PHYSICAL REVIEW B, VOLUME 64, 205404 Rate-equation approach to island size distributions and capture numbers in submonolayer irreversible growth Mihail N. Popescu,1 Jacques G. Amar,2 and Fereydoon Family1 1Department of Physics, Emory University, Atlanta, Georgia 30322 2Department of Physics & Astronomy, University of Toledo, Toledo, Ohio 43606 Received 14 December 2000; revised manuscript received 25 April 2001; published 19 October 2001 We present a quantitative rate-equation approach to irreversible submonolayer growth on a two-dimensional substrate. Our method explicitly takes into account the existence of a denuded ``capture'' zone around every island and the correlations between the size of an island and the corresponding average capture zone. The evolution of the capture-zone distributions is described by a set of Voronoi-area evolution equations, which are coupled to the usual rate equations for the island densities through local rates of monomer capture. The combined set of equations leads to a fully self-consistent calculation of the size- and coverage-dependent capture numbers. The resulting predictions for the average capture-zone and capture-number distributions are in excellent agreement with experimental results and Monte Carlo simulations. As a result, the corresponding scaled island size distributions and their dependence on coverage and deposition rate are also accurately predicted in the precoalescence regime. DOI: 10.1103/PhysRevB.64.205404 PACS number s : 81.15.Aa, 68.55. a, 68.35.Bs, 82.40.Bj I. INTRODUCTION but neither for reversible nor for irreversible growth does it lead to correct predictions for the island-size Molecular-beam epitaxy MBE offers the possibility of distributions.20,21,24,33 The reason is that it is based on a atomic-scale controlled production of thin films, high-quality mean-field approximation, which ignores spatial and tempo- crystals, and nanostructures.1 The submonolayer growth in ral correlations in the growth of islands.23,30,31 MBE involves nucleation, aggregation, and coalescence of In this paper we present a rate-equation approach to two- islands leading to a distribution of islands of various sizes dimensional irreversible submonolayer growth in which the and morphologies. The morphology and the spatial distribu- existence of a denuded ``capture'' zone and the correlations tion of the islands determines the quality of the desired nano- between the size of the island and the corresponding average structure quantum dots/wires or of the multilayer growth capture zone are explicitly taken into account. A second set thin films . These nanoscale features of the surface in the of equations is used to describe the evolution of the island- early stages of MBE growth can now be measured in real- size-dependent capture zones, leading to explicit size- and time with atomic-scale resolution experimental methods, coverage-dependent capture numbers s( ) in good agree- such as scanning tunneling microscopy STM and reflection ment with experimental31 and simulation results.27 A numeri- cal solution of the resulting island-density rate equations high-energy electron diffraction RHEED .1 This has led to a leads to island-size distributions in good agreement with renewed experimental interest in submonolayer nucleation simulations, in contrast to the standard rate-equation ap- and growth,2­13 and has also stimulated considerable theoret- proach. ical work toward a better understanding of the mechanisms We note that a general outline of our method and results determining the scaling properties of the island density and has already been presented in Ref. 34. Here we present a island-size distribution in epitaxial growth.14­34 detailed derivation of the relevant equations along with ex- One of the standard tools used in theoretical studies plicit analytical expressions for the local capture number, of submonolayer growth is the rate-equation RE app- monomer density distribution, and self-consistency condi- roach.14,35,36 It involves a set of deterministic, coupled tions. In addition, a detailed study of the dependence of the reaction-diffusion equations describing the time coverage island-size distribution, capture-number distribution, and av- dependence of average quantities through a set of rate coef- erage capture zone ``distribution'' on the island morphology ficients usually called capture numbers.14,35 While simple is presented along with a study of the evolution of the island- mean-field choices for the capture numbers lead to correct size distribution as a function of coverage and deposition predictions for the scaling behavior of the island and mono- rate. A possible generalization of our method for the case of mer densities as a function of deposition flux and tempera- reversible growth is also discussed. ture, in order to make quantitative predictions accurate ex- The organization of this paper is as follows. Section II pressions for the rate coefficients should be used. presents the details of the self-consistent calculation of the Recently, Bales and Chrzan20 have developed a self- capture numbers. After a brief introduction to the rate- consistent RE approach that leads to quantitative predictions equation formalism, the geometry of the exclusion zones is for the average monomer and island capture numbers, as well defined and the local capture number is calculated as a func- as for the average island and monomer densities in two- tion of the corresponding Voronoi area. It is then shown that dimensional irreversible growth. This approach has also re- both the coverage dependence of the capture number and cently been extended to the case of reversible growth,21,33 island-size correlations are naturally included through the av- 0163-1829/2001/64 20 /205404 13 /$20.00 64 205404-1 ©2001 The American Physical Society POPESCU, AMAR, AND FAMILY PHYSICAL REVIEW B 64 205404 eraging over the distribution of Voronoi areas. This part con- cludes with a discussion of self-consistency conditions and of their connection with the geometry of the system. In Sec. III, evolution equations for the Voronoi areas are developed and solved in closed form. We then discuss, in Sec. IV, our rate-equation results and present comparisons with experi- mental as well as kinetic Monte Carlo KMC results. A sum- mary of our results and conclusions is presented in Sec. V. II. SELF-CONSISTENT THEORY OF SIZE-DEPENDENT CAPTURE NUMBERS A. Rate equations A rate-equation approach to submonolayer nucleation and growth involves a set of deterministic, coupled, diffusion- aggregation equations describing the time coverage depen- dence of average quantities.14,35,36 The RE variables are the FIG. 1. Schematic representation of the capture-zone geometry average densities of monomers, N for an island of size s radius R 1, and of islands of size s s). Shown are islands of different 2, N sizes big dark circles , monomers gray circles , the boundary of s , where s is the number of atoms in the island. A general form of these equations for irreversible growth may the exclusion zone the circle of radius Rex), and the nucleation and be written as mean-field decay lengths 1, respectively . dN neighbor perimeter sites. Depending on the details of the 1 2 growth, the resulting islands can have a compact shape (d d 2N1 2R 1N1 RN1 sNs , 1 f s 2 2) or a fractal morphology (df 2). For extended islands the factor is given by 1 N dN 1. s Once the capture numbers s( ) are known, Eqs. 1 and d RN1 s 1Ns 1 sNs ks 1Ns 1 ksNs 2 can be numerically solved to find the island-size distri- bution Ns( ). However, understanding and predicting the for s 2, 2 size and coverage dependence of the capture numbers has where is the coverage, is the fraction of the substrate not been the central problem of the rate-equation theory of MBE covered by islands, growth14,22,36­39 for more than three decades. In the next sec- s are the capture numbers, and ks are the rates of deposition on top of existing islands. Here, the tion we present a theoretical approach that allows an explicit calculation of the correct size- and coverage-dependent cap- kinetic constant R D/F is the ratio of the diffusion constant ture numbers. D where D Dh/4 for the case of nearest-neighbor hopping with isotropic hopping rate Dh on a two-dimensional lattice to the deposition flux F. The terms with B. Monomer-diffusion equation and local capture numbers s describe the rate of monomer capture by other monomers or by existing is- It has been argued that an island grows by collecting lands. The terms with ks , where ks s2/df and df is the frac- monomers mainly from a ``capture'' zone around the tal dimension of the islands,20 correspond to the deposition island,25,27,31 and that these capture zones are related to the of adatoms directly on islands of size s. Finally, the quantity physically and geometrically well-defined Voronoi cells23,27 2N1 corresponds to the deposition flux minus the direct corresponding to the set of points around an island closer to impingement. it than to its neighbors. As has already been noted, in order to In order to study the effects of island morphology on the quantitatively predict the coverage- and island-size depen- capture-number and island-size distributions, we consider dence of the capture numbers s , one needs to take into two different models: a point-island model and an extended- account these correlations between the size of the island and island model. In the point-island model, each island occupies its local environment. just one lattice site. Dimer nucleation occurs when two As shown in Fig. 1, these observations lead to the follow- monomers land on the same site, while any monomer that ing model for the environment of an island. An island of size lands on the site occupied by an island becomes part of the s is approximated by a circular region of radius Rs s1/df, island. Physically, this corresponds to islands that grow only where is a ``geometrical'' prefactor that accounts for the in a direction perpendicular to the substrate, or, alternatively, circular approximation of the island area and the fractal di- to islands with a very large fractal dimension (df ). For mension df depends on the morphology of the island. The point islands the fraction of the substrate not covered by area surrounding the island is divided into an inner (Rs r islands is then given by 1 N, where N s 2Ns is the Rex) and an outer region (Rex r ). The inner region total island density. corresponds to an exclusion zone in which only monomers In the extended-island model, an island occupies a num- can be found. The area of the exclusion zone Aex is assumed ber of lattice sites equal to its size s, and monomers attach to to be proportional to the Voronoi area AV of the Voronoi a growing island or to another monomer at the nearest- polygon surrounding the island, i.e., Aex AV, where the 205404-2 RATE-EQUATION APPROACH TO ISLAND SIZE . . . PHYSICAL REVIEW B 64 205404 factor typically larger than 1) is assumed to be the same and therefore in the limit of large R the second term may be for all islands. Accordingly, the radius of this zone is Rex neglected as well. This leads to the quasistatic monomer dif- RV , where RV AV / is the ``radius'' of the Voronoi fusion equation polygon. In the outer region, corresponding to r Rex , we assume a ``smeared'' uniform distribution of monomers and 2 n1 2 N1 / for Rs r Rex islands, which is independent of the size of the central island, 2n1 1 2 n as in Ref. 20. 1 N1 / for r Rex. 8 This geometry naturally leads to the definition of a ``nucleation'' length For the case of irreversible growth, the local monomer 1 and of a mean-field ``screening'' or monomer ``capture,'' length . The nucleation length density n 1 char- 1(r) must vanish at the island edge and must also acterizes the monomer decay in the exclusion zone and is match the average local density (N1 / ) far away from the defined by island. In addition, the interior and exterior solutions must match at the exclusion-zone boundary. This leads to the fol- 1/ 21 2 1N1 . 3 lowing boundary conditions: Similarly, the mean-field screening length , where n1 Rs 0, 9a 1/ 2 2 1N1 sNs , 4 n1 Rex n1 Rex , "n1 Rex "n1 Rex , 9b s 2 characterizes the monomer decay in the region surrounding lim n1 r N1 / , 9c the exclusion zone. Using these definitions, the monomer r density rate equation 1 may be rewritten in the form where the renormalized value N1 / far away from the island dN edge is due to the fact that the average local monomer den- 1 sity is actually larger than the overall monomer density N d 2N1 RN1 / 2. 5 1 by a factor of 1/ . A self-consistent calculation of the capture numbers Since the growth is isotropic, there is no angular depen- s entering into the rate equations 1 and 2 is then based on dence in the model and the general solution of Eqs. 8 sat- comparing the microscopic capture rate of monomers near an isfying the boundary condition 9c at infinity is given by island with the corresponding capture terms in the rate equations.20,21 To determine the microscopic capture rate, we consider the following diffusion equation describing the lo- N1 2/ aI0 r/ 1 bK0 r/ 1 n for Rs r Rex 1 r 10 cal monomer density n1(r , ): N1 1/ cK0 r/ for Rex r . 2 n , Rs r Rex 1 The coefficients a and b are determined by the boundary 1 2n1 R 2n1 Rn1 / 1 1 2n1 R 2n1 Rn1 / 2, r Rex. 6 conditions 9a and 9b , The first two terms on the right side of Eq. 6 correspond to 1 3 K1xK0x1 K1xK0s1 2K0xK1x1 K1xK0s1 deposition minus direct impingement of monomers on mono- a K , 11a 0x I1x1K0s1 K1x1I0s1 K1x I0x1K0s1 K0x1I0s1 mers, while the last two terms correspond to monomer dif- fusion and, respectively, loss of monomers by nucleation40,41 2/ aI0s1 (Rs r Rex) or by nucleation and aggregation (r Rex). b K , 11b Multiplying Eq. 6 by and subtracting the monomer 0s1 rate equation 5 , one obtains where Kjs1 Kj(Rs/ 1), Kjx Kj(Rex / ), Kjx1 Kj(Rex/ 1) and similarly for I 1 n N 2 js1 , I jx , and I jx1), and K j ,I j , j 0,1 are the 2n 1 1 modified Bessel functions of order j. 1 R R n1 N1 Equating the microscopic flux of atoms near the island, 2 2 R , to the corresponding macroscopic n sD dn1 /dr r R 1 2 N1 / , Rs r Rex s 1 RE-like term DN 2 n 1 s(AV), one obtains an expression for the 1 N1 / , r Rex, 7 local capture number s(AV), where 2 ( 1 / )2. Since by definition represents the 2 R fraction of sites that are not occupied by islands, the proper s Rs Rs s AV aI1 bK1 . 12 normalization of the local monomer density is n¯ 1 1 1 1 N1 / , where n¯1 denotes the average local monomer density in the Here the dependence of s on the Voronoi area AV or region between islands. This implies that the last term on the equivalently on the exclusion-zone area Aex AV) arises left-hand side of Eq. 7 may be neglected. Similarly, the from the dependence of the coefficients a and b on the quantity in the parentheses in the second term is also small exclusion-zone radius Rex AV / . 205404-3 POPESCU, AMAR, AND FAMILY PHYSICAL REVIEW B 64 205404 A similar analysis can be carried out for the monomer which corresponds to the requirement that the average capture number 1. However, since the monomers are mo- ``screening length'' embodied in the rate equations is the bile, in this case there is no ``exclusion'' zone, which implies same as that expressed by the diffusion equations. that R t ex R1. In this limit, i.e., taking Rs R1 Rex , Eqs. If n1(As) denotes the total number of monomers in a 11 and 12 lead to Voronoi area of size As surrounding an island of size s, then the self-consistency condition for the monomer density cor- 2 R K responds to the requirement that the total monomer density 1 1 R1 / 1 K . 13 in the Voronoi polygons must be equal to the average mono- 0 R1 / mer density N1, i.e., Apart from an extra factor of 1/ , which takes into account the enhancement of the local monomer density with increas- N t A ing coverage, this expression is the same as previously ob- sn1 s N1 . 16 s 2 tained by Bales and Chrzan.20 Using Eq. 12 , the size-dependent capture numbers In principle, Eqs. 15 and 16 can be solved at any given s are obtained by averaging the local capture numbers coverage for two of the three unknowns , , and once s(AV) over the distribution of Voronoi areas. Defining G the third is known and once the average Voronoi area As is s( ;AV) as the number density of Voronoi areas of size A known for all s. However, by carrying out detailed numerical V surrounding an island of size s at coverage , we can write comparisons we have found42 that a mean-field approxima- tion gives equally good results. This is not surprising, since the self-consistency conditions correspond to average quan- G tities. This approximation allows us to ``precalculate'' the s ;AV s AV A quantities , , , and V 1 as a function of coverage before s s AV G s carrying out the numerical integration of the full island- Gs ;AV density rate equations 1 and 2 coupled with the Voronoi- AV area rate equations see below . 1 In this approximation, the island sizes s in Eqs. 15 and N Gs ;AV s AV , 14 16 are replaced by the average island size S ( N1)/N, s AV while the corresponding Voronoi areas are replaced by the where *** average Voronoi area Aav 1/N. This leads to a very simple G denotes an average with respect to the distri- s form for the capture number and monomer density self- bution Gs( ;AV). consistency relations, The capture numbers s have now been expressed in terms of the Voronoi-area distribution Gs( ;AV), the geo- 2 metrical factors and , and the nucleation and screening 1N1 N S Aav 1/ 2, 17a lengths 1 and . The nucleation length 1 can be determined Rav using Eqs. 3 and 13 , assuming and are known. Once N dr 2 rn1 r N1, 17b the distributions G RS s( ;AV) and the parameters , , and are known, Eqs. 12 , 13 , and 14 can be solved in order to where RS S1/df and Rav 1/( N). obtain the size- and coverage-dependent capture numbers Using Eq. 10 for the monomer density n1(r), the s . However, the local monomer density and capture num- monomer-density self-consistency condition 17b may be bers must satisfy self-consistency conditions that we discuss rewritten for the case 1 as in the next section. Use of these self-consistency conditions allows the determination of two of the three parameters , , R2 2 av 2/ 1 RS 2/ 2RS 1 aI1S1 bK1S1 and once the third is known. 2Rav 1 aI1a1 bK1a1 0, 18 C. Self-consistency conditions for monomer density and where I1a1 I1(Rav / 1), K1a1 K1(Rav / 1), and the coef- capture numbers ficients a and b are given by Eq. 11 with Rex Rav . A The parameters and are independent of the island size similar expression can be written for the monomer-density and Voronoi areas, and thus they can be determined using an self-consistency condition for the case 1. approximation in which the Voronoi areas surrounding is- In the case of extended islands, the geometrical parameter lands of size s are replaced by their average values. Denoting was assumed to be independent of coverage, and its value the average Voronoi area corresponding to an island of size s 0.3 was selected such that numerical integration of a at a given coverage by A mean-field form of the island-density rate equations 1 and s , this leads to the approximation 2 -corresponding to replacing A V in Eq. 12 with Aav s s(As). The definition 4 of the screening length then 1/N-led to good agreement with Monte Carlo results for leads to the capture number self-consistency condition, the average monomer and island densities N1 and N. Al- though the value 0.3 is somewhat smaller than expected 2 for circular islands the value of would be 1/ 0.6), the 1N1 Ns s As 1/ 2, 15 s 2 same value works for all values of R while variations of 205404-4 RATE-EQUATION APPROACH TO ISLAND SIZE . . . PHYSICAL REVIEW B 64 205404 FIG. 2. Coverage-dependence of obtained from Eqs. 17 for point islands solid lines and compact islands dashed lines for Rh 107­109. Inset shows corresponding coverage dependence of for point islands. about 20% in lead to significant deviations from the Monte Carlo calculated densities N and N1. The two remaining vari- ables and were then determined by simultaneously solv- ing the two self-consistency conditions 17 as a function of coverage. In the case of point islands a slight variation of this method was found to lead to better results. In this case there is no explicit s dependence in s since Rs for all s. Thus, Eq. 17a may be rewritten as 1/ 21 N Aav 1/ 2 19 without any additional assumption of an average island size S. In this case, a self-consistent RE approach similar to that FIG. 3. Island density N and monomer density N1 as a function used in Ref. 20 for compact islands was first used in order to of coverage for Rh 107­109 obtained from RE's symbols and obtain the coverage-dependent lengths KMC solid lines for a point and b compact islands. 1( ) and ( ) for point islands. Equations 17b and 19 were then simulta- neously solved for ( ) and ( ). The inset in Fig. 2 shows our results for ( ) for the case Figure 2 shows the resulting coverage dependence of of point islands. As expected, ( ) is almost constant and for both point and compact islands. As expected, similar val- independent of Rh Dh /F for 0.01, and asymptotically ues of are obtained at low coverage for both point and i.e., in the limit of large Rh) approaches a constant value compact islands. In particular, we find 1 in the nucleation over the entire coverage range. Accordingly, in our calcula- regime corresponding to x , where x defined by tion of the island-capture numbers and size distributions for N1( x) N( x)] corresponds to the crossover from nucle- point islands, we have approximated ( ) by the constant ation to aggregation. However, for x , quickly ap- value 0.12 at all coverages. It is not surprising that this proaches a constant value for point islands, which is some- value is significantly smaller than the value 0.3 used for what above 1 and is slightly dependent on Rh Dh /F. In extended islands, since for point islands, monomers can only contrast, for extended islands continues to decrease in the attach by landing on the site occupied by another island or coverage range where the spatial extent of an island is sig- monomer, and not by nearest-neighbor attachment. nificant, and it eventually becomes smaller than 1 at cover- Figure 3 shows the resulting average island and monomer ages higher than 0.2. This may be considered as a natural densities N and N1 obtained using the island-density rate limit of our method since in that range coalescence effects, equations 1 and 2 , as discussed above, along with the which are not included in our rate equations, become signifi- corresponding KMC simulation results. As can be seen, there cant. It is interesting to note, however, that this decrease is excellent agreement between the RE predictions and the actually correctly describes the qualitative behavior of in simulation results for these average quantities at all cover- the presence of coalescence. ages in the precoalescence regime. Thus, the coverage- 205404-5 POPESCU, AMAR, AND FAMILY PHYSICAL REVIEW B 64 205404 dependent factors and and the screening and nucleation A. Voronoi-area distributions for point islands lengths and 1 have been calculated using the mean-field The point-island model is special because for this model self-consistency conditions. The only quantity left unknown the local capture numbers do not depend explicitly on the is the distribution of Voronoi areas. island size s. This allows an exact solution of the Voronoi- area evolution equations 21 and 22 . Changing the cover- III. VORONOI-AREA-DISTRIBUTION age variable from to EVOLUTION EQUATIONS To obtain the capture numbers and the island-size distri- xA RN1 A d , 23 bution, one has to consider the dependence of the Voronoi- A area distribution Gs( ;A) on the island size s. Taking into where (A) is the local capture number for point islands, account the change in the areas by nucleation and aggrega- leads to the following form of the Voronoi-area evolution tion of islands, and ignoring for the moment the breakup of equations: Voronoi areas when new islands are nucleated, one can write a general set of evolution equations for the functions dG2 xA ;A G G2 xA ;A BA xA , 24 s( ;A) in the following form: dxA dG dG 2 ;A s xA ;A G d dN/d A Aav RN1 2 A G2 ;A , dx s 1 xA ;A Gs xA ;A s 3 . A 20 25 The solution of Eq. 24 is G2(xA ;A) BAe xAH(xA), dGs ;A where H(z) is the step function H(z) 1,z 0;H(z) 0,z d RN1 s 1 A Gs 1 ;A s A Gs ;A 0 . It can also be shown32 that the general solution for the Voronoi-area distribution Gs for s 2 is s 3 , 21 G s 2 s xA ;A BAxA e xA/ s 2 !. 26 where At coverages beyond the nucleation regime, both x s(A) is the local capture number as given by Eq. A and the 12 . The first term on the right side of Eq. 20 corresponds average Voronoi area Aav are typically large. Thus, Eq. 26 to nucleation of dimers while the remaining terms in Eqs. corresponds to a sharply peaked distribution as a function of 20 and 21 correspond to growth of islands via aggrega- A, and the peak position A s satisfies43 tion. We note that in these equations it has been assumed that the Voronoi areas around the new dimers nucleated at cov- xA s 2. 27 s erage are equal to the average Voronoi area at that cover- age, A Therefore, keeping only the dominant terms in the sums, av 1/N, and for simplicity the ``source'' term in Eq. 20 has been assumed to take the form of a delta function. the average Voronoi area As corresponding to an island of As already noted, the breakup of larger areas due to nucle- size s and the corresponding capture numbers s satisfy ation has been neglected in Eq. 21 . However, we will ac- count for it through a uniform rescaling that will be justified a posteriori. The effects of direct impingement of atoms on A AGs xA ;A A A islands have also been neglected in Eqs. 20 and 21 since s s , 28 they are very small except for the case of extended islands at AGs xA;A high coverages. However, direct impingement is still taken into account in the full rate equations 1 and 2 . Defining A through the condition A 1/N( A), the A s A Gs xA ;A nucleation term in Eq. 20 may be rewritten as B A ( s s A s . 29 A), with BA 1/A2. Accordingly, Eq. 20 may be rewrit- AGs xA;A ten in the form However, since Eqs. 20 and 21 do not include the effects dG of the area breakup due to nucleation, the average areas cal- 2 ;A d RN1 2 A G2 ;A BA A . culated using Eq. 27 are expected to be larger than 22 the correct values. To account for this, the areas should be rescaled to the correct average Voronoi area, Aav 1/N. The We note that the initial conditions for a given A) for the correct capture numbers s are then given by functions Gs( ;A) are determined by Gs( ;A) 0 for all s A for s A . Since the solution of Eqs. 21 and 22 is some- s s As , where As , 30 what dependent on the particular growth model point or N extended islands , we discuss the two cases separately. sAs s 205404-6 RATE-EQUATION APPROACH TO ISLAND SIZE . . . PHYSICAL REVIEW B 64 205404 where the local capture number s(A) is given by Eq. 12 . dGs xA ;A Therefore, the calculation of the size-dependent capture dx Gs 1 xA ;A Gs xA ;A s 3 35 numbers A s has been reduced to solving Eq. 27 for the peak position A for the Voronoi-area distributions. The solution is again s and then rescaling using Eq. 30 to obtain the rescaled capture areas A given by s . Once the capture numbers s are known at each step, the rate equations 1 and 2 can G s 2e xA/ then be integrated to obtain the island-size distributions. s A BAxA s 2 !, 36 It is interesting to note that the variable xA given in Eq. with the peak of the distribution for a given s) correspond- 23 has a simple physical interpretation, which also implies ing to A a simple physical interpretation for the peak condition 27 . s determined by Since the quantity RN1( ) (A) is the average growth rate xA s 2, 37 at coverage of a point island surrounded by a capture zone s of area A, the coverage variable x and from Eq. 28 the average Voronoi area is A A d RN s A s . A 1( ) (A) is the average number of particles gained by a point island with As before, the effects of the breakup of Voronoi areas due Voronoi area A from the coverage to nucleation will be included through a rescaling of the A at which it is nucleated up to the current coverage . Thus the peak condition x areas to the correct average value Aav 1/N. However, due A s to the existence of a spatially extended island inside the s 2 given by Eq. 27 may be viewed as stating that the Voronoi area, one should consider the rescaling due to peak of the Voronoi-area distribution Gs(A) corresponds to breakup to apply only to the area at the exterior of the island, islands whose Voronoi area is such that they have on average rather than to the whole area. Therefore, one obtains for the gained s 2 particles since their nucleation as dimers, as- rescaling factor, the expression suming that their capture-zone area did not change since they were nucleated. f . 38 B. Voronoi-area distributions for extended islands AsNs 1 s In the case of extended islands the Voronoi-area evolution equations 21 cannot be solved analytically due to the ex- We note that for extended islands it is reasonable to define plicit s dependence of the local capture numbers the Voronoi polygon as passing through points halfway from s(A). However, using the approximation the edges of the islands rather then halfway from the s(A) S(A) where S is the average island size leads to a set of equations that can centers.44 Ignoring correlations between the size of an island be solved analytically. This approximation is expected to and the size of its neighbors, the radius of the rescaled have only a weak effect on the final island-size distributions Voronoi polygon should then include an additional correc- N tion, Rcorr (s1/df S1/df)/2. This leads to the corrected s( ) because it retains the dominant effect, i.e., that of an exclusion zone A Voronoi area ex , which depends on the island size s. Furthermore, the correct expression for s(As) will be re- tained in the rate equations 1 and 2 . As fAs / Rcorr 2, 39 Using this approximation, the Voronoi-area evolution and the capture numbers are given by equations become s s As . 40 dG2 ;A d dN/d A Aav RN1 S A G2 ;A , Therefore, the calculation of the coverage-dependent capture 31 numbers s for extended islands has been reduced to solving Eqs. 33 and 37 , and the full rate equations 1 and 2 can be integrated to find the island-size distributions. dGs ;A d RN1 S A Gs 1 ;A Gs ;A s 3 . C. Summary 32 Before presenting our results, we now summarize the key One can again transform to new variables, points and physical assumptions used in our approach with an emphasis on the case of extended islands. The environ- ment of an island is modeled as consisting of an inner ``ex- xA RN1 S A d 33 clusion'' zone in which only monomers can be found, which A is surrounded by an outer ``smeared'' zone consisting of both to obtain the evolution equations monomers and islands, while a circular approximation for the island and exclusion zone areas is used. In order to con- nect the island radius R dG s to the island size s as well as the 2 xA ;A radius of the exclusion zone Rex to the Voronoi cell area AV , dx G2 xA ;A BA xA , 34 A two geometrical parameters and were introduced. 205404-7 POPESCU, AMAR, AND FAMILY PHYSICAL REVIEW B 64 205404 By combining the linearized diffusion equation 6 satis- fied by the local monomer density surrounding an island with the contracted rate equation 5 for the average monomer density, and solving the resulting quasistatic monomer- diffusion equation 8 with the appropriate boundary condi- tions, a self-consistent expression 12 for the local capture number in terms of the island size and exclusion-zone area was derived. The geometrical parameter ( ) and the cap- ture length ( ) are precalculated as a function of coverage using the mean-field monomer density and capture-number self-consistency conditions 17 , while the geometrical pa- rameter is determined by the requirement of reproducing the KMC and/or mean-field RE results for the total island and monomer densities. In order to find the distribution of exclusion zones, a set of Voronoi-area evolution equations is used, which takes into account the change in the areas by nucleation and aggrega- tion of islands, while the effects of fragmentation are in- cluded through a uniform rescaling of the areas. By introduc- ing a change of variables and using an additional approximation for the case of extended islands in which the local capture number s(A) is approximated by S(A)] these equations can be solved in closed form. For large s corresponding to large values of Rh Dh /F) the solution in Eq. 36 is a sharply peaked distribution as a function of the Voronoi area A, and the peak position A s satisfies the condi- tion xA s 2 given by Eq. 37 . Thus, the most important s and time-consuming aspect of our approach involves solving Eq. 37 numerically for each value of s at each integration step. In calculating the coverage variable xA for each value FIG. 4. Scaled island-size distribution f (s/S) for point islands of A needed in the numerical routine we have used Rid- calculated using RE's solid lines , along with corresponding KMC ders's method , Eq. 12 is used for the local capture number results symbols and MF theory dashed lines for Rh 107­109. S(A), which enters in Eq. 33 . We note that calculating xA using Eq. 33 also requires knowing N 2 1( ), 1( ), ( ), av (1/ 2 1/ )/N was used in the island-density rate and S( ) or N( )] for all coverages up to the present 1 equations. However, at coverages just beyond value . While in principle these values can be calculated x) such that the average island size S is sufficiently large we chose as and stored during the integration, for convenience they are criterion S 10) and the peak in the Voronoi-area distribu- precalculated using a mean-field approach as described in tion G Sec. II C specific values are obtained via interpolation . s( ;A) is well defined, the appropriate Voronoi-area evolution equation results for s , i.e., Eq. 30 for point Once the values A s have been calculated for all s, they are islands and Eq. 40 for extended islands, were used. rescaled following Eqs. 38 and 39 to obtain the areas As . As described in Eq. 40 , As is then used in Eq. 12 to obtain the capture numbers A. Point islands s . The resulting capture num- bers s( ) are then used to advance the full rate equations Figure 4 shows our rate-equation results for the scaled 1 and 2 in order to obtain the island-size distribution. island-size distribution for point islands solid lines in the aggregation regime ( 0.1­0.5) for Rh 107­ 109. Also IV. RESULTS shown are the corresponding KMC simulation results sym- bols and, for comparison, mean-field MF RE results (Rh Using the methods described in the previous sections, the 109) obtained using the approximation s av dashed island-density rate equations 1 and 2 were numerically lines . As can be seen, there is excellent agreement between integrated along with the Voronoi-area evolution equations in the RE predictions obtained using the Voronoi-area evolution order to obtain the size- and coverage-dependent capture equations and the corresponding kinetic Monte Carlo simu- numbers s( ) as well as the scaled island-density distribu- lation results. In contrast, the mean-field results are much tions f (s/S) (S2/ )Ns( ). The rate equations were numeri- more sharply peaked and appear to be approaching the cally integrated starting at very low coverage 0 x with divergent asymptotic form16,28,30 f MF(u) (1/3)(1 initial conditions N1( 0) 0 and Ns( 0) 0 for s 2. At 2u/3) 1/2. low coverage, for which both the average island size S and The good agreement between the results obtained using typical values of xA are small, the average capture number the Voronoi-area evolution equations and the kinetic Monte 205404-8 RATE-EQUATION APPROACH TO ISLAND SIZE . . . PHYSICAL REVIEW B 64 205404 FIG. 5. RE results symbols for scaled capture-number distri- butions s / av for point islands at coverages 0.1 circles and 0.5 squares for Rh 108 and 109. Lines are KMC results from Ref. 30 at 0.2 for Rh 108 dashed and 109 dotted . Inset shows RE results symbols for dependence of s / av on scaled Voronoi area at Rh 109, 0.5. Carlo simulations is due to the inclusion of explicit correla- tions, which are not taken into account in the mean-field calculations. We note, however, that for 0.5 and high D/F the predicted distributions are shifted slightly to the right of the simulation results. This is most likely due to the fact that the uniform rescaling of the Voronoi areas does not FIG. 6. Scaled island-size distribution f (s/S) for a ­ c com- exactly take into account the effects of breakup of exclusion pact islands (df 2) and d ­ f fractal islands (df 1.72) for Rh zones due to nucleation. 108 109 and 0.06 0.3, obtained from RE's lines and KMC Figure 5 shows the corresponding results for the scaled symbols , along with MF theory dashed lines . capture number distribution s / av as a function of the However, as already noted, this difference does not appear to scaled island size for Rh 108 and 109. As can be seen, the have a significant effect on the scaled capture numbers scaled capture-number distribution is essentially independent of coverage and R s / av . h , but depends strongly on the scaled is- land size for s/S 1. Also shown dashed lines are KMC B. Extended islands simulation results at 0.2 for Rh 108 109 from Ref. 30. As can be seen, there is good agreement with the simula- Figure 6 shows our rate-equation results for the scaled tions, although for large s/S and R island-size distribution for both compact islands (d h the RE results are f 2) and slightly below the KMC results. fractal islands (df 1.72) in the aggregation regime ( The strong island-size dependence of the scaled capture 0.06­0.3) for Rh 108 and 109, along with the corre- numbers shown in Fig. 5 is due to the fact that the average sponding KMC simulation results and mean-field predictions Voronoi areas also depend strongly on the island size. As at Rh 109 dashed lines . Similar results have also been shown by the inset in Fig. 5, the capture numbers are to a obtained for Rh 107 not shown . As can be seen, the pre- good approximation linearly dependent on the Voronoi areas, dicted island-size distributions are in good agreement with the simulation results, while the mean-field distributions are much too sharply peaked and diverge rapidly with increasing coverage. In addition, we note that the predicted position of s / av a0 a1 As /Aav . 41 the peak in the size distribution is almost coverage indepen- dent for compact islands, while for fractal islands it is shift- The linear fit shown in the inset dotted line gives46 a0 ing slightly to the left with increasing coverage, in very good 0.85, a1 0.15. These values are significantly different agreement with the behavior shown by the KMC results. from the values a0 0.3, a1 0.7 obtained from simulations However, for both compact and fractal islands there is a of a two-dimensional point-island model in Ref. 30. The dif- small ``overshooting'' of the peak value, which increases ference in a0 and a1 indicates a possible difference between with coverage and/or Rh . This is most probably due to the the Voronoi areas obtained in our area-evolution plus rate- use of the mean-field approximation s S in the Voronoi- equation results and those obtained in the simulations.44 area evolution equations.45 205404-9 POPESCU, AMAR, AND FAMILY PHYSICAL REVIEW B 64 205404 FIG. 8. RE results for scaled average capture zone A s /A av for compact open circles and fractal open squares islands (Rh 109, 0.18) along with experimental results filled symbols at 0.23 from Ref. 31. Ref. 31. The RE results show excellent scaling with both Rh and coverage, and there is good agreement within statistical fluctuations with the experimental data. Also shown in Fig. 7 a are lines corresponding to the asymptotic forms31 s / av zs/S, where z 2/3 for point islands and z 1 with logarithmic corrections for compact islands. Both the experimental and the RE results lie between these two lines, thus showing an effective value of z smaller than 1, but greater than 2/3, in agreement with the experimental results of Ref. 31. FIG. 7. RE results for scaled capture-number distributions for In contrast to the results for compact islands, for fractal a compact and b fractal islands for R islands the scaled capture-number distribution is independent h 108 and 109 at 0.06 open circles and 0.3 open squares , along with experimen- of Rh , but depends on coverage, as shown in Fig. 7 b . With tal results filled symbols at 0.23 from Ref. 31. increasing coverage the distribution s / av ``rotates'' coun- terclockwise around the point s/S 1, and this may explain Another important difference between compact and frac- the differences in the island-size distributions. The decrease tal islands is the behavior of the size distribution at small s/S in the scaled capture numbers for s/S 1 with increasing values. While for both point and compact islands f (0) ap- coverage and the corresponding increase for s/S 1) is con- pears to approach a common limiting value f (0) 1/3 in sistent with both the decrease of f (0) and the shift of the agreement with the MF rate-equation prediction16,28,30 see fractal-island size-distribution peak toward s/S 1, and also Figs. 4 and 6 , for fractal islands our rate-equation results explains the incomplete scaling shown by the fractal island- correctly follow the decrease of f (0) with increasing size distributions. It can also be seen that, in contrast to the coverage,47 as shown in Fig. 6. This indicates that the de- compact case, the asymptotic behavior of the scaled capture crease in f (0) with increasing coverage is not due to coales- number at large s seems to be well described by the cence, since coalescence is not taken into account in the asymptotic value z 1. island-density rate equations or Voronoi-area evolution equa- Figure 8 shows the scaled ``capture zone'' A s /A av for tions, but is most likely due to the increase in the average both compact and fractal islands as a function of the scaled capture number island size. We note here that in agreement with Ref. 31, we av with coverage for fractal islands. The difference in the behavior of f (0) for compact and have defined A s As s, thus A av Aav , as corresponding fractal islands is somewhat surprising, since the only change to the part of the capture zone outside the island for a very in the rate equations from compact to fractal islands is the detailed discussion of this point see Ref. 44 . Also shown in value of the fractal dimension df . In order to understand this Fig. 8 are the experimental results at 0.23 for Cu/Co on result, as well as the good predictions for the island size Ru 0001 from Ref. 31. Again, there is good agreement be- distributions and for the peak position, we have analyzed the tween the predicted capture zones and the experimental re- behavior of the capture numbers for each case. Figure 7 a sults for s/S 1.75. However, for s/S 1.75, the RE predic- shows the scaled capture-number distribution s / av for tions are somewhat above the experimentally measured compact islands as a function of s/S for Rh 108­109 and values. This may be due in part to insufficient statistics in the 0.06­0.3 along with the experimentally measured experiment for large island sizes or to a slight breakdown of capture-number distribution for Cu/Co on Ru 0001 from the uniform rescaling assumption used in our RE's. In any 205404-10 RATE-EQUATION APPROACH TO ISLAND SIZE . . . PHYSICAL REVIEW B 64 205404 case, as already noted these differences have little effect on the island-size distributions since in that range the island density is already very small and decreasing, as can be seen from Fig. 6. Finally, we briefly discuss and present results for the scaled Voronoi-area distribution g(A/Aav) (1/N) s 2Gs( ;A), which may be obtained using our ap- proach. While the uniform rescaling of the Voronoi areas appears to be sufficient to obtain good results for the capture numbers and island densities, a more complicated rescaling is necessary in order to obtain the Voronoi-area distribution G(A) s 2Gs( ;A). The reason is easy to understand: re- scaling the areas shifts the distribution and thus corrects the position of the peak, but leaves the amplitudes unchanged. Proper normalization of the Voronoi-area distributions Gs( ;A) requires that they satisfy the condition 0dA Gs( ;A) Ns while the overall Voronoi-area distribu- tion G(A) must satisfy 0dA G(A) N. While the latter nor- malization condition is automatically satisfied,48 since the correct nucleation rate is included in the Voronoi-area evolu- FIG. 9. Scaled Voronoi-area distribution g(A/Aav) obtained tion equations, the ``individual'' normalization conditions on from Eq. 44 for compact islands solid line at 0.18 for Rh the G 108, along with KMC results symbols for compact islands from s( ;A) as given by Eq. 36 are not, since the area rescaling is only carried out at the end of the calculation, Ref. 25 (Rh 4 106, 0.1 0.3. rather than continuously during the integration involved in xA . is somewhat lower than obtained in simulations. The corre- While it is possible to rescale the Gs( ;A) in Eq. 36 in sponding RE results for the point-island distribution not order to obtain a prediction for the Voronoi-area distribution shown are similar, but the peak of the point-island distribu- G(A), calculation of the correct normalization factors which tion is shifted slightly to the right compared to the compact- involve integrals of the form s 2 island case , consistent with KMC results obtained for circu- 0 dA BAxA e xA) is rather tedious. Fortunately, a very simple approximation is possible. lar islands at low coverage.44 However, in both cases the tails Since the distributions G of the predicted distributions are much wider than in the s( ;A) given by Eq. 36 are sharply peaked around the rescaled areas A simulations and the RE results exhibit also a sharp cutoff for s , it is reasonable to replace them, in a ``zeroth order'' approximation, by small A/Aav , in contrast to the smooth approach to zero of functions. Taking into account the proper normalization, we the simulation results. These discrepancies are the result of obtain the neglect of the nonlinearity of the fragmentation process, which favors the breakup of large areas over small areas, and G which can lead to the formation of small Voronoi areas with s ;A Ns A As , 42 a nonvanishing probability. Overall, the qualitative agree- where As is the rescaled peak area obtained from the ment with the KMC results shown by the very simple ap- Voronoi-area evolution equations see Eq. 39 , while Ns are proximate form Eq. 44 is surprisingly good, but it is clear the island densities obtained from our island-density rate that if quantitative predictions for the area distributions are equations. needed, then the breakup should be accounted for in detail in With this approximation, the Voronoi-area distribution the area evolution equations. G(A) which implicitly satisfies the correct normalization 0dAG( ;A) N( )], may be written as V. DISCUSSION G ;A N We have developed a self-consistent rate-equation ap- s A As , 43 s proach to two-dimensional irreversible submonolayer growth in which the existence of a denuded ``capture'' zone with a and thus the scaled Voronoi distribution g(A/Aav) is fluctuating area around every island and the correlations be- tween the size of the island and the corresponding average g A/A capture zone are explicitly taken into account. To obtain the av Ns /N A As . 44 s capture numbers and the island size distribution, we have proposed a general set of evolution equations for the Figure 9 shows results for the scaled Voronoi-area distri- Voronoi-area distributions, which takes into account the bution g(A/Aav), Eq. 44 , for compact islands (Rh 108, change in the areas by nucleation and aggregation of islands, 0.18) along with kinetic Monte Carlo simulation results while the effects of fragmentation have been included (Rh 4 106, 0.1 0.3) from Ref. 25. As expected, the through a uniform rescaling of the average Voronoi areas. peak position is correctly predicted although the peak height This second set of equations has been solved analytically and 205404-11 POPESCU, AMAR, AND FAMILY PHYSICAL REVIEW B 64 205404 the solution has been used to self-consistently determine the well as the dependence on the island morphology and Rh . As size- and coverage-dependent capture numbers s( ). The a result, our method leads to a self-consistent prediction for resulting scaled island-size distributions were found to be in the evolution of the scaled capture-number and island-size excellent agreement with KMC simulations, although a small distributions with coverage, which is in good agreement with ``overshooting'' of the peak value was noted for extended both simulations and experiments. islands. In addition, our results were shown to accurately Finally, we note that it should be possible to extend the predict the dependence of the scaled island-size distribution coupled-evolution-equation method presented here to the on the island morphology as well as on the coverage and case of reversible growth, in order to predict the scaled deposition rate. The island-size dependence of the capture island-size distribution as a function of the critical island size numbers was also found to be in good agreement with i. For the case i 1, one may simply replace the equation for simulation30 and experimental31 results. G2( ;A) with the corresponding equation for Gi 1( ;A) We note that in previous work by Blackman and while the Voronoi-area evolution equations for higher island Mulheran24 reasonably accurate asymptotic island-size distri- sizes s i 1 will remain the same. A self-consistent mean- butions have been obtained in one dimension,24 by using field approach see Refs. 21,33 can be used to obtain the Monte Carlo simulation results for the scaled-gap distribu- nucleation rate dN/d , as well as the densities and capture tion coupled with rate equations and assuming scaling. More numbers of islands smaller than or equal to the critical island recently, Mulheran and Robbie25 have carried out a numeri- size. As a result, the quasistatic monomer diffusion equation cal calculation of both the asymptotic scaled Voronoi-area inside the exclusion zone Eq. 8 and all other expressions distribution G(s/S;A/ A ) and the asymptotic scaled island- for the monomer density and capture numbers will remain size distribution f (s/S) for compact islands in two- the same, except that the monomer nucleation length 1 dimensions by assuming scaling as well as a linear relation should be replaced by the exclusion-zone capture length i , between the Voronoi area and the island capture number.25 In where 1/ 2 2 contrast, our method involves a fully self-consistent calcula- i 1N1 2 s i sNs . The basic idea of coupled evolution of the capture zones and densities in- tion of the coverage-dependent capture numbers s( ) with- cluded in the present approach may also prove useful in the out any assumptions regarding the relation between the is- rate-equation modeling of a variety of other problems in- land capture number and the capture-zone area. In addition, volving growth by diffusion and aggregation, such as het- scaling of the island-size and capture-number distributions eroepitaxial growth and Ostwald ripening. was not assumed. In this connection we note that in previous work25,44 it has been pointed out that for irreversible growth the actual scaling is only approximate. Since we do not as- ACKNOWLEDGMENT sume exact scaling for a general discussion see Ref. 49 , we are able to reproduce this approximate scaling including the This research was supported by a grant from the Office of coverage dependence of the scaled island-size distribution, as Naval Research. 1 J. Y. 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