On the Uniform Convergence of the Scharfetter-Gummel Discretization in One Dimension

A convergence analysis is given for the Scharfetter-Gummel discretization of proto-type one-dimensional continuity equations as arise in the drift-diffusion system modeling semiconductors. These are linear, second-order, boundary-value problems whose coefficient functions are O(1) but can have deriv...

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Bibliographic Details
Published in:SIAM journal on numerical analysis 1993-06, Vol.30 (3), p.749-758
Main Author: Gartland, Eugene C.
Format: Article
Language:English
Subjects:
Approximation
Boundary conditions
Boundary value problems
Continuity equations
Convergence of numerical methods
Difference equations
Diffusion
Doping (additives)
Electrons
Error rates
Exact sciences and technology
Mathematical functions
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Ordinary differential equations
Partial differential equations
Sciences and techniques of general use
Semiconductor devices
Semiconductor materials
Semiconductors
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Summary:A convergence analysis is given for the Scharfetter-Gummel discretization of proto-type one-dimensional continuity equations as arise in the drift-diffusion system modeling semiconductors. These are linear, second-order, boundary-value problems whose coefficient functions are O(1) but can have derivatives that are O(1/λ), for a small positive parameter λ. For such problems, discretizations "of Scharfetter-Gummel type" are proved to be first-order accurate on general meshes uniformly in λ in a strong global sense. These results improve upon previous analyses, where discrete convergence of O(h + λ |ln λ|) was believed to be the best possible.
ISSN:0036-1429
1095-7170
DOI:10.1137/0730037