Monitoring Virtual Metrology Reliability in a Sampling Decision System Daniel Kurz, Cristina De Luca, Jürgen Pilz Alpen-Adria University (AAU) of Klagenfurt, Infineon Technologies Austria Daniel.Kurz@aau.at, Cristina.DeLuca@infineon.com, Juergen.Pilz@aau.at Abstract— In semiconductor manufacturing, metrology op- erations 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. I. I NTRODUCTION In semiconductor manufacturing, wafer measurements are performed in order to control production quality and process stability. With the growing number of products and technolo- gies, and increasing wafer diameters (e.g. 300mm wafers), an increment of measurement needs has been observed. For this reason, it is necessary to develop sampling designs deciding which wafer or lot has to be measured. The aim is to achieve a cost reduction without loosing process quality and stability. By means of a Virtual Metrology (VM) system, we are provided with predictions of wafer measurement outcomes directly after wafer processing. Based on obtained infor- mation from the process equipment (e.g. APC (Advanced Process Control) data streams and additional logistic infor- mation) a virtual wafer measurement in terms of a predictive probability distribution is computed. The interested readers find recent works on VM in [1]-[6]. In [7], a sampling decision system (SDS) relying on VM data is presented. It evaluates whether the virtual measure- ment is informative enough in order to decide on the status of the process. The output of the SDS is then a decision This work has been supported by Infineon Technologies Austria. whether to skip the corresponding physical metrology oper- ation or not. Clearly, SDS reliability strongly depends on the performance of the underlying VM system. In this paper, we present concepts for dynamically as- sessing VM reliability in a SDS using real metrology data. Primarily, we aim to update prediction uncertainty when- ever real measurements become available. This can be i.a. achieved by extending a Bayesian dynamical linear VM model [8]-[9] or by introducing VM trust factors which are based on assessing the squared Mahalanobis distance be- tween the real and virtual measurement. More efficiently, we apply a Bayesian Wishart model to the prediction precision matrix [10] which allows to update prediction uncertainty with respect to the single prediction components. Addition- ally, VM performance is quantified by means of VM predic- tion scores. Different methods including probability integral transform (PIT) [11], log- and significance scoring as well as Kullback-Leibler (KL) distance [12]-[13] are investigated. Apart from that, we address the usage of equipment health factors (EHF) [14] preventing bad sampling decisions due to unreliable VM predictions. In short, this paper should outline an implementation of methods for monitoring and updating VM prediction precision in a SDS. Therefore, it should contribute to an increased acceptance of the VM concept. II. SDS USING VIRTUAL METROLOGY A. Virtual Metrology prior information In semiconductor manufacturing, the output of a process step (e.g. layer thickness etc.) is usually controlled per- forming physical measurements for p points on the wafer. Measurement outputs correlate with process equipment pa- rameters (e.g. gas flow, pressure etc.). Machine learning techniques allow to capture those relationships forming a prediction model for real metrology outcomes. It provides the basis for the VM system. In our case, virtual measurements are predictive probability distributions indicating the more probable measurement outcomes. Expected values of p in- spection points of some wafer k summarized in μ VM k ∈ R p together with a covariance matrix Σ VM k ∈ R p×p (providing information on the VM precision) are predicted. Therefore, the virtual measurement defines a p-variate normal prior distribution for the unknown measurement vector Y k ∈ R p , Y k ∼ N p ( μ VM k , Σ VM k ) . (1) Parameters of the associated univariate virtual measurement Y k ∼ N p ( μ VM k ,σ 2 VM k ) (2) 2013 IEEE International Conference on Automation Science and Engineering (CASE) SuBPl.4 978-1-4799-1515-6/13/$31.00 ©2013 IEEE 20