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.