Continuum Model for Chaotic Pattern Dynamics on Au Surfaces Sputtered by Focused Ion Beam

Under bombardment by a rastered 30keV Ga$^+$ ion beam, a flat Au surface is found to exhibit the well known sputter ripple instability with a characteristic lateral length scale on the order of 100nm and an RMS saturation height on the order of 10nm. Using {\it in situ} SEM imaging, we are able observe the dynamics of these ripples as they form and evolve. Accurate topography data is also gathered using {\it ex situ} AFM. These experimental data are compared to 2D numerical solutions of the dimensionless partial differential equation $\partial_t{h}=-\nabla^2{h}-\nabla^4{h}-\alpha\left|\nabla{h}\right|^2+\beta\nabla^2\left|\nabla{h}\right|$, which capture the essential features of the sputter ripples. A semi-implicit spectral method is used to solve the equation on a $128\times128$ grid covering a $20(2\pi)\times20(2\pi)$ periodic domain. A length scale near $2\pi\sqrt{2}$, consistent with linearized stability is observed, as is a saturation height of order $1$ when the constraint $\alpha^2+\beta^2=1$ is enforced. Interestingly, the ratio $\alpha/\beta$ is found to control the timescale of the chaotic post-saturation dynamics in addition to fine tuning the length and height scales.