Compatibility Conditions for Semiconductor

The basic drift{diiusion model for semiconductors gives rise to a system of three coupled partial diierential equations: one potential eqation and two continuity equations. GG artner 6] was the rst to point out that, using diierent discretization schemes for both types of equations, the coupling might not be represented correctly. This paper further elaborates his ideas: in an abstract setting we explain how a dicretization scheme should take into account the interaction of the equations. To that end we formulate two compatibility conditions. Then we develop a technique for quickly assessing the viability of a particular scheme. It relies on purely local examinations and a few heuristic rules. Next we investigate several existing schemes to what extent they meet the compatibility conditions. We show that only the classical Scharfetter{Gummel scheme passes the tests. All other methods under examination are rejected.