Reliability of correlation-length measurements: A comparative study.

Abstract

Reliability and accuracy of lateral correlation length measurements are studied and quantified from the analysis of synthetic correlated profiles. First, we review available methods to characterize self-affine profiles; we discuss the autocorrelation, Hurst, interface width, and maximum difference functions as widely applicable. We demonstrate that the functions are essentially related and give comparable predictions of the scaling behavior. Second, we consider definitions of correlation length employed in practice. Roughly, they might be cast into two groups, selecting by whether they do not require or require fitting. Approaches from the first group are straightforward and easy to implement; however, they may over- or underestimate the value of correlation length on a small data set due to a size effect. Approaches from the second group may smooth possible oscillations; nevertheless, their precision highly depends on the fit quality. We demonstrate that correlation length found with different approaches might be easily rescaled one onto another. In addition, we introduce a parameter which is further used to estimate accuracy of measurements. Third, we estimate accuracy of all functions and methods on synthetic correlated profiles. We demonstrate influence of the size effect on statistical functions. We show that different methods have different accuracy and different convergence rate to the true value. Fourth, we estimate a parameter saying how much profile length should be larger than the measured value of correlation length in order to get the desirable accuracy of measurements. Finally, we conclude our paper with the recommendation to define correlation length as an abscissa of the e^{-1} point of the autocorrelation function, and note that the accuracy of 10% is achieved if 100 studied profiles are at least 50 times larger than the measured value of correlation length.