Quantum interference and the formation of the proximity effect in chaotic normal-metal/superconducting structures

We discuss a number of basic physical mechanisms relevant to the formation of the proximity effect in superconductor/normal-metal (SN) systems. The physics of this effect, most notably the phenomenon of density of states depletion in normal metals adjacent to superconductors, sensitively depends on various aspects of quantum interference and chaotic charge carrier dynamics in mesoscopic systems. Specifically, we review why the proximity effect sharply discriminates between systems with integrable and chaotic dynamics, respectively, and how this feature can be incorporated into theories of SN systems. Turning to less well investigated terrain, we discuss the impact of quantum diffractive scattering on the structure of the density of states in the normal region. We consider ballistic systems weakly disordered by pointlike impurities as a test case and demonstrate that diffractive processes akin to normal metal weak localization lead to the formation of a hard spectral gap—a hallmark of SN systems with chaotic dynamics. Turning to the more difficult case of clean systems with chaotic boundary scattering, we argue that semiclassical approaches, based on classifications in terms of classical trajectories, cannot explain the gap phenomenon. Employing an alternative formalism based on elements of quasiclassics and the ballistic σ model, we demonstrate that the inverse of the so-called Ehrenfest time is the relevant energy scale in this context. We discuss some fundamental difficulties related to the formulation of low-energy theories of mesoscopic chaotic systems in general and how they prevent us from analyzing the gap structure in a rigorous manner. Given these difficulties, we argue that the proximity effect represents a basic and challenging test phenomenon for theories of quantum chaotic systems.