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

Abstract

Under specific experimental circumstances, sputter erosion on semiconductor materials exhibits highly ordered hexagonal dot-like nanostructures. In a recent attempt to theoretically understand this pattern forming process, Facsko et al. [Phys. Rev. B 69, 153412 (2004)] suggested a nonlocal, damped Kuramoto-Sivashinsky equation as a potential candidate for an adequate continuum model of this self-organizing process. In this study we theoretically investigate this proposal by (i) formally deriving such a nonlocal equation as minimal model from balance considerations, (ii) showing that it can be exactly mapped to a local, damped Kuramoto-Sivashinsky equation, and (iii) inspecting the consequences of the resulting non-stationary erosion dynamics.

Surface structuring by multiple ion beams

“…Modeling stochastic fluctuations in the sputtering process by Gaussian white noise with covariance 2D, the general form of a balance equation reads ‫ץ‬ t H = ١ · J H + F + . In our former contributions 19,24 we expanded the detachment contribution F to lowest order in linear and nonlinear terms. From a systematic point of view, it is more stringent to expand the linear terms up to fourth order in ١, since ١ · J H is of order four in ١.…”

“…͑1͒, i.e., with a 0 =0,a 1x = a 1y , a 3x = a 3y , has already been successfully employed to describe pattern formation under normal incidence of the ion beam. [23][24][25] In a shifted frame of reference, Eq. ͑1͒ becomes local and reads in dimensionless variables ‫ץ‬ t h = − ͑␥ + ‫ץ‬ x 2 + ‫ץ␣‬ y 2 + ٌ 4 ͒h + ‫ץ͑‬ x h͒ 2 + ‫ץ͑␤‬ y h͒ 2 + .…”

How ripples turn into dots: Modeling ion-beam erosion under oblique incidence

“…As a consequence of [15,16] Possible anisotropies stemming from ion-induced surface diffusion are at this stage neglected.…”

Continuum modeling of particle redeposition during ion-beam erosion

“…R into the model leads to a vast number of coefficients in combination with the Kuramoto-Sivashinsky equation by what it cannot be considered as minimal model anymore. However, the derivation of the Kuramoto-Sivashinsky equation as erosion model is typically achieved by a systematic expansion in terms compatible with the symmetries to the lowest order [22,38,59]. If F app.…”

“…To that end, a variety of generalizations of equation (1) have been proposed and investigated [32][33][34][35][36][37][38][39][40][41][42][43]. The outcome can be concluded as follows: in order to reproduce hexagonally arranged dots with a generalized Kuramoto-Sivashinsky equation, one either has to incorporate a non-local term or couple in an additional surface composition equation [36,37].…”