Bandgap narrowing in moderately to heavily doped silicon

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 × 10¹⁷to 1.5 × 10²⁰/cm³at room temperature. An analytic expression for the bandgap reduction in nondegenerate material is obtained kT)^1/2having a square-root dependence on the majority carrier concentration. At room temperature this becomes meV. In degenerate material, the bandgap reduction is independent of temperature, following the relationship 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.