Application to Fault Detection and Diagnosis in Semiconductor Etch

Monitoring and fault detection of batch chemical processes is complicated by stretching of the time axis, resulting in batches of different length. This paper offers an approach to the unequal time axis problem using the Parallel Factor Analysis 2 (PARAFAC2) model. In part I of this series an algorithm for PARAFAC2 was developed and extended to N-way arrays. Unlike PARAFAC, the PARAFAC2 model does not assume parallel proportional profiles, but only that the matrix of profiles preserve its 'inner product structure' from sample to sample. PARAFAC2 also allows each matrix in the multi-way array to have different numbers of rows. Part II of this series demonstrated how the PARAFAC2 model could be used to model chromatographic data with retention time shifts. Fault detection, and to a lesser extent diagnosis, in a semiconductor etch process is considered in this paper. It is demonstrated that PARAFAC2 can effectively model process data with unequal dimension in one of the orders such as the unequal batch length problem. It is shown that the PARAFAC2 model has approximately the same sensitivity to faults as other competing methods, including principal components analysis (PCA), unfold PCA (often referred to as Multi-way PCA), Tri-Linear Decomposition (TLD), and conventional PARAFAC. The advantage of PARAFAC2 is that it is easier to apply than MPCA, TLD and PARAFAC because unequal batch lengths can be handled directly, rather than through preprocessing methods. It also provides additional diagnostic information: the recovered batch profiles. It is likely, however, that it is less sensitive to faults than PARAFAC2.