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

The anisotropic antisymmetric Dzyaloshinskii-Moriya (DM) interactions between local magnetic moments ${\mu }_i$ and ${\mu }_j$, 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 $D_{ij}$ on the EF strength $E$ is shown to be approximately linear for $\left|E\right|\le 0.6$ V/Å, 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 $p^0$ that are already present in the field-free dimers whenever their local magnetic moments are not collinear. The symmetry rules governing $p^0$ 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 $\theta$ between the local moments. While the electronic calculations show that the changes in the effective $D_{ij}$ can be quite important for arbitrary $\theta$, one also observes that $D_{ij}$ depends weakly on $\theta$ and is thus transferable within a limited range of noncollinear magnetic arrangements, provided that they are not too far from the lowest-energy configuration.