A symmetry-based approach to the characterization of complex surface morphologies: Application in CuO and NiO nanostructures

In recent years, bottom-up nanofabrication techniques and especially solution-based ones have gained significant popularity due to their relatively facile implementation, low-cost and design flexibility that allows the development of a plethora of three-dimensional nanostructures and nanoarchitectures. Nonetheless, despite the apparent simplicity of the methods, the resulting nanostructures are often morphologically and hierarchically complex with no apparent way of appropriately describing their features in a quantifiable and methodological way. In this work, we propose a symmetry-based approach for the quantitative characterization of the morphology of nanostructured surfaces. The key idea is to characterize the morphology of nanostructures by means of the fundamental spatial symmetry metrics and the deviations from the fully symmetrical counterparts. The focus of the proposed methodology is on the translational and scaling symmetry, which are investigated through the mathematical tools of Fourier and (multi)fractal analysis, respectively. The calculated metrics characterizing these symmetries are the period and the fractal dimension of surfaces while the deviations are quantified by the disorder parameter ωd related to the FWHM of Fourier peak and by the asymmetry of the multifractal spectrum for the translational and scaling symmetry respectively. As an initial study case and in order to demonstrate its suitability, the developed symmetry-based methodology was applied to nickel oxide (NiO) and copper oxide (CuO) nanostructures hydrothermally-grown on silicon substrates. The results show that even in the cases where the surfaces display similar symmetry metrics (period, fractal dimension) the symmetry-based approach reveals subtle, but quantifiable differences in the deviations from full symmetries which are not easily identified by other approaches.