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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