To investigate the accuracy of the six moments (SM) model we
consider a series of one-dimensional -
-
test-structures. Although these structures are
not of practical relevance, they still display similar features as contemporary
MOS and bipolar transistors like velocity overshoot and a mixture of a hot and
a cold distribution function in the 'drain' region.
Here we present results of numerical solutions of six moments models (SM) and compare them to self-consistent Monte Carlo data (SCMC). Relaxation times and mobilities are extracted from bulk Monte Carlo data as a function of the carrier temperature and the doping.
In addition to the SM model we consider the
corresponding energy-transport (ET) model formulated in moments
(see Equations 2.24, 2.25),
where the equation for
is kept but the equation
for the energy-flux
is closed with
,
corresponding to a heated Maxwellian distribution.
This decouples the equation for
from the lower order equations and provides
an estimate for
and thus
[GKGS01], [SYT+96].
The doping concentrations
were taken to be
and
. The channel length was
varied from
down to
while maintaining a
maximum electric field of
.
A comparison of the average
velocity
and the kurtosis
obtained from the SM and ET models
with the SCMC simulation is shown in Fig. 4.4 and
Fig. 4.5 for the
nm device. The
spurious velocity overshoot is significantly reduced in the SM model,
consistent with previous results [GKS01], while the kurtosis produced by
the ET model is only a poor approximation to the MC results.
Despite
the fact that SM models provide the kurtosis of the distribution function, they
do not require a heated Maxwellian closure in the energy flux relation. This
has a significant impact on the resulting device currents for channel lengths
smaller
where the ET models show the well-known
overestimation of the device currents (see Figure 4.6). The results of the SM
model, on the other hand, stay close to the SCMC results which makes the SM
model a good choice for TCAD applications.
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