Surface structuring by multiple ion beams

We study surface patterns produced by low energy ion sputtering of amorphous targets based on a generalized version of the damped Kuramoto-Sivashinsky equation with the focus on the simultaneous incidence of four identical ion beams. This setup allows us to quantitatively investigate the relevance of anisotropic ion-induced effective surface diffusion in theory and in experiments. We find that hexagonal, square, and superstructure patterns can develop depending on the strength of that anisotropy. Squares and superstructures serve as clear indicators for the presence of ion-induced effective surface diffusion. We also discuss the possibility of simulating multiple ion beam incidence by stepwise rotation of the target.

Figure 1

Stability diagram of the flat front solution $h_{FF}=0$ for $\eta =0$. In the gray shaded regions $h_{FF}$ is linearly stable, otherwise unstable. Markers indicate parameter values used for the surface morphology plots in Fig. 2 and Fig. 11.

Figure 2

Surface patterns close to instability $(\gamma =0.24)$ for increasing anisotropy. From left to right column: $\delta =2$, $\delta =2.8$, $\delta =3.46$, $\delta =5$, $\delta =7$, at times: top row $t=200$ each, middle row $t=400$, $t=800$, $t={10}^3$, $t={10}^3$, $t={10}^3$, lower row $t={10}^4$ each. The lateral size is $x=200$. Lines indicate where the surface cuts in Fig. 7 were taken.

Figure 3

Power spectral density for the late time morphologies $t={10}^4$ in Fig. 2. From left to right, top to bottom $\delta =2.8$, $\delta =3.46$, $\delta =5$, $\delta =7$. Arbitrary units, logarithmic gray scale, only the relevant selection between $\max \{\mid k_x\mid ,k_y\mid \}<3\pi ∕4$ is shown.

Figure 4

Sketch of the temporal evolution of the surface morphology for $\gamma =0.24$. Gray shaded regions indicate transitional regimes between squares $\left(sq\right)$, superstructures $\left(su\right)$, and hexagons $\left(h\right)$.

Figure 5

Temporal evolution of the surface roughness. Averaged over 20 runs (left-hand panel) and the representative individual runs (right-hand panel) from the surface morphology plots in Fig. 2. Damping fixed to $\gamma =0.24$, the anisotropy varies from the highest to the lowest roughness at final time $t={10}^4$: $\delta =2$, $\delta =2.8$, $\delta =3.46$, $\delta =7.0$, $\delta =5$.

Figure 6

Surface roughness $w_f$ for late stages at $t={10}^4$. Averaged over 20 runs. $\gamma =0.24$ (stars, ×) and $\gamma =0.23$ (crosses +). Lines derived from a Galerkin-model, cf. Sec. 5. Upper line, stable superstructures for $\gamma =0.23$; lower line, stable superstructures and squares for $\gamma =0.24$.

Figure 7

Surface cuts for $\gamma =0.24$ taken at intervals of $\Delta t={10}^3$ along the lines in Fig. 2. From left to right, top to bottom $\delta =2$, $\delta =3.46$, $\delta =5$, $\delta =7$.

Figure 8

Regions of stability for surface patterns derived from Galerkin model. Flat front solution $\left(f\right)$, squares $\left(sq\right)$, superstructures $\left(su\right)$, hexagons $\left(h\right)$, and bistable regions $(h+su)$, $(h+sq)$, $(h+f)$. The thick solid line indicates the location of the saddle-node bifurcation of the hexagonal branch.

Figure 9

Bifurcation diagrams for fixed anisotropies $\delta =2$ and $\delta =2.8$. Surface roughness values belonging to ripple $\left(r\right)$ in $k_x$ or $k_y$ direction, square $\left(sq\right)$, superstructure $\left(su\right)$, and hexagonal $\left(h\right)$ branches in dependence on $\gamma$. Solid (stable branches) and dashed (unstable branches) lines are derived from the Galerkin model. Crosses and diamonds stem from numerical simulations of the full field Eq. (8). See also the explanation in the text.

Figure 10

Bifurcation diagrams. Same as Fig. 9, except for $\delta =3.46$ and $\delta =5$.

Figure 11

Surface morphologies for reduced damping $\gamma$ at $t={10}^4$. The lateral size is $x=200$. First three rows from top to bottom $\delta =2.8,3.46,5$ for $\gamma =0.2$ (left-hand column) and $\gamma =0.16$ (right-hand column). Last row $\delta =7$, $\gamma =0$, morphology and corresponding power spectral density.

Figure 12

Surface roughness evolution calculated from Eq. (3) for $F_0=0\mathrm{nm}{\mathrm{s}}^{-1}$, $b=-0.249{\mathrm{s}}^{-1}$, $a_0=0\mathrm{nm}{\mathrm{s}}^{-1}$, $a_{1x}=-1.15{\mathrm{nm}}^2{\mathrm{s}}^{-1}$, $a_{1y}=-0.85{\mathrm{nm}}^2{\mathrm{s}}^{-1}$, $a_2+a_{2x}=-1.2{\mathrm{nm}}^4{\mathrm{s}}^{-1}$, $2a_2+a_{2xy}=-2.6{\mathrm{nm}}^4{\mathrm{s}}^{-1}$, $a_2+a_{2y}=-0.8{\mathrm{nm}}^4{\mathrm{s}}^{-1}$, $a_{3x}=1.2\mathrm{nm}{\mathrm{s}}^{-1}$, $a_{3y}=0.8\mathrm{nm}{\mathrm{s}}^{-1}$, with rotation period $T=4\mathrm{s}$, $T=400\mathrm{s}$, $T=1200\mathrm{s}$, no rotation (from top to bottom, left to right). For $T=4\mathrm{s}$ the superimposed oscillations are not visible at this resolution.

Figure 13

Surface morphologies for a target rotated with $T=400\mathrm{s}$ (other parameters are the same as for Fig. 12). At times $t=9200\mathrm{s}$ (left-hand panel) and $t=9250\mathrm{s}$ (right-hand panel). The lateral size is $x=200\mathrm{nm}$. At $t=9300\mathrm{s}$ the pattern looks like the one at $t=9200\mathrm{s}$ but with ripples aligned along the $y$ axis.

Figure 14

Basic modes ● and first excited modes ×. (a) The superposition of two ripples in the $x$ and the $y$ direction yields squares; (b) for superstructures an additional basic mode is added to the square pattern plus the corresponding first excited modes; (c) hexagons which contain the ripple mode in the $k_x$ direction; (d) the combination of all basic modes and the resulting first excited modes—only the modes in the first quadrant are relevant for the system of ordinary differential equations $D$.