Monitoring wafers’ geometric quality using an additive Gaussian process model

ABSTRACT

The geometric quality of a wafer is an important quality characteristic in the semiconductor industry. However, it is difficult to monitor this characteristic during the manufacturing process due to the challenges created by the complexity of the data structure. In this article, we propose an Additive Gaussian Process (AGP) model to approximate a standard geometric profile of a wafer while quantifying the deviations from the standard when a manufacturing process is in an in-control state. Based on the AGP model, two statistical tests are developed to determine whether or not a newly produced wafer is conforming. We have conducted extensive numerical simulations and real case studies, the results of which indicate that our proposed method is effective and has potentially wide application.

Keywords:

• Geometric quality
• Gaussian process
• T² test
• generalized likelihood ratio test

Acknowledgement

We would like to thank the reviewers and the editor for their constructive comments.

Funding

Kaibo Wang is partially supported by the National Natural Science Foundation of China under grants 71471096 and 71072012. Nan Chen is partially supported by the Singapore AcRF grant R-266-000-078-112.

Appendixes

A. Maximum profile likelihood of the AGP model

To estimate the parameters of the AGP model from in-control measurements, we need to maximize the likelihood function (9(9) ). However, direct optimization is easily trapped in local optima. The scales of each dimension are also generally quite different, making the optimization more difficult. To improve the optimization performance, we can reduce its dimension by maximizing the profile likelihood.

In more details, given and , the two correlation matrices are completely determined. The first correlation matrix is denoted as S, with elements . Because of the independence of ε_i(x) for different i, the second correlation matrix is a block diagonal matrix , where V_i has in each entry. According to the physics of the process and observed data, often σ² > τ² and , where the inequality between the two vectors is interpreted using an element-wise comparison. This is because the standard profile often has a larger variation but smoother transitions compared with the deviation profile due to process variations. Consequently, we can define τ² = ρ × σ², with 0 ≤ ρ ≤ 1.

Using these notations, the log-likelihood with respect to μ, σ² and ρ can be expressed as (A1) Taking the partial derivative of l_r and setting the gradient to zero results in (A2) where Tr(S) denotes the trace of the matrix S. The first two expressions in Equation (A2(A2) ) are self-explanatory, and the third one can be transformed to (A3) by plugging in the expression for σ². When M₀ is large, the inverse of S + ρV may still take some time for each different ρ. We can significantly shorten the computational time by noting that , where is the identity matrix of dimension M₀ × M₀, and V = V^1/2 × V^1/2. Taking the singular value decomposition , we have . As a result, ρ only appears in the diagonal matrix , and all of the computationally intensive operations such as Cholesky decomposition, singular value decomposition, and most matrix multiplications only need to be calculated once for different ρ.

Using this computationally efficient procedure, we can find the solution to Equation (A3(A3) ) in the interval [0, 1]. If no solution exists in this interval, one of the end points ρ = 0, 1 with largest likelihood value will be selected. Denoting as the value selected that maximizes l_r, we can obtain based on the first two expressions in Equation (A2(A2) ). Then the maximum profile log-likelihood becomes (up to a constant) (A4)

where all of the quantities depend on explicitly or implicitly. As a result, the MLE estimator can be found by (A5) This optimizationproblem is much easier because the variables have similar scales, and the dimension is reduced. Thus, , and can be calculated using Equation (A2(A2) ) with plugged in.

B. Additional figures

Figure B1. Thickness profiles of the wafers that failed the tests: (a) wafer no. 12; (b) wafer no. 23; (c) wafer no. 24; (d) wafer no. 26; (e) wafer no. 28; and (f) wafer no. 30.

Additional information

Notes on contributors

Linmiao Zhang

Linmiao Zhang is currently a Ph.D. student in the Department of Industrial & Systems Engineering, National University of Singapore. He received his B.Eng. degree in Industrial Engineering from Nanjing University, China. His research topic is statistical modeling of complex engineering data. He is a student member of INFORMS.

Kaibo Wang

Kaibo Wang is an Associate Professor in the Department of Industrial Engineering, Tsinghua University, Beijing, China. He received his B.S. and M.S. degrees in Mechatronics from Xi’an Jiaotong University, Xi’an, China, and his Ph.D. in Industrial Engineering and Engineering Management from the Hong Kong University of Science and Technology, Hong Kong. He has published papers in journals such as IEEE Transactions on Automation Science and Engineering, Journal of Quality Technology, IIE Transactions, Quality and Reliability Engineering International, International Journal of Production Research, and others. His research is devoted to statistical quality control and data-driven complex system modeling, monitoring, diagnosis, and control, with a special emphasis on the integration of engineering knowledge and statistical theories to solve problems from industry. He is a member of INFORMS and IIE and a senior member of ASQ.

Nan Chen

Nan Chen is an Assistant Professor in the Department of Industrial and Systems Engineering at the National University of Singapore. He obtained his B.S. degree in Automation from Tsinghua University, M.S. degree in Computer Science, M.S. degree in Statistics, and Ph.D. degree in Industrial Engineering from the University of Wisconsin–Madison. His research interests include statistical modeling and surveillance of service systems, simulation modeling design, condition monitoring, and degradation modeling. He is a member of INFORMS, IIE, and IEEE.