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Irreversible island nucleation and growth with anomalous diffusion in d>2
Physica A 508 (2018) 567–576
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Physica A journal homepage: www.elsevier.com/locate/physa
Irreversible island nucleation and growth with anomalous diffusion in d > 2 Ehsan H. Sabbar, Nitun N. Poddar, Jacques G. Amar
∗
Department of Physics & Astronomy, University of Toledo, Toledo, OH 43606, USA
highlights • • • • •
Island growth with anomalous diffusion is studied in d = 2, 3, and 4 dimensions. Both the case of subdiffusion and superdiffusion are studied. Good agreement is obtained with a recently developed general rate-equation theory. Results are presented for the scaled capture-number distribution in d = 2. Results for the scaled island-size distribution in d = 2 are also presented.
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Article history: Received 29 December 2017 Received in revised form 23 April 2018 Available online xxxx Keywords: Island nucleation Submonolayer growth Anomalous diffusion Capture-number distribution Island-size distribution
a b s t r a c t Recently, there has been significant interest in the effects of anomalous diffusion on island nucleation and growth. Of particular interest are the exponents χ and χ1 which describe the dependence of the island and monomer density on deposition rate as well as the dependence of these exponents on the anomalous diffusion exponent µ and critical island size. While most simulations have been focused on growth on a 2D and/or quasi-1D substrate, here we present simulation results for the irreversible growth of ramified islands in three-dimensions (d = 3) for both the case of subdiffusion (µ < 1) and superdiffusion (1 < µ ≤ 2). Good agreement is found in both cases with a recently developed theory (Amar and Semaan, 2016) which takes into account the critical island-size i, island fractal dimension df , substrate dimension d, and diffusion exponent µ. In addition, we confirm that in this case the critical value of µ is given by the general prediction µc = 2/d = 2/3. We also present results for the irreversible growth of point-islands in d = 3 and d = 4 for both monomer subdiffusion and superdiffusion, and good agreement with RE predictions is also obtained. In addition, our results confirm that for point-islands with d ≥ 3 one has µc = 1 rather than 2/d. Results for the scaling of the capture-number distribution (CND), island-size distribution (ISD), and average capture number for the case of irreversible growth with monomer superdiffusion in d = 2 are also presented. Surprisingly, we find that both the scaled ISD and CND depend very weakly on the monomer diffusion exponent µ. These results indicate that – in contrast to the scaling of the average capture number which depends on the monomer diffusion exponent µ – both the scaled ISD and CND are primarily determined by the capture-zone distribution, which depends primarily on the ‘‘history’’ of the nucleation process rather than the detailed mechanisms for monomer diffusion. © 2018 Published by Elsevier B.V.
∗ Corresponding author. E-mail addresses: [email protected] (E.H. Sabbar), [email protected] (N.N. Poddar), [email protected] (J.G. Amar). https://doi.org/10.1016/j.physa.2018.05.123 0378-4371/© 2018 Published by Elsevier B.V.
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1. Introduction Recently, there has been significant interest in the effects of anomalous monomer diffusion [1–14] including its effects on submonolayer island growth [15–20]. This interest has been partially stimulated by recent experiments [16–18] in which values of the exponent χ which describes the dependence of the peak island density Npk on deposition rate F (e.g. Npk ∼ F χ ) were obtained which were significantly larger than 1. This result is in contrast to the standard rate-equation (RE) theory prediction [21,22] that in the case of deposition on a 2D substrate with ordinary monomer diffusion – and assuming the existence of a critical cluster size i such that clusters larger than i are stable while clusters of size i and below are unstable – one has χ = i/(i + 2). While this result applies in the case of ordinary diffusion such that the dependence of the mean-square monomer displacement on time t satisfies ⟨r 2 (t)⟩ ∼ t µ with µ = 1, it does not apply in the case of anomalous diffusion. In particular, in Ref. [17] it was suggested that the large value of the exponent χ found in the case of submonolayer growth of parahexaphenyl [16] on amorphous mica may be explained by the existence of transient hyperthermal behavior which leads to ballistic monomer diffusion (µ = 2). Similar results, e.g. χ > 1, have also been obtained in the case of submonolayer growth of pentacene on amorphous mica [18]. In addition, in Ref. [19] a RE approach was used to show that for the case of compact islands on a 2D substrate, ballistic diffusion implies that χ = 2i/(i + 3). In order to obtain a better understanding of the effects of anomalous diffusion on island nucleation and growth, a rateequation (RE) theory has recently been developed [20] which leads to general expressions for the exponent χ as a function of the critical island size i, substrate dimension d, island fractal dimension df , and diffusion exponent µ, where 0 ≤ µ ≤ 2. General expressions were also obtained [20] for the exponent χ1 which describes the dependence of the monomer density N1 on deposition rate at fixed coverage θ (e.g. N1 (θ; F ) ∼ F χ1 ) in the pre-coalescence or aggregation regime in which the island density remains constant. Here θ is an effective coverage which is equal to the fraction of the convex envelope surrounding φ stable islands, which may be approximated as θ ≃ N( N )d/df where N is the stable island density and the dose φ = Ft corresponds to the equivalent coverage if all deposited particles are placed on the d-dimensional substrate. General expressions were also obtained [20] for the exponents χ ′ and χ1′ which describe the deposition-rate dependence ′ ′ of the island and monomer densities at fixed dose φ = Ft, e.g. N(φ; F ) ∼ F χ and N1 (φ; F ) ∼ F χ1 . We note that φ is equal to the coverage θ if the island fractal dimension df is equal to the substrate dimension d. In addition, two distinct cases were identified – one corresponding to µ < µc and the other corresponding to µ > µc – where µc = 2/d for finite df as well as for point-islands with d ≤ 2, while µc = 1 for point-islands with d ≥ 3. However, these theoretical predictions have only been tested in the case of growth on a 2D substrate (d = 2) [20,23] and or quasi-1D substrate (d = 1) [24]. Here we consider the effects of anomalous diffusion on irreversible island nucleation in higher dimensions, e.g. d = 3 and d = 4 and compare with the predictions of Ref. [20]. We note that the case of irreversible nucleation in d = 3 with normal diffusion has been previously studied in Ref. [25]. In particular, results are presented for both ramified islands (df ≃ 2.5) and point-islands (df = ∞) in d = 3 for the case of both monomer subdiffusion (µ < 1) and superdiffusion (1 < µ ≤ 1.5). In general, excellent agreement is found between our simulation results for the exponents χ (µ), χ ′ (µ) and χ1′ (µ) in d = 3 and the RE predictions. We also find good agreement for the case of point-islands in d = 4 between our simulation results for the exponent χ ′ (µ) and RE predictions for the case of subdiffusion (µ < 1). Our results also confirm that for islands with finite df the critical value of µ is given by the general prediction µc = 2/d, e.g. µc = 2/3 for d = 3, while for point-islands µc = 1. In addition to these results for d ≥ 3 - since previous work for the case of substrate dimension d = 2 with anomalous monomer diffusion has focused almost exclusively on the average island and monomer densities — we also present results for the dependence of the scaled island-size and capture-number distributions on the monomer diffusion exponent µ for the case of superdiffusion with irreversible growth in d = 2. Somewhat surprisingly, we find that in this case the scaled capture number distribution (CND) does not depend on µ. This result also explains the relatively weak dependence of the scaled island-size distribution (ISD) on the value of µ found in our simulations in the case of irreversible growth. It also indicates that – in contrast to the scaling of the average capture number which depends on the monomer diffusion exponent µ – the capture number distribution is primarily determined by the capture-zone distribution [26–32], which depends primarily on the ‘‘history’’ of the nucleation process and not on the detailed mechanisms for monomer diffusion. We also present results for the scaling of the average capture number σav at fixed coverage as a function of island density for the case of ballistic diffusion (µ = 2) which directly confirm the assumptions made in the analyses of Ref. [19] and Ref. [20]. This paper is organized as follows. In Section 2 we first briefly review the RE theory, while in Section 3 we discuss the details of our simulations. We then present our simulation results in Section 4 and compare with the corresponding RE theory predictions. Finally, in Section 5, we present our conclusions and discuss possible future work. 2. Rate-equation theory By combining the steady-state assumption that in the aggregation regime the rate of monomer deposition is balanced by the rate of island attachment with an expression for the average monomer lifetime as a function of the island density N and diffusion exponent µ it was found [20] that the average capture number σav at fixed coverage θ (dose φ ) is proportional to
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a power of the average cluster size, e.g. σav (φ; N) ∼ N −δ/df and σav (θ; N) ∼ N −δ/d where,
δ = d′ − 2/µ
(1)
′
′
with d = d for finite df or for point-islands with d ≤ 2, and d = 2 for point-islands (corresponding to df = ∞) with d ≥ 3 [33]. This implies the existence of a critical value of µ (µc = 2/d′ ) such that for µ = µc one has δ = 0 and the standard RE theory (corresponding to ordinary diffusion in d = 2) applies, while for µ > µc (µ < µc ) one has δ > 0 (δ < 0) [33]. Since δ/df increases with increasing µ and d and decreases with increasing df , Eq. (1) also implies that the exponents δ and χ will also increase with increasing µ and d and decrease with increasing df . 2.1. Case µ ≥ µc In this case, assuming that the monomer capture number σ1 is independent of D/F , leads to the following results for the case of irreversible growth with i = 1 [20],
χ=
χ′ =
2 + µdf µd , χ1 = 2 + 2 + µdf 2 + 2 + µdf
µdf 2 + 2 + µ(df + 2η)
, χ1′ = 1 −
(2)
2 + µη
µdf
χ′
(3)
where η ≡ df − d. 2.2. Case µ ≤ µc In this case we assume, by analogy with the case of irreversible growth with ordinary diffusion and d = 1 [34], that both ′ σ1 and σav scale in the same way, e.g. σ1 (θ; N) ∼ σav (θ; N) ∼ N −δ . However, in Ref. [20] it was found that assuming that ′ δ = δ/d leads to poor agreement with simulations for i = 1, d = 2, and µ < µc . As a result it was conjectured [20] that for µ ≤ µc , one instead has δ ′ = δ . Taking into account the special case [20] of point-islands (df = ∞) with d ≥ 3, for which µc = 1 rather than 2/d and δ ′ = 2 − 2/µ rather than d − 2/µ, this leads to the following results for the case of irreversible growth with i = 1,
χ=
χ′ =
µd (µ ≤ µc ) d(2 − (d − 1)µ) + µ(d + df )
(2 −
(d′
µ (µ ≤ µ c ) − 1)µ) + 2µ
(4)
(5)
with χ1 = 1 −χ (1 −δ ) = (d + df )χ/d and χ1′ = 1 −χ ′ (1 −δ ) = 2χ ′ . Here d′ = d for finite df or for df = ∞ with d ≤ 2, while d′ = 2 for the case of point-islands (df = ∞) and d ≥ 3. Using these expressions, excellent agreement with simulations was found for both the case of point islands (df = ∞) and ramified islands (df ≃ 2) for µ < µc with i = 1 and d = 2 [20].
3. Simulations 3.1. Subdiffusion (µ < 1) As in previous work [20,23,24], our simulations for the case of subdiffusion were carried out assuming random substrate deposition (with rate F per site per unit time) as well as monomer diffusion using a continuous time random walk [6] to nearest-neighbor sites with a power-law distribution of waiting times P(τ ′ ) ∼ (D0 τ ′ )−1−µ . In order to avoid finite-size effects and ensure good statistics, simulations were carried out on hypercubic lattices of size Ld with relatively large system-sizes (L = 600 in units of the lattice constant for d = 3, and L = 100 for d = 4) and averaged over several runs. In addition, to determine the asymptotic value of the exponents, relatively large values of D0 /F were used, ranging from 109 to a maximum value of 1012 –1015 depending on the value of µ. In all cases periodic boundary conditions were assumed. Simulations in d = 3 were carried out for both ramified islands (df ≃ 2.5) corresponding to a model in which monomers irreversibly attach to occupied nearest-neighbor sites as well as for a point-island model (df = ∞) in which all of the particles in a cluster occupy the same site. For the case of ramified islands, simulations were carried out up to a maximum dose φmax = 0.1 which was well past the pre-coalescence regime, while for point islands (for which there is no coalescence) the submonolayer growth was
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extended up to φmax = 0.3. In both cases no deposition was allowed on already occupied sites. Due to memory limitations, our simulations in d = 4 were only carried out for point islands. 3.2. Superdiffusion (µ > 1) In contrast to the case of subdiffusion in which a random hopping direction is randomly chosen at each step, as in previous work [20] in our simulations with superdiffusion monomers were assumed to execute Lévy walks [35] with hopping rate D. n this case, the initial hopping direction for a freshly deposited monomer is chosen randomly from one of the nearest-neighbor directions, and remains constant over a persistence length l with power-law distribution P(l) ∼ l−(4−µ) , before a new random direction and persistence length are chosen [15,36]. As in the case of subdiffusion, our simulations in d = 3 were carried out using a cubic lattice with periodic boundary conditions, while each hop corresponds to a nearest-neighbor distance. In √ order to determine the asymptotic scaling behavior, simulations were carried out with D/F ranging (in steps of 10) from 106 to either 109 or 1010 , depending on the value of µ. Since finite-size effects are significant for the case of superdiffusion and become increasingly important with increasing µ and D/F , for the case of ramified islands in d = 3 our simulations were carried out using relatively large system sizes (L = 2400). In order to save memory, bit-mapping was used such that the height of each lattice-site was represented by a single bit (0 for a substrate site or 1 for a monomer or island site). Since for large D/F the number of monomers and dimers is much smaller than the number of lattice sites a hash table was also used to reduce the size of the ‘‘inverse’’ array which associates each lattice site occupied by a monomer with its corresponding location in the monomer list. However, despite the relatively large system sizes used, there were still significant finite-size effects for µ > 1.5. As a result, simulations were only carried out for values of µ between µ = 1 and µ = 1.5. In order to study the dependence of the scaled island-size and capture number distributions on µ in d = 2, as well as the scaling of the average capture number for µ = 2, simulations were also carried out on a triangular lattice for the case of ramified islands. In this case, to avoid finite-size effects simulations were carried out using very large system sizes (L = 36,000–54,000) with R = D/F ranging from 107 to 1010 . In order to calculate capture numbers we have used the method outlined in Ref. [37] which is a variation of the original method described in Ref. [38]. In this method, the capture number (at fixed dose φ ) is calculated by stopping the simulation at the desired dose, and then continuing monomer diffusion for a time ∆t with the additional rule that if a monomer attaches to an island of size s (monomer) a counter Ms (M1 ) is incremented and the monomer repositioned at a random site on the bare substrate. After averaging over several runs the capture numbers σs are then given by,
σs =
Ms L2 ∆tDNs N1
(6)
where L is the system-size. 4. Results 4.1. d = 3 4.1.1. Superdiffusion (µ > 1) We first consider the case of superdiffusion. As shown in Fig. 1, we have measured the island fractal dimension df for the case of ramified islands by studying the dependence of the island-size s on radius of gyration rG for D/F = 108 and several different values of µ ≥ 1 at the dose (φpk ≃ 0.04–0.06, depending on the value of µ) corresponding to the peak island density. As can be seen, df ≃ 2.5–2.6 for 1 ≤ µ ≤ 1.5, in good agreement with well-known results for diffusion-limited aggregation (DLA) in d = 3 [39,40]. We now consider the dependence of the asymptotic exponents χ and χ1′ on µ in the limit of large D/F . While the effective exponent χ1′ (µ) does not change significantly with increasing D/F , for the case of ramified islands the effective value of χ increases with increasing D/F , as was also found in Ref. [20] for √ the case d = 2. Accordingly, in this case we first determined N (aR) the effective value χ (R) = − log[ N pk(R/a) ]/log(a2 ) with a = 10, where Npk (u) corresponds to the peak island-density for pk D/F = u. To determine the asymptotic value of χ , we then assumed that for finite R one may write χ (R) = χ (∞) − c /log(R)p where c and p are unknown constants. In particular, for each value of µ, we have carried out linear fits of χ (R) as a function of log(R)−p for various values of p and have used the value of p which gave the best fit with the simulation data to estimate χ (∞). A similar fit was used for the case i = 1 and d = 2, and good agreement was found with the RE predictions for χ (µ) with µ > 1. While this slow crossover is not completely understood, it appears to be related to the φ -dependence of the average capture number σav , which in general is not known and may be complex. As an illustration of this, in Ref. [20] a similar slow logarithmic crossover for χ was obtained from a contracted RE calculation for µ = d = 2 in which it was φ assumed, in agreement with the expression just before Eq. (1), that σav = ( N )δ/df . In contrast, if the same dependence on N was assumed but no φ -dependence was included, e.g. the simple assumption σav = ( N1 )δ/df was used, then there was no crossover. Fig. 2 shows the corresponding effective exponents and asymptotic fits for some typical values of µ (µ = 1.25, 1.35, and 1.5). As can be seen the asymptotic value of χ , e.g. χ (∞) increases with increasing µ. The corresponding results for χ (µ) and
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Fig. 1. Island-size s as function of radius of gyration rG for ramified islands in d = 3.
Fig. 2. Effective exponent χ (R) for R = 106 –3.16 × 109 obtained from simulations (symbols) as function of 1/log(R)p for ramified islands, where values of p correspond to best fits (dashed lines). Here p = 1.4 for µ = 1.5, p = 1.1 for µ = 1.35, and p = 0.45 for µ = 1.25.
χ1′ (µ) are shown in Fig. 4. We have also carried out simulations with µ > 1 for the case of point-islands (df = ∞) in d = 3. In good agreement with the prediction of Eq. (3) with df = ∞, in this case we find that the exponent χ ′ which describes the dependence of the island density at fixed dose on D/F is equal to the value for ordinary monomer diffusion, e.g. χ ′ = 1/3. 4.1.2. Subdiffusion (µ < 1) We have also carried out simulations for the case of subdiffusion (µ < 1) with both ramified islands and point-islands. We note that in this case the island fractal dimension df for ramified islands remains equal to 2.5 since the monomers carry out ordinary, unbiased random walks while the peak dose remains relatively small. Typical results for the dependence of the peak island-density Npk on D0 /F for ramified islands with µ ≤ 1 are shown in Fig. 3. As can be seen, the value of χ increases from a value of approximately 0.09 for µ = 0.2 to 0.46 for µ = 1. Similarly, results were also obtained for the exponent χ ′ (µ) for the case of point islands. 4.1.3. Summary of results for d = 3 Fig. 4 shows a summary of our results for the exponents χ1′ (µ) and χ (µ) for ramified islands (filled symbols) as well as for the exponent χ ′ (µ) for point-islands (open symbols) over the entire range of µ (0 ≤ µ ≤ 1.5) in d = 3. Also shown are the
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Fig. 3. Log–log plots of peak island-density Npk as function of D0 /F for typical values of µ ≤ 1 for the case of ramified islands in d = 3.
Fig. 4. Exponents χ1 (µ), χ (µ), and χ ′ (µ) obtained from simulations of irreversible growth of ramified islands with anomalous diffusion in d = 3 (symbols). Solid curves correspond to Eqs. (4) and (5) with df = 2.5 (ramified islands) and d′ = 2 (point islands). Dashed curves correspond to Eqs. (2) and (3) with df = 2.5 (ramified islands) and df = ∞ (point islands).
corresponding theoretical predictions for µ < µc (Eq. (4) and Eq. (5), solid lines) and µ > µc (Eq. (2) and Eq. (3), dashed lines) where µc = 2/3 for ramified islands and µc = 1 for point-islands. We note that for the theoretical curves for ramified islands we have assumed df = 2.5. As can be seen, there is excellent agreement in both the subdiffusive and superdiffusive regimes, and for both ramified islands and point islands, between our simulation results and the corresponding theoretical predictions in d = 3. 4.2. d = 4 We now consider the case of irreversible growth in d = 4 which was studied using point islands (df = ∞). Since we expect that, as was found in d = 3, χ ′ = 1/3 for µ > 1, we have only studied the case of subdiffusion. Fig. 5 shows a summary of our results for the exponents χ ′ over the range 0 ≤ µ ≤ 1. As can be seen there is very good agreement between our simulation results and the corresponding theoretical prediction Eq. (5) with df = ∞.
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Fig. 5. Exponent χ ′ (µ) obtained from simulations (symbols) of irreversible growth of point islands (df = ∞) in d = 4. Dashed curve corresponds to Eq. (5) with d′ = 2.
4.3. d = 2 4.3.1. Scaled island-size and capture number distributions We first ∑ consider the scaled island-size distribution (ISD) f (s/S) = Ns S 2 /θ where Ns is the density of islands of size s ∞ and S = N1 s=2 sNs is the average island-size. In previous work [15] the scaled ISD for the case of ramified islands with µ = 5/3 was compared with that obtained for ordinary diffusion (µ = 1), and it was found that the peak of the scaled ISD was somewhat lower while f (u) was larger for small u. However, these results were for only one particular value of µ and were only averaged over 30 runs. In addition, they were for a relatively small system-size (L = 1000) and so may have been affected by finite-size effects. In contrast, here we present results which were obtained using a much larger system-size (L = 45,000) with D/F = 109 thus eliminating finite-size effects and also providing much better statistics. Our results for the scaled ISD for the case of ramified islands with µ = 1, 1.5, 1.75, and 2.0 (dose φ = 0.12) are shown in Fig. 6(a). Also shown in Fig. 6(a) (dashed line) is the analytical expression for the scaled ISD given in Ref. [41] for the case of ordinary monomer diffusion. As can be seen there is fair agreement between this expression and our simulation results for the case µ = 1 although the agreement is significantly better at higher coverage [41,42]. As can also be seen in Fig. 6(a), for s/S > 1 there is very little dependence of the scaled ISD on µ. However, for s/S ≤ 1 there is a weak but noticeable dependence on µ. In particular, for the case of ordinary diffusion with µ = 1, the peak of the scaled ISD is somewhat higher than for µ ≥ 1.5 while f (u) is significantly smaller for small u. Our results for µ = 1 are also consistent with previous results [43] for the case of ramified islands with ordinary diffusion. The observed decrease (for small u) in the value of f (u) with decreasing µ may be explained by a decrease in the island fractal dimension with decreasing µ which leads to increased coalescence of small islands [42]. It may also be related to the decrease in the scaled capture number (see below) with increasing µ which leads to the decreased growth of small islands. Fig. 6(b) shows the corresponding results for the scaled capture-number distribution (CND) C (s/S) = σs /σav . Somewhat surprisingly – but consistent with our results for the scaled ISD – the scaled CND also exhibits very little dependence on the diffusion exponent µ, although an examination of the behavior for small u indicates a small but definite decreasing trend in C (u) with increasing µ for u < 0.25. This behavior is consistent with the expectation that the average capture number σav increases (for fixed D/F ) with increasing µ and is also consistent with the observed decrease in f (s/S) for small s/S with decreasing µ. In addition, as indicated by the dashed line in Fig. 6, for s/S > 1.25 the scaled capture number distribution C (u) is linear with slope 1. However, for s/S < 1 the dependence of the scaled capture-number on the scaled island-size is more complicated and appears to be approximately fit by the form C (u) = a + buγ where γ varies between 0.3 and 0.45 depending on the value of µ. These results are consistent with those obtained in Ref. [37] for the case of ramified islands on a square lattice with ordinary diffusion (µ = 1). Thus, our results indicate that the presence of anomalous diffusion has a relatively small effect on the scaled capture number distribution. This may be explained by the fact that the capture number is determined by the size of each island’s surrounding capture zone, which in turn is determined by the average island-density at the time when that island nucleated, rather than the detailed diffusion mechanism. Finally, as a test of the RE theory in the case of superdiffusion, we have also studied the dependence of the average capture number σav on island-density for the case of ramified islands with ballistic diffusion (µ = 2). In particular, we have directly calculated the average capture number σav at different fixed coverages for different values of D/F . We note that for
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Fig. 6. Scaled island-size and capture-number distributions obtained from simulations of ramified islands in d = 2. Dashed curve in Fig. 6(a) corresponds to analytical expression given in Ref. [41]. Dashed line in Fig. 6(b) has slope 1.
µ = d = 2 one has df = 2, which implies that the coverage θ is equal to the dose φ . As a result, using Eq. (1) we obtain σ (θ; N) ∼ N −ν where ν = δ/d = 1/2. Fig. 7 shows our results for σav for the case of ramified islands with µ = 2 as a function of island-density N for different fixed values of the coverage in the pre-coalescence regime and for D/F = 107 , 108 , 109 , and 1010 . As can be seen, there is excellent agreement between our simulation results and the theoretical prediction that ν = 1/2 for µ = 2. 5. Conclusion In order to further understand the effects of anomalous diffusion on island nucleation and growth, and also test the RE predictions [20] for d > 2, we have carried out kinetic Monte Carlo simulations for the case i = 1 (d = 3) for both monomer subdiffusion (µ < 1) and superdiffusion (µ > 1) as well as for both ramified islands (df ≃ 2.5) and point-islands (df = ∞). In general, excellent agreement is found between our simulation results for the exponents χ (µ), χ ′ (µ) and χ1′ (µ) in d = 3 and the RE predictions. In particular, we find good agreement between our simulation results for ramified islands with µ < 2/3 and the corresponding RE predictions Eqs. (4) and (5) with df = 2.5. Similarly, we also find good agreement between our simulation results for ramified islands with 2/3 ≤ µ ≤ 1.5 and the corresponding RE predictions Eqs. (2) and (3) with df = 2.5. These results also confirm that for ramified islands in d = 3, µc is equal to 2/3. We have also obtained excellent agreement between our simulation results for χ ′ (µ) in d = 3 for the case of point-islands corresponding to df = ∞. In particular, we find good agreement between our simulation results for point-islands for µ < 1 and the corresponding RE prediction Eq. (5) with d′ = 2, as well as between our simulation results for point-islands for 1 ≤ µ ≤ 1.5 and the RE prediction that χ ′ = 1/3. We also find good agreement for the case of point-islands in d = 4
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Fig. 7. Log–log plots of σav as function of island-density N for µ = 2 and D/F = 107 –1010 (see text for details).
between our simulation results for the exponent χ ′ (µ) and the RE prediction Eq. (5) with d′ = 2 for the case of subdiffusion (µ < 1). These results also confirm that µc is equal to 1 for point islands in d ≥ 3. In addition to these results for d > 2, we have also presented results for the scaling of the average capture number σav at fixed coverage as a function of island density for the case of ballistic diffusion (µ = 2) which directly confirm the assumptions made in the analyses of Ref. [19] and Ref. [20]. In addition, we have studied the dependence of the scaled island-size and capture-number distributions on the monomer diffusion exponent µ for the case of superdiffusion with irreversible growth in d = 2. Somewhat surprisingly, we find that both the scaled ISD and CND depend very weakly on the monomer diffusion exponent µ. In particular, the scaled ISD exhibits almost no dependence on µ for s/S > 1.25 and – except for a noticeable dependence for very small s/S due to island coalescence – depends only weakly on µ for s/S < 1.25. Similarly, we find that the scaled CND exhibits almost no dependence on µ for s/S > 1.25 and depends only very weakly on µ for s/S < 1.25. These results also indicate that for the case of irreversible growth – and in contrast to the scaling of the average capture number which depends on the monomer diffusion exponent µ – both the scaled ISD and CND are primarily determined by the capture-zone distribution, which depends primarily on the ‘‘history’’ of the nucleation process rather than the detailed mechanisms for monomer diffusion. Since in recent experiments [16,18] it is believed that ballistic monomer diffusion (µ = 2) plays a crucial role, while the critical island-size is significantly larger than 1, in the future it may also be of interest to study the dependence of the scaled ISD and CND on µ in the case of reversible growth. Acknowledgments This research was supported by NSF, USA grant DMR-1410840 as well as a grant of computer time from the Ohio Supercomputer Center. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
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Irreversible island nucleation and growth with anomalous diffusion in d > 2 Ehsan H. Sabbar, Nitun N. Poddar, Jacques G. Amar
∗
Department of Physics & Astronomy, University of Toledo, Toledo, OH 43606, USA
highlights • • • • •
Island growth with anomalous diffusion is studied in d = 2, 3, and 4 dimensions. Both the case of subdiffusion and superdiffusion are studied. Good agreement is obtained with a recently developed general rate-equation theory. Results are presented for the scaled capture-number distribution in d = 2. Results for the scaled island-size distribution in d = 2 are also presented.
article
info
Article history: Received 29 December 2017 Received in revised form 23 April 2018 Available online xxxx Keywords: Island nucleation Submonolayer growth Anomalous diffusion Capture-number distribution Island-size distribution
a b s t r a c t Recently, there has been significant interest in the effects of anomalous diffusion on island nucleation and growth. Of particular interest are the exponents χ and χ1 which describe the dependence of the island and monomer density on deposition rate as well as the dependence of these exponents on the anomalous diffusion exponent µ and critical island size. While most simulations have been focused on growth on a 2D and/or quasi-1D substrate, here we present simulation results for the irreversible growth of ramified islands in three-dimensions (d = 3) for both the case of subdiffusion (µ < 1) and superdiffusion (1 < µ ≤ 2). Good agreement is found in both cases with a recently developed theory (Amar and Semaan, 2016) which takes into account the critical island-size i, island fractal dimension df , substrate dimension d, and diffusion exponent µ. In addition, we confirm that in this case the critical value of µ is given by the general prediction µc = 2/d = 2/3. We also present results for the irreversible growth of point-islands in d = 3 and d = 4 for both monomer subdiffusion and superdiffusion, and good agreement with RE predictions is also obtained. In addition, our results confirm that for point-islands with d ≥ 3 one has µc = 1 rather than 2/d. Results for the scaling of the capture-number distribution (CND), island-size distribution (ISD), and average capture number for the case of irreversible growth with monomer superdiffusion in d = 2 are also presented. Surprisingly, we find that both the scaled ISD and CND depend very weakly on the monomer diffusion exponent µ. These results indicate that – in contrast to the scaling of the average capture number which depends on the monomer diffusion exponent µ – both the scaled ISD and CND are primarily determined by the capture-zone distribution, which depends primarily on the ‘‘history’’ of the nucleation process rather than the detailed mechanisms for monomer diffusion. © 2018 Published by Elsevier B.V.
∗ Corresponding author. E-mail addresses: [email protected] (E.H. Sabbar), [email protected] (N.N. Poddar), [email protected] (J.G. Amar). https://doi.org/10.1016/j.physa.2018.05.123 0378-4371/© 2018 Published by Elsevier B.V.
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1. Introduction Recently, there has been significant interest in the effects of anomalous monomer diffusion [1–14] including its effects on submonolayer island growth [15–20]. This interest has been partially stimulated by recent experiments [16–18] in which values of the exponent χ which describes the dependence of the peak island density Npk on deposition rate F (e.g. Npk ∼ F χ ) were obtained which were significantly larger than 1. This result is in contrast to the standard rate-equation (RE) theory prediction [21,22] that in the case of deposition on a 2D substrate with ordinary monomer diffusion – and assuming the existence of a critical cluster size i such that clusters larger than i are stable while clusters of size i and below are unstable – one has χ = i/(i + 2). While this result applies in the case of ordinary diffusion such that the dependence of the mean-square monomer displacement on time t satisfies ⟨r 2 (t)⟩ ∼ t µ with µ = 1, it does not apply in the case of anomalous diffusion. In particular, in Ref. [17] it was suggested that the large value of the exponent χ found in the case of submonolayer growth of parahexaphenyl [16] on amorphous mica may be explained by the existence of transient hyperthermal behavior which leads to ballistic monomer diffusion (µ = 2). Similar results, e.g. χ > 1, have also been obtained in the case of submonolayer growth of pentacene on amorphous mica [18]. In addition, in Ref. [19] a RE approach was used to show that for the case of compact islands on a 2D substrate, ballistic diffusion implies that χ = 2i/(i + 3). In order to obtain a better understanding of the effects of anomalous diffusion on island nucleation and growth, a rateequation (RE) theory has recently been developed [20] which leads to general expressions for the exponent χ as a function of the critical island size i, substrate dimension d, island fractal dimension df , and diffusion exponent µ, where 0 ≤ µ ≤ 2. General expressions were also obtained [20] for the exponent χ1 which describes the dependence of the monomer density N1 on deposition rate at fixed coverage θ (e.g. N1 (θ; F ) ∼ F χ1 ) in the pre-coalescence or aggregation regime in which the island density remains constant. Here θ is an effective coverage which is equal to the fraction of the convex envelope surrounding φ stable islands, which may be approximated as θ ≃ N( N )d/df where N is the stable island density and the dose φ = Ft corresponds to the equivalent coverage if all deposited particles are placed on the d-dimensional substrate. General expressions were also obtained [20] for the exponents χ ′ and χ1′ which describe the deposition-rate dependence ′ ′ of the island and monomer densities at fixed dose φ = Ft, e.g. N(φ; F ) ∼ F χ and N1 (φ; F ) ∼ F χ1 . We note that φ is equal to the coverage θ if the island fractal dimension df is equal to the substrate dimension d. In addition, two distinct cases were identified – one corresponding to µ < µc and the other corresponding to µ > µc – where µc = 2/d for finite df as well as for point-islands with d ≤ 2, while µc = 1 for point-islands with d ≥ 3. However, these theoretical predictions have only been tested in the case of growth on a 2D substrate (d = 2) [20,23] and or quasi-1D substrate (d = 1) [24]. Here we consider the effects of anomalous diffusion on irreversible island nucleation in higher dimensions, e.g. d = 3 and d = 4 and compare with the predictions of Ref. [20]. We note that the case of irreversible nucleation in d = 3 with normal diffusion has been previously studied in Ref. [25]. In particular, results are presented for both ramified islands (df ≃ 2.5) and point-islands (df = ∞) in d = 3 for the case of both monomer subdiffusion (µ < 1) and superdiffusion (1 < µ ≤ 1.5). In general, excellent agreement is found between our simulation results for the exponents χ (µ), χ ′ (µ) and χ1′ (µ) in d = 3 and the RE predictions. We also find good agreement for the case of point-islands in d = 4 between our simulation results for the exponent χ ′ (µ) and RE predictions for the case of subdiffusion (µ < 1). Our results also confirm that for islands with finite df the critical value of µ is given by the general prediction µc = 2/d, e.g. µc = 2/3 for d = 3, while for point-islands µc = 1. In addition to these results for d ≥ 3 - since previous work for the case of substrate dimension d = 2 with anomalous monomer diffusion has focused almost exclusively on the average island and monomer densities — we also present results for the dependence of the scaled island-size and capture-number distributions on the monomer diffusion exponent µ for the case of superdiffusion with irreversible growth in d = 2. Somewhat surprisingly, we find that in this case the scaled capture number distribution (CND) does not depend on µ. This result also explains the relatively weak dependence of the scaled island-size distribution (ISD) on the value of µ found in our simulations in the case of irreversible growth. It also indicates that – in contrast to the scaling of the average capture number which depends on the monomer diffusion exponent µ – the capture number distribution is primarily determined by the capture-zone distribution [26–32], which depends primarily on the ‘‘history’’ of the nucleation process and not on the detailed mechanisms for monomer diffusion. We also present results for the scaling of the average capture number σav at fixed coverage as a function of island density for the case of ballistic diffusion (µ = 2) which directly confirm the assumptions made in the analyses of Ref. [19] and Ref. [20]. This paper is organized as follows. In Section 2 we first briefly review the RE theory, while in Section 3 we discuss the details of our simulations. We then present our simulation results in Section 4 and compare with the corresponding RE theory predictions. Finally, in Section 5, we present our conclusions and discuss possible future work. 2. Rate-equation theory By combining the steady-state assumption that in the aggregation regime the rate of monomer deposition is balanced by the rate of island attachment with an expression for the average monomer lifetime as a function of the island density N and diffusion exponent µ it was found [20] that the average capture number σav at fixed coverage θ (dose φ ) is proportional to
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a power of the average cluster size, e.g. σav (φ; N) ∼ N −δ/df and σav (θ; N) ∼ N −δ/d where,
δ = d′ − 2/µ
(1)
′
′
with d = d for finite df or for point-islands with d ≤ 2, and d = 2 for point-islands (corresponding to df = ∞) with d ≥ 3 [33]. This implies the existence of a critical value of µ (µc = 2/d′ ) such that for µ = µc one has δ = 0 and the standard RE theory (corresponding to ordinary diffusion in d = 2) applies, while for µ > µc (µ < µc ) one has δ > 0 (δ < 0) [33]. Since δ/df increases with increasing µ and d and decreases with increasing df , Eq. (1) also implies that the exponents δ and χ will also increase with increasing µ and d and decrease with increasing df . 2.1. Case µ ≥ µc In this case, assuming that the monomer capture number σ1 is independent of D/F , leads to the following results for the case of irreversible growth with i = 1 [20],
χ=
χ′ =
2 + µdf µd , χ1 = 2 + 2 + µdf 2 + 2 + µdf
µdf 2 + 2 + µ(df + 2η)
, χ1′ = 1 −
(2)
2 + µη
µdf
χ′
(3)
where η ≡ df − d. 2.2. Case µ ≤ µc In this case we assume, by analogy with the case of irreversible growth with ordinary diffusion and d = 1 [34], that both ′ σ1 and σav scale in the same way, e.g. σ1 (θ; N) ∼ σav (θ; N) ∼ N −δ . However, in Ref. [20] it was found that assuming that ′ δ = δ/d leads to poor agreement with simulations for i = 1, d = 2, and µ < µc . As a result it was conjectured [20] that for µ ≤ µc , one instead has δ ′ = δ . Taking into account the special case [20] of point-islands (df = ∞) with d ≥ 3, for which µc = 1 rather than 2/d and δ ′ = 2 − 2/µ rather than d − 2/µ, this leads to the following results for the case of irreversible growth with i = 1,
χ=
χ′ =
µd (µ ≤ µc ) d(2 − (d − 1)µ) + µ(d + df )
(2 −
(d′
µ (µ ≤ µ c ) − 1)µ) + 2µ
(4)
(5)
with χ1 = 1 −χ (1 −δ ) = (d + df )χ/d and χ1′ = 1 −χ ′ (1 −δ ) = 2χ ′ . Here d′ = d for finite df or for df = ∞ with d ≤ 2, while d′ = 2 for the case of point-islands (df = ∞) and d ≥ 3. Using these expressions, excellent agreement with simulations was found for both the case of point islands (df = ∞) and ramified islands (df ≃ 2) for µ < µc with i = 1 and d = 2 [20].
3. Simulations 3.1. Subdiffusion (µ < 1) As in previous work [20,23,24], our simulations for the case of subdiffusion were carried out assuming random substrate deposition (with rate F per site per unit time) as well as monomer diffusion using a continuous time random walk [6] to nearest-neighbor sites with a power-law distribution of waiting times P(τ ′ ) ∼ (D0 τ ′ )−1−µ . In order to avoid finite-size effects and ensure good statistics, simulations were carried out on hypercubic lattices of size Ld with relatively large system-sizes (L = 600 in units of the lattice constant for d = 3, and L = 100 for d = 4) and averaged over several runs. In addition, to determine the asymptotic value of the exponents, relatively large values of D0 /F were used, ranging from 109 to a maximum value of 1012 –1015 depending on the value of µ. In all cases periodic boundary conditions were assumed. Simulations in d = 3 were carried out for both ramified islands (df ≃ 2.5) corresponding to a model in which monomers irreversibly attach to occupied nearest-neighbor sites as well as for a point-island model (df = ∞) in which all of the particles in a cluster occupy the same site. For the case of ramified islands, simulations were carried out up to a maximum dose φmax = 0.1 which was well past the pre-coalescence regime, while for point islands (for which there is no coalescence) the submonolayer growth was
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extended up to φmax = 0.3. In both cases no deposition was allowed on already occupied sites. Due to memory limitations, our simulations in d = 4 were only carried out for point islands. 3.2. Superdiffusion (µ > 1) In contrast to the case of subdiffusion in which a random hopping direction is randomly chosen at each step, as in previous work [20] in our simulations with superdiffusion monomers were assumed to execute Lévy walks [35] with hopping rate D. n this case, the initial hopping direction for a freshly deposited monomer is chosen randomly from one of the nearest-neighbor directions, and remains constant over a persistence length l with power-law distribution P(l) ∼ l−(4−µ) , before a new random direction and persistence length are chosen [15,36]. As in the case of subdiffusion, our simulations in d = 3 were carried out using a cubic lattice with periodic boundary conditions, while each hop corresponds to a nearest-neighbor distance. In √ order to determine the asymptotic scaling behavior, simulations were carried out with D/F ranging (in steps of 10) from 106 to either 109 or 1010 , depending on the value of µ. Since finite-size effects are significant for the case of superdiffusion and become increasingly important with increasing µ and D/F , for the case of ramified islands in d = 3 our simulations were carried out using relatively large system sizes (L = 2400). In order to save memory, bit-mapping was used such that the height of each lattice-site was represented by a single bit (0 for a substrate site or 1 for a monomer or island site). Since for large D/F the number of monomers and dimers is much smaller than the number of lattice sites a hash table was also used to reduce the size of the ‘‘inverse’’ array which associates each lattice site occupied by a monomer with its corresponding location in the monomer list. However, despite the relatively large system sizes used, there were still significant finite-size effects for µ > 1.5. As a result, simulations were only carried out for values of µ between µ = 1 and µ = 1.5. In order to study the dependence of the scaled island-size and capture number distributions on µ in d = 2, as well as the scaling of the average capture number for µ = 2, simulations were also carried out on a triangular lattice for the case of ramified islands. In this case, to avoid finite-size effects simulations were carried out using very large system sizes (L = 36,000–54,000) with R = D/F ranging from 107 to 1010 . In order to calculate capture numbers we have used the method outlined in Ref. [37] which is a variation of the original method described in Ref. [38]. In this method, the capture number (at fixed dose φ ) is calculated by stopping the simulation at the desired dose, and then continuing monomer diffusion for a time ∆t with the additional rule that if a monomer attaches to an island of size s (monomer) a counter Ms (M1 ) is incremented and the monomer repositioned at a random site on the bare substrate. After averaging over several runs the capture numbers σs are then given by,
σs =
Ms L2 ∆tDNs N1
(6)
where L is the system-size. 4. Results 4.1. d = 3 4.1.1. Superdiffusion (µ > 1) We first consider the case of superdiffusion. As shown in Fig. 1, we have measured the island fractal dimension df for the case of ramified islands by studying the dependence of the island-size s on radius of gyration rG for D/F = 108 and several different values of µ ≥ 1 at the dose (φpk ≃ 0.04–0.06, depending on the value of µ) corresponding to the peak island density. As can be seen, df ≃ 2.5–2.6 for 1 ≤ µ ≤ 1.5, in good agreement with well-known results for diffusion-limited aggregation (DLA) in d = 3 [39,40]. We now consider the dependence of the asymptotic exponents χ and χ1′ on µ in the limit of large D/F . While the effective exponent χ1′ (µ) does not change significantly with increasing D/F , for the case of ramified islands the effective value of χ increases with increasing D/F , as was also found in Ref. [20] for √ the case d = 2. Accordingly, in this case we first determined N (aR) the effective value χ (R) = − log[ N pk(R/a) ]/log(a2 ) with a = 10, where Npk (u) corresponds to the peak island-density for pk D/F = u. To determine the asymptotic value of χ , we then assumed that for finite R one may write χ (R) = χ (∞) − c /log(R)p where c and p are unknown constants. In particular, for each value of µ, we have carried out linear fits of χ (R) as a function of log(R)−p for various values of p and have used the value of p which gave the best fit with the simulation data to estimate χ (∞). A similar fit was used for the case i = 1 and d = 2, and good agreement was found with the RE predictions for χ (µ) with µ > 1. While this slow crossover is not completely understood, it appears to be related to the φ -dependence of the average capture number σav , which in general is not known and may be complex. As an illustration of this, in Ref. [20] a similar slow logarithmic crossover for χ was obtained from a contracted RE calculation for µ = d = 2 in which it was φ assumed, in agreement with the expression just before Eq. (1), that σav = ( N )δ/df . In contrast, if the same dependence on N was assumed but no φ -dependence was included, e.g. the simple assumption σav = ( N1 )δ/df was used, then there was no crossover. Fig. 2 shows the corresponding effective exponents and asymptotic fits for some typical values of µ (µ = 1.25, 1.35, and 1.5). As can be seen the asymptotic value of χ , e.g. χ (∞) increases with increasing µ. The corresponding results for χ (µ) and
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Fig. 1. Island-size s as function of radius of gyration rG for ramified islands in d = 3.
Fig. 2. Effective exponent χ (R) for R = 106 –3.16 × 109 obtained from simulations (symbols) as function of 1/log(R)p for ramified islands, where values of p correspond to best fits (dashed lines). Here p = 1.4 for µ = 1.5, p = 1.1 for µ = 1.35, and p = 0.45 for µ = 1.25.
χ1′ (µ) are shown in Fig. 4. We have also carried out simulations with µ > 1 for the case of point-islands (df = ∞) in d = 3. In good agreement with the prediction of Eq. (3) with df = ∞, in this case we find that the exponent χ ′ which describes the dependence of the island density at fixed dose on D/F is equal to the value for ordinary monomer diffusion, e.g. χ ′ = 1/3. 4.1.2. Subdiffusion (µ < 1) We have also carried out simulations for the case of subdiffusion (µ < 1) with both ramified islands and point-islands. We note that in this case the island fractal dimension df for ramified islands remains equal to 2.5 since the monomers carry out ordinary, unbiased random walks while the peak dose remains relatively small. Typical results for the dependence of the peak island-density Npk on D0 /F for ramified islands with µ ≤ 1 are shown in Fig. 3. As can be seen, the value of χ increases from a value of approximately 0.09 for µ = 0.2 to 0.46 for µ = 1. Similarly, results were also obtained for the exponent χ ′ (µ) for the case of point islands. 4.1.3. Summary of results for d = 3 Fig. 4 shows a summary of our results for the exponents χ1′ (µ) and χ (µ) for ramified islands (filled symbols) as well as for the exponent χ ′ (µ) for point-islands (open symbols) over the entire range of µ (0 ≤ µ ≤ 1.5) in d = 3. Also shown are the
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Fig. 3. Log–log plots of peak island-density Npk as function of D0 /F for typical values of µ ≤ 1 for the case of ramified islands in d = 3.
Fig. 4. Exponents χ1 (µ), χ (µ), and χ ′ (µ) obtained from simulations of irreversible growth of ramified islands with anomalous diffusion in d = 3 (symbols). Solid curves correspond to Eqs. (4) and (5) with df = 2.5 (ramified islands) and d′ = 2 (point islands). Dashed curves correspond to Eqs. (2) and (3) with df = 2.5 (ramified islands) and df = ∞ (point islands).
corresponding theoretical predictions for µ < µc (Eq. (4) and Eq. (5), solid lines) and µ > µc (Eq. (2) and Eq. (3), dashed lines) where µc = 2/3 for ramified islands and µc = 1 for point-islands. We note that for the theoretical curves for ramified islands we have assumed df = 2.5. As can be seen, there is excellent agreement in both the subdiffusive and superdiffusive regimes, and for both ramified islands and point islands, between our simulation results and the corresponding theoretical predictions in d = 3. 4.2. d = 4 We now consider the case of irreversible growth in d = 4 which was studied using point islands (df = ∞). Since we expect that, as was found in d = 3, χ ′ = 1/3 for µ > 1, we have only studied the case of subdiffusion. Fig. 5 shows a summary of our results for the exponents χ ′ over the range 0 ≤ µ ≤ 1. As can be seen there is very good agreement between our simulation results and the corresponding theoretical prediction Eq. (5) with df = ∞.
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Fig. 5. Exponent χ ′ (µ) obtained from simulations (symbols) of irreversible growth of point islands (df = ∞) in d = 4. Dashed curve corresponds to Eq. (5) with d′ = 2.
4.3. d = 2 4.3.1. Scaled island-size and capture number distributions We first ∑ consider the scaled island-size distribution (ISD) f (s/S) = Ns S 2 /θ where Ns is the density of islands of size s ∞ and S = N1 s=2 sNs is the average island-size. In previous work [15] the scaled ISD for the case of ramified islands with µ = 5/3 was compared with that obtained for ordinary diffusion (µ = 1), and it was found that the peak of the scaled ISD was somewhat lower while f (u) was larger for small u. However, these results were for only one particular value of µ and were only averaged over 30 runs. In addition, they were for a relatively small system-size (L = 1000) and so may have been affected by finite-size effects. In contrast, here we present results which were obtained using a much larger system-size (L = 45,000) with D/F = 109 thus eliminating finite-size effects and also providing much better statistics. Our results for the scaled ISD for the case of ramified islands with µ = 1, 1.5, 1.75, and 2.0 (dose φ = 0.12) are shown in Fig. 6(a). Also shown in Fig. 6(a) (dashed line) is the analytical expression for the scaled ISD given in Ref. [41] for the case of ordinary monomer diffusion. As can be seen there is fair agreement between this expression and our simulation results for the case µ = 1 although the agreement is significantly better at higher coverage [41,42]. As can also be seen in Fig. 6(a), for s/S > 1 there is very little dependence of the scaled ISD on µ. However, for s/S ≤ 1 there is a weak but noticeable dependence on µ. In particular, for the case of ordinary diffusion with µ = 1, the peak of the scaled ISD is somewhat higher than for µ ≥ 1.5 while f (u) is significantly smaller for small u. Our results for µ = 1 are also consistent with previous results [43] for the case of ramified islands with ordinary diffusion. The observed decrease (for small u) in the value of f (u) with decreasing µ may be explained by a decrease in the island fractal dimension with decreasing µ which leads to increased coalescence of small islands [42]. It may also be related to the decrease in the scaled capture number (see below) with increasing µ which leads to the decreased growth of small islands. Fig. 6(b) shows the corresponding results for the scaled capture-number distribution (CND) C (s/S) = σs /σav . Somewhat surprisingly – but consistent with our results for the scaled ISD – the scaled CND also exhibits very little dependence on the diffusion exponent µ, although an examination of the behavior for small u indicates a small but definite decreasing trend in C (u) with increasing µ for u < 0.25. This behavior is consistent with the expectation that the average capture number σav increases (for fixed D/F ) with increasing µ and is also consistent with the observed decrease in f (s/S) for small s/S with decreasing µ. In addition, as indicated by the dashed line in Fig. 6, for s/S > 1.25 the scaled capture number distribution C (u) is linear with slope 1. However, for s/S < 1 the dependence of the scaled capture-number on the scaled island-size is more complicated and appears to be approximately fit by the form C (u) = a + buγ where γ varies between 0.3 and 0.45 depending on the value of µ. These results are consistent with those obtained in Ref. [37] for the case of ramified islands on a square lattice with ordinary diffusion (µ = 1). Thus, our results indicate that the presence of anomalous diffusion has a relatively small effect on the scaled capture number distribution. This may be explained by the fact that the capture number is determined by the size of each island’s surrounding capture zone, which in turn is determined by the average island-density at the time when that island nucleated, rather than the detailed diffusion mechanism. Finally, as a test of the RE theory in the case of superdiffusion, we have also studied the dependence of the average capture number σav on island-density for the case of ramified islands with ballistic diffusion (µ = 2). In particular, we have directly calculated the average capture number σav at different fixed coverages for different values of D/F . We note that for
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Fig. 6. Scaled island-size and capture-number distributions obtained from simulations of ramified islands in d = 2. Dashed curve in Fig. 6(a) corresponds to analytical expression given in Ref. [41]. Dashed line in Fig. 6(b) has slope 1.
µ = d = 2 one has df = 2, which implies that the coverage θ is equal to the dose φ . As a result, using Eq. (1) we obtain σ (θ; N) ∼ N −ν where ν = δ/d = 1/2. Fig. 7 shows our results for σav for the case of ramified islands with µ = 2 as a function of island-density N for different fixed values of the coverage in the pre-coalescence regime and for D/F = 107 , 108 , 109 , and 1010 . As can be seen, there is excellent agreement between our simulation results and the theoretical prediction that ν = 1/2 for µ = 2. 5. Conclusion In order to further understand the effects of anomalous diffusion on island nucleation and growth, and also test the RE predictions [20] for d > 2, we have carried out kinetic Monte Carlo simulations for the case i = 1 (d = 3) for both monomer subdiffusion (µ < 1) and superdiffusion (µ > 1) as well as for both ramified islands (df ≃ 2.5) and point-islands (df = ∞). In general, excellent agreement is found between our simulation results for the exponents χ (µ), χ ′ (µ) and χ1′ (µ) in d = 3 and the RE predictions. In particular, we find good agreement between our simulation results for ramified islands with µ < 2/3 and the corresponding RE predictions Eqs. (4) and (5) with df = 2.5. Similarly, we also find good agreement between our simulation results for ramified islands with 2/3 ≤ µ ≤ 1.5 and the corresponding RE predictions Eqs. (2) and (3) with df = 2.5. These results also confirm that for ramified islands in d = 3, µc is equal to 2/3. We have also obtained excellent agreement between our simulation results for χ ′ (µ) in d = 3 for the case of point-islands corresponding to df = ∞. In particular, we find good agreement between our simulation results for point-islands for µ < 1 and the corresponding RE prediction Eq. (5) with d′ = 2, as well as between our simulation results for point-islands for 1 ≤ µ ≤ 1.5 and the RE prediction that χ ′ = 1/3. We also find good agreement for the case of point-islands in d = 4
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Fig. 7. Log–log plots of σav as function of island-density N for µ = 2 and D/F = 107 –1010 (see text for details).
between our simulation results for the exponent χ ′ (µ) and the RE prediction Eq. (5) with d′ = 2 for the case of subdiffusion (µ < 1). These results also confirm that µc is equal to 1 for point islands in d ≥ 3. In addition to these results for d > 2, we have also presented results for the scaling of the average capture number σav at fixed coverage as a function of island density for the case of ballistic diffusion (µ = 2) which directly confirm the assumptions made in the analyses of Ref. [19] and Ref. [20]. In addition, we have studied the dependence of the scaled island-size and capture-number distributions on the monomer diffusion exponent µ for the case of superdiffusion with irreversible growth in d = 2. Somewhat surprisingly, we find that both the scaled ISD and CND depend very weakly on the monomer diffusion exponent µ. In particular, the scaled ISD exhibits almost no dependence on µ for s/S > 1.25 and – except for a noticeable dependence for very small s/S due to island coalescence – depends only weakly on µ for s/S < 1.25. Similarly, we find that the scaled CND exhibits almost no dependence on µ for s/S > 1.25 and depends only very weakly on µ for s/S < 1.25. These results also indicate that for the case of irreversible growth – and in contrast to the scaling of the average capture number which depends on the monomer diffusion exponent µ – both the scaled ISD and CND are primarily determined by the capture-zone distribution, which depends primarily on the ‘‘history’’ of the nucleation process rather than the detailed mechanisms for monomer diffusion. Since in recent experiments [16,18] it is believed that ballistic monomer diffusion (µ = 2) plays a crucial role, while the critical island-size is significantly larger than 1, in the future it may also be of interest to study the dependence of the scaled ISD and CND on µ in the case of reversible growth. Acknowledgments This research was supported by NSF, USA grant DMR-1410840 as well as a grant of computer time from the Ohio Supercomputer Center. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
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