The analytical extraction of doping profile from differential capacitance measurements have been first proposed in [50]. The method is based on the deep depletion approximation which divides the the semiconductor into two distinct regions with a sharp transition. The first region is completely depleted of mobile charge carriers, whereas perfect charge neutrality conditions prevail in the second. In the case of a MOS device, these assumptions result in the following standard equations:
and
where q is the electron charge, 
the semiconductor
permitivity, 
and 
are the gate and oxide capacitance per unit
area respectively. The limitations of the above equations and their inability
to determine the doping near the interface and for rapidly changing profiles
are well documented [74]. Various improvements and correction
schemes have also been suggested. In the inverse modeling method,
Poisson's equation is solved numerically without the approximations that
are needed to derive (4.6) and (4.7)
(i.e. deep depletion approximation). Indeed, the use of parameter optimization
techniques and numerical solution of Poisson's equation have been shown to be
more accurate than the analytical
extraction [77][45].
Using deep depletion data taken on a large MOS capacitor, the
profile is first extracted using (4.6) and
(4.7). The analytical profile is used as an initial guess for the
inverse modeling procedure. As stated earlier an algorithm for the
determination of the number of B-splines knots and their location is
used [57]. It is based on ideas similar to [106]
and is described next.
