Nature of driving force on an isolated moving vortex in dirty superconductors

TL;DR

This paper reexamines the driving force on an isolated vortex in dirty superconductors using the TDGL framework, revealing hydrodynamic and magnetic contributions depending on the spatial scale of the current distribution.

Contribution

It derives a local momentum balance relation and clarifies the nature of the driving force considering spatially varying transport currents in superconductors.

Findings

Hydrodynamic nature of the force when the region size is between coherence length and penetration depth.

Equal contribution of hydrodynamic and magnetic parts for regions comparable to the penetration depth.

Force-balance relation accounts for spatial variation of transport current in superconductors.

Abstract

We reconsider the force-balance relation on an isolated vortex in the flux flow state within the scheme of time-dependent-Ginzburg-Landau (TDGL) equation. We define force on the vortex by the total force on superconducting electrons in the region $S$ surrounding the vortex. We derive the local momentum balance relation of superconducting electrons and then find the force-balance relation on isolated vortex with taking account of the fact that the transport current in charged superconductors are inherently spatially varying with the scale of penetration depth $\lambda$. We also find that nature of the driving force is hydrodynamic when $S$ is the disk with radius $R$ satisfying $\xi\ll R\ll \lambda$ ($\xi$ is the coherence length) while the hydrodynamic and magnetic parts contribute equally to the driving force for $\lambda \lesssim R$.