# 6.5 Analysis of the MOS capacitor

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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 *x*_{d} is the depletion layer width and
*N*_{a} is the acceptor
density in the substrate. The depletion layer
width is related to the
surface potential
f_{s} 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
f_{F}. The depletion
layer then reaches its maximum value yielding the maximal depletion
layer charge. The corresponding expressions are:
(mc3)

(mc4)

with the bulk potential
f_{F} 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
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© Bart J. Van Zeghbroeck, 1996, 1997