Nature of Driving Force on an Isolated Moving Vortex in Dirty Superconductors

We reconsider the force-balance relation on an isolated vortex in the flux flow state within the scheme of the time-dependent Ginzburg–Landau (TDGL) equation. We define the force on a 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 an isolated vortex while taking account of the fact that the transport current in charged superconductors is inherently spatially varying with the scale of the penetration depth λ. We also find that the nature of the driving force is hydrodynamic when S is a disk with radius R satisfying ξ ≪ R ≪ λ (ξ is the coherence length) while the hydrodynamic and magnetic parts contribute equally to the driving force for \(\lambda \lesssim R\).