Enhanced thermoelectric performance of ${\mathrm{Mg}}_2{\mathrm{Si}}_{1-x}{\mathrm{Sn}}_x$ codoped with Bi and Cr

Magnesium silicides are favorable thermoelectric materials considering resource abundance and cost. Chromium (Cr) doping in magnesium silicides has not yet been explored. Using first-principles calculations, we have studied the stability of ${\mathrm{Mg}}_2\mathrm{Si}$ with chromium (1.85, 3.7, 5.55, and 6.25% Cr) and tin (12.5 and 50% Sn). Three ${\mathrm{Mg}}_2\mathrm{Si}$ compounds doped with Sn, (Sn + Bi), and (Sn + Bi + Cr) are used to explain doping effects on thermoelectric performance. Notably, Cr behaves nonmagnetically for $\le 2\%\mathrm{Cr}$, after which ferromagnetic ordering is favored ($\le 12.96\%$ Cr), despite its elemental antiferromagnetic state. With alloying of Sn (70.4%), ${\mathrm{Mg}}_2\mathrm{Si}$ remains an indirect-band-gap semiconductor, but adding small amounts of Bi (3.7%) increases the carrier concentration such that electrons occupy conduction bands, making it a degenerate semiconductor. ${\mathrm{Mg}}_2{\mathrm{Si}}_{0.296}{\mathrm{Sn}}_{0.666}{\mathrm{Bi}}_{0.037}$ is found to give the highest thermoelectric figure of merit (ZT) and power factor (PF) at 700 K, i.e., 1.75 and 7.04 $\text{mW}{\text{m}}^{-1}{\text{K}}^{-2}$, respectively. Adding small %Cr decreases ZT and PF to 0.78 and 4.33 $\text{mW}{\text{m}}^{-1}{\text{K}}^{-2}$, respectively. Such a degradation in thermoelectric (TE) performance is attributed to two factors: (i) uniform doping acting as an electron acceptor, decreasing conduction, and (ii) the loss of low-lying conduction band degeneracy with doping, decreasing the Seebeck coefficients. A study of configurations of Cr doping suggests that Cr has a tendency to form clusters inside the lattice, which play a crucial role in tuning the magnetic and TE performance of doped ${\mathrm{Mg}}_2\mathrm{Si}$ compounds.

Figure 1

Primitive cell of ${\mathrm{Mg}}_2\mathrm{Si}$ (left) and its first Brillouin zone (right) showing the high-symmetry lines.

Figure 2

(top) PBE electronic dispersion with orbital character (left) and atom-projected density of states (DOS) (right) and (bottom) HSE06 dispersion. Orbital contribution of individual atoms is given by colored symbols. The Fermi energy is given by the dashed line.

Figure 3

PBE electronic structure of (top) ${\mathrm{Mg}}_2{\mathrm{Si}}_{0.296}{\mathrm{Sn}}_{0.704}$; (middle) ${\mathrm{Mg}}_2{\mathrm{Si}}_{0.296}{\mathrm{Sn}}_{0.666}{\mathrm{Bi}}_{0.037}$; and (bottom) ${\mathrm{Mg}}_{1.981}{\mathrm{Cr}}_{0.019}{\mathrm{Si}}_{0.296}{\mathrm{Sn}}_{0.666}{\mathrm{Bi}}_{0.037}$. Various atomic contributions are shown by different color symbols. Band gap is scaled to the gap value obtained from HSE06 calculations. The dashed line indicates the Fermi energy.

Figure 4

Seebeck coefficient ($S$), power factor ($S^2\sigma$), and ZT vs chemical potential $\mu$ at $T$'s for ${\mathrm{Mg}}_2{\mathrm{Si}}_{0.296}{\mathrm{Sn}}_{0.704}$.

Figure 5

Seebeck coefficient ($S$), power factor ($S^2\sigma$), and ZT vs $\mu$ at $T$'s for ${\mathrm{Mg}}_2{\mathrm{Si}}_{0.296}{\mathrm{Sn}}_{0.666}{\mathrm{Bi}}_{0.037}$.

Figure 6

$S$, $S^2\sigma$, and ZT vs $\mu$ at $T$'s for ${\mathrm{Mg}}_{1.981}{\mathrm{Cr}}_{0.019}{\mathrm{Si}}_{0.296}{\mathrm{Sn}}_{0.666}{\mathrm{Bi}}_{0.037}$.

Figure 7

Temperature dependence of Seebeck coefficients ($S$), power factor ($S^2\sigma$), and ZT for three systems. Here 1, 2, and 3 denote the ${\mathrm{Mg}}_2{\mathrm{Si}}_{0.296}{\mathrm{Sn}}_{0.704}$, ${\mathrm{Mg}}_2{\mathrm{Si}}_{0.296}{\mathrm{Sn}}_{0.666}{\mathrm{Bi}}_{0.037}$, and ${\mathrm{Mg}}_{1.981}{\mathrm{Cr}}_{0.019}{\mathrm{Si}}_{0.296}{\mathrm{Sn}}_{0.666}{\mathrm{Bi}}_{0.037}$, respectively. Optimal carrier concentration for compounds 1, 2, and 3 are $8.93\times {10}^{19}{\text{cm}}^{-3},5.57\times {10}^{20}{\text{cm}}^{-3}$, and $6.25\times {10}^{20}{\text{cm}}^{-3}$ for $n$-type carriers, and $9.05\times {10}^{19}{\text{cm}}^{-3},4.64\times {10}^{20}{\text{cm}}^{-3},$ and $3.87\times {10}^{20}{\text{cm}}^{-3}$ for $p$-type carriers, respectively.

Figure 8

$\sigma$ vs $T$ for ${\mathrm{Mg}}_2{\mathrm{Si}}_{0.296}{\mathrm{Sn}}_{0.704}$, ${\mathrm{Mg}}_2{\mathrm{Si}}_{0.296}{\mathrm{Sn}}_{0.666}{\mathrm{Bi}}_{0.037}$, and ${\mathrm{Mg}}_{1.981}{\mathrm{Cr}}_{0.019}{\mathrm{Si}}_{0.296}{\mathrm{Sn}}_{0.666}{\mathrm{Bi}}_{0.037}$ denoted by $1,2$, and 3, respectively. $\sigma$ values are plotted for carrier concentration ($n$) at which ZT attains a maxima. The $n$ values are mentioned in Table 3.

Figure 9

Relative DFT formation energies of various amounts of Cr in ${\mathrm{Mg}}_2\mathrm{Si}$ in different magnetic ordered states. The reference energy ($E_{\mathrm{ref}}$) for each amount of Cr is chosen to be different from others to keep the most stable magnetic configuration for a particular level of Cr at lowest energy (0 eV). Abbreviations: FM-F, ferromagnetic far; FM-N, ferromagnetic near; AFM-F, antiferromagnetic far; and AFM-N, antiferromagnetic near. A schematic diagram of the near and far configurations for each level of Cr (showing only Cr atoms) is shown at the bottom and top, respectively.