Economical sampling algorithm using Fourier analysis for mapping wafer critical dimension variations

Abstract

Characterizations of critical dimension (CD) errors on wafers are usually based on the assumption that the errors are random and normally distributed, and that they therefore can be fully described by a mean and 3σ value. However many sources of CD errors are known to be primarily systematic and are often spatially correlated. In order to establish an efficient measurement sampling scheme and data analysis algorithm that accounts for such nonrandom errors, we collected 900 CD data points using electrical test structures on each of nine wafers that were fabricated in two different fabrication facilities. Wafers using both phase-shifting mask and conventional mask lithography were measured. Fourier analysis was then used to study the errors in the spatial frequency domain. It was found that in all cases systematic and spatially correlated errors, in particular a set of errors that were largely repetitive with stepper field periodicity, dominated over random CD errors. By taking advantage of the correlated nature of these errors, an economical two-step sampling algorithm is defined that significantly reduces the amount of sampling needed (by approximately fourfold). By combining the results of the two-step sampling in the spatial frequency domain, an accurate map of actual error spatial distribution can be extracted. The accuracy of the scheme is verified by using subsets of the 900 data point measurements, and comparing the results to those from the full set of data.