Given a TCAD model and a set of observations (i.e. data) that relate the
model independent variables (
) to its calculated responses
(
), the problem of technology characterization consists of
finding the model parameter vector (
) that will result
in the best possible fit between measurements and model predictions.
In the present work, the weighted sum of squares (F) is used as
a measure of the goodness of fit:
where 
and 
are the weight and the residual for the 
-th
response at the 
-th data point respectively. The residual is defined as
the relative error at that data point:
where 
is the measured -th response at 
.
The weight is a user-defined positive value that increases or decreases the
significance of a data point in the overall fit. This can be related to the
user desire to achieve higher accuracy for a certain input or output variable
range or to the prior knowledge about the expected accuracy of the
measurements. It is noted that this definition of the objective definition is
a variation on the Chi-Square fitting criterion where the standard
deviation of each measurement data point is used in the formulation.
By defining 
, a vector of the 
residuals,
(2.7) can be written as:
where W is a 
diagonal matrix whose elements 
are the weights 
.
The sum of squares objective is a functional mapping in the
m-dimensional parameter space: 
. To
compute the solution of the nonlinear least squares problem reduces to
minimizing (2.7). In general, the nonlinearity of TCAD models
dictates the use of gradient based iterative methods. The present
implementation is based on the Levenberg-Marquardt (LM) [67][61][16]
algorithm which has become the standard of nonlinear least squares
routines [86][84][25]. The Levenberg-Marquardt method combines the
inherent stability of steepest descent with the quadratic convergence rate
of the Gauß -Newton method as described in the next section.
