- Email: [email protected]
Finite size effects in nucleation processes
surface science Surface Science 355 (1996) L259-L263
ELSEVIER
Surface Science Letters
Finite size effects in nucleation processes K.R. Roos a, M.C. Tringides b,. a Physics Department, Bradley University b Ames Laboratory and Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA Received 15 September 1995;accepted for publication 10 February 1996
Abstract
We have used Monte Carlo simulations to study the scaling relations describing submonolayer nucleation NocP/DxOz as a function of system size L (where N is the nucleated island density, D the diffusion coefficient, F the flux rate, 0 the coverage). For island size distributions that obey scaling, the island density N can be easily related to the step density, SD. A simple model of critical size cluster ic= 1 is used. For a given system size L there is a minimum (DIE)rain ratio where scaling breaks down with x=y=0. These considerations are relevant to systems with low terrace diffusion barrier or small substrate terraces. Keywords: Computer simulations; Molecular beam epitaxy; Nucleation; Surface diffusion
The morphology of epitaxially grown ultrathin films, described by the island size distribution is expected [ 1] to follow simple relations as a function of the growth parameters, diffusion coefficient D, flux rate F, and deposited coverage 0. The stable island density for sizes above a critical size (islands with i>ic do not dissociate) in the steady state regime obeys a scaling relation: N oc Fr/DxO ~,
( 1)
where x,y,z are universal scaling exponents that depend only on ic. This simple relation has been used as a general method to measure the terrace diffusion activation energy, Et, from the temperature-dependence of the nucleated island density N ( T ) and the exponent x. The single atom diffusion activation energy, E t , c a n be simply determined *Corresponding author. Fax: +1 515 2940689; e-mail: [email protected]. 0039-6028/96/$15.00 © 1996Elsevier ScienceB.V. All rights reserved PII S0039-6028 (96) 00610-3
from the measured activation energy of the island density, EN, and the scaling exponent x, E t = EN/X, with the exponent x measured from the flux rate variation since nucleation theory predicts x = y . There is a characteristic range for the ratios D / F where this scaling relation is valid. For D/F ratios below this range, diffusion is so low that a large number of small islands (of 2-3 atoms) nucleate, while for large D/F values, the number of nucleated islands is so small that it is statistically inaccurate. Since in real systems, terraces have limited size, L (500-4000 ,A), depending on the method of preparation, in this Letter we would like to study the regime of large D / F ratios and how the scaling relation is modified because of finite size effects. In this regime, the number of nucleated islands on terraces is so low that it cannot be used to test the scaling relation, Eq. (1), which requires at least a variation of the island density by some minimum factor. The system is not yet in the step flow
K.R. Roos, M. C. Tringides/SurJiTceScience355 (1996) L259 L263 should expect the breakdown of Eq. ( 1 ) and no information can be obtained about the terrace diffusion coefficient. The incoming atoms can either reach the edges of the islands or the few islands that have nucleated on the terraces. If the fraction of atoms joining the islands can be measured, this can be related to the number of atoms hopping over descending steps and can provide a new method of measuring the step edge barrier at descending steps. These effects should be relevant in systems with small terrace diffusion barrier (Et<0.2eV) at room temperature or in substrates prepared initially with small average terraces. Finite size effects have been previously studied on flat [2] and stepped surfaces [3,4]. The low island density regime has been identified in Ref. [2] on square lattices and with periodic boundary conditions. The main goal in Ref. [2] was to vary the system size for fixed D/F ratio and explain why the scaling exponents deduced from the simulations (for ic = 2) are lower than the predicted ones based on nucleation theory (this is explained by the smaller island density measured in the simulations because of finite size effects). Our study has focused on the role of finite size effects as D/F increases, which experimentally corresponds to growth experiments at higher temperatures. The role of finite size effects on stepped surfaces is different because the island density is limited only in the direction normal to the steps. This was clearly shown in Ref. [4] where the step density was calculated during epitaxial growth both parallel and normal to the step direction. As will be seen shortly, the step density (i.e., island perimeter) can be easily related to the island density. In the direction normal to the steps, a sharp transition to a constant step density at sufficiently low flux rates (i.e., high D/F ratios) was found; in the direction parallel to the steps, the step density variation exhibits an intermediate regime before saturation, with the scaling exponents higher than the ones predicted by Eq.(1). In Ref. [3], a generalized form of scaling for the full island size distribution was derived on stepped surfaces, which incorporates both the normal scaling regime of Eq. (1) (for large terrace widths, low D/F ratios, high coverage) and the regime where capture of atoms by steps is
the dominant aggregation process (for small terrace widths, high D/F ratios, low coverage). Our study is closer to Ref. 1-2] than Refs. I-3,4] not only because we have used flat rather than stepped surfaces but because we are interested in the breakdown of the scaling equation as a result of the competition between capture of atoms by steps versus islands, and not in deriving a modified version of scaling. Previous simulations [5] modeling the deposition process have confirmed the scaling relationship for irreversible growth (i.e., critical size io= 1) and the scaling of the domain size distribution P(s,t) (i.e., the same functional form describes the film morphology, where P(s,t) is the fraction of islands of size s), with coverage (for fixed D/F ratio) or with the ratio D/F (for fixed 0). Most of the simulations carried out so far have used a "normal" window, 105
SDocf P(s,t)s'/2ds=N~/2fo~ P(x)x~/2dx,
(2)
where x = s/g=(const )O1/2N1/2, since g = O1/2/N1/2. For non-compact island size distributions that obey scaling, an analogous relation is derived but with the exponent relating island to step density depending on the fractal dimensionality of the island. The program follows the standard algorithm of interrogating, on the average, each monomer D/F
KR. Roos, M. C Tringides/Surface Science 355 (1996) L259 L263
times to decide whether it diffuses to a neighboring site before a new atom is deposited on the surface. If a monomer encounters an island it joins it irreversibly and is removed from the list of monomers to be interrogated further. At different intervals of time we have calculated the full island size distribution, P (s,t). We have assumed compact islands, formed by instantaneous island restructuring into circular shapes, which is equivalent to a zero edge diffusion barrier, E¢. All conclusions about the role of finite size effects apply equally well to other island shapes and finite edge diffusion. We have verified several well-known results of the model which corresponds to critical size cluster ic = 1. Fig. 1 shows the step density S D v e r s u s 0 for fixed D/F= 106. It can be fitted to a parabolic dependence on coverage (,,~01/2), which implies a constant island density z = 0, as can be seen from Eq. (2). Deviations from the parabolic dependence for 0 >0.4, towards smaller island step densities, signal the onset of coalescence, since the perimeter of the resulting island is smaller than the sum of the separate perimeters after two islands coalesce. Fig. 2a shows the different size distributions, P(s,t), obtained at fixed 0=0.1 and different D/F ratios. The scaled data, P(s,t) versus s/g, are shown in Fig. 2b, verifying the time invariance of the distribution into a sharply peaked function. Fig. 3 shows the measured step density, SD,
0.04
D/F=10
6
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•
•
./ ,
011
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°
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:
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D/F=108 D/F=109
300
400
.
0.01
106
D/F=107
0.00 100
0
200
500
600
S
(a) 1.5
0.0 .
D/F=106
D / F = 107 D / F = 108 D/F=109
~,
~
~ i 1 ~
•
r
~
1.0
^
•
~
k
,
, 1
, 2
S/Say e
(b) Fig. 2. (a) Island size distribution, N~, versus s obtained at fixed coverage, 0=0.1, for a range of D/F ratios, 106-109, for a model with irreversible growth ic= 1. (b) Scaled data Ns/N versus s/s,v, where N is the total island density ( N = E Ns) s>! and say the average island size sav=(E sNs)/N.
e
0.02
0.00 0.0
0.03
014
0 Fig. 1. The step density, SD, versus deposited coverage, 0, for fixed D/F= 106, obtained with Monte Carlo simulations in a model with irreversible growth (ic= 1). It can be fitted to 0 ~/2 dependence, which implies constant island density according to Eq. (2).
versus D/F for fixed coverage 0 = 0.1 and different system sizes, L = 20-200. Our simulations span 16 orders of magnitude in the variation of the D/F ratio, well below and well above, the "normal" window 5
K.R. Roos, M. C. Tringides/Surface Science 355 (1996) L259-L263
|
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log(D/F) Fig. 3. Log (St)) versus log D/F where S D is the step density for a very wide range in D/F ratios, 10 2-1014, and different lattice sizes. L = 2 0 (~), 30 (O), 40 (A), 50 ( i ) , 60 (+), 70 (O), 80 (x), 90 ( . ) , 100 (V), 140 (*), 200 ([]). Three regimes are observed: for low D/F, the step density is very high because only small islands are formed; this is followed by the scaling x = 1/3 regime, x = 1/3, and finally the regime where finite size effects, x = y = 0, become important.
decreasing in a power law dependence on (D/F) -1/6, as expected from Eq. (2) relating S D to N and the exponent x = 1/3 (for the case ic= 1). The island densities in the scaling regime take the same values for different lattice sizes. This provides another consistency test of the program, since a smaller lattice can be thought of as a subsection of the larger one and the island density is uniform over the whole lattice. Finally, a third regime is present at low SD values (i.e., low density because of the high diffusion), which is independent of the ratio D/F with exponents x = y = 0 . This is the regime where diffusion is so high that very few 1-2 islands have nucleated in the middle of the terrace. It is the same regime that finite size effects become important in Ref. I-2]. Eventually, at even higher D/F ratios, all incoming atoms will be captured by the steps (if non-periodic boundary conditions discussed later are used) with the step density remaining constant, without any islands nucleating on the terrace. The characteristic ratio (D/F)min, where finite size effects become important, depends on the system size we have used. As L increases (D/F)min also increases. For (D/F)rain = 101° a value of L = 300 was found in Ref. [2] with one island present, SD
106
i
107
........
i
108
........
i
109
........
i
........
1010
i
,
.,a
1011 '-1'012.
i iH.
1013
(D/F)mi n Fig. 4. Plot of the lattice size L versus (D/F)~, obtained from Fig. 3. (D/F)mi, is determined from the minimum value of the ratio where finite size effects (1-2 islands present) become important. The scaling exponent is -1/6, consistent with the expected exponent, x=y= 1/3, for the island density. Such a relation can be used to relate the onset of finite size effects in experiments performedon substrates with different terrace sizes. in good agreement with our results. A coarse graining algorithm was used in Ref. 1,2] for computational efficiency. Fig. 4 shows a plot of the system size L ~ (D/F)miln/6, indicating that it follows a power law with a scaling exponent 1/6. This can be understood in terms of the island separation that, once it becomes comparable to the size of the system, L, the scaling Eq. (1) breaks down. Since the island separation scales with an exponent - x / 2 , this explains why the scaling exponent in Fig. 4 is - 1 / 6 , half the value of x for it---1. In Ref. I-3], it was found that the nucleated island density, N, on stepped surfaces of terrace width L, in the regime where step capture is significantly important, obeys the modified scaling equation N oc (D/F)- 1/3f(w) where w = (O/F)- 2/30]~. In the limit of high D/F ratios f ( w ) ~ w , one can simply derive the relation, Loc (D/F)miln/3 at fixed coverage, which is consistent with the result of Fig. 4. In Ref. [3], only the island separation normal to the steps is bounded so the island density scales NoeL -1 with the bounded terrace width (instead of the relation NocL -2 for terraces bounded in both directions). It provides an additional consistency test of the program but it can have practical value as well: experimental studies carried out on substrates with different average terraces can be
K.R. Roos, M. (2 Tringides/ Surface Science 355 (1996) L259-L263
easily compared to determine where finite size effects become important in one study, if finite size effects are already determined in the other study. Our results can first be used to determine the temperature range where scaling measurements according to Eq. (1) can be safely carried out. For Ag(lll), which has a low terrace diffusion, Et= 0.1 eV E7] for typical substrate sizes L ~ 1 0 0 0 A growth experiments can be carried out at a maximum temperature T = 150 K for typical flux rates F=0.01ML/s, as can be seen from Fig. 3. Experiments at higher temperatures should result in the formation of very few islands and zero scaling exponents x = y = 0 . We have carried out our simulations for ic = 1 but we expect that the crossover values (D/F)min t o be weakly dependent o n ie.
In the simulations, we have used periodic boundary conditions because they are commonly used in the literature [2,6] so it would be easier to evaluate the importance of finite size effects. On a real system, boundary effects are expected to be different along different terrace edges, depending on whether neighboring terraces are separated by an ascending or a descending step. In this postscaling regime where very few islands are present on the terrace, a deposited atom eventually is captured by either an ascending step, the nucleated island or the descending step. Deposition experiments at constant substrate temperature as a function of flux rate should result in flux-independent growth, y = 0, as observed in the simulations, since the diffusion length is so large that the distribution of atoms towards ascending, descending steps, and the island remains the same. However, if the temperature is varied, the number of atoms falling off the descending step is expected to increase since the probability of crossing the step barrier, Es, is thermally activated. This additional barrier is responsible for confining the atoms on the terrace in the first place but it is also important in determining the type of multilayer growth, whether 2D (for small Es) or 3D (for large E~), by controlling interlayer diffusion. Experiments in this regime, measuring the nucleated island step density SD as
a function of temperature, can be use, directly a rate, Em, which is monotomcally relateo to the step edge barrier, Es: the size of the island (and So) will be reduced as more atoms hop over a descending step. This has the potential of measuring the step edge barrier in the submonolayer regime [6]. In summary, we have studied how the scaling relation of nucleation, Eq. (1), breaks down for sufficiently high D/F ratios or small substrate terraces by measuring the step density SD, which is probed in RHEED diffraction experiments. We have used Monte Carlo simulations on the simple model of irreversible growth (ic = 1) to observe the crossover region as the scaling exponents, x,y, change from 1/3 to 0. In this region, very few islands nucleate on the terrace since the diffusion length is sufficiently high. These effects are relevant for the Ag/Ag(111) system, which has a low terrace diffusion barrier, E t = 0.1 eV.
Acknowledgements Ames Laboratory is operated by the US Department of Energy by Iowa State University under Contract No. W-7405-Eng-82. This work was supported by the Director for Energy Research, Office of Basic Energy Sciences.
References [1] A. Zangwill, in: Evolution of Surface and Thin Films Microstructure, Eds. H.A. Atwater, E.H. Chason, M.L. Grabow and M.G. Lagally (Mater. Res. Soc. Sympos. Proc. 121 (1993)). [2] M. Schroeder and D.E. Wolf, Phys. Rev. Lett. 74 (1995) 2062. [3] G.S. Bales, preprint (1995). [-4] A. Pimpinelli and P. Peyla, preprint (1995). [-5] M.C. Bartelt and J.W. Evans, Phys. Rev. B (1992) 12675. [-6] M. Stanley, C. Papageorgopoulos, K.R. Roos and M.C. Tringides, Surf. Sci. 355 (1996) L264. [7] K. Bromann, H. Brune, H. Roder and K. Kern, Phys. Rev. Lett. 75 (1995) 677.
ELSEVIER
Surface Science Letters
Finite size effects in nucleation processes K.R. Roos a, M.C. Tringides b,. a Physics Department, Bradley University b Ames Laboratory and Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA Received 15 September 1995;accepted for publication 10 February 1996
Abstract
We have used Monte Carlo simulations to study the scaling relations describing submonolayer nucleation NocP/DxOz as a function of system size L (where N is the nucleated island density, D the diffusion coefficient, F the flux rate, 0 the coverage). For island size distributions that obey scaling, the island density N can be easily related to the step density, SD. A simple model of critical size cluster ic= 1 is used. For a given system size L there is a minimum (DIE)rain ratio where scaling breaks down with x=y=0. These considerations are relevant to systems with low terrace diffusion barrier or small substrate terraces. Keywords: Computer simulations; Molecular beam epitaxy; Nucleation; Surface diffusion
The morphology of epitaxially grown ultrathin films, described by the island size distribution is expected [ 1] to follow simple relations as a function of the growth parameters, diffusion coefficient D, flux rate F, and deposited coverage 0. The stable island density for sizes above a critical size (islands with i>ic do not dissociate) in the steady state regime obeys a scaling relation: N oc Fr/DxO ~,
( 1)
where x,y,z are universal scaling exponents that depend only on ic. This simple relation has been used as a general method to measure the terrace diffusion activation energy, Et, from the temperature-dependence of the nucleated island density N ( T ) and the exponent x. The single atom diffusion activation energy, E t , c a n be simply determined *Corresponding author. Fax: +1 515 2940689; e-mail: [email protected]. 0039-6028/96/$15.00 © 1996Elsevier ScienceB.V. All rights reserved PII S0039-6028 (96) 00610-3
from the measured activation energy of the island density, EN, and the scaling exponent x, E t = EN/X, with the exponent x measured from the flux rate variation since nucleation theory predicts x = y . There is a characteristic range for the ratios D / F where this scaling relation is valid. For D/F ratios below this range, diffusion is so low that a large number of small islands (of 2-3 atoms) nucleate, while for large D/F values, the number of nucleated islands is so small that it is statistically inaccurate. Since in real systems, terraces have limited size, L (500-4000 ,A), depending on the method of preparation, in this Letter we would like to study the regime of large D / F ratios and how the scaling relation is modified because of finite size effects. In this regime, the number of nucleated islands on terraces is so low that it cannot be used to test the scaling relation, Eq. (1), which requires at least a variation of the island density by some minimum factor. The system is not yet in the step flow
K.R. Roos, M. C. Tringides/SurJiTceScience355 (1996) L259 L263 should expect the breakdown of Eq. ( 1 ) and no information can be obtained about the terrace diffusion coefficient. The incoming atoms can either reach the edges of the islands or the few islands that have nucleated on the terraces. If the fraction of atoms joining the islands can be measured, this can be related to the number of atoms hopping over descending steps and can provide a new method of measuring the step edge barrier at descending steps. These effects should be relevant in systems with small terrace diffusion barrier (Et<0.2eV) at room temperature or in substrates prepared initially with small average terraces. Finite size effects have been previously studied on flat [2] and stepped surfaces [3,4]. The low island density regime has been identified in Ref. [2] on square lattices and with periodic boundary conditions. The main goal in Ref. [2] was to vary the system size for fixed D/F ratio and explain why the scaling exponents deduced from the simulations (for ic = 2) are lower than the predicted ones based on nucleation theory (this is explained by the smaller island density measured in the simulations because of finite size effects). Our study has focused on the role of finite size effects as D/F increases, which experimentally corresponds to growth experiments at higher temperatures. The role of finite size effects on stepped surfaces is different because the island density is limited only in the direction normal to the steps. This was clearly shown in Ref. [4] where the step density was calculated during epitaxial growth both parallel and normal to the step direction. As will be seen shortly, the step density (i.e., island perimeter) can be easily related to the island density. In the direction normal to the steps, a sharp transition to a constant step density at sufficiently low flux rates (i.e., high D/F ratios) was found; in the direction parallel to the steps, the step density variation exhibits an intermediate regime before saturation, with the scaling exponents higher than the ones predicted by Eq.(1). In Ref. [3], a generalized form of scaling for the full island size distribution was derived on stepped surfaces, which incorporates both the normal scaling regime of Eq. (1) (for large terrace widths, low D/F ratios, high coverage) and the regime where capture of atoms by steps is
the dominant aggregation process (for small terrace widths, high D/F ratios, low coverage). Our study is closer to Ref. 1-2] than Refs. I-3,4] not only because we have used flat rather than stepped surfaces but because we are interested in the breakdown of the scaling equation as a result of the competition between capture of atoms by steps versus islands, and not in deriving a modified version of scaling. Previous simulations [5] modeling the deposition process have confirmed the scaling relationship for irreversible growth (i.e., critical size io= 1) and the scaling of the domain size distribution P(s,t) (i.e., the same functional form describes the film morphology, where P(s,t) is the fraction of islands of size s), with coverage (for fixed D/F ratio) or with the ratio D/F (for fixed 0). Most of the simulations carried out so far have used a "normal" window, 105
SDocf P(s,t)s'/2ds=N~/2fo~ P(x)x~/2dx,
(2)
where x = s/g=(const )O1/2N1/2, since g = O1/2/N1/2. For non-compact island size distributions that obey scaling, an analogous relation is derived but with the exponent relating island to step density depending on the fractal dimensionality of the island. The program follows the standard algorithm of interrogating, on the average, each monomer D/F
KR. Roos, M. C Tringides/Surface Science 355 (1996) L259 L263
times to decide whether it diffuses to a neighboring site before a new atom is deposited on the surface. If a monomer encounters an island it joins it irreversibly and is removed from the list of monomers to be interrogated further. At different intervals of time we have calculated the full island size distribution, P (s,t). We have assumed compact islands, formed by instantaneous island restructuring into circular shapes, which is equivalent to a zero edge diffusion barrier, E¢. All conclusions about the role of finite size effects apply equally well to other island shapes and finite edge diffusion. We have verified several well-known results of the model which corresponds to critical size cluster ic = 1. Fig. 1 shows the step density S D v e r s u s 0 for fixed D/F= 106. It can be fitted to a parabolic dependence on coverage (,,~01/2), which implies a constant island density z = 0, as can be seen from Eq. (2). Deviations from the parabolic dependence for 0 >0.4, towards smaller island step densities, signal the onset of coalescence, since the perimeter of the resulting island is smaller than the sum of the separate perimeters after two islands coalesce. Fig. 2a shows the different size distributions, P(s,t), obtained at fixed 0=0.1 and different D/F ratios. The scaled data, P(s,t) versus s/g, are shown in Fig. 2b, verifying the time invariance of the distribution into a sharply peaked function. Fig. 3 shows the measured step density, SD,
0.04
D/F=10
6
~ 01/2 o
•
•
•
./ ,
011
'
0~.2
'
013
'
O--O.1 D/F
°
0.02
:
~
k
• ~ ~
D/F=108 D/F=109
300
400
.
0.01
106
D/F=107
0.00 100
0
200
500
600
S
(a) 1.5
0.0 .
D/F=106
D / F = 107 D / F = 108 D/F=109
~,
~
~ i 1 ~
•
r
~
1.0
^
•
~
k
,
, 1
, 2
S/Say e
(b) Fig. 2. (a) Island size distribution, N~, versus s obtained at fixed coverage, 0=0.1, for a range of D/F ratios, 106-109, for a model with irreversible growth ic= 1. (b) Scaled data Ns/N versus s/s,v, where N is the total island density ( N = E Ns) s>! and say the average island size sav=(E sNs)/N.
e
0.02
0.00 0.0
0.03
014
0 Fig. 1. The step density, SD, versus deposited coverage, 0, for fixed D/F= 106, obtained with Monte Carlo simulations in a model with irreversible growth (ic= 1). It can be fitted to 0 ~/2 dependence, which implies constant island density according to Eq. (2).
versus D/F for fixed coverage 0 = 0.1 and different system sizes, L = 20-200. Our simulations span 16 orders of magnitude in the variation of the D/F ratio, well below and well above, the "normal" window 5
K.R. Roos, M. C. Tringides/Surface Science 355 (1996) L259-L263
|
-1.5
ill
0
|
t ,/ y
100
I
#
#
*
*
#
#
#
/
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i
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-2.5
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5
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........
1~0
15
log(D/F) Fig. 3. Log (St)) versus log D/F where S D is the step density for a very wide range in D/F ratios, 10 2-1014, and different lattice sizes. L = 2 0 (~), 30 (O), 40 (A), 50 ( i ) , 60 (+), 70 (O), 80 (x), 90 ( . ) , 100 (V), 140 (*), 200 ([]). Three regimes are observed: for low D/F, the step density is very high because only small islands are formed; this is followed by the scaling x = 1/3 regime, x = 1/3, and finally the regime where finite size effects, x = y = 0, become important.
decreasing in a power law dependence on (D/F) -1/6, as expected from Eq. (2) relating S D to N and the exponent x = 1/3 (for the case ic= 1). The island densities in the scaling regime take the same values for different lattice sizes. This provides another consistency test of the program, since a smaller lattice can be thought of as a subsection of the larger one and the island density is uniform over the whole lattice. Finally, a third regime is present at low SD values (i.e., low density because of the high diffusion), which is independent of the ratio D/F with exponents x = y = 0 . This is the regime where diffusion is so high that very few 1-2 islands have nucleated in the middle of the terrace. It is the same regime that finite size effects become important in Ref. I-2]. Eventually, at even higher D/F ratios, all incoming atoms will be captured by the steps (if non-periodic boundary conditions discussed later are used) with the step density remaining constant, without any islands nucleating on the terrace. The characteristic ratio (D/F)min, where finite size effects become important, depends on the system size we have used. As L increases (D/F)min also increases. For (D/F)rain = 101° a value of L = 300 was found in Ref. [2] with one island present, SD
106
i
107
........
i
108
........
i
109
........
i
........
1010
i
,
.,a
1011 '-1'012.
i iH.
1013
(D/F)mi n Fig. 4. Plot of the lattice size L versus (D/F)~, obtained from Fig. 3. (D/F)mi, is determined from the minimum value of the ratio where finite size effects (1-2 islands present) become important. The scaling exponent is -1/6, consistent with the expected exponent, x=y= 1/3, for the island density. Such a relation can be used to relate the onset of finite size effects in experiments performedon substrates with different terrace sizes. in good agreement with our results. A coarse graining algorithm was used in Ref. 1,2] for computational efficiency. Fig. 4 shows a plot of the system size L ~ (D/F)miln/6, indicating that it follows a power law with a scaling exponent 1/6. This can be understood in terms of the island separation that, once it becomes comparable to the size of the system, L, the scaling Eq. (1) breaks down. Since the island separation scales with an exponent - x / 2 , this explains why the scaling exponent in Fig. 4 is - 1 / 6 , half the value of x for it---1. In Ref. I-3], it was found that the nucleated island density, N, on stepped surfaces of terrace width L, in the regime where step capture is significantly important, obeys the modified scaling equation N oc (D/F)- 1/3f(w) where w = (O/F)- 2/30]~. In the limit of high D/F ratios f ( w ) ~ w , one can simply derive the relation, Loc (D/F)miln/3 at fixed coverage, which is consistent with the result of Fig. 4. In Ref. [3], only the island separation normal to the steps is bounded so the island density scales NoeL -1 with the bounded terrace width (instead of the relation NocL -2 for terraces bounded in both directions). It provides an additional consistency test of the program but it can have practical value as well: experimental studies carried out on substrates with different average terraces can be
K.R. Roos, M. (2 Tringides/ Surface Science 355 (1996) L259-L263
easily compared to determine where finite size effects become important in one study, if finite size effects are already determined in the other study. Our results can first be used to determine the temperature range where scaling measurements according to Eq. (1) can be safely carried out. For Ag(lll), which has a low terrace diffusion, Et= 0.1 eV E7] for typical substrate sizes L ~ 1 0 0 0 A growth experiments can be carried out at a maximum temperature T = 150 K for typical flux rates F=0.01ML/s, as can be seen from Fig. 3. Experiments at higher temperatures should result in the formation of very few islands and zero scaling exponents x = y = 0 . We have carried out our simulations for ic = 1 but we expect that the crossover values (D/F)min t o be weakly dependent o n ie.
In the simulations, we have used periodic boundary conditions because they are commonly used in the literature [2,6] so it would be easier to evaluate the importance of finite size effects. On a real system, boundary effects are expected to be different along different terrace edges, depending on whether neighboring terraces are separated by an ascending or a descending step. In this postscaling regime where very few islands are present on the terrace, a deposited atom eventually is captured by either an ascending step, the nucleated island or the descending step. Deposition experiments at constant substrate temperature as a function of flux rate should result in flux-independent growth, y = 0, as observed in the simulations, since the diffusion length is so large that the distribution of atoms towards ascending, descending steps, and the island remains the same. However, if the temperature is varied, the number of atoms falling off the descending step is expected to increase since the probability of crossing the step barrier, Es, is thermally activated. This additional barrier is responsible for confining the atoms on the terrace in the first place but it is also important in determining the type of multilayer growth, whether 2D (for small Es) or 3D (for large E~), by controlling interlayer diffusion. Experiments in this regime, measuring the nucleated island step density SD as
a function of temperature, can be use, directly a rate, Em, which is monotomcally relateo to the step edge barrier, Es: the size of the island (and So) will be reduced as more atoms hop over a descending step. This has the potential of measuring the step edge barrier in the submonolayer regime [6]. In summary, we have studied how the scaling relation of nucleation, Eq. (1), breaks down for sufficiently high D/F ratios or small substrate terraces by measuring the step density SD, which is probed in RHEED diffraction experiments. We have used Monte Carlo simulations on the simple model of irreversible growth (ic = 1) to observe the crossover region as the scaling exponents, x,y, change from 1/3 to 0. In this region, very few islands nucleate on the terrace since the diffusion length is sufficiently high. These effects are relevant for the Ag/Ag(111) system, which has a low terrace diffusion barrier, E t = 0.1 eV.
Acknowledgements Ames Laboratory is operated by the US Department of Energy by Iowa State University under Contract No. W-7405-Eng-82. This work was supported by the Director for Energy Research, Office of Basic Energy Sciences.
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