C. J. Spanos
Multivariate Control and Model-Based SPC
T2, evolutionary operation, regression
chart.
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C. J. Spanos
Multivariate Control
Often, many variables must be controlled at the same time.
Controlling p independent parameters with parallel charts:
α' = 1 - (1 - α)p
P{all in control} =(1 - α)p
If the parameters are correlated, the type I (false alarms)
and type II (missed alarms) rates change.
We need is a single comparison test for many variables. In
one dimension, this test is based on the student t statistic:
t=
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(x - µ 0) (x - µ 0)
~ t(n-1)
=
2
sx
s
n
2
C. J. Spanos
IR Charts
3
1
UCL=2.5
2
CD
1
0
Avg=-0.2
-1
-2
-3
LCL=-2.9
50
100
150
200
Monitoring Multiple Un-correlated
Variables
CD
UCL=3.3
3
Thick
2
1
Avg=0.1
0
-1
-2
-3
100
150
200
1.0
Thick
UCL=3.0
3
Angle
2
1
Avg=-0.1
0
-1
-2
1
-3
50
LCL=-3.1
100
150
200
0.9
0.8
0.7
0.6
Angle
UCL=3.0
3
0.5
2
1
X-miss
Type I Error vs Number of Variables
LCL=-3.2
50
Avg=0.0
0
-1
-2
LCL=-3.0
1
-3
50
100
150
0.4
200
0.3
X-miss
1
3
UCL=2.9
0.2
Avg=-0.1
0.1
Y-Miss
2
1
0
-1
-2
LCL=-3.0
-3
50
100
150
200
0.0
0
Y-Miss
1
3
5
10 15 20 25 30 35 40 45 50
UCL=3.0
2
Refl
1
Avg=0.0
0
-1
-2
-3
LCL=-2.9
50
100
150
200
type I, 3sigma
type I, 0.05
Refl
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C. J. Spanos
Multivariate Control (cont.)
To compare p mean values to an equal number of targets we
use the T2 statistic:
T2 = n (x - x)' S -1 (x - x)
p(n - 1)
T2α, p, n - 1 = n - p Fα, p, n - p
x1
x1
with x =
x2
...
xp
, x=
x2
...
xp
S is the covariance matrix, x are the means for the last
sample and x the global means. We get an alarm when the
T2 exceeds a critical value (set by the F-statistic).
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C. J. Spanos
Example: Center and left temps are correlated
610
608
606
604
602
600
610
608
606
604
602
600
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0
20
40
60
80
100
5
C. J. Spanos
Examining the two Variables Together
Correlations
Cent Temp
1.0000
0.6094
Variable
Cent Temp
Left Temp
Left Temp
0.6094
1.0000
608
607.5
607
606.5
606
605.5
605
604.5
604
603.5
603
602.5
602
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601
603
605
607
609
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C. J. Spanos
Example (cont.)
Since left and center are correlated, with estimated σ =1.15,
ρ=0.61, their deviation from the target 605 can be
determined by a single plot:
30
α = 0.05
20
10
0
0
20
40
60
80
100
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C. J. Spanos
Example (cont.)
For
two
parameters,
representation is possible:
another
graphical
610
608
606
604
602
600
600
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602
604
606
center
608
610
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C. J. Spanos
Example - Multivariate Control of Plasma Etch
Five strongly correlated parameters* can be collected during
the process:
Haifang's Screen Dump
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* Tune vane, load coil, phase error, plasma imp. and peak-to-peak voltage.
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C. J. Spanos
Example - Multivariate SPC of Plasma Etch (cont.)
2200
2000
1800
1600
1400
1200
1000
800
600
400
200
0
1
2
3
4
5
UCL 55.00
0
5
10
15
20
25
30
The first 24 samples were recorded during the etching of 4
"clean" wafers. The last 6 are out of control and they were
recorded during the etching of a "dirty" wafer.
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C. J. Spanos
Evolutionary Operation - An SPC/DOE Application
A process can be optimized on-line, by inducing small
changes and accepting the ones that improve its quality.
EVOP can be seen as an on-line application of designed
experiments.
Example: Assume a two-parameter process:
y = f (x 1, x 2) + e
and assume the following approximate model:
y - a x 1 + b x 2 + c x 1x 2
If we knew a, b, and c, we would know how to change the
process in order to bring y closer to the specifications.
Of course this model will only be applicable for a narrow
range of the input parameters.
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C. J. Spanos
Evolutionary Operation (cont.)
2
design a 2 factorial experiment
x2
y5
+1
y3
H
y1
L
-1
-1
y2
y4
L
H
+1
x1
(Note that x1 and x2 are scaled so that they take the
values -1, +1, at the edges of the experiment).
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C. J. Spanos
Evolutionary Operation (cont.)
Once the effects are known, choose the best corner of the
box and start a new experiment around it:
x2
x1
The process terminates when I find a box whose corners are
no better than its center.
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C. J. Spanos
Evolutionary Operation (cont.)
The values of a, b and c (or the respective "effects" and
"interactions") can be estimated:
a = Effx1 = 1 [(y3+y4) - (y2+y5)]
2
b = Effx2 = 1 [(y3+y5) - (y2+y4)]
2
c = Intx1x2 = 1 [y2+y3 - (y4+y5)]
2
This calculation is repeated for n-cycles until one of the
effects emerges as a significant factor.
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C. J. Spanos
Evolutionary Operation (cont.)
To decide whether an effect is significant, we need a good
estimate of the process sigma.
The sigma of the process can be estimated from the
difference of the last average and the new value at each of
the experimental points. This value is distributed as:
N ( 0, σ2 n )
n-1
The 95% confidence interval of each effect is:
+/- 2n s
and of the change-in-mean effect is:
+/-
1.78 s
n
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C. J. Spanos
Example - Use EVOP on a cleaning solution
The yield from the first 4 cycles of a chem. process is shown
below. The variables are % conc. (x1) at 30 (L), 31 (M), 32 (H)
and temp. (x2) at 140 (L), 142 (M), 144 F (H).
Cycle Y1 M-M
1
60.7
2
69.1
3
66.6
4
60.5
avg Y 64.2
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Y2 L-L
69.8
62.8
69.1
69.8
67.9
Y3 H-H
60.2
62.5
69.0
64.5
64.1
Y4 H-L
64.2
64.6
62.3
61.0
63.0
Y5 L-H
57.5
58.3
61.1
60.1
59.3
conc. effect = 1 [(y 3+y 4) - (y 2+y 5)] = -0.025
2
temp. effect = 1 [(y 3+y 5) - (y 2+y 4)] = -3.800
2
interaction = 1 [y 2+y 3 - (y 4+y 5)] = 4.825
2
chng in mean = 1 [y 2+y 3+y 4+y 5 - 4y 1] = -0.540
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C. J. Spanos
Example - EVOP on a cleaning solution (cont.)
We use the range of consecutive differences in order to
estimate the sigma of the process:
Take range of ( y ji - y j-1
i ) for i = 1,2,3,4,5
ave rage for j = 2,3,4 R D = 7.53
and
RD
= σ n/(n-1) , i.e. σ =2.787
d2
The 95% confidence limits for concentration, temperature
and their interaction are:
+/- 2 σ / n = +/- 2.787
So, temperature and interaction are significant. Their signs
dictate moving to point 2 (L-L).
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C. J. Spanos
EVOP Monitoring in the Fab
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C. J. Spanos
Regression Chart - Model Based SPC
In typical SPC, we try to establish that certain process
responses stay on target.
What happens if there is one assignable cause that we know
and we can quantify?
If, for example, the deposition rate of poly is a function of the
time since the last tube cleaning, it will never be "in control".
In cases like this, we build a regression model of the
response vs the known effect, and we try to establish that the
regression model remains valid throughout the operation.
Limits around the regression line are set according to the
prediction error of the model.
A t-statistic is used to update the model whenever necessary.
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C. J. Spanos
Regression chart (cont.)
Regression Chart
Polysilicon Deposition Rate
1100
1400
UCL 1348.96
1200
1000
1000
900
797.24
800
2σ
800
600
700
400
600
LCL 245.51
500
200
0
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Sample Count
20
0
10
20
30
# of runs after clean
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Regression Chart (cont.)
C. J. Spanos
• The regression chart can be generalized for complex
equipment models.
• An empirical model is built to describe the changing
aspects of the process.
• The difference between prediction and observation
can be used as the control statistic
• If the control statistic becomes consistently different
than zero, its value can be used to update the model.
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Model Test and Adaptation
C. J. Spanos
LPCVD Model
ln(Ro) = A + B ln (P) + C(1/T) + D (1/Q)
After substitution, the equation used is:
Y = A + BxB + CxC + DxD
Control Limits: Y +/- (t * s)
n
Cumulative
t=
Σ
i=1
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(Y i-y i)
sy
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C. J. Spanos
Actual Rate [ln(Å/min)]
Regression Chart (Example)
Actual Rate [ln(Å/min)]
5.6
5.5
5.4
5.3
Actual Rat
UCL
LCL
5.2
5.1
5.0
4.9
4.8
4.8 4.9 5.0 5.1 5.2 5.3 5.4 5.5 5.6
Model Rate [ln(Å/min)]
6.0
5.9
5.8
5.7
5.6
Actual
5.5
UCL
5.4
5.3
LCL
5.2
5.1
5.0
4.9
4.8
4.84.95.05.15.25.35.45.55.6
Model Rate [ln(Å/min)
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C. J. Spanos
Regression Chart (Example)
Outdated Model
Updated Model
5.6
5.6
5.5
5.5
5.4
5.3
Actual Rate
UCL
5.2
LCL
5.1
5.0
4.9
4.8
4.8
5.3
Actual Rate
5.2
Revised UCL
Revised LCL
5.1
5.0
4.9
4.9
5.0
5.1
5.2
5.3
5.4
Model Rate [ln(Å/min)]
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Actual Rate [ln(Å/min)]
Actual Rate [ln(Å/min)]
5.4
5.5
5.6
4.8
4.8
4.9
5.0
5.1
5.2
5.3
5.4
5.5
5.6
Revised Model Rate [ln(Å/min)]
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C. J. Spanos
Summary so far...
As we move from classical, human operator oriented
techniques, to more automated CIM based approaches:
• We need to increase sensitivity (reduce type II error),
without increasing type I error. (CUSUM, EWMA).
• We need to distinguish between abrupt and gradual
changes. (Choice of EWMA shape).
• We need to accommodate multiple sensor readings
(T2 chart).
• We need to accommodate multiple recipes and products
in each process (EVOP, model-based SPC).
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