Generalization of the Scharfetter-Gummel scheme

For Blakemore-type distribution functions F(η) = 1/(exp(-η)+γ) describing the carrier density in semiconductors a generalization of the classical Scharfetter-Gummel scheme can be derived resulting in a nonlinear equation per edge to calculate the edge current. This approach provides a good approximation of the carrier density in degenerate semiconductors for values of the chemical potential η <; 1.3k_BT. We discuss an extension of this approach based on a piecewise approximation of the distribution function by functions of that type in order improve the approximation for larger values of the chemical potential.