Electron energy bands in tellurium

Abstract

The method of pseudopotentials is used to calculate the electron band structure of tellurium along the kz axis. The calculated band structure is then used to analyse certain infrared absorption phenomena in terms of the selection rules governing optical transitions between bands. The crystal potential is approximated by a superposition of Thomas-Fermi potentials for free neutral atoms. Granted this approximation, it is shown that the low-lying eigenfunctions of the crystal Hamiltonian are closely approximated by Bloch sums of of the atomic wave functions calculated by Herman and Skillman. Thus, the crystal pseudopotential is taken to be a superposition of atomic pseudopotentials calculated from the data of Herman and Skillman. The calculated band structure shows a direct energy gap of 0.065 Ry at kz=pi/c and an indirect gap of about the same magnitude between the valence band edge at kz = 0 and the conduction band edge at kz=pi/c. The experimental gap is about 0.024 Ry; the disagreement with experiment is attributed to the approximate potential used in the calculation. It is shown that the valence bands of tellurium, which were assumed to be associated with 5p orbitals, actually contain significant mixtures of 5s, 5d, and possibly, higher energy orbitals. Because of this mixing, the spin-orbit splitting of the valence bands is much smaller than had been anticipated. The effects of uncertainties in the crystal potential are studied using first order perturbation theory. It is found that the experimental energy gap can be obtained by making the crystal potential more attractive in the regions between the atomic cores. An adjusted band structure, which would result from a more attractive potential, is proposed. There are no significant differences in the qualitative features of the computed and adjusted band structures. The selection rules governing optical transitions between bands are used to show that the polarization dependence of the fundamental absorption edge is consistent with the symmetry properties of the calculated valence and conduction band edges. The polarization-dependent, 11 micron absorption peak observed by Caldwell and Fan is interpreted as a transition between two closely spaced valence bands of the same symmetry. It is concluded that the most important features of the tellurium band structure are determined by the symmetry of the crystal and are rather insensitive to approximations to the crystal potential. Thus, the simplifying assumptions employed in the calculation do not have a significant effect on the proposed band structure and the resulting interpretations of the infrared absorption properties of tellurium.