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Multivariate Modeling of Batch Processes Using Summary

Variables

Barry M. Wise and Neal B. Gallagher

EIGENVECTOR RESEARCH, INC. Wenatchee, WA USA

bmw@eigenvector.com

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Outline

•  The process monitoring data problem •  Semiconductor Etch Data Set •  Approaches to preprocessing

•  Variable means •  Warping for multi-way •  Data summary

•  Comparison of results •  Polymerization Data Set •  Conclusions

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Batch Process Monitoring Data Problem

•  Messy-typically includes start-up and shut-down phases that are not of interest

•  Periods of “steady-state” where not much is changing

•  Variable record lengths •  Lots of data! •  Reduce to a set of more compact descriptors?

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Plasma Metal Etch

•  Linewidth (Critical Dimension) Control •  Constant linewidth reduction run to run and across wafer •  Constant linewidth reduction for every material in stack

•  Minimal damage to oxide

Silicon

Oxides500Å Ti1000Å TiN

6000Å AlCu (.5%)

500Å TiNResist Resist

Silicon

OxidesTi

TiN

AlCu

TiNResist Resist

TiN

CD

TiN

TiEtch in

Cl2/BCl3Plasma

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Metal Etch Data Set

•  Barna, G.G., White, D., Wise, B.M., Gallagher, N.B., Sofge, D., “Development of Robust Fault Detection and Classification Techniques/SEMATECH J-88E Project at TI”, SEMATECH AEC/APC Workshop VIII, Santa Fe, New Mexico, Oct. 27-30, 1996.

•  White, D., Barna, G.G., Butler, S.W., Wise B., and Gallagher, N., "Methodology for Robust and Sensitive Fault Detection," Electrochemical Society Meeting, Montreal, May, 1997.

•  Gallagher, N.B.,Wise, B.M., Butler, S.W., White, D., and Barna, G.G., "Development and Benchmarking of Multivariate Statistical Process Control Tools for a Semiconductor Etch Process: Improving Robustness Through Model Updating", IFAC ADCHEM'97, Banff, Canada, 78–83, June, 1997.

•  Wise, B.M., Gallagher, N.B., Butler, S.W., White, D., and Barna, G.G., "A Comparison of Principal Components Analysis, Multi-way Principal Components Analysis, Tri-linear Decomposition and Parallel Factor Analysis for Fault Detection in a Semiconductor Etch Process," J. Chemometr., 13, 379–396 (1999).

•  Wise, B.M., Gallagher, N.B., Butler, S.W., White, D., and Barna, G.G., "Development and Benchmarking of Multivariate Statistical Process Control Tools for a Semiconductor Etch Process: Impact of Measurement Selection and Data Treatment on Sensitivity", IFAC SAFEPROCESS'97, 35–42, Kingston Upon Hull, U.K., Aug., 1997.

•  Wise, B.M., Gallagher, N.B., and Martin, E.B., “Application of PARAFAC2 to Fault Detection and Diagnosis in Semiconductor Etch,” J. Chemometr., 15(4), 285–298 (2001).

•  Wise, B.M., Gallagher, N.B., "Multi-way Analysis in Process Monitoring and Modeling," AIChE Symposium Series, 93(316), 271–274 (1997).

•  Warren, J. and Gallagher, N.B., “Heuristic and Statistical Methods for Fault Detection: Complementary or Competing Approaches?”, SEMATECH AEC/APC Symposium XVIII, Westminster, CO, Sept. 30-Oct. 5 (2006).

This data has been used many times and is available at www.eigenvector.com. A few examples:

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Available Measurements •  Regulatory controller setpoints & controlled variable

measured values •  gas flows, pressure, plasma powers

•  Regulatory controller manipulated variables •  exhaust throttle valve, capacitors •  mass flow controller do not provide valve position

•  Additional process measurements •  endpoint intensity (plasma emission at particular frequency) •  impedance measurements

•  Optical emission spectra •  RF plasma variables

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Sensitivity of MSPC Models

•  Three experiments performed with 21 “induced” faults on: •  TCP top power •  RF bottom power •  Cl2 flow •  BCl3 flow •  Chamber pressure •  Helium chuck pressure

•  Goal: Compare ability of models considered for detecting faults

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Global and Local Models

•  Global models based on data over long period of time with considerable variance due to drift •  Tend to be less sensitive to faults

•  Local models build over narrower time windows, less drift variance •  More sensitive to faults

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Generating Faults

•  Setpoints were changed for controlled process variables •  Data for the controlled variable was adjusted to have the

original desired mean •  Result is data that looks like a sensor has developed a bias

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Mean of Each Variable over Wafer Processing Time

•  Take mean of each variable •  Pros:

•  Easy, conceptually simple •  Large data reduction •  No problem with variable record length

•  Cons: •  May miss some types of fault •  Completely lose time information

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Mean Procedure

Can add length of batch variable

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Arrangement of Data for Multiway Analysis

•  Records must be the same length (except PARAFAC2) •  Requires some combination of alignment, truncation or

warping •  Pros

•  Retains time information

•  Cons •  Results in many variables (parsimony problem) •  Complex preprocessing step, requires interpolation •  Model interpretation issues

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Alignment for Multi-way

Problem: How to do first step?

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Alignment and Warping Methods

•  A veritable smorgasbord of methods available •  Dynamic Time Warping (DTW) •  Correlation Optimized Warping (COW) •  Indicator variable/step number •  Linear interpolation •  Align and truncate •  Combinations and variations of the above •  Etc.

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Correlation Optimized Warping

•  Piecewise preprocessing method •  Allows limited changes in segment lengths,

controlled by slack parameter •  Linear interpolation over segments •  Dynamic programming used to optimize

correlation between warped sample and reference •  Less flexible than DTW (unless constrained)

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COW References

•  N.P.V Nielsen, J.M. Carstensen and J. Smedsgaard, “Aligning of single and multiple wavelength chromatographic profiles for chemometric data analysis using correlation optimized warping,” J. Chromatogr. A, 805, 17-35, 1998.

•  G. Tomasi, F. van den Berg and C. Andersson, “Correlation Optimized Warping and Dynamic Time Warping as Preprocessing Methods for Chromatographic Data,” J. Chemometrics, 18, 231-241, 2004.

•  G. Tomasi, T. Skov and F. van den Berg, Warping Toolbox, see: http://www.models.life.ku.dk/source/DTW_COW/index.asp

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Example: COW COW breaks signals into segments and linearly expands or contracts them to optimize correlation

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Hints on COW May be better to calculate warp with 2nd derivative Apply calculated warp to other variables Calculate warp on PCA scores or other latent variable

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Data Summary Approach •  Convert data into alternate set of descriptors •  If process has multiple steps, calculate parameters

that describe each step

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Summary Variables

•  Pros •  Conceptually simple •  Some time information retained •  Noise reduction •  Reduces number of variables (vs. MPCA)

•  Cons •  Further from original data •  May not have step numbers to work with

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Summary Variables Step 1

Step 2

Step 3 Step 1 Step 2 Step 3

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Creation of Pseudo-steps

•  Break reference process trace into “sensible” segments (manually)

•  Assign step numbers •  Warp new data onto reference •  Reverse warp reference step numbers into new

data •  Use summary variables

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Example of Step Creation

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Variation in Step Length

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PCA on Summary Variables

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Loadings for PCA Model

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Compare to MPCA Loadings

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Contribution Plots

Pressure +3 Fault

Pressure -2 Fault

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Results

TLD PARAFAC MPCA PCA/Means PARAFAC2 Summary!Global 11 12 10 10 7 14!Local 14 17 11 16 13 18!

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Faults Caught by PCA on Summary Variables Compared with TLD, PARAFAC, MPCA, PCA on the Means and PARAFAC2. 21 total faults possible

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Polymerization Process

•  10 process variables (sanitized) •  100 time intervals each •  TempR 1, TempR 2, TempR 3, Press 1, Flow 1, TempC

1, TempC 2, Press 2, Press 3, Flow 2

•  Calibration: 1 to 36 (normal batches) •  Test: 37 to 55 (one normal and seven faults) •  Described in

*Nomikos, P. and J.F. MacGregor, “Multivariate SPC charts for monitoring batch processes,” Technometrics, 37(1), 1995.

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Compare Summary Variables to MPCA

•  Summary Variable •  Divided records into 7 steps based on transitions in

variable Flow 1 (control variable) •  Calculated mean value and step length each segment •  10 variables x 7 steps + 7 step lengths = 77 variables

•  MPCA •  Block Scaling •  10 variables x 100 time points = 1000 variables

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Unusual Batches Identified

0 10 20 30 40 50 60101

102

103

104

105

106

107

Sample

Q R

esid

uals

(46.

72%

)

37 39 43 44 45

46 47

48

49

50 51 52 53 54 55

Samples/Scores Plot of Summary of trs & Summary of trs,

Same as in Nomikos & MacGregor

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Compare Loadings

0.3 0.2 0.1 0 0.1 0.2 0.3 0.40.25

0.2

0.15

0.1

0.05

0

0.05

0.1

0.15

0.2

0.25

PC 1 (24.77%)

PC 2

(15.

75%

)

Variables/Loadings Plot for Summary of trs

PC 2 (15.75%)TempR 1TempR 2TempR 3Press 1TempC 1Flow 1TempC 2Press 2Press 3Flow 2Step Length

0.3 0.2 0.1 0 0.1 0.2 0.3 0.40.25

0.2

0.15

0.1

0.05

0

0.05

0.1

0.15

0.2

0.25

PC 1 (24.77%)

PC 2

(15.

75%

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Variables/Loadings Plot for Summary of trs

PC 2 (15.75%)Step 1Step 2Step 3Step 4Step 5Step 6Step 7

0.1 0.08 0.06 0.04 0.02 0 0.02 0.04 0.06 0.08 0.10.06

0.04

0.02

0

0.02

0.04

0.06

0.08

0.1

PC 1 (38.55%)PC

2 (1

1.68

%)

PC 2 (11.68%) TempR 1PC 2 (11.68%) TempR 2PC 2 (11.68%) TempR 3PC 2 (11.68%) Press 1PC 2 (11.68%) Flow 1PC 2 (11.68%) TempC 1PC 2 (11.68%) TempC 2PC 2 (11.68%) Press 2PC 2 (11.68%) Press 3PC 2 (11.68%) Flow 2

Summary PCA MPCA

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Conclusions

•  Digestion into summary variables creates smaller, more manageable and interpretable data sets

•  Segments can be based on process steps, or inflection points in control variables, etc.

•  Warping techniques such as COW can be used to create pseudo-steps

•  Models based on summary variables at least as sensitive as MPCA but easier to work with

•  Less chance of overfitting