On the stability of normal states for a generalized Ginzburg–Landau model

Abstract

We formulate a spectral problem related to the onset of superconductivity for a generalized Ginzburg–Landau model, where the order parameter and the magnetic potential are defined in the whole space. This model is devoted to the ‘proximity effect’ for a superconducting sample surrounded by a normal material. In the regime when the Ginzburg–Landau parameter (of the superconducting material) is large, we estimate the critical applied magnetic field for which the normal state will lose its stability, a result that has some roots in the physical literature. In some asymptotic situations, we recover results related to the ‘standard’ Ginzburg–Landau model, where we mention in particular the two-term expansion for the upper critical field obtained by Helffer–Pan.