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

Andreev Billiards
C.W.J. Beenakker
Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden,
The Netherlands
Summary. This is a review of recent advances in our understanding of how Andreev
reflection at a superconductor modifies the excitation spectrum of a quantum
dot. The emphasis is on two-dimensional impurity-free structures in which the
classical dynamics is chaotic. Such Andreev billiards differ in a fundamental way
from their non-superconducting counterparts. Most notably, the difference between
chaotic and integrable classical dynamics shows up already in the level density, instead
of only in the level–level correlations. A chaotic billiard has a gap in the
spectrum around the Fermi energy, while integrable billiards have a linearly vanishing
density of states. The excitation gap Egap corresponds to a time scale �/Egap
which is classical (�-independent, equal to the mean time τdwell between Andreev
reflections) if τdwell is sufficiently large. There is a competing quantum time scale,
the Ehrenfest time τE, which depends logarithmically on �. Two phenomenological
theories provide a consistent description of the τE-dependence of the gap, given
qualitatively by Egap � min(�/τdwell, �/τE). The analytical predictions have been
tested by computer simulations but not yet experimentally.
1 Introduction
Forty years ago, Andreev discovered a peculiar property of superconducting
mirrors [1]. As illustrated in Fig. 1, an electron that tries to enter a superconductor
coming from the Fermi level of a normal metal is forced to retrace its
path, as if time is reversed. Also the charge of the particle is reversed, since the
negatively charged electron is converted into a positively charged hole. The
velocity of a hole is opposite to its momentum, so the superconducting mirror
conserves the momentum of the reflected particle. In contrast, reflection at
an ordinary mirror (an insulator) conserves charge but not momentum. The
unusual scattering process at the interface between a normal metal (N) and
a superconductor (S) is called Andreev reflection.
Andreev reflection is the key concept needed to understand the properties
of nanostructures with NS interfaces [2]. Most of the research has concentrated
on transport properties of open structures, see [3, 4] for reviews. There
C.W.J. Beenakker: Andreev Billiards, Lect. Notes Phys. 667, 131–174 (2005)
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