Designing steep, sharp patterns on uniformly ion-bombarded surfaces

Joy C. Perkinson, Michael J. Aziz, Michael P. Brenner, Miranda Holmes-Cerfon

Research output: Contribution to journalArticlepeer-review


We propose and experimentally test a method to fabricate patterns of steep, sharp features on surfaces, by exploiting the nonlinear dynamics of uniformly ion-bombarded surfaces. We show via theory, simulation, and experiment that the steepest parts of the surface evolve as one-dimensional curves that move in the normal direction at constant velocity. The curves are a special solution to the nonlinear equations that arises spontaneously whenever the initial patterning on the surface contains slopes larger than a critical value; mathematically they are traveling waves (shocks) that have the special property of being undercompressive. We derive the evolution equation for the curves by considering long-wavelength perturbations to the one-dimensional traveling wave, using the unusual boundary conditions required for an undercompressive shock, and we show this equation accurately describes the evolution of shapes on surfaces, both in simulations and in experiments. Because evolving a collection of one-dimensional curves is fast, this equation gives a computationally efficient and intuitive method for solving the inverse problem of finding the initial surface so the evolution leads to a desired target pattern. We illustrate this method by solving for the initial surface that will produce a lattice of diamonds connected by steep, sharp ridges, and we experimentally demonstrate the evolution of the initial surface into the target pattern.

Original languageEnglish (US)
Pages (from-to)11425-11430
Number of pages6
JournalProceedings of the National Academy of Sciences of the United States of America
Issue number41
StatePublished - Oct 11 2016


  • Fib
  • Ion bombardment
  • Shock wave
  • Simulations
  • Steep structure design

ASJC Scopus subject areas

  • General


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