Flux flow solution as a boundary value problem of the Time-dependent Ginzburg-Landau equation

We study force on vortices in type II superconductors (SC) with finite Ginzburg-Landau parameter κ within the generalized time-dependent-Ginzburg-Landau (TDGL) equations developed by Watts-Tobin, Krähenbühl Kramer (1981) for dirty s-wave SC.While earlier studies addressed the flow conductivity only via the solvability condition for the flux flow problem, we calculate the full solution for boundary value problem with the Meissner current state as the asymptotic state in a distant region from the vortex core. We find transport current j_t which appears in the force-balance equation is not the local (actual) current through the vortex core, but the extrapolated value at the vortex center defined from the boundary condition. We also calculate the hydrodynamic force, magnetic Lorentz force, environmental force and the driving force, which is the sum of the hydrodynamic force and magnetic Lorentz force, on the basis of the full solution of TDGL equation. We find that only the sum of the hydrodynamic and magnetic Lorentz force is eligible as a well-defined driving force on vortex. We also identify the radius of dissipation region around the vortex core as “the radius of vortex” from the viewpoint of defining the force on vortex.