Nucleation theory and the early stages of thin ﬁlm...

Nucleation theory and the early stages of thin film growthC. Ratscha)

Department of Mathematics, University of California, Los Angeles, California 90095-1555

J. A. VenablesDepartment of Physics and Astronomy, Arizona State University, Tempe, Arizona 85287-1504and CPES, University of Sussex, Brighton BN1 9QJ, United Kingdom

~Received 9 June 2003; accepted 17 June 2003; published 2 September 2003!

A review is given of nucleation and growth models as applied to the earliest stages of thin filmgrowth. Rate equations, kinetic Monte Carlo, and level set simulations are described in some detail,with discussion of remaining uncertainties, in particular the functional form of the so-called capturenumbers in rate equations. Recent examples are given of sub-monolayer nucleation at surfacedefects, attachment-limited capture, and Ostwald ripening. The experimental literature is cited, andexperiment–theory comparisons are made where possible. Emphasis is given to fast computationalmodels that can span a large range of length and time scales, which might be further developed inthe direction of on-line process control. ©2003 American Vacuum Society.@DOI: 10.1116/1.1600454#

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I. INTRODUCTION

Nucleation and growth of thin films include processestime and length scales that span many orders of magnitAtomic motion occurs on length scales of the order of Å, atime scales that reflect the typical atomic vibration frequcies ~i.e., 10213 s). On the other hand, a typical optoelectronic device might be up to several microns in size,its growth can take minutes or even hours. Thus, modenucleation and growth of thin films presents a substanchallenge to theoretical physicists and material scientiMoreover, some of the phenomena that occur are inherestochastic in nature, and an ideal model would seamlecombine the different time and length scales, but incluonly the necessary fluctuations.

The models typically used in nucleation theory are eitcompletely stochastic or completely deterministic. Mefield rate equations~REs! are a set of coupled ordinary differential equations~ODEs!, that were developed for thiproblem1,2 more than 30 yr ago. They are easy to formulaand relatively easy to solve. Several results of nucleatheory have been successful in elucidating basic aspecepitaxial growth. In particular, scaling results derived froRE nucleation models have, under the appropriate circstances, been used to deduce microscopic parameters sudiffusion constants, adsorption and binding energies frcomparison with experimental measurements. Howethese equations contain no explicit spatial information, athus do not readily yield information on surface morphologOne of the challenges is to include the spatial informatproperly into a model that is mean-field by construction;cent progress in this area and limitations are discussedetail in this article.

Continuum models based on partial differential equatio~PDEs! are appropriate mainly at large time and lengscales.3,4 By construction, features on the atomic scale

a!Electronic mail: [email protected]

S96 J. Vac. Sci. Technol. A 21 „5…, SepÕOct 2003 0734-2101 Õ200

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neglected, so they are poorly suited to describe growththis scale, and we do not discuss continuum models inarticle. However, we note that since continuum models,well as rate equations, are based on differential equatithey are amenable to analytic treatments that can elucide.g., asymptotic or stability properties.

An alternative to completely analytic approaches are amistic models that explicitly take into account the stochasnature of each microscopic process that may occur dunucleation and growth of thin films. They are typicalimplemented in the form of molecular dynamics~MD!5 orkinetic Monte Carlo~KMC!6 simulations. MD simulationsare very useful for identifying relevant microscopic prcesses, such as the detailed steps during nucleation. Butand size limitations make them unfeasible for studyigrowth on technologically relevant time and length scalKMC simulations, on the other hand, have been used scessfully to study qualitative, and in limited cases, quanttive, behavior of growth. They allow for easy implementtion of a large number of microscopic processes, whose rare ideally obtained from first principles calculations.7–9

However, the occurrence of very fast rates~which is particu-larly relevant at higher temperatures! ultimately limits theapplicability of these methods to larger systems.

Recent work has attempted to develop new modelsare hybrid models between continuum, PDE-based methand atomistic, stochastic methods. One approach, terconfigurational continuum, which appears to be very proising, has been developed by Kandel and co-workers;refer to Ref. 10 for further details. Another approach that hbeen developed by one of us~C.R.! in the past few years isan island dynamics model, based on the level-set methThis approach will be discussed in more detail in Sec. IIThis model allows us to describe thin film growth as cotinuous in the plane of the surface, yet each atomic layediscretely resolved. Moreover, different sources of fluctutions can be isolated and studied individually.

S963Õ21„5…ÕS96Õ14Õ$19.00 ©2003 American Vacuum Society

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S97 C. Ratsch and J. Venables: Early stages of thin film growth S97

One generally distinguishes between growth on singu~nominally flat! surfaces, and vicinal, or stepped surfacesvicinal surface can be considered as a number of flat terrthat are separated by steps of atomic height. Growth on thsurfaces proceeds either via step-flow, where atoms diftoward a step before meeting another atom, or via nucleaand growth, where islands nucleate and grow on the terrGrowth on vicinal surfaces was first discussed in a sempaper by Burton, Cabrera, and Frank~BCF!.11

In this article we focus on modeling of growth on singulsurfaces, where one distinguishes the following three gromodes: Frank–van der Merwe~FM! ~layer-by-layer! growth,Volmer–Weber ~VW!, and Stranski–Krastanov~SK!growth.12–14 During VW growth three-dimensional~3D! is-lands form on the surface. The competition between FMVW growth can easily be understood based on energeticguments as a competition between surface and interfaceergies. The case of SK growth is more complicated, anintimately connected to~elastic! strain energy, which arisein most heteroepitaxial systems. During SK growth, onemore layers form initially~the so-called wetting layer!, fol-lowed by the formation of 3D islands. Understanding ttransition between growth modes is of great interest. Tfocus of this article is on submonolayer growth and will naddress these issues, but the methods described are alsplicable to multilayer growth.

The remainder of this article is organized as follows:Sec. II B we discuss mean-field rate equations. In particuwe focus on scaling laws, and how these equations can perly describe mean-field quantities. In Sec. II C we descrthe kinetic Monte Carlo method, and discuss some ofapplications. The level-set method is described in Sec. IIt is well known that rate equations do not properly predthe entire island size distribution, for the reason that requations do not contain any explicit spatial information. Rcent attempts and progress in including this spatial informtion implicitly through particular forms of capture numbeis discussed in Sec. III A. Recent models of nucleationdefect sites are described in Sec. III B. In Sec. III C,address the problem of attachment-limited, and timdependent capture numbers. Section III D highlights issrelated to coarsening and Ostwald ripening, and annealingeneral. Finally, in Sec. IV we give an assessment ofrelation of all the theoretical methods introduced, and a psonal impression of future directions.

II. MATHEMATICAL AND COMPUTATIONALMETHODS

A. General considerations

Typical processes that may occur during epitaxial groware illustrated schematically in Fig. 1. Atoms are deposionto a perfect substrate surface with a deposition fluxF ~a!;in the older literature, this same quantity has been termeddeposition rateR, both measured typically in monolayers psecond. Once atoms are on the surface as adatoms, thediffuse with a diffusion constantD ~b!. Adatoms can meeother adatoms to form a dimer~c!, or attach to existing is-

JVST A - Vacuum, Surfaces, and Films

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lands~d!. Once adatoms are attached to an island, theydetach from the island edge~e! or diffuse along the islandedge~f!. Deposition of adatoms on top of islands and tcorresponding processes have to be considered as wel~g!,~h!; at high temperatures some adatoms can re-evaporat~i!.In this section we describe different methods that mothese processes at different levels of detail.

The most detailed description is a MD5 simulation. In aMD simulation one calculates the forces on all atoms, athen moves the atoms according to the equations of motThe most crucial aspect of an MD simulation is the knowedge of the correct potentials. The timestep is required tosmall enough to resolve the vibrational frequency of theoms, so that there is a natural limit on the time that cansimulated. Moreover, even with simple pairwise potentithe evaluation of forces is rather time-consuming, so tsimulations of realistic system sizes on realistic time scato describe nucleation and growth is currently not possiwith MD simulations. This is still true, despite some impresive recent advances in speeding up MD simulations.15 How-ever, these simulations are very useful to identify some ofrelevant microscopic pathways during nucleation and tfilm growth.

One can make significant progress by using transitstate theory~TST!.16,17 In TST rates are associated with mcroscopic events such as adatom diffusion, and all the irevant atomic vibrations are neglected. This way, the simution timestep and thus the computational efficiencyincreased by 5-10 orders of magnitude for realistic growconditions. One can usually express the rates in the formn5n0 exp(2DE/kT), whereDE is an activation energy barrierk is the Boltzman constant, andT is the temperature. Theprefactorn0 is typically of the order of the atomic vibrationfrequency, and is set to 1012– 1013 s21 in many simulations.TST is the basis of RE, KMC, and many other approachwhich are the subject of the remainder of this section.

B. Mean-field rate equations

1. Basic concepts

The time evolution of mean-field quantities, such asdensity of adatomsn1 and of islands of sizes, ns , can bedescribed by a set of coupled ODEs known as REs. Adatocan either meet another adatom with a capture efficiencys1

FIG. 1. Typical atomistic processes during epitaxial growth.

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to form a dimer, an island of size 2, or get capturedislands of sizes with a capture efficiencyss .

At high substrate temperature, these adatoms canevaporate. For example, evaporation is clearly requiredestablish equilibrium with the 3D vapor phase, and mamodels have been developed for the low supersaturationgime, following the pioneering work more than half a cetury ago by BCF,11 much of which is now in the textbookand monograph literature.18,19 If the adsorption energyEa islow, then the evaporation timeta is short, and this evaporative quasiequilibrium is quickly established at low adatodensity, with n15Fta . RE models for nucleation angrowth in this situation were established in the 1960s,1 andexact solutions for the capture efficiency, the so-called cture numbers, were obtained in this case.2

This high temperature problem is soluble in closed forbecause the islands that form are isolated from each odue to the short BCF diffusion length,xs5(Dta)1/2, themean adatom surface diffusion distance before evaporaThe approach becomes more complex if there are sevcompeting processes at work. For this more general cashas been argued20,21 that one can develop the idea of competitive capture, in which characteristic times for differeprocesses add inversely, and the shortest time dominatethis formulation, the RE forn1 at low coverage is essentiall

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which has a steady state solutionn15Ft. The compositeterm n1 /t represents all the loss terms, adsorption (ta),nucleation (tn), capture by stable clusters (tc) and maybeother processes@ . . . in Eq. ~1!#, all of which add like resis-tances in parallel. One clearly may envisage many othercesses that might take place on, or close to, the subssurface. However, as in any other modeling situation, copleteness is bought at the price of loss of simplicity aclarity; thus one only adds new terms when compelled toso by the~experimental! evidence. Some of these situatioare developed in Sec. III. In addition to Eq.~1! for the singleadatoms, we need a complete set of REs for the larger cters, sizes>2. Assuming initially that we are concerned onwith single adatom processes, these can most conveniebe written as

dns /dt5Us212Us , ~2!

whereUs is the net rate of conversion of clusters of sizesinto size (s11). As in Eq.~1!, one can add other processemost naturally coalescence between islands,Uc , which willreduce the island density at small sizes and increase thesity at large sizes. This topic has been extensively conered, both in the early literature2,22,23and more recently.24,25

Some comments are made below.The above discussion implies that the models develo

for the different growth modes are going to be differentdetail. However, all RE models have certain features in comon, and we discuss these features here. First, we nee

J. Vac. Sci. Technol. A, Vol. 21, No. 5, Sep ÕOct 2003

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pressions for the net rate,Us21 and Us , in the set of Eqs.~2!. Each of these is the difference between two termscapture and a decay term, such that when these termsequal, we have local equilibrium. Ignoring direct impingment for the moment, the capture term is given bys1Dn1

2 forforming a dimer, or isssDn1ns for forming an (s11) clus-ter from ans cluster, where the capture numbersss remainto be determined. The decay terms can be written in the fonsGs , whereGs is the decay rate ofs clusters forming (s21) clusters. Details of the decay rate expressions are gin the Appendix.

There are two main types of RE models in the literatuThe first emphasizes the role of the critical nucleus of sizei .The second approach does not include a critical islandexplicitly. Rather, attachment and detachment rates forlands of all sizes are in principle included. Those two aproaches are described in the following two subsections.

2. Rate equations with explicit critical island size

The idea of a critical nucleus has been explored in mapapers in the literature, and has several related consequeThe main ideas are illustrated in Fig. 2. The left-hand sidethis diagram indicates that, because of equal forwardback reaction rates~full lines!, small clusters may be in locaequilibrium with the instantaneous adatom densityn1 . Usingthe detailed balance arguments set out in the Appendix,are able to write that, for subcritical clustersj < i the corre-sponding net ratesU j are zero, and the Walton relation26 @Eq.~A1!# can be used to express the density of critical nucleini ,in terms of the adatom densityn1 and the energy of thecritical nucleusEi . The right-hand side indicates that ‘stablclusters eventually grow, and the back reaction rate becoless important~dashed line!. These stable clusters, sizes. i , grow by diffusive capture and, maybe at a later stagedirect impingement.

As developed by one of us~J.V.! and others, this approaccan be combined with compacting the REs for all clussizess. i into one RE for the stable cluster densitynx , suchthat

dnx /dt5(s. i

dns /dt5Ui2Uc , ~3!

FIG. 2. Critical size in nucleation and growth models.

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where, from Eqs.~2!, all the otherUs cancel out in pairs, andwe have included the coalescence termUc , which may limitnx at higher coverage. Thus in this treatment we haveduced an infinite set of REs to just two, one forn1 @essen-tially Eq. ~1!# and one fornx @Eq. ~3!#. We have abandoneany serious attempt to calculate the island size distribuns(s), and we need to assume specific island shapes to mfurther progress. But we can also calculate other mean-fiquantities, e.g., mean island size (wx), substrate coverag(Z), differential or integrated condensation coefficients~a orb! with this approach, using an auxiliary growth raequation.2,14,20,21This equation for the number of atomsstable clusters,nxwx , has the form

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where all relevant growth [email protected]., not includingta21

in Eq. ~1!# are included in the first term. The second terincluded at this stage for illustration, corresponds to dirimpingement onto alls clusters of areaas , andks>1 is ageometrical correction factor, which may be needed tocount for impingement immediately next to a cluster.25 With-out this factor, the sum(s. insas is just equal toZ, thecoverage of the substrate by stable and critical islands. Tthe expressionF(12Z) has been used to correct Eq.~1! forn1 , for the effects of finite coverage.20,21 The coalescencerate Uc at low coverage is simply proportional t2nxdZ/dt;2,23 high Z changes to this formula are given morecently.25

The above scheme is relatively simple, and using itcan make scaling predictions for all the mean field quantitnotably the densitiesn1 andnx , as a function of the variousmaterial parameters and the critical nucleus sizei . In theversion that the second author has developed, this invoscaling with the adsorption energyEa ~which governsta),Ed ~which governs the diffusion coefficientD), and the bind-ing energyEi of the critical cluster. The scaling relationsthe complete condensation, initially incomplete, and extreincomplete condensation regimes have been documentetables for both 2D and 3D island shapes,14,21 and some ofthese regimes, notably complete condensation, havethoroughly tested, both by simulation and by experimeThese relationships, originally conceived to be useful for heroepitaxy, have been revisited on occasion, and the 2Dtreme incomplete condensation case has been modified27 tomake contact with the classic BCF case of homoepitaxylarge coverage.

When lateral bonds are suitably strong, at low tempetures, we havei 51, so that adatom pairs~dimers! are alreadystable nuclei, andEi , or E2 in this special case, is therebalso not relevant until higher temperatures. This simptype of equation focuses on the formation of pairs and thsubsequent growth as stable nuclei, a case that hastermed irreversible nucleation by the first author28 and manyothers. In the complete condensation regime, with the oenergyEd left in the problem, in the form of the diffusioncoefficientD5(nd/4)exp(2Ed /kT), it has become customar


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to emphasize the scaling ofnx with the lumped paramete(D/F)2x, with x51/3. This scaling is the same as in thei51 case for 2D islands, as discussed and tabulated prously, where the general form follows the power law wiexponent x5 i /( i 12). For 3D islands, this is modifiedslightly to x5 i /( i 12.5)20,21 under steady state conditionThese expressions are valid when the island shape is cpact; small corrections are needed for fractal islands.

Further progress cannot be made without specific assutions about bonding, and in particular about how the eneEi can be expressed in terms of lateral ‘‘bonds’’ of strengEb within the critical cluster. With this assumption, one ccompute both the densities and deduce the critical sizei asan output of the calculation; this size is the size that~self-consistently! gives the lowest nucleation rate and density,all possible sizesj considered.20 With the adoption of theEinstein model of lattice vibrations, contact was also mawith the equilibrium vapor pressure at low supersaturat~high temperature!, and so this model spans the complerange of behavior from low to high supersaturation, usjust the three energy parameters,Ea , Ed andEb . This is thesimplest three-parameter model to achieve this result, ansuch it is valuable as a base for further exploration of mcomplex models, as described here in Sec. III.

3. Rate equations without explicit critical island size

The second type of RE model takes a somewhat differapproach. At high supersaturation, re-evaporation is negible, soEa is irrelevant; this is the complete condensatiregime. Within this regime, at higher temperatures, wweak lateral bonds, atoms can leave the clusters by detment and subsequent diffusion. In this case, nucleationbeen termed reversible, and we need a suitable formulato describe the rate of such detachment processes. Aexample, the first author28 has used the set of equations, inoring direct impingement and coalescence, as

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Here, thess are the capture numbers as before, and the tethat involve Ddetgs describe the rate of detachment of aatom from an island of sizes. Note that these equations awritten here are valid only in the submonolayer growthgime, but coverage effects, coalescence of islands or destion of adatoms can also be included.

The second and third terms in Eq.~5! are explicit forms ofEq. ~1!, such that

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while the brackets in Eq.~6! emphasize the possibility osteady-state rather than local equilibrium, which was hilighted in the discussion following Eq.~2!. In both Eqs.~5!and ~6! the terms involvingDdetgs involve detachment froms clusters, as discussed in more detail in the Appendix.

The capture of adatoms by other adatoms and islandscurs with an efficiency that is given by the capture numbss . If chosen properly thesess do encode spatial information. Many recent studies have the goal of finding a formthe ss that properly accounts for the spatial correlationstween adatoms and islands. Some recent progress widiscussed below in Sec. III A. There has been less focusthe detachment ratesgs . One reason is that at low tempertures detachment is negligible, and often one assumesgs50 for all s. This is of course the regime that is termeirreversible aggregation. At higher temperatures, the detament should be related to the number of~attached! edge at-oms, and simple expressions for this have been deranalytically.29 But we would like to point out that the reasituation might be more complicated. For example, the pruct Ddetgs might reflect the fact that detachment can be strdependent, and thus this term also has some spdependence.

C. Atomistic kinetic Monte Carlo simulations

An alternative to completely analytic approaches are amistic models that explicitly take into account the spatinformation and stochastic nature of thin film growth. Oway to implement such models is in the form of KMsimulations.6,30,31 The main difficulty and challenge inKMC simulation is to identify which, and how many, microscopic processes need to be included. Different philosopexist to address the latter question. One philosophy issuch KMC models will never include all the microscopdetails, and that one should keep the model as simplepossible. This approach has been used to a large extent ipast 20 yr, and has helped to study and understand mgeneral trends during growth. In this section we will refersuch models as generic growth models. On the other enthe spectrum one might want to include all the relevantcroscopic processes, with the goal of making detailed pretions for a specific material system. We will refer to sumodels as high resolution growth models.

In a typical KMC simulation, one first specifies the prcesses that are included, and associates each processrate. At every timestep one identifies all the sites whereof the processes might occur. One of these events isexecuted with a probability that corresponds to the ratespractice, this is implemented in a way that one alwaysecutes an event~so there are no rejected moves!, and afterevery event the simulation clock is updated appropriately

A model that has been used with considerable successtudy general trends is a simple cubic, solid-on-solid mowith nearest neighbor interactions. The processes thatallowed are~random! adatom deposition, and hops to a neest neighbor site, where the rate for a hop is given byn5n0 exp(2(Ed1nEn)/kT). Ed is the energy barrier for sur


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face diffusion, and in fact forn50 it is n54D; in the KMCliteratureEd is often referred to as the surface bond enerEn is a nearest neighbor bond energy,n0 is the relevant pref-actor ~typically 10212– 10213 s21), and n50,1,2,3,4 is thenumber of in-plane nearest neighbors. Additional micscopic parameters such as enhanced diffusion along aedge or reduced~or enhanced! diffusion over a step edge~theso-called Ehrlich–Schwoebel barrier! can easily be addedGeneric KMC models have been used successfully to stqualitative trends during growth, such as scaling laws.limited cases they have also been successful for quantitapredictions such as the occurrence and decay of the reflechigh energy electron diffraction signal. Note that the ratesall possible events at every site only depend on the loenvironment; a dependence of the rates on more long rainteractions is possible, but at a significant additional coputational cost.

In the literature one often finds a distinction betwemodels for irreversible and reversible aggregation. Tmodel just described is a reversible model, as it allowsatoms with lateral nearest neighbors to move. The rate foatom with one nearest neighborn51 to move~and detachform an island boundary! then is Ddet5n0 exp(2(Ed

1En)/kT). In the limit thatEn is large, and/orT is small, thismodel then effectively becomes an irreversible growmodel. Thus, a continuous change of the parameterEn in-duces a continuous change from irreversible to reversgrowth. It has been demonstrated by one of us~C.R.! that thescaled ISD changes its shape continuously as a functiothe reversibility.28,32 This is illustrated in Fig. 3, where thescaled ISD as obtained from a KMC simulation32 is shown incomparison with the ISD as obtained from experiment33 andalso from the level-set method~discussion below!. It is evi-dent that the ISD sharpens as the reversibility increa~which is realized in the simulation by decreasingEn), inexcellent agreement with the experimental ISD that sharpas the growth temperature increases. We stress that no vfor a critical island size has to be specified. Rather, the inplay between attachment and detachment kinetics descthe degree of reversibility, which one might associate wit~noninteger! effective critical island size.32 We note that asimilar sharpening of the ISD has been obtained by Amand Family34 in a KMC model where islands larger than thcritical size~that has to be specified in advance! are assumedto be stable against breakup.

We conclude this section with some remarks abouttailed high resolution KMC models. The work of Madhukand co-workers31 spearheaded the development of growmodels for III/V compound semiconductor systems. Hoever, the large number of microscopic parameters wereficult to obtain, since for example not even the exact surfreconstructions were known 15 yr ago. But the situationchanged in the last few years with the advances in denfunctional theory~DFT! calculations, together with the morreadily available comparison to detailed experimental dain particular scanning tunneling microscopy~STM! images.We are aware of several detailed high resolution KMC mo

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However, one problem with such high resolution growmodels is that the range of magnitudes of all the differprocesses is rather large. Since the simulation timestep hbe defined by the fastest process, these models are oftenslow. Thus, it has to be the goal to develop growth modthat allow us to describe the fast processes in some averamean-field approach. For example, repeated detachmensubsequent reattachment of atoms from and to island ecan be described by the overall net attachment~or detach-ment!. One way to realize this is contained in a model thhas been developed over the past few years, as describthe next section.

D. Level set „island dynamics … model

In the past few years, the island dynamics model, ancorresponding level set~LS! method for its numerical simulation have been developed.38,39 This model is essentiallycontinuum in the x-y-plane, but it resolves individual atom

FIG. 3. Island size distribution, as given by KMC~squares! and LS~circles!methods, in comparison with STM experiments~triangles! on Fe/Fe~001!~Ref. 33!. The reversibility increases from top to bottom.


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All the physical information is in the normal componentv ofthe velocity function. Islands grow because atoms diffutoward and attach to island boundaries, and shrink becathey can detach from an island boundary. Thus, the velocan be written as

v5vatt2vshrink5D@n•¹r#2Ddetpescl. ~9!

In this equation, the first term on the right represents attament of adatoms to the step from the upper and the loterrace,40 in which @ # denotes the jump across the interfacThe second term represents detachment of adatoms fromland boundaries,41 in which Ddet is the microscopic detachment rate,pesc is the probability for an atom to escape frothe capture zone of the island it just detached from, andl isthe density of edge atoms that can detach. The adatomsity r(x,t) solves the diffusion equation

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with the boundary conditionr50 for irreversible aggregation. The last term in Eq.~10! is the rate of nucleation of newislands, where& denotes the spatial average.42

The model as described so far is almost completely deministic. However, it was found that certain fluctuations arequired to complete the model. The last term in Eq.~10!prescribeswhen a new island is to be nucleated. But thlocation of a new island has to be chosen with a probabiweighted by the local value ofn1

2 as obtained from the solution of the diffusion equation.43 Moreover, it was also foundthat randomness is essential in the thermal dissociationsmall islands.41 The idea of the LS method is illustrated iFig. 4, where a snapshot of a typical LS simulation is showThe stochastic prescriptions described above have beenfully validated by direct comparison with an atomistic KMmodel. This is also illustrated in Fig. 3. Again, we see ththe ISD becomes narrower and sharper as the detachrate increases.

Significantly, and in contrast to atomistic simulatiomethods, the inclusion of fast microscopic processes coat essentially no additional computational cost within thmethod. The reason for this is the following. In a typicatomistic simulation~where every event is resolved!, thecomputational time increases dramatically when events wlargely different rates are allowed. For example, frequatomic detachment and reattachment at island boundariecomputationally expensive, even though the morpholomight not change. In the LS approach, such fast eventsaccounted for in a quasi mean-field approach, without beexplicitly resolved. This leaves the numerical time-step uchanged, resulting in essentially no increase in overall coputational time.

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This new method seems well-suited to model problemepitaxial growth that may be difficult to describe with othmethods. One current example is the inclusion of strain:long range nature of the elastic field presents a challenginclude strain realistically in a KMC model~where the ratesare typically determined by the local environment!. How-ever, recent progress in this area is intriguing and mpresent a way forward, as discussed in Secs. III C and IIMoreover, solving the elastic equations is computationavery expensive, so that a method that allows for a latimestep is essential. As mentioned above, the levelmethod is also able to describe problems where the imporevents have vastly different rates. Work in both these areacurrently in progress.

III. RECENT PROGRESS: EXPERIMENTAL ANDCOMPUTATIONAL EXAMPLES

A. Capture numbers

The rate equations as formulated in Sec. II B are equatfor mean field densities, and thus by construction doinclude any explicit spatial information. Spatial informatio

FIG. 4. Typical level set simulation. Shown is a snapshot of the levelfunction ~a!, and the corresponding adatom concentration~b!. The islandboundaries corresponding to this level set function are shown in~c!.


n

eto

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nst

can however be included implicitly: consider the REs in tform of Eqs.~5! and ~6!. As mentioned already above, cature of adatoms by other adatoms and islands occurs witefficiency that is given by the capture numbersss . If chosenproperly thesess do encode spatial information. For the pupose of the discussion in this section, we assume that theno detachment from islands~i.e., gs50). Many recent stud-ies have the goal of finding a form for thess that properlyaccount for the spatial correlations between adatoms anlands. It has become customary to focus on the scaled Ito judge the validity of a certain choice for the capture nubers. In particular, simple RE approaches lead to a singuity of the ISD in the asymptotic scaling~largeD/F) regime,and one test of any approximation of thess is whether or notit leads to scaling of the ISD.

In this section we discuss the different choices that hbeen introduced forss , and summarize some recent progrein understanding the form of the capture numbers. The splest choice isss51 ~i.e., a simple constant! for all s. Thischoice is often referred to as the point-island approximatiThe only justification for this choice is that the equations aeasier to analyze, and that certain analytic results caneasily obtained. Another simple choice is ans-dependencess5sp. In 2D, p50.5 reflects the idea that capture is prportional to the perimeter of the island. Similarly, one cargue for the choicep51/3 for 3D islands; both of thesechoices were assumed in the initial work on rate equatio1

and have been made on occasion since then. However,well established that none of these choices forss gives thecorrect scaling of the ISD; in particular, the scaled ISD bcomes singular in the scaling limit.

There were several early attempts to account for the loenvironment of an island; this work was summarized aextended in Ref. 2. The uniform depletion approximatideveloped in that work is based on the mean-field assution that at every point outside of an island the densitiesislands of sizes takes on an average valuens(t). This ap-proach then gives an analytic formula for thess in terms ofmodified Bessel functions. Bales and Chrzan42 showed, forthe irreversible nucleation (i 51) case, that the mean-fieladatom and island densitiesn1 and N5(s>2ns obtainedfrom integrating the REs with these self-consistent captnumbers are in excellent agreement with the ones obtafrom KMC simulations, for a wide range ofD/F values.However, their approximation fails to reproduce properly tscaled ISD, which is a more stringent test for the spainformation in thess .

A completely different approach was taken by Bartelt aEvans in Ref. 44. In this paper, the authors performed exsive KMC simulation of a point island model tomeasurenumerically the capture numbers as a function of the islasize. They suggest thatss depends linearly ons for s.sav, wheresav is the average island size, and is essentiaa constant fors,sav. In a later article45 they relaxed thepoint-island constraint and instead considered spatiallytended islands, and obtained similar results~even though the

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plateau is less pronounced in this case!. The data is also inagreement with capture numbers that have been measvery carefully in an experiment where first Co islands aformed on Ru~001!, and additional Cu is deposited subsquently. Because of the differences of Cu and Co inSTM, the additional~Cu! mass of each island can then bmeasured in a careful STM experiment. Similar data wobtained in a later experiment for Ag islands on Ag~001!.46

Bartelt and Evans also established44 that capture effi-ciency of islands, as described by the capture numbersss , isintimately connected to the so-called capture areasAs . Thecapture area of an island can be defined as the area surroing an island, with the property that an adatom withincapture area, or zone, will on average diffuse towardcorresponding island~and will be captured by it!. Geometri-cally, these capture areasAs can be approximated by peforming a Voronoi tessellation. In the complete condensatregime, the shape of the scaled capture area distributionAs issimilar to the shape of the scaled capture number distribuss , a fact which has been known experimentally for a lotime.47,48 But in either case, capture numbers are only msured for one particular coverage, and these capture numare never used to integrate the REs. From more rework43,49 it is clear that these capture numbers can not pduce the correct form of the ISD in the scaling limit.

Gibou et al.49,50 determined capture numbers from simlations using the level set method. In contrast to the KMapproach, the dependence of thess on s could be obtained adifferent times. The reason is that, due to the mean fitreatment of the adatom density, a meaningful value forss

can be measured at any time, and growth of the islands dnot have to be artificially suppressed. It was found thatfunctional form of thess changes as a function of timeThere is essentially nos-dependence in the nucleation phasand an~almost50! linear dependence in the aggregation ph~cf. Fig. 5, upper panel!. The rate equations can be integratwith these~time dependent! capture numbers, and the shaof the scaled ISD in fact does agree with the one obtaifrom LS or KMC simulations, and in particular does ndisplay a singularity. This is shown in Fig. 5.

But there is no analytic form forss that allows an inte-gration of the REs fromt50, to get the proper ISD. Thereason is that there is substantial cross-correlation betwthe capture numbers and related capture areas, which leasignificant numerical noise in the results in Refs. 49 andIt is not clear at the moment whether better simulation dwill solve this problem, or whether there is a fundamenlimitation within this approach.

The ultimate reasons why it has not been possible to dto find a functional form for all thess(t) are the following:Different islands of a particular sizes have different captureareas. This is so because of spatial fluctuation duringnucleation of islands. As a result, these different islandssize s grow differently, which is hard to incorporate intosingle ss . Moreover, the correlation between capture eciency and the spatial distribution of islands is complicadue to nucleation. More precisely, the capture area~and thus


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the capture number! of a particular island may change~dra-matically! if a new island is nucleated nearby.

From the foregoing it is clear that any successful analytreatment of the problem needs to consider the capture nbersss as well as the capture areasAs . There has been somprogress in the past few years, but we believe that the plem has not yet been solved, and remains an area of accurrent research. Evans and Bartelt51 developed a formalismwhere rate equations for the island densitiesns were comple-mented by rate equations for the average capture areasAs . Ina different approach it was suggested by Mulheran aRobbie52 that the problem might be solved by consideringjoint probability distribution~JPD! for islands and captureareas. Equations were introduced to describe the time evtion of the island density as a function of the island sizes,and the size of capture areas,A. One complication that ariseis: ‘How are the capture areas treated in the case of anucleation event?’ In Ref. 52 the evolution of the captuzones is modelled as a fragmentation process when newlands are nucleated. The JPD then obtained agrees withone obtained from a KMC simulation.

Amar et al.53,54 also utilized the idea of the JPD. Themade some significant simplifications about the effectnucleation on the capture areas. In essence they assumethe capture area of thenth island is simply proportional tothe average capture area at the time. For intermediate vaof D/F their analytic model does indeed give better scalof the scaled ISD. However, as mentioned in Ref. 55, andis also evident from the simulation data presented by Poscuet al.,54 the scaled ISD still exhibits singular behavior folarger values ofD/F. It was pointed out by Evans anBartelt56 that the reason for this might be an unphysical de

FIG. 5. Capture numbers in the aggregation phase as determined withlevel set approach~upper panel!. Shown in the lower panel is the scaled ISobtained from the time dependent~measured! ss , in comparison with theoriginal LS data~Ref. 50!. The solid line in the lower panel is an analytiprediction from Ref. 45.

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function scaling form for the JPD, which results from thfact that capture areas are not properly divided upon nuation of new islands.

A recent, and perhaps more realistic treatment of thepact of nucleation on capture areas is given in Ref. 56. Inarticle, Evans and Bartelt study the JPD with a point islaKMC simulation. Scaling of the JPD of the typF(s/sav,A/Aav) is established. But whether or not this lateapproach yields capture numbers that give proper scalinthe ISD in the asymptotic limit, as (D/F) tends to infinity, iscurrently still an open question that needs to be answere

B. Nucleation on surface defects

All the above discussion has been concerned with nuation and growth on a perfect substrate. But it is well knothat many substrates are far from perfect, and indeedcontain impurities, surface point defects and/or steps, alwhich may promote nucleation. Early examples, especiin island growth systems, which are particularly sensitivesuch effects, are given in several reviews.13,14,57 More re-cently, attention has turned to nucleation of small particlesAg, Pd, and Pt on oxides such as MgO~001! andAl2O3(0001), which are of interest as model catalysts. Thare now several reviews of these systems that can besulted for background information and detailebehavior.58–61 There is an extensive theoretical literaturbased on cluster chemical,62 ionic pair potentials,63,64or den-sity functional65 calculations. For the most part, these calclations are not yet at the stage where definitive reviewsbe given, largely because of their extreme sensitivitycharge imbalance at the oxide surface, and at surface defwhich may also be charged.

Here we discuss how defects have been incorporatedthe framework of RE, KMC, and LS models. In all of themodels, the introduction of defect traps requires thatspecify their density,nt , and the energyEt with which dif-fusing adatoms are trapped. Within RE models, we can csider either irreversible trapping, such that adatoms canleave the traps~so thatEt is irrelevant!, or reversible trap-ping as illustrated in Fig. 6, where we can qualitatively ide

FIG. 6. Model for nucleation at attractive random point defects~densitynt),which can be occupied by adatoms~densityn1t), clusters~densitynxt) orcan be empty~Refs. 19 and 67!.


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tify three regimes. This three-regime behavior has beenproduced in a steady state rate equation model developethe second author,64,66,67which focuses on the rate equatiofor the trapped adatoms, densityn1t , and considers nucleation on both trap and normal sites.

The rate equation forn1t is approximately

dn1t /dt5s1tDn1nte2n1tnd exp~2~Et1Ed!/kT!, ~11!

wherente5nt2n1t2nxt is the number of empty traps. Fosimplicity we have assumed that the capture numbersempty and filled traps are the same. In steady state, this etion is zero, and inserting the usual form forD;nd exp3(2Ed /kT), we deduce

n1t /~nt2nxt!5A/~11A!,

with

A5n1Ct exp~Et /kT!, ~12!

whereCt is an entropic constant, which has been put equa1 in the illustrative calculations performed to date. Equat~12! shows that the traps are full (n1t5nt2nxt) in the strongtrapping limit, whereas they depend exponentially onEt /kTin the weak-trapping limit, as expected. This equation isLangmuir-type isotherm for the occupation of traps; the traping time constant@t t , in analogy to Eq.~1!# to reach thissteady state is very short, unlessEt is very large; but ifEt islarge, then all the traps are full anyway.

The total nucleation rate is the sum of the nucleation ron the terraces and at the defects. The nucleation rate etion without coalescence, analogous to Eq.~2!, is

dnx /dt5s iDn1ni1s itDn1nit , ~13!

where the second term is the nucleation rate on defects,nit is the density of critical clusters attached to defects,s it

being the corresponding capture number. In the simplest cwhere the traps only act on the first atom which joins theand entropic effects are ignored, we have

At5n1t /n15~nt2nxt!A/@n1~11A!#. ~14!

Typically, there are three regions: a high-temperaturegion where adatoms visit the traps but can become detacfrom them; a low-temperature region where the traps are fbut the nucleation density is largely unaffected, sincenx

.nt . In between, there is a plateau region wherenx5nt ;this plateau is longer ifEt is higher andEd lower. The firstrequirement is obvious, and the latter is required so that atoms reach the traps before finding each other. This stestate model calculation, originally intended for Fe/CaF2 ,66,67

is shown for particular energy parameters in Fig. 7.This defect nucleation model contains several subca

depending on values of the parameters. An interestingample is Pd/MgO~001!, studied with atomic forcemicroscopy,64,68 where a single set of experiments has beanalyzed to put bounds on four energies; these data requhigh trapping energyEt and a low value ofEd , while alsobeing sensitive toEb andEa . In this case, the high temperature portion of the data corresponds to the transition ti53, so that individual adatoms remain attached to traps,

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subsequent adatoms can become detached. These featurin semi-quantitative agreement with calculations62,64for trap-ping of Pd in oxygen ion vacancies. The role of surfacharges in stabilizing both surface vacancies on insulatorfaces, and small clusters attached to such point defectvery marked. Currently, different calculations agree that seffects are strong, but disagree on their exmagnitude;62,64,65 more comparative work is needed in tharea.

The RE model described above predicts the number dsities of islands, but~by construction! does not yield anyspatial information, as for example the shape of the ISD.this purpose, nucleation on defects has also been investigusing KMC and LS simulations. An example is the workLee and Baraba´si,69 who showed in a KMC simulation thaan ordered array of defect trapping centers can leadregular array of islands on the surface. A recentsimulation70 showed that the scaled ISD in the case of relarly spaced defect is a very sharp function~essentially adfunction!. In the same study it was shown that in the opposlimit, when the defects are randomly distributed on the sface, the ISD has the form of aG distribution.

The ISD is controlled by the spatial distribution of defecwhen the mean diffusion length is comparable with or greathan the distance between defect traps. This correspondthe upper end of the plateau regime, shown here in Figwhere the adatom catchment area is roughly the same aregular Voronoi polyhedron around each defect site. Whthe diffusion length is~much! less than the average distanbetween defects, both the nucleation density and theconverges to that obtained during nucleation on defect-terraces. The narrow ISD suggests that if one can conexperimentally the distribution of defect traps on the surfaone can control the ISD. The goal of a uniform ISD is cleadesirable for applications; we note that this may alsoaided by stress and diffusion fields; this topic is also dcussed in the next section.

FIG. 7. Algebraic solution to rate equations for trapping energyEt50.5,Ea51.16, Eb51.04, and a range ofEd values between 0.1 and 0.6 eVRecalculated~Ref. 67!, after original model for Fe/CaF2(111) ~Ref. 66!.


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C. Diffusion versus attachment-limited capture

The problem of determining capture numbers, and recprogress, has been described in Sec. III A, especially fortermining size and spatial distributions in complete condsation. Most of this effort has gone into finding diffusiosolutions fors1 , s i andsx ~or ss in general!, especially forthe case of irreversible nucleation (i 51), which is appropri-ate for STM experiments conducted at low temperatures.24,25

Low temperature deposition, even onto smooth clopacked metal surfaces, can be conducted in a regime wthere is very little diffusion of adatoms during deposition.such a case, essentially all motion occurs after depositand this regime, sometimes denoted byi 50, has been ob-served experimentally for the case of Cu/Ni~001!,71 see alsoRefs. 24 and 19. This regime implies no dependence ornucleation densitynx on the fluxF, but nx increases andn1

decreases following annealing at either higher temperatand/or longer times.

Immediately following deposition there are no spatial corelations between adatoms on the surface, but diffusioning annealing establishes the spatial correlations, betwadatoms and clusters, and between the adatoms themseThere is an initial, transient, regime in which the captunumber is time-dependent, before the diffusion solutioncomes established. As discussed below, this transienlonger if there are~repulsive! interactions between adatomleading to attachment-limited behavior.

Two sets of STM experiments have been conducted ondeposition and annealing of Cu adatoms on Cu~111! at lowtemperatures.72,73 After deposition and subsequent diffusiobefore observation, the spatial distribution is nonrandowith a preferred spacing between adatoms. This featurebeen analyzed quantitatively to determine the long-rangecillatory interaction between Cu adatoms as a functionradial separation. In the second of these experiments, Cudeposited at 16.5 K, to sub-ML doses (;1.431023 ML),followed by annealing at various temperatures around 20for times up to 20 min. At short distances, there is repulsbetween adatoms, and this repulsion forms a barrier todimer formation; but once formed, dimers are completstable and do not diffuse.

These experiments test capture number models, as apulsive barrier of heightEB , or a repulsive energy landscapof heightV0 , changes the form of the diffusion field arounadatoms and clusters, and reduces the capture number medly if EB /kT or V0 /kT.0.2. As shown recently,74 the fulltime-dependent form of the capture numbers is requiredobtain agreement between RE solutions and KMC simutions in the earliest stages of low coverage~sub-ML! anneal-ing. The diffusion solution is almost sufficient when thbarrier is zero, but for finite barriers the diffusion solutiois quite wrong, and the attachment-limited~barrier!solution, ss52p(r s11)exp(2EB /kT) or ss52p(r s11)3exp(2V0 /kT) is much closer.

Surprisingly, this form ofss can be appropriate, even fobarriers smaller than the diffusion energy. Note also thatcapture solution is similar to the form originally used b

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Zinsmeister,1 but now reduced exponentially by the Boltmann factor for the barrier. A related RE–KMC study bOvesson75 was applied to the deposition of Al/Al~111! aswell as Cu/Cu~111!. Fichthorn et al.76 have found similareffects for Ag/Ag~111! and related systems. As emphasisabove, these low temperature, smooth surface systems,respond to the case where there are small, close-range, rsive interactions prior to attachment, which determineprogress of nucleation. These repulsive interactions maycoupled with~even smaller! long-range oscillations, thoughto be due to surface state interactions, which determinepreferred adatom spacings after deposition.

A full solution for annealing, appropriate to Cu/Cu~111!,is shown in Fig. 8. As a result of having the agreementtween the KMC and RE solutions, one can extrapolate wsome confidence to other conditions, and compare withexperimental results.72,73These results showed no dimer fomation during 20 min at;17 K and the completion of dimeformation after 20 min at 22 K. As a result, Venables aBrune74 were able to deduce that the barrier heightEB , oralternatively the repulsive energy maximumV0 , for Cu/Cu~111!, lies between 10 and 14 meV, as illustrated in Fig.This figure is based on an integration of the REs for eachV0

value, up to the end of annealing~2 or 20 min! using theknownEd value, which is (4061) meV for Cu/Cu~111!.25,74

The comparison with KMC simulations is excellent, but tRE computation is much faster. This again points to a rfor RE solutions to summarize large amounts of computadone by other methods.

D. Nucleation and Ostwald ripening

In 1900, Ostwald published a famous paper on theproach to equilibrium for a solution where a dense phasea dilute phase coexist.77 When the dense phase is presentthe form of a distribution of compact clusters with differesizes, he argued that the Gibbs–Thomson effect78 provides a

FIG. 8. RE solutions forn1 and nx annealing curves as a function o(D1t)0.5, for annealing at 16.5 K with attachment barriersEB50, 5, and 10meV, compared to KMC simulations. The capture numbers used are bon an interpolation scheme between attachment barrier and diffusiontions, showing essential agreement with the KMC simulations. See texdiscussion of how these curves apply to STM experiments on Cu/Cu~111!.


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thermodynamic driving force for large clusters to grow at texpense of small clusters. This phenomenon is called riping or coarsening. The basic physics is the desire ofsystem to minimize the free energy associated with the infaces between the two phases. Often ripening is a late-sphenomenon, related to degradation of microstructures olong times, and thereby unrelated to nucleation and growtearly stages, the topic of this article. However, theresome cases where coarsening needs to be considered, arelationship between nucleation, growth and ripening isinterest in its own right. Some comments on this topic agiven below.

An authoritative review of the subject in the contextthin film growth is given by Zinke-Allmanget al.79 In a se-ries of articles, this group has studied island coarsening, bdue to cluster mobility and to island instability. The terripening is generally reserved for the latter phenomenon;usually formulated in terms of the~adatom! diffusion coeffi-cient and the edge energies of islands.

In an atomistic context, the rate of Ostwald ripening feach island depends on the difference between attachmand detachment rates to/from the island. The first quandepends on the number of adatoms on the surface andmobility D, while the latter is related to the detachment raDdet and density of atoms that can detach. Ostwald ripencan be incorporated into nucleation and growth modelsseveral ways. For rate equations, there are at least twoSec. II B, we have not relied in Eqs.~5! or ~6! on the idea ofa critical nucleus size, and can consider the possibility tall clusters are to some extent unstable, both during deption, and particularly during annealing. Then the differenbetweenssDn1 and Ddetgs is the relevant quantity to describe ripening. Calculations with specific forms of the latterm are described in Ref. 80.

An essentially equivalent RE approach, used by one o

edlu-or

FIG. 9. RE solutions for annealing curves as a function of barrier heightV0 ,at temperatures 17 K<Ta<23 K. Plotted is the ratio (n11nx) after a 2 minanneal, divided by the initial valuentot5(n11nx) after deposition. Thesecurves use the time-dependent capture number expression as in Fig. 8curves for 19 and 21 K are also compared with the KMC simulatioAdditionally a curve for annealing at 22 K for 20 min is given. See textdiscussion of how these curves apply to STM experiments on Cu/Cu~111!.

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~J.V.!, is to modify the capture time (tc) in Eq. ~1!, to allowfor decay of ‘stable’ clusters. Integration of rate, or rate dfusion, equations containing such terms does allow for copetition between nucleation, growth and ripening durideposition, and also gives a good overall description ofnealing. This method has been applied to experimentsboth Ag/Fe~001!81 and Ag/Ge~111! and Ag/Si~111!,82 and rel-evant energies have been extracted. But for the semicontor systems82 there are details left to sort out, notably thoconcerned with small particle mobility and interdiffusion. Sfar this rate-diffusion equation approach has only been tein one-dimensional geometries.

One might imagine that KMC simulations would bepowerful tool to study 2D Ostwald ripening. However, ware aware of only a single paper devoted to this subjec83

The reason is, for realistic values of the physical parametisland detachment processes are so slow that good statrequire an inordinate amount of computer time. Conquently, simulationists tend to focus on coarsening by isladiffusion and coalescence,84,85 a scenario that is known tooccur for Ag~100!.86 Another approach that has been usedstudy Ostwald ripening is based on the boundary integmethod.87 This method is extremely efficient for solving thdiffusion equation~and hence island growth velocities! for asurface morphology in the absence of nucleation and me

In recent studies the LS method described in Sec. II Dbeen used to simulate Ostwald ripening.88,89 The LS methodallows us to simulate nucleation and growth and subseqcoarsening within one unified approach, which is in contrto the boundary integral method, where only the coarsenof islands can be simulated efficiently. It was shown in R88 that the predicted scaling behavior of the ISD, andtime evolution of the average island size, can be descrivery well within the LS approach.

One additional complication is that, in many systemspractical interest, we also would like to include the effectsstress in the islands, which gets larger during growth ulimited by the introduction of misfit dislocations. We apresently quite a way from a fully quantitative model thincludes all competing effects such as nucleation, gro~initial and stress-limited!, ripening or coarsening, clusteshape fluctuations and transitions, stress-influenced intefusion, and so on. But many pieces of the argument arplace. For semiconductor systems such as Ge/Si~001! thehigh ad-dimer concentration and the small dimer–dimerteraction relates to a high critical nucleus size~small super-saturation! for nucleation and initial growth. The high density of mobile species makes both Ostwald ripening acluster shape fluctuations relatively easy. For further discsions of this topic, and the role of stress and interdiffusionthe formation of quantum dots in semiconductor systems,refer to recent reviews.90–93

IV. DISCUSSION AND FUTURE PROJECTION

Despite the large amount of work performed over the l30 yr, even on submonolayer growth, there are many avenstill be to be explored. Some recent progress has been


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tese-

scribed in Sec. III. From these examples, it is clear that enew level of complexity requires new variables and mateparameters for an adequate description of the model. Thea clear trend towards using at least two of the three maematical methods, i.e., REs, KMC, and LS methods thathave described in Sec. II, for a comparison with experimeThere is also a clear trend, which we have not yet empsized strongly in this article, to combine the results of themethods, or the parameters needed as input for such mods, with MD simulations, and withab initio calculationssuch as DFT or cluster chemistry computations. But to cstruct all such models, and do all of these calculations, efor one system, represents a large amount of effort, whictypically well beyond the capacity of one research grouThus the system has to be important enough, and the posresults decisive enough, to warrant the investment, bothmoney and time.

Our main argument for investing time and moneygrowth modeling is that we need to understand the baprocesses in detail in order to be able to make worthwhpredictions. But, from an applied or industrial viewpoint, thargument in favor of modeling is simply that the costexperiment by trial and error is rising even faster. Therenow so many process steps in producing a device that eneers usually insist that experiments, even to introduce smchanges, are performed under conditions essentially idento those used in production. In the semiconductor indusfor example, this means that large wafers, few and infrequexperiments, and incremental change are the rule; modeprovided it is well-grounded enough to extrapolate to nsituations realistically, can be a much less expensive optThere are several calculations in the recent literature whhave this as an aim, and many process simulators for whcollaborations and web sites are available.94

The challenge for the scientists working on such topicsto know enough about the system to model it with sufficieconfidence. It is well known, for example, that the wideused chemical vapor deposition~CVD! technique presentsparticular challenge to modeling, since a whole sequencprocesses occur, in the gas phase, on the surface, and ddesorption of the reactants. By contrast molecular beamitaxy ~MBE!, which is the context for much of the wordescribed here, is much less used in production, and isused for the most demanding applications. The reason isit cannot match CVD for speed of growth and selectivityreaction, and because it is more costly to install and matain. Somehow, these gaps have to be overcome in the fuby the modeling methods described here, as emphasizerecent articles95 and funding initiatives.96

In this article we have not dealt in detail witcoalescence25 nor with subsequent nucleation and growthhigher layers, which can in principle be formulated.97 Thereare also more recent papers on this topic98–100but the num-ber of effective parameters becomes rather large, somost such work has focused on the submonolayer regimedescribed here. A particular current theoretical interestcuses on the adatom statistics on the second and higher

dththiccu

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ers being different from the first layer, because of the neeconsider adatoms as belonging to a particular island, sothey cannot in general be considered to roam all oversubstrate. We have not considered such effects in this artand refer the reader to the above papers for ongoing dission.

An ultimate goal for workers in this field would be thave models that are robust enough, and fast enough, soproduction processes could be directly controlled frommodel predictions. As far as we know, no atomistic mohas actually been developed in this direction, and impmented to date. Clearly, measurement of system paramesuch as gas pressure, flow rate, source temperatures andsequences are already used routinely for real time, in-scontrol of CVD and MBE growth. Of the three types omodel discussed here, only REs are fast enough to accplish real time prediction, and it seems worthwhile to cosider further development of such tools in the directionon-line control.

ACKNOWLEDGMENTS

The authors would like to thank R. Caflisch, D. Cahill,Drucker, E. Hall, and R. Vardavas for stimulating discusions. C.R. acknowledges financial support from the Nfocused research group Grant No. DMS-0074152.

APPENDIX: DECAY AND DETACHMENT RATES

This Appendix amplifies the remarks made in Sec. II Bdecay and detachment rates in the various rate equationmulations. For example, we know that if single adatoms adimers are in local equilibrium, we haven25C2n1

2

3exp(2E2 /kT), a result originally derived by Walton,26 andused in all models that contain the idea of a critical nuclesizei greater than one. The general form of the Walton eqlibrium relation for sizej is, in ML units

nj5n1j (

mCj~m!exp~Ej~m!/kT!, ~A1!

where we take account of the possibility of different configrations (m) at each sizej , each with a statistical weighCj (m), which is an equilibrium property. Thus we can sthat for small clustersj < i , if we make the equilibrium assumption,U j50, and we have

s jDn1nj 215njG j . ~A2!

Thus the complete expression forG j must be

G j5s jD(m21Cj 21~m!exp~Ej 21~m!/kT!

(mCj~m!exp~Ej~m!/kT!. ~A3!

If we restrict consideration to the configuration that hashighest binding energy, and drop the configuration labelm,we obtain

G j5~s jCj 21 /Cj !D exp~2DEj /kT!. ~A4!

This Eq.~A4! has the correct form as discussed above intext for the general sizes, but also has the correct preexponential (s jCj 21 /Cj ) appropriate for subcritical clusters


toatele,s-

hatel-rs,imeu,

m--f

.-F

or-d

si-

-

e

e

each of which has a dominant configuration. Assumingsingle configuration is good for large supersaturation, buvery questionable at high temperatures, close to full~3D!equilibrium, when many fluctuations can be important. Treason for proceeding with such assumptions in practicthat, under typical growth conditions, one or more of treaction rates are far from equilibrium, thus renderingcorresponding decay terms negligible.

The discussion in the text following Eq.~7! can be under-stood as follows. By comparing Eq.~6! with Eq. ~A4!, wesee that local equilibrium for small clusters would corrspond to

Ddetgs5~ssCs21 /Cs!D exp~2DEs /kT!, ~A5!

whereDEs5Es2Es21 is the same definition as previouslThus the two rate equation formulations both contain determs, which describe equivalent physical phenomena.

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