Conditioning of the Steady State Semiconductor Service Problem

When solving numerically the steady state semiconductor device problem using appropriate discretizations, extremely large condition numbers are often encountered for the linearized discrete device problem. These condition numbers are so large that, if they represented a sharp bound on the amplification of input errors, or even of roundoff errors, then the obtained numerical solution would be meaningless. As it turns out, one major reason for these large numbers is due to poor row and column scaling, which is essentially harmless and/or can be fixed. But another reason could be an ill-conditioned device, which yields a true loss of significant digits in the numerical calculation. In this paper we carry out a conditioning analysis for the steady state device problem. We consider various quasilinearizations as well as Gummel-type iterations and obtain stability bounds which may indeed allow ill-conditioning in general. These bounds are exponential in the potential variation, and are sharp e.g. for a thyristor. But for devices where each smooth subdomain has an Ohmic contact, e.g. a pn-diode, moderate bounds guaranteeing well-conditioning are obtained. Moreover, the analysis suggests how various row and column scalings should be applied in order for the measured condition numbers to correspond more realistically to the true loss of significant digits in the calculations.