Statistical-noise effect on discrete power spectrum of line-edge and line-width roughness

The control of line-edge roughness (LER) and line-width roughness (LWR) is a key issue in addressing the growing challenge of device variability in large-scale integrations. The accurate characterization of LER and LWR forms a basis for this effort and mostly hinges on reducing the effects of noise inherent in experimental results. This article reports how a power spectral density (PSD) is affected by a statistical noise that originates from the finiteness of the number N_L of available samples. To achieve this, the authors numerically generated line-width data using the Monte Carlo (MC) method and assuming an exponential autocorrelation function (ACF). By analyzing the pseudoexperimental PSDs obtained using the MC data, they found that the standard deviation of normalized analysis errors was determined by the total number NALL of width data used in each analysis, regardless of N_L and the number N of width data in each line segment. The authors found that decreased with NALL approximately in inverse proportion to N ALL 3/4. It is noteworthy that they could obtain accurate results even in the case of N _L =1 as long as NALL was sufficiently large, although the distribution of PSDs was large due to a large statistical noise. This resulted from the fact that the PSD distribution was not completely irregular, but centered at the true value and that the best-fitted PSD accordingly approached the true one with an increasing N. On the other hand, at a fixed NALL decreased with the ratio Δy/ξ of an interval Δy of width data to a correlation length, approximately in inverse proportion to (Δy/) 3/8. As a result, NALL at a specified decreased with Δy/ξ in inverse proportion to the square root of Δy/ξ in the case when Δy/ξ was 0.3 or smaller. Beyond this threshold of Δy/, the authors needed to increase NALL markedly to achieve the same accuracy of analyses. This comes from a decrease in the range of the PSD with an increasing Δy/ξ and a subsequent loss of sensitivity of the PSD to the change of . Based on these results, they established guidelines for accurate analyses as follows: Δy/0.3 and NALL A -4/3 (Δy/) -1/2, where A is 1.8× 10² for and 7.2× 10¹ for the variance of widths, respectively. Equivalently in terms of the total measurement length LALL, instead of NALL, the guidelines are given in Δy/0.3 and LALL /A -4/3 (Δy/) 1/2 using the same A 's as those of NALL. Being expressed in universal forms like these, the guidelines of this study can be applied to many practical problems beyond LER and LWR to accurately analyze PSDs, as long as the stochastic processes have exponential ACFs.