Eigenmodes of magnetic skyrmion lattices

We explore the interplay between topology and eigenmodes by changing the stabilizing mechanism of skyrmion lattices (skX). We focus on two prototypical ultrathin films hosting a hexagonal [Pd/Fe/Ir(111)] and a square [Fe/Ir(111)] skyrmion lattice, which can both be described by an extended Heisenberg Hamiltonian. We first examine whether the Dzyaloshinkskii-Moriya, or the exchange interaction as the leading energy term affects the modes of the hexagonal skX of Pd/Fe/Ir(111). In all cases, we find that the lowest-frequency modes correspond to internal degrees of freedom of individual skyrmions, and suggest a classification based on azimuthal and radial numbers (l, p), with up to l = 6 and p = 2. We also show that the gyration behavior induced by an in-plane field corresponds to the excitation of l = 1 deformation modes with varying radial numbers. Second, we examine the square lattice of skyrmions of Fe/Ir(111). Its stabilization mechanism is dominated by the four-spin interaction. After relaxation, the unit cell does not carry a topological charge, and the eigenmodes do not correspond to internal skyrmion deformations. By reducing the four-spin interaction, the integer topological charge is recovered, but the charge carriers do not possess internal degrees of freedom, nor are they separated by energy barriers. We conclude that a four-spin dominated Hamiltonian does not yield skyrmion lattice solutions and that, therefore, a nontrivial topology does not imply the existence of skyrmions.