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A model of bandgap reduction in silicon through the stored electrostatic energy of majority-minority carrier pairs is developed and compared with experimental results in the doping range from 3 × 10 17 to 1. 5 × 10 20 /cm 3 at room temperature. An analytic expression for the bandgap reduction in nondegenerate material is obtained ₆ = 3q^2/ (16) (q^2n/ kT) 1/2 having a square-root dependence on the majority carrier concentration. At room temperature this becomes ₆ = 22. 5 (n/10^18) ^1/2 meV. In degenerate material, the bandgap reduction is independent of temperature, following the relationship ₆ = 162 (n/10^20) ^1/6 meV. The experimental data at room temperature are in excellent agreement with this theory. Plots of bandgap narrowing as a function of doping level are presented for a number of temperatures.
Lanyon et al. (Sun,) studied this question.