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
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: