Diagrammatic theory of random scattering matrices for normal-metal-superconducting mesoscopic junctions

The planar-diagrammatic technique of large-N random matrices is extended to evaluate averages over the circular ensemble of unitary matrices. It is then applied to study transport through a disordered metallic “grain” attached through ideal leads to a normal electrode and to a superconducting electrode. The latter enforces boundary conditions which coherently couple electrons and holes at the Fermi energy through Andreev scattering. Consequently, the leading order of the conductance is altered, and thus changes much larger than (Formula presented)/h are observed when, e.g., a weak magnetic field is applied. This is in agreement with existing theories. The approach developed here is intermediate between the theory of dirty superconductors (the Usadel equations) and the random-matrix approach involving transmission eigenvalues (e.g., the Dorokhov-Mello-Pereyra-Kumar equation) in the following sense: Even though one starts from a scattering formalism, a quantity analogous to the superconducting order parameter within the system naturally arises. The method can be applied to a variety of mesoscopic normal-metal-superconducting (N-S) structures, but for brevity we consider here only the case of a simple disordered N-S junction.