Debye-Waller Factors and the PbTe Band-Gap Temperature Dependence

Abstract

Measurements of the temperature variation of x-ray diffraction peak intensities have been used to show that the mean-square thermal displacement of Pb in PbTe is approximately twice that of Te. This phenomenon is explained, using a calculation of the mean-square displacements from vibrational eigenvalues and eigenvectors, in terms of the heavy thermal weighting of the acoustic modes relative to the optical. The Pb motion dominates in the acoustic modes, whereas the Te motion dominates in the optic modes. Debye-Waller factors obtained from the calculated mean-square displacements have been associated with a modified Lin-Kleinman pseudopotential to obtain a theoretical estimate of the temperature dependence of the band gap. The modifications are necessary because of errors associated with their treatment of spin-orbit interaction. The calculated band-gap increase of 0.008 Ry between 100 and 300 °K is in fair agreement with an experimental estimate of the explicit increase. It is shown that the magnitude of the shift is quite sensitive to adjustable parameters in the gap calculation as well as to the Debye-Waller factors. The sign of the shift is dependent on the relative ordering of levels and the relative mean-square displacements of the ions but not, otherwise, on the details of the calculation.