Semiclassical limit in a simplifiedquantum energy-transport model for semiconductors

Li Chen; Xiu-Qing Chen; Ansgar Jüngel

doi:10.3934/krm.2011.4.1049

Article Contents

2011, Volume 4, Issue 4: 1049-1062. Doi: 10.3934/krm.2011.4.1049

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Semiclassical limit in a simplified quantum energy-transport model for semiconductors

1.
Department of Mathematical Sciences, Tsinghua University, Beijing, 100084
2.
School of Sciences,, Beijing University of Posts & Telecommunications, Beijing, 100876
3.
Institute for Analysis and Scientiﬁc Computing, Vienna University of Technology, Wiedner Hauptstr. 8-10, 1040 Wien

Received: May 31, 2011

Revised: June 30, 2011

Published: November 2011

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Abstract

The semiclassical limit in a quantum energy-transport model for semiconductors is proved. The system consists of a nonlinear parabolic fourth-order equation for the electron density, including temperature gradients; a degenerate elliptic heat equation for the electron temperature; and the Poisson equation for the electric potential. The equations are solved in a bounded domain with periodic boundary conditions. The asymptotic limit is based on a priori estimates independent of the scaled Planck constant, obtained from entropy functionals, on the use of Gagliardo-Nirenberg inequalities, and weak compactness methods.

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Mathematics Subject Classification: Primary: 35B25, 35J40, 35Q40; Secondary: 82D37.

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