Producing nanodot arrays with improved hexagonal order by patterning surfaces before ion sputtering

When the surface of a nominally flat binary material is bombarded with a broad, normally incident ion beam, disordered hexagonal arrays of nanodots can form. Shipman and Bradley have derived equations of motion that govern the coupled dynamics of the height and composition of such a surface [Shipman and Bradley, Phys. Rev. B 84, 085420 (2011)]. We investigate the influence of initial conditions on the hexagonal order yielded by integration of those equations of motion. The initial conditions studied are hexagonal and sinusoidal templates, straight scratches, and nominally flat surfaces. Our simulations indicate that both kinds of templates lead to marked improvements in the hexagonal order if the initial wavelength is approximately equal to or double the linearly selected wavelength. Scratches enhance the hexagonal order in their vicinity if their width is close to or less than the linearly selected wavelength. Our results suggest that prepatterning a binary material can dramatically increase the hexagonal order achieved at large ion fluences.

Figure 1

(a) A section of the simulation result shown in Fig. 3 with blue dots indicating the nanodot peaks. (b) Around each peak, a circle of radius $p=5$ has been drawn. (c) The connectivity parameter $p$ has increased to 10.5 and red edges have been placed between the centers of circles that enclose each other's centers. Only the circles that enclose each other's centers are shown for clarity. (d) When $p=11$, there are four filled in triangles where three circles all enclose each other's centers. Only the circles related to the filled in faces are shown. (e) When $p=12.5$, many faces have been filled in and holes have emerged. The eight white polygonal regions with red edge boundaries are identified as holes at this value of $p$. (f) A plot of $p_{\mathrm{end}}$ versus $p_{\mathrm{start}}$ for the holes found by our persistent homology analysis of (a). The green squares (red circles) are holes which are present (absent) in (e). The line $p_{\mathrm{end}}=p_{\mathrm{start}}$ is also shown. A point's vertical distance above the line is the corresponding hole's persistence interval length.

Figure 2

The height (a) and the magnitude of the Fourier transform (b) of a nontemplated surface after integrating to time $t={10}^4$. In order to prevent the central peak from dominating the plot, the gray scale in (b) is capped at the maximum value of the magnitudes of the Fourier modes within the annulus of linearly unstable wave vectors. This is also done in all subsequent figures showing Fourier transforms.

Figure 3

The height (a) and the magnitude of the Fourier transform (b) of a nontemplated surface after integrating to time $t={10}^4$.

Figure 4

The height (a) and the magnitude of the Fourier transform (b) of a hexagonally templated surface after integrating to time $t={10}^4$. The initial wavelength was ${\lambda }_1=20\simeq 2{\lambda }_T$.

Figure 5

The height (a) and the magnitude of the Fourier transform (b) of a hexagonally templated surface after integrating to time $t={10}^4$. The initial wavelength was ${\lambda }_1=400/38\simeq {\lambda }_T$.

Figure 6

The height (a) and the magnitude of the Fourier transform (b) of a hexagonally templated surface after integrating to time $t={10}^4$. The initial wavelength was ${\lambda }_1=400/18\simeq 22.2$.

Figure 7

The deviation of the surface composition from its steady-state value for the hexagonally templated surface after integrating to time $t={10}^4$. The initial wavelength was ${\lambda }_1=400/38\simeq {\lambda }_T$. The simulation is the same as the one which produced the surface height seen in Fig. 4.

Figure 8

The $H_1$ sum versus the ratio $k_1/k_T$ for the hexagonal templates after integrating to time $t={10}^4$, averaged over 10 realizations. The two horizontal blue lines show the $H_1$ sum averaged over 10 nontemplated initial surfaces after integrating to time $t={10}^4$, plus or minus the standard deviation.

Figure 9

The mean of the nearest-neighbor distribution $\mathrm{\Lambda }\left(n\right)$ versus the ratio $k_1/k_T$ for the hexagonal templates after integrating to time $t={10}^4$, averaged over 10 realizations. The two horizontal blue lines show the mean of $\mathrm{\Lambda }\left(n\right)$ averaged over 10 nontemplated initial surfaces after integrating to time $t={10}^4$, plus or minus the standard deviation.

Figure 10

The variance of the nearest-neighbor distribution $\mathrm{\Lambda }\left(n\right)$ versus the ratio $k_1/k_T$ for the hexagonal templates after integrating to time $t={10}^4$, averaged over 10 realizations. The two horizontal blue lines show the variance of $\mathrm{\Lambda }\left(n\right)$ averaged over 10 nontemplated initial surfaces after integrating to time $t={10}^4$, plus or minus the standard deviation.

Figure 11

The surface height (a) and the magnitude of the Fourier transform (b) after integrating to time $t={10}^4$ for a sinusoidal template with ${\lambda }_2=20\simeq 2{\lambda }_T$.

Figure 12

The surface height (a) and the magnitude of the Fourier transform (b) after integrating to time $t={10}^4$ for a sinusoidal template with ${\lambda }_2=400/39\simeq {\lambda }_T$.

Figure 13

Surface height (a) and the magnitude of the Fourier transform (b) after integrating to time $t={10}^4$ with a scratch of width $2\sigma =4$. The red vertical lines indicate the approximate boundary of the initial scratch.

Figure 14

Surface height (a) and the magnitude of the Fourier transform (b) after integrating to time $t={10}^4$ with a scratch of width $2\sigma \simeq 13.86$. The red vertical lines indicate the approximate boundary of the initial scratch.

Figure 15

Surface height (a) and the magnitude of the Fourier transform (b) after integrating to time $t={10}^4$ with a scratch of width $2\sigma \simeq 20.40$. The red vertical lines indicate the approximate boundary of the initial scratch.

Figure 16

The $H_1$ sum per unit area versus strip width for the initial conditions with a scratch of width 4 (a) and with a scratch of width approximately $2{\lambda }_T$ (b) after integrating to time $t={10}^4$, averaged over 10 realizations.