Validity of the rigid band approximation in the study of the thermopower of narrow band gap semiconductors

Theoretical studies of thermoelectric properties using ab initio electronic structure calculations help not only to understand existing experimental data but also to predict new materials which can be potentially good thermoelectrics. However, in these studies it is inevitable to employ some approximations. It is therefore important to verify their reliability. To this end, we have investigated the validity of the rigid band approximation (RBA), commonly used in calculating the thermopower ($S$) in doped (sometimes heavily) narrow band gap semiconductors. We have considered two important systems: half-Heusler HfCoSb and PbTe. We calculate band structures of pure and doped systems [using quasiperiodic approximation (QPA)] by employing the density-functional method. We then use Boltzmann transport theory to calculate the thermopower using both RBA and the band structure with QPA. We find that band structures do not change significantly when isovalent impurities are present excepting in specific cases. However, charged impurities (relevant to the doping case) providing carriers can change the host band structure appreciably. We find that impurities in general remove existing degeneracies which tend to reduce the RBA value of $\mid S\mid$. The reduction is significant in both HfCoSb and PbTe when charged defects are present.

Figure 1

Band structure of HfCoSb with different BZ schemes: (a) FCC, (b) simple cubic (SC), and (c) $2\times 2\times 2$ cubic supercell (CSC). For SC, we show the contribution of Co $d$ orbital (blue triangles) and Hf $d$ orbital (red circle) to the bands. The size of the symbols represents the strength of the contributions. The Fermi level is set to be at zero energy.

Figure 2

Band structure with SC BZ for (a) Hf${}_{0.75}$Zr${}_{0.25}$CoSb, (b) HfCo${}_{0.75}$Ir${}_{0.25}$Sb, (c) HfCoSb${}_{0.75}$Bi${}_{0.25}$. The Fermi level is set to be at zero energy.

Figure 3

Band structure with SC BZ for (a) Hf${}_{0.75}$Y${}_{0.25}$CoSb, (b) HfCo${}_{0.75}$Fe${}_{0.25}$Sb, (c) HfCoSb${}_{0.75}$Ge${}_{0.25}$, (d) HfCoSb${}_{0.75}$Sn${}_{0.25}$, (e) HfCoSb${}_{0.75}$Pb${}_{0.25}$. The Fermi level is set to be at zero energy.

Figure 4

Band structure using $2\times 2\times 2$ CSC BZ for HfCoSb${}_{1-x}$Sn${}_x$ with (a) $x=0.03125$, (b) $x=0.125$, and (c) $x=0.25$. The Fermi level is set to be at zero energy.

Figure 5

Calculated thermopower of (a) isovalent impurity, (b) $p$ doping along with that of pure HfCoSb as a function of temperature. The thermopower of pure and isovalent-impurity HfCoSb is obtained using RBA. The corresponding carrier concentration is $4.5\times {10}^{21}$/cm${}^3$.

Figure 6

Band structure of (a) PbTe with FCC BZ, (b) PbTe with CSC BZ, (c) K-doped PbTe with CSC BZ, and (d) Na-doped PbTe with CSC BZ. The Fermi level is set to be at zero energy.

Figure 7

Comparison of the thermopower with and without impurity in PbTe. It is calculated without SOI using a $2\times 2\times 2$ supercell. The corresponding carrier concentration is $4.4\times {10}^{20}$/cm${}^3$.