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One important step in the fabrication of silicon-based integrated circuits is the creation of semiconducting areas by diffusion of dopant impurities into silicon. Complex models have been developed to investigate the redistribution of dopants and point defects. In general, numerical analysis of the resulting PDEs is the central tool to assess the modelling process. We present an adaptive approach which is able to judge the quality of the numerical approximation and which provides an automatic mesh improvement. Using linearly implicit methods in time and multilevel finite elements in space, we are able to integrate efficiently the arising reaction-drift-diffusion equations with high accuracy. Two different diffusion processes of practical interest are simulated.

Numerical Simulation of Single Species Dopant Diffusion in Silicon under Extrinsic Conditions
(1997)

In this article we consider a general model for phosphorus diffusion in silicon under extrinsic doping conditions. At such high concentrations we have to include the charged species and the internal electric field of the crystal, both of which can have profound effects on diffusion. In principle, this leads to a very large number of drift--diffusion--reaction equations: one for each charge state of every species, plus one Poisson equation to describe the internal electric field (in terms of the electron/hole concentration). The number of equations can be reduced substantially by making additional assumptions on the distribution of charge states and local equilibrium assumptions concerning the reaction terms. The resulting model turns out to be very interesting for numerical investigation. We solve the problem numerically in two space dimensions with the adaptive finite element program KARDOS and describe the numerical method used here to treat the resulting drift--diffusion--reaction problem.