4.1.1 Variants of the Scharfetter Gummel Discretization

In MINIMOS-NT a Scharfetter-Gummel (SG) style discretization [Tan84] was initially used for the discretization of the six moments model. Simulations using this variant of the Scharfetter Gummel discretization did not converge for closures with parameter $c \leq 2$.

In the literature there are several suggestions for discretizations which generalize and claim to improve on the Scharfetter-Gummel method. These were implemented and tried as a possible cure for the convergence problems. Methods of this kind which we studied are: Meinerzhagen's model [ME88], Forghieri's method [CAP⁺94] and variants thereof. All these methods can be unified in the framework of an optimal artificial diffusion method [TI95].

A variety of variants of Scharfetter-Gummel was implemented and tested laboriously, but the achieved improvements were minute. In the rare cases where the Newton solver converged, the results were of bad quality, showing traces of central differencing. We suspect that these are caused by interpolating quantities from the odd to the even grid and vice versa.

In Scharfetter-Gummel style discretization interpolation occurs at the following steps: First, the mobility $\mu$ is needed on the odd grid, but depends on temperature $T_n$ which is defined on the even grid. Secondly, the electric field $E$ which is naturally an odd quantity is needed on the even grid in Equation 2.57. Here the term $Q = E_{x_3}M_{2i-1}$ is defined on the odd grid, but is needed on the even grid.