Spatial Variance Spectrum Analysis and Its Application to Unsupervised Detection of Systematic Wafer Spatial Variations

Investigation of wafer spatial variations is critical for semiconductor process/equipment optimization and circuit design. The objective of spatial variation study is to differentiate the systematic variation component from the random component. This is usually done by contrasting with a set of known systematic patterns based on engineering knowledge. However, there could exist unknown systematic components remaining in the unexplained residuals and overlooked by the conventional spatial variation study. In this paper, we develop a novel spatial variance spectrum (
SVS
) to analyze the systematic variations without any priori information of the systematic patterns. The
SVS
is a series of spatial variations over a range of spatial moving window sizes from the smallest spatial moving window consisting of only two metrology sites to the largest one covering all metrology sites of the entire wafer. The
SVS
can be used to characterize the wafer spatial variations and to detect existence of systematic variations by a proposed hypothesis test. We also propose an index to summarize from the
SVS
the systematic proportion of the spatial variation. The proposed test and index of systematic variations will be demonstrated and validated through both hypothetical examples and actual cases of wafer critical dimension (CD) metrology data.