Models of polaron fluctuations in $\mathrm{LuF}{\mathrm{e}}_2{\mathrm{O}}_4$

The structure of $\mathrm{LuF}{\mathrm{e}}_2{\mathrm{O}}_4$ below the charge ordering temperature of $T_{\mathrm{CO}}=320$ K has been studied immensely due to the question of whether charge ordering on the mixed valence Fe sublattice leads to a macroscopic electric polarization. In contrast, the local structure associated with polaron fluctuations above $T_{\mathrm{CO}}$, from which the charge ordered structure emerges, is largely unknown. In this work, we characterize this local structure using the x-ray three-dimensional difference pair distribution function ($3\mathrm{D}\text{−}\mathrm{\Delta }\mathrm{PDF}$). Atomic correlations extracted directly from the $3\mathrm{D}\text{−}\mathrm{\Delta }\mathrm{PDF}$ are used to propose models for the local structure, which are simulated using Monte Carlo methods. It is found that the primary Fourier components of the diffuse scattering can be described by the large, correlated Lu displacements. The local structure of Lu displacements draws conceptual parallels to the classical models of triangular Ising antiferromagnets with the added complexity of interplane interactions. The weaker features of the $3\mathrm{D}\text{−}\mathrm{\Delta }\mathrm{PDF}$ are modeled by displacement correlations involving Fe and O, the latter resulting in an increased coordination number of Lu compared with the average structure. Analysis of the Fe-coordination environments in the atomic configurations obtained from Monte Carlo simulations reveals a strong correlation between distortion of the coordination environment and the +II oxidation state of Fe, which suggests a Jahn-Teller polaronic nature of the charge carriers.

Figure 1

(a) Average structure of $\mathrm{LuF}{\mathrm{e}}_2{\mathrm{O}}_4$ at 350 K as refined against single crystal Bragg diffraction data. The inset shows a close-up of the $\mathrm{Lu}{\mathrm{O}}_6$ coordination environment to highlight the split-site Lu position. The figure was drawn using vesta [20]. (b) Single crystal x-ray scattering in the (hhl) plane of reciprocal space. The scattering data is displayed on an arbitrary scale. The negative part of the color bar is included to show that the background subtraction does not result in significant unphysical negative intensities.

Figure 2

In (a) the 350 K $3\mathrm{D}\text{−}\mathrm{\Delta }\mathrm{PDF}$ in the $xy0$ and $x0z$ planes of Patterson space is shown. Red is positive while blue is negative; the data are shown on an arbitrary scale. The Cartesian $x$ axis is parallel to the $\left[\overline{2}\overline{1}0\right]$ direction and the $z$ axis is parallel to the $\left[001\right]$ direction. (b) An illustration of the correlation between the displacements of Lu (green) and ${\mathrm{O}}_{\mathrm{A}}$ (red). The solid-black/dashed-gray vectors are vectors that are more/less frequent in the real structure compared to the average structure. (c) A sketch of the $3\mathrm{D}\text{−}\mathrm{\Delta }\mathrm{PDF}$ that would result from the correlated displacements illustrated in panel (b). (d)–(f) show the details of the $3\mathrm{D}\text{−}\mathrm{\Delta }\mathrm{PDF}$ features corresponding to (d) the Lu-${\mathrm{O}}_{\mathrm{A}}$ correlation, (e) the Lu-Fe correlation, and (f) the Lu-${\mathrm{O}}_{\mathrm{B}}$ correlation discussed in the text.

Figure 3

(a) Illustration of the effective interactions in the model Hamiltonian. $J_{xy}$ is a positive interaction within the individual Lu layers leading to negative correlations for nearest neighbors. $J_z$ is a weak negative interaction between nearest neighbors along the $z$ direction leading to positive correlations between these Lu-Lu pairs. (b) Diffuse scattering resulting from two different values of $J_{\mathrm{L}}$ and three different Monte Carlo temperatures. For all the simulations in this figure $J_z=-1$ and $J_{xy}=1000$. Scattering simulated with other parameter values is shown in the Supplemental Material [35]. The scattering data is shown on an arbitrary scale. In (c) and (d) two examples cut from representative Lu layers for the simulations with $T=5$ are shown for $J_{\mathrm{L}}=0$ and $J_{\mathrm{L}}=10$, respectively. Dark green and light blue distinguish different Lu-displacement states, either up or down. Regions of partial honeycomb ordering are clearly visible in the $J_{\mathrm{L}}=10$ layer in (d). In (e) the experimental scattering in the plane spanned by ${\mathbf{a}}^{*}$ and ${\mathbf{b}}^{*}$ with $l=17$ is compared to the scattering simulated based on the $J_{\mathrm{L}}=0$ and $J_{\mathrm{L}}=10$ models (both with $J_z=-1, J_{xy}=1000$, and $T=5$). The diffuse maxima observed in the experimental data indicated by arrows are only reproduced by the $J_{\mathrm{L}}=10$ model.

Figure 4

(a) A sketch of the effective spring interactions between Fe-Lu, Fe-Fe, and ${\mathrm{O}}_{\mathrm{A}}$-Lu pairs. Red lines indicate classical Hooke's law springs. (b) A sketch of the effective spring interactions between ${\mathrm{O}}_{\mathrm{B}}-\mathrm{L}{\mathrm{u}}_{\mathrm{NN}z}, {\mathrm{O}}_{\mathrm{B}}-\mathrm{L}{\mathrm{u}}_{\mathrm{NNN}z}$, and ${\mathrm{O}}_{\mathrm{B}}-{\mathrm{O}}_{\mathrm{B}}$ pairs. The ${\mathrm{O}}_{\mathrm{B}}-\mathrm{L}{\mathrm{u}}_{\mathrm{NN}z}$ and ${\mathrm{O}}_{\mathrm{B}}-\mathrm{L}{\mathrm{u}}_{\mathrm{NNN}z}$ interactions (indicated by blue lines) are modified by an effective size-effect parameter that favors negative displacement correlations for these pairs. (c) $x0z$ planes of the simulated $3\mathrm{D}\text{−}\mathrm{\Delta }\mathrm{PDFs}$ based on the $J_{\mathrm{L}}=0$ and $J_{\mathrm{L}}=10$ Lu-displacement models and the experimental $3\mathrm{D}\text{−}\mathrm{\Delta }\mathrm{PDF}$.

Figure 5

(a) Illustration of a perfect $\mathrm{Fe}{\mathrm{O}}_5$ trigonal bipyramid and an orbital diagram illustrating the splitting of the $d$-orbital energies in such a crystal field. The black arrows indicate the expected electron configuration for a $d^5$ ion such as $\mathrm{F}{\mathrm{e}}^{3+}$. The combination of the red and black arrows shows the expected electron configuration for a $d^6$ ion such as $\mathrm{F}{\mathrm{e}}^{2+}$. The figure was drawn using vesta [20]. (b) Histograms of calculated CShM and BVS for the Fe-coordination environments of the big boxes. The partitioning into $\mathrm{F}{\mathrm{e}}^{3+}$ and $\mathrm{F}{\mathrm{e}}^{2+}$ ions is based on the median of the CShM or BVS of all coordination environments. (c) Comparison of representative charge distributions in an Fe layer (orange is $\mathrm{F}{\mathrm{e}}^{3+}$ while blue is $\mathrm{F}{\mathrm{e}}^{2+}$) estimated based on the CShM (left) and BVS (right) for both the $J_{\mathrm{L}}=0$ and the $J_{\mathrm{L}}=10$ models. The number above the arrow is the Pearson correlation calculated for the full big box models.