Dipole-exchange spin waves in two-dimensional van der Waals ferromagnetic films and stripes

Abstract

A spin-wave (SW) theory that includes the long-range dipole–dipole interactions is presented for monolayers of van der Waals (vdW) ferromagnets for which the magnetic ions lie on a two-dimensional honeycomb lattice. The dipolar interactions provide an additional anisotropy in these materials, along with the Ising exchange interaction and/or single-ion anisotropies that typically stabilize the two-dimensional magnetic ordering. Analytical results for the linearized SW energies are obtained for the ferromagnets in two geometries: complete films and finite-width stripes (or ribbons). In both cases it is found that the inclusion of the dipole–dipole interactions leads to a shift and sometimes a splitting of the magnetic modes in the vdW structure. Also, in the latter case, where the edges are assumed to be along the zigzag lattice directions, the dipole–dipole interactions are found to play a role, as well as the exchange interactions, in modifying the localized edge SWs. Numerical examples are given, including applications to the ferromagnet CrI3.