STATISTICS OF PATTERN PLACEMENT ERRORS IN LITHOGRAPHY

The placement of pattern features in lithography must be accurate to within some small fraction of the minimum feature size or critical dimension (CD). As CDs decrease over the years, driven by the demand for high density memory, the requirements on pattern placement become accordingly more stringent. A practical problem exists: after a wafer or mask is written, a binary decision must be made, based on a set of measurements, whether the pattern placement is deemed acceptably accurate. Based on this decision, the substrate will either be retained for further processing, or discarded. The large number of pattern features, together with the slowness of present-day metrology, cause the acceptance decision to be based on measurement of a tiny fraction of all features in the pattern. It is a classic problem in sampling statistics. A proper analysis relies on the probability that any feature, were it to be measured, would have a placement error exceeding some predetermined limit. In practice errors contain both systematic and random components. Systematic errors include magnification, orthogonality, and distortion, to name a few. Random errors arise from noise and jitter in the lithography or metrology tool, for example. The purpose of this paper is to derive a criterion for the binary acceptance decision, which is useful in the presence of combined systematic and random errors. The distribution of systematic errors is shown to be proportional to the area between the lines of a contour map of the error. The method is applied to an e-beam written mask for x-ray lithography. It is shown that conventional variance analysis is inadequate for predicting the error rate.