Estimating the Superconducting Superheating Field in Time-Dependent Ginzburg-Landau Theory using Bifurcation Analysis

The expulsion of magnetic fields is a hallmark of superconductivity known as the Meissner effect.~ In the presence of an applied magnetic field, the Meissner state is thermodynamically stable up to a critical magnetic field (Hc for type I superconductors and Hc1 for type II superconductors).~ However, the Meissner state may persist as a metastable state up to the so-called "superheating field", Hsh.~ Understanding the dependence of Hsh on material and geometry is an important question for improving performance of particle accelerators.~ We numerically study the superheating transition in time-dependent Ginzburg-Landau theory using finite-element methods.~ At the superheating field, the equations exhibit a saddle-node bifurcation.~ We use techniques from numerical analysis of dynamical systems to estimate Hsh.~ We estimate the time for the system to equilibrate at small values of the applied field and extrapolate to where the equilibration time diverges.~ We explore the dependence on Hsh on material and geometric properties of interest in accelerator physics.