Upload
allen-ball
View
218
Download
0
Tags:
Embed Size (px)
Citation preview
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 1
ldquoNew trends in statistical methods applied
in a semiconductor companyrdquo
ldquoNew trends in statistical methods applied
in a semiconductor companyrdquo
Workshop on
ldquoStatistical methods applied in microelectronicsrdquoCatholic University of Milan and University of Milan-Bicocca
June 13th 2011
Luigi Radaelli Statistical Methods eng
PC amp Robustness group - Micron
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 2
1048708 Incorporated 1978 Boise Idaho
1048708 20794 worldwide
1048708 Stock Information
Traded on the NYSE - Symbol MU
1048708 Developer of DRAM NAND and Image Sensors
Only US producer of DRAM
1048708 Company Divisions
Crucial Technology ndash Meridian ID
SpecTek ndash Nampa ID
Lexar Media ndash San Jose CA
Aptina Imaging ndash San Jose CA
1048708 Related Joint Ventures
TECH Semiconductor ndash DRAM
IM Flash Technologies ndash Flash
IM Flash Singapore ndash Flash
Inotera Memories ndash DRAM
MP Mask Technology Center
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 3
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 4
Introduction
bull The continuous technology development requires complex
manufacturing processes with consequent increasing of the
difficulties in monitoring their evolution over time
bull Most process monitoring involve several quality characteristics
A large number of variables often strongly correlated must be
kept under control to guarantee the effectiveness of the
manufacturing process
bull For these reasons although the usual univariate techniques are
adequate for the single variable the use of multivariate
approaches jointly considering the variables avoid inefficient
and erroneous conclusions
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 5
Introduction
bull Because of the complexities of the process the standard
methodologies do not always give a satisfying answer so new
approaches conform to more strict requirements must be
developed
bull A further frequent drawback looking at the data describing
manufacturing processes is the evidence that they show
distributions different from the Gaussian
bull Seldom the characteristics are independent and
heteroschedasticity is often observed As a consequence
most of the classical statistical techniques must be integrated
with modern non parametric statistics inferential procedures
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 6
Some new trendshellipbull Among the non standard methodologies geostatistic was
implemented in the development of new and interesting applications for process control
bull Also about the LogVariance methodology interesting application can be performed
bull In this presentation some case studies will be showed
bull the study of the spatial distribution of the defectivity over the wafer surface
bull the optimization of the size of the maps used to measure parameters on wafer
bull joint modelling of mean and variance surface to monitor Critical Dimension parameter
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 7
The spatial distribution defectivity over the wafer surface
bull The aim is to investigate if the defects on the surface are dislocated in clusters after a washing step process
bull Are any defectivity clusters present on the surface of the wafer And if yes where are they dislocated
bull How can we decide if any clusters are present
Cluster Cluster
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 8
The spatial distribution defectivity over the wafer surface
bull A situation where no clusters are present is desiderable
because it means the process is in control (eg no
ldquospecialrdquo source of particles is present)
bull This situation is statistically modellized by a
Homogeneous Poisson Process (HPP) The process is
defined Complete Spatial Randomness (CSR)
bull The defectivity pattern is evaluated by the estimation of
the distribution function of the euclidean distances
between pairs of defects
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 9
The spatial distribution defectivity over the wafer surface
bull A comparison between the empirical cumulative distribution function (EDF) and the CSR cumulative distribution function is performed by a graphical test
bull The CSR hypothesis should be rejected if the EDF (the continuous black colored line) lies outside the envelope identified by the dotted lines The EDF is far below the lower bound of the envelope we can assume that there is a presence of clusters
Clusters No Clusters
0 5000 10000 15000 20000 25000
00
02
04
06
08
10
envpp
r
F(r
)obstheohilo
0 5000 10000 15000 20000 25000 30000
00
02
04
06
08
10
envpp
r
F(r
)
obstheohilo
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |10
The optimization of the size of the measurement maps
bull Starting from an in-use monitoring map the goal is to consider all the
possible configurations of a reduced map to evaluate for each of them
the fitness function and to select the optimum by some criteria
bull This means to consider a very high
number of possible configurations that is
impossible to evaluate by enumeration
The simulating annealing a combinatorial
optimization algorithm allows to draw the
optimum reduced map
bull Because technical constraints only a
sub-grid of the n starting point grid can
be selected
bull This means to consider a very high
number of possible configurations that is
impossible to evaluate by enumeration
The simulating annealing a combinatorial
optimization algorithm allows to draw the
optimum reduced map
bull Because technical constraints only a
sub-grid of the n starting point grid can
be selected
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |11
The optimization of the size of the measurement maps
bull Universal kriging was used to predict the deposition surface
assuming a complete polynomial of second order for the mean
function of the process
bull To select the optimum reduced map the fitness function
should be related to the kriging variance evaluated at the
current sample configuration
bull The algorithm starts from a random configuration of sample
points and sequentially updates it At each step i the current
configuration is modified by replacing one point by one point
The candidate point for replacement is selected randomly and
it is accepted if this determines an improvement in the value of
the fitness function
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |12
The optimization of the size of the measurement maps
bull The procedure is iterated until the value of the fitness function gets stable and cannot be further reduced
Optimal submap - 20sites
-100 -50 0 50 100
-100
-50
050
100
campione ottimo fitness= 499911
X
Y
x
x
x x
x
x
x
x
x
x
x x
x
x
x
xx
x
x
x
xx x
x
x
x
x
x
x
49 sites starting map
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |13
The LogVariance methods
bull The early development phase for a technological step is the study of
a working point able to both center the specification limits and
minimize the unavoidable differences among positions on wafer
area
bull In this phase the measurement time and data analysis are only
possible for a small number of samples If the system response
match the specification limits the technological step is implemented
on an experimental WIP in order to monitor its behavior
bull Data produced are very important to estimate whether the process
will be stable enough for production volumes or not
bull The aim of the method is to give a quantitative evaluation of the
degree of stability of a technological step
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |14
The LogVariance methods
bull The wafers come from different lots processed in different times and they are randomly selected and supposed to be stochastically independent each other and by time
bull The aim is to evaluate the homogeneity in mean and variance on the whole wafer
Mean RS
Standard deviation RS
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35 -35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
-35
-3
-25
-2
-15
-1
-05
0
05
1
15
2
25
3
35
-35 -25 -15 -05 05 15 25 35 -3
5 -25 -15 -05 05 15 25 35
-35
-15
05
25
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |15
The LogVariance methods
bull Using the mean and the standard deviation estimated over the whole wafer surface by the L-RS model it is possible both to point out whether a region is far from the target and to find the distribution of the Cpk over the surface
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
-35 173 191 204 211 211 205 194
-3 159 186 21 231 246 253 246 233 216 196 175
-25 157 189 221 249 272 279 276 267 254 236 216 193 169
-2 184 22 256 285 294 298 295 287 273 255 234 21 185
-15 169 209 25 283 299 31 314 312 305 291 273 251 226 2 173
-1 189 232 271 293 31 322 328 327 32 307 289 267 241 214 186
-05 205 249 276 299 318 331 338 338 332 32 302 28 254 226 198
0 218 25 278 302 322 336 344 346 341 33 313 291 265 237 207
05 218 248 276 301 322 338 347 35 347 336 32 299 273 245 215
1 214 244 272 298 319 336 347 351 349 339 324 304 279 251 221
15 207 236 265 291 313 331 343 348 347 339 325 305 281 253 217
2 227 255 281 304 322 335 342 342 335 322 303 28 249
25 215 243 269 292 31 324 331 333 327 315 298 273 229
3 229 254 277 296 31 318 32 316 305 284 246
35 26 279 293 302 304 301 282
Region expected to be not on the target
Cpk contour plot
-35
-3
-25
-2
-15
-1
-05
0
05
1
15
2
25
3
35
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
1125
151752
225
252753
325
353754
425
4-425
375-4
35-375
325-35
3-325
275-3
25-275
225-25
2-225
175-2
15-175
125-15
1-125
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |16
The LogVariance methods
Possible other applications
The algorithm proposed may be used in other applications
bull to check the alignment between two equipments that
produce wafers (are they aligned regarding average and
variance)
bull in a DOE experiment to find the area with the smallest
variance and the best average value or to understand what
are the most relevant factors
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |17
Conclusions
bull The application of the approaches here presented
allows to save measurement time and to reduce
equipment access time
bull The use of non appropriate methods can yield
misleading results with consequent erroneous
decisions and waste of time and materials
bull The study and the development of adequate statistical
approaches are fundamentals also for a better
understanding of the process
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |18
References
bull Diggle PJ 2003 Statistical Analysis of Spatial Point Patterns 2-nd edition Arnold London
bull Illian J Penttinen A Stoyan D 2008 Statistical Analysis and Modeling of Spatial Point Patterns Wiley New York
bull Aarts E Korst J Simulated Annealing and Boltzmann Machines - A Stochastic Approach to Combinatorial Optimization and Neural Computing Wiley New York (1990)
bull Chiles JP Delfiner P Geostatistics Modeling Spatial Uncertainty John Wiley amp Sons New York (1999)
bull Aitkin M (1987) Modelling Variance Heterogeneity in Normal Regression Using GLIM Applied Statistics 36 332-339
bull Faraway J J (2006) Extending the Linear Model with R Chapmanand Hall
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |19
Thank you for your kind attention
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 2
1048708 Incorporated 1978 Boise Idaho
1048708 20794 worldwide
1048708 Stock Information
Traded on the NYSE - Symbol MU
1048708 Developer of DRAM NAND and Image Sensors
Only US producer of DRAM
1048708 Company Divisions
Crucial Technology ndash Meridian ID
SpecTek ndash Nampa ID
Lexar Media ndash San Jose CA
Aptina Imaging ndash San Jose CA
1048708 Related Joint Ventures
TECH Semiconductor ndash DRAM
IM Flash Technologies ndash Flash
IM Flash Singapore ndash Flash
Inotera Memories ndash DRAM
MP Mask Technology Center
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 3
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 4
Introduction
bull The continuous technology development requires complex
manufacturing processes with consequent increasing of the
difficulties in monitoring their evolution over time
bull Most process monitoring involve several quality characteristics
A large number of variables often strongly correlated must be
kept under control to guarantee the effectiveness of the
manufacturing process
bull For these reasons although the usual univariate techniques are
adequate for the single variable the use of multivariate
approaches jointly considering the variables avoid inefficient
and erroneous conclusions
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 5
Introduction
bull Because of the complexities of the process the standard
methodologies do not always give a satisfying answer so new
approaches conform to more strict requirements must be
developed
bull A further frequent drawback looking at the data describing
manufacturing processes is the evidence that they show
distributions different from the Gaussian
bull Seldom the characteristics are independent and
heteroschedasticity is often observed As a consequence
most of the classical statistical techniques must be integrated
with modern non parametric statistics inferential procedures
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 6
Some new trendshellipbull Among the non standard methodologies geostatistic was
implemented in the development of new and interesting applications for process control
bull Also about the LogVariance methodology interesting application can be performed
bull In this presentation some case studies will be showed
bull the study of the spatial distribution of the defectivity over the wafer surface
bull the optimization of the size of the maps used to measure parameters on wafer
bull joint modelling of mean and variance surface to monitor Critical Dimension parameter
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 7
The spatial distribution defectivity over the wafer surface
bull The aim is to investigate if the defects on the surface are dislocated in clusters after a washing step process
bull Are any defectivity clusters present on the surface of the wafer And if yes where are they dislocated
bull How can we decide if any clusters are present
Cluster Cluster
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 8
The spatial distribution defectivity over the wafer surface
bull A situation where no clusters are present is desiderable
because it means the process is in control (eg no
ldquospecialrdquo source of particles is present)
bull This situation is statistically modellized by a
Homogeneous Poisson Process (HPP) The process is
defined Complete Spatial Randomness (CSR)
bull The defectivity pattern is evaluated by the estimation of
the distribution function of the euclidean distances
between pairs of defects
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 9
The spatial distribution defectivity over the wafer surface
bull A comparison between the empirical cumulative distribution function (EDF) and the CSR cumulative distribution function is performed by a graphical test
bull The CSR hypothesis should be rejected if the EDF (the continuous black colored line) lies outside the envelope identified by the dotted lines The EDF is far below the lower bound of the envelope we can assume that there is a presence of clusters
Clusters No Clusters
0 5000 10000 15000 20000 25000
00
02
04
06
08
10
envpp
r
F(r
)obstheohilo
0 5000 10000 15000 20000 25000 30000
00
02
04
06
08
10
envpp
r
F(r
)
obstheohilo
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |10
The optimization of the size of the measurement maps
bull Starting from an in-use monitoring map the goal is to consider all the
possible configurations of a reduced map to evaluate for each of them
the fitness function and to select the optimum by some criteria
bull This means to consider a very high
number of possible configurations that is
impossible to evaluate by enumeration
The simulating annealing a combinatorial
optimization algorithm allows to draw the
optimum reduced map
bull Because technical constraints only a
sub-grid of the n starting point grid can
be selected
bull This means to consider a very high
number of possible configurations that is
impossible to evaluate by enumeration
The simulating annealing a combinatorial
optimization algorithm allows to draw the
optimum reduced map
bull Because technical constraints only a
sub-grid of the n starting point grid can
be selected
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |11
The optimization of the size of the measurement maps
bull Universal kriging was used to predict the deposition surface
assuming a complete polynomial of second order for the mean
function of the process
bull To select the optimum reduced map the fitness function
should be related to the kriging variance evaluated at the
current sample configuration
bull The algorithm starts from a random configuration of sample
points and sequentially updates it At each step i the current
configuration is modified by replacing one point by one point
The candidate point for replacement is selected randomly and
it is accepted if this determines an improvement in the value of
the fitness function
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |12
The optimization of the size of the measurement maps
bull The procedure is iterated until the value of the fitness function gets stable and cannot be further reduced
Optimal submap - 20sites
-100 -50 0 50 100
-100
-50
050
100
campione ottimo fitness= 499911
X
Y
x
x
x x
x
x
x
x
x
x
x x
x
x
x
xx
x
x
x
xx x
x
x
x
x
x
x
49 sites starting map
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |13
The LogVariance methods
bull The early development phase for a technological step is the study of
a working point able to both center the specification limits and
minimize the unavoidable differences among positions on wafer
area
bull In this phase the measurement time and data analysis are only
possible for a small number of samples If the system response
match the specification limits the technological step is implemented
on an experimental WIP in order to monitor its behavior
bull Data produced are very important to estimate whether the process
will be stable enough for production volumes or not
bull The aim of the method is to give a quantitative evaluation of the
degree of stability of a technological step
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |14
The LogVariance methods
bull The wafers come from different lots processed in different times and they are randomly selected and supposed to be stochastically independent each other and by time
bull The aim is to evaluate the homogeneity in mean and variance on the whole wafer
Mean RS
Standard deviation RS
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35 -35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
-35
-3
-25
-2
-15
-1
-05
0
05
1
15
2
25
3
35
-35 -25 -15 -05 05 15 25 35 -3
5 -25 -15 -05 05 15 25 35
-35
-15
05
25
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |15
The LogVariance methods
bull Using the mean and the standard deviation estimated over the whole wafer surface by the L-RS model it is possible both to point out whether a region is far from the target and to find the distribution of the Cpk over the surface
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
-35 173 191 204 211 211 205 194
-3 159 186 21 231 246 253 246 233 216 196 175
-25 157 189 221 249 272 279 276 267 254 236 216 193 169
-2 184 22 256 285 294 298 295 287 273 255 234 21 185
-15 169 209 25 283 299 31 314 312 305 291 273 251 226 2 173
-1 189 232 271 293 31 322 328 327 32 307 289 267 241 214 186
-05 205 249 276 299 318 331 338 338 332 32 302 28 254 226 198
0 218 25 278 302 322 336 344 346 341 33 313 291 265 237 207
05 218 248 276 301 322 338 347 35 347 336 32 299 273 245 215
1 214 244 272 298 319 336 347 351 349 339 324 304 279 251 221
15 207 236 265 291 313 331 343 348 347 339 325 305 281 253 217
2 227 255 281 304 322 335 342 342 335 322 303 28 249
25 215 243 269 292 31 324 331 333 327 315 298 273 229
3 229 254 277 296 31 318 32 316 305 284 246
35 26 279 293 302 304 301 282
Region expected to be not on the target
Cpk contour plot
-35
-3
-25
-2
-15
-1
-05
0
05
1
15
2
25
3
35
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
1125
151752
225
252753
325
353754
425
4-425
375-4
35-375
325-35
3-325
275-3
25-275
225-25
2-225
175-2
15-175
125-15
1-125
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |16
The LogVariance methods
Possible other applications
The algorithm proposed may be used in other applications
bull to check the alignment between two equipments that
produce wafers (are they aligned regarding average and
variance)
bull in a DOE experiment to find the area with the smallest
variance and the best average value or to understand what
are the most relevant factors
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |17
Conclusions
bull The application of the approaches here presented
allows to save measurement time and to reduce
equipment access time
bull The use of non appropriate methods can yield
misleading results with consequent erroneous
decisions and waste of time and materials
bull The study and the development of adequate statistical
approaches are fundamentals also for a better
understanding of the process
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |18
References
bull Diggle PJ 2003 Statistical Analysis of Spatial Point Patterns 2-nd edition Arnold London
bull Illian J Penttinen A Stoyan D 2008 Statistical Analysis and Modeling of Spatial Point Patterns Wiley New York
bull Aarts E Korst J Simulated Annealing and Boltzmann Machines - A Stochastic Approach to Combinatorial Optimization and Neural Computing Wiley New York (1990)
bull Chiles JP Delfiner P Geostatistics Modeling Spatial Uncertainty John Wiley amp Sons New York (1999)
bull Aitkin M (1987) Modelling Variance Heterogeneity in Normal Regression Using GLIM Applied Statistics 36 332-339
bull Faraway J J (2006) Extending the Linear Model with R Chapmanand Hall
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |19
Thank you for your kind attention
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 3
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 4
Introduction
bull The continuous technology development requires complex
manufacturing processes with consequent increasing of the
difficulties in monitoring their evolution over time
bull Most process monitoring involve several quality characteristics
A large number of variables often strongly correlated must be
kept under control to guarantee the effectiveness of the
manufacturing process
bull For these reasons although the usual univariate techniques are
adequate for the single variable the use of multivariate
approaches jointly considering the variables avoid inefficient
and erroneous conclusions
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 5
Introduction
bull Because of the complexities of the process the standard
methodologies do not always give a satisfying answer so new
approaches conform to more strict requirements must be
developed
bull A further frequent drawback looking at the data describing
manufacturing processes is the evidence that they show
distributions different from the Gaussian
bull Seldom the characteristics are independent and
heteroschedasticity is often observed As a consequence
most of the classical statistical techniques must be integrated
with modern non parametric statistics inferential procedures
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 6
Some new trendshellipbull Among the non standard methodologies geostatistic was
implemented in the development of new and interesting applications for process control
bull Also about the LogVariance methodology interesting application can be performed
bull In this presentation some case studies will be showed
bull the study of the spatial distribution of the defectivity over the wafer surface
bull the optimization of the size of the maps used to measure parameters on wafer
bull joint modelling of mean and variance surface to monitor Critical Dimension parameter
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 7
The spatial distribution defectivity over the wafer surface
bull The aim is to investigate if the defects on the surface are dislocated in clusters after a washing step process
bull Are any defectivity clusters present on the surface of the wafer And if yes where are they dislocated
bull How can we decide if any clusters are present
Cluster Cluster
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 8
The spatial distribution defectivity over the wafer surface
bull A situation where no clusters are present is desiderable
because it means the process is in control (eg no
ldquospecialrdquo source of particles is present)
bull This situation is statistically modellized by a
Homogeneous Poisson Process (HPP) The process is
defined Complete Spatial Randomness (CSR)
bull The defectivity pattern is evaluated by the estimation of
the distribution function of the euclidean distances
between pairs of defects
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 9
The spatial distribution defectivity over the wafer surface
bull A comparison between the empirical cumulative distribution function (EDF) and the CSR cumulative distribution function is performed by a graphical test
bull The CSR hypothesis should be rejected if the EDF (the continuous black colored line) lies outside the envelope identified by the dotted lines The EDF is far below the lower bound of the envelope we can assume that there is a presence of clusters
Clusters No Clusters
0 5000 10000 15000 20000 25000
00
02
04
06
08
10
envpp
r
F(r
)obstheohilo
0 5000 10000 15000 20000 25000 30000
00
02
04
06
08
10
envpp
r
F(r
)
obstheohilo
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |10
The optimization of the size of the measurement maps
bull Starting from an in-use monitoring map the goal is to consider all the
possible configurations of a reduced map to evaluate for each of them
the fitness function and to select the optimum by some criteria
bull This means to consider a very high
number of possible configurations that is
impossible to evaluate by enumeration
The simulating annealing a combinatorial
optimization algorithm allows to draw the
optimum reduced map
bull Because technical constraints only a
sub-grid of the n starting point grid can
be selected
bull This means to consider a very high
number of possible configurations that is
impossible to evaluate by enumeration
The simulating annealing a combinatorial
optimization algorithm allows to draw the
optimum reduced map
bull Because technical constraints only a
sub-grid of the n starting point grid can
be selected
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |11
The optimization of the size of the measurement maps
bull Universal kriging was used to predict the deposition surface
assuming a complete polynomial of second order for the mean
function of the process
bull To select the optimum reduced map the fitness function
should be related to the kriging variance evaluated at the
current sample configuration
bull The algorithm starts from a random configuration of sample
points and sequentially updates it At each step i the current
configuration is modified by replacing one point by one point
The candidate point for replacement is selected randomly and
it is accepted if this determines an improvement in the value of
the fitness function
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |12
The optimization of the size of the measurement maps
bull The procedure is iterated until the value of the fitness function gets stable and cannot be further reduced
Optimal submap - 20sites
-100 -50 0 50 100
-100
-50
050
100
campione ottimo fitness= 499911
X
Y
x
x
x x
x
x
x
x
x
x
x x
x
x
x
xx
x
x
x
xx x
x
x
x
x
x
x
49 sites starting map
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |13
The LogVariance methods
bull The early development phase for a technological step is the study of
a working point able to both center the specification limits and
minimize the unavoidable differences among positions on wafer
area
bull In this phase the measurement time and data analysis are only
possible for a small number of samples If the system response
match the specification limits the technological step is implemented
on an experimental WIP in order to monitor its behavior
bull Data produced are very important to estimate whether the process
will be stable enough for production volumes or not
bull The aim of the method is to give a quantitative evaluation of the
degree of stability of a technological step
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |14
The LogVariance methods
bull The wafers come from different lots processed in different times and they are randomly selected and supposed to be stochastically independent each other and by time
bull The aim is to evaluate the homogeneity in mean and variance on the whole wafer
Mean RS
Standard deviation RS
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35 -35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
-35
-3
-25
-2
-15
-1
-05
0
05
1
15
2
25
3
35
-35 -25 -15 -05 05 15 25 35 -3
5 -25 -15 -05 05 15 25 35
-35
-15
05
25
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |15
The LogVariance methods
bull Using the mean and the standard deviation estimated over the whole wafer surface by the L-RS model it is possible both to point out whether a region is far from the target and to find the distribution of the Cpk over the surface
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
-35 173 191 204 211 211 205 194
-3 159 186 21 231 246 253 246 233 216 196 175
-25 157 189 221 249 272 279 276 267 254 236 216 193 169
-2 184 22 256 285 294 298 295 287 273 255 234 21 185
-15 169 209 25 283 299 31 314 312 305 291 273 251 226 2 173
-1 189 232 271 293 31 322 328 327 32 307 289 267 241 214 186
-05 205 249 276 299 318 331 338 338 332 32 302 28 254 226 198
0 218 25 278 302 322 336 344 346 341 33 313 291 265 237 207
05 218 248 276 301 322 338 347 35 347 336 32 299 273 245 215
1 214 244 272 298 319 336 347 351 349 339 324 304 279 251 221
15 207 236 265 291 313 331 343 348 347 339 325 305 281 253 217
2 227 255 281 304 322 335 342 342 335 322 303 28 249
25 215 243 269 292 31 324 331 333 327 315 298 273 229
3 229 254 277 296 31 318 32 316 305 284 246
35 26 279 293 302 304 301 282
Region expected to be not on the target
Cpk contour plot
-35
-3
-25
-2
-15
-1
-05
0
05
1
15
2
25
3
35
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
1125
151752
225
252753
325
353754
425
4-425
375-4
35-375
325-35
3-325
275-3
25-275
225-25
2-225
175-2
15-175
125-15
1-125
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |16
The LogVariance methods
Possible other applications
The algorithm proposed may be used in other applications
bull to check the alignment between two equipments that
produce wafers (are they aligned regarding average and
variance)
bull in a DOE experiment to find the area with the smallest
variance and the best average value or to understand what
are the most relevant factors
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |17
Conclusions
bull The application of the approaches here presented
allows to save measurement time and to reduce
equipment access time
bull The use of non appropriate methods can yield
misleading results with consequent erroneous
decisions and waste of time and materials
bull The study and the development of adequate statistical
approaches are fundamentals also for a better
understanding of the process
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |18
References
bull Diggle PJ 2003 Statistical Analysis of Spatial Point Patterns 2-nd edition Arnold London
bull Illian J Penttinen A Stoyan D 2008 Statistical Analysis and Modeling of Spatial Point Patterns Wiley New York
bull Aarts E Korst J Simulated Annealing and Boltzmann Machines - A Stochastic Approach to Combinatorial Optimization and Neural Computing Wiley New York (1990)
bull Chiles JP Delfiner P Geostatistics Modeling Spatial Uncertainty John Wiley amp Sons New York (1999)
bull Aitkin M (1987) Modelling Variance Heterogeneity in Normal Regression Using GLIM Applied Statistics 36 332-339
bull Faraway J J (2006) Extending the Linear Model with R Chapmanand Hall
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |19
Thank you for your kind attention
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 4
Introduction
bull The continuous technology development requires complex
manufacturing processes with consequent increasing of the
difficulties in monitoring their evolution over time
bull Most process monitoring involve several quality characteristics
A large number of variables often strongly correlated must be
kept under control to guarantee the effectiveness of the
manufacturing process
bull For these reasons although the usual univariate techniques are
adequate for the single variable the use of multivariate
approaches jointly considering the variables avoid inefficient
and erroneous conclusions
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 5
Introduction
bull Because of the complexities of the process the standard
methodologies do not always give a satisfying answer so new
approaches conform to more strict requirements must be
developed
bull A further frequent drawback looking at the data describing
manufacturing processes is the evidence that they show
distributions different from the Gaussian
bull Seldom the characteristics are independent and
heteroschedasticity is often observed As a consequence
most of the classical statistical techniques must be integrated
with modern non parametric statistics inferential procedures
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 6
Some new trendshellipbull Among the non standard methodologies geostatistic was
implemented in the development of new and interesting applications for process control
bull Also about the LogVariance methodology interesting application can be performed
bull In this presentation some case studies will be showed
bull the study of the spatial distribution of the defectivity over the wafer surface
bull the optimization of the size of the maps used to measure parameters on wafer
bull joint modelling of mean and variance surface to monitor Critical Dimension parameter
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 7
The spatial distribution defectivity over the wafer surface
bull The aim is to investigate if the defects on the surface are dislocated in clusters after a washing step process
bull Are any defectivity clusters present on the surface of the wafer And if yes where are they dislocated
bull How can we decide if any clusters are present
Cluster Cluster
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 8
The spatial distribution defectivity over the wafer surface
bull A situation where no clusters are present is desiderable
because it means the process is in control (eg no
ldquospecialrdquo source of particles is present)
bull This situation is statistically modellized by a
Homogeneous Poisson Process (HPP) The process is
defined Complete Spatial Randomness (CSR)
bull The defectivity pattern is evaluated by the estimation of
the distribution function of the euclidean distances
between pairs of defects
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 9
The spatial distribution defectivity over the wafer surface
bull A comparison between the empirical cumulative distribution function (EDF) and the CSR cumulative distribution function is performed by a graphical test
bull The CSR hypothesis should be rejected if the EDF (the continuous black colored line) lies outside the envelope identified by the dotted lines The EDF is far below the lower bound of the envelope we can assume that there is a presence of clusters
Clusters No Clusters
0 5000 10000 15000 20000 25000
00
02
04
06
08
10
envpp
r
F(r
)obstheohilo
0 5000 10000 15000 20000 25000 30000
00
02
04
06
08
10
envpp
r
F(r
)
obstheohilo
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |10
The optimization of the size of the measurement maps
bull Starting from an in-use monitoring map the goal is to consider all the
possible configurations of a reduced map to evaluate for each of them
the fitness function and to select the optimum by some criteria
bull This means to consider a very high
number of possible configurations that is
impossible to evaluate by enumeration
The simulating annealing a combinatorial
optimization algorithm allows to draw the
optimum reduced map
bull Because technical constraints only a
sub-grid of the n starting point grid can
be selected
bull This means to consider a very high
number of possible configurations that is
impossible to evaluate by enumeration
The simulating annealing a combinatorial
optimization algorithm allows to draw the
optimum reduced map
bull Because technical constraints only a
sub-grid of the n starting point grid can
be selected
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |11
The optimization of the size of the measurement maps
bull Universal kriging was used to predict the deposition surface
assuming a complete polynomial of second order for the mean
function of the process
bull To select the optimum reduced map the fitness function
should be related to the kriging variance evaluated at the
current sample configuration
bull The algorithm starts from a random configuration of sample
points and sequentially updates it At each step i the current
configuration is modified by replacing one point by one point
The candidate point for replacement is selected randomly and
it is accepted if this determines an improvement in the value of
the fitness function
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |12
The optimization of the size of the measurement maps
bull The procedure is iterated until the value of the fitness function gets stable and cannot be further reduced
Optimal submap - 20sites
-100 -50 0 50 100
-100
-50
050
100
campione ottimo fitness= 499911
X
Y
x
x
x x
x
x
x
x
x
x
x x
x
x
x
xx
x
x
x
xx x
x
x
x
x
x
x
49 sites starting map
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |13
The LogVariance methods
bull The early development phase for a technological step is the study of
a working point able to both center the specification limits and
minimize the unavoidable differences among positions on wafer
area
bull In this phase the measurement time and data analysis are only
possible for a small number of samples If the system response
match the specification limits the technological step is implemented
on an experimental WIP in order to monitor its behavior
bull Data produced are very important to estimate whether the process
will be stable enough for production volumes or not
bull The aim of the method is to give a quantitative evaluation of the
degree of stability of a technological step
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |14
The LogVariance methods
bull The wafers come from different lots processed in different times and they are randomly selected and supposed to be stochastically independent each other and by time
bull The aim is to evaluate the homogeneity in mean and variance on the whole wafer
Mean RS
Standard deviation RS
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35 -35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
-35
-3
-25
-2
-15
-1
-05
0
05
1
15
2
25
3
35
-35 -25 -15 -05 05 15 25 35 -3
5 -25 -15 -05 05 15 25 35
-35
-15
05
25
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |15
The LogVariance methods
bull Using the mean and the standard deviation estimated over the whole wafer surface by the L-RS model it is possible both to point out whether a region is far from the target and to find the distribution of the Cpk over the surface
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
-35 173 191 204 211 211 205 194
-3 159 186 21 231 246 253 246 233 216 196 175
-25 157 189 221 249 272 279 276 267 254 236 216 193 169
-2 184 22 256 285 294 298 295 287 273 255 234 21 185
-15 169 209 25 283 299 31 314 312 305 291 273 251 226 2 173
-1 189 232 271 293 31 322 328 327 32 307 289 267 241 214 186
-05 205 249 276 299 318 331 338 338 332 32 302 28 254 226 198
0 218 25 278 302 322 336 344 346 341 33 313 291 265 237 207
05 218 248 276 301 322 338 347 35 347 336 32 299 273 245 215
1 214 244 272 298 319 336 347 351 349 339 324 304 279 251 221
15 207 236 265 291 313 331 343 348 347 339 325 305 281 253 217
2 227 255 281 304 322 335 342 342 335 322 303 28 249
25 215 243 269 292 31 324 331 333 327 315 298 273 229
3 229 254 277 296 31 318 32 316 305 284 246
35 26 279 293 302 304 301 282
Region expected to be not on the target
Cpk contour plot
-35
-3
-25
-2
-15
-1
-05
0
05
1
15
2
25
3
35
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
1125
151752
225
252753
325
353754
425
4-425
375-4
35-375
325-35
3-325
275-3
25-275
225-25
2-225
175-2
15-175
125-15
1-125
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |16
The LogVariance methods
Possible other applications
The algorithm proposed may be used in other applications
bull to check the alignment between two equipments that
produce wafers (are they aligned regarding average and
variance)
bull in a DOE experiment to find the area with the smallest
variance and the best average value or to understand what
are the most relevant factors
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |17
Conclusions
bull The application of the approaches here presented
allows to save measurement time and to reduce
equipment access time
bull The use of non appropriate methods can yield
misleading results with consequent erroneous
decisions and waste of time and materials
bull The study and the development of adequate statistical
approaches are fundamentals also for a better
understanding of the process
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |18
References
bull Diggle PJ 2003 Statistical Analysis of Spatial Point Patterns 2-nd edition Arnold London
bull Illian J Penttinen A Stoyan D 2008 Statistical Analysis and Modeling of Spatial Point Patterns Wiley New York
bull Aarts E Korst J Simulated Annealing and Boltzmann Machines - A Stochastic Approach to Combinatorial Optimization and Neural Computing Wiley New York (1990)
bull Chiles JP Delfiner P Geostatistics Modeling Spatial Uncertainty John Wiley amp Sons New York (1999)
bull Aitkin M (1987) Modelling Variance Heterogeneity in Normal Regression Using GLIM Applied Statistics 36 332-339
bull Faraway J J (2006) Extending the Linear Model with R Chapmanand Hall
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |19
Thank you for your kind attention
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 5
Introduction
bull Because of the complexities of the process the standard
methodologies do not always give a satisfying answer so new
approaches conform to more strict requirements must be
developed
bull A further frequent drawback looking at the data describing
manufacturing processes is the evidence that they show
distributions different from the Gaussian
bull Seldom the characteristics are independent and
heteroschedasticity is often observed As a consequence
most of the classical statistical techniques must be integrated
with modern non parametric statistics inferential procedures
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 6
Some new trendshellipbull Among the non standard methodologies geostatistic was
implemented in the development of new and interesting applications for process control
bull Also about the LogVariance methodology interesting application can be performed
bull In this presentation some case studies will be showed
bull the study of the spatial distribution of the defectivity over the wafer surface
bull the optimization of the size of the maps used to measure parameters on wafer
bull joint modelling of mean and variance surface to monitor Critical Dimension parameter
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 7
The spatial distribution defectivity over the wafer surface
bull The aim is to investigate if the defects on the surface are dislocated in clusters after a washing step process
bull Are any defectivity clusters present on the surface of the wafer And if yes where are they dislocated
bull How can we decide if any clusters are present
Cluster Cluster
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 8
The spatial distribution defectivity over the wafer surface
bull A situation where no clusters are present is desiderable
because it means the process is in control (eg no
ldquospecialrdquo source of particles is present)
bull This situation is statistically modellized by a
Homogeneous Poisson Process (HPP) The process is
defined Complete Spatial Randomness (CSR)
bull The defectivity pattern is evaluated by the estimation of
the distribution function of the euclidean distances
between pairs of defects
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 9
The spatial distribution defectivity over the wafer surface
bull A comparison between the empirical cumulative distribution function (EDF) and the CSR cumulative distribution function is performed by a graphical test
bull The CSR hypothesis should be rejected if the EDF (the continuous black colored line) lies outside the envelope identified by the dotted lines The EDF is far below the lower bound of the envelope we can assume that there is a presence of clusters
Clusters No Clusters
0 5000 10000 15000 20000 25000
00
02
04
06
08
10
envpp
r
F(r
)obstheohilo
0 5000 10000 15000 20000 25000 30000
00
02
04
06
08
10
envpp
r
F(r
)
obstheohilo
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |10
The optimization of the size of the measurement maps
bull Starting from an in-use monitoring map the goal is to consider all the
possible configurations of a reduced map to evaluate for each of them
the fitness function and to select the optimum by some criteria
bull This means to consider a very high
number of possible configurations that is
impossible to evaluate by enumeration
The simulating annealing a combinatorial
optimization algorithm allows to draw the
optimum reduced map
bull Because technical constraints only a
sub-grid of the n starting point grid can
be selected
bull This means to consider a very high
number of possible configurations that is
impossible to evaluate by enumeration
The simulating annealing a combinatorial
optimization algorithm allows to draw the
optimum reduced map
bull Because technical constraints only a
sub-grid of the n starting point grid can
be selected
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |11
The optimization of the size of the measurement maps
bull Universal kriging was used to predict the deposition surface
assuming a complete polynomial of second order for the mean
function of the process
bull To select the optimum reduced map the fitness function
should be related to the kriging variance evaluated at the
current sample configuration
bull The algorithm starts from a random configuration of sample
points and sequentially updates it At each step i the current
configuration is modified by replacing one point by one point
The candidate point for replacement is selected randomly and
it is accepted if this determines an improvement in the value of
the fitness function
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |12
The optimization of the size of the measurement maps
bull The procedure is iterated until the value of the fitness function gets stable and cannot be further reduced
Optimal submap - 20sites
-100 -50 0 50 100
-100
-50
050
100
campione ottimo fitness= 499911
X
Y
x
x
x x
x
x
x
x
x
x
x x
x
x
x
xx
x
x
x
xx x
x
x
x
x
x
x
49 sites starting map
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |13
The LogVariance methods
bull The early development phase for a technological step is the study of
a working point able to both center the specification limits and
minimize the unavoidable differences among positions on wafer
area
bull In this phase the measurement time and data analysis are only
possible for a small number of samples If the system response
match the specification limits the technological step is implemented
on an experimental WIP in order to monitor its behavior
bull Data produced are very important to estimate whether the process
will be stable enough for production volumes or not
bull The aim of the method is to give a quantitative evaluation of the
degree of stability of a technological step
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |14
The LogVariance methods
bull The wafers come from different lots processed in different times and they are randomly selected and supposed to be stochastically independent each other and by time
bull The aim is to evaluate the homogeneity in mean and variance on the whole wafer
Mean RS
Standard deviation RS
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35 -35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
-35
-3
-25
-2
-15
-1
-05
0
05
1
15
2
25
3
35
-35 -25 -15 -05 05 15 25 35 -3
5 -25 -15 -05 05 15 25 35
-35
-15
05
25
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |15
The LogVariance methods
bull Using the mean and the standard deviation estimated over the whole wafer surface by the L-RS model it is possible both to point out whether a region is far from the target and to find the distribution of the Cpk over the surface
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
-35 173 191 204 211 211 205 194
-3 159 186 21 231 246 253 246 233 216 196 175
-25 157 189 221 249 272 279 276 267 254 236 216 193 169
-2 184 22 256 285 294 298 295 287 273 255 234 21 185
-15 169 209 25 283 299 31 314 312 305 291 273 251 226 2 173
-1 189 232 271 293 31 322 328 327 32 307 289 267 241 214 186
-05 205 249 276 299 318 331 338 338 332 32 302 28 254 226 198
0 218 25 278 302 322 336 344 346 341 33 313 291 265 237 207
05 218 248 276 301 322 338 347 35 347 336 32 299 273 245 215
1 214 244 272 298 319 336 347 351 349 339 324 304 279 251 221
15 207 236 265 291 313 331 343 348 347 339 325 305 281 253 217
2 227 255 281 304 322 335 342 342 335 322 303 28 249
25 215 243 269 292 31 324 331 333 327 315 298 273 229
3 229 254 277 296 31 318 32 316 305 284 246
35 26 279 293 302 304 301 282
Region expected to be not on the target
Cpk contour plot
-35
-3
-25
-2
-15
-1
-05
0
05
1
15
2
25
3
35
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
1125
151752
225
252753
325
353754
425
4-425
375-4
35-375
325-35
3-325
275-3
25-275
225-25
2-225
175-2
15-175
125-15
1-125
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |16
The LogVariance methods
Possible other applications
The algorithm proposed may be used in other applications
bull to check the alignment between two equipments that
produce wafers (are they aligned regarding average and
variance)
bull in a DOE experiment to find the area with the smallest
variance and the best average value or to understand what
are the most relevant factors
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |17
Conclusions
bull The application of the approaches here presented
allows to save measurement time and to reduce
equipment access time
bull The use of non appropriate methods can yield
misleading results with consequent erroneous
decisions and waste of time and materials
bull The study and the development of adequate statistical
approaches are fundamentals also for a better
understanding of the process
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |18
References
bull Diggle PJ 2003 Statistical Analysis of Spatial Point Patterns 2-nd edition Arnold London
bull Illian J Penttinen A Stoyan D 2008 Statistical Analysis and Modeling of Spatial Point Patterns Wiley New York
bull Aarts E Korst J Simulated Annealing and Boltzmann Machines - A Stochastic Approach to Combinatorial Optimization and Neural Computing Wiley New York (1990)
bull Chiles JP Delfiner P Geostatistics Modeling Spatial Uncertainty John Wiley amp Sons New York (1999)
bull Aitkin M (1987) Modelling Variance Heterogeneity in Normal Regression Using GLIM Applied Statistics 36 332-339
bull Faraway J J (2006) Extending the Linear Model with R Chapmanand Hall
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |19
Thank you for your kind attention
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 6
Some new trendshellipbull Among the non standard methodologies geostatistic was
implemented in the development of new and interesting applications for process control
bull Also about the LogVariance methodology interesting application can be performed
bull In this presentation some case studies will be showed
bull the study of the spatial distribution of the defectivity over the wafer surface
bull the optimization of the size of the maps used to measure parameters on wafer
bull joint modelling of mean and variance surface to monitor Critical Dimension parameter
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 7
The spatial distribution defectivity over the wafer surface
bull The aim is to investigate if the defects on the surface are dislocated in clusters after a washing step process
bull Are any defectivity clusters present on the surface of the wafer And if yes where are they dislocated
bull How can we decide if any clusters are present
Cluster Cluster
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 8
The spatial distribution defectivity over the wafer surface
bull A situation where no clusters are present is desiderable
because it means the process is in control (eg no
ldquospecialrdquo source of particles is present)
bull This situation is statistically modellized by a
Homogeneous Poisson Process (HPP) The process is
defined Complete Spatial Randomness (CSR)
bull The defectivity pattern is evaluated by the estimation of
the distribution function of the euclidean distances
between pairs of defects
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 9
The spatial distribution defectivity over the wafer surface
bull A comparison between the empirical cumulative distribution function (EDF) and the CSR cumulative distribution function is performed by a graphical test
bull The CSR hypothesis should be rejected if the EDF (the continuous black colored line) lies outside the envelope identified by the dotted lines The EDF is far below the lower bound of the envelope we can assume that there is a presence of clusters
Clusters No Clusters
0 5000 10000 15000 20000 25000
00
02
04
06
08
10
envpp
r
F(r
)obstheohilo
0 5000 10000 15000 20000 25000 30000
00
02
04
06
08
10
envpp
r
F(r
)
obstheohilo
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |10
The optimization of the size of the measurement maps
bull Starting from an in-use monitoring map the goal is to consider all the
possible configurations of a reduced map to evaluate for each of them
the fitness function and to select the optimum by some criteria
bull This means to consider a very high
number of possible configurations that is
impossible to evaluate by enumeration
The simulating annealing a combinatorial
optimization algorithm allows to draw the
optimum reduced map
bull Because technical constraints only a
sub-grid of the n starting point grid can
be selected
bull This means to consider a very high
number of possible configurations that is
impossible to evaluate by enumeration
The simulating annealing a combinatorial
optimization algorithm allows to draw the
optimum reduced map
bull Because technical constraints only a
sub-grid of the n starting point grid can
be selected
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |11
The optimization of the size of the measurement maps
bull Universal kriging was used to predict the deposition surface
assuming a complete polynomial of second order for the mean
function of the process
bull To select the optimum reduced map the fitness function
should be related to the kriging variance evaluated at the
current sample configuration
bull The algorithm starts from a random configuration of sample
points and sequentially updates it At each step i the current
configuration is modified by replacing one point by one point
The candidate point for replacement is selected randomly and
it is accepted if this determines an improvement in the value of
the fitness function
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |12
The optimization of the size of the measurement maps
bull The procedure is iterated until the value of the fitness function gets stable and cannot be further reduced
Optimal submap - 20sites
-100 -50 0 50 100
-100
-50
050
100
campione ottimo fitness= 499911
X
Y
x
x
x x
x
x
x
x
x
x
x x
x
x
x
xx
x
x
x
xx x
x
x
x
x
x
x
49 sites starting map
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |13
The LogVariance methods
bull The early development phase for a technological step is the study of
a working point able to both center the specification limits and
minimize the unavoidable differences among positions on wafer
area
bull In this phase the measurement time and data analysis are only
possible for a small number of samples If the system response
match the specification limits the technological step is implemented
on an experimental WIP in order to monitor its behavior
bull Data produced are very important to estimate whether the process
will be stable enough for production volumes or not
bull The aim of the method is to give a quantitative evaluation of the
degree of stability of a technological step
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |14
The LogVariance methods
bull The wafers come from different lots processed in different times and they are randomly selected and supposed to be stochastically independent each other and by time
bull The aim is to evaluate the homogeneity in mean and variance on the whole wafer
Mean RS
Standard deviation RS
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35 -35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
-35
-3
-25
-2
-15
-1
-05
0
05
1
15
2
25
3
35
-35 -25 -15 -05 05 15 25 35 -3
5 -25 -15 -05 05 15 25 35
-35
-15
05
25
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |15
The LogVariance methods
bull Using the mean and the standard deviation estimated over the whole wafer surface by the L-RS model it is possible both to point out whether a region is far from the target and to find the distribution of the Cpk over the surface
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
-35 173 191 204 211 211 205 194
-3 159 186 21 231 246 253 246 233 216 196 175
-25 157 189 221 249 272 279 276 267 254 236 216 193 169
-2 184 22 256 285 294 298 295 287 273 255 234 21 185
-15 169 209 25 283 299 31 314 312 305 291 273 251 226 2 173
-1 189 232 271 293 31 322 328 327 32 307 289 267 241 214 186
-05 205 249 276 299 318 331 338 338 332 32 302 28 254 226 198
0 218 25 278 302 322 336 344 346 341 33 313 291 265 237 207
05 218 248 276 301 322 338 347 35 347 336 32 299 273 245 215
1 214 244 272 298 319 336 347 351 349 339 324 304 279 251 221
15 207 236 265 291 313 331 343 348 347 339 325 305 281 253 217
2 227 255 281 304 322 335 342 342 335 322 303 28 249
25 215 243 269 292 31 324 331 333 327 315 298 273 229
3 229 254 277 296 31 318 32 316 305 284 246
35 26 279 293 302 304 301 282
Region expected to be not on the target
Cpk contour plot
-35
-3
-25
-2
-15
-1
-05
0
05
1
15
2
25
3
35
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
1125
151752
225
252753
325
353754
425
4-425
375-4
35-375
325-35
3-325
275-3
25-275
225-25
2-225
175-2
15-175
125-15
1-125
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |16
The LogVariance methods
Possible other applications
The algorithm proposed may be used in other applications
bull to check the alignment between two equipments that
produce wafers (are they aligned regarding average and
variance)
bull in a DOE experiment to find the area with the smallest
variance and the best average value or to understand what
are the most relevant factors
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |17
Conclusions
bull The application of the approaches here presented
allows to save measurement time and to reduce
equipment access time
bull The use of non appropriate methods can yield
misleading results with consequent erroneous
decisions and waste of time and materials
bull The study and the development of adequate statistical
approaches are fundamentals also for a better
understanding of the process
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |18
References
bull Diggle PJ 2003 Statistical Analysis of Spatial Point Patterns 2-nd edition Arnold London
bull Illian J Penttinen A Stoyan D 2008 Statistical Analysis and Modeling of Spatial Point Patterns Wiley New York
bull Aarts E Korst J Simulated Annealing and Boltzmann Machines - A Stochastic Approach to Combinatorial Optimization and Neural Computing Wiley New York (1990)
bull Chiles JP Delfiner P Geostatistics Modeling Spatial Uncertainty John Wiley amp Sons New York (1999)
bull Aitkin M (1987) Modelling Variance Heterogeneity in Normal Regression Using GLIM Applied Statistics 36 332-339
bull Faraway J J (2006) Extending the Linear Model with R Chapmanand Hall
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |19
Thank you for your kind attention
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 7
The spatial distribution defectivity over the wafer surface
bull The aim is to investigate if the defects on the surface are dislocated in clusters after a washing step process
bull Are any defectivity clusters present on the surface of the wafer And if yes where are they dislocated
bull How can we decide if any clusters are present
Cluster Cluster
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 8
The spatial distribution defectivity over the wafer surface
bull A situation where no clusters are present is desiderable
because it means the process is in control (eg no
ldquospecialrdquo source of particles is present)
bull This situation is statistically modellized by a
Homogeneous Poisson Process (HPP) The process is
defined Complete Spatial Randomness (CSR)
bull The defectivity pattern is evaluated by the estimation of
the distribution function of the euclidean distances
between pairs of defects
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 9
The spatial distribution defectivity over the wafer surface
bull A comparison between the empirical cumulative distribution function (EDF) and the CSR cumulative distribution function is performed by a graphical test
bull The CSR hypothesis should be rejected if the EDF (the continuous black colored line) lies outside the envelope identified by the dotted lines The EDF is far below the lower bound of the envelope we can assume that there is a presence of clusters
Clusters No Clusters
0 5000 10000 15000 20000 25000
00
02
04
06
08
10
envpp
r
F(r
)obstheohilo
0 5000 10000 15000 20000 25000 30000
00
02
04
06
08
10
envpp
r
F(r
)
obstheohilo
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |10
The optimization of the size of the measurement maps
bull Starting from an in-use monitoring map the goal is to consider all the
possible configurations of a reduced map to evaluate for each of them
the fitness function and to select the optimum by some criteria
bull This means to consider a very high
number of possible configurations that is
impossible to evaluate by enumeration
The simulating annealing a combinatorial
optimization algorithm allows to draw the
optimum reduced map
bull Because technical constraints only a
sub-grid of the n starting point grid can
be selected
bull This means to consider a very high
number of possible configurations that is
impossible to evaluate by enumeration
The simulating annealing a combinatorial
optimization algorithm allows to draw the
optimum reduced map
bull Because technical constraints only a
sub-grid of the n starting point grid can
be selected
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |11
The optimization of the size of the measurement maps
bull Universal kriging was used to predict the deposition surface
assuming a complete polynomial of second order for the mean
function of the process
bull To select the optimum reduced map the fitness function
should be related to the kriging variance evaluated at the
current sample configuration
bull The algorithm starts from a random configuration of sample
points and sequentially updates it At each step i the current
configuration is modified by replacing one point by one point
The candidate point for replacement is selected randomly and
it is accepted if this determines an improvement in the value of
the fitness function
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |12
The optimization of the size of the measurement maps
bull The procedure is iterated until the value of the fitness function gets stable and cannot be further reduced
Optimal submap - 20sites
-100 -50 0 50 100
-100
-50
050
100
campione ottimo fitness= 499911
X
Y
x
x
x x
x
x
x
x
x
x
x x
x
x
x
xx
x
x
x
xx x
x
x
x
x
x
x
49 sites starting map
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |13
The LogVariance methods
bull The early development phase for a technological step is the study of
a working point able to both center the specification limits and
minimize the unavoidable differences among positions on wafer
area
bull In this phase the measurement time and data analysis are only
possible for a small number of samples If the system response
match the specification limits the technological step is implemented
on an experimental WIP in order to monitor its behavior
bull Data produced are very important to estimate whether the process
will be stable enough for production volumes or not
bull The aim of the method is to give a quantitative evaluation of the
degree of stability of a technological step
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |14
The LogVariance methods
bull The wafers come from different lots processed in different times and they are randomly selected and supposed to be stochastically independent each other and by time
bull The aim is to evaluate the homogeneity in mean and variance on the whole wafer
Mean RS
Standard deviation RS
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35 -35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
-35
-3
-25
-2
-15
-1
-05
0
05
1
15
2
25
3
35
-35 -25 -15 -05 05 15 25 35 -3
5 -25 -15 -05 05 15 25 35
-35
-15
05
25
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |15
The LogVariance methods
bull Using the mean and the standard deviation estimated over the whole wafer surface by the L-RS model it is possible both to point out whether a region is far from the target and to find the distribution of the Cpk over the surface
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
-35 173 191 204 211 211 205 194
-3 159 186 21 231 246 253 246 233 216 196 175
-25 157 189 221 249 272 279 276 267 254 236 216 193 169
-2 184 22 256 285 294 298 295 287 273 255 234 21 185
-15 169 209 25 283 299 31 314 312 305 291 273 251 226 2 173
-1 189 232 271 293 31 322 328 327 32 307 289 267 241 214 186
-05 205 249 276 299 318 331 338 338 332 32 302 28 254 226 198
0 218 25 278 302 322 336 344 346 341 33 313 291 265 237 207
05 218 248 276 301 322 338 347 35 347 336 32 299 273 245 215
1 214 244 272 298 319 336 347 351 349 339 324 304 279 251 221
15 207 236 265 291 313 331 343 348 347 339 325 305 281 253 217
2 227 255 281 304 322 335 342 342 335 322 303 28 249
25 215 243 269 292 31 324 331 333 327 315 298 273 229
3 229 254 277 296 31 318 32 316 305 284 246
35 26 279 293 302 304 301 282
Region expected to be not on the target
Cpk contour plot
-35
-3
-25
-2
-15
-1
-05
0
05
1
15
2
25
3
35
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
1125
151752
225
252753
325
353754
425
4-425
375-4
35-375
325-35
3-325
275-3
25-275
225-25
2-225
175-2
15-175
125-15
1-125
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |16
The LogVariance methods
Possible other applications
The algorithm proposed may be used in other applications
bull to check the alignment between two equipments that
produce wafers (are they aligned regarding average and
variance)
bull in a DOE experiment to find the area with the smallest
variance and the best average value or to understand what
are the most relevant factors
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |17
Conclusions
bull The application of the approaches here presented
allows to save measurement time and to reduce
equipment access time
bull The use of non appropriate methods can yield
misleading results with consequent erroneous
decisions and waste of time and materials
bull The study and the development of adequate statistical
approaches are fundamentals also for a better
understanding of the process
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |18
References
bull Diggle PJ 2003 Statistical Analysis of Spatial Point Patterns 2-nd edition Arnold London
bull Illian J Penttinen A Stoyan D 2008 Statistical Analysis and Modeling of Spatial Point Patterns Wiley New York
bull Aarts E Korst J Simulated Annealing and Boltzmann Machines - A Stochastic Approach to Combinatorial Optimization and Neural Computing Wiley New York (1990)
bull Chiles JP Delfiner P Geostatistics Modeling Spatial Uncertainty John Wiley amp Sons New York (1999)
bull Aitkin M (1987) Modelling Variance Heterogeneity in Normal Regression Using GLIM Applied Statistics 36 332-339
bull Faraway J J (2006) Extending the Linear Model with R Chapmanand Hall
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |19
Thank you for your kind attention
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 8
The spatial distribution defectivity over the wafer surface
bull A situation where no clusters are present is desiderable
because it means the process is in control (eg no
ldquospecialrdquo source of particles is present)
bull This situation is statistically modellized by a
Homogeneous Poisson Process (HPP) The process is
defined Complete Spatial Randomness (CSR)
bull The defectivity pattern is evaluated by the estimation of
the distribution function of the euclidean distances
between pairs of defects
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 9
The spatial distribution defectivity over the wafer surface
bull A comparison between the empirical cumulative distribution function (EDF) and the CSR cumulative distribution function is performed by a graphical test
bull The CSR hypothesis should be rejected if the EDF (the continuous black colored line) lies outside the envelope identified by the dotted lines The EDF is far below the lower bound of the envelope we can assume that there is a presence of clusters
Clusters No Clusters
0 5000 10000 15000 20000 25000
00
02
04
06
08
10
envpp
r
F(r
)obstheohilo
0 5000 10000 15000 20000 25000 30000
00
02
04
06
08
10
envpp
r
F(r
)
obstheohilo
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |10
The optimization of the size of the measurement maps
bull Starting from an in-use monitoring map the goal is to consider all the
possible configurations of a reduced map to evaluate for each of them
the fitness function and to select the optimum by some criteria
bull This means to consider a very high
number of possible configurations that is
impossible to evaluate by enumeration
The simulating annealing a combinatorial
optimization algorithm allows to draw the
optimum reduced map
bull Because technical constraints only a
sub-grid of the n starting point grid can
be selected
bull This means to consider a very high
number of possible configurations that is
impossible to evaluate by enumeration
The simulating annealing a combinatorial
optimization algorithm allows to draw the
optimum reduced map
bull Because technical constraints only a
sub-grid of the n starting point grid can
be selected
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |11
The optimization of the size of the measurement maps
bull Universal kriging was used to predict the deposition surface
assuming a complete polynomial of second order for the mean
function of the process
bull To select the optimum reduced map the fitness function
should be related to the kriging variance evaluated at the
current sample configuration
bull The algorithm starts from a random configuration of sample
points and sequentially updates it At each step i the current
configuration is modified by replacing one point by one point
The candidate point for replacement is selected randomly and
it is accepted if this determines an improvement in the value of
the fitness function
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |12
The optimization of the size of the measurement maps
bull The procedure is iterated until the value of the fitness function gets stable and cannot be further reduced
Optimal submap - 20sites
-100 -50 0 50 100
-100
-50
050
100
campione ottimo fitness= 499911
X
Y
x
x
x x
x
x
x
x
x
x
x x
x
x
x
xx
x
x
x
xx x
x
x
x
x
x
x
49 sites starting map
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |13
The LogVariance methods
bull The early development phase for a technological step is the study of
a working point able to both center the specification limits and
minimize the unavoidable differences among positions on wafer
area
bull In this phase the measurement time and data analysis are only
possible for a small number of samples If the system response
match the specification limits the technological step is implemented
on an experimental WIP in order to monitor its behavior
bull Data produced are very important to estimate whether the process
will be stable enough for production volumes or not
bull The aim of the method is to give a quantitative evaluation of the
degree of stability of a technological step
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |14
The LogVariance methods
bull The wafers come from different lots processed in different times and they are randomly selected and supposed to be stochastically independent each other and by time
bull The aim is to evaluate the homogeneity in mean and variance on the whole wafer
Mean RS
Standard deviation RS
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35 -35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
-35
-3
-25
-2
-15
-1
-05
0
05
1
15
2
25
3
35
-35 -25 -15 -05 05 15 25 35 -3
5 -25 -15 -05 05 15 25 35
-35
-15
05
25
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |15
The LogVariance methods
bull Using the mean and the standard deviation estimated over the whole wafer surface by the L-RS model it is possible both to point out whether a region is far from the target and to find the distribution of the Cpk over the surface
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
-35 173 191 204 211 211 205 194
-3 159 186 21 231 246 253 246 233 216 196 175
-25 157 189 221 249 272 279 276 267 254 236 216 193 169
-2 184 22 256 285 294 298 295 287 273 255 234 21 185
-15 169 209 25 283 299 31 314 312 305 291 273 251 226 2 173
-1 189 232 271 293 31 322 328 327 32 307 289 267 241 214 186
-05 205 249 276 299 318 331 338 338 332 32 302 28 254 226 198
0 218 25 278 302 322 336 344 346 341 33 313 291 265 237 207
05 218 248 276 301 322 338 347 35 347 336 32 299 273 245 215
1 214 244 272 298 319 336 347 351 349 339 324 304 279 251 221
15 207 236 265 291 313 331 343 348 347 339 325 305 281 253 217
2 227 255 281 304 322 335 342 342 335 322 303 28 249
25 215 243 269 292 31 324 331 333 327 315 298 273 229
3 229 254 277 296 31 318 32 316 305 284 246
35 26 279 293 302 304 301 282
Region expected to be not on the target
Cpk contour plot
-35
-3
-25
-2
-15
-1
-05
0
05
1
15
2
25
3
35
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
1125
151752
225
252753
325
353754
425
4-425
375-4
35-375
325-35
3-325
275-3
25-275
225-25
2-225
175-2
15-175
125-15
1-125
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |16
The LogVariance methods
Possible other applications
The algorithm proposed may be used in other applications
bull to check the alignment between two equipments that
produce wafers (are they aligned regarding average and
variance)
bull in a DOE experiment to find the area with the smallest
variance and the best average value or to understand what
are the most relevant factors
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |17
Conclusions
bull The application of the approaches here presented
allows to save measurement time and to reduce
equipment access time
bull The use of non appropriate methods can yield
misleading results with consequent erroneous
decisions and waste of time and materials
bull The study and the development of adequate statistical
approaches are fundamentals also for a better
understanding of the process
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |18
References
bull Diggle PJ 2003 Statistical Analysis of Spatial Point Patterns 2-nd edition Arnold London
bull Illian J Penttinen A Stoyan D 2008 Statistical Analysis and Modeling of Spatial Point Patterns Wiley New York
bull Aarts E Korst J Simulated Annealing and Boltzmann Machines - A Stochastic Approach to Combinatorial Optimization and Neural Computing Wiley New York (1990)
bull Chiles JP Delfiner P Geostatistics Modeling Spatial Uncertainty John Wiley amp Sons New York (1999)
bull Aitkin M (1987) Modelling Variance Heterogeneity in Normal Regression Using GLIM Applied Statistics 36 332-339
bull Faraway J J (2006) Extending the Linear Model with R Chapmanand Hall
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |19
Thank you for your kind attention
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc | 9
The spatial distribution defectivity over the wafer surface
bull A comparison between the empirical cumulative distribution function (EDF) and the CSR cumulative distribution function is performed by a graphical test
bull The CSR hypothesis should be rejected if the EDF (the continuous black colored line) lies outside the envelope identified by the dotted lines The EDF is far below the lower bound of the envelope we can assume that there is a presence of clusters
Clusters No Clusters
0 5000 10000 15000 20000 25000
00
02
04
06
08
10
envpp
r
F(r
)obstheohilo
0 5000 10000 15000 20000 25000 30000
00
02
04
06
08
10
envpp
r
F(r
)
obstheohilo
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |10
The optimization of the size of the measurement maps
bull Starting from an in-use monitoring map the goal is to consider all the
possible configurations of a reduced map to evaluate for each of them
the fitness function and to select the optimum by some criteria
bull This means to consider a very high
number of possible configurations that is
impossible to evaluate by enumeration
The simulating annealing a combinatorial
optimization algorithm allows to draw the
optimum reduced map
bull Because technical constraints only a
sub-grid of the n starting point grid can
be selected
bull This means to consider a very high
number of possible configurations that is
impossible to evaluate by enumeration
The simulating annealing a combinatorial
optimization algorithm allows to draw the
optimum reduced map
bull Because technical constraints only a
sub-grid of the n starting point grid can
be selected
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |11
The optimization of the size of the measurement maps
bull Universal kriging was used to predict the deposition surface
assuming a complete polynomial of second order for the mean
function of the process
bull To select the optimum reduced map the fitness function
should be related to the kriging variance evaluated at the
current sample configuration
bull The algorithm starts from a random configuration of sample
points and sequentially updates it At each step i the current
configuration is modified by replacing one point by one point
The candidate point for replacement is selected randomly and
it is accepted if this determines an improvement in the value of
the fitness function
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |12
The optimization of the size of the measurement maps
bull The procedure is iterated until the value of the fitness function gets stable and cannot be further reduced
Optimal submap - 20sites
-100 -50 0 50 100
-100
-50
050
100
campione ottimo fitness= 499911
X
Y
x
x
x x
x
x
x
x
x
x
x x
x
x
x
xx
x
x
x
xx x
x
x
x
x
x
x
49 sites starting map
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |13
The LogVariance methods
bull The early development phase for a technological step is the study of
a working point able to both center the specification limits and
minimize the unavoidable differences among positions on wafer
area
bull In this phase the measurement time and data analysis are only
possible for a small number of samples If the system response
match the specification limits the technological step is implemented
on an experimental WIP in order to monitor its behavior
bull Data produced are very important to estimate whether the process
will be stable enough for production volumes or not
bull The aim of the method is to give a quantitative evaluation of the
degree of stability of a technological step
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |14
The LogVariance methods
bull The wafers come from different lots processed in different times and they are randomly selected and supposed to be stochastically independent each other and by time
bull The aim is to evaluate the homogeneity in mean and variance on the whole wafer
Mean RS
Standard deviation RS
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35 -35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
-35
-3
-25
-2
-15
-1
-05
0
05
1
15
2
25
3
35
-35 -25 -15 -05 05 15 25 35 -3
5 -25 -15 -05 05 15 25 35
-35
-15
05
25
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |15
The LogVariance methods
bull Using the mean and the standard deviation estimated over the whole wafer surface by the L-RS model it is possible both to point out whether a region is far from the target and to find the distribution of the Cpk over the surface
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
-35 173 191 204 211 211 205 194
-3 159 186 21 231 246 253 246 233 216 196 175
-25 157 189 221 249 272 279 276 267 254 236 216 193 169
-2 184 22 256 285 294 298 295 287 273 255 234 21 185
-15 169 209 25 283 299 31 314 312 305 291 273 251 226 2 173
-1 189 232 271 293 31 322 328 327 32 307 289 267 241 214 186
-05 205 249 276 299 318 331 338 338 332 32 302 28 254 226 198
0 218 25 278 302 322 336 344 346 341 33 313 291 265 237 207
05 218 248 276 301 322 338 347 35 347 336 32 299 273 245 215
1 214 244 272 298 319 336 347 351 349 339 324 304 279 251 221
15 207 236 265 291 313 331 343 348 347 339 325 305 281 253 217
2 227 255 281 304 322 335 342 342 335 322 303 28 249
25 215 243 269 292 31 324 331 333 327 315 298 273 229
3 229 254 277 296 31 318 32 316 305 284 246
35 26 279 293 302 304 301 282
Region expected to be not on the target
Cpk contour plot
-35
-3
-25
-2
-15
-1
-05
0
05
1
15
2
25
3
35
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
1125
151752
225
252753
325
353754
425
4-425
375-4
35-375
325-35
3-325
275-3
25-275
225-25
2-225
175-2
15-175
125-15
1-125
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |16
The LogVariance methods
Possible other applications
The algorithm proposed may be used in other applications
bull to check the alignment between two equipments that
produce wafers (are they aligned regarding average and
variance)
bull in a DOE experiment to find the area with the smallest
variance and the best average value or to understand what
are the most relevant factors
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |17
Conclusions
bull The application of the approaches here presented
allows to save measurement time and to reduce
equipment access time
bull The use of non appropriate methods can yield
misleading results with consequent erroneous
decisions and waste of time and materials
bull The study and the development of adequate statistical
approaches are fundamentals also for a better
understanding of the process
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |18
References
bull Diggle PJ 2003 Statistical Analysis of Spatial Point Patterns 2-nd edition Arnold London
bull Illian J Penttinen A Stoyan D 2008 Statistical Analysis and Modeling of Spatial Point Patterns Wiley New York
bull Aarts E Korst J Simulated Annealing and Boltzmann Machines - A Stochastic Approach to Combinatorial Optimization and Neural Computing Wiley New York (1990)
bull Chiles JP Delfiner P Geostatistics Modeling Spatial Uncertainty John Wiley amp Sons New York (1999)
bull Aitkin M (1987) Modelling Variance Heterogeneity in Normal Regression Using GLIM Applied Statistics 36 332-339
bull Faraway J J (2006) Extending the Linear Model with R Chapmanand Hall
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |19
Thank you for your kind attention
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |10
The optimization of the size of the measurement maps
bull Starting from an in-use monitoring map the goal is to consider all the
possible configurations of a reduced map to evaluate for each of them
the fitness function and to select the optimum by some criteria
bull This means to consider a very high
number of possible configurations that is
impossible to evaluate by enumeration
The simulating annealing a combinatorial
optimization algorithm allows to draw the
optimum reduced map
bull Because technical constraints only a
sub-grid of the n starting point grid can
be selected
bull This means to consider a very high
number of possible configurations that is
impossible to evaluate by enumeration
The simulating annealing a combinatorial
optimization algorithm allows to draw the
optimum reduced map
bull Because technical constraints only a
sub-grid of the n starting point grid can
be selected
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |11
The optimization of the size of the measurement maps
bull Universal kriging was used to predict the deposition surface
assuming a complete polynomial of second order for the mean
function of the process
bull To select the optimum reduced map the fitness function
should be related to the kriging variance evaluated at the
current sample configuration
bull The algorithm starts from a random configuration of sample
points and sequentially updates it At each step i the current
configuration is modified by replacing one point by one point
The candidate point for replacement is selected randomly and
it is accepted if this determines an improvement in the value of
the fitness function
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |12
The optimization of the size of the measurement maps
bull The procedure is iterated until the value of the fitness function gets stable and cannot be further reduced
Optimal submap - 20sites
-100 -50 0 50 100
-100
-50
050
100
campione ottimo fitness= 499911
X
Y
x
x
x x
x
x
x
x
x
x
x x
x
x
x
xx
x
x
x
xx x
x
x
x
x
x
x
49 sites starting map
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |13
The LogVariance methods
bull The early development phase for a technological step is the study of
a working point able to both center the specification limits and
minimize the unavoidable differences among positions on wafer
area
bull In this phase the measurement time and data analysis are only
possible for a small number of samples If the system response
match the specification limits the technological step is implemented
on an experimental WIP in order to monitor its behavior
bull Data produced are very important to estimate whether the process
will be stable enough for production volumes or not
bull The aim of the method is to give a quantitative evaluation of the
degree of stability of a technological step
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |14
The LogVariance methods
bull The wafers come from different lots processed in different times and they are randomly selected and supposed to be stochastically independent each other and by time
bull The aim is to evaluate the homogeneity in mean and variance on the whole wafer
Mean RS
Standard deviation RS
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35 -35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
-35
-3
-25
-2
-15
-1
-05
0
05
1
15
2
25
3
35
-35 -25 -15 -05 05 15 25 35 -3
5 -25 -15 -05 05 15 25 35
-35
-15
05
25
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |15
The LogVariance methods
bull Using the mean and the standard deviation estimated over the whole wafer surface by the L-RS model it is possible both to point out whether a region is far from the target and to find the distribution of the Cpk over the surface
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
-35 173 191 204 211 211 205 194
-3 159 186 21 231 246 253 246 233 216 196 175
-25 157 189 221 249 272 279 276 267 254 236 216 193 169
-2 184 22 256 285 294 298 295 287 273 255 234 21 185
-15 169 209 25 283 299 31 314 312 305 291 273 251 226 2 173
-1 189 232 271 293 31 322 328 327 32 307 289 267 241 214 186
-05 205 249 276 299 318 331 338 338 332 32 302 28 254 226 198
0 218 25 278 302 322 336 344 346 341 33 313 291 265 237 207
05 218 248 276 301 322 338 347 35 347 336 32 299 273 245 215
1 214 244 272 298 319 336 347 351 349 339 324 304 279 251 221
15 207 236 265 291 313 331 343 348 347 339 325 305 281 253 217
2 227 255 281 304 322 335 342 342 335 322 303 28 249
25 215 243 269 292 31 324 331 333 327 315 298 273 229
3 229 254 277 296 31 318 32 316 305 284 246
35 26 279 293 302 304 301 282
Region expected to be not on the target
Cpk contour plot
-35
-3
-25
-2
-15
-1
-05
0
05
1
15
2
25
3
35
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
1125
151752
225
252753
325
353754
425
4-425
375-4
35-375
325-35
3-325
275-3
25-275
225-25
2-225
175-2
15-175
125-15
1-125
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |16
The LogVariance methods
Possible other applications
The algorithm proposed may be used in other applications
bull to check the alignment between two equipments that
produce wafers (are they aligned regarding average and
variance)
bull in a DOE experiment to find the area with the smallest
variance and the best average value or to understand what
are the most relevant factors
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |17
Conclusions
bull The application of the approaches here presented
allows to save measurement time and to reduce
equipment access time
bull The use of non appropriate methods can yield
misleading results with consequent erroneous
decisions and waste of time and materials
bull The study and the development of adequate statistical
approaches are fundamentals also for a better
understanding of the process
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |18
References
bull Diggle PJ 2003 Statistical Analysis of Spatial Point Patterns 2-nd edition Arnold London
bull Illian J Penttinen A Stoyan D 2008 Statistical Analysis and Modeling of Spatial Point Patterns Wiley New York
bull Aarts E Korst J Simulated Annealing and Boltzmann Machines - A Stochastic Approach to Combinatorial Optimization and Neural Computing Wiley New York (1990)
bull Chiles JP Delfiner P Geostatistics Modeling Spatial Uncertainty John Wiley amp Sons New York (1999)
bull Aitkin M (1987) Modelling Variance Heterogeneity in Normal Regression Using GLIM Applied Statistics 36 332-339
bull Faraway J J (2006) Extending the Linear Model with R Chapmanand Hall
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |19
Thank you for your kind attention
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |11
The optimization of the size of the measurement maps
bull Universal kriging was used to predict the deposition surface
assuming a complete polynomial of second order for the mean
function of the process
bull To select the optimum reduced map the fitness function
should be related to the kriging variance evaluated at the
current sample configuration
bull The algorithm starts from a random configuration of sample
points and sequentially updates it At each step i the current
configuration is modified by replacing one point by one point
The candidate point for replacement is selected randomly and
it is accepted if this determines an improvement in the value of
the fitness function
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |12
The optimization of the size of the measurement maps
bull The procedure is iterated until the value of the fitness function gets stable and cannot be further reduced
Optimal submap - 20sites
-100 -50 0 50 100
-100
-50
050
100
campione ottimo fitness= 499911
X
Y
x
x
x x
x
x
x
x
x
x
x x
x
x
x
xx
x
x
x
xx x
x
x
x
x
x
x
49 sites starting map
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |13
The LogVariance methods
bull The early development phase for a technological step is the study of
a working point able to both center the specification limits and
minimize the unavoidable differences among positions on wafer
area
bull In this phase the measurement time and data analysis are only
possible for a small number of samples If the system response
match the specification limits the technological step is implemented
on an experimental WIP in order to monitor its behavior
bull Data produced are very important to estimate whether the process
will be stable enough for production volumes or not
bull The aim of the method is to give a quantitative evaluation of the
degree of stability of a technological step
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |14
The LogVariance methods
bull The wafers come from different lots processed in different times and they are randomly selected and supposed to be stochastically independent each other and by time
bull The aim is to evaluate the homogeneity in mean and variance on the whole wafer
Mean RS
Standard deviation RS
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35 -35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
-35
-3
-25
-2
-15
-1
-05
0
05
1
15
2
25
3
35
-35 -25 -15 -05 05 15 25 35 -3
5 -25 -15 -05 05 15 25 35
-35
-15
05
25
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |15
The LogVariance methods
bull Using the mean and the standard deviation estimated over the whole wafer surface by the L-RS model it is possible both to point out whether a region is far from the target and to find the distribution of the Cpk over the surface
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
-35 173 191 204 211 211 205 194
-3 159 186 21 231 246 253 246 233 216 196 175
-25 157 189 221 249 272 279 276 267 254 236 216 193 169
-2 184 22 256 285 294 298 295 287 273 255 234 21 185
-15 169 209 25 283 299 31 314 312 305 291 273 251 226 2 173
-1 189 232 271 293 31 322 328 327 32 307 289 267 241 214 186
-05 205 249 276 299 318 331 338 338 332 32 302 28 254 226 198
0 218 25 278 302 322 336 344 346 341 33 313 291 265 237 207
05 218 248 276 301 322 338 347 35 347 336 32 299 273 245 215
1 214 244 272 298 319 336 347 351 349 339 324 304 279 251 221
15 207 236 265 291 313 331 343 348 347 339 325 305 281 253 217
2 227 255 281 304 322 335 342 342 335 322 303 28 249
25 215 243 269 292 31 324 331 333 327 315 298 273 229
3 229 254 277 296 31 318 32 316 305 284 246
35 26 279 293 302 304 301 282
Region expected to be not on the target
Cpk contour plot
-35
-3
-25
-2
-15
-1
-05
0
05
1
15
2
25
3
35
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
1125
151752
225
252753
325
353754
425
4-425
375-4
35-375
325-35
3-325
275-3
25-275
225-25
2-225
175-2
15-175
125-15
1-125
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |16
The LogVariance methods
Possible other applications
The algorithm proposed may be used in other applications
bull to check the alignment between two equipments that
produce wafers (are they aligned regarding average and
variance)
bull in a DOE experiment to find the area with the smallest
variance and the best average value or to understand what
are the most relevant factors
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |17
Conclusions
bull The application of the approaches here presented
allows to save measurement time and to reduce
equipment access time
bull The use of non appropriate methods can yield
misleading results with consequent erroneous
decisions and waste of time and materials
bull The study and the development of adequate statistical
approaches are fundamentals also for a better
understanding of the process
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |18
References
bull Diggle PJ 2003 Statistical Analysis of Spatial Point Patterns 2-nd edition Arnold London
bull Illian J Penttinen A Stoyan D 2008 Statistical Analysis and Modeling of Spatial Point Patterns Wiley New York
bull Aarts E Korst J Simulated Annealing and Boltzmann Machines - A Stochastic Approach to Combinatorial Optimization and Neural Computing Wiley New York (1990)
bull Chiles JP Delfiner P Geostatistics Modeling Spatial Uncertainty John Wiley amp Sons New York (1999)
bull Aitkin M (1987) Modelling Variance Heterogeneity in Normal Regression Using GLIM Applied Statistics 36 332-339
bull Faraway J J (2006) Extending the Linear Model with R Chapmanand Hall
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |19
Thank you for your kind attention
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |12
The optimization of the size of the measurement maps
bull The procedure is iterated until the value of the fitness function gets stable and cannot be further reduced
Optimal submap - 20sites
-100 -50 0 50 100
-100
-50
050
100
campione ottimo fitness= 499911
X
Y
x
x
x x
x
x
x
x
x
x
x x
x
x
x
xx
x
x
x
xx x
x
x
x
x
x
x
49 sites starting map
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |13
The LogVariance methods
bull The early development phase for a technological step is the study of
a working point able to both center the specification limits and
minimize the unavoidable differences among positions on wafer
area
bull In this phase the measurement time and data analysis are only
possible for a small number of samples If the system response
match the specification limits the technological step is implemented
on an experimental WIP in order to monitor its behavior
bull Data produced are very important to estimate whether the process
will be stable enough for production volumes or not
bull The aim of the method is to give a quantitative evaluation of the
degree of stability of a technological step
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |14
The LogVariance methods
bull The wafers come from different lots processed in different times and they are randomly selected and supposed to be stochastically independent each other and by time
bull The aim is to evaluate the homogeneity in mean and variance on the whole wafer
Mean RS
Standard deviation RS
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35 -35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
-35
-3
-25
-2
-15
-1
-05
0
05
1
15
2
25
3
35
-35 -25 -15 -05 05 15 25 35 -3
5 -25 -15 -05 05 15 25 35
-35
-15
05
25
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |15
The LogVariance methods
bull Using the mean and the standard deviation estimated over the whole wafer surface by the L-RS model it is possible both to point out whether a region is far from the target and to find the distribution of the Cpk over the surface
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
-35 173 191 204 211 211 205 194
-3 159 186 21 231 246 253 246 233 216 196 175
-25 157 189 221 249 272 279 276 267 254 236 216 193 169
-2 184 22 256 285 294 298 295 287 273 255 234 21 185
-15 169 209 25 283 299 31 314 312 305 291 273 251 226 2 173
-1 189 232 271 293 31 322 328 327 32 307 289 267 241 214 186
-05 205 249 276 299 318 331 338 338 332 32 302 28 254 226 198
0 218 25 278 302 322 336 344 346 341 33 313 291 265 237 207
05 218 248 276 301 322 338 347 35 347 336 32 299 273 245 215
1 214 244 272 298 319 336 347 351 349 339 324 304 279 251 221
15 207 236 265 291 313 331 343 348 347 339 325 305 281 253 217
2 227 255 281 304 322 335 342 342 335 322 303 28 249
25 215 243 269 292 31 324 331 333 327 315 298 273 229
3 229 254 277 296 31 318 32 316 305 284 246
35 26 279 293 302 304 301 282
Region expected to be not on the target
Cpk contour plot
-35
-3
-25
-2
-15
-1
-05
0
05
1
15
2
25
3
35
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
1125
151752
225
252753
325
353754
425
4-425
375-4
35-375
325-35
3-325
275-3
25-275
225-25
2-225
175-2
15-175
125-15
1-125
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |16
The LogVariance methods
Possible other applications
The algorithm proposed may be used in other applications
bull to check the alignment between two equipments that
produce wafers (are they aligned regarding average and
variance)
bull in a DOE experiment to find the area with the smallest
variance and the best average value or to understand what
are the most relevant factors
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |17
Conclusions
bull The application of the approaches here presented
allows to save measurement time and to reduce
equipment access time
bull The use of non appropriate methods can yield
misleading results with consequent erroneous
decisions and waste of time and materials
bull The study and the development of adequate statistical
approaches are fundamentals also for a better
understanding of the process
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |18
References
bull Diggle PJ 2003 Statistical Analysis of Spatial Point Patterns 2-nd edition Arnold London
bull Illian J Penttinen A Stoyan D 2008 Statistical Analysis and Modeling of Spatial Point Patterns Wiley New York
bull Aarts E Korst J Simulated Annealing and Boltzmann Machines - A Stochastic Approach to Combinatorial Optimization and Neural Computing Wiley New York (1990)
bull Chiles JP Delfiner P Geostatistics Modeling Spatial Uncertainty John Wiley amp Sons New York (1999)
bull Aitkin M (1987) Modelling Variance Heterogeneity in Normal Regression Using GLIM Applied Statistics 36 332-339
bull Faraway J J (2006) Extending the Linear Model with R Chapmanand Hall
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |19
Thank you for your kind attention
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |13
The LogVariance methods
bull The early development phase for a technological step is the study of
a working point able to both center the specification limits and
minimize the unavoidable differences among positions on wafer
area
bull In this phase the measurement time and data analysis are only
possible for a small number of samples If the system response
match the specification limits the technological step is implemented
on an experimental WIP in order to monitor its behavior
bull Data produced are very important to estimate whether the process
will be stable enough for production volumes or not
bull The aim of the method is to give a quantitative evaluation of the
degree of stability of a technological step
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |14
The LogVariance methods
bull The wafers come from different lots processed in different times and they are randomly selected and supposed to be stochastically independent each other and by time
bull The aim is to evaluate the homogeneity in mean and variance on the whole wafer
Mean RS
Standard deviation RS
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35 -35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
-35
-3
-25
-2
-15
-1
-05
0
05
1
15
2
25
3
35
-35 -25 -15 -05 05 15 25 35 -3
5 -25 -15 -05 05 15 25 35
-35
-15
05
25
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |15
The LogVariance methods
bull Using the mean and the standard deviation estimated over the whole wafer surface by the L-RS model it is possible both to point out whether a region is far from the target and to find the distribution of the Cpk over the surface
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
-35 173 191 204 211 211 205 194
-3 159 186 21 231 246 253 246 233 216 196 175
-25 157 189 221 249 272 279 276 267 254 236 216 193 169
-2 184 22 256 285 294 298 295 287 273 255 234 21 185
-15 169 209 25 283 299 31 314 312 305 291 273 251 226 2 173
-1 189 232 271 293 31 322 328 327 32 307 289 267 241 214 186
-05 205 249 276 299 318 331 338 338 332 32 302 28 254 226 198
0 218 25 278 302 322 336 344 346 341 33 313 291 265 237 207
05 218 248 276 301 322 338 347 35 347 336 32 299 273 245 215
1 214 244 272 298 319 336 347 351 349 339 324 304 279 251 221
15 207 236 265 291 313 331 343 348 347 339 325 305 281 253 217
2 227 255 281 304 322 335 342 342 335 322 303 28 249
25 215 243 269 292 31 324 331 333 327 315 298 273 229
3 229 254 277 296 31 318 32 316 305 284 246
35 26 279 293 302 304 301 282
Region expected to be not on the target
Cpk contour plot
-35
-3
-25
-2
-15
-1
-05
0
05
1
15
2
25
3
35
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
1125
151752
225
252753
325
353754
425
4-425
375-4
35-375
325-35
3-325
275-3
25-275
225-25
2-225
175-2
15-175
125-15
1-125
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |16
The LogVariance methods
Possible other applications
The algorithm proposed may be used in other applications
bull to check the alignment between two equipments that
produce wafers (are they aligned regarding average and
variance)
bull in a DOE experiment to find the area with the smallest
variance and the best average value or to understand what
are the most relevant factors
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |17
Conclusions
bull The application of the approaches here presented
allows to save measurement time and to reduce
equipment access time
bull The use of non appropriate methods can yield
misleading results with consequent erroneous
decisions and waste of time and materials
bull The study and the development of adequate statistical
approaches are fundamentals also for a better
understanding of the process
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |18
References
bull Diggle PJ 2003 Statistical Analysis of Spatial Point Patterns 2-nd edition Arnold London
bull Illian J Penttinen A Stoyan D 2008 Statistical Analysis and Modeling of Spatial Point Patterns Wiley New York
bull Aarts E Korst J Simulated Annealing and Boltzmann Machines - A Stochastic Approach to Combinatorial Optimization and Neural Computing Wiley New York (1990)
bull Chiles JP Delfiner P Geostatistics Modeling Spatial Uncertainty John Wiley amp Sons New York (1999)
bull Aitkin M (1987) Modelling Variance Heterogeneity in Normal Regression Using GLIM Applied Statistics 36 332-339
bull Faraway J J (2006) Extending the Linear Model with R Chapmanand Hall
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |19
Thank you for your kind attention
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |14
The LogVariance methods
bull The wafers come from different lots processed in different times and they are randomly selected and supposed to be stochastically independent each other and by time
bull The aim is to evaluate the homogeneity in mean and variance on the whole wafer
Mean RS
Standard deviation RS
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35 -35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
-35
-3
-25
-2
-15
-1
-05
0
05
1
15
2
25
3
35
-35 -25 -15 -05 05 15 25 35 -3
5 -25 -15 -05 05 15 25 35
-35
-15
05
25
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |15
The LogVariance methods
bull Using the mean and the standard deviation estimated over the whole wafer surface by the L-RS model it is possible both to point out whether a region is far from the target and to find the distribution of the Cpk over the surface
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
-35 173 191 204 211 211 205 194
-3 159 186 21 231 246 253 246 233 216 196 175
-25 157 189 221 249 272 279 276 267 254 236 216 193 169
-2 184 22 256 285 294 298 295 287 273 255 234 21 185
-15 169 209 25 283 299 31 314 312 305 291 273 251 226 2 173
-1 189 232 271 293 31 322 328 327 32 307 289 267 241 214 186
-05 205 249 276 299 318 331 338 338 332 32 302 28 254 226 198
0 218 25 278 302 322 336 344 346 341 33 313 291 265 237 207
05 218 248 276 301 322 338 347 35 347 336 32 299 273 245 215
1 214 244 272 298 319 336 347 351 349 339 324 304 279 251 221
15 207 236 265 291 313 331 343 348 347 339 325 305 281 253 217
2 227 255 281 304 322 335 342 342 335 322 303 28 249
25 215 243 269 292 31 324 331 333 327 315 298 273 229
3 229 254 277 296 31 318 32 316 305 284 246
35 26 279 293 302 304 301 282
Region expected to be not on the target
Cpk contour plot
-35
-3
-25
-2
-15
-1
-05
0
05
1
15
2
25
3
35
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
1125
151752
225
252753
325
353754
425
4-425
375-4
35-375
325-35
3-325
275-3
25-275
225-25
2-225
175-2
15-175
125-15
1-125
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |16
The LogVariance methods
Possible other applications
The algorithm proposed may be used in other applications
bull to check the alignment between two equipments that
produce wafers (are they aligned regarding average and
variance)
bull in a DOE experiment to find the area with the smallest
variance and the best average value or to understand what
are the most relevant factors
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |17
Conclusions
bull The application of the approaches here presented
allows to save measurement time and to reduce
equipment access time
bull The use of non appropriate methods can yield
misleading results with consequent erroneous
decisions and waste of time and materials
bull The study and the development of adequate statistical
approaches are fundamentals also for a better
understanding of the process
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |18
References
bull Diggle PJ 2003 Statistical Analysis of Spatial Point Patterns 2-nd edition Arnold London
bull Illian J Penttinen A Stoyan D 2008 Statistical Analysis and Modeling of Spatial Point Patterns Wiley New York
bull Aarts E Korst J Simulated Annealing and Boltzmann Machines - A Stochastic Approach to Combinatorial Optimization and Neural Computing Wiley New York (1990)
bull Chiles JP Delfiner P Geostatistics Modeling Spatial Uncertainty John Wiley amp Sons New York (1999)
bull Aitkin M (1987) Modelling Variance Heterogeneity in Normal Regression Using GLIM Applied Statistics 36 332-339
bull Faraway J J (2006) Extending the Linear Model with R Chapmanand Hall
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |19
Thank you for your kind attention
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |15
The LogVariance methods
bull Using the mean and the standard deviation estimated over the whole wafer surface by the L-RS model it is possible both to point out whether a region is far from the target and to find the distribution of the Cpk over the surface
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
-35 173 191 204 211 211 205 194
-3 159 186 21 231 246 253 246 233 216 196 175
-25 157 189 221 249 272 279 276 267 254 236 216 193 169
-2 184 22 256 285 294 298 295 287 273 255 234 21 185
-15 169 209 25 283 299 31 314 312 305 291 273 251 226 2 173
-1 189 232 271 293 31 322 328 327 32 307 289 267 241 214 186
-05 205 249 276 299 318 331 338 338 332 32 302 28 254 226 198
0 218 25 278 302 322 336 344 346 341 33 313 291 265 237 207
05 218 248 276 301 322 338 347 35 347 336 32 299 273 245 215
1 214 244 272 298 319 336 347 351 349 339 324 304 279 251 221
15 207 236 265 291 313 331 343 348 347 339 325 305 281 253 217
2 227 255 281 304 322 335 342 342 335 322 303 28 249
25 215 243 269 292 31 324 331 333 327 315 298 273 229
3 229 254 277 296 31 318 32 316 305 284 246
35 26 279 293 302 304 301 282
Region expected to be not on the target
Cpk contour plot
-35
-3
-25
-2
-15
-1
-05
0
05
1
15
2
25
3
35
-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35
1125
151752
225
252753
325
353754
425
4-425
375-4
35-375
325-35
3-325
275-3
25-275
225-25
2-225
175-2
15-175
125-15
1-125
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |16
The LogVariance methods
Possible other applications
The algorithm proposed may be used in other applications
bull to check the alignment between two equipments that
produce wafers (are they aligned regarding average and
variance)
bull in a DOE experiment to find the area with the smallest
variance and the best average value or to understand what
are the most relevant factors
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |17
Conclusions
bull The application of the approaches here presented
allows to save measurement time and to reduce
equipment access time
bull The use of non appropriate methods can yield
misleading results with consequent erroneous
decisions and waste of time and materials
bull The study and the development of adequate statistical
approaches are fundamentals also for a better
understanding of the process
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |18
References
bull Diggle PJ 2003 Statistical Analysis of Spatial Point Patterns 2-nd edition Arnold London
bull Illian J Penttinen A Stoyan D 2008 Statistical Analysis and Modeling of Spatial Point Patterns Wiley New York
bull Aarts E Korst J Simulated Annealing and Boltzmann Machines - A Stochastic Approach to Combinatorial Optimization and Neural Computing Wiley New York (1990)
bull Chiles JP Delfiner P Geostatistics Modeling Spatial Uncertainty John Wiley amp Sons New York (1999)
bull Aitkin M (1987) Modelling Variance Heterogeneity in Normal Regression Using GLIM Applied Statistics 36 332-339
bull Faraway J J (2006) Extending the Linear Model with R Chapmanand Hall
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |19
Thank you for your kind attention
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |16
The LogVariance methods
Possible other applications
The algorithm proposed may be used in other applications
bull to check the alignment between two equipments that
produce wafers (are they aligned regarding average and
variance)
bull in a DOE experiment to find the area with the smallest
variance and the best average value or to understand what
are the most relevant factors
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |17
Conclusions
bull The application of the approaches here presented
allows to save measurement time and to reduce
equipment access time
bull The use of non appropriate methods can yield
misleading results with consequent erroneous
decisions and waste of time and materials
bull The study and the development of adequate statistical
approaches are fundamentals also for a better
understanding of the process
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |18
References
bull Diggle PJ 2003 Statistical Analysis of Spatial Point Patterns 2-nd edition Arnold London
bull Illian J Penttinen A Stoyan D 2008 Statistical Analysis and Modeling of Spatial Point Patterns Wiley New York
bull Aarts E Korst J Simulated Annealing and Boltzmann Machines - A Stochastic Approach to Combinatorial Optimization and Neural Computing Wiley New York (1990)
bull Chiles JP Delfiner P Geostatistics Modeling Spatial Uncertainty John Wiley amp Sons New York (1999)
bull Aitkin M (1987) Modelling Variance Heterogeneity in Normal Regression Using GLIM Applied Statistics 36 332-339
bull Faraway J J (2006) Extending the Linear Model with R Chapmanand Hall
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |19
Thank you for your kind attention
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |17
Conclusions
bull The application of the approaches here presented
allows to save measurement time and to reduce
equipment access time
bull The use of non appropriate methods can yield
misleading results with consequent erroneous
decisions and waste of time and materials
bull The study and the development of adequate statistical
approaches are fundamentals also for a better
understanding of the process
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |18
References
bull Diggle PJ 2003 Statistical Analysis of Spatial Point Patterns 2-nd edition Arnold London
bull Illian J Penttinen A Stoyan D 2008 Statistical Analysis and Modeling of Spatial Point Patterns Wiley New York
bull Aarts E Korst J Simulated Annealing and Boltzmann Machines - A Stochastic Approach to Combinatorial Optimization and Neural Computing Wiley New York (1990)
bull Chiles JP Delfiner P Geostatistics Modeling Spatial Uncertainty John Wiley amp Sons New York (1999)
bull Aitkin M (1987) Modelling Variance Heterogeneity in Normal Regression Using GLIM Applied Statistics 36 332-339
bull Faraway J J (2006) Extending the Linear Model with R Chapmanand Hall
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |19
Thank you for your kind attention
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |18
References
bull Diggle PJ 2003 Statistical Analysis of Spatial Point Patterns 2-nd edition Arnold London
bull Illian J Penttinen A Stoyan D 2008 Statistical Analysis and Modeling of Spatial Point Patterns Wiley New York
bull Aarts E Korst J Simulated Annealing and Boltzmann Machines - A Stochastic Approach to Combinatorial Optimization and Neural Computing Wiley New York (1990)
bull Chiles JP Delfiner P Geostatistics Modeling Spatial Uncertainty John Wiley amp Sons New York (1999)
bull Aitkin M (1987) Modelling Variance Heterogeneity in Normal Regression Using GLIM Applied Statistics 36 332-339
bull Faraway J J (2006) Extending the Linear Model with R Chapmanand Hall
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |19
Thank you for your kind attention
copy2009 Micron Technologies Inc All rights reserved Products are warranted only to meet Micronrsquos production data sheet specifications Information products andor specifications are subject to change without notice All information is provided on an ldquoAS ISrdquo basis without warranties of any kind Dates are estimates only Drawings are not to scale Micron and the Micron logo are trademarks of Micron Technology Inc All other trademarks are the property of their respective owners
Company Confidential | copy2009 Micron Technology Inc |19
Thank you for your kind attention