Dissipative continuum model for self-organized pattern formation during ion-beam erosion

Abstract

A continuum model, based on the damped Kuramoto-Sivashinsky equation, is shown to reproduce the morphology evolution during ion sputtering quite successfully. In a very narrow range of the damping parameter ␣, the alignment of the structures into hexagonal domains is obtained under normal incidence of ions with striking resemblance to the experimentally observed dot patterns. The origin of this damping factor is discussed. DOI: 10.1103/PhysRevB.69.153412 PACS number͑s͒: 68.55.Ϫa, 05.45.Ϫa, 79.20.Rf The detailed knowledge of the surface morphology induced by ion sputtering is of central importance for the formation of self-organized periodic patterns of nanostructures 1 and for the exploration of fundamental limits in top down processes based on lithography and ion etching.Recent attempts to explain the temporal evolution of dot patterns during ion sputtering by the undamped KuramotoSivashinsky ͑KS͒ equation failed in two decisive points. 2,3Firstly, they could not reproduce the hexagonal ordering of dots under normal incident ion sputtering ͑or off normal sputtering with simultaneous rotation of the sample͒ reported in the last years on III-V semiconductor and Si surfaces. 1,4 -6 Second, they did not predict the stabilization of the periodic patterns for long sputtering times. 2,6 Since the ordering arises equally on different surface orientations and even on amorphous surfaces it can be concluded that the ordering into hexagonal domains does not depend on the initial crystal structure of the surface. Therefore, the ordering mechanism is inherent to the pattern formation process. 7In an extended numerical analysis of the long-time behavior of the two-dimensional ͑2D͒ damped KuramotoSivashinsky ͑DKS͒ equation Paniconi and Elder presented a stationary hexagonal ordered solution in the long-time limit that bears a striking resemblance with the dot patterns obtained by ion sputtering. The DKS equation applied for the surface morphology evolution during the erosion by ion beam sputtering under normal incidence is a partial differential equation for the surface height h (x,y,t), with x and y lying in the surface plane:Here, v 0 is the constant erosion velocity of the plane surface, is the ''effective surface tension,'' which is caused by the erosion process and usually has a negative value leading to a primary ͑linear͒ surface instability.9,10 The diffusion coefficient D eff , which is assumed isotropic, stands for the sum of all diffusion coefficients, i.e., the thermal diffusion 9 and the erosion induced diffusion. 5,11 Further on, the nonlinear term /2("h) 2 accounts for the tilt dependent sputtering yield and brings forth the saturation of the surface roughness in time. The time, where the surface roughness starts to saturate is referred to as the crossover time t c . In the following, times smaller than t c are called early time ͑linear regime͒ and times larger than t c late time regime ͑nonlinear regime͒. The term accounts for the fluctuations introduced by the stochastic nature of the sputtering process ...

Continuum modeling of sputter erosion under normal incidence: Interplay between nonlocality and nonlinearity

“…Second, we show that the re-sulting non-local model equation can be mapped to the damped KS equation by the use of a temporally nonlocal transformation. By that we show how the stabilized KS equation suggested by Facsko et al 15 systematically fits into the theoretical framework of a continuum description of eroded surfaces obeying all desired invariances. Third, we discuss in detail the subtle interplay between potentially nonlocal terms in the evolution equation for the eroding surfaces and the role of the KS-nonlinearity (∇H) 2 .…”

“…(1) (1) stands, it violates the translation invariance in the erosion direction and, moreover, the physical meaning of the dissipation term −αH is not directly obvious. Facsko et al 15 briefly argued that (i) translational invariance can be restored by assuming that −αH has to be replaced by −α(H − H) with H being the erosion depth averaged over the sample area and (ii) that the term −α(H − H) might be interpreted as an approximation of a newly suggested redeposition effect of the sputtered target particles. Due to the nonlocal character of this term, however, the systematic connection between this nonlocally extended damped KS equation and Eq.…”

“…The functional form of the term representing the relaxational currents at the surface, ∇ · J H in (2) remains to be specified. Following Facsko et al 15 , we suppose…”

“…(1) is far from being obvious and the main reason for our paper. First, we argue on general grounds, by using balance considerations, symmetries and allowing for nonlocal dependencies, what the simplest functional form of the spatio-temporal dynamics of the morphology of ion sputtered surfaces under normal incidence of the ions might be, if nonlocal terms as suggested by Facsko et al 15 are taken into account. Second, we show that the re-sulting non-local model equation can be mapped to the damped KS equation by the use of a temporally nonlocal transformation.…”

“…To theoretically explain the hexagonal ordering and taking advantage of earlier work by Elder et al 12,13 (cf. also an even earlier study 14 on this subject), Facsko et al 15 have recently suggested a (stochastically extended) stabilized or damped Kuramoto-Sivashinsky (KS) equation for the dynamics of the surface height H.…”