Distribution of Capture Zone Areas of Just-Nucleated Islands
Once a nucleation event occurs, the distribution of CZs (and the growth rates) of the islands neighboring the nucleation site is modified. The typical area assigned to the new island is an essential quantity in rate-equation formulations of the evolution of the CZ area distribution during island formation.[2,8,9,18] Typical distributions of CZ areas for a new island, for tessellations of ECs, are shown in Fig. 9, for a selected set of simulation parameters. It is interesting to compare these distributions with those of the CZ areas for all dimers. First, note that the average CZ area of a dimer must be strictly smaller than the average CZ area of a just-nucleated dimer (and both of these smaller than the average CZ area for all islands) because areas of just-nucleated dimers can only decrease with continued island nucleation. However, the basic statistical properties of the two distributions, including the area fluctuations, are similar (Fig. 9 and Table 1). Second, there is a noticeable difference for point-islands and near-square islands in the average CZ areas of dimers or just-nucleated islands relative to the total average CZ area (for all islands). For near-square islands, the ratio is smaller than 0.6 (and only slightly higher, if taking the ratio to the average free area, defined as Aav — sav, instead) compared with 0.9 for point-islands. Finally, for contrast, Fig. 10 shows the CZ area distribution for all islands. Note that the (normalized) envelope of this distribution is the function g(a).
Table 1 Statistical properties of the area distributions for VCs of point-islands (PI) and ECs of near-square islands (NSI) of size s = 2 (just-nucleated dimers and all dimers) at 6=0.1 ml
|
System |
h/F |
Ajust-nucleated |
A2 |
njust-nucleated |
V2 |
kjust-nucleated |
k2 |
|
PI |
107 |
284.7 |
268.3 |
88.6 |
87.7 |
0.40 |
0.45 |
|
108 |
572.7 |
537.5 |
182.8 |
179.2 |
0.42 |
0.45 |
|
|
109 |
1165.0 |
1098.1 |
381.3 |
374.2 |
0.42 |
0.45 |
|
|
NSI |
107 |
268.7 |
251.8 |
93.9 |
94.8 |
0.39 |
0.40 |
|
108 |
560.6 |
515.9 |
198.7 |
199.5 |
0.37 |
0.43 |
|
|
109 |
1125.0 |
1076.1 |
455.5 |
427.8 |
0.40 |
0.42 |
Fig. 10 Scaled area distributions (a = A/Aav) for (a) VCs of point-islands and (b) ECs of near-square islands of all sizes (h/F =10 , 6=0.1 ML). The lines are Gaussian fits. The standard deviation and skewness of the data are n ~ 0.33Aav, k ~ 0.46 for point-islands, and n ~0.38Aav, k~0.46 for near-square islands. The skewness for a random Voronoi tessellation in two dimensions.
The Impact of Nucleation on the Capture Zones of Existing Islands
Some of the natural measures of the impact of nucleation on a configuration of islands, and in particular on its CZ area distribution, are 1) the probability that the CZ of a given island will be overlapped (and thus reduced) by the CZ of the just-nucleated island, 2) the typical number of such overlapped islands, per nucleation event, sorted by island size or CZ area, and 3) the typical fraction of the area that each overlapped island contributes to the CZ area of the just-nucleated island (see the schematic in Fig. 11). Simulation results for each of these measures are presented below.
These three measures are not independent. If Ps>A denotes the probability that a nucleation event reduces the CZ of area A belonging to an island of size s and (M0) denotes the average number of existing CZs overlapped by the CZ of a just-nucleated island, then one has the normalization condition
In addition, if Asubset(s,A) denotes the average area of the portion of the CZ of the just-nucleated island that overlaps the existing CZ of area A belonging to an island of size s and (Asubset) denotes the average area of such individual portions, then one also has the constraint
where Ajust-nucleated is the average area of a just-nucleated island (Fig. 9 and Table 1).
Fig. 11 Schematic showing the rearrangement of VC boundaries for a point-island distribution (h/F =1010, 6=0.001 ML) following a M0 = 4 nucleation event. The new island is marked with a x, and existing islands are marked with dots. The shifted boundaries (dashed lines) enclose the CZ of the new island. Little or no rearrangement occurs for cells beyond the nearest neighbors of the new island. The four area portions contributed to the cell of the new island are highlighted with a pattern.
The probability that a nucleation event reduces the capture zone of an existing island
Although the probability Ps,A that a nucleation event impacts (i.e., reduces) a CZ of area A belonging to an island of size s includes contributions from nucleation events occurring inside that area-A cell and inside neighboring cells, generally of different area, one anticipates that Ps,A scales directly with the fraction of islands of size s and CZ area A, Ns,A/Nav. One can then write
where the scaling factor q(s,A) reflects the actual propensity for overlap. The simulation data shown in Fig. 12 for the reduced probabilities Ps=SaPSaA that a nucleation event impacts the CZ of some island of size s and PA=SsPsa that a nucleation event reduces the CZ of some island of CZ area A indeed have a shape similar to Ns and Na, respectively. If the above factorization holds, this behavior of the reduced probabilities implies that the factor q(s,A) is a weak function of both s and A, in the range where Ns,A is significant. In fact, q(s,A) is probably a stronger function of A than of s because the likelihood of overlap should depend primarily on the area A.[9]
The distribution of M0 values
Fig. 13 shows distributions of M0 values for two distinct coverages, one near the onset of the steady state, where point-islands and near-square islands behave similarly, and the other within the steady state. In either case, note that no nucleation events with M0= 1 were recorded in the simulations; that is, the CZ of a just-nucleated island never overlapped just one existing CZ. Thus despite the dramatic simplification of theory that this scenario brings,[2,8] such an assumption is not well justified. Other features are interesting to note. With increasing coverage, there is a large shift in the distribution for near-square islands toward smaller M0. For example, far fewer nucleation events with M0 > 7 and many more events with M0=3 occur at 0.1 ML than at 10— 3 ML. Such shift would be consistent with an increasing preference to nucleate closer to triple points (assuming the average number of neighboring cells of a CZ, which is about 6,[13] and other topological properties of the tessellation do not change significantly with increasing coverage).
The portions, Asubset, of existing capture zone areas overlapped by the capture zone of a new island
The distribution of Asubset values, distinguished by size and area (A) of the overlapped CZs, is well described by an exponential distribution, as shown in Fig. 14. (Note that for an exponential distribution, the mean and variance are equal, and the skewness is equal to 2.) It exhibits a significant population at small area portions and exponentially small populations for Asubset above Aav. In fact, Asubset is, on average, only 0.14A and only 0.17Aav. However, it does follow, as anticipated, that Asubset(s,A) is mainly determined by A (that is, it is a weaker function of s) (see the insets in Fig. 14 and Ref. [9] for more details). Note that the relation Ajust-nucleated= (Asubnuc)(M0) holds, e.g., for h/F =107, 6=0.1 ML, one finds Ajust-nucleated«285 (Table 1), (Asubnuc)«52 (Fig. 14), and (M0)«5.5 (Ref. [9]).
Fig. 12 Simulation results for (a) Ps and (b) PA. Data are for point-islands, h/F =107, 6 = 0.1 ML (Aav« 307, sav « 30.7). The thinner lines are the average density of islands of size s, in (a), and the average density of islands of CZ area A, in (b). The data were normalized so that the area under all the curves is unity.
Fig. 13 The distribution of M0 values for (a)-(b) point-islands and (c)-(d) near-square islands, for h/F =108. In (a) and (c), 6 =10 3 ML; in (b) and (d), 6=0.1 ML. The lines are Gaussian fits. The average, (M0), and the standard deviation, nMo, are (a) (M0) =5.9, nM= 1.2, (b) (M0) =5.6, nM=1.2, (c) (M0) = 5.8, nM=1.3, (d) (M0) =4.8, nM=1.2. Statistics were obtained "for 10,000-100,000 nucleation events.
The Island Nucleation Positions
A natural expectation is that most nucleation events that occur in the steady-state regime must occur near the boundaries of CZs where the diffusing ad-species density (and thus the nucleation rate) is relatively high. (Some caution is justified when using geometric tessellations, as these do not exactly reflect the spatial profile of the adspecies density.) Small sets of nucleation events do show such bias.[20] However, for statistically more relevant sets of nucleation events, this feature is less obvious (Figs. 4 and 15). Fig. 15 shows not only some clustering of nucleation events near boundaries and triple points, but also many nucleation events occurring inside the larger cells. This, no doubt, reflects the feature that the ad-species density does not vary sharply across the CZ boundaries, as is often assumed.
Fig. 14 Simulation data for Asubset, in (a), scaled by the total area, A, of the overlapped CZs, in (b). Data are for 100,000 point-island nucleation events, with h/F=107, 6 = 0.1 ML (Aav ~ 307, sav ~ 30.7): (Asubset)~ 52, nsubset ~ 51, Ksubset ~ 1.5, (Asubset/A)~0.14, nsubset/A ~ 0.13, Ksubset/A ~ 1.1. The lines are fits with an exponential distribution. The insets show Asubset(s), averaged over A, in (a), and Asubset(A), averaged over s, in (b). Here, statistics were obtained from nearly 500,000 nucleation events. The lines in the insets are guides to the eye only.
Fig. 15 One thousand nucleation events (small dots) for a fixed configuration of point-islands (large dots) obtained at (a) 0.001 ML (sav= 12.0) and (b) 0.01 ML (sav = 53.8), for h/F = 1010. The tessellation of VCs is also shown. The average VC area is (a) 12,100 and (b) 5375 lattice sites. Both panels are a 380 x 380 site zoom of a larger simulation lattice.
CONCLUSION
The selection of simulation results discussed above underscores two important features of the behavior of canonical lattice-gas models of thin film epitaxy. First, the exact behavior of even simple quantities, including those describing spatial aspects of island nucleation, is highly nonintuitive, making phenomenological approximations of their behavior often unreliable. Second, most quantities are better described by broad probability distributions, reflecting strong fluctuations about the average environment of an island, and the stochastic nature of the island nucleation process. Moments of these distributions are key input to refined rate-equation formulations.[8,9]
