On the stabilization of ion sputtered surfaces

The classical theory of ion beam sputtering predicts the instability of a flat surface to uniform ion irradiation at any incidence angle. We relax the assumption of the classical theory that the average surface erosion rate is determined by a Gaussian response function representing the effect of the collision cascade, and consider the surface dynamics for other physically motivated response functions. We show that although instability of flat surfaces at any beam angle results from all Gaussian and a wide class of non-Gaussian erosive response functions, there exist classes of modifications to the response that can have a dramatic effect. In contrast to the classical theory, these types of response render the flat surface linearly stable, while imperceptibly modifying the predicted sputter yield vs incidence angle. We discuss the possibility that such corrections underlie recent reports of a “window of stability” of ion-bombarded surfaces at a range of beam angles for certain ion and surface types, and describe some characteristic aspects of pattern evolution near the transition from unstable to stable dynamics. We point out that careful analysis of the transition regime may provide valuable tests for the consistency of any theory of pattern formation on ion sputtered surfaces.

Figure 1

(a) Plot of sputter yield curve $I\left(\theta \right)$, normalized by $I\left(0\right)$. [(b) and (c)] Plots of Bradley-Harper coefficients $S_x\left(\theta \right)$ and $S_y\left(\theta \right)$, normalized by $\mid S_x\left(0\right)\mid =\mid S_y\left(0\right)\mid$. The parameters used are $a=1.5\mathrm{nm}$, $\sigma =0.9\mathrm{nm}$, and $\mu =0.5\mathrm{nm}$.

Figure 2

Normalized yield curve and BH coefficients $S_x$ and $S_y$ for two sets of parameters of the two-Gaussian model, Eq. (21). The parameters $a_1$, ${\sigma }_1$, and ${\mu }_1$ are the “Sigmund parameters” taken as in Fig. 1 and the same normalization factors are used. The new parameters are (top row) $\alpha =0.03$, $a_2=0.5\mathrm{nm}$, ${\sigma }_2=0.5\mathrm{nm}$, and ${\mu }_2=1\mathrm{nm}$, and (bottom row) $\alpha =0.03$, $a_2=0.9\mathrm{nm}$, ${\sigma }_2=0.2\mathrm{nm}$, and ${\mu }_2=1.5\mathrm{nm}$.

Figure 3

Normalized coefficients $S_x$, ${\nu }_x$, $S_y$, and ${\nu }_y$, comparing the effect of induced surface currents [$\nu$, Eq. (27)] on erosion from Gaussian ellipsoids [$S$, Eq. (1)]. The relative magnitude of $\nu ∕S$ at normal incidence varies with conditions and is chosen arbitrarily here for illustrative purposes.

Figure 4

Schematic plots depicting the transition between stable and unstable surface dynamics for three dispersion relations. (a) Generalized Bradley-Harper [Eq. (31)], where the transition occurs at $S^{\mathrm{eff},*}=0$ with diverging wavelength. (b) With Facsko nonlocal “damping term,” transition occurs at $S^{\mathrm{eff},*}<0$ with finite wavelength. (c) With Asaro-Tiller nonlocal elastic energy mechanism, transition occurs at $S^{\mathrm{eff},*}>0$ with finite wavelength.