Crater function approach to ion-induced nanoscale pattern formation: Craters for flat surfaces are insufficient

Abstract

In the crater function approach to the erosion of a solid surface by a broad ion beam, the average crater produced by the impact of an ion is used to compute the constant coefficients in the continuum equation of motion for the surface. We extend the crater function formalism so that it includes the dependence of the crater on the curvature of the surface at the point of impact. We then demonstrate that our formalism yields the correct coefficients for the Sigmund model of ion sputtering if terms up to second order in the spatial derivatives are retained. In contrast, if the curvature dependence of the crater is neglected, the coefficients can deviate substantially from their exact values. Our results show that accurately estimating the coefficients using craters obtained from molecular dynamics simulations will require significantly more computational power than was previously thought.

Model for roughening and ripple instability due to ion-induced mass redistribution [Addendum to H. Hofsäss, Appl. Phys. A 114 (2014) 401, “Surface instability and pattern formation by ion-induced erosion and mass redistribution”]

“…This includes the prediction of the pattern orientation with respect to the direction of the incident ion beam and also the prediction of the ripple propagation velocity. Recently, Harrison and Bradley pointed out that an exact treatment of the BH model within the crater function approach requires also the evaluation of erosion crater functions on curved surfaces [2]. Pattern formation due to directed mass redistribution in the collision cascade volume parallel to the local surface was first introduced by Carter and Vishnyakov (CV) [3] and recently evaluated by Madi et al [4], Davidovitch et al [5,6], Norris et al [7,8] and Hossain et al [9] using the crater function approach.…”

“…Both effects often lead to the formation of spatially periodic ripple patterns. Several aspects of pattern formation due to curvature-dependent ion beam erosion can be described by the linear theory of Bradley and Harper (BH) [1,2]. This includes the prediction of the pattern orientation with respect to the direction of the incident ion beam and also the prediction of the ripple propagation velocity.…”

Self-organized nanopatterning of silicon surfaces by ion beam sputtering

“…On the other hand, also surface redistribution effects take place, through the generation of permanent displacements of the atoms near the surface, also known as craters. The potential contribution of these to the dynamics of the target surface in macroscopic time scales has started to be assessed by Molecular Dynamics (MD) simulations only very recently [131][132][133][134], although limitations to the procedure seem to exist [135][136][137]. In Fig.…”

Neon ion beam induced pattern formation on amorphous carbon surfaces

“…For other conditions, the surfaces remain flat. Using the Monte Carlo Simulation program SDTrimSP [27][28][29] we calculated the curvature coefficients C 11 , taking into account the BH theory, including the curvature dependence of the crater function introduced by Harrison and Bradley,9 the CV model, the dynamic layer thickness dependence introduced by Hofsäss 10,11 as well as the effect of ion implantation. 13,14 The C 11 coefficients become negative in the experimentally observed regime and are thus able to quantitatively predict pattern formation by irradiation with low energy Ne ions.…”

“…[3][4][5][6][7][8] Recently Harrison and Bradley have shown that the crater function models used up to now are incomplete and must be extended to include the surface curvature dependence of the erosion crater function. 9 This curvature dependence is already accounted in the original Bradley-Harper model using Sigmund's ellipsoidal energy deposition and contributes to the stabilization of the surface under ion irradiation for most ion incidence angles. However, for large angles of incidence this curvature dependence leads to a strong destabilizing contribution.…”