PHYSICAL REVIEW B 68, 195407 ~2003!

Relaxation kinetics in two-dimensional structures

JoseL. Iguain and Laurent J. LewisDepartement de Physique et Groupe de Recherche en Physique et Technologie des Couches Minces (GCM), Universite´ de Montreal,

Case Postale 6128, Succursale Centre-Ville, Montre´al, Quebec, Canada H3C 3J7~Received 17 June 2003; published 14 November 2003!

We have studied the approach to equilibrium of islands and pores in two dimensions. The two-regimescenario observed when islands evolve according to a set of particular rules, namely, relaxation by steps at lowtemperature and smooth at high temperature, is generalized to a wide class of kinetic models and the two kindsof structures. Scaling laws for equilibration times are analytically derived and confirmed by kinetic MonteCarlo simulations.

DOI: 10.1103/PhysRevB.68.195407 PACS number~s!: 61.46.1w, 68.35.2p, 68.35.Md

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I. INTRODUCTION

A large amount of work has been done in the domainatomic structure kinetics since Burton, Cabrera, and Frpresented, in a seminal paper, the first serious attempmodel the behavior of atoms adsorbed onto a vicisurface.1 Nevertheless, several problems remain unsolvMany important advances in the techniques for the obsetion and manipulation of atoms now allow for dealing wistructures of decreasing size and, with the help of powecomputation, gaining insight into the basic mechanisms gerning atomic dynamics.~For a review, see, for examplRefs. 2–4 and references therein!.

In the past few years, considerable efforts have been dcated to understanding how nanostructures relax. A classdescription of structure shape equilibration was developedHerring, Nichols, and Mullins~HNM!.5 In this theory, thetransport of mass responsible for relaxation occurs insmooth and continuous way via migration of border adatofrom regions of high curvature~high chemical potentialm)to regions of low curvature~low m). It is assumed that structure border is rough enough to allow a continuous~coarse-grained! description of it. However, as this is not valid fosurfaces below the roughening temperature, the low tempture decay has been the subject of research btheoretically6–8 and experimentally.9–12

Recently, studies of two-dimensional island shape relation with a simple model have revealed an unexpecphenomena.6,7 It was shown that two qualitatively differenrelaxation regimes exist. At high temperature, islands evotoward equilibrium according to HNM. At low temperaturislands become faceted and another mechanism appeawas demonstrated that, in this case, shape relaxation ocby steps, where the limiting process consists in the nuation of new adatom rows on the flat edges of islands. Inabove model adatoms lie on a triangular lattice and bequilibrium and kinetic properties depend on the singlerametere0 /kBT ~wherekB is the Boltzmann constant andTis the temperature! since, for an adatom withni nearestneighbors~NN’s!, nie0 is both the potential energy and thkinetic barrier controlling migration.

This two-regime scenario is manifested in the dependeof the equilibration timeteq on temperature and island sizN. At high temperature,teq;N2exp(3be0)(b51/kBT), while

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at low T, teq;N exp(4be0). The regime to which relaxationof a given island belongs is determined by border roughnand depends not only on temperature but also on island sIn the T-N plane, the separation line between the twogimes is provided by the crossover island sizeNc(T);exp(be0), a rough indication of the size of the largest iland that is fully faceted atT. Interestingly, if thetemperature-dependent factors appearing in the leading teof teq are rewritten in terms ofNc , the properly scaledequilibration time becomes a function ofN/Nc only, satisfy-ing

S teq

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forN

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~1!

A first problem of interest is the universality of this scaing law. The model described above is at best a first apprmation to real situations and many details can be modifieorder to improve it. For example, such a well-known effeas the Ehrlich-Schwoebel barrier13 can be accounted for byintroducing new kinetic barriers in a phenomenological wain an even more accurate approach, minimum energy tration paths could be calculated from interatomic forces.14 As adifferent set of kinetic barriers will entail, to the least, neactivation energies, the question remains whether the abscaling behavior is a consequence of the model or, tocontrary, would carry over to more general situations.

Another closely related problem is the relaxation of twdimensionalpores, i.e., islands of vacancies. In some senislands and pores are ‘‘specular images’’ of one anothegiven pore has the same boundary as the correspondinland, and so would be faceted or rough dependingwhether the number ofvacanciesin it is smaller or largerthan Nc(T). It would be desirable to know if pore shaprelaxation also exhibits a two-regime scenario and to anathe possibility of scaling laws for the corresponding equbration times.

As far as we know, the questions above have not binvestigated, even in the case of simple models. In this wwe study the main properties of shape relaxation in twdimensional structures of adatoms. Our approach is twof

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JOSEL. IGUAIN AND LAURENT J. LEWIS PHYSICAL REVIEW B 68, 195407 ~2003!

theoretical analysis and numerical simulations. In the alytic part, we treat the shape relaxation problem in a genway. Our study suggests the presence of two regimes, regless of the specific set of kinetic barriers and, baseddetailed-balance conditions, we show that similar scallaws apply for both islands and pores. Since most ofarguments we invoke are heuristic, some independent comation of the derived properties is desirable. This is the gof the numerical part. We analyze, using standard kinMonte Carlo ~KMC! simulations, the relaxation of islandand pores according to two models considered in this paand compare the results with our theoretical predictions.

The paper is organized as follows. In Sec. II we discthe causes for scaling behavior and derive the asymplaws of scaling functions. The models we use are defineSec. III and the outcome of the KMC simulations are psented in Sec. IV. Finally, we give our conclusions in Sec

II. ANALYTIC APPROACH

In this section we investigate a generic model with atoms lying on a triangular lattice where the total bindienergy per NN pair ise0. Equilibrium properties depend oa single parameter, namelybe0, but kinetics involve in gen-eral a greater number of them. Our aim here is to estabscaling laws for both island and pore relaxation times.

Let us consider islands first. It is easy to see that manthe arguments presented in Refs. 6 and 7 remain applicregardless of the specific set of kinetic barriers. First ofNc(T), which separates rough and faceted islands at a gtemperature, is the size at which the perimeter equalsaverage distance between border kinks. This does not deon the transition barriers but only on the binding energiesthe different configurations, so the formNc;exp(be0) is stillvalid. Next, two qualitatively different modes of relaxatioshould exist. On the one hand, for islands greater thatNc ,shape relaxation dynamics is adequately described byHNM theory. It consists of a set of equations for the tempoevolution of the border curvature, where the kinetics is donated by perimeter diffusion and always leads to ateq;N2

power law. We will call this regimerough relaxation mode~RRM!. On the other hand, mechanisms driving islansmaller thanNc toward equilibrium are also quite general.this scenario, islands spend most of the time in highfaceted configurations. We will call this regimefaceted re-laxation mode~FRM!. Evolution occurs by steps consistinin nucleation and stabilization of new adatom rows. The tiassociated to each of these steps is proportional to flength (;N1/2) and, given an initial configuration, a numb;N1/2 of rows needs to be created in order to attainequilibrium shape; thusteq;N.

The size exponents for equilibration time thus appeabe universal~1 in the FRM, 2 in the RRM!. The activationenergies howeverdo depend in general on the kinetic barrers. The activation energy in the FRM corresponds toenergy characteristic of row nucleation, while in the RRMwill be equivalent to the characteristic energy for diffusialong the island border. If we call these energiesEF andERfor the FRM and RRM, respectively, the arguments p

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EF5ER1e0 , ~3!

otherwise, the exponential factors in Eq.~2! would be differ-ent and nonatural time scale@such as;N5 in Eq. ~1!#,allowing scaledteq to be expressed as a function ofN/Nc ,would exist. Let us, then, analyze the connection betweentwo energies.

As noted above, the limiting step in the RRM is adatoborder diffusion. This is in turn limited by the elementaprocesses sketched in Fig. 1, in which adatoms pass fone kink to another by jumping around a corner; thus wereadily equalER and the energy of this process.

For the FRM, the dynamics is determined by row nucation, which occurs when two adatoms meet on a flat islaedge. The nucleation rate may thus be calculated byproduct of two factors: the rate of adatoms entering a faand the probability that another adatom lies on the safacet. In this regime, the main sources of adatoms are fends, where the most weakly bound adatoms are fouThose adatoms move onto flat facets by basically the proA sketched in Fig. 1; the first factor is thus;exp(2bER).The second factor is;exp(2be0) because the system gainan energye0 when an adatom moves from a kink to a fledge. Thus, we haveEF5ER1e0, i.e., Eq.~3!, and the scal-ing form

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We consider now the case of pores. Since, as mentiobefore, a pore will also be rough or faceted dependingwhether it is larger or smaller thanNc , it is worth analyzingeach situation separately.

FIG. 1. Schematic of the limiting processes for diffusion alongrough border. Adatom detaches from a kink and go into adjacrows after jumping around a corner. All the sites below the full linwhich represents the border, are occupied.

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RELAXATION KINETICS IN TWO-DIMENSIONAL . . . PHYSICAL REVIEW B 68, 195407 ~2003!

For the relaxation of rough pores, the solution is immeate: the kinetic path is the same as that followed bycorresponding island~via a vacancy-atom exchange mechnism!. This pore-island symmetry is clearly evident in thHNM theory ~see, for example, Ref. 6!, which involves onlyborder curvature and perimeter atomic diffusion and hedoes not depend on the kind of particles enclosed withinborder. As a corollary, the equilibration time for pores shoalso satisfy the asymptotic law established by the secequation in Eq.~2!; this will be verified numerically in SecIV.

For faceted pores it is clear that relaxation will occur,in the case of faceted islands, by steps consisting ofnucleation and stabilization of new rows. However, despthis qualitative analogy the two processes are not equivabut, rather, complementary; adatom migration occurs frshort to long facets in islands, while the opposite is truepores. For the latter, row nucleation occurs when, as shschematically in Fig. 2, an adatom that lies on a long famoves to an internal corner of the pore. The new row wgrow with the arrival of more adatoms from the now ‘‘openlong facet, and become stable when fully filled.

The similarity of the relaxation mechanisms allows usapply, in the case of pores, the same kind of argumentsinvoked for the dependence ofteq on the size of facetedislands, which again yields the power lawteq;N. In order tocalculate the activation energy—which corresponds tonucleation process illustrated in Fig. 2—we consider firstinverse process, i.e., the migration of an adatom from sito site 1. In this case, the adatom first leaves a kink, tdiffuses along a flat border and, finally, goes into an adjacrow by jumping around a corner. This is similar to the prcess B sketched in Fig. 1 and must therefore have an action energyER . Knowledge of this last energy allows usfind that corresponding to nucleation. They are related bydetailed-balance condition which, taking into account tthe binding energy difference between sites 1 and 2 ise0,finally leads to the result that the activation energies for pshape relaxation are also connected by relation~3!.

As an interesting consequence, the equilibration timeislands or pores, when properly scaled withNc

a , becomes afunction of N/Nc only that satisfies the asymptotic rules~4!.We stress that even though the HNM theory impliesequivalent relaxation mode for both kinds of structures inRRM, the same symmetry cannot be expected to hold inFRM. The asymptotic forms give the exponents of the leing terms but different preexponential factors may affectNc .Thus, in general, there will be one scaling function corsponding to islands andanotherscaling function for pores

FIG. 2. Schematic of row nucleation in a pore. All adatomsoutside the line which represents the border of a fully faceted p

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For very small values of their arguments (N!Nc) they willbe parallel~on a log scale! but not necessarily equal, while athe other extreme (N@Nc) the curves should collapse arequired by HNM theory. Using KMC we will show, in SecIV, that these properties of the scaling functions are verifiWe will also see that a perfect collapse of the curves canobtained if theNc prefactors are adjusted separately forlands and pores.

III. THE MODELS

In order to assess the analytical predictions above,investigated two different models, denoted as I and II, whhave the following common characteristics: The substratrepresented by a triangular lattice and the total bindingergy per NN pair ise0. Adatom hoppings are only possiblbetween NN sites and a given adatom in an initial siteajumps to a final empty siteb with a probability per unit timePab5nexp(2bEab), wheren is a constant frequency andEabthe kinetic barrier. In order to avoid detachment, the jumare forbidden if the number of NN adatoms in the final site0. As an additional simplification, the motion of adatomwith initially more than four NN’s is also forbidden. Thiapproximation is justified because of the high energy barrof the corresponding processes; it is introduced in ordeaccelerate the simulations. Differences between models III lie only in the set of kinetic barriers.

For Model I, which is the same as the one used in Refand 7,Eab depends only on the numberna of NN adatoms inthe initial sitea. It takes the valueEab5nae0 when 1,na,5 andEab5e/10 whenna51, regardless of the number oNN adatoms inb.

For Model II, we use a more accurate set of activatienergies, obtained from a detailed study of the kinetic baers for adatoms lying on a triangular lattice and interactvia a Lennard-Jones potential.15 It is useful to define thismodel with the help of Fig. 3, where the different barriers arepresented. In the top part of the figure, adatoms are resented by circles while crosses indicate possible bind

ee.

FIG. 3. The energy landscape for Model II.

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JOSEL. IGUAIN AND LAURENT J. LEWIS PHYSICAL REVIEW B 68, 195407 ~2003!

sites. The bottom part shows the corresponding energy lascape, i. e., kinetic barriers separating adjacent local miniThus, in this model, we take into account the final-state cfiguration when evaluating the transition-state energy.~Note,however, that the height of the barrier depends on the nber of NN in the initial and final states but not on theprecise position!. Model II thus incorporates such importafeatures as the Ehrlich-Schwoebel barrier. This is apparewe look at Fig. 3: the barrier from 2 to 1 is lower than thbarrier from 2 to 3; the barrier from 4 to 3 is higher than tbarrier from 4 to 5, etc. The final parameters in this moare E150, E250.5e0 , E35e0 , E451.5e0 , E552e0, andE654e0. In order to facilitate comparison, Table I lists, foboth models, the kinetic barriers corresponding to a givelementary jump as a function of the number of NN befo(ni) and after (nf) the hopping.

Before closing this section, we calculate the activatenergies for each model as derived from the analysis of SII. Remember thatER is the energy characteristic of the prcess sketched in Fig. 1~a!. For Model I, this process corresponds to an adatom with three NN’s before the jump,ER

I 53e0, and, using Eqs.~2! and ~5!, EFI 54e0 anda I55,

as obtained in Refs. 6 and 7~but not yet tested with pores!.For Model II, the process in question is~for example! thetransition from 4 to 2 in Fig. 3, which leads toER

II 52e0 ,EF

II 53e0, anda II 54.

IV. NUMERICAL RESULTS

As mentioned in Sec. I, a numerical confirmation of tresults of Sec. II, obtained in a heuristic manner, is in ordIn this section, we do this using KMC simulations to exploshape relaxation according to models I and II. We consiislands and pores with sizes between 400 and 6500 partand perform, for each model, standard KMC simulatioThe values ofb ~in 1/e0 units! range between 2 and 17. Thresults shown in this section correspond to averages ovnumber of samples, varying from 4 for the largest structuto 30 for the smallest ones.

Following Refs. 6 and 7, we use the aspect ratioa, de-fined as the ratio of thex andy gyration radii, to characterizethe state of the structures. We start each simulation withaspect ratio of around 10 and define the equilibration timeteq

TABLE I. Kinetic barrier for Model I (DEI) and Model II(DEII ) as a function of the number of NN before (ni) and after (nf)the jump. For Model I, the barriers depend only onni . Detach-ments, as well as jumps of adatoms withni.4, are forbidden inboth the models.

ni DEI@e0# DEII @e0#

1 0.1 0.02 2.0 0.5, fornfÞ1

1.0, for nf513 3.0 1.5, fornfÞ1

2.0, for nf514 4.0 4.0

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as the time at whicha first becomes less than 1. First of awe analyze the dependence ofteq on N ~for fixed b) and,separately, onb ~for fixed N). Later in this section, we willcheck the scaling laws.

Let us start with Model I. In order to test the behaviorteq as a function ofN, we plot in Fig. 4~a! ln teq against lnNfor several temperatures. Filled and empty symbols cospond to islands and pores, respectively. At high~low!enough temperature,teq is expected to scale asN2 (N). Thisscaling behavior is represented in Fig. 4~a! by the lines withslopes of 2~lower! and 1~upper!, which bracket the KMCresults corresponding to the highest and lowest temperatused in our simulations, respectively. Using these lines aguide, it is indeed clear that the size exponent changes f1 to 2 when the temperature is increased, consistent withnumerical findings. Note that the agreement is as goodpores as it is for islands.

We check now our predictions for the activation energiTo do so, in Fig. 4~b!, ln teq is plotted againstb for differentsizes. The asymptotic behaviors are represented bystraight lines which have slopes of 3e0 ~low b) and 4e0~high b). The KMC data again show excellent agreemewith the predicted behavior, within a trivial prefactor~i.e.,slopes agree!. Let us remark that the results for island relaation are given here to allow comparison with the resultspores. We note however a small discrepancy betweennumerical results and those of Ref. 7: although the size

FIG. 4. Equilibration time for Model I.~a! teq as a function ofNfor b53.9 ~circles!, 5.8 ~pentagons!, 11.6 ~down-triangles!, 12.9~squares!, 14.5 ~diamonds!, and 16.6~up-triangles!; the lines haveslopes of 1~upper! and 2 ~lower!. ~b! teq againstb for N5 490~squares!, 1000~triangles!, and 4000~circles!; the lines have slopesof 3 ~left! and 4~right!. Filled symbols correspond to islands anempty ones to pores.

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RELAXATION KINETICS IN TWO-DIMENSIONAL . . . PHYSICAL REVIEW B 68, 195407 ~2003!

ponents and activation energies coincide, our equilibratimes are found to be lower~by a factor of;10) than thosereported in Ref. 7.

The corresponding results for Model II are presentedFig. 5. Note that, also in this case, the theory-simulatagreement is good for both types of structures. Thus, Figand 5 confirm our predictions that, on the one hand theexponents are universal~1 and 2! and, on the other handERand EF depend on the specific set of microscopic kinebarriers.

FIG. 5. Equilibration time for Model II.~a! teq as a function ofN for b53.9 ~circles!, 4.6 ~pentagons!, 5.8 ~down-triangles!, 10.6~squares!, 11.6 ~diamonds!, and 12.2~up-triangles!; the lines haveslopes of 1~upper! and 2 ~lower!. ~b! teq againstb for N5490~squares!, 1000~triangles!, and 4000~circles!; the lines have slopesof 2 ~left! and 3~right!. Filled symbols correspond to islands anempty ones to pores.

FIG. 6. Model I scaling of log10(teq /Nc5) against log10(N/Nc),

for islands~filled symbols! and pores~empty symbols!. The lineshave slopes of 1 and 2.

19540

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Finally, in order to test the specific scaling laws, we plon a log-log scale, (teq /Nc5) against (N/Nc) for Model I~Fig. 6! and (teq /Nc4) against (N/Nc) for Model II ~Fig. 7!.We have useda I anda II as calculated at the end of Sec. IIAccording to the analysis presented in Sec. II,teq /Nc

a shouldscale with exponents 1 and 2 for very small and very lavalues of its argument (N/Nc). In Figs. 6 and 7, we havedrawn lines with slopes of 1 and 2 to indicate theasymptotic behaviors, which are evidently closely verifieFor concreteness, in these plots, we use the sameNc50.25exp(be0) as in Ref. 6.

It is interesting to note that, besides confirming the epected asymptotic power laws, in both figures all the poicorresponding to islands collapse on one curve, and allpoints corresponding to pores collapse on another one~de-pending viaa on the microscopic details of the model!, thusgiving additional support to the scaling ansatz. Furthermoalthough for each model the island and pore scaling futions are different in the FRM, they become equivalent inRRM, as required by the perfect pore-island symmetryrelaxation in the latter regime.

Considering Figs. 6 and 7, the shift in the functions dscribing islands and pores indicates that the same valuthe scaling parameterNc cannot describe both the situationThis is a consequence of the fact that islands and poresnot perfectly symmetric, as alluded to in Sec. II. It is easysee that corner atoms detach much more easily in the caislands than in the case of pores. Thus, it is easier to fokinks on a faceted islands than it is on a faceted poredifferent values ofNc are used for islands and pores, a pfect collapse of the curves is obtained, i.e., a single scafunction describes both situations.

V. CONCLUSIONS

By considering a generic model of adatoms lying ontriangular lattice, we have studied the problem of shapelaxation of islands and pores. The arguments employedthis work allows the main properties of the equilibration timteq to be calculated as a function of temperature and sBecause our arguments are somewhat heuristic, KMC silations, using two different kinetic models, were also carrout. The numerical results confirm our theoretical predtions. For both islands and pores two qualitatively differe

FIG. 7. Model II collapse of data for both islands~filled sym-bols! and pores~empty symbols!. The lines have slopes of 1 and 2

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JOSEL. IGUAIN AND LAURENT J. LEWIS PHYSICAL REVIEW B 68, 195407 ~2003!

modes of relaxation~FRM and RRM! are found, as well asthe line Nc(T) that separates these regimes in ttemperature-size plane. It was shown that, although sizeponents are universal~1 in FRM, 2 in RRM!, the activationenergies corresponding to each mode depend on the mscopic details of the kinetic model. For a given model, inRRM the kinetics of relaxation for islands and pores issame because of the symmetry which is built into the theo5

This symmetry is broken in the FRM; however, becausethe detailed-balance condition the relaxation time for islaand pores differ only by a constant factor. A scaling behavwas indeed found for the relaxation times; the scaling futions for islands and pores can be made to coincide ifassumes different values of the scaling parameterNc , re-flecting the peculiarities of both situations. The prope

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scaled timeteq depends also, viaa, on the set of elementarykinetic barriers for the model.

ACKNOWLEDGMENTS

We are grateful to N. Combe and P. Jensen for interesdiscussions. We thank F. Bussie`res, M. de la Chevrotie`re, J.Richer, M. Simard, and C. Hudon for help with the numecal calculations, and P. Thibault who provided the dneeded for Model II. This work was supported by granfrom the Natural Sciences and Engineering Research Cou~NSERC! of Canada and the Fonds Que´becois de la recher-che sur la nature et les technologies~FQRNT! of the Prov-ince of Quebec. We are indebted to the Re´seau que´becois decalcul de haute performance~RQCHP! for generous alloca-tions of computer resources.

nd

Sci.

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