Abstract:
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 E_gap corresponds to a time scale h/E_gap which is classical (h-independent, equal to the mean time t_dwell between Andreev reflections) if t_dwell is sufficiently large. There is a competing quantum time scale, the Ehrenfest time t_E, which depends logarithmically on h. Two phenomenological theories provide a consistent description of the t_E-dependence of the gap, given qualitatively by E_gap min(h/t_dwell,h/t_E). The analytical predictions have been tested by computer simulations but not yet experimentally.

Abstract:
We studied the energy levels of graphene based Andreev billiards consisting of a superconductor region on top of a monolayer graphene sheet. For the case of Andreev retro-reflection we show that the graphene based Andreev billiard can be mapped to the normal metal-superconducting billiards with the same geometry. We also derived a semiclassical quantization rule in graphene based Andreev billiards. The exact and the semiclassically obtained spectrum agree very well both for the case of Andreev retro-reflection and specular Andreev reflection.

Abstract:
A new type of classical billiard - the Andreev billiard - is investigated using the tangent map technique. Andreev billiards consist of a normal region surrounded by a superconducting region. In contrast with previously studied billiards, Andreev billiards are integrable in zero magnetic field, {\it regardless of their shape}. A magnetic field renders chaotic motion in a generically shaped billiard, which is demonstrated for the Bunimovich stadium by examination of both Poincar\'e sections and Lyapunov exponents. The issue of the feasibility of certain experimental realizations is addressed.

Abstract:
We present a classical and quantum mechanical study of an Andreev billiard with a chaotic normal dot. We demonstrate that in general the classical dynamics of these normal-superconductor hybrid systems is mixed, thereby indicating the limitations of a widely used retracing approximation. We show that the mixed classical dynamics gives rise to a wealth of wavefunction phenomena, including periodic orbit scarring and localization of the wavefunction onto other classical phase space objects such as intermittent regions and quantized tori.

Abstract:
We give an optical physicist view of the problem of the trajectories in a polygonal billiard using only basic facts of Optics and the theory of functions of a complex variable. This approach allow us to stablish a certain correspondence between n-gon billiards and one-holed 2n-punctured tori. Therefore the existence of periodic trajectories in a certain polygon becomes the problem of the existence of closed geodesics in its associated torus.

Abstract:
We present numerical results for the local density of states in semiclassical Andreev billiards. We show that the energy gap near the Fermi energy develops in a chaotic billiard. Using the same method no gap is found in similar square and circular billiards.

Abstract:
Quantum cavities or dots have markedly different properties depending on whether their classical counterparts are chaotic or not. Connecting a superconductor to such a cavity leads to notable proximity effects, particularly the appearance, predicted by random matrix theory, of a hard gap in the excitation spectrum of quantum chaotic systems. Andreev billiards are interesting examples of such structures built with superconductors connected to a ballistic normal metal billiard since each time an electron hits the superconducting part it is retroreflected as a hole (and vice-versa). Using a semiclassical framework for systems with chaotic dynamics, we show how this reflection, along with the interference due to subtle correlations between the classical paths of electrons and holes inside the system, are ultimately responsible for the gap formation. The treatment can be extended to include the effects of a symmetry breaking magnetic field in the normal part of the billiard or an Andreev billiard connected to two phase shifted superconductors. Therefore we are able to see how these effects can remold and eventually suppress the gap. Furthermore the semiclassical framework is able to cover the effect of a finite Ehrenfest time which also causes the gap to shrink. However for intermediate values this leads to the appearance of a second hard gap - a clear signature of the Ehrenfest time.

Abstract:
A detailed description of topology of integrable billiard systems is given. For elliptical billiards and geodesic billiards on ellipsoid, the corresponding Fomenko graphs are constructed.

Abstract:
We study the effect on the density of states in mesoscopic ballistic billiards to which a superconducting lead is attached. The expression for the density of states is derived in the semiclassical S-matrix formalism shedding insight into the origin of the differences between the semiclassical theory and the corresponding result derived from random matrix models. Applications to a square billiard geometry and billiards with boundary roughness are discussed. The saturation of the quasiparticle excitation spectrum is related to the classical dynamics of the billiard. The influence of weak magnetic fields on the proximity effect in rough Andreev billiards is discussed and an analytical formula is derived. The semiclassical theory provides an interpretation for the suppression of the proximity effect in the presence of magnetic fields as a coherence effect of time reversed trajectories, similar to the weak localisation correction of the magneto-resistance in chaotic mesoscopic systems. The semiclassical theory is shown to be in good agreement with quantum mechanical calculations.

Abstract:
The energy spectrum and the eigenstates of a rectangular quantum dot containing soft potential walls in contact with a superconductor are calculated by solving the Bogoliubov-de Gennes (BdG) equation. We compare the quantum mechanical solutions with a semiclassical analysis using a Bohr--Sommerfeld (BS) quantization of periodic orbits. We propose a simple extension of the BS approximation which is well suited to describe Andreev billiards with parabolic potential walls. The underlying classical periodic electron-hole orbits are directly identified in terms of ``scar'' like features engraved in the quantum wavefunctions of Andreev states determined here for the first time.