Statistical Analysis of Overlay Errors for Steppers

In photolithography for ULSI (Ultra Large Scale Integration) manufacturing, highly accurate overlay between photomask layers has been required. Therefore, it is important to analyze the overlay errors and to reduce them. In this paper, we show an analytical method to derive their probability distribution function. The overlay errors consist of interfiled errors and intrafield errors on exposed wafers. The overlay errors were assumed as linear models. We found that the probability density function of the interfield errors has a semi-ellipse distribution and the probability density function of the intrafield errors has a trapezoid distribution, in a special case, triangle distribution. An actual overlay error from wafer measurement is a sum of the interfiled errors and intrafield errors. We call the sum of the two errors as a total overlay error. The total overlay error's distribution is derived from a convolution of these two distributions. In experiment, we analyzed interfiled errors and intrafield errors from actual exposed wafers. The convolution of two errors approximated to the total overlay error's distribution from measurement result.