Plasma-induced energy shifts of the conduction band minimum and of the valence band maximum have been calculated for 3C-, 2H-, 4H-, 6H-, 6H-SiC and Si. The resulting narrowing of the fundamental band gap and of the optical band gap are presented. The method utilized is based on a zero-temperature formalism within the random phase approximation. Electron-electron, hole-hole, electron-hole, electron-optical phonon and hole-optical phonon interactions have been taken into account. The calculations are based on band structure data from a relativistic, full-potential band structure calculation.