Calculated electronic structure of Pb${}_{1-x}$Mn${}_x$Te $(0⩽x<11\%)$: The role of $L$ and $\Sigma$ valence band maxima

Density functional theory (DFT) calculations of the electronic structure of Pb${}_{1-x}$Mn${}_x$Te were performed using the OpenMX package with fully relativistic pseudopotentials. Because of the underestimation of the band gap in DFT, “bare” DFT calculations give a wrong order of bands at the $L$ point. Correct band structures of both PbTe and Pb${}_{1-x}$Mn${}_x$Te  as well as correct pressure coefficients of the fundamental gap are obtained by tuning the strength of the spin-orbit coupling for $6p$(Pb) orbitals. The results confirm the critical role of the valence band maximum at the $\Sigma$ point in the transport properties of $p$-PbTe and $p$-Pb${}_{1-x}$Mn${}_x$Te. With the increasing content of Mn in Pb${}_{1-x}$Mn${}_x$Te  the band gap at $L$ increases while the indirect $L$-$\Sigma$ band decreases. The energies of $L$ and $\Sigma$ maxima are equal for $9\%$ of Mn. These results confirm previous empirical interpretations of experimental data. The role of internal distortions due to the lattice mismatch and the influence of Mn on spin properties of free carriers are analyzed.

Figure 1

PbTe band structure for two different strength of the spin-orbit coupling for Pb $6p$ electrons, see text, and a two-atom primitive cell. Note that the two band structures are almost identical. The $\Sigma$ maxima are denoted by arrows. The horizontal bars schematically show the contributions of different atomic orbitals to the wave functions at the L point for valence (VB) and conduction (CB) bands.

Figure 2

(a)–(c) The dependence of bands’ order on the spin-orbit coupling strength. Motion of the bands’ extrema induced by external pressure for the (d) correct and the (e) inverted bands’ order.

Figure 3

Energy bands along the $\Gamma -X-R-M-\Gamma$ path for 216-atom supercells containing from 0 to 12 Mn atoms. The $R$ point of the supercell Brillouin zone corresponds to the $L$ point of the rock salt PbTe. The lines connecting band extrema at $R$ are for the eye guide. For the two lowest Mn concentrations we show positions of the bands corresponding to $\Sigma$ band in the rock salt lattice. For higher Mn concentrations the top of this band is at the $\Gamma$ point.

Figure 4

Dependencies of the direct $L$-$L$ (circles) and indirect $L$-$\Sigma$ (diamonds) energy gaps on number of Mn atoms in the supercell. Filled points—energy gaps for Mn-dependent lattice constant, open points—for the lattice constant fixed to 6.46 Å.

Figure 5

PbMnTe band structure along $\Gamma -X-R-M-\Gamma$ path for three different distributions of four Mn atoms in the 216 atom supercell.

Figure 6

Dependence of the electronic band structure along the $\Gamma -X-R-M-\Gamma$ path for 216 atom supercells containing four Mn atoms on the deformation $\delta$. The lines connecting tops of the bands are for eye guide.

Figure 7

Dependence of the electronic band structure along $\Gamma -X-R-M-\Gamma$ path for 216 atom supercells containing four Mn atoms on their total spin. The lines connecting tops of the bands are for eye guide.