Doping effects on the band structure in n-type silicon at 300 K

Based on two screened donor potential models and taking into account the electron-electron interaction effect in the electron-donor interaction, the band-gap narrowing (BGN) in n-type doped silicon at 300 K is investigated. The BGN effect is expressed in terms of physical BGN (ΔE_g), a change in density of conduction-band states (Γ_n) and the change in effective electron (hole) mass (γ_n (γ_p)). In fact, electrical BGN is found to be given by $\text{ΔE}{}_{\mathrm{<mtext>g,\; elec.</mtext>}}{}^1\text{= ΔE}{}_{\mathrm{<mtext>g</mtext>}}\text{+ Γ}{}_{\mathrm{n}}\text{γ}{}_{\mathrm{n}}\text{+ Γ}{}_{\mathrm{p}}\text{+ ΔE}{}_{\mathrm{<mtext>g</mtext><mtext>,\; fd</mtext>}}\text{? ΔE}{}_{\mathrm{<mtext>g</mtext>}}\text{+ Γ}{}_{\mathrm{n}}$, where ΔE_g,fd represents the Fermi-Dirac statistics effect. Our results of $\text{ΔE}{}_{\mathrm{g}}\text{and}\text{ΔE}{}_{\mathrm{tg}}\text{,}\text{elec.}{}^1$ agree fairly with corresponding observed results, and the large values of $\text{Γ}{}_{\mathrm{n}}\text{? ΔtE}{}_{\mathrm{g}}\text{,}\text{elec.}{}^1\text{? ΔE}{}_{\mathrm{g}}$ obtained at high donor concentration are in good agreement with a qualitative discussion given by Marshak and Van Vliet. Finally, the present results for the effective intrinsic-carrier concentration n_ie are compared with corresponding results observed by Tang.