A Bayesian parallel site methodology with an application to uniformity modeling in semiconductor manufacturing

The increasing use of standard machines in manufacturing processes provides opportunities for sharing information from similar operations to improve the accuracy of process model characterization for supporting root cause analysis and process monitoring activities. This article investigates how to integrate data from parallel sites that are alike in some ways but dissimilar in others. The parallel sites could be several machines that accomplish the same process step, several industrial locations that produce the same product or even several different time windows for the same machine. The proposed Bayesian parallel site model flexibly allows for a compromise between pooling the data completely and treating sites as completely unrelated. One of the key features in the hierarchical model is the quasi-common parameters that are different from site to site, but have some commonality between sites. The similarities between the individual quasi-common parameters are modeled through common global hyperparameters which determine the prior distribution for the quasi-common parameters. A case study on the uniformity modeling (across the different dies on a silicon wafer, across slots in a furnace, etc.) illustrates how the hierarchy of a Bayesian model can be used to incorporate correlation structure. The Bayesian approach provides the flexibility for handling many other possible parallel data source scenarios that might be encountered in many applications.