Fourier Expansion for the Electronic Energy Bands in Silicon and Germanium

Abstract

Using a Fourier expansion for the coupled valence and conduction bands in the diamond structure, an effective-mass Hamiltonian valid throughout the Brillouin zone is developed in terms of certain Fourier-expansion coefficients or band parameters which are determined from experimental data. Explicit values for the band parameters are obtained for silicon and germanium, and the resulting energy bands are used to calculate the frequency dependence of the complex dielectric constant. Quantitative agreement is obtained between the calculated and observed dielectric constants. Results are also given for the effective masses for all $s$ and $p$ bands in silicon and germanium at the $\Gamma$, $L$, and $X$ points. The Fourier-expansion technique is computationally simple and rapid, and provides an energy-band model in quantitative agreement with a large number of experimental data.