Vibrational dynamics of the filled skutterudites $M_{1-x}$Fe${}_4$Sb${}_{12}$ ($M=\mathrm{Ca}$, Sr, Ba, and Yb): Temperature response, dispersion relation, and material properties

The vibrational dynamics of the ternary Ca${}_{1-x}$Fe${}_4$Sb${}_{12}$, Ba${}_{1-x}$Fe${}_4$Sb${}_{12}$, Sr${}_{1-x}$Fe${}_4$Sb${}_{12}$, and Yb${}_{1-x}$Fe${}_4$Sb${}_{12}$ compounds is studied for its temperature dependence, dispersion relations, and material properties by experiments and ab initio powder-averaged lattice-dynamics (PALD) calculations. The experimental techniques used are high-resolution inelastic neutron scattering (INS) and inelastic x-ray scattering (IXS). Vibrational properties of polycrystalline material have been mapped out in an extensive energy-momentum phase space facilitating the study of powder-averaged phonon-dispersion relations within the first and second Brillouin zones. Qualitatively, the experimental dynamic structure factor $S(Q,\omega )$ matches, to a high degree, PALD calculations allowing different schemes of collective vibrational eigenmodes to be evaluated. From the PALD calculations, quantitative values of velocity of sound, Debye temperatures, and elastic, bulk, and Young’s moduli are computed. The variation of the generalized density of states $G\left(\omega \right)$ with temperature is studied experimentally in the range from 2 to 600 K. No particular anomalies in the collective dynamics are observed apart from a global softening of $G\left(\omega \right)$ upon heating.

Figure 1

Phase-space coverage by the neutron time-of-flight spectrometers IN4@ILL (right, Stokes side) and IN6@ILL (left, anti-Stokes side) as they have been exploited for the data analysis. Light-shaded area indicates the IN4@ILL coverage reached with the incident neutron energy of 25.4 meV. Dark-shaded area indicates the coverage with 12.9 meV. Dotted lines sketch the constant $Q$ scans performed with the IXS spectrometer ID28@ESRF.

Figure 2

Temperature dependence of the generalized density of states, $G\left(\omega \right)$, of Ca-, Sr-, Ba-, and Yb-filled Fe${}_4$Sb${}_{12}$ compounds measured at IN4@ILL (symbols) and IN6@ILL (lines). Temperatures are color-coded and reported in the figures. Left, a close-up look at the low-energy range of $G\left(\omega \right)$. Note the different energy scale of the YbFe${}_4$Sb${}_{12}$ data. At 2 K the strong elastic contribution from the manifold of Bragg reflections of the polycrystalline samples obscures the inelastic signal of Ca-, Sr-, and Ba-filled skutterudites below 4 meV. Right, entire energy range of $G\left(\omega \right)$. Vertical arrows indicate the second-order elastic scattering from the monochromator of IN6@ILL becoming detectable at low temperatures.

Figure 3

(Color online) Low-energy range of $G\left(\omega \right)/{\omega }^2$ of YbFe${}_4$Sb${}_{12}$ at different temperatures indicated in the figures. Black full lines indicate fit results as discussed in the text. Blue full lines highlight the partial contribution by the two applied Gaussians. Fit parameters are reported in Fig. 4.

Figure 4

Fit parameters, energy $E$, full width at half maximum $\gamma$, amplitude $A$, and intensity $I$ of the two Gaussians fitted to the double-peak feature in $G\left(\omega \right)/{\omega }^2$. Full and empty squares characterize the low-energy (p1) and high-energy (p2) peaks, respectively. Full line represents the total intensity.

Figure 5

From top to bottom, phonon dispersion relation of CoSb${}_3$ and Ca-, Sr-, Ba-, and Yb-filled iron antimonides calculated along the main symmetry directions.

Figure 6

Contrast plot of the dynamic structure factor of BaFe${}_4$Sb${}_{12}$ (a) and YbFe${}_4$Sb${}_{12}$ (c) measured at 300 K. Figures (b) and (d) report the corresponding PALD-calculated signals. The low-energy part of the spectra, dominated by eigenmodes with a strong contribution of the $M$ cations, is highlighted. Red vertical arrows indicate peaks identified in the inelastic response of the compounds.[6] Yellow horizontal arrows highlight $\Gamma$ points at which the symmetry-avoided anticrossing of phonon modes is observable in the PALD signal. Black dashed lines span the limits of the energy-momentum phase space of the IN6@ILL experiment. Note that the intensity of the Yb-dominated modes has been chosen to saturate the color resolution, enabling a better visualization of the overall signal.

Figure 7

(Color online) Measured (left) and PALD-calculated (right) constant $Q$ spectra $S(Q,\omega )$ of YbFe${}_4$Sb${}_{12}$. Wave numbers $Q$ are reported in the figures. Lines represent fits to the apparent double-peak feature with the bulk Gaussian response peaks (blue). The insets highlight the line shape of the strong peaks.

Figure 8

Measured (left) and PALD-calculated (right) constant $Q$ spectra $S(Q,\omega )$ of BaFe${}_4$Sb${}_{12}$. Wave numbers $Q$ are reported in the figures.

Figure 9

(Color online) $Q$-resolved energy $E\left(Q\right)$, full width at half maximum $\gamma \left(Q\right)$, intensity $I\left(Q\right)$, and relative intensity $I_{\mathrm{rel}}\left(Q\right)$ parameters of the two Gaussian peaks (p1 and p2) fitted to the low–energy signal of YbFe${}_4$Sb${}_{12}$ measured at IN6@ILL. Symbols and lines applicable to $E\left(Q\right)$, $\gamma \left(Q\right)$, and $I\left(Q\right)$ are defined in the top figures. Symbols for $I_{\mathrm{rel}}\left(Q\right)$ data are defined in the corresponding figure. Red and blue lines represent the structure factor $S\left(Q_{\mathrm{elastic}}\right)$ as detected at IN6@ILL and IN4@ILL, respectively.

Figure 10

Figures on top display contrast plots of IXS-weighted PALD-caclulated $S(Q,\omega )$ of BaFe${}_4$Sb${}_{12}$ and YbFe${}_4$Sb${}_{12}$ skutterudites. Panels below report constant $Q$ cuts through the measured (left) and PALD-calculated (right) data. Wave numbers $Q$ are reported in the figures. YbFe${}_4$Sb${}_{12}$ data are represented by full symbols with error bars (left) and full lines (right). BaFe${}_4$Sb${}_{12}$ data are represented by the grey shaded area. The statistical accuracy of the experimental results is comparable to the one of YbFe${}_4$Sb${}_{12}$. Solid (blue) lines demonstrate fitting results approximating the signal by a set of Gaussian lines convoluted with the resolution function. The resulting energy parameters of the fits are indicated as white circles in the top figures. See text for details.

Figure 11

Contrast plots of PALD results of BaFe${}_4$Sb${}_{12}$ [(a) and (b)] and YbFe${}_4$Sb${}_{12}$ [(c) and (d)], $Z\left(\omega \right)$ (e) and $Z\left(\omega \right)/{\omega }^2$ (f) of compounds indicated in the figures. Color codes characterizing the intensity in (a) and (c) and (b) and (d) are shown to the right of (c) and (d), respectively. Plots (a) and (b) and (c) and (d) display same data sets on a linear intensity scale, however, intensity in (b) and (d) has been limited to unity to highlight the weak signal of tranverse acoustic phonons (T) in comparison to modes of longitudinal polarization (L). Dashed lines in (a) and (b) indicate lines along which the first moments $\langle \omega \rangle$ and $\langle Q\rangle$ have been calculated. Dashed lines in (e) and (f) represent the results of the Debye fit to the data.

Figure 12

(Color online) Left, scaled mode energy of the two prominent peaks. Full and empty squares represent peaks with $E^{*}=4.8$ and 5.7 meV, respectively. Blue dashed line indicates the effect of the anharmonic term $B{x}^4$ in the Yb potential derived from DFT calculation. Inset, displays the DFT-calculated Yb potential and dashed and solid lines correspond to fit results with harmonic $A{x}^2$ and anharmonic $A{x}^2+B{x}^4$ models, respectively. Right, relative changes in the inelastic intensity due to the temperature dependence of the Debye-Waller factor computed within the harmonic approximation.