| Semiclassical approaches to inhomogeneous superconductivity are explored within three settings. The first is that of Andreev billiards, which are finite, arbitrarily-shaped, normal-state regions, surrounded by superconductor. A theoretical frame-work for the investigation of the short-wave quantal properties of quasiparticles in Andreev billiards is developed. Among the central results are two semiclassical trace formulas for the density of states. The first, a lower-resolution formula, corresponds to the well-known quasiclassical approximation. The second, a higher-resolution formula, allows the density of states to be expressed in terms of: (i) An explicit formula, valid in the short-wave limit, for billiards of arbitrary shape and dimensionality; and (ii) Higher-resolution corrections, expressed as a sum over periodic orbits that creep around the billiard boundary. The second involves quasiparticle motion near an extended impurity in a d-wave superconductor. Via an extension of the quasiclassical approximation, the idea that sign-variations in the superconducting pair-potential lead to low-energy states is extended beyond its original setting of boundary scattering to the broader context of scattering by general single-particle potentials. The index-theoretic origin of these states is exhibited via a connection with Witten's supersymmetric quantum-mechanical model. The third concerns an extended impurity in a cuprate superconductor in the pseudogap regime. It is shown that if the d-wave correlations persist into the pseudogap regime, but with pair-potential phase-fluctuations that destroy long-range ordering, then the low-energy states near extended impurities in the superconducting state should persist as resonances in the pseudogap regime. Thus, such states can serve as a probe of d-wave correlations in the pseudogap regime. |
