Direct Observation of Charge Order and an Orbital Glass State in Multiferroic ${\mathrm{LuFe}}_2{\mathbf{O}}_4$

Geometrical frustration of the Fe ions in ${\mathrm{LuFe}}_2{\mathrm{O}}_4$ leads to intricate charge and magnetic order and a strong magnetoelectric coupling. Using resonant x-ray diffraction at the Fe $K$ edge, the anomalous scattering factors of both Fe sites are deduced from the $(h/3k/3l/2)$ reflections. The chemical shift between the two types of Fe ions equals 4.0(1) eV corresponding to full charge separation into ${\mathrm{Fe}}^{2+}$ and ${\mathrm{Fe}}^{3+}$. The polarization and azimuthal angle dependence of the superlattice reflections demonstrate the absence of differences in anisotropic scattering revealing random orientations of the ${\mathrm{Fe}}^{2+}$ orbitals characteristic of an orbital glass state.

Figure 1

Integrated resonant diffraction intensity of the charge order $(2/3-1/3l/2)$ reflections recorded with 2D detector at $T=15\mathrm{K}$ (see inset top left for $l=11$), corrected for background, polarization, absorption and sample geometry, normalized (shifted for clarity) and compared to charge order model described in the text. The inset (top right) shows that the superlattice intensity gradually disappears at the ferroelectric ordering temperature. For $A$, $B$, $C$, and $D$, see text.

Figure 2

(a) Intensity of the $(1/31/331/2)$ reflection as function of azimuthal angle $\Psi$. (b) Intensity of the $(1/31/33/2)$ reflection recorded with $\sigma {\sigma }^{\prime }$ and $\sigma {\pi }^{\prime }$ radiation. The latter intensity is multiplied by 10 for clarity.

Figure 3

(a) $f_1^{\prime \prime }(E)$ and $f_2^{\prime \prime }(E)$ deduced from the refinement of RXD (Fig. 1) via the method described in the text and (b) its average compared with FDMNES calculation in $R\overline{3}m$ (dashed curve) and $[f_1^{\prime \prime }(E)+f_2^{\prime \prime }(E)]/2$ from $\mu (E)$ (open circles).

Figure 4

RXD compared to three models of charge order as described in the text. The solid curve corresponds to $f_{1,2}^{\prime \prime }(E)$ as presented in Fig. 3. The dashed curve and the dotted curve correspond to identical $f_1(E)$ and $f_2(E)$ except for a chemical shift. The dashed curve is fitted to the RXD data whereas the dotted curve corresponds to $f_{1,2}(E)$ calculated with FDMNES.