SPC for Particle Counts
William A. Levinson, Frank Stensney, Ray Webb and Ronald Glahn, Fairchild Semiconductor, Mountaintop, Pa. -- Semiconductor International, 10/1/2001
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Traditional statistical process control (SPC) relies on the assumption that data follow a normal distribution. As undesirable random arrivals, particle counts in semiconductor process equipment are more likely to follow the Poisson distribution. When the average count rate is relatively high, the gamma distribution, which is the continuous-scale analogue of the Poisson, is often a good model.1
(1)
a is the shape parameter;
g is the scale parameter;
d is the threshold parameter (similar to guarantee time in reliability applications; it is the minimum possible particle count).
The gamma distribution is not, however, convenient for everyday use. Snedecor and Cochran2 show that the square root transformation is appropriate for data that follow the Poisson distribution. The square root transformation is a variance-stabilizing transformation that makes a population's variance independent of its mean. Transformed data may be suitable for analytical methods like analysis of variance (ANOVA) that rely on the normality assumption. (Note: ANOVA is actually robust to moderate non-normality.) This paper will show how to use the square root of particle count data on a standard Shewhart control chart, and how to use it to calculate process capability indices.
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| 1. Three graphs detail particle counts for Furnace #1. |
Table 1. Results for Furnace #1
| Raw data 120 out of 120 data | Gamma distribution (StatGraphics fit) | Control limits: median and 0.99865 quantile | |||
| Average | 25.5 | Shape a | 2.43 | Center line | 22.1 |
| Std. dev. | 16.0 | Scale g | 0.0955 | UCL | 102.3 |
Square root transformation
This section will justify the square root transformation further. It begins by showing that the square root of a response from a chi square distribution follows a normal distribution for high degrees of freedom. The chi square distribution with n degrees of freedom has a mean of n and a variance of 2n . When the mean is fairly large, Fisher's approximation applies:3
(2)
That is, the square root of a response from a chi square distribution with n degrees of freedom should follow a normal distribution with a mean of 
and a variance of 0.5 (standard deviation of 0.707).
There is no guarantee that particle data will follow a chi square distribution, but it is expected to follow a gamma distribution with shape parameter a and scale parameter g. The transformation y = 2 gx will convert it into a chi square distribution.
(3)
Therefore, (2g x)0.5 should follow approximately a normal distribution with mean and a variance of 0.5, where n=2a . Note that this procedure only applies when it is possible to get zero particles. If there is a threshold parameter, the gamma distribution is the three-parameter version, and the raw square root of the particle count cannot be used. The threshold parameter must be quantified and subtracted from each measurement first.
In practice, it will be more convenient to use x0.5 as a response variable. Then
(4)
Application to actual data: Furnace #1
As shown in Figure 1 , 120 particle counts fit a gamma distribution well, though a control chart shows a run of eight or more points below the control chart's center line (median). The histogram bars show the actual number of measurements in each cell. The curve is the fitted gamma distribution. They should ideally match, and the chi square goodness-of-fit test can be applied as a quantitative measure of how well they do.
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2. The square root of the particle counts from Furnace #1 fit a normal distribution quite well. |
The quantile-quantile plot in Figure 1 shows the ordered measurements x(1) ,x(2) , ... ,x(n) against the (i-0.5)/n quantile of the fitted distribution. If the fit is good the points should scatter randomly around the regression line with high correlation. Points that are far from the line are outliers.
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3. An equivalent capability index can be calculated from gamma distribution. |
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The control chart shows whether the data came from a stable process. Points that appear as outliers on the quantile-quantile plot will be above the chart's upper control limit.
The control chart's center line is its median, and its upper control limit is its 0.99865 quantile. The latter provides a false alarm rate of 0.00135, the same as the traditional 3 s Shewhart control chart. There is no lower control limit. The square root of the particle counts from Furnace #1 fit a normal distribution quite well, as shown in Figure 2 .
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4. Graphs detail the data points for Furnace #2. |
Table 2. Results for Furnace #2
| Raw data 120 out of 120 data | Gamma distribution (StatGraphics fit) | Control limits: median and 0.99865 quantile | |||
| Average | 23.35 | Shape a | 2.354 | Center line | 20.13 |
| Std. dev. | 15.41 | Scale g | 0.101 | UCL | 95.44 |
Process capability
There is no lower process capability index (CPL) because the process has no lower specification limit. CP=(USL-LSL)/6s has no meaning, and Cpk=Min[CPL,CPU]=CPU. Suppose the upper specification limit is 150 particles. The traditional index, CPU=Cpk=(USL-m)/3s, does not accurately reflect the chance of getting more than 150 particles.
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5. The square roots of particle counts fit a normal distribution reasonably well. |
An equivalent capability index can be calculated from the gamma distribution as shown in Figure 3 . The nonconforming fraction is mentioned because this is often of interest, e.g. as parts per million out of specification.
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6. Equivalent process capability calculation. |
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This process has the same expected performance as a normally distributed one with Cpk=1.357. The Cpk from the square root of the particle count, which is approximately normal, is (
-4.793)/(3×~1.581)=1.57. This is off by 0.21; remember that the square root approaches the normal distribution for high degrees of freedom in the chi square distribution, in this case, n=2a =4.86. Even under these conditions, however, the CPU estimate is far better than the one that relies on the assumption of a normal distribution. The raw particle average is 25.45 and the standard deviation is 16.04. CPU =(150-25.45)/(3×~16.04)=2.59 exceeds 6 s capability, and this suggests far less than one nonconformance per trillion (4×~10-15 as calculated by MathCAD). The actual nonconformance rate, as calculated from the fitted gamma distribution, is 23.5 nonconformances per million.
Application to actual data: Furnace #2
Figure 4 shows the histogram, quantile-quantile plot and control chart for 120 data points from Furnace #2. The data fit the gamma distribution well, although the control chart shows a long run of points above the median (at its right). Although this process's stability is questionable, its data fit the gamma distribution well enough to allow calculation of tentative control limits.
Figure 5 shows that the square roots of the particle counts fit a normal distribution reasonably well.
The Cpk from the square root transformation is ( -4.581)/(3×~1.543)= 1.656 vs. the correct value of 1.426 (Fig. 6).
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7. Three graphs detail 76 particle count data points for the film deposition Unit #1. |
Table 3. Results for Film Deposition Unit #1
| Raw data 120 out of 120 data | Gamma distribution (StatGraphics fit) | Control limits: median and 0.99865 quantile | |||
| Average | 20.62 | Shape a | 1.477 | Center line | 16.2 |
| Std. dev. | 20.25 | Scale g | 0.07162 | UCL | 108.4 |
Film deposition Unit #1
Figure 7 shows that the 76 data points fit the gamma distribution very well, and the control chart looks stable. Since the shape parameter is smaller, the chi square distribution has fewer degrees of freedom and the performance of the square root is not expected to be as good. The fit, however, is excellent, as seen in Figure 8 . The control chart looks reasonably stable, although there are two failures of optional Western Electric zone tests.
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8. Although the performance is not expected to be as good, the fit for square root of Unit #1 is excellent. |
Vertical furnace
Figure 9 shows that 82 data points fit the gamma distribution very well, and the control chart looks stable. Vertical tube furnaces have better temperature uniformity than horizontal ones. Figure 10 shows the results for the square root transformation.
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9. Three graphs detail 82 particle count data points for a vertical tube furnace. |
Table 4. Results for Vertical Furnace
| Raw data 82 out of 82 data | Gamma distribution (StatGraphics fit) | Control limits: median and 0.99865 quantile | |||
| Average | 69.8 | Shape a | 1.80 | Center line | 57.3 |
| Std. dev. | 48.6 | Scale g | 0.0259 | UCL | 328.9 |
Conclusion
This paper has shown that, when particle count data fit a gamma distribution, the square root transformation may allow use of the traditional Shewhart control chart. Although the gamma distribution is more accurate, the square root may be more convenient for everyday use. The normality of x0.5 improves as the gamma distribution's shape function, which corresponds to the chi square distribution's degrees of freedom, increases.
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10. The square root transformation of the vertical furnace. |
The process capability estimate from the transformed square root data is far more accurate than the one that assumes the raw data are normal. It is still overoptimistic, however, about the fraction of the population that exceeds the specification limit.
SPC for Tool Particle Counts
06/01/1999Statistics for non-ideal processes
06/01/1999
