Nanoscale pattern formation on ion-sputtered surfaces

ion-sputtered surfaes

Dissertation

zur Erlangung des mathematish-naturwissenshaftlihen Doktorgrades

"Dotor rerum naturalium"

der Georg-August-UniversitätGöttingen

vorgelegt von

Taha Yasseri

aus Teheran

.

D7

Referent: Prof. Dr. Reiner Kree

Koreferent: Prof. Dr. Alexander Karl Hartmann

Self-organized, nano-sale strutures appear on solid surfaes under ion

beam irradiationwith ion energies in the keV range. Within the last deade,

surfaeengineeringbyionbeamsputtering(IBS)hasbeomeaverypromising

andidateforbottom-upprodutiontehniquesofnano-devies. Morphologies

like ripples, and regular arrays of dots, pyramids and pits as well as ultra-

smooth surfaes have been obtained onawide variety ofsubstrates, inluding

importantsemiondutor materialslike Si,Ge, GaAs and InP.

Inspiteofmanysubstantialimprovementsofexperimentalsurfaestrutur-

ingbyIBS,thephysialmehanismsunderlyingthe patternformationarestill

poorly understood. In this work we use Kineti Monte Carlo (KMC) simula-

tionsandontinuumtheorytostudythe eetsofthe followingmehanismsin

detail: (i)the interplayofsurfaeerosionwithdierentsurfaediusionmeh-

anisms(Wolf-Villain,Hamiltonian,thermallyativatedhoppingviatransition

states, inludingbarriers depending on both initialand nal onguration in

a hop) and the rossover from erosion-driven to diusion driven patterns, (ii)

random orientational utuations of ion trajetories within the beam, lead-

ing to ionbeam divergene, (iii) o-deposited, steady-state, (sub)-mono-layer

overages of the substrate with a seond atomi speies (surfatant sputter-

ing) and (iv) multi-beam and rotated-beam (or rotated sample) setups. We

nd that all the four mehanisms under study may have a profound and

sometimesunexpetedimpatonthe patternformation. Dierentdiusion

mehanisms, whih all give rise to the same leading order terms in a on-

tinuum desription lead to rather dierent long-time behavior of patterns in

KMC simulations. Orientationalutuationshange the bifurationsenarios

of pattern formation and surfatant sputtering may give rise to qualitatively

new eets like mesosopior even marosopipatterns ontop of nano-sale

patterns, and the ordering of the surfatant on top of the strutured surfae.

Thisorderingleadstoafeedbak mehanismduetothemodulationinsputter-

ingyieldausedby the surfatant. On theother hand,many ofthe promising

proposals onerning the usage of multi-beam and rotatedbeam setups ould

not be onrmed(inaordane with reent experiments), but we an outline

some physialreasonsforthis failure, whih ouldguide animproved usageof

To Maman and Baba.

First and foremost I would like tothank Prof. Dr. Reiner Kree, who was my

thesis adviser. His sharp advie opens a wide window to all aspets of the

topi, while he always kindly let me follow my own ideas as well. I enjoyed

ooperating with himalot and (hopefully) learned muhfrom him.

I would also like to thank Prof. Dr. Alexander K. Hartmann, who super-

vised the rst 6 month of my researh in Göttingen and kept supporting me

even afterhe moved toOldenburg,and Dr. EmanuelO.Yewande, whokindly

provided the KMC odes he had writtenduring hisPhD, and letmy researh

reahthe produtivity level very soon.

As I have been working on this thesis, have been fortunate to have nie

disussions with Prof. Dr. Hans Hofsäss, Dr. Kun Zhang and Prof. Dr.

Rodolfo Cuerno.

Ithas beenagreatpleasure toworkintheInstituteforTheoretialPhysis,

and I would like to thank all its past and present members, speially, Prof.

Dr. Kurt Shönhammer (head of the institute), Prof. Dr. Annette Zippelius,

Prof. Dr. Marus Müller and Dr. Jürgen Holm, and then all my olleagues

and oemates; Dr. Bernd Burghardt, Andrea Fiege, Till Kranz, Alexander

Mann, Kristian Marx, Dr. Stefan Wolfsheimer and Martin Zumsande. Here,

my deepest thanks gotoOliverMelhert.

I appreiate all the eorts of the seretary team of the institute, Frauen

Glormann, Lütge-Hampe and Shubert, and the SFB seretary Frau Hühne,

to failitate the administrative proesses. I'm also deeply thankful to the

faulty seretary Frau Afshar.

ImustonfessthatmyresideneinGöttingenwouldnotbesopleasantwith-

out the support of my dearest Iranian friends (spread all aroundthe world) .

Therefore I would like to thank Talayeh Aledavood (Munih), Homa Ghalei,

Sara Hadji Moradlou (Saarbrüken), Dr. Nima Hamedani Radja (Leiden),

Majid Hojjat (Munih), Sona Nazari (Tehran), Mahmoudreza Saadat (Stan-

ford), Shahriar Shadkhoo (Los Angeles). I espeially appreiate all sienti

and non-sienti supports from Armita Nourmohammad(Cologne).

SohailKhoshnevis,Amgad SquiresandLishiaTeh,eahofthem haskindly

orreted parts of the manusriptof this thesis. I am very grateful for that.

This work was funded by the German researh foundation, the Deutshe

Forhungsgemeinsaft (DFG), within the Sonderforhungsbereih (SFB) 602:

ComplexStrutures in CondensedMatter fromAtomito Mesosopi Sales.

The simulationswere performedattheworkstation lusterofthe Institute

forTheoretial Physis, university Göttingen.

.

Å: Ångstrom

AES: Auger eletron spetrosopy

AFM: atomifore mirosopy

BCA: binary ollision approximation

BH: Bradley-Harper

DIBS: dual-ion-beam sputtering

DT: Das Sarma-Tamborenea

ES: Ehrlih-Shwoebel

eV: eletron volt

EW: Erdwards-Wilkinson

HKGK: Hartmann-Kree-Geyer-Kölbel

IBS: ion-beam sputtering

KMC: kineti MonteCarlo

KS: Kuramoto-Sivashinsky

LC: largerurvature

LD: Lai-Das Sarma

MBE: moleular beam epitaxy

MCB: Makeev-Cuerno-Barabási

MD: moleular dynamis

RIBS: rotationalion-beam sputtering

SEM: sanning eletron mirosopy

SIBS: sequentialion-beam sputtering

SIMS: seondary ion mass spetrometry

SOS: solid-on-solid

STM: sanning tunneling mirosopy

WV: Wolf-Villain

1 Introdution 1

2 Ion-beam sputtering experiments 5

2.1 Patterns onamorphous substrates . . . 5

2.1.1 Ripples. . . 6

2.1.2 Dots . . . 8

2.1.3 Holes . . . 11

2.1.4 Smooth surfaes . . . 11

2.2 Patterns onrystalline substrates . . . 14

2.3 Advaned patterning methods . . . 14

2.3.1 Surfatant sputtering . . . 16

2.3.2 Compound beams . . . 16

3 Methods 19 3.1 Binaryollision approximation . . . 20

3.1.1 Casade shape. . . 21

3.1.2 Defet generation . . . 22

3.1.3 Down-hillurrent . . . 22

3.2 Kinetitheory . . . 25

3.3 KinetiMonte Carlo . . . 25

3.3.1 Erosion . . . 26

3.3.2 Diusion . . . 26

3.4 Continuum theory. . . 28

3.4.1 Bradley-Harpermodel . . . 28

3.4.2 Cuerno-Barabásinon-linear model. . . 30

3.4.3 Makeev,Cuerno and Barabási model . . . 31

3.4.4 Non-loallinear stability analysis . . . 32

3.5 Data analysis . . . 33

3.5.1 Saling analysis . . . 33

3.5.2 Power spetral density . . . 34

4 Erosion and diusion-driven patterns 37 4.1 Erosion . . . 38

4.1.1 Dependene of patterns onionparameters . . . 38

4.1.2 BCA modelbased erosion . . . 41

4.2 Diusion . . . 41

4.2.1 Irreversible models . . . 43

4.2.2 Hamiltonianmodels . . . 45

4.2.3 Thermally ativated models . . . 47

4.2.4 Ehrlih-Shwoebel eets, pattern formationby diusion 49 4.2.5 Diusion inompound systems, lustering . . . 51

4.3 Erosion-Diusioninterplay . . . 52

4.3.1 Dependene of patterns ondiusion inthe erosive regime 55 4.3.2 Crossover from erosive regime todiusive regime . . . . 60

5 Surfatant Sputtering 65 5.1 Implementation . . . 66

5.1.1 Continuum theory . . . 67

5.1.2 KMC model . . . 69

5.2 Mesosopi height gradient . . . 70

5.3 Morphology modiation . . . 72

5.3.1 Ultra-smooth surfaes . . . 72

5.3.2 Arrays of nano-lusters . . . 73

5.3.3 ES indued patterns . . . 77

6 Beam-noise indued eets 79 6.1 Homogeneous sub-beams . . . 81

6.2 Temporallyutuating homogeneousbeams . . . 82

6.3 Spatio-temporallyutuatingbeams . . . 83

6.3.1 Normal inidene angle . . . 85

6.3.3 Highdiusion rate regime . . . 87

7 Compound sputtering setups 89 7.1 Simulationsetup . . . 90

7.2 Opposed ion-beams . . . 92

7.3 Crossed ion-beams . . . 92

7.4 Sputtering of rippledsurfaes . . . 98

7.5 Sputtering of ontinuously rotatingsample . . . 100

8 Conlusion 109

A Diretional noise 113

2.1 Bradly-Harperinstability . . . 7

2.2 Rotationof the ripples orientation . . . 8

2.3 Ripples oarsening . . . 9

2.4 Ripples ordering. . . 10

2.5 Ordered arrangements of dots . . . 12

2.6 Formationof ellularstrutures . . . 13

2.7 Surfae smoothing . . . 13

2.8 Transitionfrom diusionto erosive regime . . . 15

2.9 Ag dropletsformed insurfatantsputtering . . . 17

2.10 Sequential and dual sputtering . . . 18

3.1 Collisionasade and itsspatial energy distribution . . . 21

3.2 Energy distribution of asades atoms. . . 23

3.3 Down-hill urrent indued by ollisionasades . . . 24

3.4 Eetive surfae tensions . . . 30

3.5 Power spetral density analysis . . . 35

4.1 Kinetiphase diagramof patterns . . . 39

4.2 Dierent topographies mergingfromdierent erosionparameters 40 4.3 Pattern formationdependene onenergy distribution . . . 42

4.4 Ehrlih-Shwoebelbarrier . . . 50

4.5 Ehrlih-Shwoebelindued patterns inMBE . . . 51

4.6 Clustering onat template . . . 53

4.7 Clustering onsinusoidal template . . . 54

4.8 Ripples evolution inHamiltonianmodel. . . 56

4.9 Ripples evolution inArrhenius (net-bond-breaking) model . . . 57

4.10 Long time morphologiesemerge fromdierentdiusion models . 58

4.11 Normal inidene sputtering with dierentdiusion models . . . 59

4.12 ES indued patterns inMBE . . . 61

4.13 Eet of attemptrate onripples orientation . . . 62

4.14 Morphology dependene ondiusion rate . . . 63

4.15 Roughening suppression by high diusion . . . 64

5.1 Surfatant sputtering experimentalsetup . . . 66

5.2 Morphologies modied by idential surfatants . . . 70

5.3 Patternswavelength vs. the overage of idential surfatants . . 71

5.4 Roughness of prolesevolved in surfatant sputtering . . . 71

5.5 Mesosopi height gradient by surfatant sputtering . . . 72

5.6 Surfae smoothing by surfatant sputtering . . . 74

5.7 Surfae smoothing by surfatant sputtering; varying overage . . 75

5.8 Nano-wiresprodued by surfatantsputtering . . . 76

5.9 Surfatant density indierentheights . . . 77

5.10 Ehrlih-Shwoebel indued patterns in surfatant sputtering . . 78

6.1 Dierent types of beam-noise . . . 80

6.2 Simulated beam prole . . . 81

6.3 Growth rate as afuntion of beam-divergene . . . 82

6.4 Renormalaized surfae tensionsdue to the beam-noise . . . 84

6.5 Noise eets onsurfae evolution rate . . . 85

6.6 Noise indued eets in normal-inidenesputtering . . . 86

6.7 Length sale seletionby beam-noise innormal inidene . . . . 86

6.8 Beam-noise eets ingrazing inidene angle . . . 87

6.9 Beam-noise eets inhigh rate diusionregime . . . 88

7.1 Compoundion-beam setups. . . 91

7.2 Opposed ion-beam . . . 93

7.3 Ripples symmetry indued by opposed-beam . . . 93

7.4 Dual-beam sputtering. . . 97

7.5 Ripple diretionin dual-beams sputtering . . . 98

7.6 Sequentialion-beam sputtering . . . .101

7.7 Roughness timeevolution forrotating sample . . . .102

7.8 Rotating ion-beam sputtering . . . .105

7.10 Rotationfrequeny dependene of roughness inRIBS . . . 106

7.11 Bradley-Cirlintheory of RIBS . . . 106

7.12 Integrated yieldvs. rotationfrequeny inRIBS . . . 107

4.1 Salingexponents of irreversible disretediusion models . . . . 46

4.2 Hoppingattempt frequeny at dierent temperatures . . . 49

7.1 Beamparameters for the DIBS setup . . . 96

Introdution

Rihard P.Feynman attrated the attention of sientists and engineers tothe

enormous apability of the nano-sale world for the rst time. He gave an

insightto the great possibilities,provided by instruments,whih are designed

andfabriatedinthesizeofsomenanometersinhistalkattheannualmeeting

ofthe AmerianPhysissoiety.

1

During thelast50years, numerous dierent

tehniques andmethodstomanipulatematerialsonnano-sales are presented

and nowadays a multitude of nano-devies are produed and available in the

markets (Maynard etal. 2006).

Wide ranges of appliationsare proposed by nano-strutures fabriated on

solid surfaes. Among all the available tehniques for the fabriation of suh

strutures,e.g. hemiallithographyandatomiforemirosopy(AFM)teh-

niques,bottom-upself-organizedpatterningmethodsareofpartiularinterest,

beause they bear the potential of heap, large-sale prodution. Ion-beam

sputtering (IBS) was introdued by Navez et al. (1962) as a simple method

for preparing wave-like patterns (ripples) of sub-mirometer length sales on

the surfae of solids. In this method,surfae bombardment by a beam of keV

ions at normal or oblique inidene drives the system towards self-organized

formation of nano-patterns. Later on, many experimental developments have

been arried out to improve the quality of the patterns i.e. ripple alignment

and regularity. Meanwhile,by sputtering dierent kindsofsolids underdier-

entonditions,new typesof patternshavebeendisovered. The produtionof

1

Deember 29th 1959, California Institute of Tehnology (Calteh), There's Plenty of

RoomattheBottom.

regulararrays of nanometer-sizedolumns(dots)is oneprominentexample of

this kind (Faskoet al. 1999). Reent reviews summarize the state of the art

of surfae engineeringby IBS (Frost et al. 2008).

Although great improvement ahieved in experiments to produe various

highlyordered patterns on a wide range of dierent materials, a omprehen-

sive understanding of the physial mehanisms underlying this self-organized

pattern formation is not yet available. The simplest quantitative theory of

IBS-indued pattern formation has been put forward in a seminal paper by

Bradleyand Harper (1988). There,itispointedoutthatIBSimpliesageneri

urvatureinstability,whihroughens thesurfae. Theombinedationofthis

instabilityandsurfaediusionleadstotheappearaneofripples. Thisontin-

uumtheoryhasbeenextended inmanydierentways, butreent experiments

indiatethatitdoesnot ontain allthe physialmehanisms,whihdetermine

ripplepatterns onsolid surfaes (Chan and Chason2007).

In this work we aimtoaddress the IBS problem by analytial and ompu-

tational methods. We develop and use a Kineti Monte Carlo (KMC) model

for ion-beam erosion inspired by the kineti theory of Sigmund (1969). We

simulate a simple ubi lattie, whih undergoes bombardment of ions with

variable energy and inidene angle. We try to give new insights into physi-

almehanismsof IBS-driven patterns as well asexamine new possibilitiesto

improve and advane IBS experiments to ahieve more preise ontrol of the

patternformationproess.

In the next hapter we give a review of reently developed experimental

methodsin additionto lassiIBS tehniques of ion-beam surfae ething.

In Chapter 3 we introdue the analytial and numerial methods we use

to study IBS, espeially our KMC model and its basi assumptions and the

ontinuum desription of IBS.

It isthe ommonbelief thatpatterns under IBS formdue toa ompetition

between surfaeroughening(by erosion)and smoothing(by surfaediusion).

InthesimpleontinuumdesriptionofBradleyandHarper,thesemehanisms

enterinuniversalformsandarequantiedbythreeparameters,twoforerosion

and one for surfae diusion. But does this exhaust the interplay of dierent

surfae diusion mehanisms with ion-beam erosion? This question will be

ing dierent erosion and diusion models and show how by tuning the rate

of erosion and diusion events dierent types of instability leadingto various

kind ofpatterns an be indued.

In Chapter 5 we present results on IBS aompanied by the o-deposition

of a seond atom speies onto the surfae. Controlled o-deposition of (sub-)

mono-layeroveragesby o-sputteringof anearby targethas been introdued

byHofsässand Zhang(2008),whoalleditsurfatantsputtering. Meanwhile

there are many indiations that highly ordered regular patterns, whih have

been produedinexperiments,haveinfatinvolved o-deposition,whihwent

unnotied. We show some eets indued by o-deposition of metalli atoms

on the surfae of a substrate like Silion. We demonstrate the possibility

of preparing nano-lusters by this method and ontrol them in a pattern of

ripples.

In Chapter 6we study the eets indued by inludingexterior noise,orig-

inated fromutuations of the diretions of iontrajetories within the beam.

Ourextendedontinuummodelpreditsnewtransitionsforpatternmorpholo-

gieswhihdierfromthestandardsenarioofBradleyandHarper. Ourresults

obtainedbyKMCsimulationsareaboutthegenerieetsduetotheion-beam

noiseindierentsituations, e.g. normalandgrazinginideneangleorinhigh

temperatureregimes.

Thereareanumberofproposals,mostlybasedonqualitativereasoning,how

to improve or modify pattern formation due to IBS by using multi-ion-beam

setups,sequentialsputteringofthe samplefromdierentdiretions,orsample

rotation. Chapter 7 ontains a detailed simulation study of these proposals.

WeompareourresultswithreentexperimentalndingsbyJoe etal. (2008).

Finally, in Chapter 8 we give a onlusion and disussion on all presented

results and some outlooksfor future work.

Ion-beam sputtering experiments

Depending on the sputtering parameters e.g. ux, energy, type and inidene

angle of the ions, aswell as substrate properties e.g. type and substrate tem-

perature, a wide range of dierent patterns might emergevia IBS. Seondary

features,likebeam-proleandrotationofsamplemayalsohangethequalita-

tiveandquantitativeharateristisofthepatterns. Reentadvanedmethods

inIBS experiments that mightprodue moreomplex textures onthe surfae

of materials are based on setups omposed of doubled- or multi-beams, si-

multaneouslyo-sputteringofmetalliand non-metallisubstrates(surfatant

sputtering), and sputtering of pre-strutured templates. In the following se-

tions of this hapter,we briey review the experimental ahievements of IBS.

For more extended reviews see Valbusa etal. (2002), Frost etal. (2008) and

Muñoz-Garía etal. (2009).

2.1 Patterns on amorphous substrates

MostoftheIBSexperimentsareperformedonamorphoussubstratese.g. glass,

or substrates whih are amorphized under bombardment of keV ions e.g. Si,

SiO

2

^{, GaSb, InP, et. Ripples and -more reently- dots are the main types}

of patterns whih emerge on these types of substrates. Moreover, formation

of holesand pits, the appearane of ultra-smoothsurfaes and non-strutured

roughsurfaes are alsoreportedasoutomes of some IBS experiments. In the

following,the mentioned types of strutures are disussed inmore detail.

2.1.1 Ripples

A rst experiment by Navez et al. (1962) was followed by a huge number of

experiments, in whih wave-like ripple strutures were observed. The period-

iity of ripples varies from tens to hundreds of nanometers and their length

an exeed several mirometers. Some universal properties are observed in

experiments with very dierent ion-beam and substrate parameters. Ripple

orientation with respet to ion-beam diretion and the dependene of this

orientation on the inidene angle of the ion-beam with respet to the sur-

faenormal

θ

^{, areimportantuniversalfeatures, whihhavebeenexplainedby}

theory of Bradley and Harper (BH). In BH theory, dierent erosion rates at

dierent points on the surfae in relation to the loal urvatures is the main

destabilizingfator whihis shown tobe suient toexplain the formationof

ripplesand their orientation. A shemati drawing of BH theory is presented

in Fig. 2.1. The full desription of the theory is provided in setion 3.4.1.

Other universal features, like the diretion of ripple on rystalline substrates,

propagation and the oarsening of ripple patterns are not fully understood

withinBH theory, but extensions of this theory lead to partialunderstanding

of many features of the patternformation(see setion3.4).

Thequalityofripples,i.e. theirregularityandalignment,improvedtremen-

douslywithinthe lastdeade (omparethe struturesdepited inFig.2.2and

Fig. 2.4). However, so far, no omprehensive explanation on the onditions

and physial mehanisms, whih lead to the formation of suh ultra-regular

patterns exists. Two main ideas that may explain this experimental suess

are (i)ne tuning of ion-beam prole and (ii)manipulationsinthe proess of

pattern formation by o-deposited metalli atoms. Both ideas are disussed

extensively in this thesis inhapters5 and 6 respetively.

Orientation

The orientation of ripples is typially onned to be either parallel or per-

pendiular to the projetion of the ion-beam diretion onto the surfae. For

smallvaluesofinideneangle

θ

^,orientationisperpendiulartotheion-beam.

By inreasing

θ

^{towards grazing inidene, at some ritial value}

θ c

^{, ripples}

rotate by

90 ^◦

^{and align in diretion parallel to the ion-beam. One exam-}

o

o’

Figure2.1: Ionshitthesurfaewithnormalinideneangle,impingeintothesurfae,stop

atsomedistaneunderthesurfaeanddeposittheirkinetienergy. Sinetheamountofthe

depositedenergyreahingthelateralpointswithpositiveornegativeloalurvatures(Oor

O')isdierent,assumingtheerosion rateatsurfaepointsis proportionalto thereahing

energy,afaster erosionrateisexpetedin valleys(pointo'). This leadstoampliationof

theinitialsurfaeroughness(BradleyandHarper1988).

ple of the hange in orientation in the experiments on graphite samples by

Habeniht et al. (1999)is shown in Fig. 2.2.

Propagation

Insomeexperiments(Habenihtet al. 2002;Alkemade 2006)bysimultaneous

realtimemonitoringofpatternevolution,itisobserved thatripplespropagate

along the diretion of the ion-beam. Initial movements with veloity of

0.33

nm s

−1

are followed by deeleration and a dispersion in veloity for dierent

wavelength. At longer times, faster movementsfor rippleswith shorter wave-

length was reported. Ripple propagation is also predited by BH theory, but

the predited diretionof motion is apposite tothe observed diretion.

Coarsening

The oarsening of ripples, i.e. inrease in lateral size and spaing of rip-

ples, has been observed in a large number of experiments. A growth of

wavelength, following a power law in the form of

λ ∼ t ^0.5

^{is reported by}

Figure2.2: Rotationoftheripplesorientationbyinreasingtheinideneangle. Sanning

tunnelingmirosope(STM) miro-graphs(lateralsize1

µ

^m)of5keVXe

⁺

^erodedHOPG

surfaes.Fluene=

3 × 10 ¹⁷

^ions/m

²

^{;inidentangle}

θ

^(a)30

^◦

^,(b)60

^◦

^and()70

^◦

^{. Arrows}

indiatetheion-beamorientation. Adaptedfrom Habenihtetal. (1999).

InFig.2.3anexampleofoarseningofripplesformedonfusedsiliaisdepited

(Flammetal. 2001). Theoarseninganonlybeexplainedbyonsideringthe

non-linearitieswhih are absentin the BH model.

Ordering

In some experiments, ripples show a tendeny to beome more aligned and

ordered. Inmanyexperimentsthenumberofdefets(misalignmentorrossing

between ripples) dereases with time. For example Ziberiet al. (2005) have

seen ordering and derease of defets in sputtering on Si by Kr

+

ions (see

Fig. 2.4). The order of ripples an be estimated by the ounting the number

of peaksin the Fourier spetrumof the surfae prole.

2.1.2 Dots

Formation of nano-dots is another phenomenon reported in several experi-

ments. As the anisotropy indued by the diretion of the ion-beam is elimi-

nated,eitherbyrotatingthesample(Frost et al. 2000)orbynormal-inidene

sputtering (Faskoet al. 1999), formation of dots is observed. However, dot

formationunderobliqueinideneirradiationandalsowithoutsamplerotation

has alsobeen reported by Ziberietal. (2006) onGe. The dots are highlyor-

deredinsize andhaveshort-rangeorderinginplaement(see Fig.2.5). Inthe

Figure2.3: AFMimagesequene,showingtheevolutionoffusedsiliasurfaetopography

withinreasingsputtertime

t

^{at2,6,10,and60min,}respetively. Theion-beamparameters:

800eVAr, ux= 400

µ

^A/m

²

^and

θ =

⁶⁰

^◦

^{. Thelateralsize of theimages is1}

µ

^{m. The}

wavelength of ripples inreaseswith time as

λ ∼ t ^γ

^with

γ = 0.15 ± 0.01

^{. Adapted from}

Figure2.4: Self-organizedSiripple patternsproduedby1200eVKr

+

ion-beamerosion,

θ = 15 ^◦

^{,fordierentionuenes: (a)3.36}

× 10 ¹⁷

^ions/m

²

^,(b)2.24

× 10 ¹⁸

^ions/m

²

^and()

1.34

× 10 ¹⁹

^ions/m

²

^(a)-()2

µ

^m

×

²

µ

^{mAFMimages(thearrowsgivetheion-beamdire-}

tion). (d)-(f) CorrespondingFourierspetrum(image range

±

^127.5

µm ⁻¹

^{). Theirlein}

()showsadefetintheAFMimage. Thenumberofdefetsdereaseswithtime. Moreover,

theangularwidthoftheFourierpeakdereaseswitherosiontimemeaningthehomogeneity

ing, whereas some authorsreportedthe formationof dotpatterns with square

symmetry (Frost et al. 2004; Ziberiet al. 2009). The oarsening behavior of

the ripples is also observed for dots, although in the ase of dot patterns, an

early growth inwavelength saturates inlonger times (Gago et al. 2001). The

formation of ordered dot patterns isnot explained by BH theory.

2.1.3 Holes

The so-alled ellular strutures or holes an be produed in experiments

with amorphous substrates. Fig. 2.6 shows data from Frost etal. (2004) in

experimentsonSisurfaeirradiatedby 500eV Ar

+

with samplerotation. The

appearane of ellularstrutures in the experiments withsample rotationan

be very sensitive to the rotation frequeny of the sample. We disuss this in

setion 7.5.

2.1.4 Smooth surfaes

Besides itsappliations for texturingthe surfaes,IBS tehniques an alsobe

usedforsurfaesmoothingatmirosopilengthsalesandforthepreparation

of ultra-smooth surfaes. Conventionally, ontinuous rotation of the sample

or the ion-beam has been proposed to suppress ripple formation (Zalar1985;

Zalar 1986) in seondary ion mass spetrometry (SIMS) and Auger eletron

spetrosopy (AES), where ripple formationwould redue the depth proling

resolution. There are many reports on experiments with or without sample

rotation, in whih the initial roughness of a the surfae is redued as the

sputteringproessgoeson(fortworeentworkssee(Headrik and Zhou2009)

and (Frost etal. 2009)). In Fig. 2.7 (adapted from the latter work ) initial

topography of an InSb sample is ompared to snapshots after 10 and 120

minsputtering by 500 eV N

+

ions atnormal inidene angle with ionurrent

density 200

µ

^A/m

²

^.

There is a lak of theory to explain the smoothing of surfaes by ion-

beam tehniques. In low ux and high temperature experiments, similari-

ties to epitaxial layer-by-layer growth is laimed to exist in IBS experiments

(Chan and Chason2007). In medium and high ux experiment, a down-hill

Figure 2.5: Sanning eletron mirosope (SEM) images of highly ordered ones on a

(100) GaSb surfaeshow the temporal evolution of dot formation during ion sputtering.

The nano-sale patterns are depited for dierent ion uenes (exposure times) of (A) 4

×

¹⁰

¹⁷

^ion/m

²

^{(40 s), (B) 2}

×

¹⁰

¹⁸

^ions/m

²

^{(200 s), and (C) 4}

×

¹⁰

¹⁸

^ions/m

²

⁽⁴⁰⁰

s). (D) The orresponding size distributions of the dot diameters are extrated from the

images. Thedotted linesrepresentGaussian tstothedotdiameter histograms. Adapted

Figure 2.6: Silion surfaetopographiesafter 20 minbombardmentby500eV Ar

+

ion-

beam(samplerotation), ux=300

µ

^A/m

²

^,

θ

^(a)0

^◦

^and(b)75

^◦

^{. Cellularstruturesform}

in bothases. AdaptedfromFrostet al. (2004).

suppress the destabilizingeets of BHtheory. This willbedisussed inmore

detail in setion. 3.1. The rotation frequeny may have an important role in

ahievingthesmoothedsurfaes,similartotheformationofellularstrutures.

This willbedisussed insetion.7.5 aswell.

2.2 Patterns on rystalline substrates

The above mentioned harateristis are not ommonly observed in experi-

ments with metalli substrates. For example, ripples may form in normal

inidene experiments or isotropi patterns in oblique inidene experiments

may evolve withoutrotation. Foraomprehensive olletionofexperimental

results on metalli substrate see Valbusa etal. (2002). The dierent senar-

ios of pattern formation on single rystalline metalli substrates are mainly

due to the dierent energy barriers in onjuntion with the rystallographi

anisotropies in suh materials. On the other hand, Surfae diusion is not

isotropi in rystalline substrates and the Sigmund's theory of sputtering ne-

gletseetslikehanneling,whihareduetotheregularanisotropistruture.

Valbusa etal.denedadiusive and anerosive regime,inwhihtheorien-

tationofstruturesaredeterminedbytheunitelldiretionorbythediretion

Figure 2.7: Sequene of AFM images whih shows the progressive smoothing of a Si

surfaeunder 500 eV Ar

+

ion-beam erosion,

θ = 45 ^◦

^{, ux = 300}

µ

^A/m

²

(simultaneous samplerotation). (a) Initial surfae(pre-roughened by Ar

+

erosion at 75

◦

ion inidene),

(b)after10 min(orrespondingto atotalappliedion ueneof1.1

×

¹⁰

¹⁸

^ions/m

²

^)and

() after 180min (2.0

×

¹⁰

¹⁹

^ions/m

²

^{). The rms roughness wasredued from}

R q

^=2.25

nmto

R q < 0.2

^{nm. AdaptedfromFrostet al. (2009).}

substrate temperatureand the erosion rate an betuned by the ion-beampa-

rameters. In Fig. 2.8, a transition from diusive regime to erosive regime by

inreasing the temperature atxed ion-beamparameters is depited.

Thepyramid-likestruturesformedinthediusiveregimearesimilartothe

patterns whih form in moleular beam epitaxy (MBE). Here, the dierene

toMBE is that deposition of adatomsis replaed by reationof vaanies by

erosion. The main destabilizingfator in both ases is the biased diusion of

surfae defets (adatoms or vaanies). Therefore, most of the basi features

of patterns are similar in both MBE and IBS at high temperature and low

ux regime. In the erosive regime, however, the main underlying proess is

the BH instability and diusion is not the leading fator in the pattern for-

mation. Hene, the main harateristis of patterns in the erosive regime are

similar to those observed in the experiments on amorphized semiondutors

(see setion4.2.4).

2.3 Advaned patterning methods

In the lastfew years, steps toward alternativeomplex IBS experiments have

Figure 2.8: The role of surfae temperature in the transition from the diusive to the

erosivesputteringregimeforAg(001). 1keVNe

+

ions,

θ = 70 ^◦

^{,ux =2.2}

µ

^Am

²

^,

t = 20

min. Thewhitearrowshowstheion-beamsatteringplane. Imagesize 180

×

^180nm

²

^;at

T = 400

^K:360

×

^360nm

²

^{. AdaptedfromValbusaetal. (2002).}

tering of alloys, sputtering of thin deposited lms, sputtering with double or

multiple beams, sputtering of substrate previously strutured on mirometer

length sales, et. Apart from the pratial advantages, exploring this un-

known area of ion-beam sputtering tehnology poses new physial questions,

whihan beanswered onlyby extendingand developingthe presenttheoret-

ialmodels.

2.3.1 Surfatant sputtering

In surfatant sputtering, ion-beam erosion is aompanied by deposition of a

seond(surfatant)atomispeies(typiallybyo-sputteringanearbymetal-

litarget). Co-deposition is adjusted ina way that a steady state overage of

(sub-)mono-layer thikness emerges.

Surfatantsputtering has been introduedby Hofsäss and Zhang (2008)as

anovel method witha widerange ofontrollablepatternformationsenarios.

Thepotentiallywidespetrumofthe appliationsof surfatantsputteringhas

not yet been probed. One of the available examples, shown in Fig. 2.9, is an

arrangementof nano-drops of Ag onrippled Sisubstrate.

An important physial mehanism, whih inuenes pattern formation is

thatthe presene ofmetalliatomsonthe surfaeof asubstrate anonsider-

ablyhange the erosion rate of substrate atoms. Furthermore, the partiular

formof surfaediusionof metalliatomsand theirtendeny tomixordemix

with the substrate an also aet the pattern formation. More details are

presented in hapter 5.

2.3.2 Compound beams

Joeet al. (2009)performedexperimentsapplyingmultiplebeams(partiularly

dual-beams)and alsosequentialsputtering fromdierent diretions. The aim

oftheseexperimentswastoproduestruturesofsuperimposedripplesformed

in dierent diretions. Although in none of the ases a linear superposition

wasobserved, the ase of dual-beamsleads to square symmetripatterns (see

Fig.2.10). Inthesequentialsputteringofpre-struturedsurfaes, thestepwise

rotation of the sample by

90 ^◦

^{led to a fast destrution of initially formed}

Figure 2.9: SEM pitures of Si surfaes eroded with 5 keV Xe at

θ = 70 ^◦

^{and uene}

3

×

¹⁰

¹⁶

^ion/m

²

^{with Ag surfatants with dierent overages (inreasing from left to}

right and top to bottom) of up to 10

16

Ag atoms/m

2

. The ripple pattern and ripple

wavelength for dierent overages are strongly inuened by the surfatants. Ag nano-

partilesof size10 nmorlessan beseenonthetopsoftheatripple plateaus. Adapted

Figure 2.10: (a) Rippled Au(001) surfae sputtered in the erosive regime, (b) Surfae

morphologyinduedby sequentiallysputteringof thepre-rippledAu(001)with 2keV Ar

+

with ux=0.31 ions/nm

2

s, uene=84.8 ions/nm

2

and

θ = 72 ^◦

^{. The initial ripple pat-}

tern is heavily damaged suh that its order and mean oherene length are severely de-

graded. ()Nano-patternsformed bydual ion-beam sputteringwithux=3.25ions/nm

2

s,

uene=6350ions/nm

2

and

θ = 73 ^◦

^{. Arrowsindiate ion-beamprojetion. Adaptedfrom}

Joeetal. (2009).

diretion. More details are presented in setion 7.4.

Methods

Pattern formation in IBS, an be studied theoretially at dierent levels and

time-,energy-andlength-saleswithdierentapproahes. Theompleteprob-

lemofIBS overs lengthsalefromatomisizestosomemirometersandtime

sales over arange from

∼ 10 ⁻¹⁷

^{se (duration of the primaryatom-ionolli-}

sions)to

∼ 10

^{min(typialpatternformationtimesale). Thesame extension}

exists alsointhe overed rangeof energies; The upper bound is the energy of

an impat,initiatedwith some keV ionand the lowerbound isthe energy in-

volved indiusion proess atroomtemperature, i.e. the meV range. Toover

this wide range of sales, dierent methods and approahes must be applied

and at dierent sales dierent approximations are neessary. A ombination

of allthe approahes presents a multi-salepitureof the whole phenomenon.

In this hapter we introdue the following theoretial models whih we apply

to IBS:

(i) Atomisti simulations based on binary ollision approximation

(BCA): In this lass of simulations, one starts from single impats of ions

on the surfae and follows asades of atomi ollisions, aiming to provide a

statistis of sputtering eets aused by impinging ions. The typial length-

sales onsidered inthis approahrange from someÅ tosome nm.

(ii)Kinetitheory: Inthisapproahonestudiestheevolutionofthephase

spae density

f (r, p, t)

^{of displaed atoms (and of the ions) at position}

r

^and

with momentum

p

^{in the frameworkof} Boltzmann's transporttheory.

(iii) Kineti Monte Carlo simulations: Many results of this thesis are

obtained from this method. Usually, one starts from the results of kineti

theory, i.e. one usesa simplefuntionalformofthe averagedenergydeposited

byaollisionasadeforsingleionimpattodeterminetheerosionprobability

of surfae atoms. Diusion proesses an easily be added in this approah.

Simulatinglarger salesof some hundreds of nm an be studiedby KMC.

(iv) Continuum theory of surfae evolution: In the ontinuum model

of surfae evolution the height of surfae is onsidered as a ontinuous, single

valuedsmoothfuntionofplaneoordinates

h(x, y )

^{. Inthemostgeneralform,}

the time evolution of

h

^{is desribed by a non-linear stohasti partial dier-}

ential equation, the growth equation. In priniple the growth equation may

inlude all the underlying proesses whih lead to the surfae evolution and

alsoexternal noiseby randomlyshot ions. Here, length sales larger thanthe

penetration depth and atomisti sizes are onsidered and therefore small size

utuations are negleted. The main parts of ontinuum theories are erosion

andsurfae diusion. Mostof thetheories inthis framework are basedonthe

Sigmund's theory of sputtering and a thermallyativated diusion model.

3.1 Binary ollision approximation

The most mirosopi approah to pattern formation under IBS would be

a full-edged Moleular Dynamis (MD) simulation. However suh a simu-

lation has to bridge the above-mentioned sales in time, spae and energy,

whih is not possible at present. Nevertheless, the approah is used for sin-

gle ion impats to study the defet generation and also the mass transport

due to displaed atoms. But an approximate version of MD, the binary ol-

lision approximation (BCA), has beome a versatile tool in the study of ion

sputtering phenomena, inluding pattern formation. The main idea of this

method is to redue all interations to a series of binary ollisions between

pairs of partiles. In between ollisions, the trajetory of the partiles are

straight-line segments traversed with onstant veloity, initiating from a ol-

lision and ending at the next ollision. Changes in veloity and position af-

ter eah ollision an be integrated numerially (Robinson and Torrens 1974;

Robinson1994). This approahissuessfully usedtoquantitativelyalulate

sputtering yields in the muh used and well established programs TRIM and

(a) (b)

1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1

0 2 4 6 8 10 12

e

ρ

Figure 3.1: (a) Sample asade originating from an impat of a 5keV Cu ion on aCu

rystal. Theangleofinideneis

60 ^◦

^{. Theubeshown,atsjustassaleandhasvolume2.65}

nm

3

. (b)Surfaedensityof meanenergy ofsputteredCuatoms vs. distane

ρ

^(measured

inunitsof

a = 3.61

^{Å)frompointofioninidene. Thesolidlineisthebestttothedata;}

0.297(ρ ² − 0.392ρ) exp( − 1.27ρ)

^{andthedottedline,whihorrespondstoaGaussiant,is}

obviouslyinadequate. AdaptedfromFeixetal. (2005).

ples underIBS. Feix(2002)and Feix etal. (2005) have studiedthesputtering

of Cu rystals by means of BCA simulations to test some assumptions and

approximations used in less mirosopi approahes.

3.1.1 Casade shape

Feix etal. (2005) onsidered an ensemble of 6000 impinging ions and alu-

lated the averaged statistis of the indued ollision asades (see Fig. 3.1

(a)). One of the main results is about the distribution of deposited energy

by a single ionasthis quantity underlies the ontinuum theory and the KMC

approah (see below). For normal inidene, the simulations show an energy

distributionasdepitedinFig.3.1(b),whihhasaminimumneartheposition

where the ionpenetrates the surfae, and deays exponentially with distane.

This fromdeviates signiantly fromGaussian shape entered atthe loation

of primaryknok-onollision,whihisusedinthe vastmajorityofontinuum

and KMC approahes, and was proposed by Sigmund (1969) on the basis of

3.1.2 Defet generation

Apart from erosion of the substrate atoms by the energy transferred from

ions, generation of surfae defets (adatoms and vaanies) is known to be

another eet indued by ollision asades (Nordlundet al. 1998). Exited

atomsreahingthe surfaewithenergies morethan thesurfaebindingenergy

E b

^{, will be sputtered o, and leave a vaany, however a large fration of}

partiles have energies less than

E b

^{. These atoms remain on the surfae and}

beomeadatoms. At hightemperature, defets reombine and vanishrapidly,

whereasatlowtemperatureregimealarge numberofthem remainsforlonger

time(Floro etal. 1995). Feix (2002)found a distributionof the energy ofthe

partilesreahing the surfae

ǫ s

^{,whih obeysa simple power low}

p(ǫ _s ) ≈ a

(b + ǫ s ) ^γ ∼ ǫ ⁻² _s

^(3.1)

with

a = 5.26

^,

b = 5.03

^and

γ = 1.87

^{for 5 keV Cu ion hitting a Cu tar-}

get. The shape of the distribution is almostindependent of the distane from

the impat point up to a large distane (see Fig. 3.2). This nding is in a-

ordane with experimental observations and a simple theory of asades by

Farmeryand Thompson (1968).

3.1.3 Down-hill urrent

Carterand Vishnyakov (1996) observed that in o-normal inidene (up to

45

◦

) sputtering of Si with high energy (10-40 keV) Xe

+

ions, sputtering ero-

sion an indue smoothing. Using MD simulations Moseler et al. (2005) ex-

plained the irradiation-indued smoothing on diamond-like arbon surfaes.

Theyfound a down-hillurrentof atomsalong thebeam-diretionindued by

the ions. This urrent may suppress the urvature dependent BH instability

(setion2.1.1), beause they transport atoms bak to the ripple valleys. The

down-hill urrent is proportional to the surfae slope

j ∝ −∇ h

^{. By substitu-}

tion of the urrent density into the ontinuity equation

∂h/∂t = −∇ · j

^{, one}

obtainsa smoothingequation for the height eld

h

^,

∂h/∂t ∝ ∇ ² h

^.

This down-hill urrent is also easily observed in BCA as demonstrated in

0 10 20 30 E [eV] 40 50 60 70 80 90 100 0 2

4 6

8 10

12

r [a]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

p(r, E)

Figure3.2: Energydistributionextratedfrom6000independent5keVimpatingCuions

fordierentdistanesfromimpatpoint(measuredinunitsof

a = 3.61

^Å).Thedistribution isalmostindependentof

r

^{. Adaptedfrom Feix(2002).}

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12

y [a]

x [a]

Figure 3.3: Spatial distribution of ejeted Cu atoms emerging from 6000 independent

trials of hitting the (x,y) rystal surfae(oriented in (100) diretion) with asingle 5 keV

Cuionatoblique inidene. Distanesaremeasuredin units of

a = 3.61

^{Å. Adaptedfrom}

Feix(2002).

3.2 Kineti theory

To obtain statistis of the ollision asade and alulatethe sputtering yield

dependingonion-targetparameters,Sigmund(1969;1973)presented asolution

of a Boltzmann transport equation with assumption of random slowing down

ofions inaninnitemedium. Themain approximationinthisapproahisthe

negletofinterationsbetweenatoms,whihthusformatreeofnon-interating

partiles (referred to as the ollision asade). The most importantresults,

whihunderlie the approahes in subsequent sales are the following:

(i) The erosion rate at eah surfae point is proportional to the power

brought tothis pointvia ollision asades.

(ii)Thedistributionofollisionasadesisalulatedapproximately. Sigmund

found thatinsomeasesthis forman beapproximatedbyasimpleGaussian.

(iii) The sattering events originated by the penetration of energeti ions,

leading to sputtering our in a layer near to the surfae with very small

thikness. Mostof the sputtered atoms belong toathin surfae layer(

∼

^{5 Å).}

(iv)Theenergy distributionof ejetedpartilesfollowsFig.3.1. Thisresult

wasrstobtainedbyanelementaryargumentonasadesbyThompson (1968).

Theseresultsthenbeamethemainpriniplesofalmostalllatertheoretial

works onIBS.

3.3 Kineti Monte Carlo

AlltheexistingKMCsimulationmodelsofIBS(forexamplesseeChason et al. (2006),

Stepanova and Dew (2006),and Hartmann etal. (2002)), are based uponthe

results from the kineti theory, mentioned above and inlude two parts of

erosion, upon Sigmund's theory and a surfae relaxation proess. A simu-

lation run onsist of a sequene of single ion shots, a alulation of the de-

posited energy at the urrent surfae for eah ion and random disrete hop-

ping of surfae atoms orresponding to surfae diusion. Most models (with

theexeptionoftheworkby(Bartosz Liedke 2009))desribethesurfaeinthe

framework of a solid-on-solid (SOS) model, thus exluding overhangs, drops

and bulk vaanies. Furthermore,a re-deposition of sputtered partilesis not

whih was developed fromHartmann etal. (2002) to Hartmannet al. (2009)

(HKGKmodel).

Thesystemonsistofasquarelattiesofsize

L × L

^{(withperiodiboundary}

onditions, if not stated otherwise) and the SOS surfae is desribed by an

integer-valuedtime-dependentheightfuntion

h(x, y, t)

^{onthelattie. Inmost}

ases, we start from a at surfae, i.e.

h(x, y, 0) = 0

^{. The details of erosion}

and diusion trialsare asthe following.

3.3.1 Erosion

As mentioned above, the erosion proess is based on Sigmund's theory, i.e.

theSigmund formulaEq. 3.4isappliedfor every singleimpingingion. Anion

startsata randompositionin aplane paralleltothe planeof the initiallyat

surfae (

x − y

^{plane), and follows a straight trajetory inlined at angle}

θ

^to

the normal of this plane. The ion penetrates into the solid through a length

a

^{and releases its energy. Then we hek all the lateral atoms as the subjet}

for sputtering suh that an atom at a position

r = (x, y, h)

^{is eroded with}

probabilityproportionalto

E(r)

^.

We have put

ǫ

^{to be}

(2π) ^3/2 σµ ²

^{, whih leads to sputtering yields}

Y ≃ 7.0

^,

thusshouldbekeptinmindwhenomparingsimulationresultstoexperimental

data. Aording to the Bradley Harpertheory, the ripple wavelength

λ

^sales

like

λ ∼ Y ^−1/2

^{sothatloweryieldsleadto}orrespondinglylargerlengthsales.

Throughout this work we use a set of parameters as default values if not

statedotherwise. We xed

σ = 3

^,

µ = 1.5

^and

a = 9.3

^{(in lattieonstant).}

3.3.2 Diusion

Wehaveimplementeddierentmodelstodesribethesurfaemotionofatoms.

These range from simple, irreversible surfae relaxation to ativated hopping

over energy barriers, whih may depend both oninitial and nal state of the

move and inlude Ehrlih-Shwoebel non-equilibrium kineti eets. We al-

waysusefull diusionmodels,soonediusionstepreferstoaompletesweep

of the lattie. In the following, we briey introdue the three basi types of

diusion models, whih we have used throughout our simulations. Details of

setion 4.2.

(i) A simple, non-thermal, irreversible relaxation proess has been intro-

dued by Wolf and Villain(1990). Foreaholumn,it istested oneduring a

sweep, whetherthe partileatthetop oftheolumnaninrease itsoordina-

tionnumber, i.e.itsnumberof nearestneighbors, byhoppingtoaneighboring

olumn. Ifthis is thease, the partilehops tothatneighborolumnwhere it

obtains the highestoordination number(setion 4.2.1).

(ii) A lass of thermal diusion models is based upon a Hamiltonian

H

^,

whihontrols the thermalroughening of afaet. Trialmoves fromsite

i

^{to a}

nearestneighborsite

f

⁽

h _i → h _i − 1, h _f → h _f − 1

^{)areaeptedwith}probability

p(i → f ) = [1 + exp(∆ H /k B T ))] ⁻¹

^where

∆ H

^{is the hange in} Hamiltonian due to the hop. (setion4.2.2).

(iii)The Arrhenius models are basedona kinetiproedureand use hop-

ping via transition states. For eah step, a move from initial(

i

^{) to nal (}

f

⁾

ongurationishosenrandomlyfromapredenedlist. Herewerestritmoves

tonearestneighborhops fromsite

i

^tosite

f

^{. Wewould havetoinludemore}

moves, if we want tomodelmaterialspei diusion proesses. The moveis

performed with a probabilityproportionalto anArrhenius hoppingrate

k = k 0 exp

− E(i → f ) k B T

(3.2)

Valuesof the energybarriers

E(i → f )

^{havetobetaken from}experimentalor simulation data (setion4.2.3).

TheonnetionbetweentimeinKMCmodelsandreal experimentsismade

by omparing the attempt frequenies of dierent events in KMC with orre-

spondingkinetiratesinthelabondition. Inourmodeltherearetwodierent

time sales, (i) the time intervals between the shooting of two impingingions

τ i

^{and (ii) the waiting time between diusion sweeps}

τ d

^{. By tuning these two}

time sales, a wide range of experimental onditions an be overed. Our

default values orrespond to a typial ux of

0.75

^{(ion/atom seond) and a}

temperatureof

350

^K.

3.4 Continuum theory

3.4.1 Bradley-Harper model

Bradleyand Harperstartedfromthe resultsofkinetitheory,that thenormal

veloity of the eroded surfae

∂h(x, y, t)

∂t

1

q 1 + ( ∇ h) ² = − v n (x, y, t)

^(3.3)

isproportionaltothetotalenergytransferredtothepoint

(x, y, h(x, y))

^bythe

ollision asades. An arbitrary ion impingingthe surfae at point

P

^{, omes}

to rest at point

O ^′

^after penetrating into the solid by a distane of

a

^along

itstrajetory. The deposited energy of the ion at any point

O

^{at the surfae}

is a funtion of the distane vetor

R = (X, Y, Z)

^between

O

^and

O ^′

^{. The}

averagedenergy deposition funtionis taken tobea Gaussian

E(R) = 1

(2π) ^3/2 σµ ² exp( − X ² + Y ² 2µ ² − Z ²

2σ ² )

^(3.4)

as proposed by Sigmund (1969).

µ

^and

σ

^{are width of Gaussian funtion}

parallel and perpendiular to the beam trajetory. To alulate the erosion

rate, all the ontributions from homogeneously impinging ions at inidene

angle

θ

^{with respet tothe normalof the surfae shouldbe summedup;}

v n (r) = Y J ion

Z

dr ^′ E(r − r ^′ ) ˆ n · e _θ

^(3.5)

where

J ion e _θ

^{is the ionux with}

e _θ =











sin(θ) 0 cos(θ)











.

ˆ

n

^{is the unit vetor normal to the surfae and}

Y

^{is the sputter yield. The}

integral is taken over the surfae. The integral is evaluated in a gradient

expansion (i.e. in

( ∇ h) ⁿ

^{) and a small slope} approximation whih starts with the following terms:

∂h(x, y, t)

∂t = − v 0 (θ) + v ^′ ₀ (θ) ∂h(x, y, t)

∂x + ν x

∂ ² h(x, y, t)

∂x ² + ν y

∂ ² h(x, y, t)

∂y ² .

^(3.6)

v 0

^{is the average erosion veloity of a planar surfae.}

ν x

^and

ν x

^{are eetive}

surfae tensions indiretions parallel and perpendiular tothe projeted di-

retion of ion-beam onto the surfae. To solve the obtained growth equation,

we let

h(x, y, 0) = A exp[i(k x x + k y y)]

^{. The general solutionof Eq. 3.6for the}

mode

k = (k x , k y )

^{takes onthe from}

h k (x, y, t) = − v ₀ (θ)t + A exp[i(k _x x + k _y y − ωt) + Γt].

^(3.7)

substitution of the solutionsleads to

ω = − v ₀ ^′ (θ)k x

^(3.8)

and

Γ(k x , k y ) = − ν x k ² _x − ν y k ² _y .

^(3.9)

This means that an arbitrary mode

k

^{propagate along the} orientation of the beam (projeted onto the

x − y

^{surfae) with phase veloity}

− v ₀ ^′

^{and also}

grows (deays) in amplitude with the rate

Γ

^{. The}

θ

^{dependene of eetive}

surfaetensionsresultsfromthegradientexpansionandoneexampleisshown

inFig.3.4forthe defaultparametersofour KMCsimulation. Forsomevalues

of

θ

^{, both}

ν x

^and

ν y

^{are negative, leading to positive growth rate}

Γ

^{for all}

wavevetors. In experiments, it is observed that a spei wavelength grows

faster than all others and forms periodi ripple-likestrutures. A stabilizing,

i.e. smoothing mehanism, whih is laking in Eq. 3.6 is surfae diusion,

whih gives rise to a term

∝ ∇ ⁴ h

^{(see setion 4.2). Adding this term results}

in linearevolutionequation of Bradley-Harpertheory,

∂h(x, y, t)

∂t = − v ₀ (θ)+v ₀ ^′ (θ) ∂h(x, y, t)

∂x +ν _x ∂ ² h(x, y, t)

∂x ² +ν _y ∂ ² h(x, y, t)

∂y ² − B ∇ ² ∇ ² h

(3.10)

where

B

^{istheoeientofsurfaediusivity. Takingthediusionmehanism}

into aount hanges the growth rate into

Γ(k x , k y ) = − v x k _x ² − v y k _y ² − B(k _x ² + k ² _y ) ² .

^(3.11)

Nowforany value of

θ

^(exept

θ = 0

^and

θ = θ c

^where

ν x = ν y

^),

Γ

^{has amax-}

imum value forasingle

(k _x ² , k ² _y )

^{. Sinethe inluded diusionterm isisotropi,}

the maximum of

Γ

^{ours always for}

k

^{whih is either in}

x

^or

y

^diretion,

i.e.

k = (k ^max _x , 0)

^or

k(0, k _y ^max )

^{. The maximum lies inthe diretion, for whih}

Figure3.4: Eetivesurfaetensionsin twodiretions,parallel andperpendiulartothe

ion-beam diretion as a funtion of inidene angle

θ

^for

σ = 3 µ = 1.5

^and

a = 9.33

^.

For

0 < θ < θ c

^,

ν x

^,

ν y < 0

^and

| ν x | > | ν y |

^{, therefore, the growth of} instabilities with the wavevetorsparallel to thebeam diretionis fasterand ripples perpendiular tothe beam

diretionform. For

θ c < θ < θ c ^′

^,

| ν y | > | ν x |

^{andtherefore ripplesparallel tothe ion-beam}

diretionform. For

θ c ^′ < θ

^,

ν x

^{beomespositiveand}perturbationswiththewavevetorin

x

diretiondampandagainformationoftheripplesparalleltothebeamdiretionexpeted.

the negative surfae tension has the larger negative value. This predits for a

wide range of materials and ion parameters, ripples with wavevetor aligned

parallelto the projetion of ion-beam for

θ < θ c

^{and ripples with wavevetor}

perpendiulartothe ion-beamprojetionfor

θ > θ c

^{. This preditionhas been}

onrmedinnumerous experimentsandmakesthe BHtheoryreliablefor sur-

faetexturing by ion-beam. The typial length sale of patterns predited by

lineartheory of BH is

ℓ = (2π)

s 2B

| ν m |

^(3.12)

where

ν m = min[ν x , ν y ]

^.

3.4.2 Cuerno-Barabási non-linear model

non-linear orretions to the Eq. 3.3 They also took into aount the shot

noise i.e., the randomarrivalof ions tothe surfae asaGaussian whitenoise

η(x, y, t)

^{with zero mean and variane} proportional to the ux. The growth equation then beomes

∂h(x, y, t)

∂t = − v ₀ (θ)+v ₀ ^′ (θ) ∂h

∂x +ν _x ∂ ² h

∂x ² +ν _y ∂ ² h

∂y ² + λ x

2 ( ∂h

∂x )

2

+ λ y

2 ( ∂h

∂y )

2

− B ∇ ² ( ∇ ² h)+η.

(3.13)

This equation is an anisotropi version of the Kuramuto-Sivashinsky (KS)

equation,whihiswellknowninpatternformationtheories(Kuramoto and Tsuzuki 1976;

Sivashinsky 1977).

A rossover time

t c

^{is dened as the time in whih the nonlinear eets}

beome dominant and the system leaves the validity region of the linear ap-

proximation. From the linear equation, the amplitude of ripples at

t c

^is

∼ exp( | ν _m | t _c /ℓ ² )

^{, whereas from}

∂ _t h ∼ λ( ∇ h) ²

^{the amplitude is estimated in}

order of

ℓ ² /λt _c

^{. Combining these two relations,the rossover time is}

t c ∼ ( B

ν _m ² ) ln( | ν m |

λ ).

^(3.14)

Depending on the signs of

ν x

^,

ν y

^,

λ x

^and

λ y

^{, dierent} morphologies are expeted from non-linear theory. Typially for small values of

θ

^where

ν x

and

ν y

^{are both negative,}

λ x

^and

λ y

^{are also negative. For short time sales}

(

t ≪ t c

^{), the same ripples as predited by BH form, but ripples get blurred}

anddisappear graduallyforlongtimes(

t ≫ t c

^{). Thepatternsshowthetypial}

Kuramoto-Sivashinskytypeofspatio-temporalhaos. Inreasingtheinidene

angle,

λ x

^and

λ y

^{obtaindierent signswhere}

ν x

^and

ν y

^{arestillboth negative.}

Park et al. (1999) have shown that two transitions our in this regime. In

early stage of pattern formation, standard ripples from linear theory form;

At the rst transition, ripples disappear and the surfae beomes rough; At

the seond transition, stable ripples with rotated orientation by an angle of

tan ^{−1 q} − λ x /λ y

^{form. The stability of these ripples an be understood as a}

onsequene of a non-linear anellationof modes. (Rost and Krug1995).

3.4.3 Makeev, Cuerno and Barabási model