Capture numbers and island size distributions in models of submonolayer surface growth

The capture numbers entering the rate equations (RE) for submonolayer film growth are determined from extensive kinetic Monte Carlo (KMC) simulations for simple representative growth models yielding point, compact, and fractal island morphologies. The full dependence of the capture numbers ${\sigma }_s(\Theta ,\Gamma )$ on island size $s$ and on both the coverage $\Theta$ and the $\Gamma =D/F$ ratio between the adatom diffusion coefficient $D$ and deposition rate $F$ is determined. Based on this information, the RE are solved to give the RE island size distribution (RE-ISD), as quantified by the number $n_s(\Theta ,\Gamma )$ of islands of size $s$ per unit area. The RE-ISDs are shown to agree well with the corresponding KMC-ISDs for all island morphologies. For compact morphologies, however, this agreement is only present for coverages smaller than $\Theta \simeq 5\%$ due to a significantly increased coalescence rate compared to fractal morphologies. As found earlier, the scaled KMC-ISDs $n_s{\overline{s}}^2/\Theta$ as a function of scaled island size $x=s/\overline{s}$ approach, for fixed $\Theta$, a limiting curve $f_{\infty }(x,\Theta )$ for $\Gamma \to \infty$. Our findings provide evidence that the limiting curve is independent of $\Theta$ for point islands, while the results for compact and fractal island morphologies indicate a dependence on $\Theta$.

Figure 1

Capture numbers as a function of $s$ for $\Gamma ={10}^7$ and four different fixed coverages for the models representing (a) point, (b) compact, and (c) fractal island morphologies. The filled symbols refer to the ${\sigma }_s$ obtained from the KMC simulations and the open symbols to the results of the self-consistent theory according to Eqs. (10) (with the $R_s$ taken from the simulations, see text and Fig. 3).

Figure 2

Mean capture number $\overline{\sigma }$ and mean island size $\overline{s}$ as a function of $\Theta$ for the four simulated $\Gamma$ values and the models representing point, compact, and fractal island morphologies.

Figure 3

Mean radii of gyration $R_s$ of islands of size $s$ for the models representing compact and fractal island morphologies. The straight lines in the double-logarithmic plot indicate the power-law behavior for large $s$.

Figure 4

Scaling plot of $(1-\Theta ){\sigma }_s$ as a function of $R_s/\xi$ for the model representing compact island morphologies, with both $R_s$ and ${\xi }^{-2}=2{\sigma }_1{n}_1+{\sum }_{s\ge 2}{\sigma }_s{n}_s$ determined from the KMC simulations, (a) for various $\Gamma$ and fixed $\Theta =0.2$ and (b) for various $\Theta$ and fixed $\Gamma ={10}^8$. The insets in (a) and (b) show the corresponding results for the model representing fractal island morphologies. The solid lines represent the specific functional form in Eq. (10) predicted by the self-consistent theory.

Figure 5

KMC results (symbols) for the adatom density $n_1$ and island density $N$ as function of $\Theta$ at $\Gamma ={10}^7$ for (a) point, (b) compact, and (c) fractal island morphologies in comparison with the RE solutions, when using the direct capture areas ${\kappa }_s$ and capture numbers ${\sigma }_s$ from Sec. 3 (solid lines) and when using the capture numbers ${\sigma }_s^{\mathrm{sc}}$ from the self-consistent theory according to Eq. (10) (dashed lines).

Figure 6

Simulated island size distribution (circles) in comparison with the RE solution (lines) for three different $\Theta$ at $\Gamma ={10}^7$ for the models representing (a) point and (b) fractal island morphologies.

Figure 7

(a) ISD and (b) sub-ISD for compact islands obtained from the KMC simulations (circles) in comparison with the RE solution (lines) for $\Gamma ={10}^7$ and three different coverages $\Theta$.

Figure 8

Fraction of coalesced islands as a function of the coverage $\Theta$ for $\Gamma ={10}^7$.

Figure 9

Scaled island size distributions $n_s{\overline{s}}^2/\Theta$ as function of the scaled island size $x=s/\overline{s}$ for fixed $\Gamma ={10}^8$ and four coverages $\Theta$, for (a) point and (b) fractal island morphologies. The insets show the scaled ISDs at fixed $\Theta =0.2$ and the four simulated $\Gamma$ values. The line in (a) is a fit to the data for $\Theta =0.2$ and $\Gamma ={10}^8$ and agrees with the analytical result Eq. (7), when using the line in Fig. 10a as the estimate for $C_{\infty }\left(x\right)$.

Figure 10

Scaled capture numbers ${\sigma }_s/\overline{\sigma }$ as function of the scaled island size $s/\overline{s}$ for fixed $\Gamma ={10}^8$ and four coverages $\Theta$, for (a) point and (b) fractal island morphologies. The insets show the scaled capture numbers at fixed $\Theta =0.2$ and the four simulated $\Gamma$ values. The line in (a) marks the solution obtained from a numerical integration of Eq. (11) when using the fit (line) to the scaled ISDs curve for $\Gamma ={10}^8$ and $\Theta =0.2$ in Fig. 9a.

Figure 11

(a) Scaled island size distributions and (b) scaled capture numbers of subislands as function of the scaled island size in the case of compact island morphologies for fixed $\Gamma ={10}^8$ and four coverages $\Theta$. The inset shows the corresponding data at fixed $\Theta =0.2$ and the four simulated $\Gamma$ values.

Figure 12

(a) Mean fraction $\Phi$ of distance vectors with antiparallel orientation (with respect to the center of mass distance vector) and (b) mean minimal distance $d$ between islands as a function of the time lag $\Delta t<0$ before a coalescence event at zero time. The times are given in units of $F^{-1}$ and the data were determined from the KMC simulations for $\Gamma ={10}^7$.