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An analytical model of the band gap narrowing (BGN) in silicon was derived from a non-self-consistent finite-temperature full random-phase approximation (RPA) formalism. Exchange-correlation self-energy of the free carriers and correlation energy of the carrier-dopant interaction were treated on an equal basis. The dispersive quasi-particle shift (QPS) in RPA quality was numerically calculated for a broad range of densities and temperatures. The dispersion was found to be smooth enough for the relevant energies to justify the rigid shift approximation in accordance with the non-self-consistent scheme. A pronounced temperature effect of the BGN only exists in the intermediate density range. The contribution of the ionic part of the QPS to the total BGN decreases from 1/3 at low densities to about 1/4 at very high densities. Based on the numerical results, Padé approximations in terms of carrier densities, doping, and temperature with an accuracy of 1 meV were constructed using limiting cases. The analytical expression for the ionic part had to be modified for device application to account for depletion zones. The model shows a reasonable agreement with certain photoluminescence data and good agreement with recently revised electrical measurements, in particular for p-type silicon. The change of BGN profiles in a bipolar transistor under increasing carrier injection is demonstrated.
Andreas Schenk (Thu,) studied this question.