A qualitative analysis of the fundamental semiconductor device equations

Peter Markowich*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

We present a qualitative analysis of the fundamental static semiconductor device equations which is based on singular perturbation theory. By appropriate scaling the semiconductor device equations are reformulated as singularly perturbed elliptic system (the Laplacian in Poisson’s equation is multiplied by a small parameter λ2, the so-called singular perturbation parameter). Physically the singular perturbation parameter is identified with the square of the normed minimal Debye length of the device under consideration. Using matched asymptotic expansions for small A we characterize the behaviour of the solutions locally at pn junctions, Schottky contacts and oxide-semiconductor interfaces and demonstrate the occurrence of exponential internal/boundary layers at these surfaces. The derivatives of the solutions blow up within these layer regions (as λ2 decreases) and they remain bounded away from the layers. We demonstrate that the solutions of the ‘zero-space charge approximation’ are close to the solutions of the “full’ semiconductor problem (when A is small) away from layer regions and derive a second-order ordinary differential equation which (when subjected to appropriate boundary/interface conditions) ‘describes’ the solutions within layer regions.

Original languageEnglish (US)
Pages (from-to)97-115
Number of pages19
JournalCOMPEL - The International Journal for Computation and Mathematics in Electrical and Electronic Engineering
Volume2
Issue number3
DOIs
StatePublished - Mar 1 1983

ASJC Scopus subject areas

  • Computer Science Applications
  • Computational Theory and Mathematics
  • Electrical and Electronic Engineering
  • Applied Mathematics

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