First-principles study of the spin-orbit coupling contribution to anisotropic magnetic interactions

Abstract

Anisotropic magnetic exchange interactions lead to a surprisingly rich variety of magnetic properties. Considering the spin-orbit coupling (SOC) as perturbation, we extract the general expression of a bilinear spin Hamiltonian, including isotropic exchange interaction, antisymmetric Dzyaloshinskii-Moriya (DM) interaction, and symmetric $\mathrm{\ensuremath{\Gamma}}$ term. We derive the expressions for the second-order SOC contribution to DM interaction, and reveal that the essential distinction between the DM and $\mathrm{\ensuremath{\Gamma}}$ term is from their different hopping processes, rather than the different orders of SOC. Based on combining the magnetic force theorem and linear-response approach, we present a method of calculating anisotropic magnetic interactions, which now has been implemented in the open source software WienJ. Furthermore, we introduce another method which could calculate the first- and second-order SOC contribution to the DM interaction separately, and overcome some shortcomings of previous methods. Our methods are successfully applied to several typical weak ferromagnets for $3d$, $4d$, and $5d$ transition-metal oxides. We also predict the conditions where the DM interactions proportional to $\ensuremath{\lambda}$ approximately vanish while the DM interactions proportional to ${\ensuremath{\lambda}}^{2}$ are nonzero, and believe that it may exist in certain magnetic materials.