Magnetic structure of bixbyite $\alpha$-Mn${}_2$O${}_3$: A combined DFT$+U$ and neutron diffraction study

First-principles density functional theory DFT$+U$ calculations and experimental neutron diffraction structure analyses were used to determine the low-temperature crystallographic and magnetic structure of bixbyite $\alpha$-Mn${}_2$O${}_3$. The energies of various magnetic arrangements, calculated from first principles, were fit to a cluster-expansion model using a Bayesian method that overcomes a problem of underfitting caused by the limited number of input magnetic configurations. The model was used to predict the lowest-energy magnetic states. Experimental determination of magnetic structure benefited from an optimized sample synthesis, which produced crystallite sizes large enough to yield a clear splitting of peaks in the neutron powder diffraction patterns, thereby enabling magnetic-structure refinements under the correct orthorhombic symmetry. The refinements employed group theory to constrain magnetic models. Computational and experimental analyses independently converged to similar ground states, with identical antiferromagnetic ordering along a principal magnetic axis and secondary ordering along a single orthogonal axis, differing only by a phase factor in the modulation patterns. The lowest-energy magnetic states are compromise solutions to frustrated antiferromagnetic interactions between certain corner-sharing [MnO${}_6$] octahedra.

Figure 1

Topology of bixbyite phase. Mn in dark blue; O in medium red; unfilled tetrahedral interstitials indicated in light yellow. The four nearest Mn to a given O are drawn in the upper left.

Figure 2

Cubic $\alpha$-Mn${}_2$O${}_3$ viewed along the $c$ axis; regular octahedra surrounding high-symmetry Mn sites are shown in columns with two octahedra per repeat unit along $c$. Blue balls are Mn; red balls O.

Figure 3

Orthorhombic $\alpha$-Mn${}_2$O${}_3$: octahedra shown in Fig. 2 have undergone Jahn-Teller distortion.

Figure 4

Lowest-energy collinear magnetic structure found computationally; also the collinear model that gives the best fit $({\chi }^2=2.54)$ to experimental results at 2 K. Only Mn atoms shown; dark spheres represent spin “up” and light spheres spin “down.”

Figure 5

Calculated density of states (DOS) for $\alpha$-Mn${}_2$O${}_3$ in ground-state collinear magnetic structure.

Figure 6

Integrated density of states (DOS) for $\alpha$-Mn${}_2$O${}_3$ in ground-state collinear magnetic structure, within spheres of radius 1.24 Å centered on each Mn site. Integrated DOS of local majority spins above the line and local minority spins below the line.

Figure 7

A trace of the 622 cubic reflection at three different temperatures. This reflection appears as a single peak at 300 K but exhibits pronounced splitting at subambient temperatures. Similar trends are observed for other reflections.

Figure 8

Temperature dependence of the (a) orthorhombic lattice parameters and (b) unit-cell volume and $b/c$ ratio which characterizes the magnitude of the orthorhombic distortion. The behavior of the orthorhombic distortion changes across the magnetic transition at approximately 80 K. The error bars are within the size of the symbols.

Figure 9

Experimental (red dots) and calculated (blue line) neutron diffraction profiles (BT-1) for $\alpha$-Mn${}_2$O${}_3$ at 100 K. The residual is indicated below (green line). The agreement factors are ${\chi }^2=1.16$ and $R_{\text{wp}}=4.79\%$.

Figure 10

A low-angle portion of the neutron diffraction pattern collected at 2 K showing experimental (red dots) and calculated (blue line) neutron diffraction profiles. Note that all the reflections with $2\theta \le {32}^{\circ }$ were absent at 100 K (Fig. 9); these reflections, all indexable according to the primitive nuclear-structure unit cell, originate from magnetic ordering. The calculated profiles correspond to (top) the ${\Gamma }_3$ model by Regulski et al. with magnetic moments aligned with the $c$ axis, (middle) the ${\Gamma }_1$ model with magnetic moments collinear with the $a$ axis, and (bottom) the ${\Gamma }_1$ model with noncollinear magnetic moments residing in the $ac$ plane. The ${\Gamma }_1$ models provide a superior fit relative to the ${\Gamma }_3$ model. A noncollinear alignment of magnetic moments significantly improves the fit for several magnetic reflections (outlines using a dashed line).

Figure 11

Experimental (red dots) and calculated (blue line) neutron diffraction profiles (BT-1) for $\alpha$-Mn${}_2$O${}_3$ at 40 K (top) and 2 K (bottom). The calculated profiles correspond to the ${\Gamma }_1$ model with the noncollinear array of magnetic moments in the $ac$ plane. The agreement factors are ${\chi }^2=1.22$ and $R_{\text{wp}}=5.21\%$ (40 K) and ${\chi }^2=2.53$ and $R_{\text{wp}}=6.89\%$ (2 K).

Figure 12

Calculated magnetic interaction parameters between Mn ions in $\alpha$-Mn${}_2$O${}_3$. Positive values for $J$ favor antiferromagnetic alignment. Mn subscripts refer to the different Mn sites in the orthorhombic phase. Error bars indicate plus and minus one standard deviation of the parameter, based on cross-validation calculations.

Figure 13

Linkages between Mn in $\alpha$-Mn${}_2$O${}_3$ limited to a subset of those with strong antiferromagnetic interactions as determined in this work give the lowest-energy collinear magnetic structure on the Mn(3), Mn(4), and Mn(5) sites.

Figure 14

Similar to Fig. 13, with the Mn-Mn pairs with the five strongest AFM interactions shown. One additional interaction, not shown in Fig. 13, is indicated by dotted lines here, and leads to frustrated triangles, which presumably is the origin of the noncollinear ground-state magnetism.