Bandgap narrowing in 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 the experimental results of other workers in the doping range from 3×10¹⁷to 3.3×10¹⁹/cc. at room temperature. An analytic expression for the bandgap reduction is obtained: Δεg= 3q²/(16πε) . (q²N/εkT)^½having a square root dependence on the doping concentration. At room temperature the bandgap narrowing follows the relationship Δεg= 22.5 (N/10¹⁸)^½meV The experimental data are in excellent agreement with this expression, the experimental coefficient being within 1% of the theoretical expression.