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C. J. Spanos
Design of Experiments in
Semiconductor Manufacturing
Costas J. Spanos
Department of Electrical Engineering
and Computer Sciences
University of California Berkeley, CA 94720,
U.S.A.
tel (510) 643 6776, fax (510) 642 2739
email spanos@eecs.berkeley.edu
http://bcam.eecs.berkeley.edu
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C. J. Spanos
Design of Experiments
• Comparison of Treatments
– which recipe works the best?
• Simple Factorial Experiments
– to explore impact of few variables
• Fractional Factorial Experiments
– to explore impact of many variables
• Regression Analysis
– to create analytical expressions that “model” process
behavior
• Response Surface Methods
– to visualize process performance over a range of
input parameter values
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C. J. Spanos
Design of Experiments
• Objectives:
– Compare Methods.
– Deduce Dependence.
– Create Models to Predict Effects.
• Problems:
– Experimental Error.
– Confusion of Correlation with Causation.
– Complexity of the Effects we study.
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C. J. Spanos
Problems Solved
• Compare Recipes
• Choose the recipe that gives the best results.
• Organize experiments to facilitate the analysis of
the data.
• Use experimental results to build process
models.
• Use models to optimize the process.
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C. J. Spanos
Comparison of Treatments
•
•
•
•
Internal and External References
The Importance of Independence
Blocking and Randomization
Analysis of Variance
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The BIG Question in comparison of treatments:
• How does a process compare with other
processes?
– Is it the same?
– Is it different?
– How can we tell?
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Using an External Reference to make a Decision
• An external reference can be used to decide whether a
new observation is different than a group of old
observations.
• Example: Create a comparison procedure for lot yield
monitoring. Do it without "statistics".
• Here, I use "reference data":
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C. J. Spanos
Example: Using an External Reference
To compare the difference between the average of
successive groups of ten lots, I build the histogram from
the reference data:
• Each new point can then be judged on the basis of the
reference data.
• The only assumption here is that the reference data is
relevant to my test!
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C. J. Spanos
Using an Internal Reference...
• We could generate an "internal" reference distribution
from the data we are comparing.
• Sampling must be random, so that the data is
independently distributed.
• Independence would allow us to use statistics such as
the arithmetic average or the sum of squares.
• Internal references are based on Randomization.
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C. J. Spanos
Example in Randomization
• Is recipe A different than recipe B?
660
A
B
Etch Rate
Recipe Type
A
650
640
B
620
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630
640 650
Etch Rate
660
630
0
2
4
6 8 10 12
Sample
10
C. J. Spanos
Example in Randomization - cont.
• There are many ways to decide this...
1.External reference distribution based on old data.
2. Approximate external reference distr. (either t or
normal).
3. Internal reference distribution.
4. "Distribution free" tests.
• Options 2, 3 and 4 depend on the assumption
that the samples are independently distributed.
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C. J. Spanos
Example in Randomization - cont.
• If there was no difference between A and B, then let me
assume that I just have one out of the 10!/5!5! (252)
arrangements of labels A and B.
• I use the data to calculate the differences in means for
all the combinations:
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C. J. Spanos
The Origin of the t Distribution
The student-t distribution was, in fact, defined to
approximate randomized distributions!
(yB - yA) - (µA - µB)
t0 =
s n1 + n1
A
B
• For the etch example, t0 = 0.44 and Pr (t > t0) = 0.34
• Randomized Distribution = 0.33
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C. J. Spanos
Example in Blocking
• Compare recipes A and B on five machines.
• If there are inherent differences from one machine to the
other, what scheme would you use?
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Random
Blocked
AA
AB
ABA
BA
BA
BA
BB
AB
B
BA
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C. J. Spanos
Example in Blocking - cont.
• With the blocked scheme, we could calculate the A-B
difference for each machine.
• The machine-to-machine average of these differences
could be randomized.
d=
±d1±d2±d3±d4±d5
5
d - δ ~ tn-1
sd/ n
In
randomize what
Ingeneral,
general, randomize
whatyou
youdon't
don't know
know
block
and
what
you
do
know.
and block what you do know.
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C. J. Spanos
Analysis of Variance
Recipe Type
5
D4
C3
B2
A
1
610
620
630
640
Etch Rate
650
660
Your
same or
YourQuestion:
Question:Are
Arethe
the four
fourtreatments
treatments the
thesame
ornot?
not?
The
between
TheStatistician's
Statistician's Question:
Question:Are
Are the
thediscrepancies
discrepancies between
the
groups
greater
than
the
variation
withineach
the groups greater than the variationwithin
eachgroup?
group?
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C. J. Spanos
Calculations for our Example
i=1
650
645
623
645
1:
2:
3:
4:
i=2
648
650
628
640
i=3
632
638
630
648
i=4
645
643
620
642
i=5
641
640
618
638
Avg
643.20
643.20
623.80
642.60
s t2
202.80
86.80
104.80
63.20
νt
4
4
4
4
(yt - y)2
25.00
25.00
207.36
19.36
s2R =
s2T =
s2T
=
s2R
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C. J. Spanos
Variation Within Treatment Groups
First, lets assume that all groups have the same spread.
Lets also assume that each group is normally distributed.
The following is used to estimate their common σ:
nt
St = Σ (ytj - yt)2
s2t = St
nt - 1
j=1
2
ν s +ν s2+...+ νks2k
s2R = 1 1 2 2
ν1 + ν2 +...+ νk
= SR = SR
N - k νR
• This is an estimate of the unknown, within group s square.
• It is called the within treatment mean square
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C. J. Spanos
Variation Between Treatment Groups
• Let us now form Ho by assuming that all the groups have
the same mean.
• Assuming that there are no real differences between
groups, a second estimate of sT2 would be:
k
Σ
s2T = t=1
nt(yt - y)2
k-1
= ST
νT
This is the between treatment mean square
IfIf all
all the
the treatments
treatments are
are the
the same,
same, then
then the
the within
within and
and
between
treatment
mean
squares
are
estimating
the same
same
between treatment mean squares are estimating the
number!
number!
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C. J. Spanos
What if the Treatments are different?
If the treatments are different then:
s2T estimates σ2 +
k
Σ
nt τ2t / (k - 1)
t=1
where τt ≡ µt - µ
• In other words, the between treatment mean square is
inflated by a factor proportional to the difference between
the treatments!
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C. J. Spanos
Final Test for Treatment Significance
Therefore, the hypothesis of equivalence is rejected if:
s2T
is significantly greater than 1.0
s2R
This can be formalized since:
s2T
~ Fk-1, N-k
s2R
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C. J. Spanos
More Sums of Squares
A measure of the overall variation:
k
SD = Σ
nt
Σ
t=1 j=1
(ytj - y)2
s2D = SD = SD
N - 1 νD
Obviously (actually, this is not so obvious, but it can be proven):
SD = ST + SR and ν D = ν T + ν R
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C. J. Spanos
ANOVA Table
Source
of Var
Sum
of sq
DFs
Mean sq
between
within
ST
SR
vT (k-1)
vR (N-k)
sT
sR2
total
SD
vD (N-1)
sD
2
2
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C. J. Spanos
ANOVA Table (full)
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Source
of Var
Sum
of sq
DFs
average
between
within
SA
ST
SR
vA ( 1 )
vT (k-1)
vR (N-k)
total
S
v (N)
Mean sq
2
sA
s2T
s2R
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C. J. Spanos
Anova for our example...
Data File: CompEtch
Sum of Deg. of
Squares Freedom
Source
Between
Recipe
Mean
Squares
1.3836e+3
3
4.6120e+2
Error
4.5760e+2
16
2.8600e+1
Total
1.8412e+3
19
F-Ratio
Prob>F
1.6126e+1 4.29e-5
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C. J. Spanos
Decomposition of Observations
Y=A+T+R
In Vector Form:
yti
.
.
.
N
y
= .
.
.
1
yt - y
+
.
.
.
k-1
yti - yt
+
.
.
.
N-k
The term degrees of freedom refers to the dimensionality
of the space each vector is free to move into.
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C. J. Spanos
Geometric Interpretation of ANOVA
Y=A+D
Easy to prove that A ⊥ D.
D = R +T
Easy to prove that R ⊥ T and A ⊥ R.
Y
D
R
Y
T
A
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C. J. Spanos
Model and Diagnostics
yti = µt + eti
So, the "sufficient statistics" are:
s2R, y1, y2,..., yk
as estimators of:
σ 2, µ1, µ2,..., µk
yt
For our example:
A:
B:
C:
D:
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eti ~ N (0, σ 2)
643.20
643.20
623.80
642.60
According
Accordingto
to this
thismodel,
model,the
the
residuals
residuals are
areIIND.
IIND.How
Howdo
doyou
you
verify
that?
verify that?
s2R 28.6
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C. J. Spanos
Anova Example: Poly Deposition
t
h
i
c
k
200
190
180
170
160
150
140
130
120
110
100
90
80
70
60
50
E
Recipe
B
C
D
A
F
Are these recipes significantly different?
Analysis of Variance
Source
Model
Error
C Total
DF
5
227
232
Sum of Squares
26969.525
58418.758
85388.283
F Ratio
20.9593
Prob > F
0.0000
Mean Square
5393.91
257.35
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C. J. Spanos
Residual Plots:
Residual thick.
100
0.4
75
0.3
50
25
-40
-30
-20
-10
0
10
20
30
40
50
60
70
80
0.2
0.1
90
Residual thick.
25
0.75
20
15
0.5
10
0.25
5
-40
-30
-20
-10
0
10
20
30
40
Residual thick.
40
0.6
30
0.4
20
0.2
10
-40
-20
-10
0
10
20
30
40
50
60
Residual thick.
30
20
10
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-30
-20
-10
0
10
20
30
40
50
0.6
0.4
0.2
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C. J. Spanos
Residual Plots (cont):
R
e
s
i
d
u
a
l
t
h
i
c
k
.
90
80
70
60
50
40
30
20
10
0
-10
-20
-30
-40
E
B
C
D
Deposition Recipe
E
F
V1
V2
Wafer Vendor
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C. J. Spanos
Anova Summary
•
•
•
•
Plot Originals
Construct ANOVA table
Are the treatment effects significant?
Plot residuals versus:
–
–
–
–
treatment
group mean
time sequence
other?
• ANOVA is the basic tool behind most empirical
modeling techniques.
(chapter 6 in BHH)
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