Vorticity and vortex-core states in type-II superconductors

The origin of the vortex-core states in $s$-wave and $d_{x^2-y^2}$-wave superconductors is investigated by means of some selected numerical experiments. By relaxing the self-consistency condition in the Bogoliubov–de Gennes equations and tuning the order parameter in the core region, it is shown that the suppression of the superfluid density in the core is not a necessary condition for the core states to form. This excludes “potential well” types of interpretations for the core states. The topological defect in the phase of the order parameter, however, plays a crucial role. This fact is explained by considering the effect of the vortex supercurrent on the Bogoliubov quasiparticles and illustrated by comparing conventional vortices to multiply quantized vortices and vortex-antivortex pairs. The core states are also found to be extremely robust against random disorder of the phase.

Figure 1

Subgap energy spectrum of a singly quantized $s$-wave vortex as a function of angular momentum $\mu$ for various order-parameter profiles in the core: nearly self-consistent profile (black circles), “no-core” profile (white circles), and steplike profile (squares). The triangles show the energy spectrum of a normal region analogous to a vortex core embedded in a uniform superconductor (no supercurrent).

Figure 2

Local density of states for the systems shown in Fig. 1: conventional singly quantized vortex (left), vortex without normal core (center), and normal core without supercurrent (right). A thermal broadening was used with $k_{\mathrm{B}}T={\Delta }_{\infty }∕10$.

Figure 3

Energy spectra for multiply quantized $s$-wave vortices. The black circles correspond to vortices having a normal core given by Eq. (6), and the white circles correspond to vortices without normal core. The dashed vertical lines delimit the region $\mid \mu \mid <\mid \nu \mid g$ where the levels are sensitive to the profile of the order parameter.

Figure 4

Electron $\left({\psi }_{+}\right)$ and hole $\left({\psi }_{-}\right)$ BdG amplitudes for the lowest angular momentum in the singly (top) and doubly (bottom) quantized vortex with $r_c=\xi$. The integrand of the pairing matrix element, $\rho \tanh {(\rho ∕\sqrt{\mid \nu \mid })}^{\mid \nu \mid }{\psi }_{+}\left(\rho \right){\psi }_{-}\left(\rho \right)$, is displayed in the bottom panels.

Figure 5

Local density of states in the core of $d$-wave vortices with increasing winding numbers. The solid lines correspond to vortices having a normal core given by Eq. (6), and the dashed lines correspond to vortices without normal core. The case $\nu =0$ corresponds to a normal core without supercurrent. The arrows show the displacement of the peak maxima when the system size increases from $M=41$ to $M=51$. The bulk DOS is shown in gray and provides a common scale to compare the spectra at different $\nu$. Note that the bulk coherence peaks are not located at $E=\pm {\Delta }_{\infty }$ due to the large value of the chemical potential $\mu =t$.

Figure 6

LDOS along the nodal direction for $d$-wave vortices of increasing winding numbers. The curves show the LDOS at all sites between (0, 0) and (20, 20), and are offset vertically for clarity.

Figure 7

LDOS along the nodal direction (indicated by a thick gray line in the top part) for a vortex-antivortex pair in a $d$-wave superconductor for various vortex-antivortex separations. The antivortex (white dot, only visible at the shortest distance) is at position $\mathit{b}$ with respect to the vortex (black dot). The order-parameter phase in the region of the vortex center (excluding the $d_{x^2-y^2}$ symmetry factor for clarity) is represented in the form of a small arrow on each bond of the square lattice.

Figure 8

LDOS along the nodal direction for a $d$-wave singly-quantized vortex subject to random phase disorder. $W$ characterizes the strength of disorder. The phase field in the core region is shown in the top part. The spectra corresponding to the ordered case $(W=0)$ are shown in grey for comparison.