Linear response based theories for Dzyaloshinskii-Moriya interactions

We investigate abilities of various linear response based techniques for extracting parameters of antisymmetric Dzyaloshinskii-Moriya (DM) interactions from first-principles electronic structure calculations. For these purposes, we further elaborate the idea of Sandratskii [Phys. Rev. B 96, 024450 (2017)], which states that the $z$ component of the DM vector can be computed by retaining only the spin-diagonal part of the spin-orbit (SO) interaction. This approximation, which becomes exact to the first order in the SO coupling, greatly simplifies the calculations as it requires only minor extensions in comparison to isotropic parameters in the nonrelativistic case. We start our analysis with the magnetic force theorem (MFT), which relies on additional approximations resulting in the linear dependence of the exchange interactions on the response tensor, and compare it with the exact approach formulated in terms of the inverse response. For the ligand states, which are not primarily responsible for the magnetism but magnetized from the localized states, we propose the downfolding procedure transferring the effect of these ligand spins into parameters of effective interactions between the localized spins. These techniques are applied for the series of ${\mathrm{CrCl}}_{3}$- and ${\mathrm{CrI}}_{3}$-based materials, including bulk, monolayer, bilayer, and three-layer systems. Particularly, we discuss how the DM interactions are induced by the inversion symmetry breaking at the surface or by the electric field. As long as the response tensor between the Cr $3d$ states is calculated by taking into account the SO interaction on the heavy ligand sites, the MFT appears to be a good approximation for the DM interactions, being in contrast with the isotropic exchange, for which MFT and the exact method provide quite a different description. Finally, we discuss the relevance of our approach to other techniques proposed for calculations of the DM interactions. Particularly, we argue that the spin-current model for the DM interactions can be derived from the MFT-based expression and is the relativistic counterpart of the double exchange, occurring in metallic systems in the limit of infinite exchange splitting.