Andreev scattering in anisotropic superconductors

Abstract

Three new classes of superconductors have been discovered in the past decade: the organic superconductors, the heavy-fermion superconductors, and the oxide superconductors. All of them show characteristic anomalies that point to the possibility that they are anisotropic superconductors with a directionally dependent (k-dependent) gap function that vanishes in points or lines on the Fermi surface. The problem to identify the symmetry type of an anisotropic superconductor has not found a satisfactory solution yet. Although a number of experiments have been proposed that allow one in principle to distinguish between different symmetry types, most of them are ambiguous because they do not couple to the order parameter directly. Here we propose a new experiment: Andreev scattering, i.e., scattering of low-energy normal quasiparticles off the spatially varying order parameter when the quasiparticles approach a normal-metal--superconductor interface from the normal side.The idea is investigated in detail for anisotropic even-parity superconductors. To describe the quasiparticle dynamics, the Bogoliubov--de Gennes equations for anisotropic superconductors are introduced and approximated by the Andreev equations. The nonideality of the interface is taken into account by an interface potential parametrized by a reflection coefficient. This leads to a boundary condition for the Andreev equations at the interface. The pair potential \ensuremath{\Delta}(k^,r), i.e., the directionally and space-dependent order parameter that occurs as a scattering potential in the Andreev equations, is determined self-consistently for various nonideal interfaces to d-wave superconductors. This is equivalent to solving the proximity effect for interfaces with a finite reflection coefficient, and it is done using the quasiclassical formalism. Once \ensuremath{\Delta}(k^,r) has been obtained, the Andreev equations are integrated numerically, and the k-dependent Andreev reflection and transmission coefficients as well as the corresponding conductivities are computed.The theory predicts a directional dependence of the conductivities from which the k dependence of the order parameter can be reconstructed. For this effect to be a useful tool, new experiments will have to be devised. A double-point-contact experiment is proposed for an experimental realization of the idea.