Thermoelectric Properties of ${\mathrm{Mg}}_2(\mathrm{Ge},\mathrm{Sn})$: Model and Optimization of $ZT$

We report an investigation of the thermoelectric properties of ${\mathrm{Mg}}_2{\mathrm{Ge}}_{1-x}{\mathrm{Sn}}_x$ solid solution with $x=0.5$ using models based on first-principles calculations and experimental data. The model gives transport properties including the figure of merit $ZT$ as functions of carrier concentration and temperature. The model for the $n$-type predicts high $ZT$ at optimized doping, and suggests that the $ZT$ value can exceed 2 at $T=1000\mathrm{K}$.

Figure 1

Crystal structures of ${\mathrm{Mg}}_2{\mathrm{Ge}}_{1-x}{\mathrm{Sn}}_x$ (a) and doubled cell (b) with formula ${\mathrm{Mg}}_4\mathrm{GeSn}$. Solid line indicates the unit cell.

Figure 2

Calculated band structure of ${\mathrm{Mg}}_2{\mathrm{Ge}}_{0.5}{\mathrm{Sn}}_{0.5}$. Energy zero is set at the valence-band maximum.

Figure 3

Total and projected density of states of ${\mathrm{Mg}}_2{\mathrm{Ge}}_{0.5}{\mathrm{Sn}}_{0.5}$ with (a) showing the comparison of two different crystal structures and (b) the projected density of states of the structure used in this study. Density of states are shown per ${\mathrm{Mg}}_2{\mathrm{Ge}}_{0.5}{\mathrm{Sn}}_{0.5}$ formula unit (f.u.).

Figure 4

Calculated isoenergy surfaces of ${\mathrm{Mg}}_2{\mathrm{Ge}}_{0.5}{\mathrm{Sn}}_{0.5}$ near the VBM and CBM.

Figure 5

Calculated $S(T,n/p)$ for both $n$- and $p$-type ${\mathrm{Mg}}_2{\mathrm{Ge}}_{0.5}{\mathrm{Sn}}_{0.5}$. Solid horizontal line at $200\mu \mathrm{V}/\mathrm{K}$ represents the limitation of Seebeck coefficients for good TE materials.

Figure 6

Calculated electrical conductivity for both (a) $n$- and (b) $p$-type ${\mathrm{Mg}}_2{\mathrm{Ge}}_{0.5}{\mathrm{Sn}}_{0.5}$.

Figure 7

Calculated power factor for both $n$- (a) and $p$-type (b) ${\mathrm{Mg}}_2{\mathrm{Ge}}_{0.5}{\mathrm{Sn}}_{0.5}$.

Figure 8

Calculated $L_0$ versus temperature and doping concentration for $p$-type ${\mathrm{Mg}}_2{\mathrm{Ge}}_{0.5}{\mathrm{Sn}}_{0.5}$.

Figure 9

Calculated $ZT$ versus $T$ for both $n$- (a) and $p$-type (b) ${\mathrm{Mg}}_2{\mathrm{Ge}}_{0.5}{\mathrm{Sn}}_{0.5}$ using the model discussed in the text.

Figure 10

Calculated $ZT$ versus doping concentration for both $n$- and $p$-type ${\mathrm{Mg}}_2{\mathrm{Ge}}_{0.5}{\mathrm{Sn}}_{0.5}$ using the model discussed in the text.

Figure 11

Calculated $ZT$ versus doping concentration for $n$-type ${\mathrm{Mg}}_2{\mathrm{Ge}}_{0.25}{\mathrm{Sn}}_{0.75}$.

Figure 12

Calculated $ZT$ versus doping concentration for $n$-type ${\mathrm{Mg}}_2{\mathrm{Ge}}_{0.5}{\mathrm{Sn}}_{0.5}$ using the fitted $\tau \prime$ (see text).