A finite-volume scheme for the multidimensional quantum drift-diffusion model for semiconductors

Chainais-Hillairet, Claire; Gisclon, Marguerite; Jüngel, Ansgar

doi:10.1002/num.20592

Cited by 9 publications

(19 citation statements)

References 39 publications

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“…(32). Since the induced potential is stationary in the frame of reference moving with the external charged particle, / ind ðR;tÞ ¼ / ind ðx;y;z À vtÞ, we set the origin of the z axis to be at the instantaneous position of that particle.…”

Section: Resultsmentioning

confidence: 99%

“…(3), or the normal component of the associated Bohm force should vanish on @V. It is gratifying to recognize that the same types of ABCs often arise in the computational schemes that use variations of the QHD model in computer simulations of semiconductor devices. 32,33 We now specify V to be a region defined by jxj a in a cartesian coordinate system where R ¼ fx; y; zg, representing a metal slab with an infinite area and finite thickness of 2a, which is surrounded by a dielectric material with the relative dielectric constant that only depends on the position x. We further assume that an external point charge Z 1 e moves inside and parallel to the slab at some position x 0 2 ðÀa; aÞ with a velocity v directed along the z axis, so that the external charge density becomes q ext ðR; tÞ ¼ Z 1 e dðx À x 0 ÞdðyÞdðz À vtÞ.…”

Section: Model Descriptionmentioning

confidence: 97%

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Plasmon excitation in metal slab by fast point charge: The role of additional boundary conditions in quantum hydrodynamic model

Zhang

Song

et al. 2014

Physics of Plasmas

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We study the wake effect in the induced potential and the stopping power due to plasmon excitation in a metal slab by a point charge moving inside the slab. Nonlocal effects in the response of the electron gas in the metal are described by a quantum hydrodynamic model, where the equation of electronic motion contains both a quantum pressure term and a gradient correction from the Bohm quantum potential, resulting in a fourth-order differential equation for the perturbed electron density. Thus, besides using the condition that the normal component of the electron velocity should vanish at the impenetrable boundary of the metal, a consistent inclusion of the gradient correction is shown to introduce two possibilities for an additional boundary condition for the perturbed electron density. We show that using two different sets of boundary conditions only gives rise to differences in the wake potential at large distances behind the charged particle. On the other hand, the gradient correction in the quantum hydrodynamic model is seen to cause a reduction in the depth of the potential well closest to the particle, and a reduction of its stopping power. Even for a particle moving in the center of the slab, we observe nonlocal effects in the induced potential and the stopping power due to reduction of the slab thickness, which arise from the gradient correction in the quantum hydrodynamic model.

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Section: Resultsmentioning

confidence: 99%

Section: Model Descriptionmentioning

confidence: 97%

Plasmon excitation in metal slab by fast point charge: The role of additional boundary conditions in quantum hydrodynamic model

Zhang

Song

et al. 2014

Physics of Plasmas

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“…which after multiplication with a test function and integrating over Q implies the equations (21a). In return, (21) follows (15). Indeed, from Lemma 3.3 we have c > 0 a.e.…”

Section: Proofmentioning

confidence: 86%

“…Thus, the conditions (5), equalities (52), and (53) hold again. This implies that (15) and (21) are equivalent.…”

Section: Proofmentioning

confidence: 89%

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On generalized Poisson–Nernst–Planck equations with inhomogeneous boundary conditions: a‐priori estimates and stability

Kovtunenko

Zubkova

2016

Math Methods in App Sciences

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In this paper, we consider the strongly nonlinear Nernst–Planck equations coupled with the quasi‐linear Poisson equation under inhomogeneous, moreover, nonlinear boundary conditions. This system describes joint multi‐component electrokinetics in a pore phase. The system is supplemented by the force balance and by the volume and positivity constraints. We establish well‐posedness of the problem in the variational setting. Namely, we prove the existence theorem supported by the energy and the entropy a‐priori estimates, and we provide the Lyapunov stability of the solution as well as its uniqueness in special cases. Copyright © 2016 John Wiley & Sons, Ltd.

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Finite Volume Methods: Foundation and Analysis

Barth

Herbin

Ohlberger

2017

Encyclopedia of Computational Mechanics Second Edition

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Finite volume methods are a class of discretization schemes that have proven highly successful in approximating the solution of a wide variety of conservation law systems. They are extensively used in fluid mechanics, porous media flow, meteorology, electromagnetics, models of biological processes, semi-conductor device simulation and many other engineering areas governed by conservative systems that can be written in integral control volume form.This article reviews elements of the foundation and analysis of modern finite volume methods.The primary advantages of these methods are numerical robustness through the obtention of discrete maximum (minimum) principles, applicability on very general unstructured meshes, and the intrinsic local conservation properties of the resulting schemes. Throughout this article, specific attention is given to scalar nonlinear hyperbolic conservation laws and the development of high order accurate schemes for discretizing them. A key tool in the design and analysis of finite volume schemes suitable for non-oscillatory discontinuity capturing is discrete maximum principle analysis. A number of building blocks used in the development of numerical schemes possessing local discrete maximum principles are reviewed in one and several space dimensions, e.g. monotone fluxes, E-fluxes, TVD discretization, nonoscillatory reconstruction, slope limiters, positive coefficient schemes, etc. When available, theoretical results concerning a priori and a posteriori error estimates are given. Further advanced topics are then considered such as high order time integration, discretization of dffision terms and the extension to systems of nonlinear conservation laws.

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A finite-volume scheme for the multidimensional quantum drift-diffusion model for semiconductors

Cited by 9 publications

References 39 publications

Plasmon excitation in metal slab by fast point charge: The role of additional boundary conditions in quantum hydrodynamic model

Plasmon excitation in metal slab by fast point charge: The role of additional boundary conditions in quantum hydrodynamic model

On generalized Poisson–Nernst–Planck equations with inhomogeneous boundary conditions: a‐priori estimates and stability

Finite Volume Methods: Foundation and Analysis

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