Band Edge Structure of PbS, PbSe, and PbTe

The available experimental data on PbS, PbSe, and PbTe indicate that the valence-and conduction-band extrema of these semi-conductors occur at the $L$ point of the Brillouin zone. The nearly-free-electron model predicts that the valence and conduction states in the vicinity of the forbidden gap at $L$ each consist of three simple spin-degenerate bands. These bands interact strongly with one another and are relatively well isolated from other bands at $L$. The forms of the dispersion relations, $E\left(\mathrm{k}\right)$, for these bands are determined using their symmetry and k·P perturbation theory, and depend strongly on their order and spacing. The conduction-and valence-band extrema may be either anisotropic, with small, highly concentration-dependent transverse masses, as found in PbTe, or more nearly isotropic, as found in PbS. PbSe is thought to be an intermediate case. The theoretical variation with carrier concentration of the cyclotron masses and extremal cross-sectional areas of the Fermi surface is derived from k·P perturbation theory for a simple model of the band structure in PbTe. This model is found to be in good agreement with most of the transport data on PbTe. However, the $g$ factor for the valence band of PbTe, deduced from measurements of the Shubnikov-de Haas effect, is in definite disagreement with the predictions of the simple model, and a consideration of all six bands is necessary in order to obtain complete agreement with experiment. It appears that the band edge structure of PbSe and PbS are similar to that of PbTe with only a difference in the spacing of the various valence and conduction bands.