Finite-element dynamic-matrix approach for propagating spin waves: Extension to mono- and multi-layers of arbitrary spacing and thickness

Abstract

In our recent work [Körber et al., AIP Adv. 11, 095006 (2021)], we presented an efficient numerical method to compute dispersions and mode profiles of spin waves in waveguides with translationally invariant equilibrium magnetization. A finite-element method (FEM) allowed to model two-dimensional waveguide cross sections of arbitrary shape but only finite size. Here, we extend our FEM propagating-wave dynamic-matrix approach from finite waveguides to the important cases of infinitely extended mono- and multi-layers of arbitrary spacing and thickness. To obtain the mode profiles and frequencies, the linearized equation of the motion of magnetization is solved as an eigenvalue problem on a one-dimensional line-trace mesh, defined along the normal direction of the layers. Being an important contribution to multi-layer systems, we introduce interlayer exchange into our FEM approach. With the calculation of dipolar fields being the main focus, we also extend the previously presented plane-wave Fredkin–Koehler method to calculate the dipolar potential of spin waves in infinite layers. The major benefit of this method is that it avoids the discretization of any non-magnetic material such as non-magnetic spacers in multilayers. Therefore, the computational effort becomes independent of the spacer thicknesses. Furthermore, it keeps the resulting eigenvalue problem sparse, which, therefore, inherits a comparably low arithmetic complexity. As a validation of our method (implemented into the open-source finite-element micromagnetic package TETRAX), we present results for various systems and compare them with theoretical predictions and with established finite-difference methods. We believe this method offers an efficient and versatile tool to calculate spin-wave dispersions in layered magnetic systems.