Ecole Nationale Supérieure des Mines de Saint-Etienne Generalized Moving Variance and Decompositions Ariane FERREIRA Associate Professor 2 nd ISSPC July 13th, 2011 Rio de Janeiro
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Ecole Nationale Supérieure des Mines de Saint-Etienne
Generalized Moving Variance and Decompositions
Ariane FERREIRAAssociate Professor
2nd ISSPC July 13th, 2011 Rio de Janeiro
Statistical Control Charts for Generalized Moving Variances
Outline Introduction
What’s VARIATION? Problems and Motivation
1
Temporal Moving Variance/Covariance Temporal Moving VarianceTemporal Moving CovariancePerformance Example of Application
2
Generalized Moving Variance (GMV)Distribution-based Concept: GMVGMV DecompositionMethodology for GMV calculation
3
Case Study of Semiconductor manufacturingCVD equipment: linkage detectionEWMA control chart
4
Perspectives5
Statistical Control Charts for Generalized Moving Variances
Introduction: What is variation ?
• a change or slight difference in a level, amount, or quantity ……
• Variance a measure of the amount of variation within the values of data ………
70
80
90
100
110
120
130
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99
B
7
8
9
10
11
12
13
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 101
A
s2(A) = 0.85 s2(A) = 0.85 s2(B) = 65.30s2(B) = 65.30
S 2
ni
S 2
ni
;1
)(1
2
2
n
xxs
n
ii
x1
)(1
2
2
n
yys
n
ii
y 1
))((1
n
yyxxs
n
iii
xy
;
Statistical Control Charts for Generalized Moving Variances
Introduction: Data described by the ProfilesIn real world, you’ll always have data like:
4
0
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35
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45
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99
0
5
10
15
20
25
30
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45
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 -5
0
5
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1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99
Pattern-induced VariationPattern-induced VariationSystematic VariationSystematic Variation Random VariationRandom Variation
= +
= +
Identify
Systematic
Pattern?
Identify
Systematic
Pattern?
Measure
Random
Noise?
Measure
Random
Noise?Variation
AnalysisVariation
Analysis
Total VariationTotal Variation = +
Statistical Control Charts for Generalized Moving Variances
Introduction: Practical Method for Random Variation Analysis
5
0
5
10
15
20
25
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99
0
5
10
15
20
25
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 990
5
10
15
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25
30
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40
45
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99
0
5
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15
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1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99
22.1
95.92
xs
x
95.0
96.192
ys
y
There can be hundreds or thousands of data profiles………There can be hundreds or thousands of data profiles………
Profile analysis by windowsWindow Window
Statistical Control Charts for Generalized Moving Variances
6
Introduction: Problems and Motivation
• Difficulties• pattern-dependent profiles• non-stationary data distributions
• Model-based methodologies are commonly practiced.• Costly, time-consuming
• Chen and Blue 2009 data itself should speak much more before domain knowledge is
involved.
• Concept of Moving Variance• Generalized Moving Variance
• proportional to the volume of data distributed in the multi-dimensional variable space and can be used to measure the dispersion of profiles A. Chen and J. Blue, “Recipe-independent Indicator for Tool Health
Diagnosis and Predictive Maintenance,” IEEE Transactions on Semiconductor Manufacturing, pp. 522-535, 22, 4, November, 2009
Statistical Control Charts for Generalized Moving Variances
Temporal Moving Variance (p=2)
8
10
12
14
16
18
20
22
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
SVID
Rea
ding
Time
SVID X
8
10
12
14
16
18
20
22
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
SVID
Rea
ding
Time
SVID X
21ws
22ws . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
18ws2
19ws
19
1
22
19
1ˆ
iwix s
Profiles name : Status Variable IDentification (SVID)
pX xxw ,11
112 , pX xxw
Moving Windows of size p (<n)
npnpnX xxw ,11
Chen and Blue, 2009
Statistical Control Charts for Generalized Moving Variances
Temporal Moving Covariance (p=2)
[email protected] name : Status Variable IDentification (SVID)
Moving Windows of size p (<n)
100
150
200
250
300
350
8
13
18
23
28
33
38
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
SVID
Y R
eadi
ng
SVID
X R
eadi
ng
Time
SVID X SVID Y
100
150
200
250
300
350
8
13
18
23
28
33
38
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
SVID
Y R
eadi
ng
SVID
X R
eadi
ng
Time
SVID X SVID Y
21w 2
2w . . . . . . . . . . . . . . . . . . . . . . . . . . . 218w
219w
19
1
22
19
1ˆ
iwixy
pX xxw ,11
112 , pX xxw
npnpnX xxw ,11
pY yyw ,11
112 , pY yyw
npnpnY yyw ,11
Chen and Blue, 2009
Statistical Control Charts for Generalized Moving Variances
Performance of Temporal Moving Variance/Covariance
9 0
5
10
15
20
25
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
SVID
Rea
ding
Time
SVID X
0
5
10
15
20
25
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
SVID
Rea
ding
Time
SVID X
0
5
10
15
20
25
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
SVID
Rea
ding
Time
SVID X
SVID Y
0
5
10
15
20
25
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
SVID
Rea
ding
Time
SVID X
SVID Y
Recipe A
Recipe B
sample variance = 28.01
sample variance = 37.59
sample covariance = 20.68
sample covariance = 30.77
moving variance = 2.14
moving variance = 2.32
moving covariance = 0.12
moving covariance = 0.13
Statistical Control Charts for Generalized Moving Variances
Example of Application Semiconductor manufacturing
Voltage
Pressure
Temperature
T=1, 2, 3, ………………………………, 100
Measurement Tool
Process Tool
SVID 1
SVID 2
SVID 3
Statistical Control Charts for Generalized Moving Variances
2t
22,ps
2,vR
…………………
…………………
…………………
nt
2,nps
nvR ,
1t
21,ps
1,vR
What Do Engineers Do?
Voltage
Pressure
Temperature
wafer 1wafer 1 wafer 2wafer 2 …… wafer nwafer n
10~50 SVID’s in a
process
10~50 SVID’s in a
process
22,ts ………………… 2
,nts21,ts
2p …………………np1p
More than 2 summarized
statistics a SVID
More than 2 summarized
statistics a SVID
More than 300 Process ToolsMore than 300 Process Tools
Statistical Control Charts for Generalized Moving Variances
Distribution-based Concept: Generalized Variance
12
Voltage
Pressure
Temperature
Temperature
Voltage
Pressure
Generalized VarianceGeneralized Variance
Size of DistributionSize of Distribution
wafer 1
222
222
222
det)det(
VVVPVT
PVPPPT
TVTPTT
sss
sss
sss
S
Chen and Blue, [email protected]
SV
ID 1
SV
ID 2
SV
ID 3
Statistical Control Charts for Generalized Moving Variances
222
221
212
211
ˆˆ
ˆˆ
S
13
PM1
PM2
PM3
....
PM cycle
....
step 1 step 2
....
Moving Variance/Covariance Matrix
Moving Variance/Covariance Matrix
)det(S Generalized Moving Variance
Generalized Moving Variance
wafer 1
Generation of Generalized Moving Variance
[email protected] Chen and Blue, 2009
Statistical Control Charts for Generalized Moving Variances
0.00E+00
5.00E+41
1.00E+42
1.50E+42
2.00E+42
2.50E+42
3.00E+42
3.50E+42
4.00E+42
4.50E+42
5.00E+42
1 43 85 127 169 211 253 295 337 379 421 463 505 547 589 631 673 715 757 799 841 883 925 967
Wafer ID
Tool Health
Example of GMV MonitoringPM1 PM2 PM3
EWMA Control LimitEWMA Control Limit
.... ....Equipment Condition
Gen
eral
ized
Mov
ing
Varia
nce
14
Statistical Control Charts for Generalized Moving Variances
Generalized Moving Variance decomposition
15
Moving variance/covariance matrix S
22
22
222
ˆˆ
ˆˆ
ˆˆˆ
1
2221
11211
vvv
v
S
Matrix S’ only considers moving variance: Set all the moving covariances on the off-diagonal to zeroes .
2
2
2
ˆ00
0
ˆ0
00ˆ
22
11
vv
S
Matrix S’’ consider the moving covariance only: the influence of moving variance within S, i.e., the effect of S’, should be removed.
1ˆˆ
ˆ
1ˆˆ
ˆˆˆ
ˆ
ˆˆ
ˆ1
)()(
11
21
2211
221
11
21
2211
212
2/12/1
vv
v
vv
v
SSSS
Only relationships among SVIDs
Only SVIDs variability
Statistical Control Charts for Generalized Moving Variances
Summary of generalized moving variance decomposition
16
Equipment Conditiondet(S)
SVID Variabilitydet(S’)
SVID Interrelationdet(S’’)
)det()det()det( SSS 1)det( S
)det()det( SS
Property from Chen and Blue, 2009
Moving Generalized Variance decomposition
Statistical Control Charts for Generalized Moving Variances
221
222
221
21
212
211
ˆˆ
ˆˆ
ˆˆˆ
vvv
v
S
2
222
211
ˆ00
0
ˆ0
00ˆ
'
vv
S
1ˆˆ
ˆ
1ˆˆ
ˆˆˆ
ˆ
ˆˆ
ˆ1
''
11
21
1122
221
11
21
2211
212
vv
v
vv
v
S
Equipment Condition
SVID Variability SVID Interrelation
Property :det(S) = det(S’)det(S’’)
Property :det(S) = det(S’)det(S’’)
Equipment ConditionEquipment Condition
SVID VariabilitySVID Variability
SVID InterrelationSVID Interrelation
17
Statistical Control Charts for Generalized Moving Variances
Methodology for GMV calculation
18
For each processed wafer, prepare its FDC data consisting of v SVID’s with n observations for each SVID
Calculate the moving variance/covariance matrix S with moving window size p=2 for each wafer.
Calculate det(S) after each wafer is processed
Monitor the tool health by statistical control chart of det(S)
Statistical Control Charts for Generalized Moving Variances
SVIDs profile Data
19
13 quantitative SVIDs profile dataProcess steps: 9~12
Variables TypeContext qualitative identifiant lignesTimeStamp quanatitative tempsTime Elapsed quantitative temps du stepAFC current flow (0) quantitative -1 à 999AFC current flow (1) quantitative -2 à 1498AFC current flow 20 quantitative -1 à 374AFC current flow 21 quantitative 0 à 602AFC current flow 22 quantitative 297 à 998AFC current flow 23 quantitative 0 à 223Chamber capacitive manometer (8) quantitative 12693603 à 5629053chamber heater press (8) videschbr back pres frln tc pres 08 quantitative 33073 à 183826CVD BacksPressure 08 quantitative 632 à 3501forward pwr (8) quantitative 3 à 753reflected pwr (8) quantitative 0 à 40StepID identifiant steps -2 à 20susceptor temp per wafer temp (8) quantitative 399 à 402throttle step (8) quantitative 270 à 364
StepID Variable throttle step (8)9 quantitative 362 à36410 quantitative 364 à 27011 valeur constante 27012 quantitative 270 à 287
StepID variable chbr back pres frln tc pres 089 quantitative 175155 à 18382610 quantitative 72521 à 18271911 quantitative 41857 à 6785912 quantitative 33073 à 43402
Engineering Knowledgment: Main SVIDs
Case Study of Semiconductor Manufacturing
Diffusion: CVD Equipment MSP18-3: Linkage Detection
Statistical Control Charts for Generalized Moving Variances
Case Study of Semiconductor Manufacturing
20
• Diffusion: CVD Equipment MSP18-3: Linkage Detection
• Analysis Done• Calculate the Generalized Moving Variance for all process steps.
• nothing obvious found.• Calculate the Generalized Moving Variance for Step9~12.
• abnormal pattern shown on 3rd of December• similar pattern in SVID interrelation
• Check all the interrelations for Step9~12.• Check all the temporal profiles for Step9~12.
Statistical Control Charts for Generalized Moving Variances
21
-5.37E+08
5.00E+23
1.00E+24
1.50E+24
2.00E+24
2.50E+24
3.00E+24
3.50E+24
4.00E+24
2010
/11/
30 0
7:04
:20.
791
2010
/11/
30 0
7:11
:21.
362
2010
/11/
30 0
7:50
:08.
828
2010
/11/
30 0
7:57
:09.
180
2010
/11/
30 0
8:07
:39.
474
2010
/11/
30 0
8:14
:41.
544
2010
/11/
30 0
8:21
:41.
802
2010
/11/
30 1
8:44
:01.
341
2010
/11/
30 1
8:51
:02.
817
2010
/11/
30 1
9:01
:38.
611
2010
/11/
30 1
9:08
:39.
775
2010
/11/
30 1
9:15
:40.
080
2010
/11/
30 1
9:22
:42.
135
2010
/12/
01 0
1:31
:07.
423
2010
/12/
01 0
1:38
:09.
540
2010
/12/
01 0
1:45
:11.
923
2010
/12/
01 0
1:52
:12.
432
2010
/12/
01 0
1:59
:13.
471
2010
/12/
01 0
2:09
:43.
545
2010
/12/
03 1
9:47
:43.
216
2010
/12/
04 0
8:07
:33.
586
2010
/12/
04 0
8:14
:33.
578
2010
/12/
04 0
8:22
:11.
884
2010
/12/
04 0
8:29
:14.
986
2010
/12/
04 0
8:36
:17.
103
2010
/12/
04 0
8:43
:16.
517
2010
/12/
06 0
1:11
:01.
446
2010
/12/
06 0
1:31
:54.
860
2010
/12/
06 0
2:10
:41.
749
2010
/12/
06 0
2:17
:42.
726
2010
/12/
06 0
2:31
:45.
648
2010
/12/
06 0
2:38
:46.
437
2010
/12/
06 0
2:49
:18.
137
2010
/12/
06 0
6:06
:08.
598
2010
/12/
06 0
6:13
:09.
575
2010
/12/
06 0
6:20
:09.
774
2010
/12/
06 0
6:27
:09.
343
2011
/01/
09 0
7:20
:27.
194
2011
/01/
09 0
7:30
:58.
112
2011
/01/
09 0
7:37
:59.
417
2011
/01/
09 0
7:45
:00.
019
2011
/01/
09 0
7:52
:00.
215
2011
/01/
09 0
7:58
:59.
504
2011
/01/
09 0
9:12
:59.
620
2011
/01/
09 0
9:20
:00.
379
2011
/01/
09 0
9:26
:59.
621
2011
/01/
09 0
9:37
:29.
654
2011
/01/
09 0
9:44
:29.
066
Generalized Moving Variance
3rd of December
Generalized Moving Variance on Step9~12
Statistical Control Charts for Generalized Moving Variances
Decomposition of GMV SVID Variability
22
-4.15E+34
5.00E+49
1.00E+50
1.50E+50
2.00E+50
2.50E+50
3.00E+50
3.50E+50
2010
/11/
30 0
7:04
:20.
791
2010
/11/
30 0
7:11
:21.
362
2010
/11/
30 0
7:50
:08.
828
2010
/11/
30 0
7:57
:09.
180
2010
/11/
30 0
8:07
:39.
474
2010
/11/
30 0
8:14
:41.
544
2010
/11/
30 0
8:21
:41.
802
2010
/11/
30 1
8:44
:01.
341
2010
/11/
30 1
8:51
:02.
817
2010
/11/
30 1
9:01
:38.
611
2010
/11/
30 1
9:08
:39.
775
2010
/11/
30 1
9:15
:40.
080
2010
/11/
30 1
9:22
:42.
135
2010
/12/
01 0
1:31
:07.
423
2010
/12/
01 0
1:38
:09.
540
2010
/12/
01 0
1:45
:11.
923
2010
/12/
01 0
1:52
:12.
432
2010
/12/
01 0
1:59
:13.
471
2010
/12/
01 0
2:09
:43.
545
2010
/12/
03 1
9:47
:43.
216
2010
/12/
04 0
8:07
:33.
586
2010
/12/
04 0
8:14
:33.
578
2010
/12/
04 0
8:22
:11.
884
2010
/12/
04 0
8:29
:14.
986
2010
/12/
04 0
8:36
:17.
103
2010
/12/
04 0
8:43
:16.
517
2010
/12/
06 0
1:11
:01.
446
2010
/12/
06 0
1:31
:54.
860
2010
/12/
06 0
2:10
:41.
749
2010
/12/
06 0
2:17
:42.
726
2010
/12/
06 0
2:31
:45.
648
2010
/12/
06 0
2:38
:46.
437
2010
/12/
06 0
2:49
:18.
137
2010
/12/
06 0
6:06
:08.
598
2010
/12/
06 0
6:13
:09.
575
2010
/12/
06 0
6:20
:09.
774
2010
/12/
06 0
6:27
:09.
343
2011
/01/
09 0
7:20
:27.
194
2011
/01/
09 0
7:30
:58.
112
2011
/01/
09 0
7:37
:59.
417
2011
/01/
09 0
7:45
:00.
019
2011
/01/
09 0
7:52
:00.
215
2011
/01/
09 0
7:58
:59.
504
2011
/01/
09 0
9:12
:59.
620
2011
/01/
09 0
9:20
:00.
379
2011
/01/
09 0
9:26
:59.
621
2011
/01/
09 0
9:37
:29.
654
2011
/01/
09 0
9:44
:29.
066
SVIDVariablility
Statistical Control Charts for Generalized Moving Variances
Decomposition of GMV SVID Interrelation
23
0.00E+00
2.00E+05
4.00E+05
6.00E+05
8.00E+05
1.00E+06
1.20E+06
1.40E+06
1.60E+06
1.80E+06
2.00E+06
2010
/11/
30 0
7:04
:20.
791
2010
/11/
30 0
7:11
:21.
362
2010
/11/
30 0
7:50
:08.
828
2010
/11/
30 0
7:57
:09.
180
2010
/11/
30 0
8:07
:39.
474
2010
/11/
30 0
8:14
:41.
544
2010
/11/
30 0
8:21
:41.
802
2010
/11/
30 1
8:44
:01.
341
2010
/11/
30 1
8:51
:02.
817
2010
/11/
30 1
9:01
:38.
611
2010
/11/
30 1
9:08
:39.
775
2010
/11/
30 1
9:15
:40.
080
2010
/11/
30 1
9:22
:42.
135
2010
/12/
01 0
1:31
:07.
423
2010
/12/
01 0
1:38
:09.
540
2010
/12/
01 0
1:45
:11.
923
2010
/12/
01 0
1:52
:12.
432
2010
/12/
01 0
1:59
:13.
471
2010
/12/
01 0
2:09
:43.
545
2010
/12/
03 1
9:47
:43.
216
2010
/12/
04 0
8:07
:33.
586
2010
/12/
04 0
8:14
:33.
578
2010
/12/
04 0
8:22
:11.
884
2010
/12/
04 0
8:29
:14.
986
2010
/12/
04 0
8:36
:17.
103
2010
/12/
04 0
8:43
:16.
517
2010
/12/
06 0
1:11
:01.
446
2010
/12/
06 0
1:31
:54.
860
2010
/12/
06 0
2:10
:41.
749
2010
/12/
06 0
2:17
:42.
726
2010
/12/
06 0
2:31
:45.
648
2010
/12/
06 0
2:38
:46.
437
2010
/12/
06 0
2:49
:18.
137
2010
/12/
06 0
6:06
:08.
598
2010
/12/
06 0
6:13
:09.
575
2010
/12/
06 0
6:20
:09.
774
2010
/12/
06 0
6:27
:09.
343
2011
/01/
09 0
7:20
:27.
194
2011
/01/
09 0
7:30
:58.
112
2011
/01/
09 0
7:37
:59.
417
2011
/01/
09 0
7:45
:00.
019
2011
/01/
09 0
7:52
:00.
215
2011
/01/
09 0
7:58
:59.
504
2011
/01/
09 0
9:12
:59.
620
2011
/01/
09 0
9:20
:00.
379
2011
/01/
09 0
9:26
:59.
621
2011
/01/
09 0
9:37
:29.
654
2011
/01/
09 0
9:44
:29.
066
SVIDInterrelation X 1E+30
Similar pattern as shown in Generalized Moving VarianceSimilar pattern as shown in
Generalized Moving Variance
Statistical Control Charts for Generalized Moving Variances
SVID Interrelations Check• Investigate all the “moving covariances” between throttle step (8) and other
SVID’s.• The moving variances become less fluctuant after 3/12/2010.
• AFC current flow (1)• AFC current flow 22• CVD BacksPressure 08
• The level of moving covariance drops down a little after 3/12/2010.• AFC current flow 20• AFC current flow 23
• The level of moving covariance rises up a little after 3/12/2010.• AFC current flow 21• forward pwr (8)
24
Statistical Control Charts for Generalized Moving Variances
Wafer FDC Analyser Prototype Demonstration
25
Statistical Control Charts for Generalized Moving Variances
Perspectives
26
To develop the statistical design for GMV and Decompositions
1. Advanced Parametric Control Chart
2. Non Parametric Advanced and Innovating Control Charts
Construction of Statistical Control Charts for Generalized Moving Variance
Control Charts for Generalized Moving
Variance Decompositions Non parametric Control
Charts
Parametric Control Charts
Test and validation on simulated and industrial data
Statistical Control Charts for Generalized Moving Variances
AppendixSVID Interrelations screen shots
27
Statistical Control Charts for Generalized Moving Variances
Jakey Blue28
Statistical Control Charts for Generalized Moving Variances
Jakey Blue29
Statistical Control Charts for Generalized Moving Variances
Jakey Blue30
Statistical Control Charts for Generalized Moving Variances
Jakey Blue31
Statistical Control Charts for Generalized Moving Variances
Jakey Blue32
Statistical Control Charts for Generalized Moving Variances
Jakey Blue33
Statistical Control Charts for Generalized Moving Variances
Jakey Blue34
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