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Magnetic skyrmions are magnetic quasi-particles with enhanced stability and different manipulation mechanisms using external fields and currents making them promising candidates for future applications for instance in neuromorphic computing. Recently, several measurements and simulations have shown that thermally activated skyrmions in confined geometries, as they are necessary for device applications, arrange themselves predominantly based on commensurability effects. In this simulational study, based on the Thiele model, we investigate the enhanced dynamics and degenerate non-equilibrium steady state of a system in which the intrinsic skyrmion-skyrmion and skyrmion-boundary interaction compete with thermal fluctuations as well as current-induced spin-orbit torques. The investigated system is a triangular-shaped confinement geometry hosting four skyrmions, where we inject spin-polarized currents between two corners of the structure. We coarse-grain the skyrmion states in the system to analyze the intricacies of skyrmion arrangements of the skyrmion ensemble. In the context of neuromorphic computing, such methods address the key challenge of optimizing read-out positions in confined geometries and form the basis to understand collective skyrmion dynamics in systems with competing interactions on different scales.

We present a method of calculation of the effective magnetic permeability of magnonic metamaterials containing periodically arranged magnetic inclusions of arbitrary shapes. The spectrum of spin wave modes confined in the inclusions is fully taken into account. Within the scope of the proposed method, we compare two approaches. The first approach is based on a simple semi-analytical theory that uses the numerically calculated susceptibility tensor of an isolated inclusion as input data. Within the second approach, micromagnetic packages with periodic boundary conditions (PBC) are used to calculate the susceptibility of a single 2D periodic array of such inclusions, with the whole 3D metamaterial consisting of a stack of such arrays. To calculate the susceptibility tensor of an isolated inclusion, we have implemented and compared two different methods: (a) a micromagnetic method, in which we have employed three different micromagnetic packages: the finite element package NMAG and the two finite differences packages OOMMF and MicroMagus; and (b) the modified dynamical matrix method. To illustrate the methodology, we have calculated the effective permeability of a metamaterial consisting of a stack of hexagonal arrays of magnetic nanodisks in a non-magnetic matrix. The range of geometrical parameters for which such a metamaterial is characterized by the negative permeability has been identified. The critical comparison of the different micromagnetic packages and the dynamical matrix method (based on the calculation of the susceptibility tensor of an isolated inclusion) has demonstrated that their results agree to within 3 %., Comment: submitted to PRB, 21 pages, 4 figures