Effects of quantitative disorder on the electronic structures of Si and Ge

We have attempted to model the electronic structures of glow-discharge amorphous-Si-based alloys by studying the effects of specific kinds of quantitative disorder on the band structures of crystalline Si. A realistic tight-binding Hamiltonian has been developed, including third-nearest-neighbor interactions, which reasonably describes both the valence and first three conduction bands of the crystalline materials. Quantitative disorder is investigated by transforming a disordered system of identical atoms to an ordered diamondlike system with disordered potentials. We find that, in both Si and Ge, bond-length distortions affect mainly the bottom of the valence band at ${\Gamma }_1$ and the low-density-of-states region around $X_1$, as well as the peaks of $L_{2^{\prime }}$ and $L_1$. One might expect, therefore, that localization would be present in both regions near ${\Gamma }_1$ and $X_1$. Moreover, bond-length distortions affect the bottom of the conduction-band edge in Ge but not in Si. Bond-angle distortions are found to cause shifts of band, primarily in the $p$-like regions of valence band. These shifts are to lower binding energies and are consistent with the steepening of the top of the valence-band edge, with disorder, as observed in photoemission measurements. Finally, our results also account for the merging of the $E_1$ and the $E_2$ peaks of the imaginary part of the dielectric function ${\epsilon }_2$, with disorder, as observed by spectroscopic ellipsometry and reflectivity measurements.