Surface Roughness Measurements Using Power Spectrum Density Analysis with Enhanced Spatial Correlation Length

TLDR

Atomic force microscopy typically quantifies surface roughness using vertical statistics such as RMS, peak‑to‑valley, and average roughness, but ignores lateral variation, and although roughness depends on scan scale and sampling, no method has yet been proposed to enhance the spatial correlation length. The paper aims to address AFM roughness evaluation gaps by proposing that combining PSD profiles across a wide frequency range can enhance spatial correlation length. The authors use power spectral density analysis of AFM images, applying fractal and k‑correlation models to extract RMS roughness, correlation length, fractal dimension, and Hurst exponent. The derived parameters reveal the spatial distribution of roughness, and a longer correlation length is shown to be preferable for comprehensive surface roughness measurement at a given spatial wavelength.

Abstract

Roughness of a surface as characterized by an atomic force microscope (AFM) is typically expressed using conventional statistical measurements including root-mean-square, peak-to-valley ratio, and average roughness. However, in these measurements only the vertical distribution of roughness (z-axis) is considered. Additionally, roughness of a surface as determined by AFM is a function of the scanning scale, sampling interval and/or scanning methods; therefore, the consideration and quantification of the lateral distribution (x and y) is necessary. Power spectral density (PSD) analysis provides both lateral and vertical signals captured from AFM images. By applying one of the commonly adopted models to the PSD data, the fractal model and k-correlation model, the equivalent root mean squared roughness, correlation length, fractal dimension and Hurst exponent are quantified. These parameters describe the spatial distribution of roughness and spatial length scale of the roughness values. Longer correlation length is preferred for the comprehensive measurement of roughness of surface at a given spatial wavelength. However, a method to enhance correlation length has yet to be discussed. In this paper, we discuss the state-of-the-art issues associated with roughness evaluation from AFM analysis and propose that the spatial correlation length can be enhanced through the combination PSD profiles over a wide range of spatial frequencies.