Clathrate formation in the Ba-Pd-Ge system: Phase equilibria, crystal structure, and physical properties

Phase relations at subsolidus temperatures as well as at $T=800°\mathrm{C}$, crystallographic data, electrical and thermal transport measurements, and heat capacity data are reported for several compositions within the clathrate type-I solid solution: ${\mathrm{Ba}}_8{\mathrm{Pd}}_x{\mathrm{Ge}}_{46-x-y}{◻}_y$ (◻ is a vacancy). The solid solution derives from binary clathrate ${\mathrm{Ba}}_8{\mathrm{Ge}}_{43}{◻}_3$ with a solubility limit of 3.8 Pd atoms per formula unit at $T=800°\mathrm{C}$. Structural investigations throughout the homogeneity region confirm cubic primitive symmetry consistent with the space group type $Pm\overline{3}n$ and lattice parameters ranging from $a=1.0657\left(2\right)\mathrm{nm}$ for ${\mathrm{Ba}}_8{\mathrm{Ge}}_{43}{◻}_3$ to $a=1.07741\left(2\right)\mathrm{nm}$ for ${\mathrm{Ba}}_8{\mathrm{Pd}}_{3.8}{\mathrm{Ge}}_{42.2}{◻}_{0.0}$. The primary field of clathrate crystallization has been elucidated from micrography and differential thermal analyses. Both heat capacity and inelastic neutron diffraction define a low-lying, almost localized, phonon branch. Studies of transport properties evidence electrons as the majority charge carriers for most of the homogeneity region; however, at the Pd-rich limit, holes dominate the electronic transport. The crossover between both regimes provides appropriate conditions for attractively high Seebeck values. The lattice contribution dominates the overall thermal conductivity.

Figure 1

(Color online) Lattice parameters vs Pd content for ${\mathrm{Ba}}_8{\mathrm{Pd}}_x{\mathrm{Ge}}_{46-x-y}{◻}_y$ alloys. The dashed line denotes the solubility limit of Pd at $T=800°\mathrm{C}$.

Figure 2

(Color online) Partial liquidous surface and isothermal section at $T=800°\mathrm{C}$ (a), partial reaction scheme (b), and solidus (c) of the Ba-Pd-Ge ternary system in the region $\mathrm{Ge}\text{−}\mathrm{Pd}\mathrm{Ge}\text{−}\mathrm{Ba}{\mathrm{Ge}}_2$.

Figure 3

Microstructures of ${\mathrm{Ba}}_8{\mathrm{Pd}}_6{\mathrm{Ge}}_{40}$, (a) as-cast, (b) $T=800°\mathrm{C}$, and ${\mathrm{Ba}}_{17}{\mathrm{Pd}}_{19}{\mathrm{Ge}}_{64}$, (c) $T=800°\mathrm{C}$, (d) $T=700°\mathrm{C}$, alloys.

Figure 4

Microstructures of alloys ${\mathrm{Ba}}_8{\mathrm{Pd}}_{22}{\mathrm{Ge}}_{70}$, (a) as-cast, (b) $800°\mathrm{C}$, ${\mathrm{Ba}}_8{\mathrm{Pd}}_{30}{\mathrm{Ge}}_{62}$, (c) as-cast, and (d), (e), (f) $T=800°\mathrm{C}$.

Figure 5

(Color online) The $6d$ site preference for ${\mathrm{Ba}}_8{\mathrm{Pd}}_x{\mathrm{Ge}}_{46-x-y}{◻}_y$ (a) and (b) dependences of deficiency from $M$ $(M=\mathrm{Pd},\mathrm{Zn},\mathrm{Cd})$ content in ${\mathrm{Ba}}_8{\mathrm{Pd}}_x{\mathrm{Ge}}_{46-x-y}{◻}_y$ are compared for two boundary models for the incorporation of $M$ atoms in the lattice of ${\mathrm{Ba}}_8{\mathrm{Ge}}_{43}{◻}_3$.

Figure 6

Compositional dependence of positional parameters $y$ (a) and $z$ (b) for the Ge(3) site [$24k$ $(0,y,z)$ in ${\mathrm{Ba}}_8{\mathrm{Pd}}_x{\mathrm{Ge}}_{46-x-y}{◻}_y$ $(M=\mathrm{Pd},\mathrm{Zn},\mathrm{Cd})$]. Solid symbols, our data; open symbols, Refs. 3, 12.

Figure 7

(Color online) Thermal atom displacement parameters of ${\mathrm{Ba}}_8{\mathrm{Pd}}_{3.8}{\mathrm{Ge}}_{42.2}$ vs temperature.

Figure 8

(Color online) Temperature-dependent specific heat $C_p$ of ${\mathrm{Ba}}_8{\mathrm{Pd}}_{3.3}{\mathrm{Ge}}_{42.5}{◻}_{0.2}$ and ${\mathrm{Ba}}_8{\mathrm{Pd}}_{3.82}{\mathrm{Ge}}_{42.18}$, plotted as $C_p∕T$ vs $T$ (a) and $(C_p-\gamma T)∕T^3$ vs $\ln T$ (b). The dashed line is a least-squares fit of the experimental data using the model described in the text with two Einstein-like modes (${\theta }_D=226\mathrm{K}$, ${\omega }_{\mathit{EL}1}=53.8\mathrm{K}$ with a width of $3.6\mathrm{K}$ and ${\omega }_{\mathit{EL}2}=86.6\mathrm{K}$ with a width of $3.5\mathrm{K}$). The dash-double-dotted line is the simple Debye function with ${\theta }_D^{\mathit{LT}}=268\mathrm{K}$, highlighting the substantial difference to the heat capacity observed. The solid lines (referring to the right axis) sketch the phonon spectral function $F\left(\omega \right)$ plotted as $\omega ∕4.93$ vs $(5∕4)R{\pi }^4{\omega }^{-2}2F\left(\omega \right)$ for which $\omega$ is given in degrees Kelvin.

Figure 9

(Color online) (a) Generalized density of states $G\left(\omega \right)$ of ${\mathrm{Ba}}_8{\mathrm{Pd}}_{3.8}{\mathrm{Ge}}_{42.2}$ compared with $G\left(\omega \right)$ of ${\mathrm{Ba}}_8{\mathrm{Zn}}_{7.7}{\mathrm{Ge}}_{38.3}$. (b) Close-up of the low-energy part of $G\left(\omega \right)$ determining substantially the properties of the specific heat of the samples. Vertical arrows indicate the increase of Einstein-mode energies and an intensity enhancement found in the ${\mathrm{Ba}}_8{\mathrm{Pd}}_{3.8}{\mathrm{Ge}}_{42.2}$ sample.

Figure 10

(Color online) (a) Temperature-dependent resistivity $\rho$ of various concentrations of ${\mathrm{Ba}}_8{\mathrm{Pd}}_x{\mathrm{Ge}}_{46-x-y}{◻}_y$. (b) Normalized resistivity of ${\mathrm{Ba}}_8{\mathrm{Pd}}_x{\mathrm{Ge}}_{46-x-y}{◻}_y$. The solid lines are least-squares fits as explained in the text.

Figure 11

(Color online) (a) Temperature-dependent charge carrier density $n$ and mobility $\mu$ for ${\mathrm{Ba}}_8{\mathrm{Pd}}_x{\mathrm{Ge}}_{46-x-y}{◻}_y$, $x=3.6$ and $y=0.4$, taken at 3 and $6\mathrm{T}$. (b) Field-dependent charge carrier density $n$ and mobility $\mu$ for ${\mathrm{Ba}}_8{\mathrm{Pd}}_x{\mathrm{Ge}}_{46-x-y}{◻}_y$, $x=3.6$ and $y=0.4$ taken at $2\mathrm{K}$.

Figure 12

(Color online) (a) Temperature-dependent thermal conductivity $\lambda$ of two concentrations of ${\mathrm{Ba}}_8{\mathrm{Pd}}_x{\mathrm{Ge}}_{46-x-y}{◻}_y$.

Figure 13

(Color online) Lattice $\left({\lambda }_{\mathit{ph}}\right)$ and electronic $\left({\lambda }_e\right)$ thermal conductivity of ${\mathrm{Ba}}_8{\mathrm{Pd}}_2{\mathrm{Ge}}_{42.5}{◻}_{1.5}$. The solid line is a least-squares fit according to Eq. (3). (b) Lattice $\left({\lambda }_{\mathit{ph}}\right)$ and electronic $\left({\lambda }_e\right)$ thermal conductivity of ${\mathrm{Ba}}_8{\mathrm{Pd}}_{3.8}{\mathrm{Ge}}_{42.2}$. The solid line is a least-squares fit according to Eq. (3).

Figure 14

(Color online) (a) Temperature-dependent thermopower for various concentrations of ${\mathrm{Ba}}_8{\mathrm{Pd}}_x{\mathrm{Ge}}_{46-x-y}{◻}_y$. (b) $S\left(T\right)$ of $x=3.3$, $x=3.6$, and $x=3.7$ in an extended temperature range.