Final Presentation given at the conclusion of the 2018 IMSM by the Sandia Student Working Group. Group Members: Esther Amfo, Jiahui Chen, Rasika Rajapakshage, Kladji Sinani, John Wakefield and Meng Zhang

1. Computational Techniques to Enhance
Laser-based Characterization of Thin
Film Thermomechanical Properties
Esther Amfo, Jiahui Chen, Rasika Rajapakshage, Klajdi
Sinani, John Wakefield, Meng Zhang
Problem Presenter: Dr. Jordan E. Massad
Faculty Mentor: Drs. Ralph Smith and Paul Miles
Industrial Mathematical & Statistical Modeling Workshop for Graduate Students
Raleigh, NC
July 25, 2018

2. What’s our Problem?
• How can we remove distortion from existing
measurements?
• How can we better understand the deflectometer
through mathematical modeling?
• What can we do to obtain more accurate results in
the future?
𝑅 𝑄𝑁2
𝑅 𝑁𝑄?
How does the thermal enclosure affect laser
deflectometry of thin film warpage?
Team Laserwarp 2

3. Laser Deflectometry 101
Mirror
Detector
Thermal
Enclosure
Quartz Plate
Wafer
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4. Radius from Reflected Angle?
𝑅 sin
𝛼𝑖 − 𝛼
2
= 𝑎 − 𝑥𝑖
𝑅 = −2
𝑑𝛼𝑖
𝑑𝑡𝑖
cos
𝛼𝑖 − 𝛼
2
−1
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5. Beam Path Analysis
𝑥𝑖 𝑡𝑖 = 𝑤1 + 𝑤2 + 𝑤3 + 𝑡𝑖
𝛼𝑖(𝑅 𝑁)=2 asin
𝑎−𝑥 𝑖
𝑅 𝑁
+ 𝛼
𝑅 𝑄 𝑅 𝑁 =
−2
𝐹(𝛼 𝑖( 𝑅 𝑁)) 1−
(𝑎−𝑥 𝑖 𝑡 𝑖 )2
𝑅 𝑁
2
𝐹 𝛼𝑖(𝑅 𝑁) =
𝑑𝛼 𝑖
𝑑𝑡 𝑖
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6. Computational Approach
𝑑 𝑁𝑄
= 𝑆0 ∘ 𝑅 𝛽 𝑆
∘ 𝑇𝐿 (𝛼, 𝑥)
𝑑 𝑄
= 𝑆𝑙1+𝑙2
∘ 𝐼 𝑛 𝑄,𝑛 𝐴
∘ 𝑇𝑙1
∘ 𝐼 𝑛 𝑁,𝑛 𝑄
∘ 𝑇𝑙2
∘ 𝑅 𝛽 𝑆
∘ 𝑇𝑙2
∘ 𝐼 𝑛 𝑄,𝑛 𝑁
∘ 𝑇𝑙1
∘ 𝐼 𝑛 𝐴,𝑛 𝑄
∘ 𝑇𝑙0
(𝛼, 𝑥)
𝑇𝑙(𝜙, 𝑥) = 𝜙, 𝑥 + 𝑙 tan(𝜙)
Translation
Refraction
𝐼 𝑛1,𝑛2
(𝜙, 𝑥) = asin
𝑛1
𝑛2
sin(𝜙) , 𝑥
Reflection
𝑅 𝛽 𝑆
(𝜙, 𝑥) = 𝑆0 ∘ 𝑅 𝛽 𝑆
∘ 𝑇𝐿 (𝛼, 𝑥)
Sensor Position
𝑆ℎ 𝜙, 𝑥 = 𝑥 𝑀cos 𝛽 𝑀 + 𝑏 𝑀 + 𝑊 − 𝑥 𝑀 sin
𝜋
2
− 2 𝛽 𝑀 + 𝜙
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7. Correcting Curvature
Flat
Slope approximately one
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8. Sensitivity Analysis
Height of Quartz Plate
Mirror Height
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9. Data Analysis
• 35 wafer samples measured.
• Multiple tests done on each sample.
• Data can vary with duration between tests.
• Four test configurations identified:
1. Q - uses the quartz window
2. NQ - does not use the quartz window
3. QN2 - uses the quartz window and a nitrogen flow
4. NQN2 - does not use the quartz window, uses nitrogen flow
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10. Effect of Quartz Window?
The quartz makes the wafer happier!
No Quartz Quartz
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R < 0
R > 0
10

11. Statistical Model of Quartz Effect
− 500 − 250 0 250 500 750 1000 1250 1500
RQ (m)
− 2000
− 1500
− 1000
− 500
0
500
1000
1500
2000
RNQ(m)
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𝑅 𝑁𝑄 =
𝑅 𝑄
𝑎 − 𝜀𝑅 𝑄
𝑎 = 1.0040659 𝜀 = 4.92903 × 10−3 m−1

12. Another Factor: Nitrogen Flow
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Thermal Enclosure
• Pure nitrogen flows through the enclosure to maintain
tight thermal and moisture control.
• Observation: Nitrogen flow affects measurements.

13. Quantifying N2 Flow Effect
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𝑅 𝑄 = 𝑐𝑅 𝑄𝑁2
+ 𝑏
c = 1.000443
𝑏 = 6.7 × 10−5
m
A linear response!

14. Mapping 𝑅 𝑄𝑁2
to 𝑅 𝑁𝑄
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N2 Flow Effect
Quartz Plate
Effect
𝑹 𝑵𝑸 =
𝒂 𝟏 𝑹 𝑸𝑵 𝟐
+ 𝒂 𝟐
𝒂 𝟑 − 𝒂 𝟒 𝑹 𝑸𝑵 𝟐
𝑎1 = 1.0004
𝑎2 = 6.7 × 10−5
m
𝑎3 = 1.004
𝑎4 = 4.465 × 10−3 m-1
• 𝑅 𝑄𝑁2
is the measured radius of curvature (warpage)
with quartz plate and N2 flow.
• 𝑅 𝑁𝑄 is the actual warpage of the wafer.

15. Statistical Model Validation
Known
Flat Wafer
Known
Curved Wafer
Measured Radius with
Quartz/N2 Error (m)
[254.50, 265.03] [40.57, 40.77]
Computed Actual Radius (m) [-1925.24, -1479.14] [49.32, 49.62]
Measured Actual Radius (m) [-1925.34, -1479.09] [49.31, 49.63]
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Model predictions are within 1 standard deviation or measurements.
The model is reasonably predictive.

16. Recommendations
• Use the statistical model to estimate the actual
warpage from thermal test measurements.
• To improve the model fit, acquire measurements of
wafers with 16-500 m radius of curvature.
• To continually calibrate the model fit, run a series of
measurements with/without the quartz plate and N2
flow at the beginning and/or end of each thermal test
series.
• The mirror may significantly impact quartz error:
manage uncertainty in its position and orientation.
• Continue developing mathematical models to predict
trends in data
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17. Getting an Angle…
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