“New trends in statistical methods applied in a semiconductor company”

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



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





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



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



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



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




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




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

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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






optimum reduced map



be selected




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




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

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x

49 sites starting map



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




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

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-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




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

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15

2

25

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-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35

1125

151752

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35-375

325-35

3-325

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25-275

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2-225

175-2

15-175

125-15

1-125




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



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



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



Thank you for your kind attention



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





Introduction














Introduction




developed























Cluster Cluster



















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04

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10

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F(r

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envpp

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F(r

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obstheohilo












optimum reduced map



be selected






optimum reduced map



be selected





















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050

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x








area














Mean 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

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-1

-05

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05

1

15

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25

3

35

-35 -25 -15 -05 05 15 25 35 -3

5 -25 -15 -05 05 15 25 35

-35

-15

05

25





-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


Cpk contour plot

-35

-3

-25

-2

-15

-1

-05

0

05

1

15

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35

-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35

1125

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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








variance)






Conclusions












References














Introduction














Introduction




developed























Cluster Cluster



















0 5000 10000 15000 20000 25000

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02

04

06

08

10

envpp

r

F(r

)obstheohilo

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02

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envpp

r

F(r

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obstheohilo












optimum reduced map



be selected






optimum reduced map



be selected





















-100 -50 0 50 100

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area














Mean 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

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-15

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-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





-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


Cpk contour plot

-35

-3

-25

-2

-15

-1

-05

0

05

1

15

2

25

3

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-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35

1125

151752

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4-425

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325-35

3-325

275-3

25-275

225-25

2-225

175-2

15-175

125-15

1-125








variance)






Conclusions












References












Introduction














Introduction




developed























Cluster Cluster



















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04

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envpp

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F(r

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envpp

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optimum reduced map



be selected






optimum reduced map



be selected





















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area














Mean 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

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-05

0

05

1

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3

35

-35 -25 -15 -05 05 15 25 35 -3

5 -25 -15 -05 05 15 25 35

-35

-15

05

25





-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


Cpk contour plot

-35

-3

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-1

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0

05

1

15

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-35 -3 -25 -2 -15 -1 -05 0 05 1 15 2 25 3 35

1125

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2-225

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15-175

125-15

1-125








variance)






Conclusions












References












Introduction




developed























Cluster Cluster



















0 5000 10000 15000 20000 25000

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04

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08

10

envpp

r

F(r

)obstheohilo

0 5000 10000 15000 20000 25000 30000

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envpp

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obstheohilo












optimum reduced map



be selected






optimum reduced map



be selected





















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area














Mean 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

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-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





-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


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

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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








variance)






Conclusions












References

























Cluster Cluster



















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

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obstheohilo












optimum reduced map



be selected






optimum reduced map



be selected





















-100 -50 0 50 100

-100

-50

050

100


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x x

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x








area














Mean 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

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25

3

35

-35 -25 -15 -05 05 15 25 35 -3

5 -25 -15 -05 05 15 25 35

-35

-15

05

25





-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


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

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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








variance)






Conclusions












References
















Cluster Cluster



















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

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envpp

r

F(r

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obstheohilo












optimum reduced map



be selected






optimum reduced map



be selected





















-100 -50 0 50 100

-100

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050

100


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area














Mean 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

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25

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-35 -25 -15 -05 05 15 25 35 -3

5 -25 -15 -05 05 15 25 35

-35

-15

05

25





-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


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








variance)






Conclusions












References




























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

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obstheohilo












optimum reduced map



be selected






optimum reduced map



be selected





















-100 -50 0 50 100

-100

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050

100


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x

x

xx

x

x

x

xx x

x

x

x

x

x

x








area














Mean 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





-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


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








variance)






Conclusions












References
















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












optimum reduced map



be selected






optimum reduced map



be selected





















-100 -50 0 50 100

-100

-50

050

100


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








area














Mean 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





-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


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








variance)






Conclusions












References





















optimum reduced map



be selected






optimum reduced map



be selected





















-100 -50 0 50 100

-100

-50

050

100


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








area














Mean 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





-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


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








variance)






Conclusions












References






























-100 -50 0 50 100

-100

-50

050

100


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








area














Mean 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





-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


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








variance)






Conclusions












References















-100 -50 0 50 100

-100

-50

050

100


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








area














Mean 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





-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


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








variance)






Conclusions












References
















area














Mean 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





-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


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








variance)






Conclusions












References















Mean 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





-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


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








variance)






Conclusions












References














-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


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








variance)






Conclusions












References

















variance)






Conclusions












References












Conclusions












References












References













Documents

“New trends in statistical methods applied in a semiconductor company”