Thermal conductivity of bulk and nanowire Mg${}_2$Si${}_x$Sn${}_{1-x}$ alloys from first principles

The lattice thermal conductivity ($\kappa$) of the thermoelectric materials, Mg${}_2$Si, Mg${}_2$Sn, and their alloys, are calculated for bulk and nanowires, without adjustable parameters. We find good agreement with bulk experimental results. For large nanowire diameters, size effects are stronger for the alloy than for the pure compounds. For example, in 200 nm diameter nanowires $\kappa$ is lower than its bulk value by 30$\%$, 20$\%$, and 20$\%$ for Mg${}_2$Si${}_{0.6}$Sn${}_{0.4}$, Mg${}_2$Si, and Mg${}_2$Sn, respectively. For nanowires less than 20 nm thick, the relative decrease surpasses 50$\%$, and it becomes larger in the pure compounds than in the alloy. At room temperature, $\kappa$ of Mg${}_2$Si${}_x$Sn${}_{1-x}$ is less sensitive to nanostructuring size effects than Si${}_x$Ge${}_{1-x}$, but more sensitive than PbTe${}_x$Se${}_{1-x}$. This suggests that further improvement of Mg${}_2$Si${}_x$Sn${}_{1-x}$ as a nontoxic thermoelectric may be possible.

Figure 1

Phonon dispersion for Mg${}_2$Si along different symmetry lines. The dispersion calculated using density functional perturbation theory with an $8\times 8\times 8$ q-point grid is given by the solid blue lines. The orange lines denote the phonon dispersion calculated using the real-space force constant approach with a $5\times 5\times 5$ supercell. Experimental results from Hutchings et al. (Ref. 31) taken at room temperature are denoted by black squares.

Figure 2

The Mg${}_2$Sn phonon dispersion is shown along different symmetry lines. The solid blue lines denote the dispersion calculated using density functional perturbation theory with an $8\times 8\times 8$ q-point grid. The phonon dispersion calculated using the real-space force constant approach with a $5\times 5\times 5$ supercell is given by the solid orange lines. Experimental results from Kearney et al. (Ref. 32) taken at room temperature are denoted by black squares. Experimental errors are comparable to the size of the black squares.

Figure 3

Three-phonon scattering rates for the LA branch of Mg${}_2$Si along the $\Gamma \text{−}L$ direction in the Brillouin zone calculated by using third-order IFCs obtained with the quantum espresso package (solid lines) and siesta package (dashed lines). The low-frequency scattering rates follow well the expected ${\omega }^2$ dependence.

Figure 4

Calculated lattice $\kappa$ versus temperature for (a) Mg${}_2$Si and (b) Mg${}_2$Sn, compared with experimental data of Martin et al. (Ref. 37), Tani et al. (Ref. 38), Yang et al. (Ref. 39), Akasaka et al. (Ref. 40), Nemoto et al. (Ref. 41), and Chen et al. (Ref. 42). Data of Martin et al. are the total thermal conductivity and the other experimental data are the extracted lattice thermal conductivity.

Figure 5

Solid lines: Normalized cumulative thermal conductivity of bulk Mg${}_2$Si, Mg${}_2$Sn, and Mg${}_2$Si${}_{0.6}$Sn${}_{0.4}$ at room temperature, as a function of the mean-free path. Dashed lines: Room temperature thermal conductivities of Mg${}_2$Si, Mg${}_2$Sn, and Mg${}_2$Si${}_{0.6}$Sn${}_{0.4}$ nanowires as a function of diameter, normalized by their corresponding bulk values.

Figure 6

Room temperature $\kappa$ of Mg${}_2$Si${}_x$Sn${}_{1-x}$ as a function of $x$ for both bulk materials (solid line) and 20 nm nanowires along [001] growth direction (dashed line), compared with experimental data taken from the Thermoelectrics Handbook (Ref. 4), Zhang et al. (Ref. 5), Chen et al. (Ref. 42), and Tani et al. (Ref. 38).