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The Linearized Transient Quantum Drift Diffusion Model — Stability of Stationary States
Author(s) -
Pinnau René
Publication year - 2000
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/(sici)1521-4001(200005)80:5<327::aid-zamm327>3.0.co;2-h
Subject(s) - nonlinear system , discretization , mathematics , quantum , stationary state , mathematical analysis , hilbert space , van der pol oscillator , stability (learning theory) , transient (computer programming) , physics , quantum mechanics , machine learning , computer science , operating system
The transient quantum drift diffusion model is derived from the isothermal quantum hydrodynamic model via a zero relaxation time limit. This yields a fourth‐order nonlinear parabolic equation for the electron density, which is self‐consistently coupled to the Poisson equation for the electrostatic potential. A stability analysis of the linearized system is performed by means of Hilbert space methods, which rely on a generalized Poincaré‐type inequality. In the quantum case more states are linearly stable than in the classical one. The linear stability of an implicit Euler discretization for the nonlinear equations is proven and numerical results for a diode are presented, indicating that no maximum principle holds for the linearized system.