Multistep Virtual Metrology Approaches for Semiconductor Manufacturing Processes

Presenter: Simone Pampuri (University of Pavia, Italy)Authors:

Simone Pampuri, University of Pavia, ItalyAndrea Schirru, University of Pavia, Italy

Gian Antonio Susto, University of Padova, ItalyCristina De Luca, Infineon Technologies AT, Austria

Alessandro Beghi, University of Padova, Italy Giuseppe De Nicolao, University of Pavia, Italy

Introduction Collaboration between University of Pavia (Italy), University of

Padova (Italy) and Infineon Technologies AT (Austria) Activity funded by the European project EU-

IMPROVE: Implementing Manufacturing science solutions to increase equiPment pROductiVity and fab pErformance

Introduction Collaboration between University of Pavia (Italy), University of

Padova (Italy) and Infineon Technologies AT (Austria) Activity funded by the European project EU-

IMPROVE: Implementing Manufacturing science solutions to increase equiPment pROductiVity and fab pErformance

Duration: 42 months (since Jan 2009) Global fundings: 37.7 M€ 32 partners, including

• Semiconductor fabs• Academic institutions• Research centers• Software houses

Thematic Work Packages

Motivations1

Machine Learning2

Multilevel framework3

Multistep VM4

55

Contents

Results and Conclusions

What is Virtual Metrology?

In semiconductor manufacturing, measurement operations are costly and time-consuming

Only a small part of the production is actually measured

What is Virtual Metrology?

In semiconductor manufacturing, measurement operations are costly and time-consuming

Only a small part of the production is actually measured Virtual metrology exploits sensors and logistic

information to predict process outcome

Sensor Data

Recipe Data

Logistic Data

VM

What is Virtual Metrology?

In semiconductor manufacturing, measurement operations are costly and time-consuming

Only a small part of the production is actually measured Virtual metrology exploits sensors and logistic

information to predict process outcome

Controllers

Sampling tools

Decision tasks

Sensor Data

Recipe Data

Logistic Data

VM PredictiveInformation

Motivations1

Machine Learning2

Multilevel framework3

Multistep VM4

55

Contents

Results and Conclusions

Machine learning (in a nutshell) Machine learning algorithms create models from observed data

(training dataset), using little or no prior informations about the physical system

Input(X)

Output(Y)

Modelf(X)

LearningAlgorithm

Training dataset

Machine learning (in a nutshell) Machine learning algorithms create models from observed data

(training dataset), using little or no prior informations about the physical system

The model is then able to predict patterns similar to the observed ones

Input(X)

Output(Y)

Modelf(X)

LearningAlgorithm

Training dataset

ModelInput(Xnew)

Prediction(Ynew)

Machine learning (in a nutshell) Machine learning algorithms create models from observed data

(training dataset), using little or no prior informations about the physical system

The model is then able to predict patterns similar to the observed ones

Input(X)

Output(Y)

Modelf(X)

LearningAlgorithm

Training dataset

ModelInput(Xnew)

Prediction(Ynew)

Most famous algorithm:

Ordinary Least Squares (OLS)that consists in solving the optimization

problem defined by the loss function

The curse of dimensionality Problem: the so-called “curse of dimensionality”

Consequence: the predictive power of machine learning models reduces as the number of candidate predictors increases

In semiconductor manufacturing, it is common to have hundreds of candidate predictors: how totackle the problem?

The number of selected predictors grows almost linearly with the number of candidate predictors

The curse of dimensionality Problem: the so-called “curse of dimensionality”

Consequence: the predictive power of machine learning models reduces as the number of candidate predictors increases

In semiconductor manufacturing, it is common to have hundreds of candidate predictors: how totackle the problem?

The number of selected predictors grows almost linearly with the number of candidate predictors

Regularization (or Penalization)methods

The curse of dimensionality Problem: the so-called “curse of dimensionality”

Consequence: the predictive power of machine learning models reduces as the number of candidate predictors increases

The number of selected predictors grows almost linearly with the number of candidate predictors

19431943Ridge (or Tikhonov) regression: in order to improve the least squares method, stable (“easier”) solutions are encouraged by penalizing coefficients through the parameter a

The curse of dimensionality Problem: the so-called “curse of dimensionality”

Consequence: the predictive power of machine learning models reduces as the number of candidate predictors increases

The number of selected predictors grows almost linearly with the number of candidate predictors

19431943Ridge (or Tikhonov) regression: in order to improve the least squares method, stable (“easier”) solutions are encouraged by penalizing coefficients through the parameter a

• Best value for hyperparameter is chosen via validation

• Computationally easy (closed form solution)

• No sparse solution

The curse of dimensionality Problem: the so-called “curse of dimensionality”

Consequence: the predictive power of machine learning models reduces as the number of candidate predictors increases

1996 –today

1996 –today

L1-penalized methods: by constraining the solution to belong to an hyper-octahedron, sparse models can be obtained (variable selection). Most famous example: LASSO

The number of selected predictors grows almost linearly with the number of candidate predictors

The curse of dimensionality Problem: the so-called “curse of dimensionality”

Consequence: the predictive power of machine learning models reduces as the number of candidate predictors increases

1996 –today

1996 –today

L1-penalized methods: by constraining the solution to belong to an hyper-octahedron, sparse models can be obtained (variable selection). Most famous example: LASSO

The number of selected predictors grows almost linearly with the number of candidate predictors

• Best value for hyperparameter is chosen via validation

• Sparse solution (variable selection)

• Solved by iterative algorithms (e.g. SMO)

Motivations1

Machine Learning2

Multilevel framework3

Multistep VM4

55

Contents

Results and Conclusions

The hierarchical variability We deal every day with multiple levels of variability:

Every equipment has several chambers In some cases, these chambers are splitted in sub-chambers Different process groups, recipes run on the same equipment

The hierarchical variability We deal every day with multiple levels of variability:

Every equipment has several chambers In some cases, these chambers are splitted in sub-chambers Different process groups, recipes run on the same equipment

Simple (“naive”) solution: create one model for every possible combination of factors We’ll never have enough data to that,

especially for low volume recipes

The hierarchical variability We deal every day with multiple levels of variability:

Every equipment has several chambers In some cases, these chambers are splitted in sub-chambers Different process groups, recipes run on the same equipment

Simple (“naive”) solution: create one model for every possible combination of factors We’ll never have enough data to that,

especially for low volume recipes

Better solution: handle those different levels of variability inside the model

The hierarchical variability We deal every day with multiple levels of variability:

Every equipment has several chambers In some cases, these chambers are splitted in sub-chambers Different process groups, recipes run on the same equipment

Simple (“naive”) solution: create one model for every possible combination of factors We’ll never have enough data to that,

especially for low volume recipes

Better solution: handle those different levels of variability inside the model

Multilevel Techniques:

Multilevel Ridge Regression (RR)&

Multilevel Lasso

First step is to create an extended input matrix to reflect the relationships between the j clusters. For instance, in the case of j mutually exclusive nodes,

The input matrix reflects the dependency on logistic paths

The Multilevel Transform

Motivations1

Machine Learning2

Multilevel framework3

Multistep VM4

Results and Conclusions55

Contents

Standard scenario Production flow: sequence of steps; each step

represents an operation that must be performed on a wafer in order to obtain a specific results

Each step is performed by different equipment (composed by multiple chambers): The knowledge of which wafer is processed by a specific

equipment is available (logistic information) The information about processed wafer (e.g. sensor

readings and recipe setup) might be available On some equipments a “single step” VM system is already

in place (estimated measures for each processed wafer are available)

Cascade Multistep VM This approach allow to build a pipe system in which the

predictive information is propagated forward to concur to further model estimation.

The generation of multilevel input matrix consists in replace j-th cluster’s process variables with j-th VM-j estimation

Cascade Multistep VM This approach allow to build a pipe system in which the

predictive information is propagated forward to concur to further model estimation.

The generation of multilevel input matrix consists in replace j-th cluster’s process variables with j-th VM-j estimation

Pros:

o Small overhead append to the input space

o Computational effort very similar to “single step” VM case

Cons:

o Steps without “single step” VM must be excluded

o There might be some information loss between two or more steps

Process and Logistic Multistep VM With this approach, all the relevant logistic, process and

recipe information from all the considered steps is included in the input set

In this case, the generation of input matrix fully follows the previous Multilevel Transform

Process and Logistic Multistep VM With this approach, all the relevant logistic, process and

recipe information from all the considered steps is included in the input set

In this case, the generation of input matrix fully follows the previous Multilevel Transform

Pros:

o Steps with no (or meaningless) measurements can be included

o All the available information is provided to the learning algorithm

Cons:

o Input space dimension is significantly increased by this approach

o More observations are needed to train the learning algorithm

Contents

Motivations1

Machine Learning2

Multilevel framework3

Multistep VM4

Results and Conclusions55

Production flow for methodologies validation:1. Chemical Vapor Deposition (CVD)

2. Thermal Oxidation

3. Coating

4. Lithography Target: post-litho CDs Dataset: 583 wafers anonymized Hyper-parameter tuning: 10-fold crossvalidation Multistep VM setups:

CVD-Litho Cascade CVD-Litho Process and Full Logistic

Scenario

Cascade

The cascade VM allows to further improve the VM performances using RR. This result might be related to the additional hidden knowledge provided by the intermediate CVD metrology prediction.

The cascade approach performs worse with the LASSO. It should be noted that this is the only case in which the extended input space does not improve the predictive performances.

Process and Full Logistic

Validation RMSE results for Ridge Regression: it is apparent how the full step choice allows to improve the predictive performances.

LASSO is consistently outperformed by Ridge Regression in the dataset that was used for the experiment; nevertheless, the extended input space proves to be fruitful also in this case, with respect to the Lithography based approach.

Best Lasso and Best RR

The best overall results for Ridge Regression are obtained with thecascade approach and by considering all the process steps.

For the LASSO, the best overall results are obtained by considering the extended process values for all the involved steps.

Research and design of Multistep VM strategies targeted to specific semiconductor manufacturing needs

Main features: Enhancing precision and accuracy of regular VM system Taking in account process without measurements

Tests showed promising results; however, the strategy to be implemented must be carefully designed: Sample size and relevance of the steps are fundamental

criteria to obtain the best performances

Conclusions

www.themegallery.com

Thanks for your attention!

Authors:Simone Pampuri, University of Pavia, Italy

Andrea Schirru, University of Pavia, ItalyGian Antonio Susto, University of Padova, Italy

Cristina De Luca, Infineon Technologies AT, AustriaAlessandro Beghi, University of Padova, Italy

Giuseppe De Nicolao, University of Pavia, Italy

Presenter: Simone Pampuri (University of Pavia, Italy)