Vibrational dynamics of the host framework in Sn clathrates

We use nuclear resonance inelastic x-ray scattering (NRIXS), a relatively new, synchrotron-based, isotope-specific technique in combination with a more traditional one, Raman spectroscopy, to probe the vibrational dynamics of the host frameworks in two Zintl clathrates: ${\mathrm{K}}_8{\mathrm{Zn}}_4{\mathrm{Sn}}_{42}$ (KZS) and ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Sn}}_{30}$ (BGS). From the normalized Sn vibrational density of states obtained from NRIXS, we calculate the stiffness, a mean force constant of the Sn environment, the resilience, a compact way of expressing the temperature dependence of the Sn mean square displacement, and several thermodynamic properties. The stiffness and the resilience are approximately $7\%$ lower in KZS, reflecting its larger unit cell compared to BGS. We emphasize the complementariness between NRIXS and Raman spectroscopy and establish a series of benchmarks for a more quantitative evaluation of the Raman spectra for the numerous clathrates that are still not suitable for NRIXS studies.

Figure 1

Structure of type-I clathrate ${\mathrm{K}}_8{\mathrm{Zn}}_4{\mathrm{Sn}}_{42}$. Color scheme: gray = K, yellow = Zn/Sn, red = Sn. One small (pentagonal dodecahedron) and large (tetrakaidecahedron) host framework cage are highlighted in green and blue, respectively.

Figure 2

Structure of type-VIII clathrate ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Sn}}_{30}$. Color scheme: gray = Ba, red = Sn/Ga. One host framework cage (pentagonal dodecahedron) is highlighted in blue.

Figure 3

Raman spectra for KZS and BGS. The frequencies of the peaks marked with ${}^{*}$ and ${}^{**}$ are plotted in Fig. 4.

Figure 4

(Top) Dependence of the frequency of the peaks marked with an asterisk in Fig. 3 on the number of the Sn atoms in the unit cell. (Bottom) Dependence of the frequency of the peaks marked with a double asterisk in Fig. 3 on the total mass of the substituted elements in the framework. Data for KZS (up triangle) and BGS (diamond) are from this work. The references for the other compounds are given in the text.

Figure 5

Raman (top) and NRIXS (Sn VDOS, bottom) spectra for KZS. NRIXS data points are shown as markers with error bars, while the solid curve represents three-point running average. The individual peaks are shown as gray dashed lines. Their frequencies are listed in Table 1.

Figure 6

Raman (top) and NRIXS (Sn VDOS, bottom) spectra for BGS. NRIXS data points are shown as markers with error bars, while the solid curve represents five-point running average. The individual peaks are shown as gray dashed lines. Their frequencies are listed in Table 1.

Figure 7

Temperature dependence of the vibrational contribution to the Sn msd extrapolated from the Sn VDOS measured at a single temperature from Eq. (2). Dashed lines represent the high-temperature slopes determined from Eq. (3).

Figure 8

NRIXS data for KZS and BGS shown in terms of $D\left(\overline{\nu }\right)$, $D\left(\overline{\nu }\right)/\overline{\nu }$, $D\left(\overline{\nu }\right)/{\overline{\nu }}^2$, and $D\left(\overline{\nu }\right){\overline{\nu }}^2$ to emphasize the vibrational modes that have the main contribution to the low-[Eq. (4)] and high-[Eq. (3)] temperature limiting values for Sn msd, and to the stiffness [Eq. (6)].