Orbital degrees of freedom as origin of magnetoelectric coupling in magnetite

A microscopic understanding of magnetoelectricity, i.e., the coupling between magnetic (electric) properties and external electric (magnetic) fields, is a crucial milestone for future generations of electrically controlled spintronic devices. Here, we focus on the first magnetoelectric known to mankind: magnetite. By means of density-functional simulations, we show that magnetoelectricity in charge-/orbital-ordered Fe₃O₄ in the noncentrosymmetric $Cc$ structure is driven by the interplay between a peculiar orbital-order and on-site spin-orbit coupling. The excellent agreement with available experiments confirms our theoretical picture, pointing to magnetite as a peculiar magnetoelectric system where the electric polarization is induced by two origins, noncentrosymmetric-charge-ordering- and spin-orbit-coupling-induced anisotropic p-d hybridization.