1. Sources of Variation and Error in Wafer Fab Processing
EE412

2. Data Charts

3. The Normal or Gaussian Distribution
Standard deviation = dispersion
Probabilities are often described as number of SD’s from the mean
FWHM ≠ standard deviation

4. The Normal Distribution In Control Charts
UCL = Upper Control Limit = Mean + 3 sigma
LCL = Lower Control Limit = Mean - 3 sigma

5. How can you tell if two distributions are different?

6. Null Hypothesis From Wikipedia:
Hypothesis testing works by collecting data and measuring how probable the data are, assuming the null hypothesis is true. If the data are very improbable then the experimenter concludes that the null hypothesis is false. If the data do not contradict the null hypothesis, then no conclusion is made. In this case, the null hypothesis could be true or false; the data give insufficient evidence to make any conclusion.
In short:
You can demonstrate that two groups of data are /are not different with a certain statistical confidence -- but you cannot show they are the same.

7. More data is better

8. Remember: the null hypothesis cannot be proven, only disproven...
But when you don’t have enough, there’s the Student’s t-test
Student’s t-distribution for degrees of freedom of 1 (green), 5 (red) and 10 (blue). Note the areas under the tails get smaller with increasing numbers of samples.
Remember: the null hypothesis cannot be proven, only disproven...

9. Other ways to look at probability distribution
Normal probability plots are a good way to test for normal distribution – and to identify outliers, as a method of significance testing.
Example at left is a non-normal distribution (Poisson-like).

10. Other Distributions that might be encountered in wafer processing
Poisson distribution:
Characteristic of discrete events or other counting activity. Lambda is the mean AND the standard deviation. Examples: particle or defect counting, defective parts, time-to failure, %nonuniformity (hi-lo/mean)
Chi-Square distribution:
Describes distribution of variances. Example: %nonuniformity (std dev/mean)
So: you need to use the appropriate significance test for the non-normal distribution you might encounter.

11. Sources of Variation Precision: Associated with “random” error
Repeated measurements yields better result
Accuracy:
Associated with “systematic” error
Need to identify or compensate for in experimental plan

12. Significant Digits
The last significant figure should be the same order of magnitude as the uncertainty.
Though you round the last significant figure in reporting, more significant figures should be used in calculations.
Please do not report Woollam results to four significant digits!

13. Propagation of Error: Etch Rate Example Measurement Precision
Before
After
Difference T
5923
1090
4833 C
6081
950
5131 B
5823
1050
4773 L
5850
1023
4827 R
5888
1083
4805
Average
5913
1039.2
4873.8
Stdev
101.26
56.65
145.69
%COV
1.71
5.45
2.99
Range
258
140
358
%NonUnif
2.18
6.74
3.67
Measurement Precision
118.26
20.78
139.04
%MP/Average
2.00
2.85

14. Propagation of Error Rules

15. References
An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements, John R. Taylor
Introduction to the Theory of Error, Yardley Beers.

16. Stress Test Example
The micrometer setup was used for measuring wafer thickness with ~3% accuracy (within 10um for um wafers). Nanospec and P2 were used for measuring film thickness measurement with ~5% accuracy. Assuming the curvature measurement accuracy is within 20%, the measured values are within 35% of the real value. Adjusting the measured values by 35%, the worst-case mean stress becomes ~ -23.5MPa, which can still be considered "low stress" by the ±50MPa definition.
where Es is Young's modulus, νs is Poisson's ratio of the substrate, and R and Ro are the radii of curvature of the film before and after film deposition, respectively.

17. Mistakes To Avoid!
Too many significant figures! (Especially from the Woollam.)
Using 0,0 as a data point for fitting a line.
Sources of variation not identified (repeat measurements in the same spot? Measurements on different parts of a wafer? Measurements from different wafers?)
Throwing out a “bad” data point without statistical validation.