Vortices in the three-dimensional thin-film Ginzburg–Landau model of superconductivity

Abstract

Building on the results of Chapman et al. (Z Angew Math Phys 47:410-431, 1996) on the behavior of minimizers in the Ginzburg-Landau thin-film model, we show that the vortices in the three-dimensional superconducting thin films are located in the cylinders whose cross sections coincide with the disks that contain the vortices in the two-dimensional model. To arrive at this conclusion, we prove that the three-dimensional minimizers converge to the two-dimensional counterparts in H 1 and in C α . We also give examples of regimes in which the vortex structure of the two-dimensional minimizers is well understood. Our results, in particular, provide insight into the behavior of the three-dimensional vortices in these regimes. (2000). Primary 35Q56. Mathematics Subject Classification

Superconducting properties of a parallelepiped mesoscopic superconductor: A comparative study between the 2D and 3D Ginzburg–Landau models

“…In particular, as a consequence of the demagnetization effects, when the size of the sample along the direction of the applied magnetic field is smaller than the lateral dimensions of its cross section, the local magnetic field near the edges of the sample is enhanced and interacts with the shielding currents. There are many experimental (see for instance [1][2][3][4][5]) and theoretical (see for instance [6][7][8][9][10]) studies in threedimensional (3D) systems. For example, in [11], a superconducting wire with a constriction in the middle was investigated.…”

Superconducting properties of a mesoscopic parallelepiped with anisotropic surface conditions

“…However, from the quantitative point of view it is much more advantageous than the two-dimensional (2D) TDGL model, since we obtain precisely the critical fields, the vortex configurations, energy and magnetization stability curves, etc. There are many experimental (see for instance [1][2][3][4][5][6]) and theoretical (see for instance [7][8][9][10]) studies for 3D systems. In all these theoretical studies, the Ginzburg-Landau model has been proven to give a good account of the superconducting properties in samples of several geometries, i.e., disks with finite thickness and spheres [11,12], shells [13], cone [14], thin circular sectors, thin disks and SQUID geometry [15][16][17][18].…”