Heiss W.D. (ed.) Quantum dots.. a doorway to - tiera.ru
Heiss W.D. (ed.) Quantum dots.. a doorway to - tiera.ru
Heiss W.D. (ed.) Quantum dots.. a doorway to - tiera.ru
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170 C.W.J. Beenakker<br />
that a fully microscopic approach, for example bas<strong>ed</strong> on the ballistic σ-model<br />
[81, 82], could provide a conclusive answer. At present technical difficulties<br />
still stand in the way of a solution along those lines [83].<br />
A new direction of research is <strong>to</strong> investigate the effects of a nonisotropic<br />
superconducting order parameter on the Andreev billiard. The case of d-wave<br />
symmetry is most interesting because of its relevance for high-temperature<br />
superconduc<strong>to</strong>rs. The key ingr<strong>ed</strong>ients ne<strong>ed</strong><strong>ed</strong> for a theoretical description exist,<br />
notably RMT [84], quasiclassics [85], and a numerically efficient Andreev<br />
map [86].<br />
Acknowl<strong>ed</strong>gments<br />
While writing this review, I benefitt<strong>ed</strong> from correspondence and discussions<br />
with W. Belzig, P. W. Brouwer, J. Cserti, P. M. Ostrovsky, P. G. Silvestrov,<br />
and M. G. Vavilov. The work was support<strong>ed</strong> by the Dutch Science Foundation<br />
NWO/FOM.<br />
A Excitation Gap in Effective RMT<br />
and Relationship with Delay Times<br />
We seek the <strong>ed</strong>ge of the excitation spect<strong>ru</strong>m as it follows from the determinant<br />
(87), which in zero magnetic field and for E ≪ ∆ takes the form<br />
�<br />
Det 1+e 2iEτE/� S0(E)S0(−E) †�<br />
=0. (95)<br />
The unitary symmetric matrix S0 has the RMT distribution of a chaotic<br />
cavity with effective parameters Neff and δeff given by (85) and (86). The<br />
mean dwell time associat<strong>ed</strong> with S0 is τdwell. The calculation for Neff ≫ 1<br />
follows the method describ<strong>ed</strong> in Sects. 6.1 and 6.2, modifi<strong>ed</strong> as in [37] <strong>to</strong><br />
account for the energy dependent phase fac<strong>to</strong>r in the determinant.<br />
Since S0 is of the RMT form (30), we can write (95) in the Hamil<strong>to</strong>nian<br />
form (32). The extra phase fac<strong>to</strong>r exp(2iEτE/�) introduces an energy depen-<br />
dence of the coupling matrix,<br />
W0(E) = π<br />
cos u<br />
�<br />
W0W T 0 sin u W0WT 0<br />
W0W T 0 W0W T �<br />
, (96)<br />
0 sin u<br />
where we have abbreviat<strong>ed</strong> u = EτE/�. The subscript 0 reminds us that the<br />
coupling matrix refers <strong>to</strong> the r<strong>ed</strong>uc<strong>ed</strong> set of Neff channels in the effective RMT.<br />
Since there is no tunnel barrier in this case, the matrix W0 is determin<strong>ed</strong> by<br />
(31) with Γn ≡ 1. The Hamil<strong>to</strong>nian<br />
� �<br />
H0 0<br />
H0 =<br />
(97)<br />
0 −H0