4.1.3 Discussion

All Scharfetter-Gummel discretisation variants can solve the six moments Boltzmann Poisson system reliably for the closure parameter $c=3$. By stepping the parameter $c$ we found that there seems to be a certain sharp value of $c$ below which no convergence can be reached.

Especially, for closure with $c=2$ the solver fails to converge on certain benchmark examples no matter which Scharfetter-Gummel style discretization we choose. Overall it is found that none of the (supposedly) more sophisticated methods can improve much on the original Scharfetter-Gummel style discretization.

Initially we blamed the central differencing effects to be the cause for the failure, as these effects are much more pronounced in simulations using $c=2$. However, blaming central differencing effects alone for non-convergence misses the point as is proved by the naive discretization which we termed the ``double grid discretization'' in Section 4.1.2.

Surprisingly the double grid discretization outperforms Scharfetter Gummel with respect to nonlinear convergence for $c \leq 2$. From simulation results it was found that for closures with $c \leq 2$ the system of equations can exhibit oscillations in the solution. A typical result is depicted in Figure 4.2,

Figure 4.2: Unphysical oscillations in the equilibrium electron density.

where the electron concentration in equilibrium is depicted. A closure parameter $c = 1.5$ was used. In this way the closures using $c \leq 2$ could be ruled out as unphysical. In such unphysical cases the double grid discretization performs much better than the Scharfetter-Gummel style discretizations.

Also the overall convergence behaviour is more regular and sensible. Paradoxically, we observed that in the case of Scharfetter-Gummel style discretizations global refinement often makes the convergence worse. This effect is also diminished with the double grid discretization. At the time being we do not really understand the reason for the in these respects superior performance of the double grid discretization.

The main drawback of the scheme is that it is not clear how to extend it to dimensions two and three on an unstructured grid, although it generalizes easily to equispaced meshes in higher dimensions. We see the main accomplishment of this scheme in the proof that there is still room for improving on Scharfetter-Gummel style discretizations. At least part of the blame for non-convergence has to be put on the discretization, though we found that the kind of applied highest order moment closure is a bigger factor. For closures consistent with bulk Monte Carlo data Scharfetter-Gummel style discretizations perform superiorly.

We have not provided any theoretical analysis of the numerical properties of the applied discretizations. It is an open problem to adapt a scheme which allows for estimates of the entropy (in the style of [Rin01]) to our models.