Creating sharp features by colliding shocks on uniformly irradiated surfaces

Holmes-Cerfon, Miranda; Aziz, Michael J.; Brenner, Michael P.

doi:10.1103/physrevb.85.165441

Cited by 12 publications

(17 citation statements)

References 26 publications

(23 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Another successful application of the nonlinear generalization of the BH equation occurs in a case in which, rather than assuming a small-slope expansion as in Eqs. (7) and (8), arbitrary values of the slope are allowed for [215][216][217]. This has been shown to provide a good description for the propagation of high steps or ridges on Si targets [215,216], even leading to a fabrication method for patterns made of knife-edge ridges [217].…”

Section: Refinements Of Bradley-harper Theorymentioning

confidence: 99%

Self-organized nanopatterning of silicon surfaces by ion beam sputtering

Muñoz-García

Vázquez

Castro

et al. 2014

Materials Science and Engineering: R: Reports

164

201

View full text Add to dashboard Cite

Section: Refinements Of Bradley-harper Theorymentioning

confidence: 99%

Self-organized nanopatterning of silicon surfaces by ion beam sputtering

Muñoz-García

Vázquez

Castro

et al. 2014

Materials Science and Engineering: R: Reports

164

201

View full text Add to dashboard Cite

“…These ridges were shown in [24] to be a special solution to (1) that arises whenever steep regions propagating in opposite directions collide. The steepness and radius of curvature of this solution were shown to be fixed numbers that depend on the material, ion, and energy via R(b), and we expect that certain materials can achieve much smaller length scales [23]. Because identical ridges arise spontaneously, they are a useful structure to consider for patterns as they are not sensitive to the initial condition.…”

Section: A Evolving An Initial Condition To Produce a Target Morphologymentioning

confidence: 89%

“…A general one-dimensional traveling wave solution has the form h x = s(x − ct) and solves [23]. The wave speed is found by integrating from −∞ to +∞ to be c = (R(b r ) − R(b l ))/(b r − b l ).…”

Section: Knife Edge Ridge Curving After Initial Formationmentioning

confidence: 99%

Designing steep, sharp patterns on uniformly ion-bombarded surfaces

Perkinson

Aziz

Brenner

et al. 2016

Proc. Natl. Acad. Sci. U.S.A.

Self Cite

View full text Add to dashboard Cite

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 onedimensional 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 onedimensional 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.ion bombardment | steep structure design | simulations | FIB | shock wave F abricating steep, sharp features with desired morphologies on surfaces is a major challenge of materials science. Certain methods are available by direct engraving, such as focused ion beam (FIB) or lithography (1-4), but these require enormous time and energy. A promising method to efficiently make patterns on a large scale by exploiting dynamics is to erode a surface with uniform ion bombardment (5-9). A flat surface can become unstable and develop features such as quantum dots or hexagonal patterns (10-17). Such spontaneous pattern growth could spawn high-throughput methods to manufacture periodic metamaterials such as optical antenna arrays (18) and the split ring resonators used in negative refractive index materials and optical cloaking (19).However, because linear instabilities are neither small enough nor amenable enough to control, interest has recently turned to the potential for nonlinear dynamics to create even steeper, sharper features (20). Large-amplitude, steep structures are of interest for 3D engineering applications such as micromechanics, microprocessor integration, data storage, and photonic band gap waveguides. They are also of interest in atom probe tomography, which uses samples frequently created by FIB ion irradiation (21) and which is strongly influenced by the geometry of the steep-walled final samples (22). Experiments and simulations have shown that knife-edge-like ridges, varying on scales at least an order of magnitude smaller than those accessible to linear instabilities, can arise ...

show abstract

“…In a study by Holmes-Cerfon et al (12), it was proposed that two undercompressive waves could be collided to form isolated steep ridges. A fully 2D experiment was realized in a magnesium compared well with fully 2D simulations of the nonlinear model (13).…”

mentioning

confidence: 99%

Designer shocks for carving out microscale surface morphologies

Bertozzi¹

2016

Proc. Natl. Acad. Sci. U.S.A.

View full text Add to dashboard Cite

Shockwaves are propagating disturbances with a long history of study in gas dynamics, fluid dynamics, and astrophysics. We also see examples of shocks in everyday life such as a traffic jam, in which the oncoming traffic has much lower density than the cars within the traffic jam.A classical compression wave involves propagation of a discontinuity in which information is absorbed from both sides in the shock layer. Undercompressive shocks are more unusual; they have special properties, including the transfer of information through the shock and often different stability properties than their compressive cousins. In the past decade, undercompressive waves have been studied in microscale and nanoscale applications in which surface forces dominate the physics in the shock layer. These forces can permit the existence of undercompressive waves, and their utility is just now coming to fruition. In PNAS, Perkinson et al.(1) demonstrate how to use undercompressive shocks in ion-bombarded surfaces to create patterns with steep ridges on a micrometer scale.The traffic jam example is one that can be modeled with a simple 1D, first-order, nonlinear wave equation, or "conservation law," introduced in the mid-1950s (2, 3):Here, u is the material being transported in the direction x and f is the flux of the material. A shock is a solution to such an equation with a discontinuity in u that travels with speed s given by the RankineHugoniot jump condition:Such models can be solved exactly for any choice of states u L and u R on the left and right of the discontinuity. All such shockwaves are compressive, meaning that they satisfy an entropy condition:The speed s of the shock is faster than the characteristics speed f ′(u R ) ahead of it and slower than the speed f ′(u L ) behind it. It was traditionally thought that any physical process described by such a simple 1D model could have only compressive shocks. This fact can be proved rigorously in the case where the physics in the shock layer is "diffusion" or Brownian motion, as is seen in gas dynamics. The modification of Eq. 1 to include linear diffusion in the shock layer isWith diffusion, shock discontinuities are smoothed; however, the basic shock structure, including the speed of the shock as determined by the RankineHugoniot jump condition, is exhibited in smooth traveling wave solutions of Eq. , who showed that it produced undercompressive shocks that could be reproduced in experiments. The model is a conservation law in which u is the slope of the surface and the nonlinear flux function f is the yield function, which gives the average velocity of erosion of the surface as a function of its local slope, and is, in general, nonconvex. Their model has a fourth-order term that models additional smoothing effects such as Mullins-Herring surface diffusion or ion-enhanced viscous flow confined to a thin surface layer. They showed that stable steep undercompressive waves were experimentally viable in ion-bombarded surfaces with common physical parameters.In the case of compressive...

show abstract

Creating sharp features by colliding shocks on uniformly irradiated surfaces

Cited by 12 publications

References 26 publications

Self-organized nanopatterning of silicon surfaces by ion beam sputtering

Self-organized nanopatterning of silicon surfaces by ion beam sputtering

Designing steep, sharp patterns on uniformly ion-bombarded surfaces

Designer shocks for carving out microscale surface morphologies

Contact Info

Product

Resources

About