We consider the trapezoidal gate pulse shown in Figure 3.2.
The gate top level 
is variable, whereas the gate bottom level 
can be either fixed or variable. Let us assume that the bottom level is low
sufficiently to fill all interface traps by holes during 
. The traps are
assumed to be located without the junctions to escape the effect of the spatial
variations of the charge-pumping threshold and flat-band voltages; the problem
is considered in one dimension, perpendicular to the interface. We adopted that
the electron surface concentration 
(in 
-channel MOSFET) follows
instantaneously the surface potential variations during rising edges 
and falling edges 
of the gate pulses. Three processes occur at the
top level of duration 
: electron emission with the time constant
, electron capture with 
and hole emission with
. The hole capture during the top level may be neglected
if the top level lies sufficiently high so that the hole surface concentration
is very small (if 
holds for the hole quasi-Fermi
level at the top level). Let us assume rectangular pulses for the moment;
and are very short so that the processes occurring during the pulse edges
can be neglected. All traps filled by electrons on the top level are recombined
by holes at the bottom level. Only those traps which are filled by capturing
electrons from the conduction band are active in producing the charge-pumping
current. The initial conditions 
at the beginning of the rising
edge 
are given by the steady-state occupancy function 
corresponding to the bottom level . At the beginning of the top level
the electron capture into traps occurs, obeying
equation 3.34
with the total time constant
,
for each individual level 
with density 
. Since all time
constants , 
and 
are constant during
(if the charge-potential feedback effect is neglected) the general solution
of 3.47 is
is the steady-state
occupancy function connected with the electron Fermi level 
at
the gate top level . Defining the change in the trap occupancy at the
end of the top level by 
the
charge-pumping current is given by
if the hole emission is negligible during . 
is the active
charge-pumping area and 
is the gate pulse frequency. The charge-pumping
current given by 3.49 may be considered as the theoretically
maximal current 
. It follows
The actual 
is smaller than due to the hole emission at
the top level and the electron and hole emission at the pulse edges.
These processes are considered in more detail in the following.
Rather that developing a general theory like this in [49], which yields a complex system of implicit equations, we shall present a phenomenological description, yet accurate sufficiently in most of practical applications. In our approach the band gap is divided into characteristic regions for each time interval of the gate pulse. Individual regions are associated with only one dominant generation-recombination process.
