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 ...

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Crystalline and amorphous GaSb surfaces are compared concerning their response to sputter erosion with low energy Ar ϩ ions under normal incidence. We show that the formation of regular nanostructures on GaSb is basically independent of whether the initial material is crystalline or amorphous. The similarity in the temporal and spatial evolution demonstrates that the dynamics of the morphology evolution is entirely controlled by a thin amorphous surface layer. © 2002 American Institute of Physics. ͓DOI: 10.1063/1.1429750͔The large potential of nanoelectronics has stimulated much effort to find new methods for the parallel processing of nanostructures. Promising techniques, like StranskiKrastanov growth of semiconductor heterostructures 1 and self-assembly of semiconductor nanocrystals by colloid chemistry 2 have been exploited. Recently, during the erosion of semiconductor and metal surfaces by ion sputtering, a self-organizing mechanism has been discovered leading to regular patterns of structures with dimensions of some tens of nanometers.3,4 These surface structures can form quantum dots with a high aspect ratio and are therefore particularly attractive for quantum electronic applications.The appearance of periodic ripple patterns on semiconductor and metal surfaces bombarded with ions under offnormal incidence has been known for a long time. 5 In certain metals and III-V semiconductors these ripple patterns transform into hexagonally ordered isolated nanostructures under normal incidence sputtering. 3,4,6 -8 Up to now these observations have only been made in crystalline materials. The main points addressed in this letter concern ͑i͒ the comparison of the pattern evolution from initially crystalline ͑c-GaSb͒ and amorphous GaSb ͑␣-GaSb͒ targets and ͑ii͒ the microscopic structure of the patterned surface. We find that the surface dot patterns generated are independent of the crystallinity of the initial material surface. This independence is explained by the decisive role of a thin amorphous layer that forms during the very first seconds of the sputtering process. Besides the fundamental insight into the sputtering process on semiconductor surfaces, the structuring of amorphous layers deposited on arbitrary materials provides a technologically attractive method of surface nanostructuring.Stochastic nonlinear continuum models have been introduced to explain the temporal and spatial evolution of nanoscale surface patterns during ion sputteringwhere h(x,y) is the surface height function, v 0 is the constant erosion velocity, is the ''negative surface tension,'' ͑͒ is a nonlinear coefficient attributed to the tilt dependent sputter yield with the tilt angle, D is the thermal or ion induced surface diffusion, and denotes a noise term, that describes the stochastic character of the sputtering process. This continuum equation represents an approximation to the surface dynamics because higher order derivatives are not considered. The effect of additional higher order terms in Eq. ͑1͒ is believed to be minor. 10 The m...