Concentration Behavior and Lattice Structure of 3D Surface Superconductivity in the Half Space

Abstract

We study a 3D Ginzburg-Landau model in a half-space which is expected to capture the key features of surface superconductivity for applied magnetic fields between the second critical field $H_{C_{2}}$ and the third critical field $H_{C_{3}}$. For the magnetic field in this regime, it is known from physics that superconductivity should be essentially restricted to a thin layer along the boundary of the sample. This leads to the introduction of a Ginzburg-Landau model on a half-space. We prove that the non-linear Ginzburg-Landau energy on the half-space with constant magnetic field is a decreasing function of the angle ν that the magnetic field makes with the boundary. In the case when the magnetic field is tangent to the boundary (ν = 0), we show that the energy is determined to leading order by the minimization of a simplified 1D functional in the direction perpendicular to the boundary. For non-parallel applied fields, we also construct a periodic problem with vortex lattice minimizers reproducing the effective energy, which suggests that the order parameter of the full Ginzburg-Landau model will exhibit 3 dimensional vortex structure near the surface of the sample.