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Exponentially fitted mixed finite volumes for energy balance models in semiconductor device simulation. (English) Zbl 1005.78514

Bock, Hans Georg (ed.) et al., ENUMATH 97. Proceedings of the 2nd European conference on numerical mathematics and advanced applications held in Heidelberg, Germany, September 28-October 3, 1997. Including a selection of papers from the 1st conference (ENUMATH 95) held in Paris, France, September 1995. Singapore: World Scientific. 188-197 (1998).
From the introduction: We deal with the two-dimensional numerical simulation of semiconductor devices using an advanced transport model, the so-called energy-balance (EB) equations. The fundamental contraints for the discretization are the conservation of electric field, current and energy fluxes, and the nonnegativity of the concentrations and of the temperatures of the carriers.
To this end, we propose efficient and accurate solvers for the EB equations in steady-state conditions. The well-known Gummel’s decoupled algorithm is employed to solve iteratively the full system, while the discretization is based on the use of cell-centred mixed finite volume methods. These latter are derived from the standard Raviart-Thomas finite elements of lowest degree through a suitable extension of the quadrature formula to diagonalize the element mass matrix.
In the case of linearized Poisson equation the coefficient matrix is symmetric, positive definite and diagonally dominant, while in the case of the linearized current continuity and energy-balance equations, the quadrature rule is coupled with a Scharfetter-Gummel approximation of interelement fluxes to provide the needed amount of upwinding and the stability of the approximation. Indeed, the resulting matrices turn out to be \(M\)-matrices diagonally dominant with respect to the columns and the corresponding right-hand sides can be shown to be nonnegative. These properties ensure that a discrete maximum principle holds, and, in particular, that both carrier densities and temperatures are nonnegative.
For the entire collection see [Zbl 0949.00502].

MSC:

78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
82D37 Statistical mechanics of semiconductors
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N38 Boundary element methods for boundary value problems involving PDEs
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