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

An effective two-by-two dielectric response matrix with elements correct to second order in the pseudopotential in the high-frequency limit ($\hslash \omega \gg \left|U_{\overrightarrow{\mathrm{G}}}\right|$) is derived within the self-consistent-field approximation. It is used to study the plasmon dispersion for the two lowest plasmon bands in Si along the [111] direction and the shape of the loss function in the neighborhood of the Brillouin-zone boundary. The plasmon band gap at the zone boundary in the [111] direction is evaluated to second order in the pseudopotential, and a previous theoretical calculation for the plasmon gap at the zone boundary to first order in the pseudopotential is recovered for semiconductors of the diamond and zinc-blende structures. The possibility of experimental observation of the plasmon band gaps is discussed. Lifetime effects are shown to be so strong at the Brillouin-zone boundary that there is no indication in the total loss function of a double-peaked structure. The loss dispersion along the [100] direction is also obtained and compared with the results for the [111] direction in the case of Si.