Electrical and optical bandgap narrowing due to heavy doping in silicon

Optical (real) bandgap narrowing is due to the change in bottom of conduction band and top of valence band, leading to a gap shift ℰc−ℰv = ℰ − Δℰg, where ℰ is the low doping standard band-gap. In electrical measurements, an electrical (apparent) bandgap narrowing is derived from the altered pn product n = n exp (ΔG0/kT); here n is the low doping standard pn product, being the square of the intrinsic carrier density. It is shown that ΔG0 = Δℰg + θ + θ; Δℰg is the rigid band shift, but θ and θ involve the effects of Fermi statistics as well as of the altered entire density of states in the presence of heavy doping. Optical and electrical bandgap narrowing can only be concordant if θ and θ are explicitly computed. A preliminary computation is made, using the quantum mechanical data of Lowney and Bennett for a donor doping of 9.64 × 1019 cm−3. The optical bandgap change Δℰg for that doping at 300 K is known to be close to 83 meV. The additional change due to θ + θ is computed to be 10.1 meV. Thus the electrical bandgap narrowing is larger than the optical bandgap narrowing, the total effect (≈ 93 meV) being in good agreement with the recently compiled data by del Alamo et al.