Vibrational and structural properties of the $R{\mathrm{Fe}}_4{\mathrm{Sb}}_{12}$ $(R=\mathrm{Na}, \mathrm{K}, \mathrm{Ca}, \mathrm{Sr}, \mathrm{Ba})$ filled skutterudites

Vibrational and elastic properties of the $R{\mathrm{Fe}}_4{\mathrm{Sb}}_{12}$ skutterudites are investigated by, respectively, temperature $\left(T\right)$ dependent extended x-ray absorption fine structure (EXAFS) and pressure $\left(P\right)$ dependent x-ray diffraction (XRD) experiments. The Fe $K$-edge EXAFS experiments of the $R=\mathrm{K}$, Ca, and Ba materials were performed in the $T$ interval $6<T<300\mathrm{K}$ and XRD experiments of the $R=$ Na, K, Ca, Sr, and Ba materials were performed in the $P$ interval $1\text{atm}<P<16\mathrm{GPa}$. From EXAFS, we obtained the correlated Debye-Waller parameters that were thus analyzed to extract effective spring constants connected with the Fe-$Y$ (where $Y=$ either $R$, Fe or Sb) scattering paths. Our findings suggest that in the case of the light cations, $R=\mathrm{K}$ or Ca, the $R$ atoms are relatively weakly coupled to the cage, in a scenario reminiscent to the Einstein oscillators. From the XRD experiments, we obtained the bulk modulus $B_0$ for all $R=\mathrm{Na}$, K, Ca, Sr, and Ba materials, with values ranging from 77 GPa ($R=\mathrm{K}$) to $R=99\mathrm{GPa}$ ($R=$ Ba) as well as the compressibility $\beta$ as a function of $P$. The trend in $\beta$ as a function of the $R$ filler is discussed and it is shown that it does not correlate with simple geometrical considerations but rather with the filler-cage bonding properties.

Figure 1

(a)–(c) Three representations of the ternary filled skutterudite structure (space group $Im\overline{3}$): (a) the $R{\mathrm{Fe}}_4{\mathrm{Sb}}_{12}$ conventional unit-cell highlighting the ${\mathrm{FeSb}}_3$ distorted octahedra; (b) the structural elements considered in the EXAFS analysis; (c) the vibrating elements of the 1D correlated rattling model (see text) [46]. In all cases, the $R$, Fe and Sb atoms are represented by, respectively, red, blue, and green spheres. (d)–(f) Fourier-transformed EXAFS spectra of the $R=\mathrm{K}$, Ca, and Ba materials in $T$-interval $6\text{K}<T<300\mathrm{K}$. (g)–(i) Low-$T$ (upper panel) and high-$T$ (lower panel) representative EXAFS data and data analysis (thick red line) for the $R=\mathrm{K}$, Ca, and Ba materials. In each panel, the inset shows the residuals of the presented fittings.

Figure 2

Temperature dependence of the correlated Debye-Waller parameters obtained from the Fe $K$-edge EXAFS analysis for the (a) ${\mathrm{KFe}}_4{\mathrm{Sb}}_{12}$, (b) ${\mathrm{CaFe}}_4{\mathrm{Sb}}_{12}$, and (c) ${\mathrm{BaFe}}_4{\mathrm{Sb}}_{12}$ samples.

Figure 3

Effective 1D correlated rattling model for the $R=\mathrm{K}$ (Ca) and Ba samples. In ascending frequency, we have the acoustic mode (red) and the first, (green), second (orange), and third (blue) optical phonon modes. The modes are displayed as function of $qa$ where $a$ is the material lattice constant and $q$ is the phonon wave vector. In panels (a)–(i), the mass relation between the vibrating elements is that of the $R=\mathrm{K}$ case and, in panels (j)–(o), the mass relation between the vibrating elements is that of the $R=$ Ba case. The adopted parameters are $k_{\text{cs}}^{\text{R}}=1.8$ $\mathrm{eV}/{\text{Å}}^2$ for all panels and $k_{\text{cc}}^{\text{K}}=10$ $\mathrm{eV}/{\text{Å}}^2$, in panels (a)–(i), and $k_{\text{cc}}^{\text{Ba}}=25$ $\mathrm{eV}/{\text{Å}}^2$, in panels (j)–(o), reflecting the different $K_{\text{eff}}^{R,\text{Fe-Fe}}$ parameters. The adopted $k_{\text{rs}}^R$ and $k_{\text{rc}}^R$ values are indicated in each panel in units of $\mathrm{eV}/{\text{Å}}^2$.

Figure 4

Representative $P$ dependent XRD experiments for all $R{\mathrm{Fe}}_4{\mathrm{Sb}}_{12}$-filled skutterudite, wherein we have (a) $R=$ Na, (b) $R=\mathrm{K}$, (c) $R=$ Ca, (d) $R=$ Sr, and (e) $R=$ Ba. $P$ is as indicated in the panels. The thick red lines represent the diffraction peaks of the skutterudite structure. The blue and green arrows point to diffraction peaks not ascribed to the skutterudite phase and are discussed in the main text and elsewhere [51].

Figure 5

(a) The unit-cell volume $V$ as function of $P$. Error bars for $P$ are estimated from the half-width at half maximum of the ruby fluorescence spectra and amount to about $3\%\text{–}5\%$ in this pressure interval. (b) The cage empty volume fraction $f_{\text{E}}$ (see text) as a function of $P$. The inset shows a representation of the skutterudite structure highlighting the Sb cage around the filler site (the cage volume was calculated with vesta [46]). (c) The $P$ as a function of $V$ data and their respective fittings to the Birch-Murnaghan model. (d) The compressibility $\beta$ as a function of $P$ for all samples.

Figure 6

Contour plots for the MOs describing the filler-cage bonding in the $(x,y)$ plane in the cases of the Ca-, Sr-, and Ba-filled materials as indicated. The figures are formed by 120 contours in the isovalue interval indicated in the color scale on the right side of the figure. Ligand orbitals are plotted via the gabedit program [60].