1. Lithography Diagnostics Based on Empirical Modeling
SFR Workshop
May 24, 2001
Jiangxin Wang, Costas Spanos
Berkeley, CA
This work proposes a novel approach for diagnostics of recipe
drifts using observations of CD profiles in the lithography
process based on empirical models such as Neural Networks (NN).
12/3/2018

2. System Description 130-136 (°C) PEB Temperature 54-66 (sec) PEB Time
04-0.7
Partial Coherence
(um)
Focus Position
10-15 (mJ/cm²)
Exposure Dose
Inputs Range
Inputs
Silicon Substrate
Si Oxide
Polysilicon
ARC
Resist
Mask
Outputs:
10%CD
50%CD
90%CD
SideWallAngle
Side Wall Angle (SWD)
10%CD
50%CD
90%CD
12/3/2018

3. Simulated Data from PROLITH
2000
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12000
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150
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350
# of data point
50% CD (nm)
Overall Data Set
Use PROLITH to replace experimental data collection for modeling
Advantage: convenient for generating large number of data of different recipes required by model development
Disadvantage: not exactly the same as a real process, must be tuned by experimental data in practice.
200
400
600
800
1000
1200
1400
1600
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2000
100
150
250
300
350
# of data point
50% CD (nm)
exposure dose increase
12/3/2018

4. Modeling Methodologies
Simple, accurate, empirical modeling of lithography process is desired for fast simulation
Linear Regression (LR): Simple; model formula is available: illustrative
Neural Networks (NN)
can approximate any linear/nonlinear functions
may require large number of data points to train (depends on complexity of the problem, PROLITH makes it possible in this work)
black box: not illustrative
The two methods can be used for different purposes.
Comparison of LR and NN modeling results:
Forward Model:
Input Recipes  CD Profiles
Inverse Model:
CD Profiles  Input Recipes
12/3/2018

5. Diagnostics Challenges
1. Ideally, if there exists a unique inverse model, diagnostics would be easier. But, since CD profiles and input recipes are not related in an one-to-one fashion, a unique inverse model may not exist.
Inverse Model Approach
2. The inverse model approach can still be useful when the parameter space can be divided such that each subspace has a unique inverse model. But space segmentation can be difficult due to the high-dimensionality of the parameter space.
Space Segmentation
3. There could be multiple input recipes corresponding to each CD profile. What is an effective way to find out the solution set, analyze all the results, and get the optimal estimates of input parameters.
12/3/2018

6. Diagnostics Based on Forward Modeling
Monte Carlo simulation, generate input parameters uniformly distributed in the ranges that we concern
Use the forward system model by neural networks, use the knowledge of parameter ranges, build a library of inputs-outputs pairs
Evaluate Cost Function for each element in the library
New measurements
Set of matched elements (solution set)
cost<threshold
Analyze the statistical properties of the solution set, determine the best estimation of the input parameters corresponding to the new output measurements Y N
discard
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7. Diagnostic Results I (Single Point Estimation)
Estimate Exposure when all other four inputs are fixed and known
Estimate Exposure and Focus when all inputs are fixed but unknown
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8. Diagnostic Results II (Time Series Estimation)
(1st order autoregressive model)
All other input parameters are fixed and known
Exposure use the same model as above. Focus is fixed but unknown, other input parameters are fixed and known
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9. Summary and Conclusions
Neural Networks can be used to build an accurate forward model.
Diagnostics performance depends on the number of input parameters to be detected. Estimates of input variation can be very accurate when the input monitored is the only unknown parameter.
The proposed diagnostics approach can be used to estimate the time series model parameters of the input parameter drifts and the prediction from the time series model can be helpful to diagnostics.
In the near future,
Simulate more complex time series models (higher order ARMA) for input parameter drifts.
Examine the situation when two or more input parameters are subject to time series drifts at the same time.
Examine the situation when the assumed time series model structure is different from the real one.
Design good test schemes to evaluate the diagnostic methodology.
12/3/2018