However, the initial results obtained from using a Scharfetter-Gummel style discretization, though often unstable, appeared to better reproduce the MC results. By using the double grid discretization convergence could be also achieved where the Scharfetter-Gummel style discretizations failed. The oscillations showing up in the result ruled out the closure as unphysical.
Discarding the restriction of to integer values we now take a somewhat different approach: by requiring consistency with bulk MC simulations we allow to take any real value and obtain a best match for a value of . This approach is analogous to our approach for determining bulk mobilities and relaxation times.
To investigate the influence of the highest order moment closure we consider a one-dimensional -- test-structure. The doping concentrations are taken to be and . The channel length is .
In Fig. 4.3 we show the relative error (ratio between sixth moment calculated from the MC moments , and and the real MC solution for ) of the closure for bulk and from the simulation result using the nm device. It can be seen that for high electrical field the error from the cumulant closure increases, which also explains the observed bad convergence behavior when a high bias is applied.
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 bulk data approach gives a numerically more robust closure and an accurate kurtosis, which is a prerequisite for modeling hot carrier effects.
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