Boundary Behavior of the Ginzburg–Landau Order Parameter in the Surface Superconductivity Regime

Abstract

We study the 2D Ginzburg-Landau theory for a type-II superconductor in an applied magnetic field varying between the second and third critical value. In this regime the order parameter minimizing the GL energy is concentrated along the boundary of the sample and is well approximated to leading order (in L 2 norm) by a simplified 1D profile in the direction perpendicular to the boundary. Motivated by a conjecture of Xing-Bin Pan, we address the question of whether this approximation can hold uniformly in the boundary region. We prove that this is indeed the case as a corollary of a refined, second order energy expansion including contributions due to the curvature of the sample. Local variations of the GL order parameter are controlled by the second order term of this energy expansion, which allows us to prove the desired uniformity of the surface superconductivity layer.

Surface superconductivity in presence of corners

“…The corresponding minimizing α will be denoted by α , while f will stand for the function minimizing E 1D α . The 1D functional (1.16) was known to provide a model problem for surface superconductivity since [Pa], but only in [CR2] (for disc-like samples) and [CR3] (see also [CR4] for further results in this direction) it was proven that f is a good approximation of |ψ GL |. The main idea is indeed that any minimizing GL order parameter has the structure…”

“…Before proceeding however we briefly comment on possible future perspectives. Inspired by the analysis contained in [CR3] one can indeed aim at capturing higher order corrections to the energy asymptotics (1.20) and isolate the curvature corrections to the energy. At this level the presence of corners might give rise to an additional contribution to the energy of the same order of the curvature corrections.…”

The density of superconductivity in the bulk regime

“…This corresponds to the situation of an external magnetic field with intensity strictly below the second critical field H c 2 (κ) := κ . The case where b > 1 in (1.7) is related to the phenomenon of surface superconductivity which is extensively studied by many authors [4,8,10,22].…”

On the Ginzburg–Landau energy with a magnetic field vanishing along a curve

“…That has been fairly understood for type II superconductors in the case where the magnetic field is uniform (i.e. B 0 = 1) which has allowed to distinguish between three critical values for the intensity of the applied magnetic field, denoted by H C 1 , H C 2 and H C 3 whose role can be described as follows (see [13,24,9,8,10,15]):…”