Analysis of charge and orbital order in Fe₃O₄ by Fe L_2,3 resonant x-ray diffraction

To elucidate charge and orbital order below the Verwey transition temperature TV∼125 K, a thin layer of magnetite partially detwined by growth on the stepped MgO(001) substrate has been studied by means of soft x-ray diffraction at the Fe L2,3 resonance. The azimuth angle, incident photon polarization, and energy dependence of the ( 0012)c and (001)c reflection intensities have been measured, and analyzed using a configuration-interaction FeO6 cluster model. The azimuth dependence of the (0012)c reflection intensities directly represents the space-group symmetry of the orbital order in the initial state rather than indirectly through the intermediate-state level shifts caused by the order-induced lattice distortions. From the analysis of the ( 0012)c reflection intensities, the orbital order in the t 2g orbitals of B sites below TV is proved to have a large monoclinic deformation with the value of Re[Fxy]/Re[F yz]∼2. This finding contradicts the majority of theories on the Verwey transition so far proposed. We show that the experimentally observed resonance spectra cannot be explained by orbital and charge orders obtained with recent LDA+U and GGA+U band structure calculations but by a complex-number orbital order with excellent agreement.