Fragmentation approach to the point-island model with hindered aggregation: Accessing the barrier energy

We study the effect of hindered aggregation on the island formation process in a one- (1D) and two-dimensional (2D) point-island model for epitaxial growth with arbitrary critical nucleus size $i$. In our model, the attachment of monomers to preexisting islands is hindered by an additional attachment barrier, characterized by length $l_a$. For $l_a=0$ the islands behave as perfect sinks while for $l_a\to \infty$ they behave as reflecting boundaries. For intermediate values of $l_a$, the system exhibits a crossover between two different kinds of processes, diffusion-limited aggregation and attachment-limited aggregation. We calculate the growth exponents of the density of islands and monomers for the low coverage and aggregation regimes. The capture-zone (CZ) distributions are also calculated for different values of $i$ and $l_a$. In order to obtain a good spatial description of the nucleation process, we propose a fragmentation model, which is based on an approximate description of nucleation inside of the gaps for 1D and the CZs for 2D. In both cases, the nucleation is described by using two different physically rooted probabilities, which are related with the microscopic parameters of the model ($i$ and $l_a$). We test our analytical model with extensive numerical simulations and previously established results. The proposed model describes excellently the statistical behavior of the system for arbitrary values of $l_a$ and $i=1$, 2, and 3.

Figure 1

Time evolution of (a) $N_1$ and (b) $N$ for $D/F=5\times {10}^6$ and large but finite attachment barriers $l_a=250$. Three different values of $i$ were considered: 1, 2, and 3. See text for discussion.

Figure 2

Case $i=1$: (a) Fragmentation kernel $p^{\chi }\left(\chi \right)$, (b) gap-size distribution $p^{\left(0\right)}\left(\ell \right)$, and (c) capture-zone distribution $p^{\left(1\right)}\left(\ell \right)$ for a 1D system. The parameters used in the PM are ${\ell }_c=1.7$, ${\gamma }_{\text{ex}}=4$ for $l_a=0$; and ${\ell }_c=1.75$, ${\gamma }_{\text{ex}}=3$ for $l_a=250$. For $l_a=10$, $p^{\left(0\right)}\left(\ell \right)$ and $p^{\left(1\right)}\left(\ell \right)$ (not shown) differ little from those for $l_a=250$. The insets give log-log plots of the first four spacing distributions.

Figure 3

Case $i=2$: (a) Fragmentation kernel $p^{\chi }\left(\chi \right)$, (b) gap-size distribution $p^{\left(0\right)}\left(\ell \right)$, and (c) capture-zone distribution $p^{\left(1\right)}\left(\ell \right)$ for a 1D system. The parameters used in the PM are ${\ell }_c=1.5$, ${\gamma }_{\text{ex}}=5$ for $l_a=0$; and ${\ell }_c=1.2$, ${\gamma }_{\text{ex}}=4$ for $l_a=250$. As in Fig. 2, the insets give log-log plots of the first four spacing distributions.

Figure 4

Case $i=3$: (a) Fragmentation kernel $p^{\chi }\left(\chi \right)$, (b) gap-size distribution $p^{\left(0\right)}\left(\ell \right)$, and (c) capture-zone distribution $p^{\left(1\right)}\left(\ell \right)$ for a 1D system. The parameters used in the PM are ${\ell }_c=1.5$, ${\gamma }_{\text{ex}}=7$ for $l_a=0$; and ${\ell }_c=1.0$, ${\gamma }_{\text{ex}}=3$ for $l_a=250$. As in Fig. 2, the insets give log-log plots of the first four spacing distributions.

Figure 5

Capture zone distributions for kMC simulations in 2D (dense discrete symbols) with (a) $i=1$ and (b) $i=2$ with $l_a=0$ (narrower set of curves) and $l_a=250$ (broader set). In all cases, the fragmentation (PM) approach (dash-dotted curves) gives excellent results. The insets show log-log plots of the kMC results for $P\left(\mathcal{A}\right)$. Note that Eq. (57), kMC results for $l_a=250$ and the PM with $\varphi =1$, $\delta =0$, and $\kappa =1$ are almost indistinguishable for both $i$ values.