SPC for Tool Particle Counts

William A. Levinson and Angela Polny, Harris Semiconductor, Mountaintop, Pa. -- Semiconductor International, 6/1/1999

Semiconductor manufacturing often involves the management of statistically non-ideal processes, processes whose data do not fit the normal distribution. Traditional treatment of data such as tool particle counts results in inaccurate estimation of process capabilities and excessive false alarms from control charts. This article shows how to apply the gamma distribution to estimate the impact on process capability due to particles generated in process tools.

Statistics for non-ideal processes

Fig. 1.

The histogram bars shown in Figure 1 represent a real manufacturing process, particle samples from an Applied Materials etcher. The 48 particle samples have an average of 294.6 particles and a standard distribution of 219.1 particles. The curve shows what happens when a normal distribution is assumed. Most commercial SPC software packages will treat the data this way. Unfortunately, application of the traditional model to these manufacturing situations produces misleading results. Estimates of the in-specification proportion can be off by more than an order of magnitude. In other words, the process capability estimate is inaccurate. In addition, the distribution extends below zero, suggesting the chance of obtaining negative particle counts. The presence of a one-sided specification is also a good indicator that the distribution is not normal. The same situation arises in material purity applications, where measurements are in the ppm range.¹

Harris Semiconductor's plant at Mountaintop, Pa. has had extensive experience with these non-ideal processes. For example, batch processes involve nested variation sources, within and between batch.² In our experience, the gamma distribution is a good model for material impurities.¹ Distributions like the lognormal, Weibull and gamma distributions are often good models for skewed or asymmetrical data populations. Particles and impurities are "undesirable random arrivals." When the average count is five or 10, the Poisson distribution is often the correct model. The gamma distribution is the continuous (real-number) analogue of the Poisson distribution and is often appropriate when the average count is much higher.

Fitting the gamma distribution

Table 1 lists the 48 particle data points corresponding with Figure 1. Data were taken using a Tencor Surfscan inspection tool before and after wafers were placed in an Applied Materials hexode-type plasma etcher. The test wafers were placed in the etcher with the RF power off and gas flows on. The etcher used CHF₃ and oxygen to etch oxides, boron phosphosilicate glass (BPSG) and PSG. The 48 data points in the table are calculated averages of added particles, using two particle counts per sample.

The two-parameter gamma probability density function is:
 
where a is the shape parameter, and g is the scale parameter. The distribution's mean is a/g, and its standard deviation is sqrt[a/g²]. Lawless shows how to fit a two-parameter distribution to data.³

Like real-world impurity and particle data, measurement values can range from zero to infinity. Sometimes, however, there is a lower practical limit for the measurements. For instance, in reliability testing, this is the guarantee time. No units will fail before the guarantee time, unless they are defective. In stress testing, no units will fail below a certain stress level, unless there is a flaw like a stress concentrator. Similarly, there may be a lower practical limit for impurities and particle counts. A three-parameter gamma distribution can account for this practical minimum using:

The distribution's mean is [[a/g]+d], and its variance is a/g².

d must be between zero and the smallest datum. d is selected to maximize the likelihood function for the data set. To determine d, the user chooses a value for d, calculates a and g and then computes the likelihood function.¹ For n data x_i,

and

The d value that maximizes the likelihood function is the optimum value. In our example (Table 1), the smallest particle count is 35.5. A plot of L(a,g,d) versus d (Fig. 2) shows a maximum at d= 34. With d= 34, a= 1.172 and g= 0.004498. The mean is therefore: (1.172/0.004498) + 34 = 294.6, and the standard deviation is: sqrt[1.172/(0.004498)²] = 240.7.

Fig. 2. A third parameter,d, adjusts the value of the mean for the lowest practical minimum number of particle counts. In this case. 34.

These values agree well with the actual data showing an average value of 294.583 and a standard deviation of 219.07 particles.

The traditional Shewhart control limits would be 294.583 ± 3(219.07), or [-362.63, 951.79]. There is an obvious problem with these limits, since negative particles do not exist. Also, the false alarm rate should be 0.135% for each control limit. The chance of particle counts exceeding the upper control limit (UCL) of 951.79 should be 0.135%, yet 951.79 actually covers 98.29% of this gamma distribution, making the false alarm risk for this control limit 1.71%, more than 10X what it should be (Fig. 3).

Once we fit the correct distribution, we can set proper control limits. If the traditional three-sigma limit gives the wrong control limits, how do we determine the right control limits? We must determine y such that F(y), the cumulative probability for the gamma distribution, is 0.99865. The result is ~1603 (versus 951.79). Using this as the UCL, we get a 0.135% false alarm risk. For a process with an upper specification limit (USL), the non-conforming fraction is 1 - F(USL).

Fig. 3. Using the traditional 3 s limit, the upper control limit is at 951.8 particles, leading to a false alarm risk of 1.71%, more than 10 times what it should be.

Testing goodness of fit

We can, of course, fit any probability density function to any set of data. This includes the normal distribution, which yielded the results of Figure 1, proving the model was inappropriate for this data set. Many tests can be performed to determine whether a distribution function is appropriate for a given data set.

A Q-Q plot is similar to a normal probability plot. With a Q-Q plot, the ordered data x_(i), where x₍₁₎< x₍₂₎<x...<x_n, is plotted against the (i-0.5)/n percentile of the selected distribution. In other words, find y such that:

Fig. 4. A Q-Q plot showing an intercept of zero, a slope of one and good correlation, such as this, confirms the selection of a proper SPC model.

The lower limit of zero applies for the gamma, lognormal and Weibull probability density functions, f(x). The lower limit for a normal distribution is negative infinity. For the Q-Q plot, we want a slope of one, an intercept of zero and good correlation. Figure 4 shows the Q-Q plot for our example, where the slope is 0.926, the y-intercept is 20.796 and the correlation is 0.9933. From each ordered datum, d has been subtracted.

Alternatively, a chi square test statistic measures the "badness of fit" between the fitted distribution and the histogram. For n data points and k histogram cells,

where

E_i is the expected count, and f_i is the actual count (frequency) in the cell whose limits are [L_i, U_i]. Large differences between the expected and actual counts increase the value of c².

Figure 5 shows the observed and expected cell counts for the gamma distribution and the data in Table 1. The numbers on the y-axis are the cell minima. There are a total of seven cells, so there are four degrees of freedom. c² = 3.24, which easily passes for goodness of fit for the following reasons. When c ² >c²_k-p-1;a, we estimate the probability that the distribution does not fit the data (i.e., badness of fit) with 100(1-a)% confidence. The test statistic has k-1-p degrees of freedom, when we estimate p parameters from the data. Here, p = 2, since d, the third parameter, is chosen from the likelihood plot. In this case, c²_4;0.05 = 9.49, which we compare to 3.24. The expected count in each cell should be five or more. If necessary, we can combine cells (as we did in Figure 5), to fulfill this condition.

Fig. 5. The chi square test applied to the data in Table 1 shows a good correlation between expected particle counts and actual particle counts.

Other applications

After discarding one outlying data point of 48 samples, a three-parameter gamma distribution fit the particle data for an Applied Materials 8110 etcher. We then applied the gamma distribution to particle data from a GaSonics Aura plasma etcher, which uses oxygen and a small amount of nitrogen to strip photoresist. In this case, several outlying data points had to be discarded from the particle data. The data fit a two-parameter gamma distribution. That is, zero particles could be obtained, a possibility accommodated by the model. Data from a second Aura etcher fit a three-parameter gamma distribution.

Conclusions

We successfully applied the gamma model to four separate cases. It has also proven to work well for impurity data. We recommend that users try applying the gamma distribution in situations involving "undesirable random arrivals," such as particles and impurities generated in semiconductor process tools. In any case, the statistical model should always be checked for goodness of fit prior to use in a production environment.

References

1. W. Levinson, "Watch Out for Nonnormal Distributions of Impurities," Chemical Engineering Progress, May 1997, p. 70.
2. W. Levinson, "Statistical Process Control in Microelectronics Manufacturing," Semiconductor International, November 1994, p. 95.
3. J. Lawless, Statistical Models and Methods for Lifetime Data, John Wiley & Sons, New York, 1982, p. 204.

William A. Levinson is a staff engineer at Harris Semiconducto. He is principal author of Levinson and Tumbelty, SPC Essentials and Productivity Improvement: A Manufacturing Approach (1997, ASQ Quality Press) and editor of Leading the Way to Competitive Excellence: The Harris Mountaintop Case Study (1998, ASQ Quality Press). Phone: 570-474-6761 x4325 Fax: 570-474-3279

Angela M. Polny is a process engineer at Harris Semiconductor. She has a bachelor's degree in environmental engineering from Wilkes University. Her current responsibilities include plasma and wet etch of all films, as well as photoresist removal. She is a coinventor on a patent application for a process to eliminate negative photoresist in metal lithography processes.

 

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