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 × 10
¹⁷
to 1.5 × 10
²⁰
/cm
³
at room temperature. An analytic expression for the bandgap reduction in nondegenerate material is obtained
\delta\epsilon _{g} = 3q^{2}/(16\pi \epsilon) \cdot (q^{2}n/\epsilon
kT)
^1/2
having a square-root dependence on the majority carrier concentration. At room temperature this becomes
\delta\epsilon _{g} = 22.5 (n/10^{18})^{1/2}
meV. In degenerate material, the bandgap reduction is independent of temperature, following the relationship
\delta \epsilon _{g} = 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.