From vortex molecules to the Abrikosov lattice in thin mesoscopic superconducting disks

Abstract

Stable vortex states are studied in large superconducting thin disks (for numerical purposes we considered with radius R = 50ξ). Configurations containing more than 700 vortices were obtained using two different approaches: the nonlinear Ginzburg-Landau (GL) theory and the London approximation. To obtain better agreement with results from the GL theory we generalized the London theory by including the spatial variation of the order parameter following Clem's ansatz. We find that configurations calculated in the London limit are also stable within the Ginzburg-Landau theory for up to ∼ 230 vortices. For large values of the vorticity (typically, L 100), the vortices are arranged in an Abrikosov lattice in the center of the disk, which is surrounded by at least two circular shells of vortices. A Voronoi construction is used to identify the defects present in the ground state vortex configurations. Such defects cluster near the edge of the disk, but for large L also grain boundaries are found which extend up to the center of the disk.

Multivortex and giant vortex states near the expulsion and penetration fields in thin mesoscopic superconducting squares

“…At fixed temperature, they studied well-defined shell structures containing up to 40 vortices and identified rules of shell filling and magic numbers, in agreement with Ref. 6. Also squares and triangles were studied but no evidence for giant vortex states was found.…”

“…Circular mesoscopic disks have been the most popular in this respect, both theoretically [1][2][3][4][5][6] and experimentally. [7][8][9][10][11][12][13] Two types of vortex states were found in such mesoscopic superconducting disks: ͑i͒ giant vortex states ͑GVSs͒, where the order parameter has a single zero and ͑ii͒ multivortex states ͑MVSs͒ consisting of several singly quantized vortices ͑mostly situated on shells͒.…”

Vortex shells in mesoscopic triangles of amorphous superconducting thin films

“…These structural ordering compete with each other and the resulted vortex arrangement is determined by the interplay between the geometry-induced confining potential and the mutual repulsive interaction between vortices [1,2,3,4]. For large vorticities, a piece(s) of the triangular vortex lattice remains near the disk center because of densely packed vortices [3,5], whereas for small vorticities, vortices are arranged themselves into regular polygons situated on concentric circular rings from the disk center [1,2,4]. This is called "vortex shells", of which configurations have been visualized by some experimental studies on mesoscopic superconducting disks of Nb and amorphuous MoGe thin films [6,7].…”

“…The issue of vortex shells would be interesting when vortices are confined in polygonal small superconductors like triangle [8,9,10], square [11,12] and pentagonal shaped dots [13]. These polygonal dots have discrete, natural symmetries that coincide or compete with the configurational symmetry of vortex polygons, depending on vorticity L. In triangle dots, vortices intrinsically form a triangular cluster of which symmetry is commensurate with that of the triangle geometry.…”

Electrostatic self-energies of discrete charge distributions on Jordan curves

“…The combination of Coulombic interparticle interactions and two-dimensional confining potentials of cylindrical symmetry usually produces assemblies of particles positioned on either vertices of polygons inscribed on concentric rings or nodes of triangular lattice [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. Formation of such patterns, which is observed both in experimental settings and numerical simulations, occurs in systems ranging from electrons in quantum dots [2][3][4][5][6][7] to ions in dusty plasmas [8][9][10] and triboelectrically charged objects [11].…”