Charge-Orbital Ordering and Verwey Transition in Magnetite

$\mathrm{\text{Local density approximation}}+\mathrm{\text{Hubbard}}\mathrm{U}$ ($\mathrm{L}\mathrm{D}\mathrm{A}+U$) band structure calculations reveal that magnetite (${\mathrm{F}\mathrm{e}}_3{\mathrm{O}}_4$) forms an insulating charge-orbital-ordered state below the Verwey transition temperature. The calculated charge ordering is in good agreement with that inferred from recent experiments. We found an associated $t_{2g}$ orbital ordering on the octahedral ${\mathrm{F}\mathrm{e}}^{2+}$ sublattice. Such an orbital ordering results primarily from the on-site Coulomb interaction. This finding unravels such fundamental issues about the Verwey transition as the mechanism for the charge ordering and for the formation of the insulating gap, as well as the nonobedience of the Anderson’s criterion for the charge ordering.