Quasiparticle spectra of Abrikosov vortices in a uniform supercurrent flow

We calculate the local density of states of a vortex in a two-dimensional $s$-wave superconductor in the presence of a uniform applied supercurrent. The supercurrent induces changes in the electronic structure for the isolated vortex as well as the vortex lattice, which agree with the recent measurements in $2H$-NbSe${}_2$ [Maldonado et al., Phys. Rev. B 88, 064518 (2013)]. We find that the supercurrent polarizes the core states when the vortices are pinned. This shows that the transfer of momentum from the supercurrent to the bound states and the rigidity of the wave functions must be considered for understanding the various forces acting on collectively pinned Abrikosov vortices.

Figure 1

Finite-size effects. The thick solid lines show the exact DOS calculated for each value of $t$ at zero field and without applied current. The thin lines show the LDOS calculated at the central site of a $M\times M$ mesh, setting $\Psi \left(\mathit{r}\right)=\Delta$ in Eq. (2). Finite-size effects are small for $t=2.5$ and $M=71$ but remain significant for the largest $M$ if $t=10$, even at subgap energies.

Figure 2

(a) Zero-field LDOS at the center of the $71\times 71$ mesh in the absence of superflow (dotted line), and in a uniform superflow with $q=0.05$ (solid line) and $q=0.1$ (dashed line). (b) Relative variation of the LDOS at zero energy and at the gap edge versus the applied current. The solid lines are for $t=2.5$ ($k_{\mathrm{F}}\xi \sim 3$); the dashed lines are for $t=10$ ($k_{\mathrm{F}}\xi \sim 13$). Due to finite-size effects, the gap edge is at $1.3$ for $t=10$.

Figure 3

(a) LDOS along the $(0,1)$ direction for an isolated vortex at position $(0,0)$ with $t=10$ and $r_c=4$. The dotted lines are for $q=0$ and the solid lines for $q=0.02$. The inset shows the direction of the applied current; ${\mathit{\Phi }}_0$ is the vortex magnetic flux vector. The zero-energy LDOS along the diagonal and along the $y$ axis are shown in (b) and (c), respectively, without (full symbols) and with (empty symbols) the applied current. The insets show $dN(r,0)/dr$ obtained by numerical differentiation.

Figure 4

(a) Zero-field LDOS at the center of the $71\times 71$ mesh (top panel) and LDOS in a triangular vortex lattice at equal distance from three neighboring vortices (bottom panel: the data of the top panel is repeated for comparison). The dotted lines are for $q=0$ and the solid lines for $q=0.02$. (b) LDOS in real space for a vortex core in a vortex lattice at various energies without (left) and with (right) an applied current. The region shown is the $51\times 51$ central part of the $71\times 71$ mesh. The color scale is the same in all images. The green dot is the point where $\left|\Psi \right(\mathit{r}\left)\right|=0$.