In the preceeding section a model of the charge-pumping current is presented
for the large-signal trapezoidal gate pulses. A step approximation of the
non-steady-state occupancy function is assumed in the derivation. In the
engineering applications of charge pumping the most important quantity is
the maximal charge-pumping current at the upper plateau of the 
curve. In this region, the active energy interval contributing to the
current is determined by the non-steady-state emission levels of electrons
and holes 
. In order to evaluate the error
introduced by a step approximation of the occupancy function we will discuss
the three-level gate waveform shown in Figure 3.3, instead of
the trapezoidal waveform. In the three-level technique the electron emission
time is exactly known and equals to the mid-level duration 
if the falling
edges are sufficiently abrupt (
). By considering the
three-level method, an eventual error in the determination of the emission time
, which can result from an inaccurate 
and 
,
is avoided. In the three-level method, it is assumed that all available traps
are filled by electrons at the top level and recombined by holes at the bottom
level. When the falling edges are abrupt the 
is given by
where 
is the steady-state occupancy function at the mid level
and 
is the lower energy boundary
(
in the conditions of interest here).
We further assume 
for traps of interest and a level
high sufficiently that the hole capture is negligible during .
Consequently, 
.
For not too short the steady-state occupancy function 
may also
be approximated by a step-like function at 
and 
holds for traps of interest.
By introducing a level 
so that
expression 3.105 reduces to
where 
. The subintegral function
is the solution for the occupancy function
in the non-steady-state emission mode when
. A general solution, also valid for
short time intervals is shown in 3.105. The
function 
is independent of , namely of the observation time
. It is plotted by the solid line in
Figure 3.9 (left). This function is not
symmetrical with respect to . The width of the transition region
from an occupancy of 
to an occupancy of 
is given by
For example, the interval from 
to 
trap occupancy has a width
of 
, which is not negligible with respect to the band gap.
On the other hand, in a step approximation of the non-steady-state occupancy function one may write
is the emission level we are looking for. In common cases, it
may be assumed that 
and
. The emission level
can be derived by equaling given by
expressions 3.105 and 3.109.
In a standard approximation 
, where
is defined by 3.106 [206][154].
Note that a general expression for valid also for 
has
been discussed in [435][232]. However, observing the non-steady-state
occupancy function shown in Figure 3.9 (left) we
conclude that many traps below the level have emitted during ,
while the traps above are emptied almost in complete. Therefore,
. It follows the equation with respect to
Let us assume a constant 
. A substitution of
in 3.110 yields the equation
The right-hand-side in 3.111 is the 
function,
whereas the left-hand-side is the 
function ([3]). Taking
advantage of the relationship 
, where
is the Euler constant, one obtains
for the solution of 3.111.
Finally,
where is defined by 3.106. For the hole
emission level the correction becomes 
. The 
shift is small, but not negligible in comparison with the active energy
interval which is less than 
. When accounting for both, the
electron and the hole non-steady-state emission the reduction of the active
energy interval becomes 
, which is 
at room temperature.
This can result in 
reduction in .
For real Si-SiO
interfaces 
can significantly vary with energy
along the forbidden band, particularly close to the band edges. To estimate
the correction of in these cases we assume an exponential
test distribution of . A simple replacement in
equation 3.110 leads to
is the Gamma function. The factor
for small 
, which is a typical case
in practice. For example, assuming a variation of the trap density in the
upper half of the band gap so that
at the mid-gap and
at the band edge, it
follows 
and 
. Therefore, an
exponentially varying function also can be nearly considered
as a slowly varying within the interval around 
where the occupancy
function 
changes rapidly. As a conclusion, the
correction 
is a valid approximation in all cases of
practical interest.
In the previous section we have obtained that the analytical model
overestimates the maximal in Figures 3.6. When the
shift is taken into account for both, and
levels the maximal current agrees well with the numerical
calculation, as is shown for the same example in Figure 3.7.
In order to analyze the accuracy of the analytical model on large time scales,
we calculate the charge recombined per one period 
over 
decades of
the fall and rise times. The calculated family of the curves is shown in
Figure 3.10, together with the rigorous numerical
result. The assumed trap density is linearly distributed across the band gap
and is shown in
Figure 3.11. The gate signal switches the interface
from strong inversion at 
to strong accumulation at
. In this case, 
and
are fulfilled for all 
and 
, including
the shortest considered (see Figure 3.8 for a similar
device). Therefore, the is solely determined by the non-steady-state
emission levels of electrons and holes. In the numerical calculations the
traps only reside in the channel where the spatial conditions are uniform. For
the numerical results we assume the DC component of the net generation rates
, because of a very
large geometric component occurring in 
for 
in the 
-range. Note
that the frequency applied in the calculations is unacceptably high for
and low for 
and 
, regarding practical measurements.
In all analytical calculations we adopt 
and
.
When assuming a most simple approximation 
and
the analytical model remarkably overestimates
at short and 
in the -range and underestimates at long
and in the 
-range (these results are not shown here). In a
better approach and 
are calculated for a specified
and , as is explained in the preceding section. The resulting is
shown by the solid lines in Figure 3.10. We still
overestimate in the -range, but less than for
. At very
short fall times 
holds, which results in a longer
and a lower when accounting for a proper
. If the correction 
is taken into account the
agreement with the numerical result is excellent in the sub-
range
(dotted curves which are shown for some time decades). However, the
is too low at long and in the later case.
We attribute the later finding to the variation in the capture-onset levels
and 
with and , respectively. This explanation
is fully supported by the numerical simulations. By observing the evolution of
the terminal currents and the total electron and hole interface net generation
rates, 
and 
, during the fall edge of
the gate pulse we found that 
becomes dominant over
at 
for 
and at
for 
. However, while in the case for
the hole capture stops rapidly the electron emission, in the
case for the traps also continue to emit in the condition
until the level 
is
exceeded. Moreover, the study of the geometric component presented in
Section 3.4 shows that the electrons emitted in the conduction band
when a significant hole capture occurs are mostly not collected by the source
and drain junctions, but are injected into the bulk, thereby building an
additional geometric component in the measured charge-pumping current. The
analytically calculated emission-onset voltages are 
and 
for and 
, respectively. The device voltages are
and 
. Consequently, the
variation of with is of the same order as the variation of
with . Only the later effect is included in our present
analytical model. It is worth to point out that the former effect was never
considered, modeled nor included in the charge-pumping theory. The same
conclusions hold when considering the level as a function of .
Note that increasing leads to an increase in , shortening of
and consequently, increases.
We propose that the changes in with can be analytically
calculated by adopting the assumption
. It is a trivial task
to reduce the problem to a coupled system of two implicit algebraic equations,
which can be simply solved. These results will be presented in a further study.
Finally, it is concluded that for the trapezoidal waveform an exact determination of the non-steady-state emission time for electrons and holes is not possible without significant increasing of model complexity.
In the following, the extraction of the energy distribution of trap density
from the experimental characteristics is studied.
We employ the numerically calculated data shown in
Figure 3.10 instead of the experimental data. This
characteristic is obtained assuming a linear trap distribution shown by the
solid line in Figure 3.11. The deviation of the
extracted from the assumed one serves as a direct monitor for
the accuracy of the extraction methods.
When the is controlled by the electron emission level the
differentiation of expression 3.105 yields
where we adopted several approximations mentioned above. The subintegral
function 
is sharply peaked at , as is
shown in Figure 3.9. The main contribution to the
integral comes from the region around and the overall integral of this
function is 
for the interval 
. Therefore, we may
approximate
The level actually scanned is given
by 3.106. Formula 3.114 is
valid if 
is not a function of energy 
. This method has been
proposed in [154] and is in the standard use in engineering practice.
In the three-level technique, is controlled directly [377][206].
For the trapezoidal waveform, 
holds. When applying
relationship 3.115, it is usually assumed that
.
After applying method 3.115 on the numerical data shown in Figure 3.10 we have obtained the trap distributions presented in Figure 3.11. We have also carried out the same procedure for an initially assumed uniform trap distribution, which is shown in Figure 3.11 as well. In both cases, the trapezoidal waveform is applied on the gate. Inspecting the extracted versus the assumed distributions we conclude that the present technique leads to an error in the amount of the traps density. Moreover, a shift on the energy scale cannot be excluded as well. The errors could be attributed to
resulting from 3.66.
The variations in and influence ,
thereby inducing an error in the calculation of the position
which is actually scanned.
due to a step approximation of the
occupancy function.
.
The extracted trap densities depicted by 
and 
for the linear
and uniform distributions, respectively, are calculated by accounting for the
variation of with . The calculated 
is
used in obtaining the emission time which is applied to determine
the scanned position . For the data shown by 
and
the level is calculated assuming resulting
from the approximation 
. Note that
is used in all calculations. We conclude that the error in the
currently scanned position due to an inaccurate is small and cannot
account for the disagreement between the extracted and the assumed linear
distribution.
To estimate the error due to a finite width of the transition region in the non-steady-state occupancy function we introduce a differential charge-pumping current
like the signal in DLTS measurements. The parameter 
determine the
width of the time steps used in the charge-pumping measurements. Typically,
the current is measured in the subsequent moments 
and 
and
formula 3.115 is applied on finite differences.
The normalized subintegral function
determines the width of the scanned region. This universal function is
independent of the actual time , but depends only on the time window.
The integral of 
in the interval equals to
. By subtracting the occupancy functions in the two closely set moments
(e.g. 
) the long tails in the region 
cancel each other, yielding a nearly symmetric function around
. The function is sharply peaked and its maximal
contribution to the integral comes from the region around the maximum at
. For a common value the
maximum is shifted by only 
towards the lower energies from
. In addition, the variations in the trap distribution can be
neglected within the window , also for rapidly changing
distributions in practice.
The disagreement in the amplitude for the uniform distribution clearly
originates from relationship 3.115. This
relationship is, however, correct, but its application to the trapezoidal
waveform introduces an error when
is assumed. In fact,
relationship 3.115 refers to the
data. The measured data are
. The relation between the later two
data-sets is given over the factor 
. For the
trapezoidal pulses it equals to
Finally, formula 3.115 yields for the trapezoidal pulses
The second term in the brackets at the right-hand-side was always neglected. This factor is negative. Therefore, the calculated trap distribution in the energy space is lower when this factor is not accounted for. This effect is responsible for an enhanced error when approaching the mid-gap and for a strong underestimation in the trap density close to the band-edges. Note that this effect does not exist in the three-level charge-pumping measurements.
