Producing virtually defect-free nanoscale ripples by ion bombardment of rocked solid surfaces

Bombardment of a solid surface with a broad, obliquely incident ion beam frequently produces nanoscale surface ripples. The primary obstacle that prevents the adoption of ion bombardment as a nanofabrication tool is the high density of defects in the patterns that are typically formed. Our simulations indicate that ion bombardment can produce nearly defect-free ripples on the surface of an elemental solid if the sample is concurrently and periodically rocked about an axis orthogonal to the surface normal and the incident beam direction. We also investigate the conditions necessary for rocking to produce highly ordered ripples and discuss how the results of our simulations can be reproduced experimentally. Finally, our simulations show that periodic temporal oscillations of coefficients in the Kuramoto-Sivashinsky equation can suppress spatiotemporal chaos and lead to patterns with a high degree of order.

Figure 1

Plots of the rocked surface in real space (inset) and in Fourier space for $r_1=2, r_2=0$, dimensionless frequency $\tilde{\omega }\equiv \omega B/A_0^2=0.15\pi$, domain length $L=100$, and $\tilde{t}={10}^3$.

Figure 2

Plots of the unrocked surface in real space (inset) and in Fourier space for $r_1=r_2=0, L=100$, and $\tilde{t}={10}^3$.

Figure 3

Space-time plots of the surface for (a) $r_1=2$ (rocked) and (b) $r_1=0$ (unrocked), respectively, with $\tilde{\omega }=0.15\pi$ and $r_2=0$. A shorter time scale was used for the unrocked case so that the finer structure is visible. For clarity, these plots show the deviation of the surface height from its average value.

Figure 4

Fourier peak width as a function of the scaled rocking frequency $\tilde{f}\equiv \tilde{\omega }/\left(2\pi \right)$ for two values of $r_1$ and with $r_2=0$. Each point represents a single simulation, and the average for each value of the frequency is shown. Note that for $r_1=0.5$ (upper, lighter line), $\lambda$ always has the same sign, while for $r_1=4$ (lower, darker line), it changes sign.

Figure 5

Fourier peak width for $r_1=4$ and $\tilde{\omega }=0.15\pi$ for opposite signs of $r_2$. For $r_2<0$ the surface continues to form nearly perfect ripples for relatively large values of $|r_2|$.

Figure 6

Real space plots of surfaces produced by (a) a sinusoidal and (b) a discrete variation of $\lambda$, respectively. For both surfaces $\tilde{\omega }=0.15\pi , r_2=0, L=100$, and $t={10}^3$. In (a) $\lambda =1+5sin\left(\omega t\right)$, whereas in (b) $\lambda =1+5\text{sgn}[sin(\omega t\left)\right]$.

Figure 7

(a)–(c) A time series of a surface for a square domain of side length $L=120$ with $A=A^{\prime }=1, B=1, \lambda =2+8\text{sgn}[sin\left(\tilde{\omega }\tilde{t}\right)], {\lambda }^{\prime }=2$, and $\tilde{\omega }=0.15\pi$ at times $\tilde{t}=106,330$, and 1840. (d) A simulation with the same parameters as in (a)–(c) but with $\lambda =2$ so that there is no rocking. The time is $\tilde{t}=1840$.

Figure 8

A plot of the surface for a square domain of side length $L=120$ with $A=A^{\prime }=1, B=1, \lambda =2+8\text{sgn}[sin\left(\tilde{\omega }\tilde{t}\right)], {\lambda }^{\prime }=2+0.2\text{sgn}[sin\left(\tilde{\omega }\tilde{t}\right)]$, and $\tilde{\omega }=0.15\pi$ at time $\tilde{t}=1840$.