Abstract
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.
Highlights
Nonlinear differential equations (NLDEs) play a pivotal role in virtually all areas of physics
We report as the main result of this paper the description of a finite element method that we have developed which is capable describing mesoscopic systems in 2D and 3D using quasiclassical theory without any linearization
We have demonstrated how the full, spin dependent, Usadel equation may be solved by the finite element method
Summary
Where α is an element of equation 13 and F (α) is a function that performs the matrix multiplications of equation 12 and extracts the appropriate element. A similar expression is found the...∼ operation to equation 15 These are Neumann boundary conditions of the type for n. The unit vector ν is an outward pointing surface normal, and is either parallel or antiparallel with the normal vector n as defined in the Kupriyanov-Lukichev boundary conditions. It may be expressed as ν = (ν ⋅ n )n. The integrals over the element domains are performed by changing coordinates to a reference element, and integrating numerically by means of a Gauss quadrature This puts restrictions on how distorted a mesh can be, as the Jacobian for the coordinate transformation has to exist. A structured mesh where the deviation from the geometry of the reference element is small will often give higher accuracy and reduce the computation time as the sparsity of the assembled matrices is increased
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