Non-local linear stability of ion beam eroded surfaces

Continuum theories of spontaneous pattern formation at solid surfaces during ion irradiation exist in many variants, but all of them are based upon low order gradient expansions of an underlying non-local theory and are formulated as partial differential equations. Here we reconsider the non-local theory based upon a simple Gaussian erosive crater function of Sigmund's theory of sputtering, which is also a basic ingredient of most of the existing continuum theories. We keep the full non-locality of the crater function in a linear stability analysis of a flat surface. Without gradient expansion the evolution of the height profile is governed by an integral equation. We show that low order gradient expansions may be misleading and that the bifurcation scenarios become significantly more complex, if the non-locality is taken into account. In a second step, we extend our analysis and include mass redistribution due to ion-induced drift currents of collision cascade atoms. The model is based upon results from kinetic theory and uses a simple phenomenology. Both erosion and mass redistribution share the same non-local features, as they are both caused by the collision cascade. If mass redistribution is the dominant pattern forming mechanism, we show that the resulting bifurcation scenarios may provide explanations for many of the recent, seemingly contradictory experimental results of pattern formation on Si surfaces.