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Author: Farrell, Patricio × Author: Koprucki, Thomas × Has files: Yes × Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik ×

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Now showing 1 - 5 of 5
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    Highly accurate quadrature-based Scharfetter-Gummel schemes for charge transport in degenerate semiconductors
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Patriarca, Matteo; Farrell, Patricio; Fuhrmann, Jürgen; Koprucki, Thomas
    We introduce a family of two point flux expressions for charge carrier transport described by drift-diffusion problems in degenerate semiconductors with non-Boltzmann statistics which can be used in Voronoi finite volume discretizations. In the case of Boltzmann statistics, Scharfetter and Gummel derived such fluxes by solving a linear two point boundary value problem yielding a closed form expression for the flux. Instead, a generalization of this approach to the nonlinear case yields a flux value given implicitly as the solution of a nonlinear integral equation. We examine the solution of this integral equation numerically via quadrature rules to approximate the integral as well as Newtons method to solve the resulting approximate integral equation. This approach results into a family of quadrature-based Scharfetter-Gummel flux approximations. We focus on four quadrature rules and compare the resulting schemes with respect to execution time and accuracy. A convergence study reveals that the solution of the approximate integral equation converges exponentially in terms of the number of quadrature points. With very few integration nodes they are already more accurate than a state-of-the-art reference flux, especially in the challenging physical scenario of high nonlinear diffusion. Finally, we show that thermodynamic consistency is practically guaranteed.
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    Computational and analytical comparison of flux discretizations for the semiconductor device equations beyond Boltzmann statistics
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Farrell, Patricio; Koprucki, Thomas; Fuhrmann, Jürgen
    For a Voronoi finite volume discretization of the van Roosbroeck system with general charge carrier statistics we compare three thermodynamically consistent numerical fluxes known in the literature. We discuss an extension of the Scharfetter-Gummel scheme to non-Boltzmann (e.g. Fermi-Dirac) statistics. It is based on the analytical solution of a two-point boundary value problem obtained by projecting the continuous differential equation onto the interval between neighboring collocation points. Hence, it serves as a reference flux. The exact solution of the boundary value problem can be approximated by computationally cheaper fluxes which modify certain physical quantities. One alternative scheme averages the nonlinear diffusion (caused by the non-Boltzmann nature of the problem), another one modifies the effective density of states. To study the differences between these three schemes, we analyze the Taylor expansions, derive an error estimate, visualize the flux error and show how the schemes perform for a carefully designed p-i-n benchmark simulation. We present strong evidence that the flux discretization based on averaging the nonlinear diffusion has an edge over the scheme based on modifying the effective density of states.
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    On thermodynamic consistency of a Scharfetter-Gummel scheme based on a modified thermal voltage for drift-diffusion equations with diffusion enhancement
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Koprucki, Thomas; Rotundo, Nella; Farrell, Patricio; Doan, Duy Hai; Fuhrmann, Jürgen
    Driven by applications like organic semiconductors there is an increased interest in numerical simulations based on drift-diffusion models with arbitrary statistical distribution functions. This requires numerical schemes that preserve qualitative properties of the solutions, such as positivity of densities, dissipativity and consistency with thermodynamic equilibrium. An extension of the Scharfetter-Gummel scheme guaranteeing consistency with thermodynamic equilibrium is studied. It is derived by replacing the thermal voltage with an averaged diffusion enhancement for which we provide a new explicit formula. This approach avoids solving the costly local nonlinear equations defining the current for generalized Scharfetter-Gummel schemes.
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    Numerical methods for drift-diffusion models
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Farrell, Patricio; Rotundo, Nella; Doan, Duy Hai; Kantner, Markus; Fuhrmann, Jürgen; Koprucki, Thomas
    The van Roosbroeck system describes the semi-classical transport of free electrons and holes in a self-consistent electric field using a drift-diffusion approximation. It became the standard model to describe the current flow in semiconductor devices at macroscopic scale. Typical devices modeled by these equations range from diodes, transistors, LEDs, solar cells and lasers to quantum nanostructures and organic semiconductors. The report provides an introduction into numerical methods for the van Roosbroeck system. The main focus lies on the Scharfetter-Gummel finite volume disretization scheme and recent efforts to generalize this approach to general statistical distribution functions.
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    Comparison of thermodynamically consistent charge carrier flux discretizations for Fermi-Dirac and Gauss-Fermi statistics
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Farrell, Patricio; Patriarca, Matteo; Fuhrmann, Jürgen; Koprucki, Thomas
    We compare three thermodynamically consistent ScharfetterGummel schemes for different distribution functions for the carrier densities, including the FermiDirac integral of order 1/2 and the GaussFermi integral. The most accurate (but unfortunately also most costly) generalized ScharfetterGummel scheme requires the solution of an integral equation. We propose a new method to solve this integral equation numerically based on Gauss quadrature and Newtons method. We discuss the quality of this approximation and plot the resulting currents for FermiDirac and GaussFermi statistics. Finally, by comparing two modified (diffusion-enhanced and inverse activity based) ScharfetterGummel schemes with the more accurate generalized scheme, we show that the diffusion-enhanced ansatz leads to considerably lower flux errors, confirming previous results (J. Comp. Phys. 346:497-513, 2017).