Let us define the charge-pumping threshold voltage 
in 
-channel
devices as the gate-source bias which induces the surface minority carrier
concentration 
sufficient to fill traps by electrons to a some
degree during the gate top level 
The parameter 
in the definition is a free parameter of order 
. It
equals to 
when the total trap filling is assumed. An analogous definition
can be introduced for the charge-pumping flat-band potential 
.
This voltage represents the gate-bulk bias sufficient to induce the majority
carrier surface concentration 
given by
Equivalent definitions may be given for 
-channel devices, with
and exchanged.
Relation 3.57 yields 
. Therefore, the levels
and 
cross each other very closely to the
charge-pumping threshold.
In the subthreshold region an increase in 
produces roughly speaking
a proportional increase in 
and, consequently, an exponentially
increasing 
. Let us denote 
.
The ratio 
increases exponentially with in the subthreshold
region until the charge-pumping threshold specified by 
. In the
subthreshold region (
) expression 3.63
yields 
. Both, the lower energy
boundary 
and the upper energy boundary
are constant with respect to .
Therefore and because of 
, the 
increases almost
linearly with 
in the lin-log scale in the region under the
charge-pumping threshold voltage, as is shown in Figure 3.6.
The voltages and must be strongly distinguish from
the levels 
and 
introduced in the following,
which determine the emission times.
