Robust estimation of line width roughness parameters

Abstract

Estimation of line width roughness (LWR) parameters is necessary for semiconductor process optimization, comparison of next-generation lithography processes as well as device performance simulation. According to previous studies [V. Constantoudis et al., J. Vac. Sci. Technol. B 21, 1019 (2003); J. Vac. Sci. Technol. B 22, 1974 (2004)], a complete description of LWR can be provided by three parameters: root-mean square roughness (σ), correlation length (ξ), and roughness exponent (α). However, the primary challenge in the estimation of the aforementioned LWR parameters is the limited availability of data in the scanning electron micrograph (SEM) image. A typical SEM image consists of 8–20 lines of 300–1500 nm length. It has been recognized that for a given line, in the presence of correlation between line widths at a given separation, the estimate of σ for a finite length of line can be significantly biased. Furthermore, for next-generation lithography (NGL) technologies such as double patterning, the number of lines available to estimate each lithography step is cut in half since alternate lines are produced from different processes. Thus, there is a need for an estimation procedure that performs robustly for arbitrarily shorter and fewer numbers of lines. Using a vectorized block or block of blocks bootstrap technique for dependent data and a weighted least-squares (WLS) fitting procedure, the authors fit a specific form of a variogram model. Block of blocks bootstrap is used to estimate the variance of a variogram, which in turn provides the WLS weights. Additionally, the bootstrap approach also allows us to estimate the error in the estimated LWR parameters, a vital requirement that has not been addressed by any of the previously reported procedures on this subject. The authors’ procedure works even in the presence of some unknown local critical dimension (CD) variation or if there is a systematic difference in CD (by design or otherwise) between the lines. The authors validate their procedure with simulated roughness profiles with deterministic LWR parameters. Lastly, as an application of their procedure, they evaluate actual profiles from a variety of different mainstream NGL processes such as litho-freeze-litho-etch double patterning lithography, self-aligned double patterning, and extreme ultraviolet, as well as alternatives such as directed self-assembly and nanoimprint lithography. It was shown that σ can be overestimated by over two times if local variation is not treated properly.