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Statistical Process Control
Implementation
in Semiconductor Manufacturing
Tzu-Cheng Lin 林資程
Advanced Control Program/ IIPD/ R&D
Taiwan Semiconductor Manufacturing Company, Ltd
[email protected], [email protected]
March 26, 2010
NCTS Industrial Statistics
Research Group Seminar
Page 1 of 56
Department of Electrical and
Control Engineering, NCTU
Electron Industrial Control Lab
Agenda:
This presentation will cover the following topics:
1. MVA application: Advanced Bi-Variate Semiconductor Process Control Chart.
2. MVA application: Yield2Equipment Events Mining.
3. PLS application: Virtual Metrology of Deep Trench Chain.
4. Time series application: KSI-Based to Predict Tool Maintenance.
5. Survival application: Advanced Queue-Time to Yield Monitoring System.
6. SPC chart application: Smart Process Capability Trend Monitoring System.
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Department of Electrical and
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Electron Industrial Control Lab
Case(1): MVA application
Advanced Bi-Variate Semiconductor
Process Control Chart.
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Page 3 of 56
Department of Electrical and
Control Engineering, NCTU
Electron Industrial Control Lab
Advanced Bi-Variate Semiconductor
Process Control Chart
Motivation:
Process Variation = (Process Metrology Value) + (Tool Healthy Quality) + (Metrology Tool Calibration)
SPC monitoring system FDC monitoring system  MSA calibration scheme
As you know, In Line Process Control is a great important task on semiconductor manufacturing.
We usually use the SPC system to monitor the process measurement data, and use the FDC
system to monitor the tool healthy index. Although engineers via theses two regular systems, they
could check the process is stable or not ?? BUT it is time consuming for engineers, ……..
Innovative idea !!
If we could build up the Bi-Variate Process Control Chart which based on the
relationships between In-Line metrology data and FDC tool parameter monitoring
data, and provide the Ellipse Control Region to real time tell engineers what’s
current status for the latest process capability is stable or not??
In this way, it will give a big hand for engineers not only to monitor the SPC chart ,
but also to monitor the FDC chart at the same time.
SPC
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FDC
Department of Electrical and
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Electron Industrial Control Lab
Advanced Bi-Variate Semiconductor
Process Control Chart
Innovative idea
profile:
Remarks:
(1)
Process_A

FDC Summary Value (Y)
(3)

(5)
(6)
(4)

(7)
(X,Y

)






 


(9)
9
(8)
8
(10)
  
In-Line Metrology Value (X)
NCTS Industrial Statistics
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(1) The box is showing the process
information on this chart.
(2) X-axis is the In-Line metrology value (X).
(3) Y-axis is the FDC summary value (Y).
(4) The light-gray area is the Ellipse Control
Region with 1 sigma.
(5) The mid-gray area is the Ellipse Control
Region with 2 sigma.
(6) The dark-gray area is the Ellipse Control
Region with 3 sigma.
(7) The red point is contributed from (X,Y)
and draw it on this specific control chart.
(8) When the point is out of 3 sigma area, it’ll
give a ‘x’ symbol to represent the OOC case.
(9) When the point is OOC, it’ll also provide
the Wafer_ID nearby it.
(10) The ‘green’, ‘yellow’, and ’red’ light will
point out the degree of stability on this
process.
(2)
Page 5 of 56
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Electron Industrial Control Lab
Full-Line
Bi-Variate Semiconductor Process Control Chart
It can integrate semiconductor full-line process & tool information into one
system, and to be a kind of real time control tool for modern 12” iFab.
Via this advanced
process control
chart, we’d be
more easily to
check the
process
status.
NCTS Industrial Statistics
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Page 6 of 56
Department of Electrical and
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Electron Industrial Control Lab
Case study
ALD NOLA Depth is a new process for new generation. So, we’re going to use this “Advanced
Bi-Variate Semiconductor Process Control Chart” to monitor this critical process:
1) In-line metrology value: Depth (nm).
2) Equipment FDC parameters: Var1-Var25.
Trial data looks like…
3) 34 raw data sets.
ALDA102 - PM4
1.25
LCL
1.2
ALDA102-PM4
Depth (nm)
1.15
1.1
Target
1.05
1
UCL
0.95
0.9
0
5
10
15
20
25
30
35
Run#
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Department of Electrical and
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Electron Industrial Control Lab
Step(1):
Select Key Steps and Parameters
Due to for ALDA equipment has so many tool parameters, we need the
engineers/ vendors to provide the key process steps (some critical steps in
the recipe) and parameters where measurements have significant effect on
product quality.
Process step
Variables
Identify the key steps
and variables.
:the key parameter in corresponding step.
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Page 8 of 56
Department of Electrical and
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Electron Industrial Control Lab
Step(2):
T-Score Transformation
TOOL: ALDA102/PM4
ChamberPressure
PumpingPressure
MFC1
GasLineHeater1Temp
StageHeaterInTemp
StageHeaterOutTemp
Source1HeaterTemp
Source2HeaterTemp
ThrottleValveHeaterTemp
PumpingLineHeaterTemp
ChamberWallHeaterTemp
ChamberBottomHeaterTe
mp
SHInletHeaterTemp
VATValveHeaterTemp
Source1_Outlet_Pressure
……….
………
…..
…
..
.
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Matrix [34X25]
…..
*Huge data reduce to only ONE index:
T 2  f ( x1 , xn )
*FDC Summary Value:
T 2  ( x  x )S 1 ( x  x )
Based on each wafer, we’d
provide the one index- FDC
summary value, which could
represents all tool
parameters healthy status.
Page 9 of 56
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Step(3):
Ellipse Control Region
 Ellipse
Equations:
An ellipse centered at the point (h,k) and having its major axis parallel to the x-axis may
be specified by the equation
This ellipse can be expressed parametrically as
where t may be restricted to the interval
So, we based on the historical raw data (w/ good wafers), to set up the Ellipse
control region, and use the Confident-Interval concept to calculate the 1 to 3 sigma
alarm region to be the SPC-like, Bi-Variate process control chart.
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Department of Electrical and
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Electron Industrial Control Lab
Step(4):
Simulation for NOLA Depth process
These four points are in
warning control region.
(4)
(3)
The 1 to 3 sigma
Ellipse control region.
(2)
These two points are OOC!!
(1)
Bi-Variate Semiconductor
Process
ControlStatistics
Chart:
NCTS
Industrial
Research
Seminar
SPC+FDCGroup
information.
Page 11 of 56
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Conclusions
 From the simulation testing, it seems that our innovative
proposal Advanced Bi-Variate Semiconductor Process Control
Chart can monitor the semiconductor process variation
successfully.
 Advanced Bi-Variate Semiconductor Process Control Chart
approach not only can be used to monitor the Process
Information (SPC Chart) , but also to monitor the Tool
Information (FDC Chart) at the same time.
 The degree of process capability (like Traffic Lights) for
specific critical process also can be known via this novel
Bi-Variate process control chart. In this way, the
engineers could control process more easily and
efficiently.
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Department of Electrical and
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Electron Industrial Control Lab
Case(2): MVA application
Yield2Equipment Events Mining.
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Yield2Equipment Events Mining
Novel Idea:
Tool-A
MVA T-Score
PM
PM
‧‧‧‧‧‧‧‧ ‧‧ ‧‧‧
‧
‧
‧
‧
‧
‧
‧
‧
‧‧
‧‧‧‧‧‧ ‧‧
T-Score ‧‧‧‧ ‧‧ ‧ ‧
‧
‧‧‧‧ ‧ ‧ Trend
Up
YB Yield
Trend Down
time/date
Another way to point
out the abnormal tool !!
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T  ( x  x )S 1 ( x  x )
T-Score is an index to represent all tool
parameters status. If the T-Score is
larger than specific limit we can say that
this data point is significant different
from the normal condition.
During this PM cycle, the Yield and T-Score
have high correlation and T-Score is bigger
than normal condition.
In this way, we can induce that this may
occur some critical issues in this specific
time period.
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Invention Program Flowchart
Variable & Key Step
selection
•Key Step: certain time period
•Variable: critical recipe/ process parameters
Data transformation
to T-Score
•MVA Principal Component Analysis
•MVA T-Score calculation
•MSPC Hotelling T2 control limit set up [0, UCL]
Correlation analysis
between
Yield & Tool Events
•Yield & T-score trend up/down monitoring
•Pearson Correlation Analysis
•Highlight the HIGH correlation PM Cycle to
conduct Yield2Equipment Events Mining
Root Cause Analysis
•Identify suspected ill-parameters
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Department of Electrical and
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Electron Industrial Control Lab
Step(1):
Select Key Steps and Parameters
Due to for each equipment has so many tool parameters, we need the
engineers/ vendors to provide the key process steps (some critical steps in
the recipe) and parameters where measurements have significant effect on
product quality.
Process step
Variables
Identify the key steps
and variables.
:the key parameter in corresponding step.
NCTS Industrial Statistics
Research Group Seminar
Page 16 of 56
Department of Electrical and
Control Engineering, NCTU
Electron Industrial Control Lab
Step(2):
T-Score Transformation
TOOL: ALDA102 / PM5
ChamberPressure
PumpingPressure
MFC1
GasLineHeater1Temp
StageHeaterInTemp
StageHeaterOutTemp
Source1HeaterTemp
Source2HeaterTemp
ThrottleValveHeaterTemp
PumpingLineHeaterTemp
ChamberWallHeaterTemp
ChamberBottomHeaterTemp
SHInletHeaterTemp
VATValveHeaterTemp
Source1_Outlet_Pressure
……….
………
…..
…
..
.
NCTS Industrial Statistics
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*Huge data reduce to
ONLY one index
T 2  f ( x1 , xn )
*T2 Score
T 2  ( x  x )S 1 ( x  x )
Page 17 of 56
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Step(3):
Correlation Analysis
In this step, we’ll conduct the Pearson’s linear correlation analysis to find
out the most important PM Cycle in this process and it will be our Highlight
issues.
Pearson linear correlation analysis equation
n
r
 ( xi  x )( yi  y )
i 1
n
 ( xi  x ) 2
i 1
n
(y
i 1
i
 y)2

Tool-A
‧ ‧‧‧‧ ‧‧‧‧PM
‧‧‧‧ ‧‧ ‧
‧
‧
‧
‧
‧
‧
‧
‧
‧
‧
‧
‧
‧
‧‧ ‧ ‧ ‧
‧
‧
‧
T-Score ‧ ‧‧ ‧ ‧ ‧


YB Yield
time/date
Correlation Analysis Table
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It has high significant correlation !! And
then we can put more emphasized eyes
on this PM Cycle!!
Page 18 of 56
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Step(4):
RCA-Root Causes Analysis
T1 Chart
Root Cause Analysis via
Multi-Variate Analysis
T 2  f ( x1 , xn )
PCA Index
Raw Data
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Highlight the suspected issued
parameter based on MVA
Index!!
Page 19 of 56
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Conclusions
 From the simulation results, it seems that our
proposal Yield2Equip Events Mining module can
monitor PM Events on Yield effects obviously.
 The Yield2Equip Events Mining approach not only
can be used to monitor PM performance, but also it
is useful to do RCA tasks when the T-Score and Yield
have HIGH correlation relationship.
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Case(3): PLS application
Virtual Metrology of Deep Trench Chain.
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Virtual metrology of deep trench chain
1.5 days
• DT Chain Process Flow:
As you know, the deep trench control is more critical for process engineers. Due to the process
time between DTMO Etch to DT Etch is about 1.5 days, during this time period no one can be
aware of the quality of DT final CD.
If we could set up the virtual metrology model according to DT Litho CD, DT PHMO CD, and DTMO
CD to predict DT final CD. It will be more helpful to assist in on-line process control.
Innovative idea !!
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DT
DT
DT
Litho
PHMO
MO
CD
CD
CD
Page 22 of 56
predicted
y  f ( x1 , x2 , x3 )
DT
ETCH
CD
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Methodology introducedPLS modeling overview
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Methodology introducedPLS modeling geometric interpretation
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Simulation(1)Predicted DT final CD via PLS/LSE
Tool: D90 OXEC103-chamber A

RMSE
Error Rate
PLSR
0.0017766
1.211%
LSER
0.0023455
1.618%
Formula:
(1) RMSE 
1 n
(Y  Yˆ ) 2

n i 1
n
(2) Error - Rate 
(
i 1
Y  Yˆ
Y
n
)
*From the chart, it seems that we could get the better DTME
predicted CD via PLS modeling technical.
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Simulation(2)SPC for virtual metrology of DT final CD
Tool: T90 OXEC107-chamber A
It can correctly catch the
process alarm message !!
Summary:
1) PLS model predicts the virtual metrology values by the pre-process metrology data.
2) At the same time, SPC scheme will monitor the prediction value of metrology parameter.
3) It will also give alarms to engineers when the prediction value is out of the specification.
→ So, via this virtual scheme, we could ensure that the process is within specification.
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Conclusions



•
•
•
Virtual Metrology of deep trench chain.
DT Chain Healthy Index set up.
Early alarm/detection system.
Process grouping for following process.
Improve throughputs for critical process.
Improve line stability.
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Appendix
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page.1
Partial least squares regression (PLSR)
• Abstract: When the number of X is large compared to the number of
observations, the multiple linear regression is no longer feasible ( because of
multicolinearity). In order to solve the problem, several approaches have been
developed. One is principal component regression (PCR) and the other is
Partial least squares regression (PLSR)
• Goal:
To solve multicolinearity problem
To reduce data dimension
To predict Y from X and to describe their common structure
To get important X variables
• Difference between PLSR and PCR: PLSR finds components from
X that are also relevant for Y
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page.2
Basic concept
 11  12 ...  1n 
1
Cov( X )   1   

 , w s.t. w T  1 w R1   
 n1  n 2  nn 
 0
(w is the eigen vect or of  1 , R1 is the eigenv value of  1 )
11 12 ... 1n 
 1
Cov(Y )   2   

 , c s.t. c T  2 c R2   
 n1  n 2  nn 
 0
(c is the eigenvecto r of  2 , R2 is the eigenv value of  2 )
0...
0...

0

0
0
 
n 
0
 
 n 
t = Xw  Cov(t) = Cov(Xw) = wTCov(X)w = 1
u = Yc  Cov(u) = Cov(Yc) = cTCov(X)c = 2
To find two sets of weights w and c in order to create (respectively) a linear
combination of the columns of X and Y such that their covariance is maximum!!
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page.3
Nonlinear Iterative Partial Least Squares Algorithm (NIPALS)
X  TP  E 
T
a
t
j 1
Y  UQ  F 
T
Y  UQ  F 
pTj  E
T is score matrix
qTj  F
The columns of T are the latent vectors
a
u
j 1
T
j
j
a
 u j qTj  F (U  TB)
P is loading matrix
j 1
j=0, E0=Xn×m , F0=Yn×p , uj=any column of Y matrix, t = Xw, u = Yc
(1) w j 
E Tj 1.u j
E Tj 1.u j
(2) t j  E j 1.w j
(3) c j 
FjT1.t j
FjT1.t j
(4) u j  Fj 1.c j
( .
(5) p j 
E Tj 1.t j
t Tj .t j
(6) p j ,new 
p j ,old
p j ,old
(10) j  j  1, to j  min(m, n)
E j  E j 1  t j pTj
u Tj t j
(7) t j ,new  t j ,old . p j ,old
Fj  Fj 1  b t c , b j 
(8) w j ,new  w j ,old . p j ,old
When E is a null matrix then stop
T
j j j
t Tj t j
is Euclidean norm or 2-norm)
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Case(4): Time series application
KSI-Based to Predict Tool Maintenance.
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KSI-Based to Predict Tool Maintenance
Due to the tool maintenance schedule is usually arranged by date, wafer run
counts, RF hours, and for the furnace process it will also consider the
equipment sidewall film thickness, but all of them are not sensitive to catch tool
real status which need to conduct PM or not?.
However, we all know that correct trend monitoring via tool signals can be
used to determine approaching timing for preventive maintenance. In this way,
our innovative idea can be described as following:
Idea of invention:
PM
Once this KSI is greater than a
pre-scribed limit (threshold).
PM
KSI
Threshold
Time
Call for engineers &
Call for tool maintenance !!
KSI: Key Sensitive Index.
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Invention Program Flowchart
Variable & Key Step
selection
Correlation analysis
Time series model
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•Key Step: certain time period
•Variable: critical recipe/ process parameters
•Correlation: the quantity of variables
•Screen out key parameter & key step
•Extract out the signal characteristics
•Time series models fit the trend of variables
•Auto-correlation: the q of MA model
•Partial Auto-correlation: the p of AR model
•Defined Time Series ARIMA(p,d,q) model
•KSI would decide when to call tool maintenance
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Step(1):
Select key steps and parameters
Due to for each equipment has so many tool parameters, we need the
engineers/ vendors to provide the key process steps (some critical steps in
the recipe) and parameters which measurements have significant effects on
product quality.
Process step
Variables
Identify the key steps
and variables.
:the key parameter in corresponding step.
NCTS Industrial Statistics
Research Group Seminar
Page 35 of 56
Department of Electrical and
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Step(2):
Extract out KSV from tool signals
The KSV (Key Sensitive Process Variables), may not be the measurements
itself in corresponding key step. However, we can transform the original
tool signals into some statistic quantity, such as slop, area, maxima and
minima…,etc., which can really represent the characteristics of tool status.
Tool signals
How to extract out the useful tool
signal information ??
1. Time Length
2. Mean
3. Stdev
4. Median
5. Max
6. Min
7. Area
8. Quantile……………….
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Step(3):
Correlation analysis
In this step, we’ll conduct the Pearson’s linear correlation analysis to find
out the most important KSV in this process and it will be our Time Series
Modeling variable.
Correlation Analysis Table
Pearson linear correlation analysis equation
n
r
 ( x  x )( y
i 1
i
n
i
n
 y)
 (x  x)  ( y
2
i 1
i
i 1
2

y
)
i
It has high significant correlation !! And
then we can use it to be modeling item.
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Step(4):
Fitted the Time Series model to get KSI
According to the previous study, we can realize that
(Step_4)+(Variable_3)+(Stdev)
is the KSV in this process, and correct trend monitoring can be used to
determine appropriate timing for preventive maintenance.
Trend chart for Variable_3 - Stdev
10
Fitted Time Series Model
Model:
ARIMA(1,1,2)
Variable 3__Stdev
5
A(q )y (t )  C (q )e(t ),   (1  q 1 )
A(q )  1  0.873q 1
0
C (q )  1  0.657 q 1  0.358q  2
How to fit this Time
Series model ??
-5
0
50
100
150
200
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250
Wafer#
300
350
400
450
500
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Step(5):
Compute KSI & Simulation
ARIMA(1,1,2) predicted model
Time Series Model can catch the
tool KSV decayed trend.
10
Variable 3__Stdev
5
In this work, the KSI (Key Sensitive Index)
based approach is proposed for process
trend monitoring.
0
-5
0
50
100
150
200
250
Wafer#
300
350
400
450
500
10
10
PM
PM
PM
KSI Index
KSI-Based
0
-5
Threshold
5
Variable 3__Stdev
5
PM
0
0
200
400
600
800
1000
1200
1400
1600
1800
Wafer#
-5
KSI
0
200
400
KSI
KSI
600
800
1000
1200
1400
1600
1800
Wafer#
Based-on KSI and Threshold limit we can predict when to do Preventive Maintenance !!
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Conclusions
 From the simulation results, it seems that our proposal KSI can
catch the tool decayed trend, and when the KSI is greater than
users defined threshold, then we can suggest engineers to do
PM jobs.
 The KSI-Based to Predict Tool Maintenance approach not only
can be used for Furnace and Etch tools to assist engineers in
when to call for Preventive Maintenance, but also it is useful to
do process trend monitoring in FDC system.
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Research Group Seminar
Page 40 of 56
Department of Electrical and
Control Engineering, NCTU
Electron Industrial Control Lab
Case(5): Survival application
Advanced Queue-Time to Yield
Monitoring System.
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Research Group Seminar
Page 41 of 56
Department of Electrical and
Control Engineering, NCTU
Electron Industrial Control Lab
Survival Function BasedAdvanced Q-Time2Yield Morning System
Motivation:
Queue Time Definition: Q-Time = [t2 - t1]
Process A end time (t1)
Process A
Process B start time (t2)
Process B
…
For chemical processes, they
usually put the criteria for Q-Time
control to avoid excursions. If the
Q-Time longer than the specific
specification, we can induce that
this may occur some critical
issues in this specific time period.
As you know, Q-Time Process Control is a great important task on semiconductor manufacturing.
In Fabs, the following processes are also involved in Q-Time issues:
1. DT ME → Change FOUP (Q-Time < 3hrs)
HSG Depo → HSG Recess (Q-Time < 10hrs)
3. RC1a → Poly1b (Q-Time < 6hrs) …,and so on.
2.
Nowadays, we usually set the Q time < k hours monitoring scheme to control these critical
processes.
If we could build up the Survival Function Model which based on the relationships
between Q-Time and Yield decayed process, and provide the probability of risksInnovative idea !!
degrees to real time tell engineers what’s the current status for yield detractor and
how long could we wait for next process starting.
In this way, it will give a big hand for not only Q-Time process control, but also
productivity scheduling and cycle time improvement.
NCTS Industrial Statistics
Research Group Seminar
Page 42 of 56
Department of Electrical and
Control Engineering, NCTU
Electron Industrial Control Lab
Survival Function Introduction
Survival analysis attempts to answer questions, such as:
1) What is the fraction of a population which will survive past a certain time?
2) What rate will they die or fail?
3) Can multiple causes of death or failure be taken into account?
4) How do particular circumstances or characteristics increase or decrease the
odds of survival?
Exponential Survival Function
Survival Function KPIs
1) Survival function:
t
S (t )  Pr(T  t )  1  F (t )  1   f (t )dt
0
2) Lifetime distribution function:
How to read it ??
F (t )  Pr(T  t )  1  S (t )
If xt=2, then
3) Hazard function:
Survival probability =0.2
f (t )
h(t ) 
S (t )
4) MTBF/ MTTF:

MTBF   t  f (t )dt
0
NCTS Industrial Statistics
Research Group Seminar
Page 43 of 56
Department of Electrical and
Control Engineering, NCTU
Electron Industrial Control Lab
Invention Program Flowchart
Q-Time and Yield data
collecting/mapping
Survival distributions
selecting
Model parameters
fitting
based on distribution
Survival function
KPIs calculating
NCTS Industrial Statistics
Research Group Seminar
•Key process selecting from engineers Know-How.
•Variable: critical WAT/ Yield data.
•RMSE/ MME/ TMSE evaluated.
•Survival model validation.
•Likelihood function to fit parameters.
•Kaplan-Meier estimator.
•Reliability theory.
•Survival function.
•Lifetime distribution function.
•Hazard Function.
•MTTF/ MTBF.
Page 44 of 56
Department of Electrical and
Control Engineering, NCTU
Electron Industrial Control Lab
Step(1):
Q-Time and Response Data Mapping
From engineers Know-How, we could collect the specific Q-Time control
processes, and related WAT (electrical testing data)/ Yield data.
And then, we are going to conduct the Rank Correlation Analysis to find out
the variables which are higher correlation between process and WAT
parameters.
Q-Time control process
WAT Variables
OP_1
OP_2
OP_3
OP_4
OP_5
Identify the sensitivity
process and variable.
WAT_1
WAT_2
WAT_3
WAT_4
WAT_5
WAT_6
:the highly correlation relationship.
NCTS Industrial Statistics
Research Group Seminar
Page 45 of 56
Department of Electrical and
Control Engineering, NCTU
Electron Industrial Control Lab
Step(2):
Survival Distribution Selecting
 Model identification:
Probability Plots for 4 Survival distributions
Four-way Probability Plot for weib

ML Estimates - Complete Data
Weibull
Lognormal base e
Weibul
distribution
99
95
90
80
70
60
50
40
30
Lognormal
distribution
99
95
Percent
Percent
For the Survival function distribution
identification, we usually choose 4
popular distributions:
1) Weibul distribution
2) Lognormal distribution
3) Exponential distribution
4) Normal distribution
to be the initial testing model,and
based on the “Anderson-Darling
value”, we could select the best
fitted distribution as the Survival
function.
20
10
5
3
2
1
Anderson-Darling (adj)
Weibull
2.172
80
70
60
50
40
30
20
Lognormal base e
2.179
Exponential
10
2.579
5
Normal
2.219
1
10
100
100
Exponential
Normal
Exponential
distribution
99
98
1000
Normal
distribution
99
95
97
Percent
95
Percent
Which one is better ??
90
80
70
60
50
10
5
30
10
1
0
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80
70
60
50
40
30
20
100
200
300
Page 46 of 56
400
500
600
700
800
0
100
200
300
Department of Electrical and
Control Engineering, NCTU
Electron Industrial Control Lab
Step(3):
Model Parameters Fitting
 Fitted parameters to data:
After identified the process decayed distribution, we need to estimate the model
parameters. Currently, there are two popular methods to figure out the model
parameter estimations:
1) Kaplan-Meier estimator:
i
(A) Mean Rank  Fˆ (t(i ) )  (
)
n 1
i  0.3
( B) Median Rank  Fˆ (t(i ) )  (
)
n  0.4
2) Maximum Likelihood estimation(MLE):
Find  * such that


L(ˆ* | x )  max{L( | x )}

ˆ* is called MLE for 
From below CDF charts, it ‘s obviously to see
that K-M estimator could estimate the survival
function from life-time data as good as original
distribution.
1
0.9
0.8
K-M cdf
Normal cdf
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
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5
Page 47 of 56
6
7
8
9
10
11
12
13
14
Department of Electrical and
Control Engineering, NCTU
Electron Industrial Control Lab
15
Step(4):
Survival Function KPIs Calculating
Survival function for Q - Time
controlled
process
1
Simulation Result:
0.9
We set the Q-Time controlled
process belongs to Exp(Θ=12)
distribution, and its Survival
function is also shown here.
Survival Function S(t)
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
10
20
30
Xt
 Survival KPIs:
50
60
We set Xt=10 to evaluate each KPI.
1) Survival probability:
0.4346
2) Lifetime distribution function:
3) Hazard function:
4) MTBF/ MTTF:
40
Q - Time ( hours )
0.0362
0.0833
So, if the Lot queuing time in critical process
is 10 hours, its Survival probability is 0.4346.
At the same time, RTD could reference this
Survival probability to dispatch FOUPs.
12
NCTS Industrial Statistics
Research Group Seminar
Page 48 of 56
Department of Electrical and
Control Engineering, NCTU
Electron Industrial Control Lab
Conclusions
 From the simulation results, it seems that our innovative
proposal Survival Function Based-Advanced Q-Time2Yield
Morning System can estimate the Q-Time controlled
process decayed behavior successfully.
 The Survival Function Based-Advanced Q-Time2Yield
Morning System approach not only can be used to monitor
queuing time between process ended to next process
starting, but also give the Survival probability function for
risks-degrees if wafer waited for a long time.
 The Cycle Time and Productivity Scheduling efficiency will
also be improved, if Fab RTD System could reference this
Survival probability value to work logistic dispatch.
NCTS Industrial Statistics
Research Group Seminar
Page 49 of 56
Department of Electrical and
Control Engineering, NCTU
Electron Industrial Control Lab
Case(6): SPC chart application
Smart Process Capability
Trend Monitoring System.
NCTS Industrial Statistics
Research Group Seminar
Page 50 of 56
Department of Electrical and
Control Engineering, NCTU
Electron Industrial Control Lab
Smart Process Capability
Trend Monitoring System
At present, we use the SPC (Statistical Process Control) to monitor process capability and so on. The SPC
charts use Western Electric rule to monitor tools real-time alert.
Western Electric rule
One point out of control limit (3 Sigma)
USL
UCL
Hold
CL
LCL
LSL
7 points increasing or decreasing
USL
UCL
Hold
CL
LCL
LSL
Problem:
Currently, we only can find problems when a tool violates Western Electric rules. We can’t provide the prealert system when tools have potential problem. Although our Yield system provide the Cpmk index to
monitor the tool health, but there are no monitoring rules like SPC in it.
How:
Here we will use the CUSUM (Cumulative sum control chart) method to transform the Cpmk value of tool in
our Yield system. We not only provide the monitoring rules and can find the trend down situation of tools.
NCTS Industrial Statistics
Research Group Seminar
Page 51 of 56
Department of Electrical and
Control Engineering, NCTU
Electron Industrial Control Lab
Method Introduction
Cusum Concept:
Small trend down situation
Cpmk
Value
Mean
Cpmk limit
Period/date
Key Concept:
The Cusum method will calculate the upper cusum value and lower cusum value which base on last
cusum value. So we just need to monitor the cusum value of each period that there is trend down
situation in condition periods.
Method equation:
upper  side cusum Ci  max[ 0, xi  (  0  k )  Ci1 ]
lower  side cusum C  max[ 0, (  0  k )  xi  C ]

i

i 1
where C 0  C 0  0
K
| 1   0 | 
  , (if 1   0   )
2
2
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Cusum(Cumulative sum control chart)
It is out of control situation, that the upper sum
higher than H (decision interval) or the lower
sum lower than H. Normally H  5
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Department of Electrical and
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Electron Industrial Control Lab
Simulation
• Tool A
• FAB: Cross Fab
• Date range: 4/15-5/13
• Condition 4 periods
• Analysis result:
There is a trend down
situation in periods 1-17 and
periods 32-50.
NCTS Industrial Statistics
Research Group Seminar
Page 53 of 56
Department of Electrical and
Control Engineering, NCTU
Electron Industrial Control Lab
Conclusions
 Provided a new monitor index (CUSUM) for tools’
health and pre-alert model when tools have potential
problems that engineers can handle tools’ health
conveniently & prevent tools from occurring
significant problems.
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Research Group Seminar
Page 54 of 56
Department of Electrical and
Control Engineering, NCTU
Electron Industrial Control Lab
Thank You.
Questions & Answers…
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Department of Electrical and
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Electron Industrial Control Lab
Published Papers
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Department of Electrical and
Control Engineering, NCTU
Electron Industrial Control Lab