Symmetry of Orbital Order in ${\mathrm{Fe}}_3{\mathbf{O}}_4$ Studied by Fe $L_{2,3}$ Resonant X-Ray Diffraction

We studied the symmetry of the Fe $3d$ wave function in magnetite below the Verwey temperature $T_V$ with resonant soft-x-ray diffraction. Although the lattice structure of the low-temperature phase of ${\mathrm{Fe}}_3{\mathrm{O}}_4$ is well described by the pseudo-orthorhombic $Pmca$ with a slight monoclinic $P2/c$ distortion, we find that the $3d$ wave function does not reflect the $Pmca$ symmetry, and its distortion toward monoclinic symmetry is by far larger than that of the lattice. The result supports a scenario in which the Verwey transition involves the ordering of $t_{2g}$ orbitals with complex-number coefficients.

Figure 1

(a) RSXD spectra of the $(00\frac{1}{2}{)}_c$ peak at Fe $L_{2,3}$ resonances for $\sigma$ (solid line) and $\pi$ (dotted line) polarized light taken at 83 K from a ${\mathrm{Fe}}_3{\mathrm{O}}_4$ film grown on a stepped MgO substrate. (b) The azimuthal-dependent polar plot of the integrated $(00\frac{1}{2}{)}_c$ peak intensity of $L$ scan at Fe $L_3$ maximum taken from (a) (circles) compared with the single-level model calculations (lines); the closed (open) circles and the solid (dotted) lines correspond to those for $\sigma$ ($\pi$) polarized light. (c) The same as (b) but for bulk sample. (d) The same as (b) but for symmetrized film data with respect to 90°–270° mirror plane to emulate fully twinned sample. For the calculations in (b), (c), and (d), the parameter set $\mathrm{Re}[F_{yz}]:\mathrm{Im}[F_{zx}]:\mathrm{Re}[F_{xy}]=1:0.63:0.63$ is used. While two monoclinic domains with equal volumes restricted by epitaxial strain imposed by the stepped MgO substrate are assumed in (b), all of four possible monoclinic domains with equal volumes are considered in (c) and (d).

Figure 2

Azimuth and polarization dependence of the $(00\frac{1}{2}{)}_c$ reflection intensities. The solid and dashed lines denote the intensities with the $\sigma$- and $\pi$-polarization light, respectively. Those for the single-domain crystal (top row) and for the ${\mathrm{Fe}}_3{\mathrm{O}}_4$ layer on the stepped MgO substrate (bottom row) are depicted. The choices of the three components of the scattering matrices in (a) and (b) are for $P\underline{mc}a$, and those in (c) are for $P\underline{2}/\underline{c}$; their ratios for $\mathrm{Re}[F_{yz}]:\mathrm{Im}[F_{zx}]:\mathrm{Re}[F_{xy}]$ are (a) $1:0:0$; (b) $1:1:0$; and (c) $1:1:1$.