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.