When a Surface Becomes a Network: SEM Reveals Hidden Scaling Laws and a Percolation-Like Transition in Thin Films

The morphology of solid surfaces encodes fundamental information about the physical mechanisms that govern their formation. Here, we reinterpret scanning electron microscopy (SEM) micrographs of oxide thin films as two-dimensional self-affine surfaces and analyze them using a multiscale statistical-physics framework that integrates spectral, multifractal, geometric, and topological descriptors. Fourier-based power spectral density (PSD) provides the spectral slope β and apparent Hurst exponent H, while multifractal scaling yields the information dimensions D_q, the singularity spectrum f(α), and its width Δα, which quantify scale hierarchy and intermittency. Lacunarity captures intermediate-scale heterogeneity, and Minkowski functionals—especially the Euler characteristic χ(θ)—probe connectivity and identify the onset of a percolation-like network structure. Two representative surfaces with contrasting morphologies are used as model systems: one exhibiting an anisotropic, porous, strongly multifractal structure with fragmented domains; the other showing a compact, nearly isotropic, and nearly monofractal organization. The porous regime displays steep PSD decay, broad multifractal spectra, and positive χ, consistent with a sub-percolated, diffusion-limited, Edwards–Wilkinson-like (EW-like) growth regime. Conversely, the compact regime exhibits gentler spectral slopes, narrower f(α), enhanced lacunarity at intermediate scales, and a χ(θ) zero-crossing indicative of a connectivity transition where a surface becomes a percolating network, consistent with a Kardar–Parisi–Zhang-like (KPZ-like) correlated growth regime. These results demonstrate that individual SEM micrographs encode quantitative fingerprints of nonequilibrium universality classes and topology-driven transitions from fragmented surfaces to connected networks, establishing SEM as a quantitative probe for testing theories of rough surfaces and kinetic growth in experimental thin-film systems.