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 ($μ_{e}$), 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 $μ_{e}$ 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 $μ_{e}$ closely matches experimental $μ_{e}$ 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 ($κ_{ph}$) 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 $κ_{ph}$ through nanostructuring and mass-difference scattering.

Figures (10)

• Figure 1: (a) DFT, ZPR correction in band gap at 0 K and temperature dependence of indirect (for Mg$_{2}$Si) and direct (for Ca$_{2}$Si) band-gaps from on-the-mass-shell (OTMS) [solid line] and linearized-QP-equation (LQE) [dashed line] methods. (b) The absolute values of coefficients $|a|$ and $|b|$ are obtained from non-linear fittings of Varshni equation of $\Delta$E$_{\mathrm{g}}$ = $a$T$^{2}$/(T+$b$).
• Figure 2: Convergence of the electron mobility at 300 K with respect to the Brillouin zone sampling of electron (k) and phonon (q) wave vectors, using methods based on SERTA and MRTA [with and without using double grid (DG)], and the iterative solution of the BTE (IBTE).
• Figure 3: Temperature-dependent electron mobility of (a) Mg$_{2}$Si and (b) Ca$_{2}$Si, calculated using the linearized (SERTA, MRTA) and iterative BTE (IBTE). Experimental data for Mg$_{2}$Si by Morris et al.morris1958semiconducting, represented by triangles in panel (a), are included for comparison.
• Figure 4: Seebeck coefficient as a function of temperature at different electron doping densities (n$_{\mathrm{e}}$) for $n$-type Mg$_{2}$Si and Ca$_{2}$Si, calculated using [(a), (c)] the CRTA and [(b), (d)] the MRTA (solid lines) and SERTA (dashed lines). Experimental data (dotted lines) for Mg$_{2}$Si in [(a), (b)] for different n$_{\mathrm{e}}$ values are shown: 9 $\times$$10^{17}$ cm$^{-3}$ (Wang $et$$al.$wang2012theoretical), 2.2 $\times$$10^{18}$ cm$^{-3}$ (Jung $et$$al.$jung2011solid), 1.8 $\times$$10^{19}$ and 1.1 $\times$$10^{20}$ cm$^{-3}$ (Tani $et$$al.$tani2005thermoelectric), 4 $\times$$10^{19}$ cm$^{-3}$ (Bux $et$$al.$bux2011mechanochemical), and 1.1 $\times$$10^{20}$ cm$^{-3}$ (Akasaka $et$$al.$akasaka2008thermoelectric).
• Figure 5: Calculated temperature-dependent electrical conductivity ($\sigma$) at similar doping densities, using all three RTAs and the same experimental references for $\sigma$ data (shown as dashed-dotted-dotted lines) as in Fig. \ref{['fig:sbk']}.