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

Andreev Billiards 171
is that of a chaotic cavity with mean level spacing δeff. We seek the gap in the
density of states
ρ(E) =− 1
�
Im Tr 1+
π dW0
� �
E + i0
dE
+ �−1 −H0 + W0 , (98)
cf. (35).
The selfconsistency equation for the ensemble-averaged Green function,
G =[E + W0 − (Mδeff/π)σzGσz] −1 , (99)
still leads to (43a), but (43b) should be replaced by
G11 + G12 sin u = −(τdwell/τE)uG12
× (G12 +cosu + G11 sin u) . (100)
(WehaveusedthatNeffδeff =2π�/τdwell.) Because of the energy dependence
of the coupling matrix, the (44) for the ensemble averaged density of states
should be replaced by
〈ρ(E)〉 = − 2
δ Im
�
G11 − u
cos u G12
�
. (101)
The excitation gap corresponds to a square root singularity in 〈ρ(E)〉,
which can be obtained by solving (43a) and (100) jointly with dE/dG11 =0
for u ∈ (0,π/2). The result is plotted in Fig. 21. The small- and large-τE
asymptotes are given by (88) and (89).
The large-τE asymptote is determined by the largest eigenvalue of the
time-delay matrix. To see this relationship, note that for τE ≫ τdwell we may
replace the determinant (95) by
�
� � �
2
τdwell
Det 1 + exp[2iEτE/� +2iEQ(0)] + O
=0. (102)
The Hermitian matrix
τE
Q(E) = 1 † d
S0(E)
i dE S0(E) (103)
is known in RMT as the Wigner-Smith or time-delay matrix. The roots Enp
of (102) satisfy
2Enp(τE + τn) =(2p +1)π�, p =0, 1, 2,... . (104)
The eigenvalues τ1,τ2,...τNeff of �Q(0) are the delay times. They are all
positive, distributed according to a generalized Laguerre ensemble [87]. In the
limit Neff →∞the distribution of the τn’s is nonzero only in the interval