Robust Relevance Vector Machine with Variational Inference for Improving Virtual Metrology Accuracy

Virtual metrology (VM) technology is an efficient and effective method of online and wafer-to-wafer process monitoring. It is realized by constructing a prediction model between real-time equipment sensor data and the quality characteristics of wafers that should be measured. The most commonly employed prediction method for VM is a neural network (NN) approach due to its flexibility and fast computation time. However, it can easily suffer from the overfitting problem and is affected by naturally occurring potential outlying observations contained in given data. Moreover, it does not provide prediction intervals for future observations that can be used to detect abnormal process problems. In this paper, an advanced prediction model for VM is developed to resolve these issues. The proposed method is a robust regression model based on relevance vector machine. The proposed method can reduce the effect of outliers by using a weight strategy. Given a prior distribution of weights, it is shown that the weight values can be determined in a probabilistic way and computed automatically during training. We employ the variational inference method to estimate the posterior distribution over model parameters. Therefore, no validation data set is needed to control the model complexity. That is, the complexity of our proposed method can be self-adjusted in the model training phase. Based on the posterior distribution, we can obtain not only point estimates but useful statistical information such as probabilistic intervals which provide us some useful information about the current status of a manufacturing process. If the actual metrology value falls outside of the intervals, it can be a signal which alerts engineers to the need for preventive maintenance or VM model adjustment. The real plasma etching process of semiconductor manufacturing is presented as a case study to compare the predictive performance of our proposed method with that of conventional VM prediction models. The experimental results demonstrate that the proposed method can improve VM prediction accuracy compared to other methods.