Electronic structure of Ga-, In-, and Tl-doped PbTe: A supercell study of the impurity bands

The physics of deep defect states associated with group-III (Ga, In, Tl) impurities in PbTe has been of great interest over the last several decades. Three different models have been proposed to understand interesting and unusual properties exhibited by these impurities. These are “impurity level” model, “mixed-valence” model, and “autocompensation” model. Recent studies that were carried out using ab initio density-functional theory and supercell models give a detailed microscopic picture of the deep defect states associated with these impurities. The computed single-particle electronic density of states does not support the mixed-valence model where the impurities are supposed to be in $3^{+}$ and $1^{+}$ charge states, rather it is in favor of the impurity level model. In this paper, we present a detailed analysis of the impurity bands, impurity levels in the dilute limit, and the hybridization between the impurity level and the host PbTe bands. We find that the impurity level overlaps the conduction band (In), is in the gap (Ga), and overlaps the valence band (Tl). The possibility of the autocompensation model where impurities in charge states $1^{+}$ and $1^{-}$ coexist (a different type of mixed valency) is briefly discussed.

Figure 1

(Color online) Band structure of undoped PbTe in different Brillouin zones: (a) fcc primitive cell (2 atoms/cell; $a=6.55\text{Å}$), (b) sc unit cell (8 atoms/cell; $a=6.55\text{Å}$), (c) sc supercell (64 atoms/cell; $a=13.1\text{Å}$), and (d) sc supercell (64 atoms/cell; $a=13.1\text{Å}$) with SOI included. The Fermi level (at 0 eV) is set to the middle of the band gap.

Figure 2

(Color online) (a) Band structures of In-doped PbTe in calculations using relaxed (solid curves) and unrelaxed (dash-dotted curves) structures, and (b) a blowup of the band structure near the $\Gamma$ point. The Fermi level for the relaxed structure (at 0 eV) is set to the highest occupied states, whereas that for the unrelaxed one it was shifted by $-0.180\text{eV}$ so that the valence bands match; SOI was not included.

Figure 3

(Color online) Partial charge densities [in the (100) plane] associated with (a) the HDS and (b) the DDS bands of In-doped PbTe. The results were obtained in calculations using the $\left(2\times 2\times 2\right)$ supercell and without SOI. The isolines in the two figures are plotted with the same scale. Only the impurity atom (large balls at the centers of the figures) and Te (small balls) atoms which are neighbors of the impurity are shown.

Figure 4

(Color online) Band structures in the sc BZ of PbTe doped with Ga, In, and Tl impurities in calculations using the $\left(2\times 2\times 2\right)$ supercell and with [(a)–(c)] and without [$\left({\text{a}}^{\prime }\right)$–$\left({\text{c}}^{\prime }\right)$] SOI. The dash-dotted curves are the DDS bands. The Fermi level (at 0 eV) is set to the highest occupied states.

Figure 5

(Color online) Band structures in the sc BZ of PbTe doped with Ga, In, and Tl impurities obtained in calculations using the $\left(3\times 3\times 3\right)$ supercell and without [(a)–(c)] and with [$\left({\text{a}}^{\prime }\right)$–$\left({\text{c}}^{\prime }\right)$] SOI. The energy levels near the conduction-band bottom (at the $\Gamma$ point) are also given in $\left({\text{a}}^{\prime }\right)$–$\left({\text{c}}^{\prime }\right)$ with their respective degeneracies (including spins) in the curly brackets. Note that the zero of energy (the Fermi level) is ill-defined in the results presented in $\left({\text{a}}^{\prime }\right)$–$\left({\text{c}}^{\prime }\right)$ because the calculations scanned only a very small part of the BZ.

Figure 6

(Color online) Band splittings due to impurity-impurity interaction in group-III-doped PbTe. The results were obtained in calculations using the $\left(2\times 2\times 2\right)$ supercell with an impurity pair: (In,In) [in (a) and $\left({\text{a}}^{\prime }\right)$], (Ga,Ga) [in (b) and $\left({\text{b}}^{\prime }\right)$], and (Tl,Tl) [in (c) and $\left({\text{c}}^{\prime }\right)$]. The two impurity atoms in a pair are either the second or the fifth nearest neighbors of one another. The dash-dotted curves are the DDS bands. SOI that was included and the Fermi level (at 0 eV) is set to the highest occupied states.

Figure 7

(Color online) Band structure and partial charge densities associated with the HDS bands [No. 63 and No. 64] and the DDS bands [Nos. 159 and 160] of PbTe doped with (In,In) where the two In atoms in the pair are the second-nearest neighbors of one another. The results were obtained in calculations using the $\left(2\times 2\times 2\right)$ supercell and without SOI. The isolines in the figures were plotted with the same scale. Only the In impurity atoms (large balls) and Te (small balls) atoms which are neighbors of the impurity atoms are shown.