Distributional fixed-point equations for island nucleation in one dimension: A retrospective approach for capture-zone scaling

The distributions of interisland gaps and captures zones for islands nucleated on a one-dimensional substrate during submonolayer deposition are considered using a novel retrospective view. This provides an alternative perspective on why scaling occurs in this continuously evolving system. Distributional fixed-point equations for the gaps are derived both with and without a mean-field approximation for nearest neighbor gap-size correlation. Solutions to the equations show that correct consideration of fragmentation bias justifies the mean-field approach, which can be extended to provide closed-from equations for the capture zones. Our results compare favorably to Monte Carlo data for both point and extended islands using a range of critical island size $i=0,1,2,3$. We also find satisfactory agreement with theoretical models based on more traditional fragmentation theory approaches.

Figure 1

The scaled GSD $\varphi$ for extended and point islands with $i=0$, 1, 2, 3 and $\theta =5$%, 20%, and, in the point-island case only, 100% obtained from MC simulations [$R={10}^7$ (black), $R={10}^8$ (blue/light gray), and $R={10}^9$ (red/dark gray)]. Note that the data for $R={10}^9$ with $i=3$ and $\theta =100$% are not included.

Figure 2

The scaled CZD $P$ for extended and point islands with $i=0$, 1, 2, 3 and $\theta =5$%, 20%, and, in the point-island case only, 100% obtained from MC simulations [$R={10}^7$ (black), $R={10}^8$ (blue/light gray), and $R={10}^9$ (red/dark gray)].

Figure 3

The islands numbered $I_1$–$I_5$ on the one-dimensional substrate. The gaps between the islands are labeled $g_1$, $g_3$, $g_5$, and $g_4$, and the capture zones of islands $I_3$, $I_5$, $I_4$ are labeled $C_3$, $C_5$, and $C_4$, respectively.

Figure 4

The evolution of gap-size distribution under iteration of Eq. (4) with $i=1$. The solid lines are for $\alpha =i+1$ in Eq. (5), and the broken lines for $\alpha =i$, where the broken lines are shifted along the abscissa for clarity.

Figure 5

The GSDs compared to histograms of MC data [21], i.e., $R=8\times {10}^6$, for various critical island size $i$, taken at nominal coverage $\theta =100$%. The solid curves are the converged solutions to Eq. (4) with $\alpha =i+1$ with $i=0,1,2,3$, and the broken lines are for $\alpha =i$ with $i=1,2,3$.

Figure 6

The evolution of gap-size distribution under iteration of Eq. (8) with $\alpha =1$, 2, 3, 4.

Figure 7

Comparison of the integral Eqs. (4), (8), and (10) and, for the $\alpha =1$ ($i=0$) case, Treat's solution Eq. (6) for the various gap-size models with $\alpha =1$, 2, 3, 4.

Figure 8

The CZDs compared to histograms of MC data [21] for various critical island size $i$, taken at nominal coverage $\theta =100$%. The solid black curves are the solutions to Eq. (11) for $\alpha =i+1$ with $i=0,1,2,3$, and the broken lines are for $\alpha =i$ with $i=1,2,3$.