Thermal conductivity of bulk and nanowire Mg<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow/><mml:mn>2</mml:mn></mml:msub></mml:math>Si<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow/><mml:mi>x</mml:mi></mml:msub></mml:math>Sn<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow/><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>x</mml:mi></mml:mrow></mml:msub></mml:math>alloys from first principles

The lattice thermal conductivity ($\ensuremath{\kappa}$) 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 $\ensuremath{\kappa}$ 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, $\ensuremath{\kappa}$ of Mg${}_{2}$Si${}_{x}$Sn${}_{1\ensuremath{-}x}$ is less sensitive to nanostructuring size effects than Si${}_{x}$Ge${}_{1\ensuremath{-}x}$, but more sensitive than PbTe${}_{x}$Se${}_{1\ensuremath{-}x}$. This suggests that further improvement of Mg${}_{2}$Si${}_{x}$Sn${}_{1\ensuremath{-}x}$ as a nontoxic thermoelectric may be possible.