On the Discretization of van Roosbroeck’s Equations with Magnetic Field
We investigate qualitative properties of the drift-diffusion model of carrier transport in semiconductors when a magnetic field is present. At first the spatially continuous problem is studied. Essentially, global stability of the thermal equilibrium is shown using the free energy as a Lyapunov function. This result implies exponential decay of any perturbation of the thermal equilibrium. Next, we introduce a time discretization that preserves the dissipative properties of the continuous system and assumes no more than the naturally available smoothness of the solution. Finally, we present a space discretization scheme based on weak and consistent definitions of discrete gradients and currents. Starting with a fundamental result on global stability (dissipativity) of the classical Scharfetter-Gummel scheme (without magnetic field), we adapt this scheme with respect to magnetic fields and study the M-property of the associated matrix. For two dimensional applications we formulate sufficient conditions in terms of the grid geometry and the modulus of the magnetic field such that our scheme is dissipative and yields positive solutions. These conditions cover fields up to |b|µv ≈ 0.5 for very fine grids. This means approximately 200 Tesla for Silicon. Sufficient for some typical semiconductor sensor applications. The grid requirements might become prohibitive for large magnetic fields and complex three dimensional structures. Our techniques of defining discrete currents can be applied to similar situations, especially if projections of currents are involved in model parameters.
KeywordsGlobal Stability Discrete Gradient Large Magnetic Field Grid Geometry Grid Requirement
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- H. Gajewski, K. Gärtner, On the discretization of van Roosbroeck’s equations with magnetic field, Technical Report 94/14 Integrated Systems Laboratory, ETH Zurich, to appear in ZAMM. and literature cited in .Google Scholar