Band gap renormalization, carrier mobility, and transport in Mg$_{2}$Si and Ca$_{2}$Si: \textit{Ab initio} scattering and Boltzmann transport equation study

Abstract

We perform first-principles electron-phonon interaction (EPI) calculations based on many-body perturbation theory to study the temperature-dependent band-gap and charge-carrier transport properties for Mg$_{2}$Si and Ca$_{2}$Si using the Boltzmann transport equation (BTE) under different relaxation-time approximations (RTAs). For a PBE band gap of 0.21 (0.56) eV in Mg$_{2}$Si (Ca$_{2}$Si), a zero-point renormalization correction of 29-33 (37-51) meV is obtained using various approaches, while the gap at 300 K is 0.15-0.154 (0.46-0.5) eV. The electron mobility ($\mu_{e}$), with a detailed convergence study at 300 K, is evaluated using linearized (self-energy and momentum RTA, or SERTA and MRTA) and iterative BTE (IBTE) solutions. At 300 K, the $\mu_{e}$ values are 351 (100), 573 (197), and 524 (163) cm$^{2}V^{-1}s^{-1}$ from SERTA, MRTA, and IBTE, respectively, for Mg$_{2}$Si (Ca$_{2}$Si). SERTA (MRTA) provides results in better agreement with IBTE at higher (lower) temperatures, while SERTA-derived $\mu_{e}$ closely matches experimental $\mu_{e}$ values for Mg$_{2}$Si. Thermoelectric (TE) transport coefficients significantly influenced by the choice of RTA, with SERTA and MRTA yielding improved agreement with experimental results compared to constant RTA (CRTA) for Mg$_{2}$Si over an electron concentration range of $10^{17}$ to $10^{20}$ cm$^{-3}$. The lattice thermal conductivity ($\kappa_{ph}$) at 300 K due to phonon-phonon interactions is estimated to be 22.7 (7.2) W m$^{-1}K^{-1}$ for Mg$_{2}$Si (Ca$_{2}$Si). The highest calculated figure of merit (zT) under CRTA is 0.35 (0.38), which decreases to 0.08 (0.085) when EPI is included using MRTA. This study clearly identifies the critical role of EPI in accurate transport predictions of TE silicides. Finally, we explore strategies to enhance zT by reducing $\kappa_{ph}$ through nanostructuring and mass-difference scattering.