High temperature thermoelectric efficiency in ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$

The high thermoelectric figure of merit $\left(zT\right)$ of ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ makes it one of the best $n$-type materials for thermoelectric power generation. Here, we describe the synthesis and characterization of a Czochralski pulled single crystal of ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ and polycrystalline disks. Measurements of the electrical conductivity, Hall effect, specific heat, coefficient of thermal expansion, thermal conductivity, and Seebeck coefficient were performed up to $1173\mathrm{K}$ and compared with literature results. Dilatometry measurements give a coefficient of thermal expansion of $16\times {10}^{-6}{\mathrm{K}}^{-1}$ up to $1175\mathrm{K}$. The trend in electronic properties with composition is typical of a heavily doped semiconductor. The maximum in the thermoelectric figure of merit is found at $1050\mathrm{K}$ with a value of 0.8. The correction of $zT$ due to thermal expansion is not significant compared to the measurement uncertainties involved. Comparing the thermoelectric efficiency of segmented materials, the effect of compatibility makes ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ more efficient than the higher $zT$ $n$-type materials SiGe or skutterudite $\mathrm{Co}{\mathrm{Sb}}_3$.

Figure 1

(Color online) (a) Using the Czochralski pulling method, a large ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ single crystal was formed. (b) ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ is a type I clathrate, a structure composed of covalently bound Ga-Ge with cages containing rattling cations. The anion framework of tetrakaidecahedral and dodecahedral polyhedra is shown in blue with red cations.

Figure 2

Resistivity of polycrystalline and single-crystalline ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$. The resistivity behavior is that of a heavily doped semiconductor, with a linear increase in resistivity with temperature until high enough temperatures are reached to increase the carrier concentration by exciting intrinsic carriers.

Figure 3

The Seebeck coefficient $(\Delta V∕\Delta T)$ rises to $-145\mu \mathrm{V}∕\mathrm{K}$ at $1050\mathrm{K}$, and then rolls over with increasing intrinsic conduction at higher temperatures. Single crystalline (closed squares) and polycrystalline (open circles) samples show nearly identical behavior.

Figure 4

The natural logarithm of the conductivity and the Seebeck coefficient are linearly related in a semiconductor with a slope of $86\mu \mathrm{V}∕\mathrm{K}$ (shown with dotted line). Prior measurements at $900\mathrm{K}$ of $n$-type single crystalline (empty squares) and polycrystalline (empty circles) ${\mathrm{Ba}}_8{\mathrm{Ga}}_x{\mathrm{Ge}}_{46-x}$ are shown alongside our single crystalline (filled square) and polycrystalline (filled circle) measurements.

Figure 5

Heat capacity $\left(C_p\right)$ values derived from thermal conductivity measurements of ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$. The $C_v$ value predicted by Dulong and Petit is shown as a dotted line.

Figure 6

The total thermal conductivity of polycrystalline and single crystalline ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ is shown with temperature. The lattice component of the thermal conductivity is extremely low (comparable to a glass) and is derived from the Wiedemann-Franz law.

Figure 7

(Color online) The lattice component of the thermal conductivity of several $n$-type high efficiency thermoelectric materials, including ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$. Values derived from the Wiedemann-Franz law.

Figure 8

(Color online) Thermoelectric efficiency $\left(zT\right)$ of single crystalline ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ plotted with other $n$-type high efficiency thermoelectric materials

Figure 9

(Color online) The thermoelectric compatibility factor of several $n$-type thermoelectric materials. Good compatibility arises when the difference is less than a factor of 2. ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$ and $\mathrm{Co}{\mathrm{Sb}}_3$ show comparable compatibility factors with their adjacent $n$-type materials while SiGe is incompatible.

Figure 10

(Color online) The compatibility factor $s$ (purple) varies between materials and across temperatures. Optimizing the reduced current density $u$ (black) for the maximum efficiency (maximized when $u=s$) results in a compromise. Three legs are shown, based on ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ (BT), PbTe, SiGe, $\mathrm{Co}{\mathrm{Sb}}_3$ (CS), and ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$. The two lines are closer in ${\mathrm{Ba}}_8{\mathrm{Ga}}_{16}{\mathrm{Ge}}_{30}$, explaining why its actual efficiency is closer to the maximum efficiency (Fig. 11).

Figure 11

(Color online) Three segmented thermoelectric leg configurations are considered across a temperature range of $300–1173\mathrm{K}$. For each of these, a maximum reduced efficiency $\left({\eta }_{\mathit{max}}\right)$ is shown in black and below, shown in blue, is the actual reduced efficiency due to the differences in the compatibility factor.