Figure 3.6 shows a comparison between the present analytical
model and the exact numerical solution. The numerical calculations are carried
out by using the rigorous transient model in MINIMOS. In the numerical
calculations the traps are placed in the middle of the channel where the
conditions at the interface are spatially uniform. Bulk is uniformly doped.
To avoid the parasitic geometric component in 
for
at 
, we consider the DC component
of the surface net recombination as the numerical solution. Care has been taken
to obtain the periodic steady-state solution. Note that the comparison shown in
Figure 3.6 does not include any fitting.
The analytical model is accurate at the rising edge of the characteristics,
including the region between 
and the upper plateau. It is also
accurate in the deep subthreshold region, as is demonstrated by plotting the
results in a lin-log scale (lower figure). The analytical results seems to
be shifted to the left with respect to the numerical results by about 
on
the voltage axis. This discrepancy can be partially attributed to the error in
determining the exact 
and 
levels, due to
a step approximation of the non-steady-state occupancy function. When
accounting for the 
correction of these emission levels , as
derived in the next section, the 
reduces and becomes close to the
numerical results in the subthreshold region, but do not match them exactly,
as shown in Figure 3.7.
The analytical model overestimates the current at the maximum of the
characteristics where the current depends on the electron and the hole
non-steady-state emission levels. This error also originates due to a step
approximation of the non-steady-state emission, where we assume that all traps
between the levels 
and 
given
by 3.66 and 3.81 are active in the charge
pumping. However, many traps below and above
also emit, which reduces , as is discussed in the next section.
When accounting for this effect the maximal current calculated analytically
closely match the numerical results.
A second observation on Figures 3.5 and 3.6
is that the threshold voltage approaches the beginning of the
upper plateau when either the top-level duration decreases, the rise increase
or the fall time increase.
The present phenomenological theory does not cover all cases
, but is provides us with
a good understanding of the charge-pumping effect. In the derivation of the
model we have explicitly accounted only for the electron capture at the top
level in the calculation of the amount of filled traps. However, the electron
capture occurs at the rising and even falling edge of the gate pulses as well.
For a high 
the amount of the electron capture at the
rising edge is insignificant, since all traps are completely filled during
, anyway. For a low 
, the electron capture at the
rising and the falling edges becomes very important if 
or
. These conditions are rarely found for the trapezoidal waveform
in the practice, but are typical for the triangular and the sawtooth waveform.
For the later two, the present model shows a shift of the 
characteristics towards the lower gate top levels in the subthreshold region.
It yields, however, an exact maximal current. Related calculations are shown
in Figure 3.7.