A Diagrammatic Theory of Random Scattering Matrices for Normal-Superconducting Mesoscopic Junctions

TL;DR

This paper develops a diagrammatic large-N random matrix technique to analyze transport in normal-superconducting mesoscopic junctions, revealing significant conductance changes due to Andreev scattering and magnetic fields.

Contribution

It introduces a novel diagrammatic approach bridging random matrix theory and superconducting boundary effects in mesoscopic systems.

Findings

Conductance is significantly affected by Andreev scattering.

Weak magnetic fields cause large conductance variations.

Method aligns with existing theories and extends analysis capabilities.

Abstract

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 {\it leading order} of the conductance is altered, and thus changes much larger than $e^2/h$ 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 DMPK 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-superconducting structures, but for brevity we consider here only the case of a simple disordered N-S junction.