To illustrate the failure of simple knot placement strategies,
Fig. 4.1
compares the analytically extracted profile with profiles determined
using 10 and 15 equally spaced knots. It is clear that the oscillations
between 0.6 
and 1.3 are an artifact of the extraction
procedure. In essence, the range and the number of available measurement
voltages, the doping level, and the number of coefficients make such
an extraction ill-conditionned.
The use of the self-adaptive scheme described in the previous section resolves these problems. New knots are only inserted where they are needed as determined from the capacitance fit errors. In this case, only six knots are sufficient to reach a good fit. Fig. 4.2 shows the extracted doping using the proposed algorithm with the analytical extracted profile. As seen, the failure of the deep depletion approximation near the interface and in rapidly changing profile regions, results in large one-dimensional profile errors in the analytical profile. The performance of the knot placement strategy is depicted in Fig. 4.3 which compares the decrease in the residual sum of squares as a function of the number of knots when using the new algorithm to the case of equally spaced knots. Finally, the excellent capacitance fit achieved is shown in Fig. 4.4.
Figure 4.1: Comparison of Analytically Extracted Profile (solid-symbols) with
Profile Extracted using 10 (dashed) and 15 (dotted) equally
spaced knots for a MOS Capacitor with 70
oxide and 4000
area.
Figure 4.2: Inverse Modeling (solid) and Analytical (dashed) extracted
profiles for a MOS Capacitor with 70 oxide and 4000 area.
Figure 4.3: Residuals Sum of Squares (SSQ) as a function of number of knots
for the new placement algorithm (solid) and equally spaced knots (dashed).
Figure 4.4: Simulated (solid) and experimental (symbols) deep depletion
C-V of a MOS Capacitor with 70 oxide and 
area.