Enhancement of superconductivity outside an Abrikosov vortex core in a tightly bound Cooper pair superconductor

TL;DR

This work shows that in the zero-temperature limit, Abrikosov vortices in tightly bound Cooper-pair superconductors acquire an outward electrostatic field that enforces global neutrality, leading to a non-monotonic superconductive-electron density with a maximum outside the vortex core. By extending Feynman’s phenomenological theory to include an explicit charged background, the authors derive a coupled set of stationary field equations with dimensionless parameters $\\gamma=\\delta/\\xi$ and $\\kappa=\\lambda/\\xi$, and obtain three nonlinear ODEs for the density $R(\\rho)$, the electrostatic potential $w(\\rho)$, and the vector-potential-related function $F(\\rho)$. Analytical results reveal weak- and strong-screening limits: in strong screening the core contracts to a scale $\\sqrt{\\gamma}$ and the density peaks above the bulk value, yielding universal field profiles; in weak screening the density maximum is located near $\\rho\\sim\\gamma$ and decays toward the bulk value from above. Numerical results for the purely electric problem corroborate the analytic limits and indicate a measurable outside-core density enhancement (order 5–10%) for realistic screening lengths, suggesting STM as a viable probe in materials like NbSe$_2$.

Abstract

Abrikosov vortices play a central role in the disruption of superconductivity in type-II superconductors. It is commonly accepted that as one moves away from the vortex's axis of an $s$-wave superconductor, the density of superconductive electrons gradually increases from zero to its bulk value. However, we demonstrate that this behavior is qualitatively altered in the zero-temperature limit provided that the Cooper pairs comprising the superconductive liquid are sufficiently tightly bound. Specifically, outside the vortex core, the density of superconductive electrons reaches a maximum surpassing its bulk value. This phenomenon has electrostatic origins: since normal electrons are absent and there exists a charged ionic background, the spatial variation of the charge density of superconductive electrons violates local neutrality, leading to the generation of an electric field. This electric field shrinks the vortex core and turns the density profile into that with a maximum, ensuring global neutrality. The effect is most pronounced in the limit of strong electrostatic screening, where the field configurations describing the vortex attain a universal form, with the electric field screened over a length scale determined by the London penetration depth.

Figures (2)

• Figure 1: The universal forms (in the strong-screening limit $\gamma\rightarrow 0$) of the magnetic field (\ref{['magnetic_field']}) (characterized by the function $F(\rho)$) and the electrostatic potential (\ref{['conjecture']}) (proportional to $w(\rho)$) associated with the Abrikosov flux line as given by Eqs.(\ref{['Abrikosov']}) and (\ref{['limiting_potential']}). The length $\rho$ is measured in units of the coherence length (\ref{['coherence_length']}).
• Figure 2: Dimensionless density of superconductive electrons $R^{2}(\rho)$ (bold) and potential $w(\rho)$ (grey scale) of an $|l|=1$ Abrikosov vortex at distance $\rho\ll\kappa$ from its axis according to Eqs.(\ref{['Bernoulli_dimensionless']}),(\ref{['Poisson_dimensionless']}) and (\ref{['Ampere_dimensionless']}) in the $\kappa\gg 1$ limit for different strengths of electrostatic screening parameterized by the dimensionless Debye screening length $\gamma$. Dotted red curve is the Ginzburg-Pitaevskii solution for the density profile $R^{2}(\rho)$ of the Onsager-Feynman vortex in neutral superfluid LL9. Solid red and dashed red curves represent asymptotic strong screening results for the density and potential given by Eqs.(\ref{['correction2local_neutrality']}) and (\ref{['universal_potential']}), respectively.