Nano-patterning of Surfaces by Ion Sputtering: Numerical Study of the Kuramoto-Sivashinsky Equation by Implicit Time Splitting

Abstract

The present study focuses in the simulation of pattern formation by ion beam sputtering on a material surface. This process is responsible for the appearance of unexpectedly organized patterns, such as ripples, nanodots and hexagonal arrays of nanoholes. A numerical analysis of preexisting patterns is proposed to investigate surface dynamics, based on a model derived from a Kuramoto-Sivashinsky anisotropic equation, in a two dimensional surface with periodic boundary conditions. While deterministic, its highly nonlinear character gives a rich range of results, making it possible to describe accurately different patterns. A finite-difference semi- implicit splitting scheme is employed on the discretization of the governing equation. Simulations were conducted with realistic coefficients related to physical parameters (anisotropies, beam orientation, diffusion). The study dealt with a random and a monomode 1D structure initial condition in order to understand how the shape and wavelength of the initial pattern evolve with time. Ripples and hexagonal patterns were observed in the results, being physically consistent with the phenomenon. Moreover, the Method of Manufactured Solution has been used as verification of the developed numerical scheme.