Modeling Fermi energy, free-carrier density, and resistivity in degenerate n-Ge

A new expression for Fermi energy vs doping is derived using the standard model for free carriers in n-type semiconductors. The new expression is composed of the Fermi energy in non-degenerate semiconductors, a doping function for bandgap narrowing (BGN), and an adjustable energy variation. In non-degenerate semiconductors, the new expression is equivalent to the standard Boltzmann expression. Calculated curves of Fermi energy are assigned in the Fermi–Dirac expression for the donor ionization ratio, and reported data of electron density and resistivity measured in heavily doped n-Ge layers are fitted. Five reported doping functions for BGN are used. One of the BGN functions allows modeling frustrated incomplete ionization. Another allows modeling bandgap widening.