Microscopic Calculation of Dzyaloshinskii-Moriya Interaction in a Rashba Ferromagnet

We theoretically study the Dzyaloshinskii-Moriya interaction (DMI) in metallic magnets with strong spin-orbit interaction. We derive a general formula for the DMI in free energy in terms of Green’s functions. Generally, the DMI coefficient in free energy can be different from that in spin-wave dispersion. We show the relationship between these coefficients. Using the general formula, we investigate the DMI in a Rashba ferromagnet. We observe that the DMI coefficient shows a strong dependence on electron density and a higher-order dependence on the spin-orbit interaction.