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End date: 2009 × Start date: 2000 × Subject: asymptotic behaviour × WGL Institute: WIAS ×

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Now showing 1 - 5 of 5
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    Electro-reaction-diffusion systems in heterostructures
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2000) Glitzky, Annegret; Hünlich, Rolf
    The paper is devoted to the mathematical investigation of a general class of electro-reaction-diffusion systems with nonsmooth data which arises in applications to semiconductor technology. Besides of a basic problem, a reduced problem is considered which is obtained if the kinetics of the free carriers is fast. For two dimensional domains we prove a global existence and uniqueness result. In addition, asymptotic properties of solutions are studied. Basic ideas are energy estimates, Moser iteration, regularization techniques and an existence result for electro-diffusion systems with weakly nonlinear volume and boundary source terms which is proved in the paper, too. The relationship between the property that the energy functional decays exponentially in time to its equilibrium value and the existence of global positive lower bounds for the densities of the species is investigated. We illustrate relations between the model and its reduced version in general and for concrete examples. Finally, we discuss the special features of heterostructures for simplified model problems.
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    Exponential decay of the free energy for discretized electro-reaction-diffusion systems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Glitzky, Annegret
    Our focus are electro-reaction-diffusion systems consisting of continuity equations for a finite number of species coupled with a Poisson equation. We take into account heterostructures, anisotropic materials and rather general statistical relations. We introduce a discretization scheme (in space and fully implicit in time) using a fixed grid but for each species different Voronoi boxes which are defined with respect to the anisotropy matrix occurring in the flux term of this species. This scheme has the special property that it preserves the main features of the continuous systems, namely positivity, dissipativity and flux conservation. For the discretized electro-reaction-diffusion system we investigate thermodynamic equilibria and prove for solutions to the evolution system the monotone and exponential decay of the free energy to its equilibrium value. The essential idea is an estimate of the free energy by the dissipation rate which is proved indirectly.
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    Analysis of a spin-polarized drift-diffusion model
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Glitzky, Annegret
    We introduce a spin-polarized drift-diffusion model for semiconductor spintronic devices. This coupled system of continuity equations and a Poisson equation with mixed boundary conditions in all equations has to be considered in heterostructures. We give a weak formulation of this problem and prove an existence and uniqueness result for the instationary problem. If the boundary data is compatible with thermodynamic equilibrium the free energy along the solution decays monotonously and exponentially to its equilibrium value. In other cases it may be increasing but we estimate its growth. Moreover we give upper and lower estimates for the solution.
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    Energy estimates for electro-reaction-diffusion systems with partly fast kinetics
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Glitzky, Annegret
    We start from a basic model for the transport of charged species in heterostructures containing the mechanisms diffusion, drift and reactions in the domain and at its boundary. Considering limit cases of partly fast kinetics we derive reduced models. This reduction can be interpreted as some kind of projection scheme for the weak formulation of the basic electro--reaction--diffusion system. We verify assertions concerning invariants and steady states and prove the monotone and exponential decay of the free energy along solutions to the reduced problem and to its fully implicit discrete-time version by means of the results of the basic problem. Moreover we make a comparison of prolongated quantities with the solutions to the basic model.
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    Energy estimates for continuous and discretized electro-reaction-diffusion systems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Glitzky, Annegret; Gärtner, Klaus
    We consider electro-reaction-diffusion systems consisting of continuity equations for a finite number of species coupled with a Poisson equation. We take into account heterostructures, anisotropic materials and rather general statistic relations. We investigate thermodynamic equilibria and prove for solutions to the evolution system the monotone and exponential decay of the free energy to its equilibrium value. Here the essential idea is an estimate of the free energy by the dissipation rate which is proved indirectly. The same properties are shown for an implicit time discretized version of the problem. Moreover, we provide a space discretized scheme for the electro-reaction-diffusion system which is dissipative (the free energy decays monotonously). On a fixed grid we use for each species different Voronoi boxes which are defined with respect to the anisotropy matrix occurring in the flux term of this species.