Electronic structure and thermoelectric properties of Sb-based semiconducting half-Heusler compounds

Electronic structure and transport properties of four Sb-based semiconducting half-Heusler compounds, $\mathit{MA}$Sb, where $\mathcal{M}=\mathrm{Hf}$ or Zr and $\mathcal{A}=\mathrm{Co}$ or Ir are studied using density-functional theory and Boltzmann transport equation in constant relaxation time ($\tau$) approximation. We find that substituting Hf with Zr does not change the band structures of these systems significantly. In contrast, replacing Co with Ir leads to drastic changes in their electronic structures. The valence band maximum occurs at the $L$ point in the Co compounds while it is at the $\Gamma$ point in the Ir compounds. The position and hybridization of an $s$-like conduction band vis-à-vis the hybridized $d$ bands of Co(Ir) determines the nature of the conduction bands near the band gap region. In addition, there is a direct band gap at the $\Gamma$ point in HfIrSb, whereas in the other three compounds, the band gap is indirect, either between $\Gamma$ and $X$ or between $L$ and $X$ points. The Co compounds usually give large thermopowers for both $p$ and $n$ dopings. However, ZrIrSb, due to its interesting conduction band structure, gives the best $n$-type thermopower at high temperatures. We discuss in detail how the subtle changes in the electronic structure near the band gap affects $S$, $\sigma /\tau$, and the power factor.

Figure 1

Band structures near the Fermi energy for (a) HfCoSb, (b) ZrCoSb, (c) HfIrSb, and (d) ZrIrSb along W-L-$\Gamma$-X-W-K. The Fermi level is set to be zero energy.

Figure 2

The contribution of each atom in the band structure of (Hf,Zr)(Co,Ir)Sb compounds: (red circles)the $d$-orbital of Hf/Zr, (blue triangles) the $d$-orbital of Co/Ir, (pink stars) the $p$ orbital of Sb, and (green diamonds) the $s$ orbital of Sb. The size of the symbols represents the strength of the contribution. The Fermi level is set to be zero energy.

Figure 3

The projected density of states (PDOS) as a function of energy. The value in parentheses is the number of degeneracy. $E_f$ denotes the Fermi level. Each symbol represents different orbitals: (square) $s$ orbital, (circle) $p$ orbital, (triangle)$e_g$ orbital, and (diamond)$t_{2g}$ orbital. Each color represents a different element: (red) Hf/Zr, (blue) Co/Ir, and (green) Sb.

Figure 4

The density of states (DOS) with the Fermi energy to be zero.

Figure 5

Seebeck coefficient as a function of carrier concentration at 300 K for (a) hole doping and (b) electron doping.

Figure 6

Transport properties for HfCoSb (red circle), ZrCoSb (blue diamond), HfIrSb (pink square), and ZrIrSb (sky blue triangle) as a function of temperature at the hole-concentration $n_h=2\times {10}^{20}$/cm${}^3$ (left panel) and $n_h=4\times {10}^{21}$/cm${}^3$ (right panel). The units for $\mu$, $S$, $\sigma /\tau$, and $S^2\sigma /\tau$ are eV, $\mu$V/K, 10${}^{17}\hspace{0.5em}{\Omega }^{-1}$cm${}^{-1}$s${}^{-1}$, and 10${}^{14}\hspace{0.5em}\mu$W cm${}^{-1}$ ${\mathrm{K}}^{-2}$ ${\mathrm{s}}^{-1}$, respectively.

Figure 7

Transport properties for HfCoSb (red circle), ZrCoSb (blue diamond), HfIrSb (pink square), and ZrIrSb (sky blue triangle) as a function of temperature at the hole-concentration $n_e=2\times {10}^{20}$/cm${}^3$(left panel) and $n_e=4\times {10}^{21}$/cm${}^3$ (right panel). The units for $\mu$, $S$, $\sigma /\tau$, and $S^2\sigma /\tau$ are eV, $\mu$V/K, 10${}^{17}\hspace{0.5em}{\Omega }^{-1}$cm${}^{-1}$s${}^{-1}$, and 10${}^{14}\hspace{0.5em}\mu$W cm${}^{-1}$ ${\mathrm{K}}^{-2}$ ${\mathrm{s}}^{-1}$, respectively.

Figure 8

Fermi-level change at the carrier concentrations of $n=2\times {10}^{20}$/cm${}^3$ (red lines) and $n=4\times {10}^{21}$/cm${}^3$ (blue lines) by (a) hole doping and (b) electron doping.

Figure 9

Seebeck coefficient with 10% doping in ZrCoSb. Theoretical values are shown with lines and experimental data with rectangles for low temperatures from Xia et al. (Ref. 14) and triangles for high temperatures from Sekimoto et al. (Ref. 18).