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.

Figure 1

Schematic of quasiparticle paths with multiple specular reflections.

Figure 2

Local density of states of a circular $d_{x^2-y^2}$-wave island without a vortex for energy (a) $\epsilon =0$, (b) $\epsilon =0.05{\Delta }_0$, and (c) $\epsilon =0.1{\Delta }_0$. The radius $r_c=5{\xi }_0$ and the smearing factor $\eta =0.01{\Delta }_0$.

Figure 3

Local density of states in a circular $d_{x^2-y^2}$-wave island with a vortex at the center for energy (a) $\epsilon =0$, (b) $0.05{\Delta }_0$, and (c) $0.1{\Delta }_0$. The radius is $r_c=5{\xi }_0$ and the smearing factor $\eta =0.01{\Delta }_0$.

Figure 4

Local density of states in a circular $d_{x^2-y^2}$-wave island with a vortex at the center along the boundary $\mathit{r}=r_c(\cos \alpha ,\sin \alpha )$ as a function of polar angle $\alpha$ and energy $\epsilon$, for radius (a) $r_c=5{\xi }_0$, (b) $r_c=10{\xi }_0$, and (c) $r_c=20{\xi }_0$.

Figure 5

Initial-value dependence of $\left|a\right(s\left)\right|$ for momentum direction (a) $\theta =0$ [$d\left({\mathit{k}}_{\mathrm{F}}\right)=1$] and (b) $\theta =7\pi /32$ [$d\left({\mathit{k}}_{\mathrm{F}}\right)\sim 0.2$] as a function of integration length $s$ (in units of ${\xi }_0$) for a circular $d_{x^2-y^2}$-wave island. The radius $r_c=5{\xi }_0$ and the smearing factor $\eta =0.01{\Delta }_0$.

Figure 6

Initial-value dependence of $\left|a\right(s\left)\right|$ for momentum direction $\theta =0$ [$d\left({\mathit{k}}_{\mathrm{F}}\right)=1$] as a function of integration length $s$ (in units of ${\xi }_0$) for a circular $d_{x^2-y^2}$-wave island, where the trajectories pass through the vortex center ($r=0$). The radius $r_c=5{\xi }_0$ and the smearing factor $\eta =1\times {10}^{-16}{\Delta }_0$.