Processes at the Top Level:

We assume that three processes occur during the top level : electron emission, electron capture and hole emission. The probabilities of these processes shall be compared.

a)

capture of electrons versus emission of electrons

 

Since , it follows . For the traps below the electron capture occurs more frequently than the electron emission. We can assume that the traps below which capture an electron do not emit the electron during , because of an exponential dependence of on , which makes the emission from the levels laying below very unlikely. More precisely, considering only the electron capture and emission and assuming for traps of interest, 3.48 yields

 

with . Below the level expression 3.52 may be approximated by

 

because of .
As opposed, the traps laying above the Fermi level emit into the conduction band with a much higher probability than they capture electrons from the conduction band. However, this fact does not mean that the traps above which are successful in capturing an electron during will really emit this electron in the interval . This problem is clarified in the following.

b)

electron emission versus top-level duration

 

 

The traps above the level are empty after although they perhaps capture some electrons during . The traps residing below the level do not emit electrons which they eventually capture during , in spite of the emission is much more probably than the capture process. Since the traps have already been covered by the analysis in a), we restrict ourselves to the interval . The relationship 3.52 reduces to

 

In deriving 3.56 we benefit from . Since holds in the interval considered, expression 3.53 is also valid above the Fermi level until the level .
A relation between and should be established

 

The upper boundary of the energy interval which is responsible for the charge-pumping current is either given by or by , whichever is higher

 

c)

electron capture versus hole emission

 

 

For traps laying under the level the hole emission is dominant, while above this level the traps are filled mostly by the capture of electrons from the conduction band. However, for our analysis, the energy interval in the band gap, which is available for the electron capture during the gate top level is of a primary interest. Its bottom boundary can be determined by considering the energy interval where the traps are totally filled by the hole emission during the top level, as is done in the following.

d)

emission of holes versus top-level duration

 

The traps below the level are filled by the hole emission during . The filling occurs quickly for all traps in the interval from to , except for the traps very close to , due to an exponential dependence of the emission time on the energy relative to the valence band edge. Thereby, the traps in this interval are not able to capture electrons from the conduction band when the later process has a low probability.
After c) and d) we conclude that the lower boundary of the energy interval available for the electron capture during is given by

 

The relation between and reads: . In the charge-pumping subthreshold region () the bottom boundary for the electron capture is determined by the emission of holes . In the strong electron-capture conditions, represents the boundary between the electron-capture dominant and the hole-emission dominant region, .

As a conclusion, at the end of the top level

• almost all traps laying over are empty,
• all traps below are filled by the hole emission (do not contribute to ) and
• the traps between the levels and are filled to a some degree by the capture of electrons from the conduction band.