LOW AND HIGH FIELD SCALING LIMITS FOR THE VLASOV– AND WIGNER–POISSON–FOKKER–PLANCK SYSTEMS

Abstract

This paper is concerned with scaling limits in kinetic semiconductor models. For the classical Vlasov–Poisson–Fokker–Planck equation and its quantum mechanical counterpart, the Wigner–Poisson–Fokker–Planck equation, three distinguished scaling regimes are presented. Using Hilbert and Chapman–Enskog expansions, we derive two drift-diffusion type approximations. The test case of a n ⁻ – n – n ⁺ diode reveals that different scaling regimes may be present at the same time in different subregions of a semiconductor device. Numerical simulations of the stationary solution illustrate the good approximation of the kinetic solution by a drift-diffusion model and by a hybrid (adaptive domain decomposition) model.

Keywords:

• Semiconductor models
• Vlasov–Poisson–Fokker–Planck equation
• Drift-diffusion models
• Scaling limits
• Domain decomposition
• AMS 1991 Subject Classification: 35B25, 76P05, 82C70, 65M99

ACKNOWLEDGMENTS

Partially supported by the grants ERBFMRXCT970157 (TMR-Network) from the EU, the bilateral DAAD-NSF Program (315/PPP/ru-ab), the DFG under Grant-No. MA 1662/1-2 and DGES (Spain) PB98-1281. The authors are grateful to the hospitality and financial support of the ESI, Vienna and TICAM, Austin, where part of this research was carried out. The second author was supported by the DGESIC MEC-Spain Perfeccionamiento de Doctores en el Extranjero fellowship. The third author is supported by the NSF DMS 9623037. The fourth author is supported by the NSF ECS-9627849 and by the Army Research Office under grant DAAG55-97-1-0318.