Diffusion of small two-dimensional Cu islands on Cu(111) studied with a kinetic Monte Carlo method

7/28/2019 Diffusion of small two-dimensional Cu islands on Cu(111) studied with a kinetic Monte Carlo method

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Diffusion of small two-dimensional Cu islands on Cu(111) studied with a kineticMonte Carlo method

Altaf Karim, Ahlam N. Al-Rawi, Abdelkader Kara, and Talat S. Rahman Department of Physics, Cardwell Hall, Kansas State University, Manhattan, Kansas 66506, USA

Oleg Trushin Institute of Microelectronics and Informatics, Academy of Sciences of Russia, Yaroslavl 150007, Russia

Tapio Ala-Nissila Laboratory of Physics, P.O. Box 1100, Helsinki University of Technology, FIN-02015 TKK, Espoo, Finland

and Department of Physics, Brown University, Providence, Rhode Island 02912-1843, USAReceived 26 September 2005; revised manuscript received 15 February 2006; published 14 April 2006

Diffusion of small two-dimensional Cu islandscontaining up to 10 atomson Cu111 has been studiedusing the newly developed self-learning Kinetic Monte CarloSLKMC method which is based on a databaseof diffusion processes and their energetics accumulated automatically during the implementation of theSLKMC code. Results obtained from simulations in which atoms hop from one fcc hollow site to another arecompared with those obtained from a parallel set of simulations in which the database is supplemented byprocesses revealed in complementary molecular dynamics simulations at 500 K. They include processes in-volving the hcpstacking-faultsites, which facilitate concerted motion of the islandssimultaneous motion of all atoms in the island. A signicant difference in the scaling of the effective diffusion barriers with island sizeis observed in the two cases. In particular, the presence of concerted island motion leads to an almost linearincrease in the effective diffusion barrier with size, while its absence accounts for strong size-dependentoscillations and anomalous behavior for trimers and heptamers. We also identify and discuss in detail the keymicroscopic processes responsible for the diffusion and examine the frequencies of their occurrence, as afunction of island size and substrate temperature.

DOI:10.1103/PhysRevB.73.165411 PACS numbers : 68.43.Jk, 68.43.Fg, 68.43.Hn, 68.47.De

I. INTRODUCTION

Acquiring a precise knowledge of the microscopic mecha-nisms responsible for island diffusion or mass transport onsurfaces is an important step towards the understanding of phenomena such as thin lm growth and its morphologicalevolution. Motivated by experimental observations, initiallyusing eld ion microscopyFIM,16 and more recently withthe use of the scanning tunneling microscopeSTM,715 thestudy of adatom and vacancy island diffusion as a function of size has been an important concern also for manytheorists1625 Because of the inherent differences in the mi-croscopic processes responsible for the diffusion and its scal-ing behavior with size, the discussion has naturally bifur-cated into those for the larger islands, usually containingmore than 20 atoms, and the smaller ones N 20 . For thelarger islands, the diffusion coefcients appear to scale as afunction of the size and the scaling exponent is expected toreect the intervening atomistic processes responsible for thediffusion.20,26 However, for the smaller islands a consistentknowledge of the variation of their mobility with size and thedetails of the responsible atomistic processes has not yetbeen fully established, especially on the111 surfaces of fccmetals.

One of the distinguishing geometrical features of thefcc 111 surface is the presence of two types of hollow sites:the so-called fcc site under which there is no atom in the

second layer and whose occupancy by an adatom maintthe crystal stacking order, and the hcp site under which this an atom in the second layer and its nucleation can leaa stacking fault. Whether or not an adatom or atoms inadatom island occupy one or the other of these two sdepends on their relative occupation energies and has signcant consequences for epitaxial growth and the morpholcal evolution of the surface. Although Ag, Cu, Pt, and Irall fcc crystals there is no guarantee that the adatom woprefer to sit in the fcc site. In fact, experiments show thathcp site is preferred on Ir111 ,3,15 while on Cu111 the fccsite appears to be favored,9 although the small difference inthe occupation energya few meV between the two sitesdoes not rule out the occupation of the hcp sites. For dimtrimers, and other larger islands, mixed occupancy of the sites is also possible. The relative probability of occupaof these two sites on fcc111 surfaces continues to be thesubject of much discussion and debate.

For small adatom islands earlier experimental studpoint to a general decrease in mobility with increasing islsize, except for some cases of anomalously large mobility16For larger islands short-range diffusion of the atoms arothe periphery, followed by adjustment of island shape, been proposed to be the dominant mechanism for diffusion.3,17 In the case of small Ir islands on Ir111 , con-certed gliding motion of the island has also been report3Subsequent molecular dynamicsMD simulations using

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many-body potentials based on the embedded atom methodEAM Ref 27 have further disclosed that in addition to

gliding, there is simultaneous motion of a portion of the is-land from the fcc to the hcp sites, creating a stacking fault.18The motion of the island could result from the rest of theatoms shifting also to the available hcp site. On Ni111 , forexample, the smaller islands reportedly nd the fcc to hcptransition to be critical to their diffusion, although gliding of

the island as a whole and periphery motion has also beenseen.18 In agreement with the experimental results of Wangand Ehrlich on the higher mobility of tetramers as comparedto trimers for Ir on Ir111 , Changet al.21 nd the barrier fordiffusion for the tetramer to be lower than that for the trimerfor a number of fcc metals. They also predict a zig-zag mo-tion to be the dominant one for the dimer and the tetramer,while predicting a concerted motion as a whole for the tri-mer. Recent theoretical studies of the energetics and dynam-ics of 17 atom Cu islands on Cu111 have once again high-lighted the role of the concerted motion of the island incontrolling its diffusion characteristics.28 In the very recentwork of Muelleret al.16 good agreement with experimentaldata on submonolayer epitaxy on Ir111 is also obtainedwith the inclusion of concerted motion of islandswith thestacking fault sites. Issues about the relative importance of the proposed diffusion mechanisms, the relevance of the oc-cupation of the hcp sites, and the observed anomalous diffu-sion for certain sizes, are striking aspects of the diffusion of small 2D islands on fcc111 surfaces and may control thesubsequent growth patterns on these surfaces.

Our purpose here is to determine the microscopic factorsthat control the diffusion of small Cu islands on Cu111 inan unbiased manner. We should hasten to mention that someresults for such systems have recently been presented byMarinicaet al.28 who focused on the application of a newerversion of the EAM potentials29 to calculate diffusion barri-

ers and pre-exponential factors for a small set of likely pro-cesses that they nd from MD simulations to be responsiblefor island diffusion. The work ignores the contribution of mechanisms associated with atom-by-atom motion in smallislands and attributes the diffusion to specic collective is-land motion. The diffusion coefcient is simply obtainedfrom application of the Arrhenius law to the activation en-ergy barrier and the diffusion prefactor calculated for thechosen diffusion process. The natural question is whethersucha priori selection of the responsible process precludesthe contributions of other processes and whether such exclu-sion makes any difference in the predicted trends in islanddiffusion. The deeper question is, of course, whether it ispossible to allow adatom islands to evolve as a function of time with mechanisms of their own choosing and therebyprovide an unbiased illustration of the rate limiting step inthe diffusion and the relative contributions of various mecha-nisms. We have recently developed a self-learning approachto kinetic Monte CarloKMC simulations SLKMC inwhich the combination of an automatic generation of a data-base of single and multiple atom2 or more atomsprocessesduring the evolution of the system, and a pattern recognitionscheme provides a possible answer to the above question.30Application26 of this method to the diffusion of 2D Cu is-lands on Cu111 containing 19100 atoms has shown the

diffusion coefcientD to scale with the number of atomsN in the island asD N 1.57. Periphery diffusion in whichsingle atoms hopped from one fcc site to another was foto be the dominant mechanism. Several types of single multiple atom processes were also revealed in the collecdatabase. However, the frequency of occurrence of multatom processes was small even for the smaller islan1920 atomsand their contribution to the diffusion proce

decreased further with increasing size and decreasing teperature. The situation may be different for islands wfewer atoms in which concerted island motion might donate the diffusion process.28 These processes necessarily in-volve occupation or transit through the hcp sites, althotheir exact nature is not knowna priori. Our aim here is toapply SLKMC to examine the trends in the diffusion of sm210 atomsCu islands on Cu111 . Since in the original

version of the code adatoms are assumed to occupy onlyhollow sites, and in light of the possible importance of hcp site from the discussion above, we have also carriedMD simulations for further insights into the mechaniscontrolling island diffusion. Indeed, the MD simulations

veal the importance of processes involving concerted islmotion. A second set of SLKMC simulations with an hanced database is then performed and comparisons ofresults of the two sets of KMC simulations provide an derstanding of the factors that control the trends in the havior, and the atomistic processes that determine the disive motion of small Cu islands on Cu111 . In particular,our results provide interesting insights into the conditithat may lead to anomalous diffusion coefcients for cersizes of islands. Also, since the issue of the relative imptance of the fcc and hcp site may be system specic,carrying out these two sets of simulations, the results psented here should have signicance for other surfaces.

II. CALCULATIONAL DETAILS

The rst set of calculations are based on the recently veloped self-learning kinetic Monte CarloSLKMC Ref.3032 technique, in which we have implemented a patterecognition scheme that assigns a unique label to the enronment of the diffusing atom up to several neighbors,efcient storage and retrieval of information on activatenergy barriers of possible processes that the system mchoose to undergo. Provisions are made for automatic calation of the energy barrier when a process is rst identiand the result is stored in a database. These energy barrare calculated using a simple method which maps out total energy of the system as the diffusing entity moves fthe initial fcc site to the aimed nal fcc site, in small steDuring the ensuing energy minimization procedure, all atin the system are allowed to relax in all directions, exceptthe diffusing atom whose motion is constrained along reaction path. Processes involving multiple atoms can tbe revealed naturally. Extensive comparisons of the resulenergy barriers30 with those obtained using the more sophiticated nudged elastic band method show only minor difences. The simpler method gives a gain of almost two orof magnitude in the time taken to acquire a comprehens

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database. To ensure that the principle of detailed balance isrigorously obeyed, the barriers of the forward processes areused, together with the total energies of the system in theinitial and nal states, to determine the energy barriers forthe reverse processes. For the calculation of the total energyof the system, interatomic potentials based on the embeddedatom methodEAM are used. The initial step in the simu-lation is the acquisition of the database. Once it has become

stable, i.e., no new processes appear for some timefor theislands under consideration the database saturated after aboutten million KMC steps, the system evolves smoothlythrough atomistic processes of its choice and statistics arecollected for calculating quantities like the mean square dis-placement of the center of mass of the island, correlationfunctions, and the frequencies of the atomistic processes.

In the SLKMC code which is extensively used here, weallow island atoms to occupy only fcc hollow sites. Also,interlayer exchange processes are not considered as the pat-tern recognition scheme for them is more complex than theone implemented here for diffusion via atomic hops. For thediffusion of 2D islands on Cu111 such exchange processesare not expected to play a major role, as their energy barriersare relatively high. Furthermore, they have not been identi-ed in either experiments or in the accompanying MD simu-lations performed at 500 K.

Assuming the validity of the transition state theory, therate for an atom to hop to a vacant site is given byr i= i exp E i / k BT . Here E i is the activation energy bar-rier, k B is the Boltzmann constant, and i is the attempt fre-quency or the so-called prefactor, which in principle can besensitive to the details of the atomic environment. The pref-actors for the various processes can thus be expected to bedifferent. However, calculation of prefactors is nontrivial al-though the methodology is well dened.22 Recent calcula-tions of the prefactors for concerted island motion containing

27 atoms show some variation with size28

particularly forthe 7-atom island, and thus we have varied the prefactor forthe 7-atom island by on order of magnitude and nd theresults reported here to remain essentially unchanged. Inprinciple it would be preferable to calculate the prefactors forall the processes present in the database. We leave these cal-culations for the future, and invoke here the often used as-sumption of a standard value of 1012 s1 for all prefactors.For further efciency in the KMC algorithm, we have em-ployed the Bortz-Kalos-LebowitzBKL updating scheme33which allows one to reach macroscopic time scales of sec-onds or even hours for simulations at, say, room temperatureas has been shown in recent works.24,34

For reasons discussed in the Introduction, MD simulationswere also carried out to identify novel processes that cannotbe automatically picked up in SLKMC runs because of itsrestrictions to fcc site occupancy. The MD simulations werecarried out using standard techniques and, naturally, with thesame EAM interatomic potentials as in the rest of the work.For efciency, these simulations were performed at 500 K.As we shall see, collective island motion was revealed in theMD simulations and the hcp sites were found to be occupiedparticularly in transiting from one set of fcc sites to another.The message from the MD simulations is clear; the occu-pancy of the hcp site has to be allowed in KMC simulations

of small islands. The rst step in the direction is to replthe original 3-shell pattern recognition scheme in SLKMwith a 9-shell one which provides labels to all fcc and sites in the vicinity of the moving atoms. The labels ofprocesses collected in the SLKMC database can then be cverted to the 9-shell scheme. For the 2-4 atom islands was not a problem and all processes in the database wrelabeled and new ones from MD simulations accordin

labeled and their energy barriers calculated and stored. Tnewer version of the code named SLKMC2 is fully equipfor the examination of the diffusion of these small Cu cters on Cu111 . For islands containing 5 or more atomhowever, the database collected by SLKMC was far too tensive to manually convert each one into the new 9-slabeling scheme. We have thus refrained from developmanually a database of all single and multiple atom and lective island diffusion processes for these larger Cu islanWork is underway to equip SLKMC2 with an efcient robust saddle point search routine such that in the futurwill be able to acquire automatically the database of all evant diffusion processes for two-dimensional islands of size. For purposes here we have used SLKMC2 to examthe diffusion 24 atom Cu islands on Cu111 in the pres-ence of all possible single and multiple atoms, and collecisland diffusion processes. The results of such simulatiare compared with those obtained from SLKMC. Note talthough concerted island motion is a type of multiple aprocess, throughout this paper we have made a distinctbetween it and other multiple atom processes.

In the case of clusters with 510 atoms, we have pceeded along another route. We have retained the origiSLKMC code and supplemented its database with procethat allow us to mimic collective island motion throughindirect procedure for the inclusion of the fcc-hcp jumThe rationale for the indirect procedure is as follows.

know that the rst step in a concerted island motion is collective jump of all island atoms from fcc to hcp sites. the island sizes in questions we calculate the activation ergy barriers E hf for all such collective processesfcc-hcp.MD simulations have been very helpful in revealing shapes of the islands before the jumps from fcc to hcp sIn particular, islands are found to assume more or less copact shapes before the collective jumps. The rate of the ccerted island motion from fcc to hcp sites is hf

exp E hf / k BT , where hf is the pre-exponential factor,as noted above. As will be seen in Figs. 7 and 9, once island atoms are in hcp sites, each one can hop to onethree equivalent fcc sites. We calculate the energy barrifor such collective hcp to fcc transitions. The relative mnitude of these barriers determines the relative probabifor the particular hcp-fcc hop. If the energy barrier forthree processes is the same, then the probability for eac1/3. In such a scenario the rate for fcc-hcp-fcc concerisland motion could be written as 1/3 hf exp E hf / k BT and in general asPhf hf exp E hf / k BT , wherePhf is theprobability of the particular hcp to fcc hop, which lies tween 1 and 1/3. We now have a recipe for including ccerted island motion from fcc to fcc sites in the databasSLKMC. To check whether the indirect procedure of incling concerted island motion in SLKMC is reliable, we h

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carried out simulations for the 24 atoms islands using thisrecipe and compared the results obtained from SLKMC2. InTable I we have summarized the results of KMC simulationsfor trimers and tetramers by using the direct and indirectmethods. As can be seen both yield almost the same diffu-sion coefcients.

As for the model system, we consider a fcc111 substratewith an adatom island on top, as shown in Fig. 1. The gray

circles are substrate atoms which stay rigid during the simu-lation, whereas the darkcolored on-linecircles are the is-land atoms, placed on fcc sites which are the hollow siteshaving no atoms underneath them in the layer below. A KMCsimulation step begins by placing an adatom island of de-sired size, in a randomly chosen conguration, on the sub-strate. The system evolves by performing a process of itschoice, from the multitude of possible single or multiple ada-tom jumps at each KMC step. We performed about 107 such

steps at 300 K, 500 K, and 700 K. Typically, at 500 K, 7KMC steps were equal to 103 s in physical time. The diffu-sion coefcient of an adatom island is calculated byD=limt RCM t RCM 0 2 /2dt , whereD is the diffusioncoefcient,RCM t is the position of the center of mass of thisland at timet , andd is the dimensionality of the system.

III. RESULTS AND DISCUSSIONS

We present rst the results that are obtained from tSLKMC method with single and multiple atom procesinvolving jumps from one fcc site to another, which are tomatically accumulated and performed during the simution. The calculated diffusion coefcients of the island300 K, 500 K, and 700 K are summarized in Table II. Thare the numerical values in the rst entry for each size tin Table II and range from 8.821010 2 /s for the dimer to

TABLE I. Diffusion coefcients of a trimer and a tetramer at different temperatures.

IslandTemperature

K SLKMCSLKMC+concerted motion

Indirect resultsSLKMC2+concerted motion

Direct results

Trimer 300 1.37 106 2.78 1010 4.89 1010

500 5.26 108 1.83 1011 3.27 1011

700 6.17 109 4.55 1011 1.22 1012

Tetramer 300 1.21104 3.40 109 4.19 109500 2.66 107 9.17 1010 1.06 1011

700 6.35 108 3.38 1011 4.60 1011

TABLE II. Diffusion coefcients of 110 atom islands at different temperatures and their effective diffusion barriers wiprefactors.

Islandsize

atoms

Diffusion coefcientD 2 / s

KMC KMC+concerted motionEffectivebarrierE a eV

Diffusionprefactor

D0 2 / s300 K 500 K 700 K

1 5.70 1011 8.50 1011 1.02 1012 0.026 1.56 1012

2 8.82 1010 5.07 1011 8.50 1011 0.104 5.14 1012

1.68 1011 6.90 1011 1.24 1012 0.091 5.58 1012

3 1.37 106 5.26 108 6.17 109 0.380 3.52 1012

4.89 1010 3.27 1011 1.22 1012 0.141 1.06 1013

4 1.21 104 2.66 107 6.35 108 0.492 2.31 1012

4.19 109 1.06 1011 4.60 1011 0.211 1.50 1013

5 1.25 105 1.16 108 2.60 109 0.440 4.13 1012

7.81 108 2.87 1010 1.40 1011 0.234 6.73 1012

6 6.66 104 5.58 107 1.19 109 0.440 1.69 1012

7.57 107 8.15 109 5.60 1010 0.300 8.22 1012

7 1.18 102 2.18 104 8.00 106 0.922 3.80 1013

2.40 107 5.80 109 7.60 1010 0.362 2.90 1013

8 9.00 102 2.53 104 5.49 106 0.800 3.70 1012

2.10 106 1.65 109 2.59 1010 0.430 3.61 1013

9 4.18 103 5.50 106 8.00 107 0.448 1.55 1011

7.72 104 7.20 107 1.45 109 0.444 2.24 1012

10 4.12 2.33 105 8.82 107 0.731 7.24 1012

1.65 103 1.37 107 7.02 108 0.580 1.06 1013


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4.12 2 /s for the 10-atom island. A log-log plot of D vsN inFig. 2a shows oscillations in the diffusion coefcient withsize. This hint for magic sizes of islands signifying reducedmobility is also seen in the Arrhenius plot of lnD vs 1/ k BT .The effective diffusion barriers extracted for each island sizefrom the Arrhenius plotFig. 3 also display oscillatory be-havior. As can be seen in Table II, the 3, 7, and 10-atomislands display higher effective barriers than the others. The

barrier for diffusion is particularly high for the perfect hexa-gon 7-atom island.As we mentioned in Sec. II, MD simulations carried out at

500 K revealed several new concerted moves of the islandswhich involved occupation of the hcp sites, too. Before dis-cussing the details of the atomistic processes let us examinethe results for the diffusion coefcients once these processesare included in the database of SLKMC. The calculated dif-fusion coefcients, effective energy barriers, and the prefac-tors for the second set of KMC simulations are summarizedin Table II. These values are given in the square bracketsunderneath the corresponding ones obtained when hcp-siteassisted processes are not included. The size dependence of the diffusion coefcients at three different temperatures withthe inclusion of concerted moves from MD are also shown inFig. 2b for comparison of the case already discussed in Fig.2 a . Further comparison of the results of the two sets of simulations is in Fig. 3, in which the effective diffusion en-ergy barrier appears to scale with the island size once thehcp-assisted concerted motion is taken into account. Thestriking result is that there is no longer any oscillation in thequantities and the 3 and 7 atom islands diffuse just like theothers, in proportion to their size. We now turn to an analysisof the details of the single and multiple atom mechanismsinvolved in the diffusion of the islands, one by one.

A. Monomer

For completeness we begin with a few comments on diffusion of an adatom on Cu111 . The primary motion for asingle atom is simply the process of hopping between theand the hcp sites. We nd an activation energy barrier oE =29 meV for the process, while with a slightly differEAM potential Marinicaet al.28 nd it to be 41 meV. Asalready mentioned, exchange processes between the adaand the substrate atoms are not included in our KMC simlations, neither do they appear in the accompanying Msimulations. The effective diffusion barrier inferred fromArrhenius plot of the monomer diffusivities from our Ksimulations isE a=263 meV, which is consistent with th

FIG. 1. Color onlineSome examples of adatom diffusion andhops on the fcc111 surface. Dark-colored atoms are active andplaced at fcc sites, whereas light-colored atoms serve as the sub-

strate. The lower edge of the layer containing active atoms forms a111 microfacet, so it is called the B-type step edge while the upperedge of the layer containing active atoms forms a100 microfacetwhich is called an A-type step.

FIG. 2. Color onlineDiffusion coefcients as a function of thisland size; a KMC results without including concerted motimechanisms; b KMC results after including concerted motimechanisms obtained from MD simulations.


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calculated energy barrier contained in our database. Thisvalue is also in agreement with that obtained from MD simu-lations E a=31.00.8 meVby Hynninenet al.23 Experi-mental results report the adatom activation energy onCu 111 to be E a=37.005 meV.9 Our results are thus inagreement with experimental data.

B. Dimer

In the case of the motion of the dimer the SLKMC codepicked up only two mechanisms which permit jumps fromfcc to fcc sites. These are labeled dimer A and dimer B inFig. 4a , and their energy barriers are 101 meV and15 meV, respectively. Results of diffusion coefcients withthese two processes at three different temperatures are inTable II. The effective diffusion barrier for the dimer fromthe Arrhenius plot is 104 meV. Of course, the dimer motionis actually nontrivial since in reality both dimer atoms could

also occupy the hcp sites or mixed sites with one atom onfcc and the other on the hcp site. The MD simulations aally revealed 13 more mechanisms for the dimer diffuswhich are shown in Fig. 4b . These illustrations show tran-sitions between the sites occupied by the dimer atoms. multaneous occupation of mixed sites is slightly favorabecause of the somewhat lower energyby 13 meV, ascompared to both atoms occupying the same type of site

Let us have a critical look at mechanisms shown in F4 b . Processes describing sliding and rotational motion,D2, D3, D4, D6, andD8, have lower energy barriers as compareto the others shown in Fig. 4b . ProcessD2 in which bothatoms are initially on hcp sites and one jumps to the fcc by crossing the bridge site, has the lowest energy barrie5 meV, The second low energy mechanism isD6 9 meV,in which both atoms occupy fcc sites initially and onethem jumps to hcp site by crossing the bridge site. The ergy barrier for the same mechanism from the experime

FIG. 3. Color onlineEffective diffusion barriers of 110 atom islands plotted as a function of island size. The dotted line wirepresents full KMC simulation results including concerted motion, whereas the dotted line with circles shows results of the KMwithout including concerted motion mechanisms. The inset shows Arrhenius plots of diffusion coefcients as a function of te


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data reported by Reppet al.9 is 183 meV, which is largerthan what we nd. Marinicaet al.28 nd this barrier to be16 meV. ProcessD4 describes dimer atoms as initially occu-pying mixed sites and nally both atoms occupying hcpsites. We nd its energy barrier to be 18 meV while Marinicaet al.28 reported it to beE =26 meV. We also observed long jump mechanisms D7, D10, D11, andD13 for dimer diffu-sion in our MD simulations.

The sliding motion between fcc and hcp sites has diffu-sion barriers of the same order of magnitude as that for thelong jump motion of the dimer. On the other hand, the rota-tional motion,D3, has a diffusion barrier20 meV closer tothe value of a single atom hopping barrier, which is 29 meV.Finally, we included all of these 13 mechanisms in ourSLKMC database that had only two mechanismsdimer Aand dimer Binitially. As we can see from Table II, with theinclusion of concerted motion the effective diffusion barrierreduces toE a=92 meV, which is closer to the value of bar-rier representing concerted motion of the dimer. Although adimer performs low energy mechanisms D2, D3, D4, D6, and D8 more frequently, the change in the center of mass posi-tion is small as compared to the long jump mechanisms D7, D10, D11, and D13 and also concerted motion mechanism D1 and D12 . Hence a small frequency of relatively highenergy mechanismslong jumpscan greatly change the cen-

ter of mass position of the dimer. This is why the effectdiffusion barrier of dimer is closer to the diffusion barrielong jumps and concerted motion mechanisms D7, D11, D13 .

C. Trimer

We have done a detailed study of trimer diffusion usSLKMC simulations. There are only nine possible atom-

atom motion mechanisms which were identied by SLKMC code. These mechanisms and their correspondenergy barriers are shown in Fig. 5a . With only fcc to fcc jumps the effective diffusion barrier for the trimer380 meV see Table II. Actually the atom-by-atom motionproduces a shape change but does not facilitate the diffuof the trimer. We obtained quite interesting results when included mechanisms describing concerted motion of trias shown in Fig. 5b . Trimer moves from one fcc site to thneighboring hcp site by performing concerted gliding rotation mechanisms. The energy barrier for concerted ging of the trimer from 3 fB to 3 hA is found to be 125 mwhereas the reverse mechanism has a barrier of 115 m

The rotation of the trimer has the lowest energy barrier of38 meV from 3 hA to 3 fA and 62 meV from 3 fA to 3 respectively. With the inclusion of these additional procesthe effective diffusion barrier is found to be 141 meV.

This is a dramatic reduction from 380 meV found earand the effect is impressively represented in Figs. 2b and3 b which shows the trimer to be relatively mobile. In Figwe plot the distribution of the frequency of events with without rotation and concerted motion, represented, resptively, by lled and open symbols. We nd that the occrence frequencies of added mechanismsconcerted motionand rotationare much higher than the occurrence frequecies of all other nine mechanisms because concerted moand rotation mechanisms have low energy barriers as copared to the mechanisms such asopening from Aand open-ing from B. Although rotation dominates, it does not playkey role in trimer diffusion because it is not responsiblethe center of mass motion of the trimer. We expect concermotion to dominate diffusion, and thus we can predict tthe value of the effective diffusion barrier should be closethe value of the concerted motion barrier, which is indtrue here. In Table II, we can clearly see the difference tween results before and after including rotation and ccerted motion mechanisms in our primary database of nprocesses.

D. Tetramer

In the case of the tetramer we have 28 possible, fcc to atom-by-atom jump processes which together with their ergy barriers are shown in Fig. 7a . As noted in Table II,KMC simulations performed with these mechanisms ledan effective diffusion barrier of 492 meV. Three mechaniexhibiting concerted motion and shearing of a diamshaped tetramer, revealed in MD simulations, and their cresponding diffusion barriers, are shown in Fig. 7b . Con-certed motion of a diamond shaped tetramer takes plthrough sliding between the fcc and the hcp sites, alongsmall and large diagonals. The ones along the small diag

FIG. 4. Color online a Illustration of two simple mechanismsfor dimer diffusion and their energy barriers, where atoms jumpfrom fcc to fcc sites.b 13 mechanisms for dimer diffusion via fccto hcp sites and their energy barriers. These mechanisms werefound from MD simulations.


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Fig. 7b 1 have lower energy barrier167 meV for fcc tohcp and 125 meV for hcp to fccthan those along the largediagonal Fig. 7b 2 . These processes have also been dis-cussed by Marinicaet al.28 However, the case of diamondshaped tetramer diffusion through shearing mechanismshown in Fig. 7b 3 with energy barrier 230 meV was nottaken into account by them. When we included these threemechanisms in our database of 28 single atom mechanismsand performed KMC simulation, we found signicantly dif-ferent values for the diffusion coefcients. In Table II, thesevalues are written in square brackets and the effective diffu-sion barrier for tetramer isE a=212 meV.

E. Islands containing 510 atoms

A few examples of single atom processes collected in thedatabase of our KMC simulations, for the islands containing510 atoms, are shown in Fig. 8 with the corresponding en-ergy barriers. The diffusion coefcients calculated fromKMC simulations based on these single atom mechanismsare very low, as shown in Table II. This is particularly thecase for the 7 and 8 atom islands whose effective diffusionbarriers are consequently the largest. This is understandablebecause we nd that processes such asAB corner detach-ment AandAB corner detachment B, shown in Fig. 8, play akey role in the island diffusion by contributing the most tothe change in the center of mass position. Processes such as

FIG. 5. Color online a Nine mechanisms for trimer diffusion with their corresponding energy barriers, where atoms are a jump from fcc to fcc sites.b Trimer diffusion mechanisms observed during MD simulations. These mechanisms are conducted tcollective motion of three atoms by rotation and gliding over the bridge sites from fcc to hcp sites, or vice versa.

FIG. 6. Color onlineDistribution of normalized frequencies oevent occurrences in the case of trimer diffusion. The lines lled symbols show the distributions of events at different temptures when only single atom mechanisms were included, and with open symbols show the distributions of all events includincollective motion of three atoms by rotation and gliding.


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step edge Aandstep edge Boccur more frequently, but theydo not contribute signicantly to the motion of the center of mass of the island; rather the atoms move around and aroundalong the periphery of the island. In Fig. 9 we show the

concerted motion processes revealed from MD simulations.Their energy barriers were determined from molecular staticcalculations by dragging the central atom of the island fromfcc to the nearest hcp site. Other atoms in the island followedits motion by gliding over the bridge sites. The differentshapes and geometries of these islands contribute to the dif-ferences in the energy barriers for the processes. For examplein our MD simulations we found that the 10-atom island canmove as a single entity from fcc to hcp sites whenever itappears into one of the three shapes shown in Fig. 9. Theenergy barriers associated with these processes are slightlydifferent. Clearly, the barriers of these concerted motionmechanismsfor 510 atom islandsare comparatively lower270590 meVthan the energy barriers of the single atom

mechanismsAB corner detachment AandAB corner detach-ment B, also considered essential for island diffusion. Afterthe inclusion of new low energy concerted motion mecha-nisms in our database, the high energy single atom mecha-nisms become less frequent in KMC simulations and highvalues of diffusion coefcients and correspondingly low val-ues of the effective diffusion barriers were obtainedseeTable II. The size-dependent oscillations of the diffusioncoefcients and the effective diffusion barriers also disap-peared from the plots shown in Fig. 2b and Fig. 3, respec-tively. We can thus conclude that the absence of the low

FIG. 7. Color online a Illustration of 28 mechanisms and their corresponding energy barriers for tetramer diffusion, where jallowed from fcc to fcc sites only.b Tetramer diffusion mechanisms revealed from MD simulations:1 diagonal glide,2 vertical glideof 4-atom island over the bridge sites and the corresponding energy barriers, and3 shearing mechanism.

FIG. 8. Color onlineA few examples of single and multiplatom mechanisms and their corresponding energy barriers useour KMC simulations for islands larger than 4 atoms. Jumpsallowed from fcc to fcc sites only.


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energy concerted motion mechanisms is responsible for theoscillatory behavior of diffusion coefcients as the functionof size. Finally, our complete KMC results show that theeffective diffusion barriers increase almost monotonicallywith increasing island size.

F. Key mechanisms and their occurrence frequencies

In Fig. 10 we show the normalized frequencies of allevents from the extended SLKMC data that were performedduring the simulations. Lines with open symbols in Fig. 10show the occurrence frequencies of the all concerted motionmechanisms, at three different temperatures, as the functionof the island size, while those with lled symbols all singleatom mechanisms. For the dimer case, most of the singleatom mechanisms have the same effective barriers as com-pared to the barriers associated with concerted motionmechanisms. Thus, the occurrence frequencies of single andmultiple atom mechanisms are almost the same for dimerdiffusion. In the case of 37 atom islands, concerted motionprocesses are associated with signicantly lower energy bar-riers as compared to the single atoms, and therefore con-certed motion occurs more frequently. A 6-atom island has aneffective barrier for concerted motion that is closer to barri-

ers of some single atom mechanisms, which play a rolethe motion of the center of mass position to some extenti.e.,step edge Aandstep edge Bprocesses. Because of the closecompetition between concerted motion and single atmechanisms, we nd a narrow gap between their occurrefrequencies in the case of a 6-atom island. A similar, nargap can be seen in the case of the 8-atom island. In this there is close competition between concerted motion andmotion of the single atom going around the periphery ofisland i.e., BB corner detachment and AA corner detach-ment processes. In the case of 9 and 10 atom islands the loenergy single atom mechanismsi.e., step edge Aand stepedge Bprocesses occur more frequently, but they do noplay a key role in island diffusion. On the other hand, sithe barriers of the concerted motion mechanisms are hig410590 meV, they occur rarely but still play an importa

role in the diffusion.

IV. CONCLUSIONS

To summarize, we have performed a systematic studythe diffusion of small Cu islands on Cu111 , using a recentlydeveloped self learning KMC simulations in which the stem is allowed to evolve through mechanisms of its chowith the usage of a self-generated database of single amultiple atom diffusion processes. Complementary molelar dynamics simulations carried out for a few cases provifurther details of several new mechanisms for small isldiffusion which were not automatically picked up by SLKMC method because of the initial restriction of fcc occupation. We found signicant changes in the sidependent variations of diffusion characteristics of thelands after including concerted motion mechanisms whwere revealed from MD simulations. We nd that thesmall-sized islands diffuse primarily through concerted mtion with a small contribution from single atom proceseven though for certain cases the frequency of single aprocesses is large because of lower activation energies.

FIG. 9. Color onlineDiffusion mechanisms found by perform-ing MD simulations for the island sizes of 510 atoms and theircorresponding energy barriers when they glide over the bridge sitesexhibiting collective motion of all atoms.

FIG. 10. Distribiution of normalized frequencies of concemotion events as a function of the island size.


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allowing the system the possibility of evolving in timethrough all types of processes of its choice, we are able toestablish the relative signicance of various types of atomis-tic processes through considerations of the kinetics and not just the energetics and/or the thermodynamics, as is oftendone. For small Cu islands on Cu111 , we nd the effectivebarriers for diffusion to scale with island size. We await ex-periments to verify our ndings.

ACKNOWLEDGMENTS

We thank James Evans for helpful discussions. This wwas supported by NSF-CRDF RU-P1-2600-YA-04, NERC 0085604, and NSF-ITR 0428826. T.A-N. has bsupported in part by a Center of Excellence grant fromAcademy of Finland.

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Diffusion of small two-dimensional Cu islands on Cu(111) studied with a kinetic Monte Carlo method

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