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

Abstract

We study the three-dimensional Ginzburg-Landau model of superconductivity for strong applied magnetic fields varying between the second and third critical fields. 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. We also study the geometric behavior of the order parameter near the surface of the sample by constructing formal solutions with lattice properties.

A 3D-Schroedinger operator under magnetic steps

“…(2.10) This is a well-known result related to the Landau bands for constant magnetic fields in R 3 (see e.g. [FMP,Proposition 2.1]).…”

“…Within this context, 3D models were also studied in the mathematical literature on unbounded/bounded domains with smooth boundary or having edges [LP2,Pan2,HM2,Pop1,Pop2,Pop3] (also see [FDM,FH3,FH4,FK2,FKP,FMP]). However, all these contributions considered magnetic fields with a constant strength.…”

Effects of corners in surface superconductivity

“…It is expected that a regime of surface superconductivity with similar features occurs also for genuine 3D samples but so far only partial results are available [38,39]. In particular, in [38,Thm 1.1] (see also [36]) it is shown that such a regime does exist and the leading order term in the energy asymptotics can be identified, although in terms of a rather implicit effective problem.…”

“…In particular, in [38,Thm 1.1] (see also [36]) it is shown that such a regime does exist and the leading order term in the energy asymptotics can be identified, although in terms of a rather implicit effective problem. In [39,Thm. 1.5] it is then proven that, when the magnetic field is parallel to the 3D boundary, the effective model is still given by the 1D functional (2.3) above.…”

On estimation of the number of eigenvalues of the magnetic Schrödinger operator in a three-dimensional layer

“…It is known that (see [1][2][3][4][5][6]) at a certain value of the magnetic field strength in superconducting materials of the second kind, a surface superconducting phenomenon occurs. This kind of magnetic field is called the third critical field.…”