A simple expression for band gap narrowing (BGN) in heavily doped Si, Ge, GaAs and GexSi1−x strained layers

Abstract This paper presents simple but accurate closed form equations for Band Gap Narrowing (BGN) for n and p type, Si, Ge, GaAs and GexSi1−x alloys and strained layers. The equations are derived by identifying the four components of BGN: exchange energy shift of the majority band edge, correlation energy shift of the minority band edge and impurity interaction shifts of the two band edges. In the simple parabolic band approximation, the BGN is determined by the effective masses of the carriers and the relative permittivity of the semiconductor. For real semiconductors, known corrections due to anisotropy of the bands, due to multi-valleys in a band and due to interactions between sub-bands are used. The values of BGN for n Si, n Ge and n and p GaAs calculated using this simple formulation agree closely with the theoretical values calculated by other authors using advanced but complex many body methods and the Random Phase Approximation for screening effects. For p Si and p Ge ours appear to be the first calculations taking all interactions into account. Experimental values of BGN for all semiconductors except for p Ge for which no data could be found, are also in very good agreement with our theory. The Fermi level for n and p Si and p GaAs is determined using the published luminescence data. In heavily doped p type semiconductors, the values are found to be considerably smaller than those calculated using the known values of the effective density of states. The values of apparent BGN for n and p Si and p GaAs calculated using experimentally determined Fermi levels are in remarkably good agreement with the experimental values derived from device measurements. All results are presented in a form which lends itself to numerical computer simulation studies.