High-frequency dielectric properties of covalent semiconductors within the nearly-free-electron approximation. I. The one-plasmon-band model

The macroscopic dielectric response function ${\epsilon }_M(\overrightarrow{\mathrm{k}}, \omega )$ is calculated analytically for covalent semiconductors in the high-frequency limit. The effects of the periodic crystal potential are included by second-order perturbation theory within the self-consistent-field approximation. A study of the plasmon line shape, i.e., of the energy-loss function $\mathrm{Im}[-\frac{1}{{\epsilon }_M}(0, \omega \left)\right]$ and of the optical absorption $\mathrm{Im}{\epsilon }_M(0, \omega )$ demonstrates the importance of local-field corrections in covalent semiconductors. In the long-wavelength limit, the theory predicts that the unknown effective pseudopotential form factors $U_{\overrightarrow{\mathrm{G}}}$ for reciprocal-lattice vectors with $\left|\overrightarrow{\mathrm{G}}\right|\ge \left|{\overrightarrow{\mathrm{G}}}_{400}\right|$ may be obtained successively for increasing $\left|\overrightarrow{\mathrm{G}}\right|$ and increasing excitation energies $\hslash \omega$ from energy-loss experiments or optical measurements. Available experimental data in two cases (Si and Ge) show absorption edges near the predicted energies associated with particular $U_{\overrightarrow{\mathrm{G}}}$. For the first time, the anisotropic plasmon dispersion has been calculated for semiconductors and is compared with recent loss experiments on single crystals of Si, GaAs, and InSb. Owing to the similarity of the crystal potential in these compounds, the theory predicts the same trends for all three materials, namely: ${\omega }_p^{\mathrm{}\left[100\right]\mathrm{}}\left(\overrightarrow{\mathrm{k}}\right)>\mathrm{}{\omega }_p^{\mathrm{}\left[110\right]\mathrm{}}\left(\overrightarrow{\mathrm{k}}\right)>\mathrm{}{\omega }_p^{\mathrm{}\left[111\right]\mathrm{}}\left(\overrightarrow{\mathrm{k}}\right)$. For Si and GaAs this is in agreement with experiment whereas unexpectedly the experiment shows that ${\omega }_p^{\mathrm{}\left[110\right]\mathrm{}}\left(\overrightarrow{\mathrm{k}}\right)>\mathrm{}{\omega }_p^{\mathrm{}\left[100\right]\mathrm{}}\left(\overrightarrow{\mathrm{k}}\right)>\mathrm{}{\omega }_p^{\mathrm{}\left[111\right]\mathrm{}}\left(\overrightarrow{\mathrm{k}}\right)$ for InSb.