Monitoring virtual metrology reliability in a sampling decision system

Abstract

In semiconductor manufacturing, metrology operations are expensive and time-consuming, for this reason only a certain sample of wafers is measured. With the need of highly reliable processes, the semiconductor industry is interested in developing methodologies covering the gap of missing metrology information. Virtual Metrology (VM) turns out to be a promising method; it aims to predict wafer and/or site fine metrology results in real time and free of costs. Using virtual measurements as the input of a sampling decision system (SDS), an optimal strategy for measuring productive wafers can be suggested. Since sampling decisions strongly depend on the accuracy of the VM system, it is a key requirement to monitor the reliability of the obtained predictions. In this paper, we present approaches for dynamically assessing VM reliability using real metrology data. A Bayesian dynamical linear model (BDLM) handles increasing VM model uncertainty over time. Model parameters are updated whenever new real measurements become available. VM prediction quality is monitored applying a probability integral transform (PIT) and scoring rules for predictive probability distributions. Inferring equipment health factors (EHF), unreliable predictions can be detected before being delivered to the SDS. Based on the likelihood of the predicted measurements, VM trust factors are introduced. A Bayesian model for the prediction precision matrix allows updating the virtual measurements' uncertainty whenever real measurements are available. Taking account of the proposed methods, one is led to an improved accuracy of the SDS.