Two-Dimensional Exponential Fitting and Applications to Drift-Diffusion Models

Brezzi, Franco; Marini, L. D.; Pietra, Paola

doi:10.1137/0726078

Cited by 152 publications

(127 citation statements)

References 12 publications

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“…Consider as an example the two-dimensional cell j in Fig. 1, for which (2.3) can be written as 4) where N (j) is the index set of neighbouring grid points of x j and where j,k is the segment or edge of the boundary j = ∂ j connecting the adjacent cells j and k . The orientation of j is counterclockwise.…”

Section: Finite Volume Discretisationmentioning

confidence: 99%

“…where the velocity u and the source term s are given by 4) respectively. The velocity is a smoothly varying function of x whereas the source term has a sharp peak at x = 1 2 , causing a steep interior layer, provided 0 < ε 1; see Fig.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…For this application these schemes are known as the ScharfetterGummel scheme; see e.g. [3,4,24]. An extension of this scheme is due to Miller [16], who included the recombination term in the fluxes.…”

Section: Introductionmentioning

confidence: 99%

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The Finite Volume-Complete Flux Scheme for Advection-Diffusion-Reaction Equations

Boonkkamp

Anthonissen

2010

J Sci Comput

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We present a new finite volume scheme for the advection-diffusion-reaction equation. The scheme is second order accurate in the grid size, both for dominant diffusion and dominant advection, and has only a three-point coupling in each spatial direction. Our scheme is based on a new integral representation for the flux of the one-dimensional advection-diffusion-reaction equation, which is derived from the solution of a local boundary value problem for the entire equation, including the source term. The flux therefore consists of two parts, corresponding to the homogeneous and particular solution of the boundary value problem. Applying suitable quadrature rules to the integral representation gives the complete flux scheme. Extensions of the complete flux scheme to two-dimensional and time-dependent problems are derived, containing the cross flux term or the time derivative in the inhomogeneous flux, respectively. The resulting finite volume-complete flux scheme is validated for several test problems.

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Section: Finite Volume Discretisationmentioning

confidence: 99%

Section: Numerical Examplesmentioning

confidence: 99%

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The Finite Volume-Complete Flux Scheme for Advection-Diffusion-Reaction Equations

Boonkkamp

Anthonissen

2010

J Sci Comput

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“…The inhomogeneous case can be transformed into the homogeneous case by substracting a special function satisfying the boundary conditions. We consider the following decoupled linearised continuity équation for the électron concentration n and the appropriate boundary conditions 2 and H x D (n) = {v e H l (£2) : i; \ hÜD = o}. We use | .…”

Section: Statement and Reformulation Of The Problemmentioning

confidence: 99%

“…Corresponding to the two meshes T h and D k , we now construct two finitedimensional spaces L h <^L 2 …”

Section: Definition 32 : the Dirichlet Tessellation D H Correspondmentioning

confidence: 99%

An analysis of the Scharfetter-Gummel box method for the stationary semiconductor device equations

Miller¹,

Wang²

1994

ESAIM: M2AN

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Mixed finite volume methods for semiconductor device simulation

Sacco

Saleri

1997

Numer. Methods Partial Differential Eq.

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We deal with the two-dimensional numerical solution of the Van Roosbroeck system, widely employed in modern semiconductor device simulation. Using the well-known Gummel's decoupled algorithm leads to the iterative solution of a nonlinear Poisson equation for the electric potential and two linearized continuity equations for the electron and hole current densities. The numerical approximation is based on the dual mixed formulation for a self-adjoint second-order elliptic operator by using the Raviart-Thomas (RT) finite elements of lowest degree on a triangular partition of the device domain. In this article, we propose a suitable variant of the RT method, based on the diagonalization of the element mass matrix. This is achieved by use of an appropriate numerical integration that eliminates the fluxes and gives rise to a cell-centered finite volume scheme for the scalar unknown with the same approximation properties of the mixed approach, but at a reduced computational cost. The above procedure suggests also a natural way to introduce in the frame of the classical Box Method (BM) suitable vector basis functions (edge elements) to represent the current field over each mesh triangle. This issue may be profitably employed both as a postprocessing tool, as well as a technique for solving the current continuity equations when source terms depending on the current itself are included in the mathematical model. Simulations of realistic semiconductor devices are then included to demonstrate the accuracy and stability of the new method. I. BASIC MATHEMATICAL MODEL FOR SEMICONDUCTOR DEVICE SIMULATIONThe classical steady-state Van Roosbroeck system [1], usually addressed as drift-diffusion equations and commonly used for modeling the electron and hole charge flow throughout a semiconductor device, is represented by the following set of conservation laws [2], [3]:

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Two-Dimensional Exponential Fitting and Applications to Drift-Diffusion Models

Cited by 152 publications

References 12 publications

The Finite Volume-Complete Flux Scheme for Advection-Diffusion-Reaction Equations

The Finite Volume-Complete Flux Scheme for Advection-Diffusion-Reaction Equations

An analysis of the Scharfetter-Gummel box method for the stationary semiconductor device equations

Mixed finite volume methods for semiconductor device simulation

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