Ginzburg-Landau vortex dynamics with pinning and strong applied currents

TL;DR

This paper analyzes the motion of vortices in a superconductor model under electric currents and pinning effects, deriving their limiting dynamics as the Ginzburg-Landau parameter becomes small.

Contribution

It provides a unified derivation of vortex dynamics considering both pinning potentials and strong applied currents in the Ginzburg-Landau framework.

Findings

Vortex velocity is the sum of Lorentz and pinning forces.

Derived limiting vortex dynamics as the Ginzburg-Landau parameter approaches zero.

Extended results to models without magnetic fields but with forcing.

Abstract

We study a mixed heat and Schr\"odinger Ginzburg-Landau evolution equation on a bounded two-dimensional domain with an electric current applied on the boundary and a pinning potential term. This is meant to model a superconductor subjected to an applied electric current and electromagnetic field and containing impurities. Such a current is expected to set the vortices in motion, while the pinning term drives them toward minima of the pinning potential and "pins" them there. We derive the limiting dynamics of a finite number of vortices in the limit of a large Ginzburg-Landau parameter, or $\ep \to 0$, when the intensity of the electric current and applied magnetic field on the boundary scale like $\lep$. We show that the limiting velocity of the vortices is the sum of a Lorentz force, due to the current, and a pinning force. We state an analogous result for a model Ginzburg-Landau equation without magnetic field but with forcing terms. Our proof provides a unified approach to various proofs of dynamics of Ginzburg-Landau vortices.