On the Discretization of van Roosbroeck's Equations with Magnetic Field

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Abstract
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 not 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‐Tummel scheme (without magnetic field), we adapt this scheme with respect to magnetic fields and study the M‐property o the associated matrix. For two dimensional applications explicit conditions on the smallest inner angle and the modulus of the magnetic field are derived.

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