Designing steep, sharp patterns on uniformly ion-bombarded surfaces

摘要

We propose and experimentally test a method to fabricate patterns of steep,sharp features on surfaces, by exploiting the nonlinear dynamics of uniformlyion bombarded surfaces. We show via theory, simulation, and experiment, thatthe steepest parts of the surface evolve as one-dimensional curves which movein the normal direction at constant velocity. The curves are a special solutionto the nonlinear equations that arises spontaneously whenever the initialpatterning on the surface contains slopes larger than a critical value;mathematically they are traveling waves (shocks) that have the special propertyof being undercompressive. We derive the evolution equation for the curves byconsidering long-wavelength perturbations to the one-dimensional travelingwave, using the unusual boundary conditions required for an undercompressiveshock, and we show this equation accurately describes the evolution of shapeson surfaces, both in simulations and in experiments. Because evolving acollection of one-dimensional curves is fast, this equation gives acomputationally efficient and intuitive method for solving the inverse problemof finding the initial surface so the evolution leads to a desired targetpattern. We illustrate this method by solving for the initial surface that willproduce a lattice of diamonds connected by steep, sharp ridges, andexperimentally demonstrating the evolution of the initial surface into thetarget pattern.