Complex-Orbital Ordering and Verwey Transition in Fe₃O₄

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The charge and orbital orderings above and below the Verwey transition in magnetite (Fe₃O₄) have been investigated using a three-band model for electrons with minority spins in the t_2g orbitals on the B sites with a mean-field approximation. For the high-temperature phase above the Verwey transition, an orbital ordered state with spontaneous breaking of the time reversal symmetry having noncollinear orbital moments is obtained as the lowest energy solution. The result is in contrast to usual magnetic materials, in which the orbital moments are induced by ordered spins via the spin–orbit interaction. For the low-temperature phase, a similar orbital ordering with longer periodicity accompanied by charge disproportionation caused by a monoclinic lattice distortion is proposed.