Statistical-noise effect on autocorrelation function of line-edge and line-width roughness

Accurate characterization of line-edge roughness (LER) and line-width roughness (LWR) is essential to cope with the growing challenge of device variability in large-scale integrations. The accuracy is affected markedly by statistical noise, which is caused by the finiteness of a number of samples. The statistical noise produces random oscillatory fluctuations of autocorrelation function (ACF) of LER/LWR. These fluctuations are obstacles to estimating LWR statistics by comparing experimental and theoretical ACFs. Using the Monte Carlo (MC) method to prepare pseudoexperimental ACFs (MC-ACFs), the authors found that an error of the estimates is minimized in the case when a ratio of a fitting-window size to a correlation length is 0.3 or smaller, being less affected by the statistical noise. under a fixed sampling interval is determined by the total number Nall of width data used to obtain the MC-ACF. This comes from the fact that the MC-ACF is obtained after averaging approximately for Nall times. The authors also investigated the case when LWR consisted of two components that had different correlation lengths. They confirmed that of both components increase with a decrease in their occupancies in the entire LWR. This, together with a large correlation length, makes it difficult to accurately characterize the longer-correlation component, which is mostly minor (small occupancy) in actual cases. This difficulty is also an obstacle to estimating the shorter-correlation component, because the statistics of the former are mostly the prerequisites for analyzing the latter. These facts make a stark contrast to a power-spectral-density (PSD) fitting method, where at least the shorter-correlation component is estimated with almost the same accuracy as in the case of a single component. Based on these results, the authors propose to investigate PSDs, rather than ACFs, in the case of multicomponent LWR.