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Characterization of submonolayer growth of Cu islands on Cu(001)
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surface science letters
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ELSEVIER
Surface Science Letters 306 (1994) L569-L574
Surface Science Letters
Characterization
of submonolayer growth of Cu islands on Cu(OO1)
G.T. Barkema a, Ofer Biham a~l,M. Breeman b, D.O. Boerma b, Gianfranco VidaWC a Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY 14853-2501, USA b Nuclear Solid State Physics, Materials Science Centre, Groningen University, Groningen, Netherlands c Department of Physics, Syracuse University, Syracuse, NY 13244-1130, USA (Received 7 January 1994; accepted for publication 14 February 1994)
Abstract Submonolayer island growth of Cu on Cu (001) is simulated using energy barriers derived by the atomembedding method of Finnis and Sinclair. We find that the island density during deposition quickly saturates and forms a plateau over a range of the coverage 8. We observe that due to high edge mobility the islands form compact shapes and that the average island size x scales like i? N 0” where n = 0.5, in agreement with recent experiments.
The formation and growth of ordered islands in deposition experiments on metallic substrates has recently attracted considerable attention both experimentally [ l-41 and theoretically [ 5-101. In the experiments atoms are deposited on a clean surface at a constant rate, from zero coverage up to a full monolayer. It has been observed that due to thermal diffusion, and possibly other effects [ 11, the atoms have considerable mobility on the surface which allows them to bunch together into islands of an ordered crystalline phase [ 111. One can identify three basic processes in these systems. At low coverage the deposited atoms hop as random walkers until they nucleate into islands. Additional atoms then aggregate into existing islands, which grow Corresponding author. Fax: + 1 3 15 443 9103; E-mail: [email protected]. ’ On leave from the Department of Physics, Syracuse University, Syracuse, NY 13244, USA. l
0039-6028/94/$07.00
as the coverage increases. At high coverage coalescence effects become important, where two or more islands merge into one large island. Related phenomena appear in deposition of liquid droplets on surfaces, where the droplets, which have no internal order, grow and coalesce [ 12 1. Recent experiments have shown that during deposition of Ag on Si( 111) for 350°C~ T < 450°C the average island size exhibits a power law dependence on the coverage [ 21. It was found that the average linear size of islands ?? is a power of the coverage 0: R N
en,
where 0.2 < n < 0.35, depending on the temperature. In experiments on the growth of Pb monolayers on Cu(OO1) it was found that Pb forms islands of an ordered phase with a large unit cell [ 3 1. Over a broad range of submonolayer coverage it was observed that the average island size
@ 1994 Elsevier Science B.V. All rights reserved
SSDZOO39-6028(94)00110-U
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G.T. Barkema et al. /Surface Science Letters 306 (1994) L569-L574
grows as a power law of the form (1) with n = 0.54 f 0.03. Ernst et al. explored the growth of Cu on Cu (001) using helium beam scattering and observed island growth for substrate temperatures above 160 K [4]. Computer simulations of island growth in the low coverage limit have recently been carried out using a variety of models [ 5-9 ] and simple rate equations which are derived from them [ 5-7,131. In these models atoms are randomly deposited on a square lattice at rate rd and hop between nearest neighbor sites. When an atom hops into a site adjacent to an atom or an island it nucleates to form a new island or aggregates into the existing island, respectively. In Ref. [ 5 ] the low coverage limit was studied using a model in which the island structure is suppressed. More complete models of the diffusion limited aggregation (DLA) type, which give rise to fractal island shapes, were studied in Refs. [ 6,8 1. In the model of Ref. [ 9 ] islands form compact square shapes while the model studied in Ref. [ 7 ] is a solid-on-solid type model. These studies seem to agree that there is a range of coverage in which the island density N during deposition scales like N N (rd/h) 1’3, where rd is the deposition rate and h is the hopping rate of atoms on the surface. In this Letter we present a simulation of island growth of Cu on Cu (00 1) using realistic deposition rates and barriers derived by the atomembedding method. We explore the island density during deposition and find that it increases rapidly in the early stages and then saturates and forms a plateau over a range of the coverage. At higher coverage coalescence effects become dominant and then the number of islands starts to decrease. We find that the formation of plateaus is in agreement with recent experimental studies of homoepitaxial [4] as well as heteroepitaxial growth [ 3,14 1. For our simulations we use a method related to the one proposed by Voter [ 15 1. We use a square lattice for the Cu (001) substrate at T = 350 K, with typical size of 100 x 100 on which we deposit Cu atoms onto random sites at a constant rate. Each atom can then hop into any vacant nearest neighbor site. The hopping rate is given by
h = uexp(-EB/kBT),
where v is the internal vibration
(2)
frequency
and
EB is the height of the energy barrier that the particle needs to cross. The barrier Eg for hopping
from an initial site to a vacant nearest neighbor site depends on the local environment of the atom and particularly on the occupancy of the 10 sites around the initial and final sites. This gives rise to 2 lo = 1024 possibilities. The complete set of barriers we use was obtained earlier [ 161 using the atom-embedding method of Finnis and Sinclair [ 171. The barrier heights range between 0.13 and 1.2 eV while the hopping barrier for an isolated atom is 0.70 eV. Since the spectrum of hopping rates is dominated by the exponential dependence on EB, we used one internal vibration frequency u = lOi Hz for all moves. The single atom activation energy was measured using atom beam scattering and lowenergy ion scattering and was found to be 0.28 f 0.06 and 0.39 f 0.06 eV, respectively [ 4,18 1. The analysis which led to the first value has been recently questioned [ 71. Some calculations [ 191 report that nearest neighbor hopping is the dominant mechanism for diffusion on the surface, while others [20] report that the exchange mechanism proposed by Feibelman [ 2 1 ] for self-diffusion on Al(001) is dominant. Our calculations indicate that the hopping mechanism is dominant, in agreement with Ref. [ 191 and recent LDA-calculations [ 221. The single atom energy barrier from Ref. [ 18 ] agrees with a recent LDA calculation [ 221 but it is 50% higher than EAM or the low energy scattering result [ 181. However, we found that the most important barriers in Ref. [ 161 scale with the same factor to a corresponding set of barriers recently calculated by Karimi using EAM [23]. We use the set of Ref. [ 161 since this is the only complete set available. The properties we study in this Letter are not very sensitive to small changes in the barriers but rather to general features such as edge mobility versus single atom mobility. Therefore, we expect the results not to be strongly affected by the shortcomings of the barrier set that we use. Although ab-initio
G.T. Barkema et al. /Surface Science Letters 306 (1994) L569-LS74
calculations may, in principle, provide more accurate data, obtaining as large a set as we need is computationally out of reach. We observe that the large number of barriers which are lower than the one for isolated atoms are associated with motion towards island edges and along these edges. Atoms hitting island edges hardly ever escape since the energy barriers for leaving are too high, but they hop back and forth along the edges. This indicates that there is enough edge mobility for islands to form compact shapes, which are typically lower in energy. However this causes practical difficulties in the simulations, since the hopping rate along an edge is overwhelmingly larger than for isolated atoms. As a result most of the simulation time is consumed with “useless” moves. To cure this problem we arbitrarily increase all the barriers which are smaller than ET = 0.6 eV in the following way. If En < ET we replace it by EL = E,+a(ET-En).Thelimita = Omeans no change in the barriers while for a = 1 all the barriers smaller than ET are replaced by ET. Increasing a! speeds up the simulation by orders of magnitude which enables us to reach realistic time scales. We explored the results for a variety of values of a and found that at least up to cr = 0.7 there is no significant change in the properties we study here. Therefore, we speeded up the simulation by choosing (Y= 0.7. This choice still leaves enough edge mobility for islands to become compact, while it lowers the frequency of useless moves in which an atom simply goes back and forth along the edge. In the simulation atoms are deposited on the surface at rate rd. They then hop on the surface and form islands. For the temperature and deposition rates reported in this Letter it is very unlikely that a second layer will form on top of the islands. The reason is that in order to form a second layer two or more atoms need to fall on top of the same island and then nucleate before one of them falls down onto the first layer. This is unlikely for low deposition rates, in particular at low coverage when islands are small. We use this property to simplify the simulation by not allowing more than one atom to stay on top of each island at any given time. An atom
0
0.1
0.2
0.3
0.4
Coverage(fraction of ML)
Fig. 1. The number of islands versus coverage averaged over ten runs for a 100x 100 lattice, T = 350 K and the deposition rates (from top to bottom) rd = 1 ML/IO s, 1 ML/20 s, 1 ML/50 s, 1 ML/100 s, 1 ML/200 s, 1 ML/500 s and 1 ML/1000 s. The number of islands N tends to saturate above some coverage and its value at saturation scales like N - ri’3.
that falls on top of an island thus hops randomly until it falls down into the lower layer before another atom is allowed to fall on top of that island. Note that the results presented here are not sensitive to this choice, which is made for computational simplicity. In experiments no second layer was observed prior to the completion of the first layer for T > 160 K [4]. Our results for the number of islands (with two or more atoms) as a function of coverage are shown in Fig. 1 for a variety of deposition rates between 1 ML/ 10 s and 1 ML/ 1000 s (ML = monolayer), at a simulation temperature of 350 K. What we observe is that at low deposition rates the number of islands reaches saturation at relatively low coverage, beyond which no new islands nucleate. The plateau is limited by coalescence that reduces the number of islands. As the deposition rate decreases, the island density at saturation decreases while the plateau becomes broader. Due to the high edge mobility islands form compact shapes as seen in Fig. 2. As the deposition rate is increased, the number of islands increases and the average island size decreases. At the highest deposition rates used, the shape of islands becomes
G.T. Barkema et al. /Surface Science Letters 306 (1994) L569-L574
0
Fig. 2. A snapshot of the islands on the surface for coverage 0 = 0.2 ML, deposition rate rd = 1 ML/100 s and T = 350 K. Note that due to the high edge mobility islands form compact shapes.
a bit more rugged [ 241. The distribution of island sizes on the plateau, forrd = 1 ML/loos, 8 = 0.1 MLand T = 350 K is shown in Fig. 3. It has the typical shape of a gamma distribution decorated by oscillations. These are due to the fact that islands with 3, 5 or 7 atoms contain an atom with only one nearest neighbor and therefore are less stable. From the same runs we obtained the structure factor S( k ) = C, eikrG( r) and the circularly averaged adatom-adatom correlation function G(r) = (n(r’)n(r’ + r)),,, in which n(r) is one for occupied sites and zero otherwise. S (k ) is normalized into the unit interval, after suppression of the specular peak (k = 0). Both S (k ) and G (r ) are depicted in Fig. 4. The peak of S (k ) gives an indication of the separation between large islands [ 4 1. The minimum in G (r ) indicates a depletion zone around an island, where formation of other islands is suppressed. In experiments, diffraction data are typically obtained for intermediate coverage range, where islands are large enough to diffract. Comparing temperatures and deposition rates we conclude that this range is likely to overlap with the range where our simulations show a plateau in the island density. Therefore, our results are consis-
10
20
30
40
50
Island size Fig. 3. The distribution of island sizes on the plateau for rd = 1 ML/100 s, 0 = 0.1 ML and T = 350 K, obtained from 205 simulations on a 100 x 100 lattice. Note that islands with 3, 5 or 7 atoms are less frequent. These islands contain an atom with only one nearest neighbor and therefore are less stable.
tent with the experiment of Pb on Cu(OO1) reported in Ref. [ 3 1, and seem to confirm the interpretation given in that paper that the island density in the experimentally accessible regime is approximately constant. The computational and experimental results can be linked by a simple conservation law for the deposited atoms 8 N iV(e);l(e).
(3)
where N is the island density and 2 is the average island size (number of atoms), which scales like 2 N (R2). Assuming that K N 6” and N(8) N 84 one can show that (R2) N (z)2 [2]. Eq. (3) thus gives rise to the scaling relation [2] q+2n=
1.
(4)
Therefore, on the plateau, where q M 0 one expects n z 0.5 which is in agreement with the experimental results of Ref. [ 31. The relation (4 ) can also be used to characterize the three stages of island growth. In the first stage, when nucleation of new islands is dominant the island density increases and therefore q > 0 and n < l/2. In the second stage the island
G.T. Barkema et al. /Surface Science Letters 306 (1994) L569-L574
and their density thus decreases. 1.0
We would like to thank G.V. Chester and M. Karimi for helpful discussions and Hong Zeng for technical assistance. G.T.B. would like to acknowledge support by NSF grant DMR9121654 through the Material Science Center and the Cornell National Supercomputing Facility. O.B. thanks the NSF for support under grants DMR-9118065 and DMR-9012974 (at Cornell) and DMR- 9217284 (at Syracuse) and G.V. under grant DMR-9119735. M.B. and D.O.B. would like to acknowledge support from the NW0 under the FOM program.
0.6
0.6
0.4
0
5
10
15
20
0.2
0.0 0.0
0.1
0.2
0.3
0.4
0.5
k Fig. 4. Structure factor S(k) = c, e*“G(r) on the plateau for rd = 1 ML,‘100 s, 6 = 0.1 ML and T = 350 K, obtained fmm 205 simulations on a 100 x 100 lattice. The circularly averaged adatom-adatom correlation function G(r) = (n(r’)n(r’ + r)&,, in which n(r) is one for occupied sites and zero otherwise, is shown in the inset. S(k ) is normalized into the unit interval, after suppression of the specular peak (k = 0). r is measured in lattice spacings and k in inverse lattice spacings.
density is approximately constant, which means that q = 0 and n = l/2. In the third stage, when coalescence effects are dominant and the island density decreases, q < 0 and n > l/2. While the first and the third stages have been observed in a variety of systems before, the second growth stage in which the island density is approximately a constant has not been studied theoretically before. In summary, we have explored submonolayer island growth during deposition of Cu on Cu(OO1) and identified three stages. First, for very low coverage, the island density increases due to nucleation of new islands. Subsequently the nucleation process is suppressed and the island density exhibits a plateau over a range of the coverage. These results are in agreement with recent experiments and support their proposed inte~retation [ 3,4]. In the third growth stage, for high coverage, islands tend to coalesce
References [ 1 J K.R. Roos and MC. Tringides, Phys. Rev. B 47 f 1993) 12705. [2] J.-KZuo and J.F.Wendelken, Phys. Rev. Lett. 66 (1991) 2227. [3] W. Li, G. Vidali and 0. Biham, Phys. Rev. B 48 (1993) 8336. [4] H.-J. Ernst, F. Fabre and J. Lapujoulade, Phys. Rev. B 46 (1992) 1929: Phvs. Rev. Lett. 69 (1992) 458: . Surf. Sci. 275 (1992) L682. 1 M.C. Bartelt and J.W. Evans, Phys. Rev. B 46 (1992) 12675; M.C. Bartelt, M.C. Tringides and J.W. Evans, Phys. Rev. B 47 (1993) 13891. L.-II. Tang, J. Phys. (Paris) I 3 (1993) 935. C. Ratsch, A. Zangwill, P. Smilauer and D.D. Vvedensky, preprint; A. Zangwill, in: Evolution of Surface and Thin Film Microstructure, Eds. H.A. Atwater, E. Chason, M. Gabmw and M. Lagally (MRS, Pittsburgh, 1993) p. 121. [ 8 ] J. G. Amar, F. Family and P.-M. Lam, preprint; in: Mechanisms of Thin Film Evolution, Eds. S.M. Yahsove, C.V. Thompson and D.J. Eaglesham, Mat. Res. Sot. Proc., 1994, in press. [9] MC. Bartelt and J.W. Evans, Surf. Sci. 298 (1993) 421. [lo] J. Villain, A. Pimpinelli and D. Wolf, Comments Cond. Mat. Phys. 16 ( 1992) 1. [ 111 In the temperature range of these experiments step Sow growth is suppressed. [ 121 F. Family and P. Meakin, Phys. Rev. Lett. 61 f 1988) 428; P. Meakin, Rep. Prog. Phys. 55 (1992) 157. I
I
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[ 131 0. Biham, G.T. Barkema and M. Breeman, in which
[ 141 [ 151 [ 161 [ 171 [ 181 [ 191
we will describe rate equations that agree with the simulation results, to be submitted. E. Kopatzki, S. Gunther, W. Nichtl-Pecher and R.J. Behm, SurfSci. 284 (1993) 154. A.F. Voter, Phys. Rev. B 34 (1986) 6819. M. Breeman, PhD thesis, Groningen University (1993). M.W. Finnis and J.E. Sinclair, Philos. Mag. A 50 (1984) 45. M. Breeman and D.O. Boerma, Surf. Sci. 269/270 (1992) 224. C.L. Liu, J.M. Cohen, J.B. Adams and A.F. Voter,
Surf. Sci. 253 (1991) 334; D.E. Sanders and A.E. DePristo, Surf. Sci. 260 ( 1992) 116. [20] L. Hansen, P. Stoltze, K.W. Jacobsen and J.K. Nsrskov, Phys. Rev. B 44 (1991) 6523; L.S. Perkins and A.E. DePristo, Surf. Sci. 294 ( 1993) [21] &. Feibelman, Phys. Rev. Lett. 65 (1990) 729. [22] C. Lee, G.T. Barkema, M. Breeman, A. Pasquarello and R. Car, preprint. [23] M. Karimi, private communication. [24] Hong Zeng, M. Karimi, R. Kennett, 0. Biham and G. Vidali, in preparation.
Js!LJ ..I
I
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surface science letters
4._
ELSEVIER
Surface Science Letters 306 (1994) L569-L574
Surface Science Letters
Characterization
of submonolayer growth of Cu islands on Cu(OO1)
G.T. Barkema a, Ofer Biham a~l,M. Breeman b, D.O. Boerma b, Gianfranco VidaWC a Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY 14853-2501, USA b Nuclear Solid State Physics, Materials Science Centre, Groningen University, Groningen, Netherlands c Department of Physics, Syracuse University, Syracuse, NY 13244-1130, USA (Received 7 January 1994; accepted for publication 14 February 1994)
Abstract Submonolayer island growth of Cu on Cu (001) is simulated using energy barriers derived by the atomembedding method of Finnis and Sinclair. We find that the island density during deposition quickly saturates and forms a plateau over a range of the coverage 8. We observe that due to high edge mobility the islands form compact shapes and that the average island size x scales like i? N 0” where n = 0.5, in agreement with recent experiments.
The formation and growth of ordered islands in deposition experiments on metallic substrates has recently attracted considerable attention both experimentally [ l-41 and theoretically [ 5-101. In the experiments atoms are deposited on a clean surface at a constant rate, from zero coverage up to a full monolayer. It has been observed that due to thermal diffusion, and possibly other effects [ 11, the atoms have considerable mobility on the surface which allows them to bunch together into islands of an ordered crystalline phase [ 111. One can identify three basic processes in these systems. At low coverage the deposited atoms hop as random walkers until they nucleate into islands. Additional atoms then aggregate into existing islands, which grow Corresponding author. Fax: + 1 3 15 443 9103; E-mail: [email protected]. ’ On leave from the Department of Physics, Syracuse University, Syracuse, NY 13244, USA. l
0039-6028/94/$07.00
as the coverage increases. At high coverage coalescence effects become important, where two or more islands merge into one large island. Related phenomena appear in deposition of liquid droplets on surfaces, where the droplets, which have no internal order, grow and coalesce [ 12 1. Recent experiments have shown that during deposition of Ag on Si( 111) for 350°C~ T < 450°C the average island size exhibits a power law dependence on the coverage [ 21. It was found that the average linear size of islands ?? is a power of the coverage 0: R N
en,
where 0.2 < n < 0.35, depending on the temperature. In experiments on the growth of Pb monolayers on Cu(OO1) it was found that Pb forms islands of an ordered phase with a large unit cell [ 3 1. Over a broad range of submonolayer coverage it was observed that the average island size
@ 1994 Elsevier Science B.V. All rights reserved
SSDZOO39-6028(94)00110-U
(1)
G.T. Barkema et al. /Surface Science Letters 306 (1994) L569-L574
grows as a power law of the form (1) with n = 0.54 f 0.03. Ernst et al. explored the growth of Cu on Cu (001) using helium beam scattering and observed island growth for substrate temperatures above 160 K [4]. Computer simulations of island growth in the low coverage limit have recently been carried out using a variety of models [ 5-9 ] and simple rate equations which are derived from them [ 5-7,131. In these models atoms are randomly deposited on a square lattice at rate rd and hop between nearest neighbor sites. When an atom hops into a site adjacent to an atom or an island it nucleates to form a new island or aggregates into the existing island, respectively. In Ref. [ 5 ] the low coverage limit was studied using a model in which the island structure is suppressed. More complete models of the diffusion limited aggregation (DLA) type, which give rise to fractal island shapes, were studied in Refs. [ 6,8 1. In the model of Ref. [ 9 ] islands form compact square shapes while the model studied in Ref. [ 7 ] is a solid-on-solid type model. These studies seem to agree that there is a range of coverage in which the island density N during deposition scales like N N (rd/h) 1’3, where rd is the deposition rate and h is the hopping rate of atoms on the surface. In this Letter we present a simulation of island growth of Cu on Cu (00 1) using realistic deposition rates and barriers derived by the atomembedding method. We explore the island density during deposition and find that it increases rapidly in the early stages and then saturates and forms a plateau over a range of the coverage. At higher coverage coalescence effects become dominant and then the number of islands starts to decrease. We find that the formation of plateaus is in agreement with recent experimental studies of homoepitaxial [4] as well as heteroepitaxial growth [ 3,14 1. For our simulations we use a method related to the one proposed by Voter [ 15 1. We use a square lattice for the Cu (001) substrate at T = 350 K, with typical size of 100 x 100 on which we deposit Cu atoms onto random sites at a constant rate. Each atom can then hop into any vacant nearest neighbor site. The hopping rate is given by
h = uexp(-EB/kBT),
where v is the internal vibration
(2)
frequency
and
EB is the height of the energy barrier that the particle needs to cross. The barrier Eg for hopping
from an initial site to a vacant nearest neighbor site depends on the local environment of the atom and particularly on the occupancy of the 10 sites around the initial and final sites. This gives rise to 2 lo = 1024 possibilities. The complete set of barriers we use was obtained earlier [ 161 using the atom-embedding method of Finnis and Sinclair [ 171. The barrier heights range between 0.13 and 1.2 eV while the hopping barrier for an isolated atom is 0.70 eV. Since the spectrum of hopping rates is dominated by the exponential dependence on EB, we used one internal vibration frequency u = lOi Hz for all moves. The single atom activation energy was measured using atom beam scattering and lowenergy ion scattering and was found to be 0.28 f 0.06 and 0.39 f 0.06 eV, respectively [ 4,18 1. The analysis which led to the first value has been recently questioned [ 71. Some calculations [ 191 report that nearest neighbor hopping is the dominant mechanism for diffusion on the surface, while others [20] report that the exchange mechanism proposed by Feibelman [ 2 1 ] for self-diffusion on Al(001) is dominant. Our calculations indicate that the hopping mechanism is dominant, in agreement with Ref. [ 191 and recent LDA-calculations [ 221. The single atom energy barrier from Ref. [ 18 ] agrees with a recent LDA calculation [ 221 but it is 50% higher than EAM or the low energy scattering result [ 181. However, we found that the most important barriers in Ref. [ 161 scale with the same factor to a corresponding set of barriers recently calculated by Karimi using EAM [23]. We use the set of Ref. [ 161 since this is the only complete set available. The properties we study in this Letter are not very sensitive to small changes in the barriers but rather to general features such as edge mobility versus single atom mobility. Therefore, we expect the results not to be strongly affected by the shortcomings of the barrier set that we use. Although ab-initio
G.T. Barkema et al. /Surface Science Letters 306 (1994) L569-LS74
calculations may, in principle, provide more accurate data, obtaining as large a set as we need is computationally out of reach. We observe that the large number of barriers which are lower than the one for isolated atoms are associated with motion towards island edges and along these edges. Atoms hitting island edges hardly ever escape since the energy barriers for leaving are too high, but they hop back and forth along the edges. This indicates that there is enough edge mobility for islands to form compact shapes, which are typically lower in energy. However this causes practical difficulties in the simulations, since the hopping rate along an edge is overwhelmingly larger than for isolated atoms. As a result most of the simulation time is consumed with “useless” moves. To cure this problem we arbitrarily increase all the barriers which are smaller than ET = 0.6 eV in the following way. If En < ET we replace it by EL = E,+a(ET-En).Thelimita = Omeans no change in the barriers while for a = 1 all the barriers smaller than ET are replaced by ET. Increasing a! speeds up the simulation by orders of magnitude which enables us to reach realistic time scales. We explored the results for a variety of values of a and found that at least up to cr = 0.7 there is no significant change in the properties we study here. Therefore, we speeded up the simulation by choosing (Y= 0.7. This choice still leaves enough edge mobility for islands to become compact, while it lowers the frequency of useless moves in which an atom simply goes back and forth along the edge. In the simulation atoms are deposited on the surface at rate rd. They then hop on the surface and form islands. For the temperature and deposition rates reported in this Letter it is very unlikely that a second layer will form on top of the islands. The reason is that in order to form a second layer two or more atoms need to fall on top of the same island and then nucleate before one of them falls down onto the first layer. This is unlikely for low deposition rates, in particular at low coverage when islands are small. We use this property to simplify the simulation by not allowing more than one atom to stay on top of each island at any given time. An atom
0
0.1
0.2
0.3
0.4
Coverage(fraction of ML)
Fig. 1. The number of islands versus coverage averaged over ten runs for a 100x 100 lattice, T = 350 K and the deposition rates (from top to bottom) rd = 1 ML/IO s, 1 ML/20 s, 1 ML/50 s, 1 ML/100 s, 1 ML/200 s, 1 ML/500 s and 1 ML/1000 s. The number of islands N tends to saturate above some coverage and its value at saturation scales like N - ri’3.
that falls on top of an island thus hops randomly until it falls down into the lower layer before another atom is allowed to fall on top of that island. Note that the results presented here are not sensitive to this choice, which is made for computational simplicity. In experiments no second layer was observed prior to the completion of the first layer for T > 160 K [4]. Our results for the number of islands (with two or more atoms) as a function of coverage are shown in Fig. 1 for a variety of deposition rates between 1 ML/ 10 s and 1 ML/ 1000 s (ML = monolayer), at a simulation temperature of 350 K. What we observe is that at low deposition rates the number of islands reaches saturation at relatively low coverage, beyond which no new islands nucleate. The plateau is limited by coalescence that reduces the number of islands. As the deposition rate decreases, the island density at saturation decreases while the plateau becomes broader. Due to the high edge mobility islands form compact shapes as seen in Fig. 2. As the deposition rate is increased, the number of islands increases and the average island size decreases. At the highest deposition rates used, the shape of islands becomes
G.T. Barkema et al. /Surface Science Letters 306 (1994) L569-L574
0
Fig. 2. A snapshot of the islands on the surface for coverage 0 = 0.2 ML, deposition rate rd = 1 ML/100 s and T = 350 K. Note that due to the high edge mobility islands form compact shapes.
a bit more rugged [ 241. The distribution of island sizes on the plateau, forrd = 1 ML/loos, 8 = 0.1 MLand T = 350 K is shown in Fig. 3. It has the typical shape of a gamma distribution decorated by oscillations. These are due to the fact that islands with 3, 5 or 7 atoms contain an atom with only one nearest neighbor and therefore are less stable. From the same runs we obtained the structure factor S( k ) = C, eikrG( r) and the circularly averaged adatom-adatom correlation function G(r) = (n(r’)n(r’ + r)),,, in which n(r) is one for occupied sites and zero otherwise. S (k ) is normalized into the unit interval, after suppression of the specular peak (k = 0). Both S (k ) and G (r ) are depicted in Fig. 4. The peak of S (k ) gives an indication of the separation between large islands [ 4 1. The minimum in G (r ) indicates a depletion zone around an island, where formation of other islands is suppressed. In experiments, diffraction data are typically obtained for intermediate coverage range, where islands are large enough to diffract. Comparing temperatures and deposition rates we conclude that this range is likely to overlap with the range where our simulations show a plateau in the island density. Therefore, our results are consis-
10
20
30
40
50
Island size Fig. 3. The distribution of island sizes on the plateau for rd = 1 ML/100 s, 0 = 0.1 ML and T = 350 K, obtained from 205 simulations on a 100 x 100 lattice. Note that islands with 3, 5 or 7 atoms are less frequent. These islands contain an atom with only one nearest neighbor and therefore are less stable.
tent with the experiment of Pb on Cu(OO1) reported in Ref. [ 3 1, and seem to confirm the interpretation given in that paper that the island density in the experimentally accessible regime is approximately constant. The computational and experimental results can be linked by a simple conservation law for the deposited atoms 8 N iV(e);l(e).
(3)
where N is the island density and 2 is the average island size (number of atoms), which scales like 2 N (R2). Assuming that K N 6” and N(8) N 84 one can show that (R2) N (z)2 [2]. Eq. (3) thus gives rise to the scaling relation [2] q+2n=
1.
(4)
Therefore, on the plateau, where q M 0 one expects n z 0.5 which is in agreement with the experimental results of Ref. [ 31. The relation (4 ) can also be used to characterize the three stages of island growth. In the first stage, when nucleation of new islands is dominant the island density increases and therefore q > 0 and n < l/2. In the second stage the island
G.T. Barkema et al. /Surface Science Letters 306 (1994) L569-L574
and their density thus decreases. 1.0
We would like to thank G.V. Chester and M. Karimi for helpful discussions and Hong Zeng for technical assistance. G.T.B. would like to acknowledge support by NSF grant DMR9121654 through the Material Science Center and the Cornell National Supercomputing Facility. O.B. thanks the NSF for support under grants DMR-9118065 and DMR-9012974 (at Cornell) and DMR- 9217284 (at Syracuse) and G.V. under grant DMR-9119735. M.B. and D.O.B. would like to acknowledge support from the NW0 under the FOM program.
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k Fig. 4. Structure factor S(k) = c, e*“G(r) on the plateau for rd = 1 ML,‘100 s, 6 = 0.1 ML and T = 350 K, obtained fmm 205 simulations on a 100 x 100 lattice. The circularly averaged adatom-adatom correlation function G(r) = (n(r’)n(r’ + r)&,, in which n(r) is one for occupied sites and zero otherwise, is shown in the inset. S(k ) is normalized into the unit interval, after suppression of the specular peak (k = 0). r is measured in lattice spacings and k in inverse lattice spacings.
density is approximately constant, which means that q = 0 and n = l/2. In the third stage, when coalescence effects are dominant and the island density decreases, q < 0 and n > l/2. While the first and the third stages have been observed in a variety of systems before, the second growth stage in which the island density is approximately a constant has not been studied theoretically before. In summary, we have explored submonolayer island growth during deposition of Cu on Cu(OO1) and identified three stages. First, for very low coverage, the island density increases due to nucleation of new islands. Subsequently the nucleation process is suppressed and the island density exhibits a plateau over a range of the coverage. These results are in agreement with recent experiments and support their proposed inte~retation [ 3,4]. In the third growth stage, for high coverage, islands tend to coalesce
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[ 131 0. Biham, G.T. Barkema and M. Breeman, in which
[ 141 [ 151 [ 161 [ 171 [ 181 [ 191
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