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4.2 Consistent Highest Order Moment Closure

We started our investigations with the Gaussian invariant closures for parameter integer values $ c=1,2,3$. We found that numerically the closures behave quite differently. Stable implementations were only obtained for $ c=3$.

However, the initial results obtained from $ c=2$ 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 $ c=2$ closure as unphysical.

Discarding the restriction of $ c$ to integer values we now take a somewhat different approach: by requiring consistency with bulk MC simulations we allow $ c$ to take any real value and obtain a best match for a value of $ c = 2.7$. 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 $ n^+$-$ n$-$ n^+$ test-structure. The doping concentrations are taken to be $ 5\times 10^{19}$ and $ 10^{17}  \mathrm{cm^{-3}}$. The channel length $ L_{\mathrm{c}}$ is $ 100   \mathrm{nm}$.

In Fig. 4.3 we show the relative error (ratio between sixth moment calculated from the MC moments $ M_0, M_2$, and $ M_4$ and the real MC solution for $ M_6$) of the closure for bulk and from the simulation result using the $ L_{{c}} =100 $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.

Figure 4.3: Relative error (ratio) of the closure for bulk and for the $ L_{{c}} =100 $nm device. Error from cumulant closure increases with high bias. Best fit for $ c = 2.7$.

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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R. Kosik: Numerical Challenges on the Road to NanoTCAD