Quasiparticle band structures and thermoelectric transport properties of Mg$_{2}$Si, Mg$_{2}$Ge, and Mg$_{2}$Sn

Mg$_{2}$Si, Mg$_{2}$Ge, and Mg$_{2}$Sn are narrow-gap semiconductors with large Seebeck coefficients and favorable thermoelectric properties. We calculated the quasiparticle band structures of Mg$_{2}$Si, Mg$_{2}$Ge, and Mg$_{2}$Sn using density functional and many-body perturbation theory in the GW approximation. The calculated band gaps are in good agreement with experiment. The inclusion of semicore states in the valence is necessary to obtain accurate band gaps for Mg$_{2}$Ge and Mg$_{2}$Sn. We used the maximally localized Wannier function method and the Boltzmann transport equation in the constant relaxation-time approximation to determine the Seebeck coefficient and the electrical and carrier thermal conductivities. We discuss the importance of quasiparticle corrections to accurately determine the Seebeck coefficients at high temperatures.