The frozen-field approximation and the Ginzburg-Landau equations of superconductivity

TL;DR

The paper studies the transition from the full time-dependent Ginzburg–Landau model to the frozen-field approximation in type-II superconductors when the Cooper-pair charge vanishes while the applied field remains near the upper critical field. It combines detailed numerical simulations with rigorous asymptotic analysis, showing that the GL solution converges to the frozen-field solution at a rate of $O(\kappa^{-2})$ as $\kappa\to\infty$. The main contributions are a precise scaling, a functional-analytic reduction, and convergence estimates that justify using the simpler frozen-field model for large $\kappa$ in vortex-dynamics studies. This provides a practical, efficient framework for simulating vortex configurations and pinning phenomena without solving the full GL system, while retaining quantitative error control.

Abstract

The Ginzburg--Landau (GL) equations of superconductivity provide a computational model for the study of magnetic flux vortices in type-II superconductors. In this article we show through numerical examples and rigorous mathematical analysis that the GL model reduces to the frozen-field model when the charge of the Cooper pairs (the superconducting charge carriers) goes to zero while the applied field stays near the upper critical field.

Figures (7)

• Figure 1: Phase diagram of a type-II superconductor.
• Figure 2: Vortex configuration in two dimensions.
• Figure 3: Kinking-induced motion of vortices through splayed columnar defects.
• Figure 4: Computational grid cell and definition of the discrete variables.
• Figure 5: Contours of the density of Cooper pairs, $|\psi|^2$, for a system with $\kappa = 200$.

Theorems & Definitions (2)

• Theorem 4.1
• Corollary 4.1