2011, 4(4): 1063-1079. doi: 10.3934/krm.2011.4.1063

The Spherical Harmonics Expansion model coupled to the Poisson equation

1. 

RICAM, Austrian Academy of Sciences, Altenbergerstrasse 69, Linz, A-4040, Austria

2. 

Courant Institute of Mathematical Sciences, New York University, 251 Mercer street, New York, 10012, United States

3. 

Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, Vienna, A-1090, Austria

4. 

Department of Mathematics, University of Tunis ElManar, Faculty of Sciences of Tunis, 2092 El-Manar, Tunisia

Received  June 2011 Revised  October 2011 Published  November 2011

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.
Citation: Jan Haskovec, Nader Masmoudi, Christian Schmeiser, Mohamed Lazhar Tayeb. The Spherical Harmonics Expansion model coupled to the Poisson equation. Kinetic & Related Models, 2011, 4 (4) : 1063-1079. doi: 10.3934/krm.2011.4.1063
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. 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. 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. 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. 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. doi: 10.3934/dcdsb.2004.4.1129.

[6]

H. Brezis, "Analyse Fonctionnelle, Théorie et Applications,'', Collection Mathématiques Appliquées pour la Maîtrise, (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.

[8]

C. Cercignani, R. Illner, M. Pulvirenti, "The Mathematical Theory of Dilute Gases,'', Applied Mathematical Sciences, 106 (1994).

[9]

P. Degond and S. Schmeiser, Macroscopic models for semiconductor heterostructures,, J. Math. Phys., 39 (1998), 4634. doi: 10.1063/1.532528.

[10]

L. C. Evans, "Partial Differential Equations,'', Graduate Studies in Mathematics, 19 (1998).

[11]

H. Grad, Asymptotic theory of the Boltzmann equation,, Phys. Fluids, 6 (1963), 147. doi: 10.1063/1.1706716.

[12]

H. Grad, Principles of the kinetic theory of gases,, in, (1958), 205.

[13]

E. Lyumkis, B. Polsky, A. Shur and P. Visocky, Transient semiconductor device simulation including energy balance equation,, COMPEL, 11 (1992), 311.

[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.

[15]

P. A. Markowich, C. Ringhofer and C. Schmeiser, "Semiconductor Equations,'', Springer-Verlag, (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. 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.

show all references

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. 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. 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. 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. 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. doi: 10.3934/dcdsb.2004.4.1129.

[6]

H. Brezis, "Analyse Fonctionnelle, Théorie et Applications,'', Collection Mathématiques Appliquées pour la Maîtrise, (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.

[8]

C. Cercignani, R. Illner, M. Pulvirenti, "The Mathematical Theory of Dilute Gases,'', Applied Mathematical Sciences, 106 (1994).

[9]

P. Degond and S. Schmeiser, Macroscopic models for semiconductor heterostructures,, J. Math. Phys., 39 (1998), 4634. doi: 10.1063/1.532528.

[10]

L. C. Evans, "Partial Differential Equations,'', Graduate Studies in Mathematics, 19 (1998).

[11]

H. Grad, Asymptotic theory of the Boltzmann equation,, Phys. Fluids, 6 (1963), 147. doi: 10.1063/1.1706716.

[12]

H. Grad, Principles of the kinetic theory of gases,, in, (1958), 205.

[13]

E. Lyumkis, B. Polsky, A. Shur and P. Visocky, Transient semiconductor device simulation including energy balance equation,, COMPEL, 11 (1992), 311.

[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.

[15]

P. A. Markowich, C. Ringhofer and C. Schmeiser, "Semiconductor Equations,'', Springer-Verlag, (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. 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.

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