Nanoscale pattern formation produced by ion bombardment of a rotating target: The decisive role of the ion energy

We study the nanoscale patterns that form on the surface of a rotating sample of an elemental material that is bombarded with a broad noble gas ion beam for angles of incidence $\theta$ just above the critical angle for pattern formation ${\theta }_c$. The pattern formation depends crucially on the ion energy $E$. In simulations carried out in the low-energy regime in which sputtering is negligible, we find disordered arrays of nanoscale mounds (nanodots) that coarsen in time. Disordered arrays of nanodots also form in the high-energy regime in which there is substantial sputtering, but no coarsening occurs close to the threshold angle. Finally, for values of $E$ just above the sputter yield threshold, nanodot arrays with an extraordinary degree of hexagonal order emerge for a range of parameter values, even though there is a broad band of linearly unstable wavelengths. This finding might prove to be useful in applications in which highly ordered nanoscale patterns are needed.

Figure 1

Results from a simulation of Eq. (51) with $\lambda =r=0$ and $\nu =1$ starting with a low-amplitude spatial white noise initial condition. The surface height at time $t=1000$ is shown in (a), along with its power spectral density in the inset. (b) shows the magnitude of the gradient of the surface shown in (a). In (c), the 2D gradient distribution is plotted for the surface shown in (a). (d) shows the probability distribution of the gradient's magnitude at three different times: $t=100$ in blue, $t=250$ in green, $t=1000$ in magenta, and $t=1500$ in black. The simulation parameters were $N=512$ and $L=30\pi$.

Figure 2

$u(x,0,t)$ versus $x$ for two simulations of Eq. (61) at three different times. In (a), the initial condition was an upright Gaussian of height 10 and width $\sigma =1$. Blue is time $t=0$, orange is $t=0.17$, and green is $t=0.178$. In (b), the initial condition was an inverted Gaussian of depth 10 and width $\sigma =1$. Blue is time $t=0$, orange is $t=0.5$, and green is $t=10$. The simulation parameters were $L=5$, $N=128$, and $\mathrm{\Delta }t=0.0001$.

Figure 3

Images taken from simulations of Eq. (50) at time $t=2000$ with a low-amplitude noise initial condition and parameters $\lambda =r=0$ and (a) $\nu =0.06$, (b) $\nu =0.05$, and (c) $\nu =0.04$. High points are light and low points are dark. The blue horizontal lines indicate the directions of line scans along which data is shown in (a′), (b′), and (c′). The graphs (a′)–(c′) show the surface height $u$ in blue, the slope $u_x$ in green, the curvature $u_{xx}$ in magenta, and the inverted Gaussian fits to the curvature in black. For these simulations, $N=256$ and $L=5\pi$.

Figure 4

Values of the (a) depth and (b) width of the Gaussian fits to the $u_{xx}$ profile along a line passing through a spike for small values of $\nu$. These results were obtained from simulations of Eq. (50) starting with a low-amplitude noise initial condition for the parameter values $\lambda =r=0$, the same values used in Fig. 3. The data displayed are for time $t=2000$. For these simulations, $N=256$ and $L=5\pi$.

Figure 5

Simulations of Eq. (50) starting from a low-amplitude spatial noise initial condition for the parameter values $\lambda =0$ and $\nu =0.1$. The nonprimed, single prime, and double primes refer to times $t=50$, $t=150$, and $t=1000$, respectively. The values of $r$ were (row a) $r=0.1$, (row b) $r=0$, (row c) $r=-0.1$, and (row d) $r=-1$. The insets show the PSD of the surface.

Figure 6

2D gradient distributions for panels (a′′) - (d′′) of Fig. 5. The values of $r$ were (a′′) $r=0.1$, (b′′) $r=0$, (c′′) $r=-0.1$ and (d′′) $r=-1$.

Figure 7

Results from a simulation of Eq. (50) with $\lambda =-0.2$, $r=0$, and $\nu =0.05$ starting from a low-amplitude spatial white noise initial condition. The surface height with the PSD in the insets is shown at times (a) $t=150$, (b) $t=500$, (c) $t=1000$, and (d) $t=2000$.

Figure 8

Results at time $t=3000$ from a simulation of Eq. (50) with the same parameters as in Fig. 7. The surface height is shown in (a) and the corresponding PSD in (b).

Figure 9

A zoomed-in view of a simulation with the same parameters used in Fig. 7. The plotted domain has a side length of 47.5. The surface height is shown at times (a) $t=500$, (b) $t=510$, (c) $t=520$, and (d) $t=530$. Blue circles are included to indicate the region that shows the fission of a nanodot into two nanodots.

Figure 10

A zoomed-in view of a simulation with the same parameters used in Fig. 7. The plotted domain has a side length of 32.8. The surface height is shown at times (a) $t=500$, (b) $t=510$, (c) $t=520$, and (d) $t=530$. Blue circles are included to indicate the region of interest. Two nanodots merge into one between (a) and (b). This nanodot then splits into two between (b) and (c).

Figure 11

Results from a simulation of Eq. (51) with $\lambda =-0.2$, $r=0$, and $\nu =0.05$ starting from a low-amplitude spatial white noise initial condition. The surface height with the PSD in the insets is shown at times (a) $t=150$, (b) $t=500$, (c) $t=1000$, and (d) $t=2000$.

Figure 12

A comparison of 2D gradient distributions for a simulation with and without the GCT. In (a), the 2D gradient distribution corresponding to Fig. 7 is shown. In (b), the 2D gradient distribution corresponding to Fig. 11 is shown.

Figure 13

Results from four simulations of Eq. (50) with different values of $\lambda$, $r$, and $\nu$ at time $t=2000$. The parameter values used were (a) $\lambda =-0.1$, $r=0$, and $\nu =0.05$; (b) $\lambda =-0.2$, $r=0$, and $\nu =0.04$; (c) $\lambda =-0.2$, $r=0.3$, and $\nu =0.04$; and (d) $\lambda =-0.2$, $r=-0.05$, and $\nu =0.04$.

Figure 14

Normalized $H_1$ values averaged over times $t=1900$ to $t=2000$ for simulations of Eq. (50) in which the three parameters were varied separately. In (a), the values $r=0$ and $\nu =0.05$ were fixed while $\lambda$ ranged from $-1$ to 1. In (b), the values $\lambda =-0.2$ and $\nu =0.05$ were fixed while $r$ ranged from $-0.5$ to 0.5. In (c), the values $\lambda =-0.2$ and $r=0$ were fixed while $\nu$ ranged from 0.04 to 0.3. The red horizontal lines correspond to the normalized $H_1$ sum for simulations with $r=\lambda =0$ and $\nu =0.05$.

Figure 15

Results from a simulation of Eq. (50) with $\lambda =-0.55$, $r=0$, and $\nu =0.05$ starting from a low-amplitude spatial white noise initial condition. The surface height with the PSD in the insets is shown at times (a) $t=150$, (b) $t=500$, (c) $t=1000$, and (d) $t=2000$.