The Spherical Harmonics Expansion model coupled to the Poisson equation

Jan Haskovec; Nader Masmoudi; Christian Schmeiser; Mohamed Lazhar Tayeb

doi:10.3934/krm.2011.4.1063

Article Contents

2011, Volume 4, Issue 4: 1063-1079. Doi: 10.3934/krm.2011.4.1063

This issue Previous Article Semiclassical limit in a simplified quantum energy-transport model for semiconductors Next Article Averaged kinetic models for flows on unstructured networks

The Spherical Harmonics Expansion model coupled to the Poisson equation

1.
RICAM, Austrian Academy of Sciences, Altenbergerstrasse 69, Linz, A-4040
2.
Courant Institute of Mathematical Sciences, New York University, 251 Mercer street, New York, 10012
3.
Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, Vienna, A-1090
4.
Department of Mathematics, University of Tunis ElManar, Faculty of Sciences of Tunis, 2092 El-Manar

Received: May 31, 2011

Revised: September 30, 2011

Published: November 2011

Abstract Related Papers Cited by

Abstract

The Spherical Harmonics Expansion (SHE) assumes a momentum distribution function only depending on the microscopic kinetic energy. The SHE-Poisson system describes carrier transport in semiconductors with self-induced electrostatic potential. Existence of weak solutions to the SHE-Poisson system subject to periodic boundary conditions is established, based on appropriate a priori estimates and a Schauder fixed point procedure. The long time behavior of the one-dimensional Dirichlet problem with well prepared boundary data is studied by an entropy-entropy dissipation method. Strong convergence to equilibrium is proven. In contrast to earlier work, the analysis is carried out without the use of the derivation from a kinetic problem.

Keywords:

Mathematics Subject Classification: Primary: 35K65, 35Q82; Secondary: 58Z05.

Citation:

Related Papers

Cited by

References

[1]	N. Ben Abdallah, Weak solutions of the initial-boundary value problem for the Vlasov-Poisson system, Math. Meth. in the Appl. Sci., 17 (1994), 451-476.doi: 10.1002/mma.1670170604.
[2]	N. Ben Abdallah and P. Degond, On a hierarchy of macroscopic models for semiconductors, J. Math. Phys., 37 (1996), 3306-3333.doi: 10.1063/1.531567.
[3]	N. Ben Abdallah and J. Dolbeault, Relative entropies for the Vlasov-Poisson system in bounded domains, C. R. Acad. Sci. Paris Ser. I Math., 330 (2000), 867-872.doi: 10.1016/S0764-4442(00)00268-8.
[4]	N. Ben Abdallah, P. Degond, P. Markowich and C. Schmeiser, High field approximations of the spherical harmonics expansion model for semiconductors, ZAMP, 52 (2001), 201-230.doi: 10.1007/PL00001544.
[5]	N. Ben Abdallah and M. L. Tayeb, Diffusion approximation for the one dimensional Boltzmann-Poisson system, DCDS-B, 4 (2004), 1129-1142.doi: 10.3934/dcdsb.2004.4.1129.
[6]	H. Brezis, "Analyse Fonctionnelle, Théorie et Applications,'' Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 1983.
[7]	D. Chen, E. C. Kan, U. Ravaioli, C. Shu and R. W. Dutton, An improved energy transport model including non-parabolicity and non-Maxwellian distribution effects, IEEE Electron Dev. Lett., 13 (1992), 235-239.
[8]	C. Cercignani, R. Illner, M. Pulvirenti, "The Mathematical Theory of Dilute Gases,'' Applied Mathematical Sciences, 106, Springer-Verlag, New York, 1994.
[9]	P. Degond and S. Schmeiser, Macroscopic models for semiconductor heterostructures, J. Math. Phys., 39 (1998), 4634-4663.doi: 10.1063/1.532528.
[10]	L. C. Evans, "Partial Differential Equations,'' Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, Rhode Island, 1998.
[11]	H. Grad, Asymptotic theory of the Boltzmann equation, Phys. Fluids, 6 (1963), 147-181.doi: 10.1063/1.1706716.
[12]	H. Grad, Principles of the kinetic theory of gases, in "Handbuch der Physik" (herausgegeben von S. Flügge), Bd. 12, Thermodynamik der Gase, Springer-Verlag, Berlin-Göttingen-Heidelberg, (1958), 205-294.
[13]	E. Lyumkis, B. Polsky, A. Shur and P. Visocky, Transient semiconductor device simulation including energy balance equation, COMPEL, 11 (1992), 311-325.
[14]	P. A. Markowich, F. Popaud and C. Schmeiser, Diffusion approximation of nonlinear electron phonon collision mechanisms, RAIRO Modél. Math. Anal. Num., 29 (1995), 857-869.
[15]	P. A. Markowich, C. Ringhofer and C. Schmeiser, "Semiconductor Equations,'' Springer-Verlag, Vienna, 1990.doi: 10.1007/978-3-7091-6961-2.
[16]	N. Masmoudi and M. L. Tayeb, Diffusion limit of a semiconductor Boltzmann-Poisson system, SIAM J. Math. Anal., 38 (2007), 1788-1807.doi: 10.1137/050630763.
[17]	M. L. Tayeb, From Boltzmann equation to spherical harmonics expansion model: Diffusion limit and Poisson coupling, Comm. Math. Sci., 9 (2011), 255-275.