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 3×10
¹⁷
to 3.3×10
¹⁹
/cc. at room temperature. An analytic expression for the bandgap reduction is obtained: Δε
g
= 3q
²
/(16πε) . (q
²
N/εkT)
^½
having a square root dependence on the doping concentration. At room temperature the bandgap narrowing follows the relationship Δε
g
= 22.5 (N/10
¹⁸
)
^½
meV The experimental data are in excellent agreement with this expression, the experimental coefficient being within 1% of the theoretical expression.