High-throughput exploration of alloying as design strategy for thermoelectrics

We explore a material design strategy to optimize the thermoelectric power factor. The approach is based on screening the band structure changes upon a controlled volume change. The methodology is applied to the binary silicides and germanides. We first confirm the effect in antifluorite ${\mathrm{Mg}}_2\mathrm{Si}$ and ${\mathrm{Mg}}_2\mathrm{Ge}$ where an increased power factor by alloying with ${\mathrm{Mg}}_2\mathrm{Sn}$ is experimentally established. Within a high-throughput formalism we identify six previously unreported binaries that exhibit an improvement in their transport properties with volume. Among these, hexagonal ${\mathrm{MoSi}}_2$ and orthorhombic ${\mathrm{Ca}}_2\mathrm{Si}$ and ${\mathrm{Ca}}_2\mathrm{Ge}$ have the highest increment in $zT$ with volume. We then perform supercell calculations on special quasirandom structures to investigate the possibility of obtaining thermodynamically stable alloy systems which would produce the necessary volume changes. We find that for ${\mathrm{Ca}}_2\mathrm{Si}$ and ${\mathrm{Ca}}_2\mathrm{Ge}$ the solid solutions with the isostructural ${\mathrm{Ca}}_2\mathrm{Sn}$ readily forms even at low temperatures.

Figure 1

Schematic illustration of volumetric band alignment for a $n$-type material. VBM stands for the valence band minimum.

Figure 2

Flowchart of the HT procedure employed to investigate the candidates for volumetric band engineering. The criterion for good thermoelectrics, i.e., $zT>0.4$, within our scheme is checked at $T=600$ K and $n=2\times {10}^{20}{\mathrm{cm}}^{-3}$ with $zT$ evaluated from Eq. (4).

Figure 3

Volumetric band alignment in ${\mathrm{Mg}}_{2}X$. (a) and (b) respectively show the change in $PF\left(V_{\mathrm{opt}}\right)-PF\left(V_0\right)$ (in red circles) and $S\left(V_{\mathrm{opt}}\right)-S\left(V_0\right)$ (blue triangles), as a function of temperature at $n=2\times {10}^{20}{\mathrm{cm}}^{-3}$. Furthermore, in both the graphs the dashed lines correspond to $n\text{doping}$ while the bold line represents the data for the $p-\mathrm{doping}$ scenario. (c) and (d) respectively show the band structures for ${\mathrm{Mg}}_2\mathrm{Si}$ and ${\mathrm{Mg}}_2\mathrm{Ge}$, at $V_0$ (red) and $V_{\mathrm{opt}}$ (black). The insets represent the zoomed view of the conduction band minimum. The conventions used in this figure will be used in the forthcoming figures presenting similar results for other candidates in this paper.

Figure 4

Formation energy vs volume for ${\mathrm{Mg}}_2{X}_{1-x}{Y}_x$ alloys. Here $X,Y$ are Si, Ge, and Sn. The insert shows the miscibility gap calculated by minimizing Eq. (6).

Figure 5

$z{T}_{\mathrm{opt}}$ vs $z{T}_{\mathrm{o}}$ for all silicides (top) and germanides (bottom) at doping $n=2\times {10}^{20}{\mathrm{cm}}^{-3}$ and $T=600$ K. Furthermore, the scatter points are colored based on the distance from the convex hull, $\mathrm{\Delta }{E}_{\mathrm{h}}^{\mathrm{Sn}}$, of the corresponding Sn compound with the same structure as the host binary. For example, in the case of the ${\mathrm{Ca}}_9{\mathrm{Ge}}_5$ compound, $\mathrm{\Delta }{E}_{\mathrm{h}}$ of the corresponding isostructural ${\mathrm{Ca}}_9{\mathrm{Sn}}_5$ is used for the color code, and so on. The candidates that exhibit VBA, $\frac{z{T}_{\mathrm{opt}}}{z{T}_0}>1.1$, or have a large value of $z{T}_0$ are labeled. Also, if a compound crystallizes in more than one phase, the corresponding structure is also indicated with a prefix o- for orthorhombic, t- for tetragonal, and af- for antifluorite phase.

Figure 6

Formation energy vs volume for ${\mathrm{Ca}}_2{X}_{1-x}{Y}_x$ solid solutions ($X$ and $Y$ are Si, Ge, and Sn).

Figure 7

Ordered alloy structure is shown here for a $2\times 2\times 2$ supercell. Here the blue balls are Ca atoms, the grey balls are $X$ atoms and the magenta ones are $Y$ atoms. Note that an unit cell contains 12 atoms, i.e., ${\mathrm{Ca}}_8{X}_2{Y}_2$ composition. The shortest crystallographic axis, b, is perpendicular to the plane. This structure is related to the ${\mathrm{Co}}_2\mathrm{Si}$ (or TiNiSi) structure type with space group $Pnma$ [41].

Figure 8

The temperature dependence of $PF\left(V_{\mathrm{opt}}\right)-PF\left(V_0\right)$ and $S\left(V_{\mathrm{opt}}\right)-S\left(V_0\right)$ for ${\mathrm{Ca}}_2\mathrm{Si}$ and ${\mathrm{Ca}}_2\mathrm{Ge}$ respectively in (a) and (b). Furthermore, in (c) and (d) we depict the corresponding band structures ($V_{\mathrm{opt}}$ in black and $V_{\mathrm{o}}$ in red; (c) for ${\mathrm{Ca}}_2\mathrm{Si}$ and (d) for ${\mathrm{Ca}}_2\mathrm{Ge}$). The orthorhombic structure for these compounds are thermodynamically stable. Both the $n-\mathrm{doped}$ compounds exhibit an increase in $PF$ and $S$ that can be justified by the increased DOS around the CB manifold. Note also the close band alignment of the first two CBs at the $\mathrm{\Gamma }$ point in BZ, for ${\mathrm{Ca}}_2\mathrm{Ge}$.

Figure 9

The BZ of orthorhombic ${\mathrm{Ca}}_2\mathrm{Si}$ at $V_{\mathrm{opt}}$, illustrating the electron pockets (in blue) at the high-symmetry $T$ and $\mathrm{\Gamma }$ points as well as the low-symmetry pocket around $Y$. The electron pocket around the $Y$ point does not exist at $V_0$.

Figure 10

$T$ dependence of $PF\left(V_{\mathrm{opt}}\right)-PF\left(V_0\right)$ and $S\left(V_{\mathrm{opt}}\right)-S\left(V_0\right)$ for the hexagonal ${\mathrm{Ca}}_9{\mathrm{Ge}}_5$ (top). The corresponding band structures at the two volumes are shown in the bottom panel. For ${\mathrm{Ca}}_9{\mathrm{Ge}}_5$, there is an opening of the band gap with volume change that is responsible for the increase in its $PF$ and $S$.

Figure 11

We depict the temperature dependence of $PF\left(V_{\mathrm{opt}}\right)-PF\left(V_0\right)$ and $S\left(V_{\mathrm{opt}}\right)-S\left(V_0\right)$ for the $\beta -{\mathrm{MoSi}}_2$ in (a). In (b), we present the band structures at the two volumes ($V_{\mathrm{opt}}$ in black and $V_{\mathrm{o}}$ in red) for $\beta -{\mathrm{MoSi}}_2$. $n-\mathrm{type}{\mathrm{MoSi}}_2$ shows an appreciable increase in both $PF$ and $S$ at $V_{\mathrm{opt}}$ up to $T=350$ K. This increase can be explained by the lowering and aligning of the first CB manifold along multiple directions in its hexagonal BZ.

Figure 12

The BZ of $\beta -{\mathrm{MoSi}}_2$ illustrating the Fermi surface at $V_{\mathrm{opt}}$. We observe electronic contributions connecting several low symmetry points across the BZ in the Fermi surface, which is responsible for the VBA effect [also see Fig. 11].