# 6.5 Analysis of the MOS capacitor

## 6.5 Analysis of the MOS capacitor

Two assumption are made in the simple model: 1) we assume that we can use the full depletion aproximation and 2) we assume that the inversion layer charge is zero below the threshold voltage. Beyond the threshold voltage we assume that the inversion layer charge changes linearly with the applied gate voltage. We refer to this is as being the basic assumption.

We start the derivation by examining the charge (per unit area) in the depletion layer. It is given by:

(mc1)
where xd is the depletion layer width and Na is the acceptor density in the substrate. The depletion layer width is related to the surface potential fs by:
(mc2)
This equation is not valid in accumulation i.e. when the surface potential is negative. In inversion the surface potential reaches its maximum value, namely 2 times fF. The depletion layer then reaches its maximum value yielding the maximal depletion layer charge. The corresponding expressions are:
(mc3)
(mc4)
with the bulk potential fF given by:
(mc5)
The total charge in the semiconductor has to balance the charge on the gate electrode or:
(mc6)
where we define the charge in the inversion layer as a quantity which needs to determined but should still be consistent with our basic assumption. This leads to the following expression for the gate voltage:
(mc7)
In depletion, the inversion layer charge is zero so that the gate voltage becomes:
(mc8)
while in inversion this expression becomes:
(mc9)
the third term in the equation above states our basic assumption, namely that any change in gate voltage beyond the threshold requires a change of the inversion layer charge. From the second equality we then obtain the threshold voltage or:
(mc10)

6.4 ¬ ­ ® 6.6

© Bart J. Van Zeghbroeck, 1996, 1997