Electric polarization driven by non-collinear spin alignment investigated by first principles calculations

We present an approach for first principles investigations on the spin driven electric polarization in type II multiferroics. We propose a parametrization of the polarization with the parameters calculated using the Korringa-Kohn-Rostoker Green function (KKR-GF) formalism. Within this approach the induced electric polarization of a unit cell is represented in terms of three-site parameters. Those antisymmetric with respect to spin permutation are seen as an ab-initio based counter-part to the phenomenological parameters used within the inverse-Dzyaloshinskii-Moriya-interaction (DMI) model. Due to their relativistic origin, these parameters are responsible for the electric polarization induced in the presence of a non-collinear spin alignment in materials with a centrosymmetric crystal structure. Beyond to this, our approach gives direct access to the element- or site-resolved electric polarization. To demonstrate the capability of the approach, we consider several examples of the so-called type II multiferroics, for which the magneto-electric effect is observed either as a consequence of an applied magnetic field (we use Cr$_2$O$_3$ as a prototype), or as a result of a phase transition to a spin-spiral magnetic state, as for instance in MnI$_2$, CuCrO$_2$ and AgCrO$_2$.