Random-matrix theory is used to show that the proximity to a superconductor opens a gap in the excitation spectrum of an electron gas confined to a billiard with a chaotic classical dynamics. In contrast, a gapless spectrum is obtained for a non-chaotic rectangular billiard, and it is argued that this is generic for integrable systems. preprint: cond-mat/9604058

The length dependence is computed of the resistance of a disordered normalmetal wire attached to a superconductor. The scaling of the transmission eigenvalue distribution with length is obtained exactly in the metallic limit, by a transformation onto the isobaric flow of a two-dimensional ideal fluid. The resistance has a minimum for lengths near l/Γ, with l the mean free path and Γ the transmittance of the superconductor interface.

To be published in Physica Scripta)We explore the effects of the proximity to a superconductor on the level density of a billiard for the two extreme cases that the classical motion in the billiard is chaotic or integrable. In zero magnetic field and for a uniform phase in the superconductor, a chaotic billiard has an excitation gap equal to the Thouless energy. In contrast, an integrable (rectangular or circular) billiard has a reduced density of states near the Fermi level, but no gap. We present numerical calculations for both cases in support of our analytical results. For the chaotic case, we calculate how the gap closes as a function of magnetic field or phase difference.There exists a way in quantum mechanics to distinguish classically chaotic systems from integrable systems, by looking at correlations between energy levels [1,2]. In a billiard with integrable dynamics, on the one hand, the energy levels are uncorrelated, and the spectrum has Poisson statistics. In a chaotic billiard, on the other hand, level repulsion leads to strong correlations and to Wigner-Dyson statistics [3]. Although the level correlations are different, the mean level density does not distinguish between chaotic and integrable billiards.In a recent paper [4] we have shown that the proximity to a superconductor makes it possible to distinguish chaotic from integrable billiards by looking at the density of states. In a chaotic billiard, on the one hand, we have found (using random-matrix theory) that the coupling to a superconductor by means of a point contact opens a gap in the density of states of the order of the Thouless energy E T = N Γδ/2π. Here N is the number of transverse modes in the point contact, Γ is the tunnel probability per mode, and δ is half the mean level spacing of the isolated billiard. In an integrable rectangular billiard, on the other hand, the density of states vanishes linearly near the Fermi level, without a gap. We have argued (using the Bohr-Sommerfeld approximation) that the absence of an excitation gap is generic for integrable systems. In these Proceedings we present numerical calculations for a chaotic billiard in support of the randommatrix theory, and we consider the effects of a magnetic field perpendicular to the billiard. We also present calculations for an integrable circular billiard, both exact and in the Bohr-Sommerfeld approximation.The system studied is shown schematically in the inset of Fig. 1. A billiard consisting of a normal metal (N) in a perpendicular magnetic field B is connected to two superconductors (S 1 , S 2 ) by narrow leads, each containing N/2 transverse modes at the Fermi energy E F . The order parameter in S 1 and S 2 has a phase difference φ ∈ [0, π].Mode n couples to a superconductor with phase φ n = φ/2 for 1 ≤ n ≤ N/2, φ n = −φ/2 for 1 + N/2 ≤ n ≤ N . For simplicity, we assume in this paper that there is no tunnel barrier in the leads (Γ = 1). (The generalization to Γ = 1 is straightforward [4].)The excitation spectrum of the billiard is discrete for energies 0 < E < ...

The resistance is computed of an NI 1 NI 2 S junction, where N = normal metal, S = superconductor, and I i = insulator or tunnel barrier (transmission probability per mode Γ i ). The ballistic case is considered, as well as the case that the region between the two barriers contains disorder (mean free path l, barrier separation L). It is found that the resistance at fixed Γ 2 shows a minimum as a function of Γ 1 , whenThe minimum is explained in terms of the appearance of transmission eigenvalues close to one, analogous to the "reflectionless tunneling" through a NIS junction with a disordered normal region. The theory is supported by numerical simulations.

A theory is presented for the statistics of the excitation spectrum of a disordered metal grain in contact with a superconductor. A magnetic field is applied to fully break time-reversal symmetry in the grain. Still, an excitation gap of the order of δ opens up provided N Γ 2 > ∼ 1. Here δ is the mean level spacing in the grain, Γ the tunnel probability through the contact with the superconductor, and N the number of transverse modes in the contact region. This provides a microscopic justification for the new random-matrix ensemble of Altland and Zirnbauer. PACS numbers: 74.50.+r, 74.80.Fp, 05.45.+b The proximity to a superconductor is known to induce a gap in the excitation spectrum of a normal metal. Semiclassical theories of this "proximity effect" show that the gap closes if time-reversal symmetry (T ) is broken (by a magnetic field or by magnetic impurities). Recently, Altland and Zirnbauer [1] argued that a gap remains in the spectrum of a metal grain surrounded by a superconductor -even if T is broken completely. (The classical mechanics of such a system had previously been studied [2].) The gap is small (of the order of the mean level spacing in the grain), but it has the fundamental implication that the level statistics is no longer described by the Gaussian unitary ensemble (GUE) of random-matrix theory [3].The GUE has a probability distribution of energy levels of the formwith some constant c > 0 depending on the mean level spacing at the Fermi level (chosen at E = 0). This ensemble was first applied to a granular metal by Gorkov and Eliashberg [4], and derived from microscopic theory by Efetov many years later [5]. A single-particle energy level E n corresponds to an excitation energy |E n |, that is to say, the excitation spectrum is obtained by folding the single-particle spectrum along the Fermi level. The folded GUE has been studied in Ref.[6]. Altland and Zirnbauer introduce a different probability distribution,for the (positive) excitation energies of a metal grain in contact with a superconductor. (The excitation spectrum is discrete for E < ∆, with ∆ the excitation gap in the bulk of the superconductor.) The distribution (2) is related to the Laguerre unitary ensemble (LUE) of randommatrix theory [7] by a change of variables. The density of states ρ(E) in this ensemble vanishes quadratically near zero energy [1,7],The gap in the excitation spectrum is of the order of the mean level spacing δ. The folded GUE, on the contrary, has no gap but a constant ρ(E) = 1/δ near E = 0. In this paper we present the first microscopic theory for the effect on the level statistics of the coupling to a superconductor. We consider the case that the conventional proximity effect is fully destroyed by a T -breaking magnetic field [8]. Assuming non-interacting quasiparticle excitations, and starting from the well-established GUE for the level statistics of an isolated metal grain, we obtain a crossover to Altland and Zirnbauer's distribution (2) as the coupling to a superconductor is increased. This provides...