Lattice vibrations in $\mathrm{La}\left(\mathrm{Ce}\right){\mathrm{Fe}}_4{\mathrm{Sb}}_{12}$ and $\mathrm{Co}{\mathrm{Sb}}_3$: Inelastic neutron scattering and theory

We present results of time-of-flight inelastic neutron scattering phonon density of states measurements on ${(\mathrm{La},\mathrm{Ce})}_{0.9}{\mathrm{Fe}}_4{\mathrm{Sb}}_{12}$ and $\mathrm{Co}{\mathrm{Sb}}_3$ as well as of a detailed comparison with lattice dynamical models in the literature. The MARI experimental setup is replicated by a theory for scattering from a polycrystalline material. The model considered for the filled materials is a local density approximation (LDA) based short-range force constant model and those considered for $\mathrm{Co}{\mathrm{Sb}}_3$ are the LDA-based model with the La-related parameters removed and a semiempirical model. We show that the presence of La significantly affects the shape of the spectrum. We also conclude that upon filling the Sb intrasquare force constants are weakened and that the transition-metal Sb bonds are almost unchanged.

Figure 1

INS results (${\mathrm{E}}_i=30\mathrm{meV}$, $Q=5–7{\mathrm{\AA }}^{-1}$) for $\mathrm{Co}{\mathrm{Sb}}_3$ and $\mathrm{La}{\left(\mathrm{Ce}\right)}_{0.9}{\mathrm{Fe}}_4{\mathrm{Sb}}_{12}$. In the case of $\mathrm{Co}{\mathrm{Sb}}_3$ the arrows correspond to the pairs of infrared and Raman modes $\left(F_g\right)$ of lowest frequencies, where the infrared modes are the outermost ones. In the case of ${\mathrm{La}}_{0.9}{\mathrm{Fe}}_4{\mathrm{Sb}}_{12}$ the two lowest frequency Raman modes are shown and they are also of $F_g$ symmetry.

Figure 2

INS results $({\mathrm{E}}_i=50\mathrm{meV})$ for $\mathrm{La}{\left(\mathrm{Ce}\right)}_{0.9}{\mathrm{Fe}}_4{\mathrm{Sb}}_{12}$ and $\mathrm{Co}{\mathrm{Sb}}_3$. Both $5–7{\mathrm{\AA }}^{-1}$ and $5–9{\mathrm{\AA }}^{-1}$ $Q$ cuts are shown. The arrows correspond to experimental Raman $A_g$ frequencies.

Figure 3

INS and model results. Results indicate possible effect of disorder. In one calculation (dashed line) the theoretical instrumental broadening, ${\Delta }_{\mathit{ins}}\left(E\right)$, is applied to the model and in the other calculation (solid line) $\Delta \left(E\right)=\surd [2^2+{\Delta }_{\mathit{ins}}{\left(E\right)}^2]$ is used for the broadening (FWHM). The arrows correspond to the lowest frequency calculated infrared and Raman $\left(F_g\right)$ modes. The lowest frequency optical mode is infrared active and the two outermost modes (among the modes above $10\mathrm{meV}$) are infrared active. The Ce contribution to the spectrum for ${\mathrm{Ce}}_{0.9}{\mathrm{Fe}}_4{\mathrm{Sb}}_{12}$ is also given.

Figure 4

INS results for $\mathrm{Co}{\mathrm{Sb}}_3$ and corresponding results for models 1 (upper panel) and 2 (lower panel) (see text). The arrows correspond to the calculated lowest frequency pairs of Raman and infrared modes, where the outer modes are infrared.

Figure 5

Theory and experiment for the INS of ${\mathrm{La}}_{0.9}{\mathrm{Fe}}_4{\mathrm{Sb}}_{12}$ corresponding to $E_i=50\mathrm{meV}$. The experimental results for the $5–9{\mathrm{\AA }}^{-1}$ $Q$ cut are given, as are the theoretical $5–9{\mathrm{\AA }}^{-1}$ (solid) and $5–7{\mathrm{\AA }}^{-1}$ (dashed) $Q$ cuts. The La contribution to the $5–9{\mathrm{\AA }}^{-1}$ $Q$ cut is also shown. The broadening employed in obtaining the theoretical results includes disorder broadening (see text).

Figure 6

Theory and experiment for the INS of $\mathrm{La}{\left(\mathrm{Ce}\right)}_{0.9}{\mathrm{Fe}}_4{\mathrm{Sb}}_{12}$. The two theory curves correspond to theoretical instrumental broadening alone (dotted) and disorder broadening included (dashed). The Ce contribution for ${\mathrm{Ce}}_{0.9}{\mathrm{Fe}}_4{\mathrm{Sb}}_{12}$ is also shown. The arrows correspond to the calculated $A_g$ frequencies.

Figure 7

Theory and experiment for the INS of $\mathrm{Co}{\mathrm{Sb}}_3$. The theory curve is based on models 1 (upper panel) and 2 (lower panel) (see text). The theoretical instrumental broadening is used. The arrows correspond to the calculated $A_g$ frequencies.

Figure 8

Partial contributions to the INS of ${\mathrm{La}}_{0.9}{\mathrm{Fe}}_4{\mathrm{Sb}}_{12}$ calculated for $Q=5–7{\mathrm{\AA }}^{-1}$ cuts for both ${\mathrm{E}}_i=30\mathrm{meV}$ (lower panel) and $E_i=50\mathrm{meV}$ (upper panel). The solid lines include disorder broadening and the dashed lines do not (see text).

Figure 9

VDOS for the force constant (FC) models used in this work. In the middle panel the Fe mass is used rather than that of Co in order to provide a direct comparison with the results of the LDA-based calculations for $\mathrm{La}{\mathrm{Fe}}_4{\mathrm{Sb}}_{12}$. The comparison shows the effect, in the VDOS, of the La contribution to the FCs (see text).

Figure 10

VDOS contributions for $\mathrm{La}{\mathrm{Fe}}_4{\mathrm{Sb}}_{12}$.

Figure 11

Fe partial VDOS weighted using components of eigenvectors along parallel (solid) and perpendicular (dashed) directions to the La-Fe “bond” direction. Upper panel corresponds to model 1 for $\mathrm{Fe}{\mathrm{Sb}}_3$ and lower panel to $\mathrm{La}{\mathrm{Fe}}_4{\mathrm{Sb}}_{12}$.

Figure 12

Difference spectrum for $30\mathrm{meV}$ incident neutron energies; ${\mathrm{La}}_{0.9}{\mathrm{Fe}}_4{\mathrm{Sb}}_{12}\text{−}{\mathrm{Ce}}_{0.9}{\mathrm{Fe}}_4{\mathrm{Sb}}_{12}$. The smooth curves show the calculated La partial intensities for two estimates of energy broadening (see text), curves $a$ and $b$, and a least square fitted Gaussian function in the $5–10\mathrm{meV}$ spectral region, curve $c$. The Gaussian parameters for the peak position and breadth (FWHM) are $7.14\left(7\right)\mathrm{meV}$ and $3.8\left(2\right)\mathrm{meV}$, respectively.

Figure 13

Dispersion curves along principle directions: Right plots are based on our LDA force constant model for $\mathrm{La}{\mathrm{Fe}}_4{\mathrm{Sb}}_{12}$. Left plots are based on that model with the absence of La and small additional force constant changes required by the conservative force condition (see text).