Abstract
The multilayer overlay lithography process is one of the most challenging steps in wafer fabrication. In a multi-level manufacturing process, errors occur at each level and are accumulated in the upstream operations. Layer to layer error propagation, along with the stochastic bias, induce a sophisticated optimization problem and turns it into one of the most demanding topics for researchers. The optimization objective will be even more critical in a high-mixed fabrication process where overlay misalignments take place. In addition, non-convexity, immeasurable uncertainties, stochastic metrology delay, and high-dimensionality are some of the unavoidable phenomena that may occur along with optimization problems. This study aims to deal with the optimization issue of compensating the high-mixed multilayer overlay error, wherein the existence of non-convexity, high-dimensionality, stochastic delay, and uncertainties in the control system cannot manipulate it. The regular exponentially weighted moving average (EWMA) method is adopted as the basement structure of the controller for the first layer, while an approximate Bayesian approach manifested in the Gibbs sampling algorithm is applied to minimize the risk function at the upper layers. The proposed approach has also acknowledged the robustness and fast convergence rate of the optimization. The result from analyzing the manufacturing data by the proposed Bayesian approach is validated by the variation reduction, in both the predicted input and output overlay factors. Apart from the overlay model, a number of simulation case studies are outlined the rising computation issues from the Gibbs sampler.
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