# A Comparison of PCA and PFA Techniques In Generating Worst-Case Model Parameter Sets using SPAYN

Introduction

In this article the techniques of using Principal Component Analysis (PCA [1]) and Principal Factor Analysis (PFA [2]) are compared in terms of their ability to separate MOS level 3 parameters into independent groups, and to identify the dominant parameter within each group. A subsequent worst-case analysis, based on the manipulation of these dominant parameters, shows excellent agreement with results measured from a 1µm CMOS process [3].

Measured Data

The data used in this article consists of n-channel and p-channel SPICE level 3 MOSFET parameter sets measured from many wafer die over a period of time from a 1µm CMOS process. The parameters were VT0, RSD, NSUB, GAMMA, PHI, NFS, LD, WD, U0, DELTA, ETA, XJ, THETA, VMAX and KAPPA. A value of TOX was also measured at each site, and assigned to both device polarities. Parameters NSUB and PHI were not actually extracted at each site, but were assigned constant process-dependent values. The RSD source/drain parasitic resistance parameter was split into separate RS and RD parameters, with RSD = RS + RD, RS = RD.

Measured p-channel saturation and n-channel
linear region currents were also recorded for both device polarities.
These were included in the * SPAYN* database for comparison
purposes. The saturation-region currents (n-channel device assumed)
were measured at VDS = VGS
= 5.0V, VBS = 0V. The
linear-region currents were measured at VGS
= 5.0V, V DS = 0.1V, and
VBS = 0V. These currents
were measured from devices with a drawn width of 20 µm and
a drawn length of 1µm.

PCA versus PFA

The p-channel parameters NFS, KAPPA, RS and RD were found to be lognormally distributed. Thus the natural log transform was performed on these parameters prior to the PCA or PFA analyses. A recursive +/- 4 standard deviation filtering algorithm was used to eliminate outlier data points, resulting in 845 accepted data sets which were used for analysis.

A PCA was performed with an 80% variance retained criterion and a VARIMAX rotation. Six components were found to explain 80.6% of the parameter variance when each parameter was expressed in terms of a linear combination of these six components.

Figure 1 shows the parameter groups resulting from
the PCA. Each parameter is placed in a group with the component
with which it is most strongly correlated. (Correlation coefficients
are included, and parameters are sorted within each group in order
of decreasing correlation with the associated principal component).
The first parameter in each group (maximum correlation) was assigned
as the "dominant" parameter for that group. These are: TOX, LD (p-channel),
KAPPA (n-channel), RS (p-channel), RS (n-channel) and WD (p-channel).
* SPAYN* allows the user the flexibility of changing
the identity of the dominant parameters if required.

Figure 1. Table of parameter groupings after the Principal Component Analysis.

A PFA was then performed with a 0.05% communality tolerance criterion and a VARIMAX rotation. Once again 6 independent factors were found and this time they accounted for 75.7% of the variance of the original correlated model parameters.

Figure 2 shows the parameter grouping found as a result of the PFA. A comparison of figure 1 and figure 2 shows that the grouping of the parameters is very similar in both cases. The first, second and sixth groups are identical, with each group having the same parameters in the same order for both cases. The fourth and fifth groups are also practically identical, but have their order interchanged. The third group also contains the same parameters in both cases, but the order of the parameters within the group differs.

Figure 2. Table of parameter groupings after the Principal Factor Analysis.

The dominant parameters, found as a result of the PFA are identical to those found by the PCA. A VARIMAX rotation was performed in both cases. The dominant parameters are related to the core process variabilities to which the model parameters were most sensitive. Variations in gate oxide thickness, polysilicon gate length, and various doping levels or junction depths were identified as being the important process variables in this example. The exact differences between PCA and PFA are described elsewhere [2] but a PCA will try to maximize the amount of parameter variability accounted for by the derived components while a PFA will try to minimize the difference between the original parameter correlations and those which can be reproduced by the derived factors.

In most cases, including the example used here, there will be little difference between the results of using a PCA or PFA. These dominant parameters were skewed to perform a worst-case analysis.

Predicted versus Measured Worst-Case Currents

A set of 64 (2) corner model sets was generated, taking all combinations of +/- 2.5 sigma values for each of the six dominant parameters listed above. For each corner set, values for the other parameters were generated using equations obtained by a multilinear regression on the dominant parameters. Some of these equations are shown in Figure 3. Simulations were performed for each of the 64 corners in order to predict the worst-case p-channel saturation region currents and the n-channel linear region currents.

Figure 3. Some equations obtained from the multi-linear regression.

Figure 4a shows the distribution of the measured saturation region currents for the p-channel device. Figure 4b shows the simulated ID-VDS curves for each of the 64 corners for this device. The minimum and maximum current values (VDS=VGS=-5V) were 3.3 and 5.1 mA respectively, which compares very well with the measured distribution shown in fig 4a. Note that the best fit to the measured distribution is a Gamma distribution, which has a longer tail to the right than to the left of the mean value.

Figure 4a. Measured histogram
of p-channel saturation

region current (VGS=VDS=5V
and VBS-0V).

Figure 4b. Predicted p-channel
IDS-VDS sweeps (VGS=-5V)

using the derived process corner models.

Figure 5a shows the distribution of the measured linear region currents for the n-channel device, with figure 5b showing the linear-region I-V characteristics. The minimum and maximum linear current values (VGS =5V, VDS=0.1V) at the appropriate bias are 485uA and 715uA respectively, again in excellent agreement with the measured distribution in figure 5a.

Figure 5a. Measured histogram
of n-channel linear

region current (VGS=5V, VDS=0.1V
and VBS-0V).

Figure 5b. Predicted n-channel
IDS-VGS sweeps
(VGS=0.1V)

using the derived process corner models.

The internal device current solver in * SPAYN*
was then used to simulate the n-channel devices in the linear region
and the p-channel devices in the saturation region for each of the
original 845 parameter sets used in the analysis. Figure 6a shows
a plot of the measured (x-axis) versus simulated linear region currents
for the n-channel devices, while figure 6b shows a plot of the measured
(x-axis) versus simulated saturation region currents for the p-channel
devices. The linear curve fit for the n-devices gives a mean relative
error of 0.2% and maximum relative error of 0.8%. The linear curve
fit for the p-devices gives a mean relative error of 0.1% and maximum
relative error of 0.4%. This means that both the MOSFET model used,
and parameter extraction techniques which were implemented, were
more than adequate for the modeling of these device currents for
the devices analyzed.

Figure 6a. Measured versus simulated

n-channel linear region currents.

Figure 6b. Measured versus simulated

p-channel saturation region currents.

Acknowledgment

Silvaco would like to thank Barry Mason and Paul Stribley of GEC Plessey Semiconductors, Roborough, England, and Rory Clancy and Kevin McCarthy of the National Microelectronics Research Centre, Cork, Ireland, for their valuable contributions. The data used in this article was supplied by GEC Plessey Semiconductors, from their 6" wafer fabrication plant in Roborough, U.K., and came from the production 1.0µm CMOS analog process.

References

[1]. J.E. Jackson, A User's Guide to Principal Components, Wiley-Interscience, 1991. [2]. R.J. Harris, A Primer of Multivariate Statistics, Academic Press, 1985. [3]. SPAYN Application Note No. SP/94/001, <169>Worst-Case Model Development with SPAYN.