Reversible electric field manipulation of the Dzyaloshinskii-Moriya interactions in transition metal dimers

The anisotropic antisymmetric Dzyaloshinskii-Moriya (DM) interactions between local magnetic moments ${\mathbit{\ensuremath{\mu}}}_{i}$ and ${\mathbit{\ensuremath{\mu}}}_{j}$, which can be induced by an external electric field (EF) are investigated in the framework of density functional theory by considering all $3d, 4d$, and $5d$ freestanding transition metal dimers. The possibilities of triggering and reversibly tuning chiral magnetic couplings by electric means are demonstrated. The dependence of the DM-coupling vector ${\mathbit{D}}_{ij}$ on the EF strength $E$ is shown to be approximately linear for $|E|\ensuremath{\le}0.6$ V/\AA{}, with only minor third-order corrections. The first- and third-order zero-field electric susceptibility of the DM couplings are determined and analyzed as a function of $d$-band filling. The correlations between them and the chirality of the spin-orbit energy are displayed. From a microscopic perspective, the EF-induced DM couplings are shown to stem from the permanent electric dipole moments ${\mathbit{p}}^{0}$ that are already present in the field-free dimers whenever their local magnetic moments are not collinear. The symmetry rules governing ${\mathbit{p}}^{0}$ and its chirality are discussed. Finally, the dependence of the EF-induced DM couplings on the degree of noncollinearity of the magnetic order is quantified by varying systematically the angle $\ensuremath{\theta}$ between the local moments. While the electronic calculations show that the changes in the effective ${\mathbit{D}}_{ij}$ can be quite important for arbitrary $\ensuremath{\theta}$, one also observes that ${\mathbit{D}}_{ij}$ depends weakly on $\ensuremath{\theta}$ and is thus transferable within a limited range of noncollinear magnetic arrangements, provided that they are not too far from the lowest-energy configuration.