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

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 depend strongly 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 we 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.