Computer experiments on ion-induced pattern formation vs. continuum equations

Ripple pattern and hexagonally ordered dot pattern are frequently found after ion irradiation under off-normal incidence and under normal or close to normal ion incidence, respectively [1]. A first consistent analytical model for the evolution of these patterns was given by Bradley and Harper [2]. They showed that a surface curvature dependent ion erosion rate can result in a surface instability, whereas a surface smoothing process by surface-curvature-driven diffusion (Mullins-Herring diffusion) competes with that instability. By this competition regular surface patterns may evolve. Lateron, this simple model was extended to more sophisticated partial differential equations like the Kuramoto-Sivashinsly equation in order to improve the agreement with experimental findings. However, it remains difficult to describe details of the pattern evolution and dynamics by continuum equations.
Here we present atomistic 3D computer simulations on ion irradiation of surfaces which unify both, the collision cascades caused by the incident ions (including sputtering, mass drift, ion beam mixing) and the thermally excited relaxation processes (including surface and bulk diffusion of defects and impurities, phase separation, Mullins-Herring diffusion,…). For that aim, the collision cascade simulations in the BCA approximation are carried out for atom densities on the 3D lattice of the kinetic Monte Carlo (kMC) simulation which contains the full history of defects, surface undulations etc. Each ion impact is followed immediately by some kMC steps. The computer experiments show that, with the exception of grazing incidence, in silicon the Ar+ ion induced patterns are dominated by defect kinetics rather than sputtering. The results are in nice agreement with recent experiments of Madi [3]. Secondly, we show how this computer experiments can be directly related to continuum equations. In the computer all atomic jumps and the time evolution of the surface heights as well as of its slopes, curvatures etc can be registered. Thus, it might be possible to find out which term(s) of the partial differential equations should dominate the pattern formation.
[1] W. L. Chan and E. Chason, Making waves: Kinetic processes controlling surface evolution during low energy ion sputtering, J. Appl. Phys. 101, 121301 (2007).
[2] R. M. Bradley and J. M. E. Harper, Theory of Ripple Topography Induced by Ion-Bombardment, J. Vac. Sci. Technol. A 6, 2390 (1988).
[3] C.S. Madi, H.B. George and M.J. Aziz, Linear stability and instability patterns in ion-sputtered silicon, J. Phys.: Cond. Mat. 21, 224010 (2009).