# The Advantages of Weighted PCA Techniques

for Statistical Parameter Analysis

**Introduction**

This article will examine a situation where limitations were noticed in the model parameter sets generated by SPAYN as a result of a Principal Component Analysis (PCA). The inaccuracies incurred as a result of using the PCA, or for that matter a Principal Factor Analysis (PFA), were due to the variability of one of the model parameters included in the analysis being inadequately accounted for. In this example the PCA produced a system of equations which related a large correlated set of NMOS and PMOS BSIM3 parameter sets to a core set of four independent dominant parameters. The PCA ensured that on average about 75% of the variability of the entire CMOS parameter set was reproduced by the derived system of equations. However, the reproducible variabilities of the individual parameters can be above or below this 75% value. Any circuit applications which are highly sensitive to parameters with low predicted variabilities may be inaccurately characterized by the parameter sets constructed using the default PCA or PFA technique. An example of such a situation will be shown here. A weighted PCA analysis is proposed as a solution to the limitation demonstrated. The PCA parameter weights, normally equal for all parameters, can be derived from the sensitivities of the circuit performance under analysis to these parameters. The example described in this article shows how such a weighted PCA analysis can successfully improve the usefulness of the derived model sets. The weights given to the parameters during a PCA or PFA may be computed from the relevant sensitivities, or totally different criteria may determine the origin of these weights. For example, a SPAYN user may wish to give the more physical model parameters a higher weighting than the more empirical model parameters.

**Experimental Example**

The model parameter data used in this example consisted of approximately 90 sets of NMOS and PMOS BSIM3 (Version 2.0) model parameters extracted by the UTMOST parameter extraction tool. SPAYN was used to delete any outliers and to simulate device characteristics for the accepted parameter sets. In this example it was decided to simulate and analyze device currents and output conductances for NMOS and PMOS devices biased in saturation with VGS = +/- 1.5V, VDS = +/- 3.0V and VBS = 0.0V. These devices had drawn dimensions (W/L) of 1/0.6µm. Histograms for these device characteristics called ids_n_sat_lowvgs, gds_n_sat_lowvgs, ids_p_sat_lowvgs, and gds_p_sat_lowvgs are shown in Figure 1. These device characteristics were then appended to the SPAYN database in use.

Figure 1. Histograms for the NMOS and PMOS saturation region currents and output conductances.

**Unweighted PCA Analysis**

A PCA of the NMOS and PMOS model parameter data determined that 76.7% of the variance of the complete combined (CMOS) parameter set was explained by just 4 principal components. SPAYN was then used to substitute 4 of the original model parameters for these components and to derive a system of equations relating all of the other CMOS parameters to the 4 so-called dominant model parameters. In doing this the retained average parameter variability was reduced to 73.4%. The dominant parameters which were identified were the NMOS low-field mobility parameter (N_U0), the oxide thickness parameter (TOX), the PMOS zero-biased threshold voltage parameter (P_VTH0), and the PMOS non-saturation parameter (P_A2). Equal weighting was given to each of the original CMOS parameters during the PCA.

SPAYN then used these equations to construct parameter sets which were used to simulate the NMOS and PMOS device characteristics under the same bias conditions detailed above. A comparison of the original device characteristics and the simulated characteristics obtained with the parameter sets generated by the PCA-related technique was then made. Figure 2 shows scatter plots of the original characteristics (x-axes) versus the characteristics simulated with the derived system of equations. Table 1 shows the correlation coefficients, the R-squared coefficients, the mean errors, and the maximum errors associated with these plots. The mean and maximum errors refer to the deviations of the points from the straight line fits shown on the plots.

Characteristic ID | Correlation Coefficient | R-squared Coefficient | Mean Error | Maximum Error |

ids_n_sat_lowvgs | 0.99 | 0.98 | 1.9% | 6.3% |

gds_n_sat_lowvgs | 0.97 | 0.95 | 4.4% | 14.5% |

ids_p_sat_lowvgs | 0.88 | 0.78 | 6.4% | 31.9% |

gds_p_sat_lowvgs | 0.68 | 0.46 | 25.6% | 113.0% |

**Table 1. Results from the unweighted
PCA analysis.**

The results shown in Figure 2 and Table 1 indicate that the technique which was utilized was successful in the case of the prediction of NMOS device current and output conductance but that the PMOS results were poor. The models created by the system of equations in terms of the derived dominant parameters were incapable of reproducing the PMOS saturation region current and especially the PMOS saturation region output conductance. The correlation coefficients and R-squared values associated with the PMOS device characteristics are not close enough to unity and the error quoted in the results table are too large. An analysis of the results of the PCA-aided dominant parameter analysis indicated that only 60% of the variability of the PMOS drain-induced barrier lowering parameter (PDIBL1) was retained in the formulation of the system of equations used. This parameter is very important in the determination of the PMOS device output conductance and to a lesser extent in the determination of the PMOS currents in saturation. It was assumed that inadequacies in the coverage of this parameter were the cause of the problems in the prediction of the PMOS device characteristics.

Figure 2. Scatter plots of the
original NMOS and PMOS device characteristics (x-axes) versus the

predictions obtained using the unweighted PCA technique (y-axis).

**Weighted PCA Analysis**

SPAYN was now used to perform a sensitivity analysis of the PMOS saturation output conductance under examination. Not surprisingly the most sensitive parameter was found to be the PMOS PDIBL1 (P_PDIBL1) parameter. SPAYN was then used to determine sensitivity coefficients for all of the parameters and to normalize them using the most sensitive parameter (i.e. P_PDIBL1) as a base.

The sensitivity coefficients were used to determine weights for the parameters so that a weighted PCA could be performed. A minimum weight of 1 was assigned to the insensitive parameters while a maximum weight of 10 was chosen for the P_PDIBL1 parameter. The weighted PCA calculated that 4 factors could be used to predict 75.7% of the variability of the weighted parameter set. The dominant parameters which were isolated were the PMOS low-field mobility parameter (P_U0), the oxide thickness parameter (TOX), the PMOS zero-biased threshold voltage parameter (P_VTH0), and the PMOS drain-induced barrier lowering parameter (P_PDIBL1). Using the dominant parameters to determine a system of equations relating the correlated parameter set to the 4 new dominant parameters resulted in 72.5% of the parameter variability being retained. Because P_PDIBL1 is now a dominant parameter, all of its variability (i.e. 100%) will be retained in any analysis using the new derived system of equations. Using the new system of equations to determine the NMOS and PMOS device characteristics under analysis resulted in the generation of the data plotted in Figure 3. Table 2 shows the correlation coefficients, R-squared coefficients, mean errors and maximum errors associated with the scatter plots displayed in Figure 3.

Figure 3. Scatter plots of the
original NMOS and PMOS device characteristics (x-axes) versus

the predictions obtained using the weighted PCA technique (y-axis).

The results of the weighted PCA analysis are a significant improvement over the results obtained from the unweighted PCA analysis as can be seen from Figure 3 and Table 2. There is excellent agreement between the original NMOS and PMOS device characteristics and the predicted characteristics obtained from the equations formulated with the aid of the weighted PCA analysis.

Characteristic ID | Correlation Coefficient | Mean Coefficient | R-squared Error | Maximum Error |

ids_n_sat_lowvgs | 0.99 | 0.97 | 2.4% | 6.0% |

gds_n_sat_lowvgs | 0.97 | 0.93 | 4.8% | 14.5% |

ids_p_sat_lowvgs | 0.98 | 0.98 | 2.0% | 31.9% |

gds_p_sat_lowvgs | 0.98 | 0.96 | 4.5% | 19.5% |

**Table 2. Results from the weighted
PCA analysis.**

**Conclusion**

Statistical circuit simulation techniques based on the use of a PCA or PFA are becoming an accepted accurate method of predicting statistical circuit performance spreads. Worst-case modeling methodogies using these techniques provide an invaluable aid to increasing circuit yield and manufacturability. However, there are situations where the effects of critical device model parameters may be misrepresented due to an insufficient amount of their variability being retained by the models used for the statistical circuit simulations. These limitations can affect models derived using either a PCA or a PFA analysis. Weighted PCA or PFA methods can be used to avoid these situations where the weighting scheme is determined using the sensitivities of the circuit performance under test to the model parameters in use. This article used certain device performance parameters to illustrate this point. Weighting and sensitivity analysis features have been added to SPAYN and will be available in the next SPAYN release.