The electronic energy levels associated with vacancies in PbTe are obtained through the Green's-function method of Koster and Slater, the unperturbed Bloch functions being obtained from a relativistic $\overrightarrow{\mathrm{K}}·\overrightarrow{\pi }$ augmented-plane-wave (APW) energy-band calculation. APW one-electron energies were obtained at $\Gamma$ and the corresponding eigenfunctions were used to obtain matrix elements of the relativistic momentum operator $\overrightarrow{\pi }$ between states at $\Gamma$. These energies and matrix elements were used in a $\overrightarrow{\mathrm{K}}·\overrightarrow{\pi }$ secular equation to obtain energies and wave functions at approximately 4300 points in the Brillouin zone. With 11 relativistic bands at $\Gamma$, excellent results were obtained. Localized Wannier functions were constructed by taking suitable linear combinations of the unperturbed Bloch functions and these Wannier functions provided the basis in which the energy levels in the presence of the perturbing impurity potential were found. We have solved the vacancy problem using Wannier functions from nine bands (five valence and four conduction) and 13 lattice sites. The results obtained from this calculation showed that Pb vacancies produce $p$-type PbTe, whereas Te vacancies produce $n$-type PbTe, and in both cases, carriers are present at all temperatures.

• Received 20 April 1970

DOI:https://doi.org/10.1103/PhysRevB.3.2042