Sampling Decision System Daniel Kurz Department of Statistics, University of Klagenfurt in semiconductor manufacturing using Virtual Metrology Cristina.

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Presentation transcript:

Sampling Decision System Daniel Kurz Department of Statistics, University of Klagenfurt in semiconductor manufacturing using Virtual Metrology Cristina De Luca Infineon Technologies Austria Jürgen Pilz Department of Statistics, University of Klagenfurt 8th IEEE International Conference on Automation, Science and Engineering (CASE) 2012 Seoul, South Korea 8/21/2012

Copyright © Infineon Technologies All rights reserved. Contents Motivation and background ¬ Sampling Design and Virtual Metrology (VM) The Sampling Decision System (SDS)  SDS Foundations ¬ The decision-theoretical control chart model ¬ Handling Virtual Metrology information  SDS Principles ¬ The Expected Value of Measurement Information (EVofMI) ¬ The Two-Stage Sampling Decision Model ¬ The Multivariate SDS  SDS Extensions ¬ Wafer Quality Risk Values ¬ SDS Reliabillity and Bayesian Covariance Updating Experimental Results Conclusion and Outline 8/21/2012Page 2

Motivation Cost-intensive and time-consuming measurement operations ¬ Only a certain part of production can actually be measured ¬ Statical measurement rules Control Chart based Sampling Designs Page 3Copyright © Infineon Technologies All rights reserved. Virtual Metrology (VM) Skipping of lot/wafer measurements Predictions of real metrology outcomes Sampling Decision System (SDS) Virtual measurement Wafer Sampling Decision 8/21/2012

Modeling a Sampling Decision System Motivation: make use of virtual measurements in order to obtain wafer-fine sampling decisions Wafer measurements are taken in order to control output-quality of some process operation Classical - control chart: (exchangeability assumption) Now: additional (prior) information on by means of VM: Consequences of wrong decisions Copyright © Infineon Technologies All rights reserved. Decision-theoretical control chart model Page 48/21/2012

The control chart in a decision-theoretical framework Actions: States of nature/wafer: Loss function : Basic questions: ¬ What do we expect to gain from a measurement with respect to a certain decision situation? ¬ Is it worth to take a measurement or is it possible to decide on the status of the wafer just based on obtained VM information? Copyright © Infineon Technologies All rights reserved.Page 58/21/2012

The Expected Value of Measurement Information (EVofMI) Obtain optimal decision based on the virtual measurement by means of prior expected loss: Copyright © Infineon Technologies All rights reserved. virtual measurement IC/OOC prior probabilities prior expected loss prior decision value (prior risk) Page 68/21/2012

The Expected Value of Measurement Information (EVofMI) Obtain optimal action after observing sample mean by means of Bayes decision rule: Copyright © Infineon Technologies All rights reserved. posterior distribution IC/OOC posterior probabilities posterior expected loss posterior decision value (Bayes risk) Page 7 8/21/2012

The Expected Value of Measurement Information (EVofMI) The Value of Measurement Information: Actually: we have to decide before is available if it is needed or not The EVofMI: Sampling rule using measurement loss factor : Copyright © Infineon Technologies All rights reserved. marginal (prior predictive) distribution EVofMI Page 88/21/2012

The Two-Stage Sampling Decision Model The decision tree: Copyright © Infineon Technologies All rights reserved. Backwards induction Sampling rule Page 98/21/2012

Extensions of the Sampling Decision System SDS for multivariate virtual measurements: ¬ Consideration of the status of several wafer sites and abnormal variations ¬ Assessment of IC probabilities with respect to multivariate control ellipsoids ¬ Multivariate generalizations of marginal and posterior distribution for Wafer Quality Risk: ¬ Computation of an expected quality loss exploiting virtual measurements and an inverted normal loss function Wafers At Risk and cumulated decision values (avoiding too long skipping periods) Copyright © Infineon Technologies All rights reserved.Page 108/21/2012

VM Reliability and Bayesian Covariance Updating Conjugate Wishart prior distribution for VM precision matrix: Copyright © Infineon Technologies All rights reserved. initially computed VM covariance matrix Updated VM precision matrix Updated degrees of freedom : Bayesian learning Wishart posterior distribution Page 11 real measurement 8/21/2012

Some experimental results Copyright © Infineon Technologies All rights reserved. PRODUCER3-A1PRODUCER3-A2 PRODUCER3-B1PRODUCER3-B2 PRODUCER3-C1 Page 12 VM precision Process parameters 8/21/2012

EVofMI Copyright © Infineon Technologies All rights reserved. PRODUCER3-A1PRODUCER3-A2 PRODUCER3-B1PRODUCER3-B2 PRODUCER3-C1 Page 138/21/2012

The Two-Stage Sampling Decision Model Copyright © Infineon Technologies All rights reserved. PRODUCER3-A1PRODUCER3-A2 PRODUCER3-B1PRODUCER3-B2 PRODUCER3-C1 Page 148/21/2012

Wafer quality risk Copyright © Infineon Technologies All rights reserved. PRODUCER3-A1 PRODUCER3-A2 PRODUCER3-B1PRODUCER3-B2 PRODUCER3-C1 Real wafer measurement information Bayesian covariance updating Effect on future decisions Page 158/21/2012

Process drift simulation Copyright © Infineon Technologies All rights reserved. Process scenario EVofMI Two-stage sampling decision model Page 168/21/2012

Conclusion and outline SDS evaluates if it is possible to decide on the status of some wafer (process) with respect to the virtual measurement ¬ Stable process phases: skipping of a large number of real wafer measurements ¬ Process abnormalities: can be detected earlier (wafer-fine VM data) real wafer measurements suggested New sampling principle: measure nothing except those wafers/lots that are really needed to be measured SDS performance improves with accuracy of VM system Copyright © Infineon Technologies All rights reserved.Page 178/21/2012

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