C = ∞ (beause N A−B inreases linearly with the system size, while N A−A

and

N B−B

^{inrease quadrati). InFig. 4.6,}

C

^{asafuntion of}temperaturefor dierent simulation time is shown. For very low temperatures, no lustering

oursevenforlong-timeruns. Inintermediatetemperature,smallsizelusters

form and grow in time slowly, whereas in high temperature regime an strong

lustering is observed. The diusivity of partiles is inreased by inreasing

thetemperatureand inthe xed simulationduration, the partileshave more

hane to meet eah other and sine their binding energy is rather high, as

they attah,it isvery unlikely todetah again.

Metalli lusters on pre-sputtered templates

Thesame simulations arenow performed usinga sinusoidallymodulated

sub-strate template. Examples of distribution of partiles at dierent run times

Figure4.6: Clusteringevolutionofpartilesonaattemplateasafuntionoftemperature.

Dierent olors orrespond to dierent simulation times. By inreasing the temperature,

partiles make largerlusters. In all asesthe size of lusters tends to inrease, although

theinreasein lowtemperatureisveryslight. Forthedenition of

c

^{seethemain text.}

Figure4.7: Clusteringofpartilesonasinusoidaltemplatedependingontemperatureand

time. Byinreasingthetemperature,partilesmakelargerlusters. Inallasesthesize of

lusters tends to inrease, although the inreasein low temperatureis veryslight. Large

lustersformed in high temperature, loatedmainly in the valleys and plateaus, although

thereisaprefereneforvalleysin omparetotheplateaus.

(i) Partiles aumulate in 1-d arrays (nano-wires) parallel to the

align-ment of templateripples;

(ii) The width of nano-wires reahes a maximum size proportional to the

wavelengthof templateripples at enoughtime sales

(iii)Nanowires have the tendeny to formatvalleysof templates, the next

preferredloationsare the hilltop ridges.

Allthreefeatureshavebeenfoundinexperiments(AgonSi)byOateset al. (2007).

4.3 Erosion-Diusion interplay

In this setion we fous onthe study of morphologies emerging in KMC

sim-4.3.1 Dependene of patterns on diusion in the erosive

regime

Diusion is often onsidered as an unspei smoothing mehanism in the

erosive regime of IBS experiments on amorphous surfaes. In the ontinuum

theory,it issubsumed asa singleterm

∼ − B ∇ ⁴ h

^{. Inthis subsetion we show}

thatontrarytothisommonbeliefdierentdiusionmehanisms,whih

would allend up ina

− B ∇ ⁴ h

^{term in the ontinuum limitdo have profound}

eets on the morphology, espeiallyin the limitof long times, i.e. high

u-enes. Figs.4.8and4.9omparethetimeevolutionofsputteredsurfaesunder

identialonditions oferosionbut with dierent diusionmodels. Fig.4.8

de-pits the evolution of surfae morphologywith a Hamiltoniandiusion model

(

n = 2

^,

T = 0.2J

^{) and Fig. 4.9 shows} orresponding results for a net-bond-breaking model. The short-time behavior of both models is very similar, an

initialroughening is followed by the formationof ripples. The dierenes

be-tween the twomodels arise after

∼ 10

^{ions/atom of} sputtering. At that time, ripples produed in the ase of the net-bond-breaking model saturate in

am-plitude and align more regularly, and the number of defets redues as time

proeeds,whereas the ripplesproduedby the Hamiltonianmodelstarttoget

blurredand shortenin length.

InFig.4.10weomparethemorphologiesofthefourmaintypesofdiusion

modelswehaveintrodued(Wolf-Villain,Hamiltonian,bond-brakingand

net-bond-braking) at very long times.

The irreversible Wolf-villain model(similar to a

T = 0

^surfae relaxation) produes an extremely ordered stable pattern of straight ripples (even at

t ∼ 10 ⁴

ions/atom), whereas the patterns of the Hamiltonian and bond-breaking models loose long-ranged ripple order after a few hundred eroded

monolayers. But note that the net-bond-breaking model shows a defet-free

ripple pattern after 400 ions/atom. Although the overall diusivity of the

net-bond-breaking is approximately equal to that of the bond-breaking and

the Hamiltonianmodels, net-bond-breaking implies omparable diusivity of

adatomsand vaanies(unlikethebond-breakingmodel)andismoresensitive

to the energetis of bonds than to surfae morphology, whih dominates the

Figure4.8: SurfaeproleorrespondingtoHamiltonianmodelofdiusionwith

n = 2

^and

substratetemperature

T = 0.2Jk _B ⁻¹

^{and defaultvaluesof thebeamparameters(}

θ = 50 ^◦

^).

Startingfromtoptobottomandlefttoright,

t = 0.5

^{,1.5,4.0,9.0,14.0and22.0ions/atom.}

Theion-beamdiretion isindiated by thebar. Lateralsizeof thesystem

L = 256

^{. After}

initialformationofripples,theystabilize andthen startto getblurred.

Figure 4.9: SurfaeproleorrespondingtoArrhenius(net-bond-breaking)modelof

dif-fusionwith substrate temperature

T = 600

^{K, anddefault valuesof thebeamparameters}

(

θ = 50 ^◦

^{). Starting from top to bottom and left to right,}

t = 0.5

^{, 1.5, 4.0, 8.0, 12.0and}

18.0ions/atom. Theion-beamdiretion isindiatedbythebar. Lateralsize ofthesystem

L = 256

^{. Ripplesform after}

∼ 3

^{ions/atomandgrowin lateralsize andgetmoreordered}

withtime. Theripplesamplitudesaturates forlongertimes.

Figure 4.10: Long time morphologies emerge from dierent diusion models (a)

Wolf-Villainfor

t ≈ 10 ⁴

^{ions/atom,extremelyorderedpatternswithtilted}orientationin respet to the ion-beam orientation, (b) Hamiltonian with

n = 2

^{, for}

t = 300

^ions/atom,

ratio-nal ordered ripples whih blur in time gradually, () Arrheniusbond-breaking for

t = 20

ions/atom, rather short stable ripples with weak ordering and (d) Arrhenius

net-bond-breakingfor

t = 400

^{ions/atom,veryorderedrippleswhit}annihilationofdefetsbytime.

Figure4.11: Morphologies(

L = 64

^{)emergingfromdierentdiusiononditionsfor}

θ = 0 ^◦

at

t = 10

^{ions/atom. (a) nodiusion, (b) Arrhenius}bond-breaking diusion with default parameters,() Arrheniusnet-bond-breakingdiusionwithdefaultparametersand(d)

Ar-rheniusnet-bondbreakingdiusionwiththeenhanedattemptrate(seethemain text).

4.3.2 Crossover from erosive regime to diusive regime

We now turn to a disussion of the aessibility of the rossover between the

erosiveandthe diusiveregimeinKMC simulations. Toestimatetheonset of

therossover weadoptanargumentfromVillainand Pimpinelli(1994)about

adatom island formation and stability in MBE, i.e. we simply assume that

erosionduringIBS isequivalenttoadepositionofvaanies ata

harateris-tirate

F

^{, whihis the ionuxtimesthe sputtering yield. The typiallength}

sale of surfae strutures emerging from diusion and deposition is given by

Villainand Pimpinelli(1994)

ℓ _d ∼ ( D

F ) ^1/6 ,

^(4.19)

Im Dokument Nanoscale pattern formation on ion-sputtered surfaces (Seite 71-79)