A Structure Data-Driven Framework for Virtual Metrology Modeling

Abstract

Virtual metrology (VM) has been widely studied in the semiconductor industry with the purpose of decreasing the cycle time and reducing the expensive metrology measurements. Ideally, a VM model should not only be able to provide accurate predictions but also present an interpretable and rational structure to accommodate fundamental restrictions and relationships that are known to be present in the process. The last aspects have been missing in the VM models proposed hitherto. Therefore, in this article, we propose a novel framework by combining in a single VM model the capability to learn from data with the ability to incorporate the domain knowledge on the process. Thus, the new methodology can use the best of both information sources: data and the subject-matter expert (SME) knowledge. The framework consists of two phases. In the first phase, a Gaussian Bayesian network (GBN) is used to extract the implicit relationships between the metrology and production/process variables. In the second phase, the target response variable is defined, the predictors are selected through the associated Markov blanket, and finally, an empirical model is estimated to accurately predict the response. The proposed framework was tested and its effectiveness was confirmed through a real industrial data from a chemical–mechanical polishing (CMP) process in semiconductor fabrication. The physical meaning of the model obtained was also scrutinized by an SME. Note to Practitioners— Unlike the conventional virtual metrology (VM) techniques that model the x–y relationship based only on process data, the proposed method aims at consolidating the process knowledge from experts into an extensive relational structure. Not only the x–y model is inferred but also the relationships among the predictors are revealed. The structure is illustrated in a form of a connected graph so that the correlation between parameters can be expressed explicitly, which is also compatible with the physical laws. With the clear visualization of the correlation structure of variables, the practitioners are able to utilize the result in different applications. The learned structure can be further integrated into the plant advanced process control system, including VM, process monitoring and diagnosis, and run-to-run (R2R) control.