# The ICICS/CS Reading Room

## UBC CS TR-86-18 Summary

- No on-line copy of this technical report is available.

- Conditioning of the Steady State Semiconductor Service Problem, January 1986 Uri Ascher, P. A. Markowich, C. Schmeiser, H. Steinruck and R.", Weiss
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.

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