Risk and cost must be balanced in the design of semiconductor processing metrology. More specifically, one needs to balance the cost of operating the metrology tool, and the loss in terms of processing cost and yield due to the limited sampling and the time lapse between the occurrence and the correction of a process fault. In virtual metrology (VM), the real-time data produced by the processing tool (e.g. plasma etching data during isolation trench formation) is used to predict an outcome of the wafer (e.g. critical dimension of the trench) utilizing an empirical model. Although VM prediction quality is not as good as that of conventional metrology, it produces an immediate, low cost prediction for each wafer going through a process. We envision that practical metrology schemes in the future will involve a synergistic blend of VM and actual metrology, the latter being used for the needed periodic recalibration of the VM empirical model. In this work, we have formulated the costs associated with type 1 and type 2 errors that result from a blended metrology scheme; the revenue, processing cost, and off-line metrology cost. In the first part of the work, the net profit as a function of proportion of samples that go through VM prediction and the prediction quality of the VM model was plotted for a given off-line metrology cost. Results showed that the prediction quality of the VM model could be relaxed and still be beneficial in the presence of process faults. The second part expands on this result by taking into account the relationship between the proportion of samples going through VM and the quality of the prediction model. This is important since as more samples from off-line metrology are used for VM recalibration, the prediction quality of the VM model is improved. However, the cost of off-line metrology would also increase. This paper formulates and explores this tradeoff between re-calibration and off-line metrology to find the optimal number of samples that maximizes the profit. A sequence of metrology samples using a regression model with linearly drifting coefficients is simulated, a model realistically applying to a manufacturing process with linearly drifting hidden variables. Three different types of statistical models, ordinary least-squares (OLS), exponentially-weighted ordinary least-squares (WOLS), and the Kalman Filter are used as VM prediction tools. To simulate a blended metrology scheme, we alternate between training sets (VM calibration using off-line metrology), and testing sets (prediction through VM model), and compare the resulting net profit, type 1, and type 2 errors as a function of varying VM prediction sample sizes. In this work, the blended metrology scheme involved a periodic pattern starting with 20 actual metrology samples used as the re-calibration set, followed by a variable number of VM-predicted samples. Results show that each VM prediction model has a different tradeoff between the Type 1 and Type 2 errors that determine the optimal sampling scheme. The ultimate goal is to create a general framework that quickly leads to the optimal design of such schemes given the characteristics of the process in question.




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