Electronic structure and polaronic charge distributions of Fe vacancy clusters in Fe$_{1-x}$O

We perform a detailed study of the electronic structure of Fe$_{1-x}$O at moderate values of $x$. Our results evidence that the Fe vacancies introduce significant local modifications of the structural, electronic and magnetic features, that serve to explain the origin of the measured dependencies of the physical properties on $x$. The final properties are determined by a complex interplay of the charge demand from O, the magnetic interactions, and the charge order at the Fe sublattice. Furthermore, polaronic distributions of charge resembling those at magnetite, Fe$_3$O$_4$, emerge for the most stable defect structures. This defines a unique scenario to understand the nature of the short-range correlations in Fe$_3$O$_4$, and unveils their intimate connection to the long-range charge order developed below the Verwey transition temperature.