Nuclear Instruments and Methods in Physics Research B 272 (2012) 178–182

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Nuclear Instruments and Methods in Physics Research B

journal homepage: www.elsevier .com/locate /n imb

Surface nanopatterning mechanisms by keV ions: Linear instability modelsand beyond

Eric Chason ⇑, Vivek ShenoyBrown University, Division of Engineering, Providence, RI 02912, USA

a r t i c l e i n f o

Article history:Available online 1 February 2011

Keywords:Self-organization and patterningSurface structureIon beam processingSputtering

0168-583X/$ - see front matter � 2011 Elsevier B.V.doi:10.1016/j.nimb.2011.01.060

⇑ Corresponding author. Tel.: +1 401 863 2317.E-mail address: Eric_Chason@brown.edu (E. Chaso

a b s t r a c t

Sputtering solid surfaces with low energy ions is well-known to induce a wide array of nanoscale patternforming behavior (sputter ripples). A simple continuum model originally developed by Bradley and Har-per (BH) provides a useful framework that can qualitatively explain multiple types of patterning behaviorseen experimentally and their relationship to the ion beam and material parameters. The basis of themodel is a dynamic competition between roughening by ion bombardment and smoothing by surfacetransport that leads to the growth of roughness with a preferred periodicity on the surface. However,there are many experimental results that cannot be accounted for within this framework and additionalphysical mechanisms are discussed that may close the gap in our understanding.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

Low energy ion bombardment of solid surfaces induces a sur-prisingly complex array of morphology evolution that dependson the ion beam, processing conditions and material being bom-barded. A compendium of different observed behavior can befound in several recent reviews [1–4]. For different materials andconditions, the surface can spontaneously develop highly uniformwaves (sputter ripples) or arrays of nanoscale quantum dots overlarge areas. The alignment of the pattern may be controlled bythe direction of the ion beam or the crystallographic orientationof the surface. The surface roughness can grow exponentially oralgebraically with time, or in some cases refuse to roughen at all.

This wide variety of behavior has created intense interest inunderstanding its origin. Because it is self-organizing, ion-inducedpatterning holds promise as an inexpensive method for creatingnanoscale structures over large areas, e.g., for magnetic storage[5] alignment of liquid crystals [6], optoelectronic materials [7]or enhanced catalysis [8]. Equally important, understanding theprocesses that control ripple formation provide a window intothe non-equilibrium kinetic processes that occur under a combina-tion of energetic particle bombardment and defect-mediatedtransport. These processes are critical for understanding the forma-tion and stability of nanoscale structures and for the behavior ofmaterials subjected to high flux environments such as in nuclearreactors.

Much of our understanding of ripple formation has come from alinear instability model proposed by Bradley and Harper (BH) [9]

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n).

that considers the surface evolution in terms of the kinetic balancebetween roughening and smoothing processes. This approach is atthe core of many models of ion-induced pattern formation and itprovides a useful and intuitive framework for considering differentforms of pattern formation. Many aspects of ripple formation, suchas the transitions between different types of behavior under differ-ent conditions, can be understood within this picture. However,systematic studies of ripple formation kinetics over recent yearshave exposed significant shortcomings of this approach, e.g., thegrowth rate of ripples has been found to be significantly faster inexperiments than predicted by the theory. Furthermore, other as-pects of ripple formation (such as amplitude saturation, the forma-tion of quantum-dots and the angular dependence of thepatterning behavior) fall outside the scope of the linear theory. Thisindicates that other mechanisms or approaches are needed beyondthe simple linear instability model.

In the current work, we review different types of ripple forma-tion behavior that are seen on metal surfaces and use a kineticphase diagram to show how the different behaviors are relatedto the regimes of flux and temperature where they are observed.The main features of the BH model are described in order to relatethe wavelength and growth rate to the underlying surface kineticparameters and the results are compared with experiments onCu(0 0 1) and kinetic Monte Carlo simulations. Finally, we discusssome alternative mechanisms or approaches that may be able toexplain observations that can’t be explained by the linear theory.

2. Types of patterning

Ion-induced surface patterns have since been seen in a largenumber of materials systems including semiconductors, insulators

Fig. 1. Kinetic phase diagram showing regimes of pattern formation observed on Cuand Ag (001) surfaces at different fluxes and temperatures.

E. Chason, V. Shenoy / Nuclear Instruments and Methods in Physics Research B 272 (2012) 178–182 179

and metals. Since the production of sputter ripples is a kinetic phe-nomenon (i.e., the pattern is not thermodynamically stable), thetype of behavior can be modified by changing the processing con-ditions. To illustrate this, we present the results from a number ofstudies on Cu and Ag in a kinetic phase diagram in Fig. 1 The pat-terning is categorized into one of several characteristic types ofbehavior that we refer to as BH instability, ES instability, athermalBH/kinetic roughening and non-roughening. The different symbolsrepresent data from different studies that are listed in Table 1[2,10–16]. Although not all the measurements were done underthe same conditions the boundaries roughly delineate regimes oftemperature and flux in which the different types of behavior areobserved

The different types of behavior have different characteristic fea-tures that give insight into the physical mechanisms that are con-trolling their formation:

2.1. BH instability

These types of ripples are named for the fact that they are ex-plained by the original BH model. They have several distinct fea-tures [3]: the characteristic surface periodicity remains constantduring the growth, the orientation of the pattern is determinedby the direction of the ion beam and the amplitude grows expo-nentially (in the early stage before saturating). In addition, the ori-entation of the surface wavevector can be parallel or perpendicularto the ion beam direction, depending on the ion’s incident angle.The dependence of the ripple orientation on the ion beam directionindicates that their formation is dominated by the ion-surfaceinteraction. Originally seen only on semiconductor and insulatorsurfaces, they have been observed on the Cu surface when the tem-perature and flux are sufficiently high, i.e., the upper right handportion of the diagram in Fig. 1 The BH instability regime is distin-

Table 1List of the experimental conditions and the source reference for the data representedby the different symbols in Fig. 1.

Symbol Surface Sputtering conditions References

Ion h Energy (eV)s Cu(0 0 1) Ar+ 70o 800 [10–12]e Cu(0 0 1) Ar+ 45o 400–3000 [13]r Cu(0 0 1) Ar+ 30o 350 [14]D Ag(0 0 1) Ar+ 0o 1000 [15,16]h Ag(0 0 1) Ne+ 70o 1000 [2]

guished from the athermal BH regime by the fact that the ripplewavelength depends on the temperature in BH regime.

2.2. ES instability

The label ES instability refers to the Ehrlich-Schwoebel [17,18]barrier to interlayer diffusion that is found on many metal surfaceand has been used in models to explain this type of patterning[2,19]. The ES instability region (also called the diffusive regime[2]) is distinguished by patterning that typically aligns with thecrystallographic direction of the surface rather than the ion beamdirection [20]. The importance of the surface crystallography indi-cates the behavior is dominated by effects of diffusion on the sur-face rather than the ion-surface interaction. These types of ripplesare found within a limited range of temperatures where the diffu-sional effects are most important and transitions to all the othertypes of behavior have been observed by changing the temperature[16,11]. In contrast with the BH behavior, the amplitude oftengrows with a power law behavior and the wavelength is not con-stant but increases with the sputtering time.

2.3. Athermal BH/kinetic roughening

At low temperature, surface diffusion decreases and thereforethe ES-type patterning no longer dominates. In this regime, thesurface morphology may be observed to kinetically roughen with-out developing any preferred periodicity. Alternatively, the ion-surface interaction can also induce a preferred periodicity that isindependent of surface diffusion and hence temperature. Thisbehavior (labeled athermal BH) is consistent with an extension tothe BH theory by Makeev et al. [21] which is discussed below.

2.4. Non-roughening

At high temperatures, if the flux is low than the smoothing dueto surface diffusion can dominate over the ion-induced effects andprevent the nucleation of surface roughness. In this regime, thesurface remains smooth even under prolonged sputtering. Thetransition to non-roughening behavior has been observed bothby decreasing the flux from the regime where BH ripples form[10] and by increasing the temperature from the regime whereES instability patterns form [16].

The diagram shows how these patterning behaviors are relatedto different regimes of flux and temperature on the Cu(1 0 0) sur-face. High temperature and high flux are the best conditions forBH instability ripples since the effect of the ion beam is relativelylarge while the barriers to diffusion are relatively low. High tem-perature with lower flux can lead to non-roughening of the surface.The ES instability occurs at lower temperatures where the effect ofdiffusion barriers can dominate over ion-induced effects. At evenlower temperature, limited mobility can lead to kinetic rougheningor possible the formation of athermal BH patterns.

3. Linear instability model of Bradley and Harper (BH)

To provide an understanding of how these observed patterningbehaviors relates to underlying kinetic processes, we describe theBH model and refinements made to it to account for additional ef-fects. Bradley and Harper (BH) [9] developed the first quantitativemodel for ripple formation by considering the combined effects ofsputtering and surface diffusion. The approach was based on amechanism proposed by Sigmund [22,23] to describe the sputterremoval of surface atoms by incident ions. Sigmund’s model re-lated the rate of atom removal to the energy deposited by the inci-dent ion into the near surface region through a series of collisions.

ion sputtered atom

Faster erosionat troughs

Slower erosionat crests

ion sputtered atom

Faster erosionat troughs

Slower erosionat crests

Fig. 2. Schematic of Sigmund mechanism leading to a curvature-dependentsputtering yield that erodes surface troughs faster than crests.

180 E. Chason, V. Shenoy / Nuclear Instruments and Methods in Physics Research B 272 (2012) 178–182

The resulting energy distribution was approximated by a Gaussianfunction around the incident ion track (shown schematically inFig. 2) and the probability of sputtering an atom from the surfacewas taken to be proportional to the energy deposited at the surfacesite. BH extended this mechanism to consider the evolution of asurface profile and showed that it resulted in a sputter yield thatis proportional to the curvature of the surface. For a surface witha sinusoidal profile, this means that the crests of the waves onthe surface sputter slower than the troughs so that surfaces be-come rougher as they are sputtered. This roughening mechanismis balanced by surface relaxation which depends on the divergenceof the surface curvature. Putting these together, they derived anequation for the evolution of the height h(x,y,t):

@h@t¼ mx

@2h@x2 þ my

@2h@y2 � Br2r2h ð1Þ

where mx and my refer to the curvature dependence of the sputteringin directions parallel (x) and perpendicular (y) to the ion beam alongthe surface. The parameter B can describe smoothing due to classi-cal surface diffusion [24,25],where B is proportional to the surfacediffusivity (Ds) and concentration of mobile species (Cs). It has alsobeen shown that a similar functional form for the smoothing canarise from ion-induced viscous flow in the surface layer [26].

This equation predicts that each Fourier component of the sur-face height (hk(t)) will grow (or shrink) exponentially with a ratethat depends on the wavevector:

hkðtÞ ¼ hkð0Þerkt ð2aÞ

where

rk ¼ �ðmxk2x þ myk2

yÞ � Bðk2x þ k2

yÞ2 ð2bÞ

The growth rate rk has a maximum value r⁄ at the wavevector k⁄:

r� ¼ m2max

4B

k� ¼ffiffiffiffiffiffiffiffiffiffimmax

2B

rð3Þ

where mmax is the larger of the two values for m. The surface developsa characteristic periodicity of k⁄ = 2p/k⁄ because the amplitude ofthis wavelength grows faster than all the others, i.e., it is maximallyunstable.

This model accounts for many of the features observed in rip-ples of the BH instability type. It predicts exponential growth ofthe ripple amplitude, a constant characteristic surface periodicityand a pattern orientation that is determined by the direction ofthe ion beam. In addition, depending on the ion’s incident angle,the pattern can be induced with the wavevector parallel or perpen-dicular to the ion beam direction as seen experimentally.

Makeev et al. [1] extended the BH sputtering mechanism to in-clude higher order effects of the ion-surface interaction within thecontext of the Sigmund mechanism. This manifests itself as addi-tional terms of the form BI@

4h=@x4 due to the ion-surface interac-tion which added a temperature-independent term BI to the termB in eq. (1). With this addition, the characteristic wavelength onthe surface is given by:

k� /

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Bðf ; TÞ þ BIðf Þ

mmaxðf Þ

s/

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDsðTÞCsðf ; TÞ

fTþ AI

sð4Þ

where AI is a constant. The dependence of the parameters on flux (f)and temperature (T) is shown explicitly to indicate how the changein these parameters will affect the ripple wavelength. The additionof the temperature-independent term AI means that ripples canform even in the absence of thermally-activated diffusion and isconsistent with the transition to athermal BH ripples at low temper-ature in some metals. It also sets a limit to the minimum wave-length that can be achieved by ion patterning.

To explain the type of behavior seen in the ES instability regime,Valbusa [2] proposed incorporating an instability mechanism de-scribed by Villain [27,28] based on the roughening effect of ES bar-riers into the BH theory. By linearizing the theory for small slopes,they determined that the addition of an ES barrier can have a sim-ilar effect as the curvature dependence of the sputter yield butwith the roughness aligned along crystallographic directions ratherthan direction of the incident ion beam. Although the conditionsfor linearization are not always met by the experimental; studies,this merging of the two mechanisms enables an understanding ofhow changing the temperature or incident ion conditions can in-duce changes from ES to BH ripples.

4. Comparison of model with experiments and simulations inthe BH regime

Measurements of ripple formation kinetics have been made inmany systems [3] using STM/AFM, light scattering and X-ray scat-tering that qualitatively display the features predicted by the BHmodel. However, the model also provides specific predictions forthe dependence of measureable experimental quantities ( k⁄, r⁄)on the underlying kinetic processes which can be probed experi-mentally (e.g., by changing f, T, ion angle). For example, systematicmeasurements of the ripple wavelength on Cu surfaces at differenttemperatures and fluxes [29] are shown in Fig. 3 The measure-ments show a complex non-Arrhenius temperature behavior anda flux dependence that is different at high T (k�f-½) and low T(k�f 0). To analyze these results, Chan et al. developed a modelfor the flux and temperature dependence of the surface defect con-centration (details found in Ref. [29]) by considering different pro-cesses of defect creation and annihilation. At low temperature thedefect concentration is determined primarily by the ion beamwhile at high temperature it is primarily determined by thermalgeneration. This concentration model was used to evaluate the rip-ple wavelength (eq. 4) and is able to explain both the temperaturedependence and the changing flux dependence with temperature.

In addition to the experimental studies, kinetic Monte Carlo(KMC) simulations have been developed that include the Sigmundmechanisms for sputtering as well as diffusion of defects [30–34].In these KMC simulations, the interaction between the ion and sur-face is identical to the one in the Sigmund model, in contrast withthe experiments in which we cannot be sure of the ion–solid inter-action. Different simulation schemes utilize different mechanismsfor modeling the defect kinetics. In the work of Chan et al. [30], thedefect kinetics (including adatoms and vacancies) are imple-mented by allowing the individual atoms to hop around with tran-sition rates that depend on the local atomic configuration so that

λ ∝ f 0

λ ∝ f -1/2

(c) T=409 Kλ ∝ f 0

(b) T=481 Kλ ∝ f -1/2

(a)

Fig. 3. Measurements of the ripple wavelength on Cu(001) as a function of a) temperature and as a function of flux at b) T = 481 K and c) T = 409 K. The solid lines are a fit to amodel based on the BH theory with a temperature- and flux-dependent defect concentration.

Fig. 4. Results from KMC simulations. (a) Simulated ripple morphology and(b) simulated ripple wavelength as a function of flux and temperature. Solid linesare fits to BH theory using simulation parameters.

E. Chason, V. Shenoy / Nuclear Instruments and Methods in Physics Research B 272 (2012) 178–182 181

the time dependence of the surface evolution can be simulated. Animage of the simulated surface morphology and results for thesimulated wavelength at different temperatures and fluxes isshown in Fig. 4 The results show that the temperature and fluxdependence of the wavelength and growth rate can be well ex-plained by the BH theory (solid lines in the figure), indicating thatthe BH model is a good continuum approximation for the surfaceevolution when the ion–solid interaction is modeled by the Sig-mund mechanism.

5. Limitations of the BH theory and alternative approaches

The BH model and its extension provide a useful intuitiveframework for understanding why ripples form and how they de-pend on the processing conditions. However, the large number ofstudies of ripple evolution point out several significant shortcom-ings of the linear stability approach and have identified featuresof ripple formation that do not fall within the linear instabilitymodel. In this section, we will discuss features of ripple formationthat are not explained by the BH model and possible alternativesthat are being explored.

5.1. Ripple time evolution and multifield approach

By definition, the linear instability model is only appropriate inthe early stages of roughening. As the roughness increases, the rip-ples cannot continue to grow exponentially and saturation of theripple amplitude is seen in many experimental studies. This behav-ior points to the importance of non-linear terms in the surface evo-lution and attempts have been made to incorporate these into theBH formalism. However, more recently an alternative multifieldapproach has been used [35–37] which considers the coupled evo-lution of the surface height and the thin layer of mobile species onthe surface. This is described as a ‘‘hydrodynamic’’ model, similarto that used to understand pattern formation in sand dunes. In par-ticular, this approach is able to include the effect of redistributionof material on the surface during the evolution. A notable successof this model is that it can predict saturation and stationary surfacefeatures as well as the formation of quantum dot-like features seenon semiconductor surfaces. In the current formulation, it uses thesame ion-roughening term as the BH theory, so in the early stagesit is expected that the results are the same as for the BH theory.

5.2. Alternative ‘‘crater’’ functions and atom redistribution

Another alternative approach is based on the recognition thatthe Sigmund mechanism for the ion-induced roughening only in-cludes the effect of atom removal due to the incoming ion. Severalgroups have recognized that this does not agree with what is seenaround ion impact craters [38] or with what is predicted by molec-ular dynamics [39,40] and that this can have a significant impacton the surface evolution [41–43]. Including the effect of atomicredistribution due to ion bombardment has enabled an under-standing of the complicated dependence of the ripple formationrate on the ion incident angle seen in experiments on Si [44] haveshown that changing the angle of incidence can prevent the forma-tion of ripples which is in disagreement with the predictions of theBH theory. They show that the inclusion of a different crater func-tion than the one used by Sigmund can account for this angulardependence.

5.3. Surface roughening rate and ion-induced stress

From eq. 3, the BH theory predicts that there is a relationshipbetween the growth rate and wavevector that depends only on

0.5 1.0 1.5 2.0

10-3

10-4

10-5

10-6

10-7

10-8

10-9

10-11

10-10

σ (GPa)

r*( /

s)

BH roughening

ATGS roughening

BH + ATGS roughening

Fig. 5. Results of continuum model for effect of stress on ripple growth rate.

182 E. Chason, V. Shenoy / Nuclear Instruments and Methods in Physics Research B 272 (2012) 178–182

the ion-dependent parameter and not on the surface kinetics(r⁄ = ½ mmax k⁄2). When this is prediction is compared with mea-surements of rippling using Ar ions on Cu [3], the measured valueof r⁄ is 240 times larger than the value predicted by the BH model,suggesting that there are other driving forces for ripple formationbeyond sputtering. One potential mechanism is the effect of stressinduced by the ion bombardment.

Stress is known to lead to a surface instability similar to the BHmechanism (referred to as the Asaro-Tiller-Grinfeld-Srolovitz(ATGS) instability [45–47]) but with a rate of surface rougheningthat depends on @3h=@x3. Combining the stress induced rougheningwith the BH mechanism for curvature-dependent sputter yield todevelop a continuum model for the surface evolution under thecombined effects of sputter removal and ion-induced stress [48].The combination leads to a linear instability model with a rate thatdepends on the wavevector as r ¼ Ajkj2 � Bjkj4 þ Cjkj3 where C is aparameter that depends on the stress. The predicted wavevectorand roughening rate are given by:

k� ¼ 12

kATG þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12

kATG

� �2

þ kBH2

sð5aÞ

r� ¼ 12

Ak�2ð1þ að4k� � 3aÞÞ ð5bÞ

where kBH = (A/(2B))1/2, kATG = 3a/4 and a = C/B. This theory pre-dicts that the ripple will grow faster in the presence of stress thanin the simple BH theory, with a rate that rises as the wavevector ap-proaches the value predicted by the ATG theory. An example of thegrowth rate enhancement predicted by the theory is shown in Fig. 5For low stresses, r⁄ is equal to the value from the BH theory but asthe stress increases then r⁄ increases rapidly and approaches theATGS rate. This may also explain why measurements on amorphousSi show better agreement with the BH model than on Cu becausethe ion-induced stress in amorphous Si is lower. Measurementsare currently being performed to determine the significance of thestress in pattern formation

6. Conclusion

In conclusion, many features of pattern formation by low en-ergy ions can be understood within the context of the BH linearinstability model. The effects of changing ion flux and temperaturecan be understood in terms of shifting the balance between rough-ening and smoothing processes occurring on the surface, leading todifferent characteristic behavior in different kinetic regimes.

However, other features of ripple formation (saturation, angulardependence and rapid growth) cannot be explained. New mecha-nisms and better understanding of the coupling between ion-induced changes and surface transport hold out hope that acomprehensive predictive model of surface patterning induced bylow energy ions can be achieved in the future.

Acknowledgements

The authors thank W. L. Chan, Y. Ishii and N. Medhekar for theircontributions to this work. The research for this work was sup-ported by the US DOE, Office of Basic Energy Sciences, Division ofMaterials Sciences and Engineering under Award DE-FG02-01ER45913.

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