General solution of 2D and 3D superconducting quasiclassical systems: coalescing vortices and nanoisland geometries

An extension of quasiclassical Keldysh-Usadel theory to higher spatial dimensions than one is crucial in order to describe physical phenomena like charge/spin Hall effects and topological excitations like vortices and skyrmions, none of which are captured in one-dimensional models. We here present a numerical finite element method which solves the non-linearized 2D and 3D quasiclassical Usadel equation relevant for the diffusive regime. We show the application of this on three model systems with non-trivial geometries: (i) a bottlenecked Josephson junction with external flux, (ii) a nanodisk ferromagnet deposited on top of a superconductor and (iii) superconducting islands in contact with a ferromagnet. In case (i), we demonstrate that one may control externally not only the geometrical array in which superconducting vortices arrange themselves, but also to cause coalescence and tune the number of vortices. In case (iii), we show that the supercurrent path can be tailored by incorporating magnetic elements in planar Josephson junctions which also lead to a strong modulation of the density of states. The finite element method presented herein paves the way for gaining insight in physical phenomena which have remained largely unexplored due to the complexity of solving the full quasiclassical equations in higher dimensions.