Persistence of superconductivity in thin shells beyond Hc1

Abstract

In Ginzburg-Landau theory, a strong magnetic field is responsible for the breakdown of superconductivity. This work is concerned with the identification of the region where superconductivity persists, in a thin shell superconductor modeled by a compact surface M ⊂ R 3 , as the intensity h of the external magnetic field is raised above Hc1. Using a mean field reduction approach devised by Sandier and Serfaty as the Ginzburg-Landau parameter κ goes to infinity, we are led to studying a two-sided obstacle problem. We show that superconductivity survives in a neighborhood of size (Hc1/h) 1/3 of the zero locus of the normal component H of the field. We also describe intermediate regimes, focusing first on a symmetric model problem. In the general case, we prove that a striking phenomenon we call freezing of the boundary takes place: one component of the superconductivity region is insensitive to small changes in the field.

The influence of magnetic steps on bulk superconductivity

“…Recently, attention has been shifted to non-uniform smooth magnetic fields in [2,3,13,17]. Such magnetic fields may arise in the study of superconducting surfaces [7] or superconductors with applied electric currents [1].…”

First Critical Field of Highly Anisotropic Three-Dimensional Superconductors via a Vortex Density Model

“…Up to o(1) asymptotics for H c 1 are derived in [9,13] for thin superconductors. Finally, a characterization of the superconducting region for much higher values of the applied field in a superconducting shell is obtained in [11] based on a reduction to a double-sided obstacle problem. In general, a big problem in extending results from 2d to 3d lies in the description of the vorticity region which in the two dimensional case corresponds to a union of points, while in higher dimensions it can be given by very complex and nonsmooth structures.…”

Decay of superconductivity away from the magnetic zero set

“…• In the study of the time-dependent Ginzburg-Landau equations [4,5], applied magnetic fields as in Assumption 1.1 naturally appear in the presence of applied electric currents. • For superconducting surfaces submitted to constant magnetic fields [11], the constant magnetic field may induce a smooth sign-changing magnetic field on the surface. • In the transition from normal to superconducting configurations [35], one meets the problem of determining H such that the ground state energy in (1.2) vanishes on a curve meeting transversally the boundary.…”