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The energy shifts of valence and conduction band states in silicon due to the interaction of electrons and holes with ionized donors have been calculated by performing a partial wave analysis. The potential is modeled by the Yukawa form with the screening radius determined self-consistently by the Friedel sum rule. The results show that this effect is an important part of the optically measured band-gap narrowing. The variation of the Fermi energy due to this phenomenon is also calculated.
Lowney et al. (1982) studied this question.