Biases and uncertainties in the use of autocovariance and height-height covariance functions to characterize roughness

摘要

Measuring the frequency response of roughness is necessary in many applications, leading to the common use of the power spectral density (PSD) of the roughness. But biases and random uncertainties in the PSD have led some to explore the use of the autocovariance function (ACF) and the height-height covariance function (HHCF) instead. These functions also entail systematic biases and random uncertainties when applied to measured roughness, requiring detailed characterization. A combination of analytical derivations and simulations of rough edges have led to a thorough characterization of these biases and uncertainties for the measurement of line-edge and linewidth roughness of lithographically produced features. The results show that ACF estimation is problematic, but that HHCF estimation is a reasonable alternative to PSD analysis under conditions typical of linewidth roughness measurement in the semiconductor industry.