Theory of nanoscale ripple topographies produced by ion bombardment near the threshold for pattern formation

Nanoscale pattern formation on the surface of a solid that is bombarded with a broad ion beam is studied for angles of ion incidence, theta, just above the threshold angle for ripple formation, theta(c). We carry out a systematic expansion in powers of the small parameter epsilon (theta - theta(c))(1/2) and retain all terms up to a given order in epsilon. In the case of two diametrically opposed, obliquely incident beams, the equation of motion close to threshold and at sufficiently long times is rigorously shown to be a particular version of the anisotropic Kuramoto-Sivashinsky equation. We also determine the long-time, near-threshold scaling behavior of the rippled surface's wavelength, amplitude, and transverse correlation length for this case. When the surface is bombarded with a single obliquely incident beam, linear dispersion plays a crucial role close to threshold and dramatically alters the behavior: highly ordered ripples can emerge at sufficiently long times and solitons can propagate over the solid surface. A generalized crater function formalism that rests on a firm mathematical footing is developed and is used in our derivations of the equations of motion for the single and dual beam cases.