Download Sources of Variation and Error in Wafer Fab Processing

Sources of Variation and Error
in Wafer Fab Processing
EE412
Data Charts
The Normal or Gaussian Distribution
Standard deviation = dispersion
FWHM ≠ standard deviation
Probabilities are often described as
number of SD’s from the mean
The Normal Distribution
In Control Charts
UCL = Upper Control Limit = Mean + 3 sigma
LCL = Lower Control Limit = Mean - 3 sigma
How can you tell if two
distributions are different?
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.
More data is better
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...
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).
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.
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
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!
Propagation of Error: Etch Rate Example
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
2.00
2.85
Propagation of Error Rules
References
• http://phys.columbia.edu/~tutorial/
• An Introduction to Error Analysis: The Study of
Uncertainties in Physical Measurements, John R.
Taylor
• Introduction to the Theory of Error, Yardley Beers.
Stress Test Example
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.
The micrometer setup was used for measuring wafer
thickness with ~3% accuracy (within 10um for 400 - 550um
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.
Mistakes To Avoid!
1. Too many significant figures! (Especially from the
Woollam.)
2. Using 0,0 as a data point for fitting a line.
3. Sources of variation not identified (repeat measurements in
the same spot? Measurements on different parts of a
wafer? Measurements from different wafers?)
4. Throwing out a “bad” data point without statistical
validation.