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11.1 The Highest Order Moment Closure

We found that the closure relation for the highest order moment is the critical issue with respect to numerical robustness of the solution algorithm for the implemented six moments model. The main part of our work consisted in studying different solutions to this closure problem.

We have shown that closure relations derived from theoretical considerations based on analytical distribution function models ([GKHS02] or maximum entropy principle) and relations derived from the cumulants of the distribution function [WSYM98] do not deliver satisfactory results. In contrast the approach consistent with bulk data which we introduced gives a numerically more robust closure and an accurate kurtosis, which is a prerequisite for modeling hot carrier effects.

Since all model parameters are obtained from bulk MC simulations the transport model is fit-parameter free and leaves us with 'no knobs to turn' [TI97]. The existence of many fit parameters is a particular inconvenience inherent in many energy-transport models [GTKS03]. This was found to be essential for higher-order models since the interplay between the various parameters is highly complex and the numerical stability of the whole transport model depends significantly on the choice of these parameters. In particular, our MC based model outperformed its counterparts based on analytical mobility models [GKGS01] significantly, both in terms of its numerical properties and in the quality of the simulation results. On the roadmap the important gate-lengths window of 100 to 25 nm may be covered by the six moments model.

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R. Kosik: Numerical Challenges on the Road to NanoTCAD