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 η 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. η under a fixed sampling interval is determined by the total number N all of width data used to obtain the MC-ACF. This comes from the fact that the MC-ACF is obtained after averaging approximately for N all times. The authors also investigated the case when LWR consisted of two components that had different correlation lengths. They confirmed that η 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.