On Optimising Spatial Sampling Plans for Wafer Profile Reconstruction

Abstract

Wafer metrology is an expensive and time consuming activity in semiconductor manufacturing, but is essential to support advanced process control, predictive maintenance and other quality assurance functions. Keeping metrology to a minimum is therefore desirable. In the context of spatial sampling of wafers this has motivated the development of a number of data driven methodologies for optimizing wafer sampling plans. Two such methodologies are considered in this paper. The first combines Principal Component Analysis and Minimum Variance Estimation (PCA-MVE) to determine an optimum subset of sites from historical metrology data from a larger candidate set, while the second employs Forward Selection Component Analysis (FSCA), an unsupervised variable selection technique, to achieve the same result. We investigate the relationship between these two approaches and show that under specific conditions a regularized extension of FSCA, denoted FSCA-R, and PCA-MVE are equivalent. Numerical studies using simulated data verify the equivalence conditions. Results for simulated and industrial case studies show that the improvement in wafer profile reconstruction accuracy with regularization is not statistically significant for the case studies considered, and that when PCA-MVE is implemented with a denoising step as originally proposed, it is outperformed by FSCA. Therefore, FSCA is the preferred methodology.