Accessibility of the surface fractal dimension during film growth.

Fractal properties on self-affine surfaces of films growing under nonequilibrium conditions are important in understanding the corresponding universality class. However, measurement of the surface fractal dimension has been intensively investigated and is still very problematic. In this work, we report the behavior of the effective fractal dimension in the context of film growth involving lattice models believed to belong to the Kardar-Parisi-Zhang (KPZ) universality class. Our results, which are presented for growth in a d-dimensional substrate (d=1,2) and use the three-point sinuosity (TPS) method, show universal scaling of the measure M, which is defined in terms of discretization of the Laplacian operator applied to the height of the film surface, M=t^{δ}g[Θ], where t is the time, g[Θ] is a scale function, δ=2β, Θ≡τt^{-1/z}, β, and z are the KPZ growth and dynamical exponents, respectively, and τ is a spatial scale length used to compute M. Importantly, we show that the effective fractal dimensions are consistent with the expected KPZ dimensions for d=1,2, if Θ≲0.3, which include a thin film regime for the extraction of the fractal dimension. This establishes the scale limits in which the TPS method can be used to accurately extract effective fractal dimensions that are consistent with those expected for the corresponding universality class. As a consequence, for the steady state, which is inaccessible to experimentalists studying film growth, the TPS method provided effective fractal dimension consistent with the KPZ ones for almost all possible τ, i.e., 1≲τ<L/2, where L is the lateral size of the substrate on which the deposit is grown. In the growth of thin films, the true fractal dimension can be observed in a narrow range of τ, the upper limit of which is of the same order of magnitude as the correlation length of the surface, indicating the limits of self-affinity of a surface in an experimentally accessible regime. This upper limit was comparatively lower for the Higuchi method or the height-difference correlation function. Scaling corrections for the measure M and the height-difference correlation function are studied analytically and compared for the Edwards-Wilkinson class at d=1, yielding similar accuracy for both methods. Importantly, we extend our discussion to a model representing diffusion-dominated growth of films and find that the TPS method achieves the corresponding fractal dimension only at steady state and in a narrow range of the scale length, compared to that found for the KPZ class.