Local magnetic cluster size identified by neutron total scattering in the site-diluted spin glass $\mathrm{S}{\mathrm{n}}_x\mathrm{F}{\mathrm{e}}_{4-x}\mathrm{N}$ $(x=0.88)$

A detailed structure analysis for the site-diluted $\mathrm{S}{\mathrm{n}}_x\mathrm{F}{\mathrm{e}}_{4-x}\mathrm{N}$ ($x=0.25$, 0.41, and 0.88) has been carried out through complex modeling of the neutron total scattering data. We present quantitative evidence showing the local ferromagnetic cluster size extending to ∼8 Å on average when $\mathrm{S}{\mathrm{n}}_{0.88}\mathrm{F}{\mathrm{e}}_{3.12}\mathrm{N}$ undergoes the spin-glass transition (the other two not showing such a transition). The modeling methodology used in this work involving the corefinement of the nuclear and magnetic structure in both real and reciprocal space can potentially be applied generally to explore a variety of spin-glass material problems.

Figure 1

The conceptual diagram showing the two different approaches for modeling the total scattering data. Apart from the pure-nuclear Rietveld refinement, all the other three approaches are used in the paper. Here short names are given for the four approaches, UN, UNM, SN, and SNM, where U represents unit-cell, S is for supercell, N is for nuclear, and M is for magnetic.

Figure 2

The PDF fitting with SN approach using RMCProfile for (a) $\mathrm{S}{\mathrm{n}}_{0.25}\mathrm{F}{\mathrm{e}}_{3.75}\mathrm{N}$, (b) $\mathrm{S}{\mathrm{n}}_{0.41}\mathrm{F}{\mathrm{e}}_{3.59}\mathrm{N}$, and (c) $\mathrm{S}{\mathrm{n}}_{0.88}\mathrm{F}{\mathrm{e}}_{3.12}\mathrm{N}$, respectively, at 10 K. The R factor given in the figure is defined as follows:$R={\sum }_i{(y_{C,i}-y_{O,i})}^2/{\sum }_i{y}_{O,i}^2$, where $y_C$ and $y_O$ represents the calculated and observed value, respectively. The same definition applies across the whole article.

Figure 3

The local correlation coefficients for (a) Fe1–Fe1 and (b) Fe1–Fe2 pairs, with respect to various distance windows. Here $R$ is defined as the center of the distance window: $R=\left(D_{min+}{D}_{max}\right)/2$. The inset of (a) shows the unit cell of $\mathrm{S}{\mathrm{n}}_x\mathrm{F}{\mathrm{e}}_{4\text{–}x}\mathrm{N}$ with the antiperovskite crystal structure in space group $pm\overline{3}m$. The body center ($1b$) is occupied by N. The face center ($3c$) is fully occupied by Fe, and the substitution occurs only on the $1a$ position according to Refs. [39, 40].

Figure 4

The PDF refinement (following the UNM approach) result for (a) $\mathrm{S}{\mathrm{n}}_{0.25}\mathrm{F}{\mathrm{e}}_{3.75}\mathrm{N}$, (b) $\mathrm{S}{\mathrm{n}}_{0.41}\mathrm{F}{\mathrm{e}}_{3.59}\mathrm{N}$, and (c) $\mathrm{S}{\mathrm{n}}_{0.88}\mathrm{F}{\mathrm{e}}_{3.12}\mathrm{N}$ data (at 10 K), respectively. In (d), the magnetic PDF is presented for all samples with the extracted magnetic moments labeled out for each. The green-dashed line and arrow indicate the region of interest for the magnetic PDF refinement.

Figure 5

The result of the RMCProfile modeling for the S(Q) data. (a), (c), and (e) are the pure nuclear modeling without considering the magnetic contribution, for $\mathrm{S}{\mathrm{n}}_{0.25}\mathrm{F}{\mathrm{e}}_{3.75}\mathrm{N}$, $\mathrm{S}{\mathrm{n}}_{0.41}\mathrm{F}{\mathrm{e}}_{3.59}\mathrm{N}$ and $\mathrm{S}{\mathrm{n}}_{0.88}\mathrm{F}{\mathrm{e}}_{3.12}\mathrm{N}$, respectively. (b), (d), and (f) are the result when magnetic scattering is taken into account, corresponding to (a), (c), and (e), respectively.

Figure 6

The average LMOP across all local clusters with various cluster radius for all three samples.