Interlayer and interfacial Dzyaloshinskii-Moriya interaction in magnetic trilayers: First-principles calculations

We determine the Dzyaloshinskii-Moriya interaction within and between two magnetic cobalt layers separated by a nonmagnetic spacer through ab initio calculations. We investigate different materials for the nonmagnetic layer, focusing on the experimentally realized Co/Ag/Co system. We laterally shift the atoms in the nonmagnetic layer to achieve the symmetry breaking required for the interlayer Dzyaloshinskii-Moriya interaction. We compare the resulting interactions with the Levy-Fert model and observe a good overall agreement between the model and the ab initio calculations for the dependence on the atomic positions. Additionally, we derive a formula for the strength of the interlayer isotropic exchange interaction depending on the position of the atoms in the nonmagnetic layer and compare it to the first-principles results. We investigate the limitations of the Levy-Fert model by turning off the spin-orbit coupling separately on the nonmagnetic and magnetic atoms and by studying the effect of band filling. Our work advances the understanding of the microscopic mechanisms of the interlayer Dzyaloshinskii-Moriya interaction and gives insight into possible new material combinations with strong interlayer Dzyaloshinskii-Moriya interaction.