Line edge roughness metrology: recent challenges and advances toward more complete and accurate measurements

Two fundamental challenges of line edge roughness (LER) metrology are to provide complete and accurate measurement of LER. We focus on recent advances concerning both challenges inspired by mathematical and computational methods. Regarding the challenge of completeness: (a) we elaborate on the multifractal analysis of LER, which decomposes the scaling behavior of edge undulations into a spectrum of fractal dimensions similarly to what a power spectral density (PSD) does in the frequency domain. Emphasis is given on the physical meaning of the multifractal spectrum and its sensitivity to pattern transfer and etching; (b) we present metrics and methods for the quantification of cross-line (interfeature) correlations between the roughness of edges belonging to the same and nearby lines. We will apply these metrics to quantify the correlations in a self-aligned quadruple patterning lithography. Regarding the challenge of accuracy, we present a PSD-based method for a noise-reduced (sometimes called unbiased) LER metrology and validate it through the analysis of synthesized SEM images. Furthermore, the method is extended to the use of the height–height correlation functions to deliver noise-reduced estimation of the correlation length and the roughness exponent of LER.