An asymptotic analysis of a unipolar junction model

Abstract

Mock in 1983 and Kerkhoven in 1991 have introduced and studied the questions related to the existence and uniqueness of solutions of a unipolar junction model for semiconductor devices. In this model one of the carrier current density is assumed to vanish, so that the original bipolar model reduces to a system of two coupled differential equations. We perform a singular perturbation analysis of this system in the two dimensional case as the singular perturbation parameter goes to zero. We obtain L² asymptotic error estimates and uniform convergence results for the difference between the solution of the system and the first order term of the asymptotic expansion under any bias conditions. We use a new uniqueness result for the associated reduced unipolar system together with perturbation arguments with respect to the reduced solution combined with variational techniques. These results extend in a certain way the results of Alabau and Moussaoui [5] which where valid for a more complex model (the bipolar one) but only for sufficiently small biases. This is due to the fact that the obtention of asymptotic estimates is related to uniqueness of solutions of the reduced equations which does not hold in general in the bipolar case (unless the applied biases are sufficiently small).

Keywords:

• Semiconductors
• singular perturbation
• uniqueness

AMS:

• 35G30
• 35J25
• 35B5
• 34E0

^∗I.R.M.A., Université Louis Pasteur et C.N.R.S.. i rue René Descartes, 67084 Strasbourg Cedex, FRANCE.

^‡Université Bordeaux I.M.A.B., 351 Cours de la Libitation, 33405 Talence Cedex. FRANCE.

^†Ecole Centrale de Lyon. Département M.I.S., UiWR 5585, BP 16369131 Ecully Cedex, FRANCE

^∗I.R.M.A., Université Louis Pasteur et C.N.R.S.. i rue René Descartes, 67084 Strasbourg Cedex, FRANCE.

^‡Université Bordeaux I.M.A.B., 351 Cours de la Libitation, 33405 Talence Cedex. FRANCE.

^†Ecole Centrale de Lyon. Département M.I.S., UiWR 5585, BP 16369131 Ecully Cedex, FRANCE

Notes

^∗I.R.M.A., Université Louis Pasteur et C.N.R.S.. i rue René Descartes, 67084 Strasbourg Cedex, FRANCE.

^‡Université Bordeaux I.M.A.B., 351 Cours de la Libitation, 33405 Talence Cedex. FRANCE.

^†Ecole Centrale de Lyon. Département M.I.S., UiWR 5585, BP 16369131 Ecully Cedex, FRANCE