When the interface is strongly accumulated during the gate bottom level the charge-pumping current is given by
This relationship models the rising edge of the 
characteristics. An analogous relationship holds when the gate top level
drives the interface into strong inversion, while a partial or a complete
capturing of holes occurs at the bottom level. The later relationship describes
the falling edge of the 
characteristics.
When the pulse edges are abrupt the current is maximal
where 
and 
.
For small amplitudes of the gate signal an incomplete capturing of both,
electrons at the top levels and holes at the bottom levels occur
simultaneously, as is the case in the small-signal charge-pumping technique
(Section 3.1.1). Let us suppose that traps with density 
at a particular energy 
are occupied with a probability 
at
the beginning of the top level 
. The number of empty traps is
. After filling by an
exponential law the total number of filled traps becomes
at the
end of the top level. During 
the traps are emptied by the hole capture,
obeying an exponential law 
. Equaling the number of the
traps which are filled after with the number of the filled traps at the
beginning of in the periodic-steady-state condition, it follows the
factor
is given by e.g. the total number of holes captured during
The discussed energy intervals associated with the top and bottom levels and
the pulse edges are shown in Figure 3.5. Moreover, we also
show the scaled and 
currents and the parameter
. To calculate the presented characteristics all
relevant quantities in the MOS structure, like 
, 
,
, 
, 
, are numerically calculated for each
gate top level by using the one-dimensional analytical model described in
Section 2.2. Very low trap density of
is assumed to avoid an influence of the charge-potential
feedback effect (the shift on the voltage axis is less than 
), in order
to make the comparison fair. The quantities 
, 
,
and 
are constant and depend on the device
parameters, 
and 
. They are calculated only ones. We have assumed
and 
in the calculation of the
non-steady-state emission times.
A first observation is that the charge-pumping threshold defined
by 3.64 assuming 
has a value of 
and lies
significantly below the device threshold 
. Moreover, the onset
voltage 
is higher than 
due to a short 
.
Therefore, 
holds in this case.
The splitting between the levels 
and 
occurs after 
exceeds the threshold 
, because of
(this fact can be understood by help of
expression 3.67). The splitting between
and takes place when 
.
As interesting, the threshold voltage does not correspond neither
to the beginning of the upper plateau of the curve nor it
relates to 
of the maximal .
The characteristics reflects three specific parts
; almost linearly increases in a lin-log scale.
(
). The
increases because the active energy interval enlarges with
increasing .
the current is strongly constant and
governed solely by the emission processes at the pulse edges.
According to the previous results, two characteristic parts should be recognized
on the rising edge of the characteristics for low pulse
frequencies. There are, however, at least two effects which cause the stretch
out of the rising edge and, eventually, merging of these two parts on the
characteristics of real MOSFETs:
is strongly dependent on the
surface concentration . The fluctuations in the surface
potential cause large fluctuations in the free carrier concentration
. Thereby, large variations in the contribution to must
be expected across the interface, yielding a stretch-out of the ideal
characteristics, as pointed out in [49].
