Efficient solution of 3D Ginzburg-Landau problem for mesoscopic superconductors

Abstract

The recently proposed approach for the solution of Ginzburg-Landau (GL) problem for 2D samples of arbitrary shape is, in this article, extended over 3D samples having the shape of (i) a prism with arbitrary base and (ii) a solid of revolution with arbitrary profile. Starting from the set of Laplace operator eigenfunctions of a 2D object, we construct an approximation to or the exact eigenfunctions of the Laplace operator of a 3D structure by applying an extrusion or revolution to these solutions. This set of functions is used as the basis to construct the solutions of the linearized GL equation. These solutions are then used as basis for the non-linear GL equation much like the famous LCAO method. To solve the non-linear equation, we used the Newton-Raphson method starting from the solution of the linear equation, i.e., the nucleation distribution of superconducting condensate. The vector potential approximations typically used in 2D cases, i.e., considering it as corresponding to applied constant field, are in the 3D case harder to justify. For that reason, we use a locally corrected Nystrom method to solve the second Ginzburg-Landau equation. The complete solution of GL problem is then achieved by solving self-consistently both equations.