B-splines and TPS were introduced in Chapter 2 as general analytical non-parameteric formulations of one- and two-dimensional functional variation. For a sufficient number of knots, or partitions, polynomial splines can accurately approximate any continuous function [14].
Spline functions are specified by selecting the order of the spline, the number of knots and their locations, and the values of the spline coefficients at the knots. Typically, quadratic or cubic splines are used to ensure the smoothness and continuity of the profile. Determining the number of knots and their locations is critical. Whereas a small number of knots does not provide sufficient accuracy, increasing the number of knots could increase the degrees of freedom in the profile representation beyond the available information in the capacitance data [106]. Similarly, locating knots in regions not probed by the capacitance measurements can lead to ill-defined parameters. Indeed, the extent of the region where the profile can be determined, and the resolution within that region are influenced by the complex interaction of various factors. Among the most important of these factors are the range and the step size of the available measurement voltages, the doping level, its variation, and the device structure. Simple partitioning strategies such as the use of equally spaced knots can lead to problems during the profile extraction. This will be illustrated later for the extraction of MOS channel doping from deep depletion capacitance. For 1D profile extractions, a self-adaptive scheme for the positioning of the B-spline knots is proposed as a solution for the spline partitioning problem. Computational demands and added complexity prohibit the extension of the algorithm to the 2D case for positioning the TPS knots. The development of an automatic algorithm for selecting the appropriate number and location of TPS knots is still actively being pursued. Meanwhile, the TPS knot sequence for MOSFET 2D extraction is defined by the user. The selection should be guided by process information such as junction depth, gate length, spacer width, and known 1D profiles.
In order to reduce the number of knots needed to represent the exponential variation of doping profiles, the logarithm of the concentration is approximated. For the 2D (1D) case, the logarithms of the donor and acceptor concentrations are each represented as a TPS (B-spline) function. Each TPS (B-spline) has its own sequence of knots to accommodate the spline: profile representation different fields of variation for each impurity type. Using fixed polynomial order and knot locations, the doping can then be formulated as a function of the B-spline or TPS coefficients. The MOSFET 2D profile can then be written as:
where 
and 
are the acceptor and donor
TPS coefficients respectively.