An extended overlay assessment model with process correlation analysis for sub-100-nm accuracy wafer bonding

Rui Wang; Sen Lu; Kaiming Yang; Yu Zhu

doi:10.1088/1674-4926/25120038

J. Semicond. > In Press

ARTICLES

An extended overlay assessment model with process correlation analysis for sub-100-nm accuracy wafer bonding

Rui Wang¹, Sen Lu^{1, 2}, Kaiming Yang^1, and Yu Zhu^{1, 2}

+ Author Affiliations

1. Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China
2. State Key Laboratory of Tribology in Advanced Equipment, Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China

Author Bio:

Rui Wang is studying for a Ph.D. in Mechanical Engineering in Tsinghua University. He received his BS degree in Mechanical Engineering from Tsinghua University in 2019. His main research direction is Accuracy Guarantee of Alignment and Bonding technology for wafer stacking equipment.

Sen Lu is an assistant professor at the Department of Mechanical Engineering at Tsinghua University. He received his PhD in mechanical engineering from Tsinghua University in 2019. His current interests include ultra-precision measurement and control technology as well as 3D IC packaging technology.

Kaiming Yang is a professor at the Department of Mechanical Engineering at Tsinghua University. He received his BS and MS degrees in Mechanical Engineering from Zhengzhou University in 1995 and 1998, respectively, and his PhD in Mechanical Manufacturing and Automation from Tsinghua University in 2005. His research areas include ultra-precision motion control, computerized numerical control, and mechatronic equipment control.

Yu Zhu graduated from the China University of Mining and Technology in 2001 with a doctoral degree. From July 2001 to September 2004, he worked as a postdoctoral fellow at Tsinghua University. He is the head of the Institute of Mechanical Electronics at the Department of Mechanical Engineering at Tsinghua University and a leader in the field of IC equipment at Tsinghua University. His research interests include dynamical system design and analysis theory for ultra-precision systems, displacement measurement and motion control technology for nano-precision systems, and development strategies for IC manufacturing equipment.

Corresponding author: Kaiming Yang, yangkm@tsinghua.edu.cn

DOI: 10.1088/1674-4926/25120038CSTR: 32376.14.1674-4926.25120038

Abstract: To address the demand for sub-100-nm overlay accuracy in wafer bonding for 3D integration, this study proposes an extended overlay assessment model integrating physical mechanisms and data-driven approaches, along with a correlation analysis methodology with process parameters. Rigid-body models inadequately characterize systematic deformations from crystalline anisotropy and process stresses. To overcome this, we construct an extended overlay model based on Zernike polynomials, incorporating physically meaningful terms for precise description of non-uniform wafer deformation. An innovative Zernike term selection strategy combining physics-guided pre-screening and AIC-optimized stepwise regression resolves overfitting/underfitting, enhancing generalizability and interpretability. Validation using Patterned Wafer Geometry (PWG) data shows the model achieves R² > 0.70 for both net bonding deformation and lithography-compensable components, demonstrating excellent deformation decomposition. Correlation analysis of multiple process experiments reveals strong correlations (|r| > 0.85) between key process parameters (e.g., peak bonding head force) and specific Zernike modes, providing evidence for suppressing detrimental deformations via process optimization. This research establishes a complete framework from theory to experimental verification and process traceability, laying a foundation for mechanism diagnosis, predictive compensation, and closed-loop control in high-precision wafer bonding.

Key words: wafer bonding, overlay assessment, anisotropic deformation, Zernike polynomials, process correlation

References

[1]	Panigrahi A K, Bonam S, Ghosh T, et al. Low temperature, low pressure CMOS compatible Cu-Cu thermo-compression bonding with Ti passivation for 3D IC integration. 2015 IEEE 65th Electronic Components and Technology Conference (ECTC), 2015: 2205 doi: 10.1109/ECTC.2015.7159909
[2]	Mitsuishi H, Mori H, Maeda H, et al. 50 nm overlay accuracy for wafer-to-wafer bonding by high-precision alignment technologies. 2023 IEEE 73rd Electronic Components and Technology Conference (ECTC), 2023: 1664 doi: 10.1109/ECTC51909.2023.00283
[3]	Rebhan B, Bernauer M, Wagenleitner T, et al. 200 nm Wafer-to-wafer overlay accuracy in wafer level Cu/SiO2 hybrid bonding for BSI CIS. 2015 IEEE 17th Electronics Packaging and Technology Conference (EPTC), 2015: 1 doi: 10.1109/EPTC.2015.7412403
[4]	McNally P J. B-Spline X-Ray Diffraction Imaging techniques for die warpage and stress monitoring inside fully encapsulated packaged chips. 2015 16th International Conference on Thermal, Mechanical and Multi-Physics Simulation and Experiments in Microelectronics and Microsystems, 2015: 1 doi: 10.1109/EuroSimE.2015.7103166
[5]	van Dijk L, Mileham J, Malakhovsky I, et al. Wafer-shape based in-plane distortion predictions using Superfast 4G metrology. Metrol Insp Process Control Microlithogr XXXI, 2017, 10145: 101452L doi: 10.1117/12.2257475
[6]	Das H, Sunkari S, Justice J, et al. Detection of dislocations using X-ray diffraction imaging (topography) KOH etching and their evolution after epitaxial growth in 4H-SiC. ECS Trans, 2021, 104(7): 141 doi: 10.1149/ma2021-02341005mtgabs
[7]	Savchuk O, Volinsky A A. Nonparametric estimation of SiC film residual stress from the wafer surface profile. Measurement, 2021, 177: 109238 doi: 10.1016/j.measurement.2021.109238
[8]	Dedkova A A, Florinsky I V, Djuzhev N A. Approaches to determining curvature of wafers by their topography. Phys Usp, 2022, 65: 706 doi: 10.3367/ufnr.2021.10.039076
[9]	Chen X W, Yue W V. Residual stress concentration due to nano-scaled particulate contamination at direct bonding interface with localized material inhomogeneity. J Elast, 2024, 156(4): 1121 doi: 10.1007/s10659-024-10089-2
[10]	Tseng M L, Gorji N E. Metrology of warpage in silicon wafers using X-ray diffraction mapping. IEEE Trans Compon, Packag Manuf Technol, 2025, 15(7): 1523 doi: 10.1109/TCPMT.2025.3557270
[11]	Zhang X D, Han Z G, Zhao L, et al. Optimized characterization model of curvature radius-stress for wafer thin films. Acta Opt Sin, 2025, 45(16): 1612001 doi: 10.3788/AOS250950
[12]	Ju J, Kim M, Lee J, et al. Application of overlay modeling and control with Zernike polynomials in an HVM environment. Metrol Insp Process Control Microlithogr XXX, 2016, 9778: 977825 doi: 10.1117/12.2219739
[13]	Zhang L B, Feng Y B, Song Z, et al. Zernike model for overlay control and tool monitor for lithography and etch process. J Vac Sci Technol B, 2022, 40(6): 062604 doi: 10.1116/6.0002239
[14]	Duclaux B, Boustheen A, Pastol A, et al. Overcoming challenges raised by wafer load grid overlay fingerprints and correction per exposure management. Metrology, Inspection, and Process Control XXXIX, 2025: 110 doi: 10.1117/12.3050691
[15]	Ohri A, Taylor T L, Temchenko V, et al. Zernike-based photolithography track modules matching for wafer CDU. Metrology, Inspection, and Process Control XXXIX, 2025: 37 doi: 10.1117/12.3051932

Fig. 1. (Color online) Comparison of the first 18 Zernike polynomial wafer distribution maps and the characteristic post-bonding residual distribution for a (100) silicon wafer.

DownLoad: Full-Size Img PowerPoint

Fig. 2. (Color online) Optimally selected Zernike model for bonding residuals under a specific process configuration. (a) Residual distribution of the traditional model; (b) residual distribution of the optimally selected extended model; (c) extracted systematic deformation field; (d) model selection process based on the AIC criterion; (e) optimal Zernike coefficients; (f) residual distribution statistics.

DownLoad: Full-Size Img PowerPoint

Fig. 3. (Color online) Example of PWG data. (a) Af; (b) Ab; (c) Bf; (d) Bb; (e) A2Bfn; (f) A2Bbn; (g) A2Bf6; (h) A2Bb6.

DownLoad: Full-Size Img PowerPoint

Fig. 4. (Color online) Occurrence frequency of the optimally selected Zernike terms across multiple wafers (N = 15).

DownLoad: Full-Size Img PowerPoint

Fig. 5. (Color online) Comparison of fitting quality for pwg measurement data.

DownLoad: Full-Size Img PowerPoint

Fig. 6. (Color online) Comparison offitting results and correlation analysis for PWG measurement data. (a). Comparison Between Systematic Deformation and Zernike-Fitting Results; (b). Zernike Term Correlation Analysis.

DownLoad: Full-Size Img PowerPoint

Fig. 7. (Color online) Schematic diagram of the upper chuck's negative pressure zones.

DownLoad: Full-Size Img PowerPoint

Fig. 8. (Color online) Heatmap of the correlation between process features and Zernike coefficients.

DownLoad: Full-Size Img PowerPoint

Table 1. Comparison of fitting results for the three zernike models.

Model	Selected Terms	k	Residual RMS (nm)	RMSE_CV (nm)	AIC
Traditional full-term	Z₁−Z₃₀	30	0.58	1.21	−707911
Empirical-term	Z₅, Z₆, Z₁₇, Z₁₈	4	3.32	3.57	−542216
Optimally selected	Z₅, Z₇, Z₈, Z₉, Z₁₀, Z₁₃, Z₁₆, Z₁₇	8	0.88	0.91	−668943

DownLoad: CSV

Table 2. Comprehensive performance comparison of the three overlay extension models across multiple wafers.

Model	Fitted residual RMS (nm)	Average cross-validation RMSE (nm)	RMSE_CV standard deviation (nm)	Average AIC	Single-pair fitting time (s)
Traditional full-term	0.62	1.19	0.24	−712333	12.5
Empirical-term	3.18	3.55	0.38	−523879	0.8
Optimally selected	0.83	0.93	0.12	−687631	5.2

DownLoad: CSV

Table 3. Correlation coefficient matrix between key process features and Zernike coefficients (partial). Each cell shows the Pearson correlation coefficient r with the corresponding p-value in parentheses.

Process feature	Z₈	Z₉	Z₁₀	Z₁₁	Z₁₅	Z₁₆	Z₁₈	Z₁₉
Peak bonding head force	0.401 (0.088)	−0.684 (0.002)	0.640 (0.004)	0.852 (<0.001)	−0.553 (0.017)	0.589 (0.010)	0.620 (0.006)	0.517 (0.025)
Zone_1 pressure hold time	−0.401 (0.088)	0.684 (0.002)	−0.640 (0.004)	−0.852 (<0.001)	0.553 (0.017)	−0.589 (0.010)	−0.620 (0.006)	−0.517 (0.025)
Max. change rate of Zone_3 pressure	−0.401 (0.088)	0.684 (0.002)	−0.640 (0.004)	−0.852 (<0.001)	0.553 (0.017)	−0.589 (0.010)	−0.620 (0.006)	−0.517 (0.025)

DownLoad: CSV

Table 4. Correlation coefficients between process features and deformation symmetry groups.

Process feature	Rotational symmetry	Four-fold symmetry	Eight-fold symmetry
Peak bonding head force	0.009	−0.064	−0.268
Zone_1 pressure hold time	−0.009	0.064	0.268
Zone_3 average pressure	0.009	−0.064	−0.268

DownLoad: CSV

[1]	Panigrahi A K, Bonam S, Ghosh T, et al. Low temperature, low pressure CMOS compatible Cu-Cu thermo-compression bonding with Ti passivation for 3D IC integration. 2015 IEEE 65th Electronic Components and Technology Conference (ECTC), 2015: 2205 doi: 10.1109/ECTC.2015.7159909
[2]	Mitsuishi H, Mori H, Maeda H, et al. 50 nm overlay accuracy for wafer-to-wafer bonding by high-precision alignment technologies. 2023 IEEE 73rd Electronic Components and Technology Conference (ECTC), 2023: 1664 doi: 10.1109/ECTC51909.2023.00283
[3]	Rebhan B, Bernauer M, Wagenleitner T, et al. 200 nm Wafer-to-wafer overlay accuracy in wafer level Cu/SiO2 hybrid bonding for BSI CIS. 2015 IEEE 17th Electronics Packaging and Technology Conference (EPTC), 2015: 1 doi: 10.1109/EPTC.2015.7412403
[4]	McNally P J. B-Spline X-Ray Diffraction Imaging techniques for die warpage and stress monitoring inside fully encapsulated packaged chips. 2015 16th International Conference on Thermal, Mechanical and Multi-Physics Simulation and Experiments in Microelectronics and Microsystems, 2015: 1 doi: 10.1109/EuroSimE.2015.7103166
[5]	van Dijk L, Mileham J, Malakhovsky I, et al. Wafer-shape based in-plane distortion predictions using Superfast 4G metrology. Metrol Insp Process Control Microlithogr XXXI, 2017, 10145: 101452L doi: 10.1117/12.2257475
[6]	Das H, Sunkari S, Justice J, et al. Detection of dislocations using X-ray diffraction imaging (topography) KOH etching and their evolution after epitaxial growth in 4H-SiC. ECS Trans, 2021, 104(7): 141 doi: 10.1149/ma2021-02341005mtgabs
[7]	Savchuk O, Volinsky A A. Nonparametric estimation of SiC film residual stress from the wafer surface profile. Measurement, 2021, 177: 109238 doi: 10.1016/j.measurement.2021.109238
[8]	Dedkova A A, Florinsky I V, Djuzhev N A. Approaches to determining curvature of wafers by their topography. Phys Usp, 2022, 65: 706 doi: 10.3367/ufnr.2021.10.039076
[9]	Chen X W, Yue W V. Residual stress concentration due to nano-scaled particulate contamination at direct bonding interface with localized material inhomogeneity. J Elast, 2024, 156(4): 1121 doi: 10.1007/s10659-024-10089-2
[10]	Tseng M L, Gorji N E. Metrology of warpage in silicon wafers using X-ray diffraction mapping. IEEE Trans Compon, Packag Manuf Technol, 2025, 15(7): 1523 doi: 10.1109/TCPMT.2025.3557270
[11]	Zhang X D, Han Z G, Zhao L, et al. Optimized characterization model of curvature radius-stress for wafer thin films. Acta Opt Sin, 2025, 45(16): 1612001 doi: 10.3788/AOS250950
[12]	Ju J, Kim M, Lee J, et al. Application of overlay modeling and control with Zernike polynomials in an HVM environment. Metrol Insp Process Control Microlithogr XXX, 2016, 9778: 977825 doi: 10.1117/12.2219739
[13]	Zhang L B, Feng Y B, Song Z, et al. Zernike model for overlay control and tool monitor for lithography and etch process. J Vac Sci Technol B, 2022, 40(6): 062604 doi: 10.1116/6.0002239
[14]	Duclaux B, Boustheen A, Pastol A, et al. Overcoming challenges raised by wafer load grid overlay fingerprints and correction per exposure management. Metrology, Inspection, and Process Control XXXIX, 2025: 110 doi: 10.1117/12.3050691
[15]	Ohri A, Taylor T L, Temchenko V, et al. Zernike-based photolithography track modules matching for wafer CDU. Metrology, Inspection, and Process Control XXXIX, 2025: 37 doi: 10.1117/12.3051932

Search

GET CITATION

shu

Export: BibTex EndNote

Article Metrics

Article views: 256 Times PDF downloads: 52 Times Cited by: 0 Times

History

Received: 20 December 2025 Revised: 21 January 2026 Online: Accepted Manuscript: 02 March 2026Uncorrected proof: 06 March 2026

Article Navigation > Journal of Semiconductors > 2026 > Uncorrected proof

Rui Wang, Sen Lu, Kaiming Yang, Yu Zhu. An extended overlay assessment model with process correlation analysis for sub-100-nm accuracy wafer bonding[J]. Journal of Semiconductors, 2026, In Press. doi: 10.1088/1674-4926/25120038 ****R Wang, S Lu, K M Yang, and Y Zhu, An extended overlay assessment model with process correlation analysis for sub-100-nm accuracy wafer bonding[J]. J. Semicond., 2026, accepted doi: 10.1088/1674-4926/25120038

Citation:

Rui Wang, Sen Lu, Kaiming Yang, Yu Zhu. An extended overlay assessment model with process correlation analysis for sub-100-nm accuracy wafer bonding[J]. Journal of Semiconductors, 2026, In Press. doi: 10.1088/1674-4926/25120038 ****

R Wang, S Lu, K M Yang, and Y Zhu, An extended overlay assessment model with process correlation analysis for sub-100-nm accuracy wafer bonding[J]. J. Semicond., 2026, accepted doi: 10.1088/1674-4926/25120038

Citation:

PDF( 13716 KB)

An extended overlay assessment model with process correlation analysis for sub-100-nm accuracy wafer bonding

DOI: 10.1088/1674-4926/25120038

CSTR: 32376.14.1674-4926.25120038

Rui Wang¹,
Sen Lu^{1, 2},
Kaiming Yang^1,,
Yu Zhu^{1, 2}

1.
Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China
2.
State Key Laboratory of Tribology in Advanced Equipment, Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China

More Information

Rui Wang is studying for a Ph.D. in Mechanical Engineering in Tsinghua University. He received his BS degree in Mechanical Engineering from Tsinghua University in 2019. His main research direction is Accuracy Guarantee of Alignment and Bonding technology for wafer stacking equipment
Sen Lu is an assistant professor at the Department of Mechanical Engineering at Tsinghua University. He received his PhD in mechanical engineering from Tsinghua University in 2019. His current interests include ultra-precision measurement and control technology as well as 3D IC packaging technology
Kaiming Yang is a professor at the Department of Mechanical Engineering at Tsinghua University. He received his BS and MS degrees in Mechanical Engineering from Zhengzhou University in 1995 and 1998, respectively, and his PhD in Mechanical Manufacturing and Automation from Tsinghua University in 2005. His research areas include ultra-precision motion control, computerized numerical control, and mechatronic equipment control
Yu Zhu graduated from the China University of Mining and Technology in 2001 with a doctoral degree. From July 2001 to September 2004, he worked as a postdoctoral fellow at Tsinghua University. He is the head of the Institute of Mechanical Electronics at the Department of Mechanical Engineering at Tsinghua University and a leader in the field of IC equipment at Tsinghua University. His research interests include dynamical system design and analysis theory for ultra-precision systems, displacement measurement and motion control technology for nano-precision systems, and development strategies for IC manufacturing equipment
Corresponding author: yangkm@tsinghua.edu.cn
Received Date: 2025-12-20
Revised Date: 2026-01-21

Available Online: 2026-03-02

Abstract

Abstract

To address the demand for sub-100-nm overlay accuracy in wafer bonding for 3D integration, this study proposes an extended overlay assessment model integrating physical mechanisms and data-driven approaches, along with a correlation analysis methodology with process parameters. Rigid-body models inadequately characterize systematic deformations from crystalline anisotropy and process stresses. To overcome this, we construct an extended overlay model based on Zernike polynomials, incorporating physically meaningful terms for precise description of non-uniform wafer deformation. An innovative Zernike term selection strategy combining physics-guided pre-screening and AIC-optimized stepwise regression resolves overfitting/underfitting, enhancing generalizability and interpretability. Validation using Patterned Wafer Geometry (PWG) data shows the model achieves R² > 0.70 for both net bonding deformation and lithography-compensable components, demonstrating excellent deformation decomposition. Correlation analysis of multiple process experiments reveals strong correlations (|r| > 0.85) between key process parameters (e.g., peak bonding head force) and specific Zernike modes, providing evidence for suppressing detrimental deformations via process optimization. This research establishes a complete framework from theory to experimental verification and process traceability, laying a foundation for mechanism diagnosis, predictive compensation, and closed-loop control in high-precision wafer bonding.
- wafer bonding,
- overlay assessment,
- anisotropic deformation,
- Zernike polynomials,
- process correlation

FullText(HTML)

References(15)