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Author: Fuhrmann, Jürgen × End date: 2009 × Start date: 2000 × Has files: Yes × Type: Text ×

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Now showing 1 - 4 of 4
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    Convergence of a finite volume scheme to the eigenvalues of a Schrödinger operator
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Koprucki, Thomas; Eymard, Robert; Fuhrmann, Jürgen
    We consider the approximation of a Schrödinger eigenvalue problem arising from the modeling of semiconductor nanostructures by a finite volume method in a bounded domain $OmegasubsetR^d$. In order to prove its convergence, a framework for finite dimensional approximations to inner products in the Sobolev space $H^1_0(Omega)$ is introduced which allows to apply well known results from spectral approximation theory. This approach is used to obtain convergence results for a classical finite volume scheme for isotropic problems based on two point fluxes, and for a finite volume scheme for anisotropic problems based on the consistent reconstruction of nodal fluxes. In both cases, for two- and three-dimensional domains we are able to prove first order convergence of the eigenvalues if the corresponding eigenfunctions belong to $H^2(Omega)$. The construction of admissible meshes for finite volume schemes using the Delaunay-Voronoï method is discussed. As numerical examples, a number of one-, two- and three-dimensional problems relevant to the modeling of semiconductor nanostructures is presented. In order to obtain analytical eigenvalues for these problems, a matching approach is used. To these eigenvalues, and to recently published highly accurate eigenvalues for the Laplacian in the L-shape domain, the results of the implemented numerical method are compared. In general, for piecewise $H^2$ regular eigenfunctions, second order convergence is observed experimentally.
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    Electronic states in semiconductor nanostructures and upscaling to semi-classical models
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Koprucki, Thomas; Kaiser, Hans-Christoph; Fuhrmann, Jürgen
    In semiconductor devices one basically distinguishes three spatial scales: The atomistic scale of the bulk semiconductor materials (sub-Angstroem), the scale of the interaction zone at the interface between two semiconductor materials together with the scale of the resulting size quantization (nanometer) and the scale of the device itself (micrometer). The paper focuses on the two scale transitions inherent in the hierarchy of scales in the device. We start with the description of the band structure of the bulk material by kp Hamiltonians on the atomistic scale. We describe how the envelope function approximation allows to construct kp Schroedinger operators describing the electronic states at the nanoscale which are closely related to the kp Hamiltonians. Special emphasis is placed on the possible existence of spurious modes in the kp Schroedinger model on the nanoscale which are inherited from anomalous band bending on the atomistic scale. We review results of the mathematical analysis of these multi-band kp Schroedinger operators. Besides of the confirmation of the main facts about the band structure usually taken for granted ...
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    A model of an electrochemical flow cell with porous layer
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Ehrhardt, Matthias; Fuhrmann, Jürgen; Linke, Alexander
    In this paper we discuss three different mathematical models for fluid-porous interfaces in a simple channel geometry that appears e.g. in thin-layer channel flow cells. Here the difficulties arise from the possibly different orders of the corresponding differential operators in the different domains. A finite volume discretization of this model allows to calculate the limiting current of the H_2 oxidation in a porous electrode with platinum catalyst particles.
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    Experimental and numerical model study of the limiting current in a channel flow cell with a cirvular electrode
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Fuhrmann, Jürgen; Zhao, H.; Holzbecher, E.; Langmach, H.; Chojak, M.; Halseid, R.; Jusys, Z.; Behm, R.J.
    We describe first measurement in a novel thin-layer channel flow cell designed for the investigation of heterogeneous electrocatalysis on porous catalysts. For the interpretation of the measurements, a macroscopic model for coupled species transport and reaction, which can be solved numerically, is feasible. In this paper, we focus on the limiting current. We compare numerical solutions of a macroscopic model to a generalization of a Leveque-type asymptotic estimate for circular electrodes, and to measurements obtained in the aforementioned flow cell. We establish, that on properly aligned meshes, the numerical method reproduces the asymptotic estimate. Furthermore, we demonstrate, that the measurements are partially performed in the sub-asymptotic regime, in which the boundary layer thickness exceeds the cell height. Using the inlet concentration and the diffusion coefficient from literature, we overestimate the limiting current. On the other hand, the use of fitted parameters leads to perfect agreement between model and experiment.