Magnetic structure of paramagnetic MnO

Using a combination of single-crystal neutron scattering and reverse Monte Carlo refinements, we study the magnetic structure of paramagnetic MnO at a temperature (160 K) substantially below the Curie-Weiss temperature $\left|\theta \right|\sim 550$ K. The microscopic picture we develop reveals a locally ordered domain structure that persists over distances many times larger than the correlation length implied by direct analysis of the spin-correlation function. Moreover, the directional dependence of paramagnetic spin correlations in paramagnetic MnO differs in some important respects from that of its incipient ordered antiferromagnetic state. Our results demonstrate that atomistic refinement to large three-dimensional neutron-scattering datasets is a practical approach, and have implications for the understanding of paramagnetic states in weakly frustrated systems, including high-temperature superconductors.

Figure 1

(a) Nearest-neighbor AFM interactions ($J_1$) are frustrated on the face-centered-cubic lattice. (b) Next-nearest-neighbor AFM interactions ($J_2$) are not frustrated and drive checkerboard ordering of the simple cubic sublattices.

Figure 2

(a) Experimental magnetic diffuse-scattering data for paramagnetic MnO at $T=160$ K. Nuclear Bragg peaks have been removed from the data. (b) RMC fit to the experimental data shown in (a). In (a) and (b), sections of the ${\left(101\right)}^{*},{\left(1\overline{1}1\right)}^{*}$, and ${\left(001\right)}^{*}$ reciprocal-space planes are shown. The ${\left(001\right)}^{*}$ plane is shifted by $-0.5$ reciprocal-lattice units along the ${\left[001\right]}^{*}$ direction in order to highlight the strongest diffuse scattering features; i.e., it is the ${(h,k,-\frac{1}{2})}^{*}$ plane.

Figure 3

(a) Radial spin-correlation function $\langle \mathbf{S}\left(0\right)·\mathbf{S}\left(r\right)\rangle$ for MnO at $T=160$ K. Black bars show results for the $J_1\text{−}{J}_2$ model described in the text and red diamonds show results from RMC refinement to single-crystal magnetic diffuse-scattering data. The dashed grey line shows the fit of an exponential envelope to the RMC $|\langle \mathbf{S}(0)·\mathbf{S}(r)\rangle |$, which yields spin-correlation length $\xi =2.258\left(1\right)$ Å. Grey squares show the $|\langle \mathbf{S}(0)·\mathbf{S}(r)\rangle |$ values included in the fit. Error bars are smaller than the symbol size in the plot. (b) 3D spin-correlation function $g\left(\mathbf{r}\right)$ (defined in the text) obtained from RMC refinement. The figure shows the $\left(xy0\right)$ plane (i.e., a cubic face). The crystallographic unit cell is shown as a black box. (c) Schematic representation of $g\left(\mathbf{r}\right)$ for MnO in the PM phase. Red areas indicate FM correlations and grey areas AFM correlations. (d) Schematic representation of $g\left(\mathbf{r}\right)$ for a single domain of the ordered low-temperature AFM structure of MnO. (e) Schematic representation of $g\left(\mathbf{r}\right)$ obtained for the AFM structure with the point symmetry of the Mn site in the PM state applied.

Figure 4

Magnetic domain structure of MnO at $T=160$ K obtained as described in the text. In (a), different domains with local periodicity $\mathbf{k}\in {\langle \frac{1}{2}\frac{1}{2}\frac{1}{2}\rangle }^{*}$ are shown in different colors. The colored regions show the threshold where $S_{\mathbf{k}}\left(\mathbf{r}\right)=80.6$ [the range of $S_{\mathbf{k}}\left(\mathbf{r}\right)$ is between approximately 0 and 200]. In (b), the four images show different $\mathbf{k}\in {\langle \frac{1}{2}\frac{1}{2}\frac{1}{2}\rangle }^{*}$. The value of $S_{\mathbf{k}}\left(\mathbf{r}\right)$ in each image is shown using a blue-to-red color map. Colored points indicate the range where $S_{\mathbf{k}}\left(\mathbf{r}\right)\ge 80.6$, using the same colors as (a). The images in (a) and (b) are shown in the same orientation. (c) Domain-occupancy correlation function described in the text (black circles) and fit to the pair-correlation function for a sphere to extract the average domain radius (red line). Error bars indicate 1 standard deviation. (d) Spin-correlation function for spin pairs within domains (blue circles) and for “matrix” spin pairs that do not belong to any domain (red diamonds). Error bars are smaller than the symbol size in the plot.