Electronic Band Structure of Selenium and Tellurium

The band structure of selenium and tellurium has been calculated according to the tight-binding scheme in which only nearest neighbor interactions are presumed to be important. As a first approximation, von Hippel's hypothetical crystal with 90° bond angles between bonds in the chain is used, and the ninth degree secular equation for the $p$ bands factors into three identical cubic equations. Next, the crystal is pulled out along the $c$ axis to the observed bond angle, and further splitting of the $p$ bands is calculated as a perturbation. The fifteen $d$ bands are also computed, and mixing of $p$ and $d$ levels estimated. In addition, matrix elements for optical transitions between bands have been worked out. The long-wavelength absorption limit for direct transitions occurs at $k_z=\frac{\pi }{c}$, and is different for light polarized parallel or perpendicular to the $c$ axis of the crystal. Both $p\to p$ and $p\to d$ transitions give the same order for the two absorption edges, the results appearing to be in accord with experiments of Loferski on tellurium. Finally, we note Bridgman's measurements on crystals of tellurium subjected to hydrostatic pressure: the bond angle apparently increases with pressure. The calculations here show a reduction in energy gap with increasing bond angle (i.e., increasing pressure), which is in the right direction to explain the simultaneous large decrease in electrical resistivity observed by Bridgman