In the standard charge-pumping theory which is considered in the preceding section, it is regularly adopted that the free minority and majority carriers behave quasi-statically in MOSFETs. The only transient effects occur due to the generation and recombination through the interface and bulk traps. It has been proposed, already at the time of discovering the charge-pumping effect, that the charge-pumping current can also originates due to a finite time for removing the minority carriers out of the channel during the MOSFET turn-off, in addition to the current due to the transient generation-recombination over the traps [43]. As the time for the removing of the minorities is found to be strongly dependent on the geometry of MOSFETs, this current component is called ``geometric component'' [43]. In the measurements of trap properties, this current represents a parasitic effect. The primary interest was always to suppress it, either by using devices with a sufficiently short channel, by avoiding the application of very steep gate pulses or by biasing the source and drain junctions reversely [154][117]. A detailed study of the geometric component in the charge-pumping current is still missing. Recently, a very sensitive technique has been proposed to measure the geometric component also in the cases when the generation-recombination component largely dominates in the charge-pumping current [98].
In this section we studied the geometric component by means of numerical two-dimensional transient model of MOSFETs. After each transient simulation, the DC component of the net generation rates and the terminal currents is calculated for the periodic-steady-state conditions. Comparing the DC components of these quantities the geometric current component can be easily extracted. Such an approach represents a unique method to clarify the geometric component in detail [169]. We have studied bulk MOSFETs and SOI devices [192][169]. In this work, we focus on the bulk MOSFETs.
The family of the characteristics shown in Figure 3.12
represents the charge-pumping current calculated for several identical MOSFETs,
but with different gate length 
. The devices contain uniformly distributed
interface states. Trapezoidal pulses are applied on the gate. The fall time
is varied from 
to 
, while other parameters of the waveform
are kept constant (with exception of 
). A dramatical increase in the
is obtained for short , particularly in the
long-channel devices. Figure 3.13 shows the same family, but
calculated assuming a large reverse bias 
applied on the source and drain
junctions with respect to the bulk.
Some conclusions follow:
applied
on the junctions suppresses the geometric component [117]. The
calculations clearly show that only reduces this component,
namely the appearance of the effect is shifted towards shorter fall
times.
in
devices [144], or that for devices with 
the
geometric component vanishes if 
[117]. Note that the relative
importance of the geometric component depends on the total number of traps in
device.
The measured charge-pumping current is the DC component of the bulk
current 
, which consists of the electron and the hole component:
. In an ideal theory 
vanishes and 
equals to the DC component of the total net recombination rate in device (for
-channel MOSFETs). The results shown in Figure 3.12
are splitted into the electron and hole components of and presented
in Figure 3.14. It is obtained that
equals to the net recombination
rate of electrons 
and holes 
.
Therefore, always contains information on the traps.
. This claim is valid including
very short fall times.
, as well. The components 
and 
are
negligible (in -channel devices).
does not vanish at long , but shows a quasi-saturation.
In this region is much smaller than the component
which yields information on the traps. By reducing under some
threshold value, the begins to increase rapidly exceeding the
component and disturbing the measurements.
It is important to clarify which kind of devices we consider. The thickness
of the substrate 
is assumed to be only about 
. This depth is much
smaller than the thickness of the substrate in the real bulk MOSFETs (several
hundred 
) and even smaller than a typical depth of the wells in
CMOS technology. The assumed value of is, however, larger than the
thickness of the active part of devices. The minority carriers injected into
the bulk from the device active area are detected by the bulk contact positioned
at in our calculations. These carriers can also be collected by a 
-
junction laying closely under the device, as is employed in the experimental
technique in [98]. For devices made directly in the bulk (no wells),
the injected electrons have to travel huge distances to arrive at the physical
bulk contact. A part of them recombine over the deep traps in the
substrate while traveling, which increases the bulk hole current and decreases
the bulk electron current. This effect is not interesting for us. We only
analyze the processes in the active part of devices and all conclusions
given in this study refer to this assumption.
In order to find the physical mechanisms responsible for the geometric
component, we considered the evolution of the surface carrier concentrations
during the turn-off of an MOSFET. Figure 3.15 presents the
numerically calculated distributions of electrons and holes in the center of
the channel as a function of the distance from the interface at different
moments. Due to a significant time for removing the carriers from the inversion
layer through the junctions, as a consequence of a very short fall time of
applied on the device which has the gate-length of
, the minority carrier concentration near the interface is
much higher than its quasi-equilibrium value for a given 
.
Moreover, the concentration of the majority carriers is also higher than the
quasi-equilibrium value which corresponds to the current gate-bias. In a short
time interval (here of about 
) high concentrations of both, electrons
and holes are presented at the interface simultaneously. We propose three
phenomena which could occur as result of a fast switching at the falling edge
of the gate pulses:
can reduce the emission
of electrons from the interface traps by keeping the quasi-Fermi level
high during the fall time. This effect is able to increase
the 
in a limited amount, since it only cancels the lost of the
trapped charge through the emission. Moreover, the traps emit from a very
narrow energy range for very short . Accordingly, this effect is
expected to be of a minor importance.
or 
(whichever is lower), the duration of the process 
and
the concentration of the trapping sites 
. The increase of the
charge-pumping current due to an enhanced recombination is
All holes recombined by this effect contribute to the charge-pumping
current. If the process lasts a long time the recombined current can
be large, which could explain a significant increase in the geometric
component at short fall times, as proposed in [479][83]. In a
limiting case at very short , all charge in the inversion layer can
be removed by the recombination with the incoming holes. However, as we
concluded in (1), the transient numerical calculations do not show an
increased hole recombination, even at very short . Therefore,
contrary to the believe in [479][83], this mechanism cannot
explain the geometric component observed in experiments. Moreover, there
is a negligible recombination over the bulk traps in the active part
of devices (as suggested in [494]), as well. The
enhanced-recombination effect is small because the surface concentrations
are too low in comparison with very short times available for the
process. In the example shown in Figure 3.15, the
time interval is only (between the steps 
and 
). For the surface concentration 
we calculate
that the increases by about 
, which is completely
negligible in comparison with . In general, when decreases
the and increase, but becomes
shorter. Even being of a minor importance, the enhanced recombination
can be observed in the 
switching characteristics of
long channel MOSFETs at very short fall times, as is demonstrated in
Figure 3.16.
. There is no recombination in the
bulk, since the transfer time is very short. In the following, the
process is explained in detail.
the total charge in semiconductor 
decreases and the inversion-layer charge 
reduces according to the
decrease of . The depletion-region charge 
is nearly constant.
Bellow the threshold voltage 
the drops significantly,
while the depletion-region width 
and the reduce so that the
charge conversion is maintained for 
. If the junctions
are not biased, equilibrium holds in the bulk.
In transient conditions, nearly follows its quasi-static
value during the falling edge for the gate voltages laying significantly
over . Close to the current through the junctions
becomes insufficient to maintain the quasi-static conditions in the
channel. The is higher than in the quasi-static case, leading to
a decrease in the surface potential 
. The reduction of is
smaller that in the quasi-static case. Both, and decrease,
since the is reduced. With further reducing 
under
the surface field 
decreases, but less than in the
quasi-static case. The is no longer removed from the channel in a
relevant amount. Both, and instantaneously respond to the
drop in because of a very small dielectric relaxation time of the
majority carriers. Due to an enlarged , both, and
are a bit smaller than their quasi-static values. There is no longer
equilibrium for the minority carriers in the direction perpendicular to
the interface. In the equilibrium the diffusion current component due
to the gradient of 
equals to the drift current component due to
. In the transient conditions the drift current is smaller than
an enlarged diffusion current and the minorities start to diffuse
towards the bulk, driven by the gradient of the minority quasi-Fermi
level 
. To summarize, the minority
carriers which have been remained in the channel inject into the bulk
by the predominant diffusion and generate in -channel
devices. Note that the diffusion is a very fast process; in absence of
any electric field the peak of the charge-wave moves by diffusion in time
obeying a law 
, where 
is the minority carrier mobility. Assuming 
we estimate
a transit time of about 
. The carriers leave the active part of
devices before the hole surface concentration increases significantly,
namely before changes the sign and becomes repulsive for the
minorities. The process of the generation of the diffusion-wave
can be recognized in Figure 3.15. From
Figures 3.16 it is obvious that a part of the minority
carriers travels towards the bulk before the main part of them does,
because the is always a bit larger in the transient conditions
than in the quasi-static conditions. This is particularly
evident in Figure 3.16 for 
.
The charge injection into the bulk has also been simulated by a
transient numerical model in [349].
