Quasiclassical numerical method for mesoscopic superconductors: bound states in a circular d-wave island with a single vortex

TL;DR

This paper introduces an efficient numerical method to solve the Eilenberger equation for mesoscopic superconductors, revealing how vortex shadow effects depend on quasiparticle energy in small d-wave superconducting islands.

Contribution

It presents a novel numerical approach for solving the Eilenberger equation in mesoscopic superconductors, specifically applied to circular d-wave islands with vortices.

Findings

Vortex shadow effect varies with quasiparticle energy.

Constructed geometry for quasiparticle trajectories with boundary reflections.

Numerical solutions remain stable even with vanishing order parameter.

Abstract

We demonstrate an efficient numerical method for obtaining unique solutions to the Eilenberger equation for a mesoscopic or nanoscale superconductor. In particular, we calculate the local density of states of a circular d-wave island containing a single vortex. The "vortex shadow" effect is found to strongly depend on the quasiparticle energy in such small systems. We show how to construct by geometry quasiparticle trajectories confined in a finite-size system with specular reflections at the boundary, and discuss the stability of the numerical solutions even in the case of vanishing order parameter as for nodal quasiparticles in a d-wave superconductor, or for quasiparticles passing through the vortex center with zero energy.