Influence of interstitial Mn on local structure and magnetism in ${\text{Mn}}_{1+\delta }\text{Sb}$

We report x-ray total scattering and pair distribution function (PDF) studies of the structural relaxation around interstitial manganese (${\mathrm{Mn}}_i$) in ferromagnetic ${\mathrm{Mn}}_{1+\delta }\mathrm{Sb}$ ($0.03\le \delta \le 0.23$) alloys, guided by density functional theory (DFT). Refinements to the experimental PDF using a crystallographically constrained structural model indicate an expansion in the equatorial plane of the ${\mathrm{Mn}}_i{\mathrm{Sb}}_5$ trigonal bipyramidal site, which introduces significant positional disorder in addition to the nominally random occupation of interstitial voids. Observation of a weak diffuse signal near the symmetry-forbidden (001) reflection position is indicative of correlated disorder from the clustering of ${\mathrm{Mn}}_i$. Density functional relaxation of supercells approximating the $\delta =0.08$, 0.15, and 0.23 compositions provides improved models that accurately describe the short-range structural distortions captured in the PDFs. Such structural relaxation increases the DFT calculated moment on ${\mathrm{Mn}}_i$, which aligns antiparallel to the primary Mn moments, but leads to insubstantial changes in the average Mn and Sb moments and moments of Mn and Sb proximal to interstitials, thus providing a more accurate description of the observed bulk magnetic properties.

Figure 1

(a) NiAs structure exemplified by the hard ferromagnetic phase MnBi [space group $P{6}_3/mmc$, with Mn on $2a$ at (0,0,0) and Bi on $2c$ at ($\frac{1}{3},\frac{2}{3},\frac{1}{4}$)], and inelastic neutron scattering data collected at 4 K from a single crystal of MnBi. (b) Structure of ${\mathrm{Mn}}_{1+\delta }\mathrm{Sb}$ highlighting the fractionally occupied, trigonal bipyramidal coordination environment of interstitial Mn $[{\mathrm{Mn}}_i$, on $2d$ site at ($\frac{1}{3},\frac{2}{3},\frac{3}{4}$)]. Note that ${\mathrm{Mn}}_i$ is also in trigonal prismatic coordination with respect to the primary (fully occupied) ${\mathrm{Mn}}_{\mathrm{Mn}}$ site. Inelastic neutron scattering data collected at 10 K on a single crystal of ${\mathrm{Mn}}_{1.13}\mathrm{Sb}$ show a pronounced diffuse component that is not observed for MnBi. Inelastic scattering figures adapted from Taylor et al., Ref. [14].

Figure 2

(a) Axial $(c/a)$ ratio of the ${\mathrm{Mn}}_{1+\delta }\mathrm{Sb}$ lattice constants as a function of the nominal interstitial content, $\delta$, showing Végard law behavior over most of the compositional range and excellent agreement between Rietveld and PDF refinements and literature values [9]. Error bars are smaller than the symbols. (b) Room-temperature saturation magnetization as a function of $\delta$, and bulk magnetization at 0 K obtained from DFT calculations of unrelaxed and relaxed supercells of ${\mathrm{Mn}}_{48}{\left({\mathrm{Mn}}_i\right)}_x{\mathrm{Sb}}_{48}$ ($x=4,7,11$); DFT moments represent the average of 10 distinct configurations at each composition, with error bars representing the standard deviation of the average. (c) Room-temperature magnetization hysteresis curves of the ${\mathrm{Mn}}_{1+\delta }\mathrm{Sb}$ series of samples. Note that the magnetization curves for $\delta =0.01$ and $0.03$ nearly overlay, which is consistent with the samples' $c/a$ ratios being nearly the same.

Figure 3

Representative Rietveld refinement of ${\mathrm{Mn}}_{1.05}\mathrm{Sb}$ against high-resolution synchrotron x-ray diffraction data (11-BM, Advanced Photon Source). To aid in visual inspection of the fit, the low- and high-$Q$ regions of the diffraction data are omitted. Reflection positions for ${\mathrm{Mn}}_{1.05}\mathrm{Sb}$ and the MnO impurity are shown as solid vertical lines in the two middle panels.

Figure 4

PDF refinements (solid lines) against data (open circles) for samples of nominal compositions ${\mathrm{Mn}}_{1.03}\mathrm{Sb}, {\mathrm{Mn}}_{1.12}\mathrm{Sb}$, and ${\mathrm{Mn}}_{1.23}\mathrm{Sb}$, employing the crystallographic structural model in which interstitial Mn (${\mathrm{Mn}}_i$) is located on the $2d$ site and partially occupied. (a) Full-range fits, from 2 to 30 Å. The vertical line at $r=6\text{Å}$ demarcates different $x$-axis scales at shorter and longer $r$. (b) Panels showing the first-nearest-neighbor region of the three full-range fits; select PDF partial contributions for ${\mathrm{Mn}}_{\mathrm{Mn}}$-Sb (dotted line) and ${\mathrm{Mn}}_i$-Sb (solid line) correlations are vertically offset for clarity. As the amount of ${\mathrm{Mn}}_i$ increases, the crystallographic structural model's deficiency becomes more apparent: there are shorter pair correlations present in the model than observed in the data, and these arise from equatorial ${\mathrm{Mn}}_i$-Sb distances.

Figure 5

Structural depictions of representative relaxed supercell configurations for ${\mathrm{Mn}}_{48}{\left({\mathrm{Mn}}_i\right)}_4{\mathrm{Sb}}_{48}$ (top), ${\mathrm{Mn}}_{48}{\left({\mathrm{Mn}}_i\right)}_7{\mathrm{Sb}}_{48}$ (middle), and ${\mathrm{Mn}}_{48}{\left({\mathrm{Mn}}_i\right)}_{11}{\mathrm{Sb}}_{48}$ (bottom). ${\mathrm{Mn}}_{\mathrm{Mn}}$ is shown as light (pink) spheres, Sb as dark (gray) spheres, and ${\mathrm{Mn}}_i$ (blue) in polyhedral rendering. An outline of the conventional unit cell is shown for reference.

Figure 6

(a) Comparison of PDF refinements for a crystallographically constrained (unit-cell) structural model and representative DFT-relaxed configuration against data collected for the sample of nominal composition ${\mathrm{Mn}}_{1.23}\mathrm{Sb}$. Refinement $R$ factors shown from fits in the range of 2–6 Å; $R_w$ for the DFT relaxed configuration represents the average for all 10 configurations, given with the standard deviation of the average. (b) Select atom-pair partials extracted from the DFT-relaxed and unit-cell models, along with PDF partials generated from a reverse Monte Carlo simulation using a 9990-atom supercell ($R_w=8.6\%$).

Figure 7

Histograms showing the amplitude of atom displacements away from the crystallographic structural positions upon DFT relaxation, compiled for all 10 configurations of each DFT supercell composition. Top row: ${\mathrm{Mn}}_{48}{\left({\mathrm{Mn}}_i\right)}_4{\mathrm{Sb}}_{48}$; middle row: ${\mathrm{Mn}}_{48}{\left({\mathrm{Mn}}_i\right)}_7{\mathrm{Sb}}_{48}$; bottom row: ${\mathrm{Mn}}_{48}{\left({\mathrm{Mn}}_i\right)}_{11}{\mathrm{Sb}}_{48}$.

Figure 8

(a) Average moment, by site, for each of the relaxed (filled symbols) and unrelaxed (open symbols) configurations among the three compositions examined by DFT. Error bars, representing one standard deviation of the average moments, are much smaller than the symbols for ${\mathrm{Mn}}_{\mathrm{Mn}}$ and Sb. Note that the average moments on ${\mathrm{Mn}}_{\mathrm{Mn}}$ and Sb are quite insensitive to structural relaxation, whereas the magnitude of the moment on ${\mathrm{Mn}}_i$ becomes significantly larger when accounting for relaxation of and around the interstitials. (b) Energy of the relaxed (unrelaxed) configurations relative to the most stable relaxed (unrelaxed) configuration for a given composition. The energy stabilization of structural relaxation is about 20 meV ${\mathrm{at}}^{-1}$ for ${\mathrm{Mn}}_{48}{\left({\mathrm{Mn}}_i\right)}_4{\mathrm{Sb}}_{48}$, 30 meV ${\mathrm{at}}^{-1}$ for ${\mathrm{Mn}}_{48}{\left({\mathrm{Mn}}_i\right)}_7{\mathrm{Sb}}_{48}$, and 40 meV ${\mathrm{at}}^{-1}$ for ${\mathrm{Mn}}_{48}{\left({\mathrm{Mn}}_i\right)}_{11}{\mathrm{Sb}}_{48}$.

Figure 9

(a) Diffuse x-ray-scattering features observed in the vicinity of the (001) reflection in ${\mathrm{Mn}}_{1+\delta }\mathrm{Sb}$ samples ($\delta =0.12$, 0.18, and 0.23); container scattering has been subtracted from the data, which were collected at $\sim 86\text{keV}$ at a sample-to-detector distance of $\sim 19\text{cm}$. (b) Simulated diffraction patterns of ${\mathrm{Mn}}_{1.23}\mathrm{Sb}$ structures built from an infinite number of infinitely wide discrete layers. When layers with a singly occupied ${\mathrm{Mn}}_i$ site per unit cell are preferentially stacked adjacent to layers with ${\mathrm{Mn}}_i$ in the same site, the correlated disorder gives rise to a diffuse feature that is consistent with the experimentally observed scattering with regard to both position (in $Q$) and relative intensity. Illustrated models of the different structures giving rise to the simulated scattering are shown as four-layer fragments: ${\mathrm{Mn}}_{\mathrm{Mn}}$ is medium gray (pink), Sb is dark gray (gray), and ${\mathrm{Mn}}_i$ is light gray (blue).