Self-consistent autocorrelation for finite-area bias correction in roughness measurement

Abstract

Scan line levelling, a ubiquitous and often necessary step in AFM data processing, can cause a severe bias on measured roughness parameters such as mean square roughness or correlation length. Although bias estimates have been formulated, they aimed mainly at assessing the severity of the problem for individual measurements. Practical bias correction methods are still missing. This work exploits the observation that the bias of autocorrelation function (ACF) can be expressed in terms of the function itself, permitting a self-consistent formulation. From this two correction approaches are developed, both with the aim to obtain convenient formulae which can be easily applied in practice. The first modifies standard analytical models of ACF to incorporate, in expectation, the bias and thus actually match the data the models are used to fit. The second inverts the relation between true and estimated ACF to realise a model-free correction. Both are tested using simulated and experimental data and found effective, reducing the total error of roughness parameters several times in the typical cases.Scan line levelling, a ubiquitous and often necessary step in AFM data processing, can cause a severe bias on measured roughness parameters such as mean square roughness or correlation length. Although bias estimates have been formulated, they aimed mainly at assessing the severity of the problem for individual measurements. Practical bias correction methods are still missing. This work exploits the observation that the bias of autocorrelation function (ACF) can be expressed in terms of the function itself, permitting a self-consistent formulation. From this two correction approaches are developed, both with the aim to obtain convenient formulae which can be easily applied in practice. The first modifies standard analytical models of ACF to incorporate, in expectation, the bias and thus actually match the data the models are used to fit. The second inverts the relation between true and estimated ACF to realise a model-free correction. Both are tested using simulated and experimental data and found effective, reducing the total error of roughness parameters several times in the typical cases.