We construct the theory of carriers confined in Si quantum dots with finite energy barriers for electrons and holes in the framework of the multiband effective mass theory. We apply this theory for theoretical modeling of the excitation of erbium inside and outside of Si nanocrystals in SiO₂ matrix due to the Auger process induced by the recombination of a confined electron-hole pair as well as the intraband transitions of "hot" confined carriers. Auger de-excitation processes of the Er⁺³ ion leading to the quenching of erbium luminescence are discussed as well.