Designing steep, sharp patterns on uniformlyion-bombarded surfacesJoy C. Perkinsona, Michael J. Aziza, Michael P. Brennera, and Miranda Holmes-Cerfonb,1

aHarvard School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138; and bCourant Institute of Mathematical Sciences, NewYork University, New York, NY 10012

Edited by Susan N. Coppersmith, University of Wisconsin, Madison, WI, and approved August 9, 2016 (received for review June 8, 2016)

We propose and experimentally test a method to fabricate patternsof steep, sharp features on surfaces, by exploiting the nonlineardynamics of uniformly ion-bombarded surfaces. We show via theory,simulation, and experiment that the steepest parts of the surfaceevolve as one-dimensional curves that move in the normal directionat constant velocity. The curves are a special solution to the nonlinearequations that arises spontaneously whenever the initial patterningon the surface contains slopes larger than a critical value; mathemat-ically they are traveling waves (shocks) that have the special propertyof being undercompressive. We derive the evolution equation for thecurves by considering long-wavelength perturbations to the one-dimensional traveling wave, using the unusual boundary conditionsrequired for an undercompressive shock, and we show this equationaccurately describes the evolution of shapes on surfaces, both insimulations and in experiments. Because evolving a collection of one-dimensional curves is fast, this equation gives a computationallyefficient and intuitive method for solving the inverse problem offinding the initial surface so the evolution leads to a desired targetpattern. We illustrate this method by solving for the initial surfacethat will produce a lattice of diamonds connected by steep, sharpridges, and we experimentally demonstrate the evolution of theinitial surface into the target pattern.

ion bombardment | steep structure design | simulations | FIB | shock wave

Fabricating steep, sharp features with desired morphologies onsurfaces is a major challenge of materials science. Certain

methods are available by direct engraving, such as focused ionbeam (FIB) or lithography (1–4), but these require enormoustime and energy. A promising method to efficiently make pat-terns on a large scale by exploiting dynamics is to erode a surfacewith uniform ion bombardment (5–9). A flat surface can becomeunstable and develop features such as quantum dots or hexag-onal patterns (10–17). Such spontaneous pattern growth couldspawn high-throughput methods to manufacture periodic meta-materials such as optical antenna arrays (18) and the split ringresonators used in negative refractive index materials and opticalcloaking (19).However, because linear instabilities are neither small enough

nor amenable enough to control, interest has recently turned to thepotential for nonlinear dynamics to create even steeper, sharperfeatures (20). Large-amplitude, steep structures are of interest for3D engineering applications such as micromechanics, micropro-cessor integration, data storage, and photonic band gap waveguides.They are also of interest in atom probe tomography, which usessamples frequently created by FIB ion irradiation (21) and which isstrongly influenced by the geometry of the steep-walled final samples(22). Experiments and simulations have shown that knife-edge-likeridges, varying on scales at least an order of magnitude smaller thanthose accessible to linear instabilities, can arise spontaneously onuniformly bombarded surfaces, provided the initial surface containsslopes beyond a critical value (23, 24). This demonstration suggeststhat one may be able to create steep, sharp features by first pre-patterning a surface on the macroscale—something that isrelatively easy to achieve—and then bombarding it uniformly tolet even steeper, sharper features form spontaneously.

In this paper we propose a theory to design patterns on asurface using their nonlinear dynamics, and we validate our ap-proach experimentally. We argue that there is a dynamical re-gime where the full nonlinear dynamics can be approximatedby evolving a collection of one-dimensional curves at constantspeed in the normal direction, and we show this model accuratelydescribes the experimental behavior of steep-walled pits propa-gating under uniform ion irradiation approximated via FIB ras-tering. This model can be used to solve the inverse problem ofdetermining the initial surface pattern that will evolve underuniform ion bombardment to a desired target pattern, and as ademonstration we numerically design and experimentally createa lattice where the scale of the lattice pattern is many timessmaller than the scale of the initial patterning.Our model has several advantages over a direct numerical sim-

ulation of the nonlinear equations for surface evolution. First, it canrapidly determine how a given initial condition will evolve, for ex-ample by directly evolving the curves or using level set methods(25, 26), so the inverse problem can also be solved efficiently, forexample using Monte-Carlo methods. Second, the full nonlinearequations require quantitative information about the macroscopiceffects of uniform ion bombardment, such as the yield function(atoms ejected per incident ion as a function of angle of incidence)and the magnitude and type of smoothing physics, and this in-formation is not always known or easy to obtain (27, 28). Our modelrequires knowing only one material-dependent parameter that canbe measured through simple experiments, so measuring the fullyield function and smoothing physics is unnecessary. Finally, this

Significance

Large-amplitude, steep structures are of interest for a varietyof applications from micromechanics to data storage. Manysuch structures must be manufactured via effort-intensive di-rect engraving. Here, we present and experimentally test amethod for designing simple structures that will evolve into adesired steep structure under uniform ion irradiation. Thisprocess is based on the mathematics of shock waves, a theoryoriginally developed to understand fluids and gases. We showthat the dynamics of uniformly bombarded surfaces are con-trolled by special, so-called undercompressive shocks, whichmanifest as steep parts of the surface that arise spontaneouslyand move predictably. Their evolution equation requires onlyone parameter that is easily measured. The proposed process isscalable with promise for high-throughput manufacturing.

Author contributions: J.C.P., M.J.A., M.P.B., and M.H.-C. designed research; J.C.P. andM.H.-C. performed research; J.C.P. performed focused ion beam/SEM experiments;M.H.-C. performed simulations; M.H.-C. contributed new reagents/analytic tools; J.C.P.and M.H.-C. analyzed data; and J.C.P. and M.H.-C. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

See Commentary on page 11384.1To whom correspondence should be addressed. Email: holmes@cims.nyu.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1609315113/-/DCSupplemental.

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method is intuitive, so an approximate guess for the initial patterncan be made without any sophisticated numerical techniques.

Results and DiscussionEvolving an Initial Condition to Produce a Target Morphology. Webegin by performing numerical simulations of the fully nonlinear,2D equations for the evolution of the surface height hðx, y, tÞ underuniform ion bombardment, commonly described by the Sigmundtheory of sputter erosion (6). When the surface slope varies overlength scales much larger than the lateral scale over which an iondeposits its kinetic energy, the sputter integral can be expanded toyield the following nonlinear partial differential equation (20, 29):

ht +RðbÞ+B0∇ ·�

1ffiffiffiffiffiffiffiffiffiffiffiffi1+ b2

p ∇κ�= 0. [1]

The first-order term RðbÞ is the average velocity of erosion of thesurface as a function of its slope b= j∇hj (or equivalently the angleof the incoming ion beam). The fourth-order term with magnitudeB0 is a function of the surface curvature κ=∇ · ðð1+ b2Þ−1=2∇hÞ,which models additional smoothing effects such as Mullins–Herring surface diffusion (30, 31) or ion-enhanced viscous flowconfined to a thin surface layer (32). Eq. 1 can also include sec-ond-order (curvature) terms (7, 33), but we neglect these becausethe nonlinear dynamics we wish to model can only occur whenthese are small (24).The erosion function RðbÞ is related to the sputter yield Y ðbÞ by a

constant of proportionality that changes its units from [atoms out/ions in] to [length/time]. This constant, as well as steep featurepropagation speed, vary with ion flux. Therefore, whereas we mea-sure simulation progress in unitless simulation time, it is more in-formative to track experimental progress with area dose, a measureof ion fluence delivered per area reckoned in a plane parallel to theaverage surface. When ion flux is held constant, area dose is pro-portional to simulation time.

We numerically simulate Eq. 1 using the erosion function for30-keV Ga+ on Si (Materials and Methods) and estimate lengthscales for this material combination. Fig. 1 A–C shows simula-tions that begin with a periodic array of shapes punched out inthe surface. The shapes are initially 2.5 μm apart (centers 5 μmapart) and are steep at their boundaries with nearly zero slopeelsewhere. As the surface evolves, the steep regions remain steepwhereas their location changes with time. Eventually the steepregions collide and form ridges that are even steeper, with widthsof ≈100 nm. The resulting pattern is a lattice of diamonds con-nected by steep, narrow ridges.These ridges were shown in ref. 24 to be a special solution to Eq.

1 that arises whenever steep regions propagating in opposite di-rections collide. The steepness and radius of curvature of this so-lution were shown to be fixed numbers that depend on the material,ion, and energy via RðbÞ, and we expect that certain materials canachieve much smaller length scales (23). Because identical ridgesarise spontaneously, they are a useful structure to consider forpatterns because they are not sensitive to the initial condition.To show that we can form a similar lattice of knife-edge ridges

experimentally, we use FIB to mill four pits matching the initialshape used in simulations and then irradiate under repeatedboustrophedonic FIB rastering, delivering a low dose on each passto approximate uniform irradiation, as described in Materials andMethods. Due to discrepancies in pit wall propagation speed(Supporting Information, Speed Calculations), we milled the initialpits with centers 4.4 μm apart—closer than in simulations. Irra-diation continued until the pits impinged (Fig. 1 D and E). Theresulting structure is close to that designed through our methodand predicted by the simulations, successfully demonstrating theformation of knife-edge ridges under uniform ion irradiation. Onedifference from the simulations worth pointing out is that whenpits evolved within 200 nm of each other the pit rim closest to theadjacent pit accelerated and “reached out” to the adjacent pits.These behaviors caused impingement to occur both sooner thanexpected and over a smaller length of the rim. Bombardment was

Fig. 1. (A–C) Numerical simulation of surface height evolution under uniform ion bombardment (Eq. 1), at area doses 0 nC/μm2, 3 nC/μm2, and 6 nC/μm2

(simulation times 0, 0.8, and 1.6) Pits initially spaced Oð1Þμm apart collide and form ridges with widths Oð100Þ nm. By changing the initial pattern one cancreate a large variety of patterns made of these steep, sharp ridges. (D) SEM image of the fabricated initial condition predicted to evolve into knife-edgeridges, and (E) the knife-edge ridges formed after ion irradiation in the FIB, both imaged at 54° off-normal. (Scale bars, 1 μm.)

11426 | www.pnas.org/cgi/doi/10.1073/pnas.1609315113 Perkinson et al.

continued after this impingement to extend the lateral dimensionsof the knife edges, and so the knife edge evolved into a curvedridge. Indeed, our simulations show that if bombardment con-tinues beyond the moment of impingement, the knife-edge ridgesmay evolve into the curved shaped similar to those observed inexperiment (Fig. S1).

One-Dimensional Curve Evolution Equations.We designed the patternof ridges by solving an inverse problem for the initial conditions,and we now explain these calculations. The basic observation isthat we can draw a curve through the steepest parts of the surfaceat every time step and watch the curve evolve in time. Our goal isto derive an evolution equation for the curve using only the curve’sintrinsic, one-dimensional geometry, and not any informationaway from the region of high gradient.To look for such an equation we recall the theory developed in

ref. 23. This theory showed that Eq. 1 has a particular travelingwave solution for the slope that is invariant in one horizontal di-mension, that is, of the form hx = sðx− ctÞ, that acts as an attractorfor the dynamics: If a surface is patterned initially to have slopesabove a critical value, then the surface steepens and locally evolvesto that traveling wave. This wave is called undercompressive in themathematical literature (34–36) because it has the nonclassicalproperty that information can propagate away from it. In oursystem it is identifiable because it connects a steep region withconstant slope b0 = sð−∞Þ to a flat region with slope 0= sð∞Þ,and it propagates at constant speed c= ðRðb0Þ−Rð0ÞÞ=ðb0 − 0Þ,where the constants b0, c depend on the material, ion, and en-ergy via RðbÞ but not on the initial condition. When we extractthe slope and speed in the simulations of shapes such as in Fig. 1we find that they are close to b0, c, suggesting that the surfaceslope can be locally approximated as a collection of under-compressive traveling waves with slowly varying phase shifts inthe transverse direction.To understand how this collection evolves we consider two

different theories. Theory 1 simply advects curves in the normaldirection with speed c. If the curve is parameterized by α asqðα, tÞ= ðxðα, tÞ, yðα, tÞÞ, it evolves according to

dqdt

= cn̂, [2]

where n̂= ∂q∂α

⊥���∂q∂α

�� is the unit normal to the curve. This theoryrequires only one parameter, the wave speed c, which can bemeasured by simple experiments. For example, one can engravea circular pit and measure the rate of change of its radius, as inref. 24. Theory 1 is a natural heuristic and is expected to bevalid when the steep parts of the surface form a curve that isnearly straight, so it can be treated locally as a one-dimensionaltraveling wave.Theory 2 seeks to describe the evolution of transverse per-

turbations to a one-dimensional traveling wave, by looking for anasymptotically consistent solution to the nonlinear, 2D equa-tions. We start by looking for a solution whose slope has the form

hx = sðx− ct+ψðy, tÞÞ+ euðx− ct, y, tÞ+O�e2�. [3]

We assume the scalings ∂y ∼Oðe1=2Þ, ∂t ∼OðeÞ, with e � 1. Wesubstitute this ansatz into the x-derivative of Eq. 1 and performa multiscale asymptotic expansion (37). At leading order is theequation for s, which is satisfied by construction. The OðeÞequation is

ut +Lu= a1ðsÞψ t + a2ðsÞψ2y + a3ðsÞψ yy + a4ðsÞψ yyyy, [4]

where aiðsÞ are functions of the traveling wave (Eqs. S5–S8) andthe linear operator L is

Lu= ∂ηððR′− cÞuÞ

+B0

∂2η

�fd∂2ηðsf Þu

�+ ∂2η

�f∂2η

��12fds2 + fs

�u��

. [5]

We write η= x− ct, f ðbÞ= ð1+ b2Þ−1=2, fdðbÞ= ddb f ðbÞ, and all func-

tions are evaluated at sðηÞ. We have included the term propor-tional to ψ yyyy from the Oðe2Þ equation because it is sometimesrequired to smooth (Supporting Information, Which Terms to In-clude in Theory 2?).The left-hand side of Eq. 4 depends only on the fast variables

η, t, so we can integrate over these to derive a solvability condi-tion. Suppose there is a function πðηÞ such that Lpπ = 0, where Lp

is the adjoint of L with respect to the L2 -inner product h ·, · i[i.e., it satisfies

RxuðLvÞdx=

RxðLpuÞvdx for all u, v with the ap-

propriate boundary conditions]. Taking the inner product withEq. 4 and requiring u to be bounded shows that hπ,RHSi= 0,where RHS is the right-hand side of the equation. Therefore, thephase will evolve on the slow timescale as

ψ t + c2ψ2y + c3ψ yy + c4ψ yyyy = 0, [6]

where ci = hπ, aiðsÞi=hπ, a1ðsÞi.To find π requires solving Lpπ = 0 with the appropriate

boundary conditions, which are πð−∞Þ= 0, πð∞Þ= 1 for theundercompressive traveling wave (35, 38–40). Using these one cancompute π numerically, and then the constants can be found bynumerical integration. The condition of decay at −∞ is unusualand is what makes this multiscale analysis novel. The conditionarises to control information that can propagate away from thetraveling wave on its undercompressive side (Supporting Informa-tion, Boundary Conditions for the Undercompressive Wave).Eq. 6 forms the basis of Theory 2. It demonstrates rigorously

that the nonlinear dynamics of an ion-bombarded surface can beapproximated (for long-wavelength pertubations) as the evolu-tion of a collection of curves on the surface, each one propa-gating at constant speed c in a certain direction and changingshape about this direction according to Eq. 6. The theory re-quires four parameters: c2, c3, c4, and c. These can be calculatednumerically if the erosion rate RðbÞ and the magnitude of thefourth-order term B0 are known for a given material. If they arenot known, they could be extracted from experiments thatmeasure the evolution of different shapes.Note that Theory 2 considers perturbations about a horizontal

reference line, so it relies on a particular coordinate system.Theory 1, however, is intrinsic: It depends only on the local ge-ometry of the traveling wave front. We expect that one could

0123

h x

1015

20 0

10

y20 4 6 8 10

-1-0.5

00.5

1

1.52

2.5

SimulationTheory 1Theory 2

5

Fig. 2. Numerical tests comparing the curve evolution Eqs. 2 and 6 to sim-ulations of the full dynamics (Eq. 1). (Left) Initial surface slope hxðx, y, 0Þ=s�x + sin

�2π10 y

��. (Right) Curve extracted from full simulation (solid line), curve

predicted by Eq. 2 (blue dashed line), and curve predicted by Eq. 6 (red dotted line),at times 2, 7.5, and 12 (corresponding to area doses 7.5, 28.125, and 45 nC/μm2).The curves at each different time have been plotted at 1/10 the actualseparation in the vertical direction.

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derive Theory 1 from the governing Eq. 1 by considering a slowlyvarying traveling wave, and additionally that one could derive thenext-order corrections to Eq. 2 as we have done in Theory 2. Wedo not do this here because, as we will show, the differencebetween the predictions of the two theories is so small as to beundetectable experimentally, but this would be an interestingquestion for future analysis.

Comparing Theories, Simulations, and Experiments. To test how wellEqs. 2 and 6 describe the propagation of steep features, wecompared them with numerical simulations of the full 2D equa-tions (Eq. 1). We started with a traveling wave computed as asteady solution to the discretized version of Eq. 1, applied a si-nusoidal perturbation to the surface slope in the transverse di-rection, evolved this surface numerically, and identified the curveby the maximum of jhxxj as a function of y at each timestep. Wecompared this curve to numerical simulations of Eqs. 2 and 6 withthe same sinusoidal initial condition (Materials and Methods).Fig. 2 shows the three curves at different times. The curves

predicted by the theories agree extremely well with the curveextracted from the simulations. This agreement is destroyedwhen the parameters are changed from their predicted values,so it is not an accident. The small discrepancies between theo-retical predictions and simulations are thought to come fromtwo sources: numerical discretization of the 2D equations andhigher-order asymptotic corrections to the theoretical curves. Itis notable that the curves predicted by both theories are alsoextremely close to each other, showing that although they makedifferent kinds of approximations they may be used roughly in-terchangeably. Therefore, Theory 1 should be preferred underthe conditions investigated here, because it is simpler.We then compared experimental pit propagation to the pre-

dictions of Theories 1 and 2. We started with a clover-shaped hole,formed by milling four overlapping circular pits (Fig. 3) with ra-dius 2.9 μm, centered at (±2.4μm, ±2.4 μm). A fifth pit with radius1.5 μm was milled at the origin to remove the extra material notremoved by the other four pits. All pits were milled in parallel tominimize the effects of Si redistribution on pit walls. The initialand final pit morphologies, imaged using SEM at 30° off-normal,are shown in Fig. 3. The horizontal pit width expands from10.8 μm to 15.7 μm after an area dose of 30 nC/μm2 Ga+.We simulated the evolution of a curve using Eqs. 2 and 6 with

initial condition set to the boundary of the clover-shaped hole. Wefit the propagation speed c to that observed experimentally, be-cause we were unable to match it quantitatively from first princi-ples (Supporting Information, Speed Calculations). The simulatedcurves are overlaid on the real-time FIB images for comparisonand shown after an area dose of 0.07 nC/μm2, 15.00 nC/μm2, and30.00 nC/μm2 in Fig. 4. Both simulations agree equally well withexperiment, overlapping at the pit walls with greatest radius of

curvature. At the four “kinks” with tighter radius of curvature, thesimulated curves vary from experiment by an average of 0.1 μm.Both models could be used essentially indistinguishably in theseexperimental systems.

Solving the Inverse Problem. Our numerical simulations and ex-perimental tests show that either Eq. 2 or Eq. 6 can be used topredict the propagation of steep regions on the surface. These areintuitive equations that make it easy to sketch by hand an ap-proximate initial condition for a surface that evolves under bom-bardment to a given final pattern. In addition, because evolving acollection of curves is fast, the inverse problem can be efficientlysolved more precisely by numerical methods, for example byMonte-Carlo simulations. To illustrate, we explain how wedesigned the lattice in Fig. 1A. Our target pattern is a periodicarray of tiles shown in Fig. 5A. The black pixels are regions wherewe wish the surface to be elevated, where the borders are intendedto be the knife-edge ridges. We will make this pattern usinga periodic array of curves, each of the polar form rðθÞ=r0ð1+RefP kmax

k=−kmaxwkeikθgÞ, where fwkgkmaxk=−kmax

are parameters tobe determined. We set kmax = 8 and require the pattern to be formedat a fixed simulation time T. These restrictions do not come fromrealistic experimental constraints, but rather are intended to illus-trate the more general principle that one can optimize over a con-strained set of initial conditions.We then used Monte-Carlo simulations to find the initial condi-

tions that lead to the desired pattern. At each Monte-Carlo step wevaried one parameter, solved Eq. 6 up to time T, and computed thecost as the sum of the absolute discrepancy between the set of pixelslying outside each closed curve and the target pattern. We discardedmoves that increased the cost with a cost-dependent probability. Fig.5B shows the optimal initial condition after a large number ofMonte-Carlo steps (blue) and the final curve that it evolves to (red).Fig. 1 A–C shows the evolution of this initial condition with a sim-ulation of Eq. 1; again there is excellent agreement.

ConclusionWe have introduced a method to make steep, sharp patterns onsurfaces by prepatterning the surface so it dynamically evolvesunder uniform bombardment to something that is much smallerin scale and more difficult to make directly. Our method is basedon the demonstration that for certain materials, ions, and ener-gies the steep parts of the surface evolve as one-dimensionalcurves that propagate at a constant speed in the normal di-rection. This simplification agrees extremely well with simula-tions of the full nonlinear dynamics, and over large enough scalesit also agrees with experimental measurements of the evolutionof steep-walled pits under rastered FIB irradiation. On smallscales, such as when steep features approach each other to formsharp ridges, the theory is not expected to apply because thegoverning equations are based on a small-curvature approxima-tion, and indeed on these scales we observed several phenomena

A

B

Fig. 3. (A) Initial milled “clover” pit and (B) milled pit after an area dose30 nC/μm2 imaged under SEM at 30° off-normal. (Scale bars, 2 μm and identical.)

Dose = 0.07 nC/µm2 Dose = 15.00 nC/µm2 Dose = 30.00 nC/µm2

Fig. 4. The evolution of the clover-shaped pit under homogeneous FIBrastering of the entire imaged area, for irradiation doses indicated. Theo-retical predictions using Eqs. 2 (blue) and 6 (red) are superimposed.

11428 | www.pnas.org/cgi/doi/10.1073/pnas.1609315113 Perkinson et al.

that we do not yet understand. In addition, we were unable toquantitatively predict the speed of the curve from first principles,but it is easy to measure and then incorporate into the theory.Therefore, we propose that this method of evolving one-dimensionalcurves may be used as a first approximation to determining a sur-face’s final structure after uniform bombardment.Evolving curves is fast, so it naturally leads to efficient meth-

ods to solve the inverse problem of determining how to patternthe surface initially so it evolves to a desired target structure. Weshowed how to make a simple lattice connecting diamonds withsteep ridges, but one can imagine making more complex patterns,for example by starting with several different shapes, nonclosedcurves, bumps instead of pits, or undulating topography, and alsoby allowing the curves to continue evolving once they intersect, toform gaps. The space of possibilities is large and we expect thatfurther understanding of the nonlinear dynamics of ion-bombardedsurfaces will lead to new methods to invent and fabricate materials.

Materials and MethodsExperimental Methods. Samples were polished (001) Si wafers from VirginiaSemiconductor, Inc. and were irradiated using a ZEISS NVision 40 FIB usingNanoPatterning and Visualization Engine (NPVE) software. Samples wereaffixed to aluminum sample stubs using silver paste, and dust was removedusing an air jet before loading into the FIB. The SEM beam was switched offduring FIB irradiation to avoid the surface carbon contamination typical ofSEM imaging. The ion beam was 30 keV Ga+, with a current of 1.5 nA, beamdiameter 200 nm, center-to-center dot spacing 100 nm, and dwell time 1.0 μs.Steep-walled pits were milled by rastering the beam repeatedly, each timedelivering a small dose to the pit area and sputtering away a thin layer of Si.The beam was rastered over the milling area at least 10,000 times to createeach pit, ensuring that the effects of nonuniform Si redistribution were min-imized. Circular pits were created by rastering the beam in circles from thecenter outward to the outer rim to create a maximally clean, steep pit wall.Circular pits were overlapped to form the “clover” shapes. To cause pit evo-lution, a square bombardment area was chosen to overlap the pits. The same1.5-nA, 30-keV FIB beam was then repeatedly rastered over this area in adouble serpentine (boustrophedonic followed by its time-reverse) scan, de-livering a small dose in each pass to approximate uniform irradiation. A depthof ∼0.5 nm of material was sputtered away with each pass of the beam. Im-ages of the pit shape were captured during irradiation using FIB imaging using

the secondary electron (SE2) detector. Initial and final morphology were im-aged using SEM.

We found that the pit wall propagation rate was slower than the theo-retically predicted rate; to counteract this discrepancy and ensure impinge-ment occurred at the correct moment during pit evolution, the initial pitswere milled closer than directed by the simulations.

Numerical Methods. All numerical simulations were performed using anerosion function for 30 keV Ga+ on Si, found in ref. 20 to be

RðbÞ=Rð0Þffiffiffiffiffiffiffiffiffiffiffiffiffi1+b2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1+b2μ2=σ2

p

× exp�−

a2�σ2

2ð1+b2μ2=σ2+a2

�σ2

2−Σ

� ffiffiffiffiffiffiffiffiffiffiffiffiffi1+b2

p− 1

�,

[7]

with parameters a=σ= 2.04, μ=σ= 0.658, and Σ= 0.0462.To simulate Eq. 1, we nondimensionalized the equation by scaling lengths

by L= ðB0=R0Þ1=3, times by T = L=R0, and the yield function by RðbÞ→RðbÞ=R0,where R0 =Rð0Þ. We then used a nondimensional parameter ~B0 = 0.02 so wecould simulate scales much large than the width of the traveling wave. Weused a semiimplicit numerical method introduced in ref. 24. The discretiza-tion changes the undercompressive slope and speed from their theoreticalnondimensional values of b0 = 3.89, c= 1.73 to b0 = 3.45, c= 1.84. We usedthe numerical values in Eqs. 2 and 6 when comparing to simulations.

We simulated Eq. 6 using the same semiimplicit method as ref. 24 butapplied to a one-dimensional equation. For radial pits we assumed the phaseψ was a perturbation to the radius rðtÞ so changed to radial coordinates bymaking the substitution ∂y → r−1∂y. Coefficients ci were calculated numeri-cally by first computing the undercompressive traveling wave solution sðηÞusing MATLAB’s bvp solver then computing aiðsÞ using centered differencesfor the derivatives, and finally computing πðηÞ as the second element in thenull space of the numerically discretized version of L in Eq. 5 (the first ele-ment of the discretized operator is always constant.) We performed an af-fine transformation on π to ensure it had the correct boundary conditionsbecause these are not required for the discretized operator. The traveling waveand coefficients were calculated for the nondimensional equation and wereredimensionalized using B0 = 0.02 to compare with simulations and experiments.The nondimensional values were c2 = 0.866, c3 =−0.245, and c4 = 0.231. Toredimensionalize we multiply c2 by L=T =R0, c3 by L2=T =R2=3

0 B1=30 , and c4 by

L4=T =B0 (note that ψ has units of length). For the experiments, we do not knowthe true value of B0, but as long as it is small it makes little difference to thecurve dynamics.

We simulated Eq. 2 by discretizing the curve, calculating tangent vectorswith centered differences, and updating each point on the curve accordingto Eq. 2. To prevent the curve from self-intersecting we added a small termeκ1 to the right-hand side, where κ1 is the curvature vector, calculated usingcentered differences on the normalized tangent vectors. When the mini-mum separation between the points parameterizing the curve droppedbelow a threshhold we reparameterized, by linear interpolation. This stepprovides a smoothing that in some cases was sufficient to prevent the curvefrom self-intersecting, so we could use e= 0. Otherwise, we chose e= 1× 10−5.This value was small enough that the evolution was indistinguishable by eyefrom a curve that evolves with e= 0 over the regions that have not yet self-intersected.

ACKNOWLEDGMENTS. This research was funded by the National ScienceFoundation (NSF) through Harvard Materials Research Science and Engineer-ing Center Grant DMR-1420570 and Division of Mathematical Sciences GrantDMS-1411694. This work was also supported by NSF Grant DMR-1409700(to J.C.P. and M.J.A.). This work was performed in part at the Center forNanoscale Systems (CNS), a member of the National Nanotechnology Infrastruc-ture Network, which is supported by NSF Grant ECS-0335765. CNS is part ofHarvard University. M.P.B. is an Investigator of the Simons Foundation.

1. Vasile MJ, Niu Z, Nassar R, Zhang W, Liu S (1997) Focused ion beam milling: Depth

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Fig. 5. (A) Target pattern used to design the lattice in Fig. 1, where blackpixels are elevated regions. (B) Initial condition with lowest cost (blue) andcurve it evolves to (red). The cost of this initial condition is the sum of thenumber of white pixels lying outside the red curve, plus the number of blackpixels lying inside it.

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11430 | www.pnas.org/cgi/doi/10.1073/pnas.1609315113 Perkinson et al.

Supporting InformationPerkinson et al. 10.1073/pnas.1609315113Speed CalculationsAccording to the theory, the speed c of the curves is uniquely de-termined by the yield curve and the type of smoothing physics (butnot its magnitude). We compared theory to experiments, startingfrom the previously measured yield curve and our best knowledgeof the smoothing physics, believed to be surface diffusion.In Fig. 4, shocks were measured moving 80 nm over a delivered

area dose of 1 nC/μm2. For the flux used in that experiment, thisarea dose corresponds to a time of 602 s. Thus, the experimentalshock propagation speed is found to be 80 nm/602 s = 0.13 nm/s.The shock speed is theoretically given as c= ðRðb0Þ−Rð0ÞÞ=

ðb0 − 0Þ and depends on the fixed slope b0 and the erosion ve-locities Rðb0Þ and Rð0Þ. The slope is b0 = 3.89 as described byChen et al. (20). Theoretical values for the erosion velocities RðbÞcan be derived using the sputter yield Y ðbÞ, a measure of atomssputtered away from the surface per incident ion, by changingdimensions. The sputter yield for normal incidence ions, Y ð0Þ=2.78, is found using SRIM simulations and reported by Giannuzziand Stevie (41), which along with the equation for the angulardependence of the normalized sputter yield modified by the em-pirical Yamamura correction factor (42, 43) yields a value ofY ðb0Þ= 21.5. The dimensional erosion velocities can be calculatedusing these sputter yields and the atomic volume of silicon,2.00× 10−29 m3:

Rð0Þ= 1  nC�μm2

602  s

�6.24× 109   ions  in

1  nC

��2.78  atoms  out

ions  in

�

×�2.00× 10−29  m3

atoms  out

��1018μm3

1 m3

��103 nm1  μm

�= 0.576  nm=s

Rðb0 = 3.89Þ= 1  nC�μm2

602  s

�6.24× 109   ions  in

1  nC

��21.5  atoms  out

ions  in

�

×�2.00× 10−29  m3

atoms  out

��1018   μm3

1 m3

��103   nm1  μm

�

= 4.456  nm=s.

The shock speed from theory is thus c= ð4.456  nm=s− 0.576  nm=sÞ=ð3.89− 0Þ= 0.997 nm/s, a factor of 8 larger than that measured fromexperiment.This discrepancy means that either there is something wrong

with the yield curve or there is smoothing physics that is not yetincorporated in the model. We do not understand this discrep-ancy and leave it as a question for future research. Note, however,that this discrepancy is not important for the theory in this paper,because the theory only requires that the speed and slope beuniquely selected. We can simply measure the shock front ve-locity by evolving circular pits and use this measured velocity toevolve shapes.

Knife-Edge Ridge Curving After Initial FormationSimulation and experiment both demonstrate that knife-edgeridges become curved if irradiation continues after the initialimpingement of steep features, as shown in Fig. S1. The shape ofthe wall, with a central high point, arises due to the slight cur-vature remaining in the shape of the propagating walls at themoment of impingement, which causes the midpoint of the ridgeto form later than the rest of the ridge. This central high point isabsent from the ridges shown in Fig. 1E because initial pits were

spaced further apart, resulting in a more uniformly round shapeat the moment of bombardment. Furthermore, because the pits“reached out” to each other when they evolved within 200 nm ofeach other, the center of the ridge formed first, thus removingthe source of the central high point.

Detailed Multiscale CalculationsIn this sectionwe record the details of themultiscale expansionused toderive Eq. 6, the basis of Theory 2. We work in the frame of refer-ence of the traveling wave by defining η= x− ct. The η-derivative ofEq. 1 is

�hη�t+ ðRðbÞÞη − chηη +B0∂η∇·ðf ðbÞ∇∇·ðf ðbÞ∇hÞÞ, [S1]

with f ðbÞ= ð1+ b2Þ−1=2, b= j∇hj=ffiffiffiffiffiffiffiffiffiffiffiffiffiffih2x + h2y

q. We make the ansatz

h= Sðx− ct+Ψðy, tÞÞ+ h0ðtÞ+ eh1 + e2h2 + . . ., where sðηÞ= S′ðηÞis the traveling wave solution, and assume the scalings ∂t ∼OðeÞ,∂y ∼Oðe1=2Þ.The Oð1Þ and OðeÞ parts of various terms are shown in the

table below. The fourth-order term has been broken up by firstcalculating kðbÞ=∇·ðf ðbÞ∇hÞ= ∂ηðfhηÞ+ ∂yðfhyÞ and then calcu-lating mðbÞ= ∂η∇·ðf ðbÞ∇kðbÞÞ.To distinguish between various derivatives, we write a subscript

“d” when we mean the pointwise derivative of a function,with nochain rule involved, that is, fdðsðηÞÞ= d

ds f jsðηÞ. We write a primesymbol to mean derivative with respect to η, that is, f ′ðsÞ=fdðsÞs′ðηÞ. All functions are evaluated at s.

Some auxiliary calculations are given below:

Collecting up terms gives the Oð1Þ equation

∂ηη�f ðsÞ∂ηηðf ðsÞsÞ

�+ ðRðsÞÞ′− cs′= 0. [S2]

Oð1Þ Oðe1=2Þ OðeÞh S+h0ðtÞ h1

ht s+ ddth0 sψ t +h1,t

hη s h1,η

hηη s′ h1,ηη

hy sψy

hyy ψ2y s′ + sψyy

b s 12 sψ

2y +h1,η

fðbÞ fðsÞ fd�12 sψ

2y +h1,η

�fy f ′ψy

ðRðbÞÞη ðRðsÞÞ′ �12Rds

�′ψ2

y + ðRdðsÞÞ′h1,η +Rdh1,ηη

kðbÞ ðfsÞ′ ∂η½ðfds+ fÞh1,η�+ fsψyy +�12fds

2 + fs�′ψ2

y

κy ðfsÞ′′ψy

mðbÞ ðfðfsÞ′′Þ′′ = ∂ηηðf0k1,η + f1k0,ηÞ+ ∂η∂yðf0k0,yÞ= ψyy · ððfðfsÞ″Þ′+ ðfsÞ″′Þ+  ψ2

y ·��

12fdsðfsÞ″

�″+ ðfðfsÞ″Þ″

+ ðfðfs′Þ′Þ″+ �f�12fds2�″�″�+  ðfððfds+ fÞh1,ηÞ″Þ″+ ðfdðfsÞ″h1,ηÞ″

f0k1,η = f∂ηη½ðfds+ fÞh1,η�+ fðfsÞ′ψyy + f�12fds

2 + fs″ψ2y

f1k0,η = 12fdsðfsÞ″ψ2

y + fdðfsÞ″h1,η

∂yðf0k0,yÞ = ðfðfsÞ″Þ′ψ2y + fðfsÞ″ψyy

Perkinson et al. www.pnas.org/cgi/content/short/1609315113 1 of 3

This equation is satisfied by construction, because sðηÞ is as-sumed to be a solution.Next we collect up terms for the OðeÞ equation, and also include

the term for ψ yyyy from theOðe2Þ equation. We obtain, with u= h1,η:

ut +Lu= a1ðsÞψ t + a2ðsÞψ2y + a3ðsÞψ yy + a4ðsÞψ yyyy. [S3]

The linear operator is

Lu= ððRd − cÞuÞ′+B0

h�fdðfsÞ″u

″+�f ððfds+ f ÞuÞ″

″i, [S4]

and the coefficients are

a1ðsÞ= s′ [S5]

a2ðsÞ= 12ðRdsÞ′+B0

12�fdsðfsÞ″

�″+ �f ðfsÞ″�″+�f�12fds2 + fs

�″�″�

[S6]

a3ðsÞ=B0

hðf ðfsÞ″Þ′+ ðf ðfsÞ′Þ″

i[S7]

a4ðsÞ=B0�f 2s�′. [S8]

Boundary Conditions for the Undercompressive WaveWe elaborate on the boundary conditions used in the solvabilitycondition in the paragraph following Eq. 6. These were justifiedrigorously in ref. 35, but we include a heuristic version of theargument here for completeness, treating all possible types oftraveling wave solutions to Eq. 1.A general one-dimensional traveling wave solution has the

form hx = sðx− ctÞ and solves

cðs− brÞ− ðRðsÞ−RðbrÞÞ−B0

1ffiffiffiffiffiffiffiffiffiffiffi1+ s2

p

s′ð1+ s2Þ3=2

!′!′= 0,

[S9]

with boundary conditions sð−∞Þ= bl, sð+∞Þ= br, where bl, br areparameters (23). The wave speed is found by integrating from −∞to +∞ to be c= ðRðbrÞ−RðblÞÞ=ðbr − blÞ. There are three types ofwaves, characterized by the relation between c and the speedR′ðblðrÞÞ of information propagation on either side of the wave:

• A compressive wave has R′ðblÞ> c>R′ðbrÞ, so that informa-tion propagates into the wave from both sides.

• An undercompressive wave has R′ðblÞhciR′ðbrÞ or R′ðblÞ>c<R′ðbrÞ, so that information propagates in on one side andaway on the other.

• A doubly undercompressive wave has R′ðblÞ< c<R′ðbrÞ, sothat information propagates away on both sides.

A useful way of identifying the type of wave is by the dimensionsof the invariant manifolds at the endpoints (23, 34). BecauseEq. S9 is third-order, a traveling wave can be thought of asa trajectory ðs, s′, s″Þ in R3 connecting point ðbl, 0,0Þ to pointðbr , 0,0Þ. It must lie in the intersection of the unstable manifold atη=−∞ [written USð−∞Þ] and the stable manifold at η=+∞[written Sð∞Þ]. Linearizing Eq. S9 and looking for exponentiallygrowing modes ∝ eμη shows that μ3 ∝ c−R′ðblðrÞÞ, so USð−∞Þ is2D when c<R′ðblÞ and one-dimensional otherwise, and Sð∞Þ is2D when c>R′ðbrÞ and one-dimensional otherwise. Therefore, acompressive wave occurs when the two invariant manifolds are2D, an undercompressive wave when one is one-dimensional and

the other is 2D, and a doubly undercompressive wave when bothare one-dimensional.To determine the boundary conditions for the equation

Lpπ = 0, we analyze the left and right eigenfunctions of the linearoperator L. We have that Ls′= 0, because this is simply the lin-earization of Eq. S9. Therefore, 0 is an eigenvalue of L with righteigenfunction s′, so there is a corresponding left eigenfunction πsuch that hπ, s′i= 1. To find the boundary conditions that make thisnormalization possible, we perturb the operator as Lδ =L+ δL1and suppose it is analytic at the origin, so that the eigenvalues andeigenfunctions also have a perturbation expansion as*

ϕδ =ϕ+ δϕ1 + δ2ϕ3 + . . . , πδ = π + δπ1 + δ2π2 + . . . ,

λδ = 0+ δλ1 + δ2λ2 + . . .[S10]

Here ϕδ is the right eigenfunction of Lδ, πδ is the left eigenfunc-tion, and λδ is the eigenvalue. Lp will have the same leading-order boundary conditions as Lp

δ, which follow by consideringthe growth of ϕδ, πδ near ±∞ and the condition hϕδ, πδi<∞.Consider each case in turn:

• Compressive: Then USð−∞Þ, Sð∞Þ are both 2D, so genericallythey still intersect in a trajectory when perturbed. Therefore, ϕδdecays exponentially on both sides, so πδ can grow on both sides.Requiring it to be bounded implies jπð−∞Þj, jπð∞Þj< const.

In this case the solution toLpπ = 0 is π = const; this solution is whatone typically expects, for example for periodic traveling waves (e.g.,refs. 44 and 45). The constants in Eq. 6 are computed analytically as

�c1 = ðbr − blÞ, �c2 =RdðbrÞbr −RdðblÞbl,�c3 = 0, �c4 = f 2ðbrÞbr − f 2ðblÞbl,

[S11]

where �ci = hπ, aiðsÞi.• Undercompressive: If one of the invariant manifolds is one-

dimensional, then generically a perturbation to the equation willdestroy the intersection. If USð−∞Þ is one-dimensional, then ϕδwill grow exponentially on the left, so πδ must decay exponen-tially on the left to satisfy hϕ, πi<∞. Requiring it to be boundedimplies the boundary conditions πð−∞Þ= 0, πð∞Þ= const; theconstant can be chosen without loss of generality to be 1.

• Doubly undercompressive: A similar discussion to the under-compressive case implies the boundary conditions are πð−∞Þ= 0,πð∞Þ= 0.

Which Terms to Include in Theory 2?Here, we justify including a fourth-order term c4ψ yyyy in Eq. 6.The multiscale expansion to OðeÞwould include only the terms

c2ψ2y , c3ψ yy. We have found that the viscous term c3ψ yy is some-

times very small—for a compressive wave c3 = 0 (this is explainedin Supporting Information, Boundary Conditions for the Under-compressive Wave), and for an undercompressive wave we have foundthat by varying RðbÞ the magnitude of c3 can sometimes be very small.In addition there is no guarantee that the sign is positive. With noviscous or fourth-order term, there is no smoothingmechanism: Eq. 6 isa Hamilton–Jacobi equation that advects ψ y while causing it to sharpenuntil its derivatives blow up. Therefore, smaller length scales are createdand terms from the second-order equation will become as important asthose in the first-order equation in the regions of high gradient.Which terms will be the first to become large? To answer this

question, we proceed as in a boundary-layer analysis and seek thelargest length scale LðeÞ such that at least one higher-order termbecomes OðeÞ under the scaling �y= ðLðeÞÞ−1y. In introducing this

*Technically this operator as defined is not analytic at 0, but it can be made analytic by anappropriate change of variables; see ref. 35.

Perkinson et al. www.pnas.org/cgi/content/short/1609315113 2 of 3

new scale we assume that ψ y maintains its original magnitude,that is, ψ y ∼Oðe1=2Þ, but that its derivatives can become large dueto the Hamilton–Jacobi dynamics. We look at the terms in theOðe2Þ equation, which has the form†

vt +Lv= ð·Þψ yyyy + ð·Þψ yψ yyy + ð·Þψ2yy + ð·Þψ2

yψ yy + ð·Þψ4y

+ ðð·ÞuÞηψ2y + ð·Þuηηηψ yy   + ð·Þuηηηyψ y + ð·Þuηηηηψ2

y

+ uηψ�t +�ð·ÞZηuyηψ y +

�ð·Þu2�η+ ð·Þuηηyy.

[S12]

Here, v is the Oðe2Þ perturbation to hx, and ð·Þ represents somefunction of sðηÞ. Somewhat surprisingly, the right-hand side in-cludes terms proportional to uð·Þ. For a compressive wave, thecoefficients of these terms after integrating over the fast vari-ables are ci = 0, but for a noncompressive wave the perturbationsevolve nonlinearly in general (38).By applying the new scaling to each of the terms, we

find that when LðeÞ= e1=6 then ψ yyyy →ψ�yy�y�y�y ∼ e1=2+3·1=6 = e

but that all other terms are higher-order. Therefore, weinclude this term and obtain a well-posed curve evolutionequation: Smaller scales can be created by the Hamilton–Jacobi dynamics, but these are subsequently suppressed bythe fourth-order smoothing before other terms become im-portant.

Fig. S1. (Left) Simulated knife-edge ridge behavior if irradiation continues after steep feature impingement. The initial hole had a nondimensional depth of−4, and the simulation was run until simulation time 1.8. (Right) SEM image of curved pit walls resulting from irradiation after steep feature impingement,viewed at 30° off-normal. Initial pits were milled with centers 3.4 μm apart and irradiated with an area dose of 1.2 nC/μm2. (Scale bar, 2 μm.)

†Eq. S12 is the expansion using a linearized fourth-order term ∝Δ2h; the expansion forthe full nonlinear smoothing is similar but has more terms.

Perkinson et al. www.pnas.org/cgi/content/short/1609315113 3 of 3