Computational Investigation of the Effects of Sample Geometry on the Superconducting-Normal Phase Boundary and Vortex-Antivortex States in Mesoscopic Superconductors

Abstract

A computational study of superconducting states near the superconductingnormal phase boundary in mesoscopic finite cylinders is presented. The computational approach uses a finite element method to find numerical solutions of the linearized Ginzburg-Landau equation for samples with various sizes, aspect ratios, and crosssectional shapes, i.e., squares, triangles, circles, pentagons, and four star shapes. The vector potential is determined using a finite element method with two penalty terms to enforce the gauge conditions that the vector potential is solenoidal and its normal component vanishes at the surface(s) of the sample. The eigenvalue problem for the linearized Ginzburg-Landau equations with homogeneous Neumann boundary conditions is solved and used to construct the superconducting-normal phase boundary for each sample. Vortex-antivortex (V-AV) configurations for each sample that accurately reflect the discrete symmetry of each sample boundary were found through the computational approach. These V-AV configurations are realized just within the phase boundary in the magnetic field-temperature phase diagram. Comparisons are made between the results obtained for the different sample shapes.

Method for the solution of the nucleation problem in arbitrary mesoscopic superconductors: Theory and application

“…Figure 4 shows the vortex patterns for the states of vorticity 1 to 6 for the different geometries. It can be seen that the results for the triangle, square, and pentagon are in agreement with the already known results [2,8,23], indicating that the present method is a natural extension of the previously reported superconducting gauge method [22]. From Fig.…”

“…Recently, a numeric method [23] was developed to solve the LGL problem for bounded geometries using the finite element method and the superconducting gauge approach. The solutions for this problem were found for different geometries, including some highly symmetric geometries, like the square, triangle, pentagon, and five-pointed star.…”

“…These solutions share a very close correspondence between them. Some of the regular polygons have already known solutions [2,8,23] for the LGL equation. These were also applied to better test the quality of the algorithm.…”

Quantum states and vortex patterns in nanosuperconductors

“…Theoretical investigation of vortex states in mesoscopic superconductors was an active research domain in the last decade . Compared to bulk superconductors, the description of mesoscopic samples is much more complicated because of a non‐negligible effect of the boundary on the superconducting condensate.…”

“…Compared to bulk superconductors, the description of mesoscopic samples is much more complicated because of a non‐negligible effect of the boundary on the superconducting condensate. This is the reason why vortices in mesoscopic superconductors have been studied first within the phenomenological Ginzburg‐Landau (GL) theory . This approach has proven to be well suited for the description of a complex superconducting/normal phase boundary line and its dependence on the geometry of the samples (see, e.g., Ref.…”