Structured regularization modeling for virtual metrology in semiconductor manufacturing processes

Virtual metrology (VM) has been developed to economically perform wafer-to-wafer control and has been used to estimate actual measurements from process data collected via sensors attached to corresponding equipment. Achieving a reliable VM model requires selecting a set of features that can capture the actual characteristics of the sensor data. The sensor data include a huge number of features that are physically connected with each other. These connected features form a group with a strong correlation. Structured sparsity regularization methods incorporate prior assumptions about the structure of the features. Therefore, they can be successfully applied to VM modeling with data in which the feature groups have certain structures. Despite great potential of structured regularization models for addressing the group structure in sensor data, little effort has been made to examine their performance in terms of VM modeling. The main objective of this study is to propose use of structured regularization models for VM modeling and compare them with unstructured regularization models in terms of predictive accuracy, feature-selection accuracy, and stability. The effectiveness of structured regularization models was demonstrated through experiments with synthetic data as well as real data obtained during semiconductor manufacturing processes. Results of these experiments demonstrate that feature-selection accuracy and stability of structured regularization models were superior to those of corresponding unstructured regularization models.