0, 1] is
the transmission probability of mode n =1, 2,...N in the point contact. For
a ballistic contact, Γn = 1, while Γn ≪ 1 for a tunneling contact. Both the
number of modes N and the level spacing δ refer to a single spin direction.
Substituting (30) into(22), one arrives at an alternative determinantal
equation for the discrete spectrum [37]:
Det [E −H+ W(E)] = 0 , (32)
�
H 0
H =
0 −H∗ �
, (33)
� �
π
T T
EWW ∆W W
W(E) = √ . (34)
∆2 − E2 The density of states follows from
∆W W T EWW T
ρ(E) =− 1
π Im Tr (1 + dW/dE)(E + i0+ −H+ W) −1 . (35)
In the relevant energy range E < ∼ ET ≪ ∆ the matrix W(E) becomes energy
independent. The excitation energies can then be obtained as the eigenvalues
of the effective Hamiltonian [38]
�
H
Heff =
−πWW T
−πWW T −H∗ �
. (36)