Role of finite probe size in measuring growth exponent in film deposition

Abstract

We use computer simulations to investigate the effects of the tip diameter of an electrostatic force microscope (EFM) operating at a constant force on the extraction of the growth exponent β during film growing in a one-dimensional substrate. Laplace’s equation is solved in the EFM simulation using the finite element method to determine the electrostatic force between the tip and the film interface. Importantly, for EFM tips with sufficiently large apex diameters, the topographies calculated with EFM and those computed with the transformed mean height profile (TMHP) method, where the interface is divided into bins of the same tip diameter size and the average height within each bin is used to transform the original interface, are almost identical. This was shown in the context of lattice models of the Kardar–Parisi–Zhang (KPZ) and Villain–Lai–Das–Sarma (VLDS) classes. The global roughness of the film surface, W, scales with the diameter of the EFM tip, ε, as W/a=(ε/a)αg[Ψ], where a is the lattice parameter, α is the KPZ/VLDS roughness exponent, and g is a universal scaling function of the argument Ψ≡t/(ε/a)z, where t and z are the reduced time of deposition and the KPZ/VLDS dynamic exponent, respectively. These results provide a limit for ε from which a KPZ/VLDS growth exponent can be reliably determined with EFM at a constant force. When the EFM tip diameter is larger than the surface correlation length, a misleading effective growth exponent consistent with uncorrelated growth is found.