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

Under specific experimental circumstances, sputter erosion on semiconductor materials exhibits highly ordered hexagonal dotlike 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.

Figure 1

Growth rate of the mean erosion depth calculated from Eq. (5), averaged over 100 runs. Parameters are the same as in the paper by Facsko et al. (Ref. 15), except for a spatial step size of $dx=1$ (mesh size $400\times 400$, $b=-0.24$, $a_1=-1$, $a_2=-1$, $a_3=0.0025$, white noise with a maximum amplitude of $\eta =0.01$). Solid line: $b=-0.22$, dotted line: $b=-0.24$. The inset shows a semilogarithmic plot of a portion of the data where linear terms determine the growth. A linear fit in this regime yields $\ln ({\partial }_t⟨\overline{H}⟩-F_0)\propto 0.018\bullet t$ for $b=-0.24$ and $\ln ({\partial }_t⟨\overline{H}⟩-F_0)\propto 0.057\bullet t$ for $b=-0.22$. This compares well to the result which can be obtained from the linearized version of Eq. (5), i.e., $2{\sigma }_{\max }=0.02$ for $b=-0.24$ and $2{\sigma }_{\max }=0.06$ for $b=-0.22$.

Figure 2

Growth rate of the mean erosion depth calculated from Eq. (5) for five different noise sequences. Parameters are the same as in Fig. 1 with $b=-0.24$. Insets show a section of the surface morphology corresponding to the thick line in the figure at times $t=1\bullet {10}^4$ and $t=2\bullet {10}^4$.