Vortex pattern in three-dimensional mesoscopic superconducting rings

Vortex structures in a mesoscopic a superconducting ring, which is in the magnetic field generated by a circular electric current, are investigated based on the phenomenological Ginzburg-Landau (G-L) theory. Due to the axial symmetry of the system, the three-dimensional problem is reduced to a two-dimensional problem. We can mesh a two-dimensional sample into grids, and discretize the first G-L equation by using the finite-difference method. Then the eigenvalues and eigenfunctions will be evaluated numerically by solving the discrete equations. With the eigenvalues and eigenfunctions we further obtain the minimum free energy of the system and the corresponding superconducting wave function. We discuss the influences of the ring size and magnetic field distribution on two kinds of the vortex structures: giant vortex state (GVS) and multivortex state (MVS). Calculations show: 1) the GVS with axial symmetric wave function exists only in a small size superconducting ring, as the GVS is a state of single vortex line that only goes through the hole at the center of the superconducting ring and carries several magnetic flux quanta with it; 2) with the increase of the ring size, the diamagnetism of superconducting ring becomes stronger, and the critical magnetic field value of a giant vortex state increases, and the maximal number of giant vortexes that the superconducting ring can accommodate is also growing; furthermore, the entrance of a flux line will cause fluctuations of critical field values; 3) when the superconducting ring size is large enough, a GVS splits into a number of MVS. The MVS is an excited state and the GVS is mostly a ground state; 4) the free energy of the system changes with the magnetic field distribution, the magnetic field provided by a central small current loop can pass through the superconducting ring easily, and produce multivortices whose formations are diverse; if the magnetic field runs parallel to the plane of the superconducting ring, it is difficult to pass through the superconducting ring and form multivortices; 5) the vortex lines are naturally bent with the magnetic field lines and can pass through the same horizontal plane twice, so that one of the two vortex states seems to be an antivortex state; generally, the magnetic field lines can go through the hole of a superconducting ring easily but can hardly penetrate through the body of a superconducting ring, the structure of multivortices is similar to that of the magnetic field distribution in a superconducting ring. We also obtain a vortex structure with coexistences of giant vortex and multivortices. This study is of significance for the application of superconducting nanomaterials.