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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">δε g = 3q²/(16π ε) · (q²n/ε</tex> kT) 1/2 having a square-root dependence on the majority carrier concentration. At room temperature this becomes <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">δε g = 22.5 (n/10¹⁸)1/2</tex> meV. In degenerate material, the bandgap reduction is independent of temperature, following the relationship <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">δ ε g = 162 (n/10²⁰)1/6</tex> 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.