Band gap renormalization, carrier mobility, and transport in Mg 2 Si and Ca 2 Si : Ab initio scattering and Boltzmann transport equation study

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 ${\mathrm{Mg}}₂\mathrm{Si}$ and ${\mathrm{Ca}}₂\mathrm{Si}$ using the Boltzmann transport equation (BTE) under different relaxation-time approximations (RTAs). For a PBE band gap of $\ensuremath{\sim}0.21$ (0.56) eV in ${\mathrm{Mg}}₂\mathrm{Si}$ (${\mathrm{Ca}}₂\mathrm{Si}$), a zero-point renormalization correction of $\ensuremath{\sim}29--33$ (37--51) meV is obtained using various approaches, while the gap at 300 K is $\ensuremath{\sim}0.15--0.154$ (0.46--0.5) eV. The electron mobility (${\ensuremath{\mu}}_e$), 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 ${\ensuremath{\mu}}_e$ values are $\ensuremath{\sim}351$ (100), 573 (197), and 524 (163) ${\text{cm}}²\phantom{\rule{0.16em}{0ex}}{\text{V}}^{\ensuremath{-}1}\phantom{\rule{0.16em}{0ex}}{\text{s}}^{\ensuremath{-}1}$ from SERTA, MRTA, and IBTE, respectively, for ${\mathrm{Mg}}₂\mathrm{Si}$ (${\mathrm{Ca}}₂\mathrm{Si}$). These respective values at 900 K decrease to $\ensuremath{\sim}33$ (11), 48 (21), and 28 (11) ${\text{cm}}²\phantom{\rule{0.16em}{0ex}}{\text{V}}^{\ensuremath{-}1}\phantom{\rule{0.16em}{0ex}}{\text{s}}^{\ensuremath{-}1}$. SERTA (MRTA) provides results in better agreement with IBTE at higher (lower) temperatures, while SERTA-derived ${\ensuremath{\mu}}_e$ closely matches experimental ${\ensuremath{\mu}}_e$ values for ${\mathrm{Mg}}₂\mathrm{Si}$. All electrical parameters related to thermoelectrics are significantly influenced by the choice of RTA, with SERTA and MRTA yielding improved agreement with experimental results compared to constant RTA (CRTA) for ${\mathrm{Mg}}₂\mathrm{Si}$. Finally, we obtain the figure of merit under various modeling conditions, in which several promising strategies such as nanostructuring (phonon-boundary scattering) and mass-difference scattering are employed to suppress the lattice thermal conductivity computed within phonon-phonon interactions. This study clearly identifies the critical role of different scattering mechanisms (mainly EPI) in accurate transport predictions of thermoelectric silicides.