Strength and limitations of continuum equations for sputter induced pattern formation

Erosion of surfaces by ion beam sputtering leads frequently to the formation of periodic patterns, which show up as a ripple pattern under off-normal incidence and as hexagonally ordered dot patterns under normal or close to normal ion incidence.1 A first successful attempt to describe the formation of these patterns theoretically has been given by Bradley and Harper.2 In their model they assumed that ion sputtering is introducing a surface instability which is proportional to the surface curvature and will increase surface roughness. In addition, thermally activated surface self-diffusion has been considered as a smoothing mechanism. Finally, the interplay between these two competing mechanisms leads to a selection of a narrow range of surface modes which grow fastest under ion irradiation. This model has been translated into a linear continuum equation, now well known as the Bradley-Harper equation. The simplified partial differential equation already predicts successfully main experimental observations, like the change in ripple orientation when going to grazing incidence and isotropic patterns under normal incidence. However, it fails to explain the long time behaviour and the detailed dynamics of the pattern evolution.
Based on the Bradley-Harper model many extensions have been proposed since then. Some of them like the damped or undamped Kuramoto-Sivashinsky (KS) equation3 show a good qualitative agreement, however they still fail to predict several important experimental observations. In addition main improvements have been achieved by including the mass rearrangement on the surface induced by the ion impact.4
Recently, new models have been proposed for the description of pattern formation, which go beyond the KS equation. They introduce coupled fields, e.g. for the mobile species or the concentrations of the elements in compound materials.5 However, there is still a gap to be closed between the continuum equations and the microscopic model describing the collision cascade, the energy deposition into the surface, or the crater function induced by the ion impact. Here, theoretical approaches combining binary collision approximation for the ion irradiation and kinetic Monte Carlo simulations for the subsequent relaxation promise a detailed description of the mechanisms involved in the pattern formation process.
In this presentation the strengths and limitations of the existing continuum equations and extended models will be discussed and will be compared to experiments and kMC simulations.