Crystal structures, atomic vibration, and disorder of the type-I thermoelectric clathrates ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Si}}_{30}$, ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$, ${\mathrm{Ba}}_8{\mathrm{In}}_{16}{\mathrm{Ge}}_{30}$, and ${\mathrm{Sr}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$

Temperature dependent synchrotron powder diffraction and single crystal neutron diffraction data are used for probing the vibrational states and disorder in type I clathrates ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Si}}_{30}$, ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$, ${\mathrm{Ba}}_8{\mathrm{In}}_{16}{\mathrm{Ge}}_{30}$, and ${\mathrm{Sr}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$. If an empirical disorder term is included, the temperature dependence of the atomic displacement factors (ADPs) of the framework and guest atoms can be described by a Debye and Einstein model, respectively. None of the guest atoms in the large cages are located in the center and the vibrational frequencies $\left({\theta }_{\mathrm{E}}\right)$ are of the order $80\mathrm{K}$ or larger for all structures, in good agreement with theoretical predictions. Even though the Sr ADPs are larger than the Ba ADPs in all the clathrates, the data show that ${\theta }_{\mathrm{E}}$ of Sr in ${\mathrm{Sr}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ is larger than for the Ba atoms. This is due to stronger guest-host chemical bonding in ${\mathrm{Sr}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$. Since ${\theta }_{\mathrm{E}}$ of Sr has been reported to be much smaller in the literature we have also measured the specific heat of ${\mathrm{Sr}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ with ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ as a reference. It is found that localized excitations with a characteristic energy of approximately $35\mathrm{K}$ exist in both compounds, however, the total number of states is too low to be associated with either tunneling states or localized vibration of each of the guest atoms.

Figure 1

The clathrate type I structure. The large dark atoms are M1 $(\mathrm{Ba}∕\mathrm{Sr})$ on the $2a$ site, the large gray atoms M2 $(\mathrm{Ba}∕\mathrm{Sr})$ on the $6d$ site. The small, black, gray and light gray atoms are $\mathrm{Ga}∕\mathrm{Si}∕\mathrm{In}$ on the $6c$ $\left({\mathrm{H}}_{6c}\right)$, $16i$ $\left({\mathrm{H}}_{16i}\right)$, and $24k$ $\left({\mathrm{H}}_{24k}\right)$ sites, respectively.

Figure 2

(Color online) Observed and calculated diffraction patterns at $300\mathrm{K}$ as a function of $2\theta$ for ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Si}}_{30}$, ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$, ${\mathrm{Ba}}_8{\mathrm{In}}_{16}{\mathrm{Ge}}_{30}$, and ${\mathrm{Sr}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$. Curves below the diffraction patterns are the observed intensities minus the model intensities. The insets show that detailed features are also observed at higher $2\theta$ angles.

Figure 3

Isotropic mass and site averaged framework atomic displacement parameters $\left(U_{\mathrm{iso}}\right)$ as a function of temperature $\left(T\right)$ for ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Si}}_{30}$ obtained from single crystal neutron diffraction. Curve (a) is a fit to a Debye model with ${\theta }_{\mathrm{D}}=387\left(11\right)\mathrm{K}$, $d=0.043\left(3\right)\mathrm{\AA }$, ${\gamma }_{\mathrm{G}}=3.5\left(10\right)$. Curve (b) is a fit to the same model but with the $900\mathrm{K}$ data point excluded [${\theta }_{\mathrm{D}}=403\left(7\right)\mathrm{K}$, $d=0.046\left(1\right)\mathrm{\AA }$, ${\gamma }_{\mathrm{G}}=6.1\left(8\right)$]. Line (c) is a high temperature approximation of the Debye model with ${\theta }_{\mathrm{D}}=336\left(3\right)$. It is seen that line (c) crosses approximately (0,0) as expected if no disorder and zero-point vibration is present. Curve (d) shows that a fit to all data points with $d=0$ is poor [${\theta }_{\mathrm{D}}=321\left(11\right)\mathrm{K}$, ${\gamma }_{\mathrm{G}}=-1.7\left(27\right)$]. The lower right inset shows $U_{\mathrm{iso}}$ for Mg as a function of temperature (Ref. 39). The solid curve is fit to a Debye model without disorder [${\theta }_{\mathrm{D}}=333\left(3\right)\mathrm{K}$ and ${\gamma }_{\mathrm{G}}=2.2\left(2\right)$]. The upper left inset shows the volume $\left(V\right)$ of the unit cell of ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Si}}_{30}$ as a function of temperature. The solid lines are a parameterization used for modeling $U_{\mathrm{iso}}\left(T\right)$, see text.

Figure 4

Temperature $\left(T\right)$ dependence of the framework and guest atomic displacement parameters (ADP) $U$. M1 are guest atoms located on the highly symmetric $2a$ site, M2 are guest atoms located on the $6d$ site. ${\mathrm{H}}_{6c}$, ${\mathrm{H}}_{16i}$, and ${\mathrm{H}}_{24k}$ are framework atoms on the $6c$, $16i$, and $24k$ sites. Note that the scale on the $y$ axis is different for $U_{22}=U_{33}$ for M2. Data points are obtained from the modeling of synchrotron powder diffraction patterns. The solid curves are the corresponding data obtained from single crystal neutron diffraction on ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Si}}_{30}$. For the upper three graphs the dotted curves are ordered Einstein models of the ADPs assuming the guest atom mass is $137.327\mathrm{g}∕\mathrm{mol}$ (Ba) and with an Einstein temperature of 130, 100, and $70\mathrm{K}$, respectively. The dotted curve in the lower graph is an ordered Debye model of the ADPs assuming that the framework atom mass is $42.57\mathrm{g}∕\mathrm{mol}$ (weighted average of Ga and Si) and with a Debye temperature of $350\mathrm{K}$. In all dotted curves $d=0$ has been used. The error bars are of the same size as the data points except for $U_{11}=U_{22}=U_{33}$ ($2a$ site) and $U_{11}$ ($6d$ site) for Ba in ${\mathrm{Ba}}_8{\mathrm{In}}_{16}{\mathrm{Ge}}_{30}$ where representative error bars at $285\mathrm{K}$ have been included.

Figure 5

Lattice parameter normalized to the $300\mathrm{K}$ value $(a∕a_{300\mathrm{K}})$ as a function of temperature $\left(T\right)$ for the four clathrates. Data has been calculated from Tables . Since $a_{300\mathrm{K}}$ for ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ and ${\mathrm{Ba}}_8{\mathrm{In}}_{16}{\mathrm{Ge}}_{30}$ are slightly larger than expected and appear to be outliers, $a_{300\mathrm{K}}$ has been obtained by extrapolating a linear fit of $a\left(T\right)$ data above $150\mathrm{K}$. $a_{300\mathrm{K},{\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Si}}_{30}}=10.5303\mathrm{\AA }$, $a_{300\mathrm{K},{\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}}=10.7643\mathrm{\AA }$, $a_{300\mathrm{K},{\mathrm{Ba}}_8{\mathrm{In}}_{16}{\mathrm{Ge}}_{30}}=11.1649\mathrm{\AA }$, and $a_{300\mathrm{K},{\mathrm{Sr}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}}=10.7282\mathrm{\AA }$ (compare values with $a_{300\mathrm{K}}$ from Tables ). The almost temperature independent coefficients of the thermal expansion $(\alpha =1∕a\bullet \partial a∕\partial T)$ are listed in the inset.

Figure 6

The main graph shows $\Delta C_p\left(T\right)=C_{p,\text{meas}}-C_{p,\text{Debye}}$. The solid curves are model calculations of the specific heat in a system with two Einstein oscillators and an electronic contribution $C_{p,\text{electronic}}=aT$. For the solid curve imposed on the ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ data Einstein temperatures $\left({\theta }_{\mathrm{E}}\right)$ of 80 and $42\mathrm{K}$ were used and the numbers of Einstein atoms $\left(N_{\mathrm{E}}\right)$ were 6.5 per u.c. and 1.5 per u.c., respectively. The corresponding numbers for ${\mathrm{Sr}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ are ${\theta }_{\mathrm{E}}=80$ and $33\mathrm{K}$, with $N_{\mathrm{E}}=6.5$ per u.c. and 1.5 per u.c., respectively. For the electronic contribution $a=0.035$ and $0.04\mathrm{J}∕\left({\mathrm{K}}^2\mathrm{mol}\right)∕\mathrm{u.c.}$ for ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ and ${\mathrm{Sr}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$, respectively. Dashed curves are Einstein contributions with ${\theta }_{\mathrm{E}}=33$, 42, and $80\mathrm{K}$ with $N_{\mathrm{E}}=1.5$ per u.c., 1.5 per u.c., and 6.5 per u.c., respectively. The inset shows the measured specific heat $\left(C_{p,\text{meas}}\right)$ of ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ (circles) and ${\mathrm{Sr}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ (squares) as a function of temperature $\left(T\right)$. The solid curve is the specific heat from a Debye model with 46 atoms per unit cell (u.c.) and a Debye temperature $\left({\theta }_{\mathrm{D}}\right)$ of $312\mathrm{K}$ $\left(C_{p,\text{Debye}}\right)$.

Figure 7

MEM difference densities, $\rho \left(\mathrm{MEM}\right)-\rho \left(\mathrm{NUP}\right)$, in the (001) plane through Ba(1) from $15\text{to}900\mathrm{K}$. The difference densities are plotted on a logarithmic scale, $0.0125\times 2^{\mathrm{N}}$ $(\mathrm{n}=0,\dots ,6)$, and solid contours are positive and dotted contours are negative.

Figure 8

MEM difference densities, $\rho \left(\mathrm{MEM}\right)-\rho \left(\mathrm{NUP}\right)$, in the (100) plane through Ba(2) from $15\text{to}900\mathrm{K}$. Contours as in Fig. 8.

Figure 9

MEM difference densities, $\rho \left(\mathrm{MEM}\right)-\rho \left(\mathrm{NUP}\right)$, in the (001) plane through Ba(2) from $15\text{to}900\mathrm{K}$. Contours as in Fig. 8.