Island-size distribution and capture numbers in three-dimensional nucleation: Comparison with mean-field behavior

The scaling of the monomer and island densities, island-size distribution (ISD), capture-number distribution (CND), and capture-zone distribution is studied as a function of the fraction of occupied sites (coverage) and ratio $D∕F$ of the monomer hopping rate $D$ to the (per site) monomer creation rate $F$ in a three-dimensional (3D) point-island model of irreversible nucleation and island growth. Our model is a 3D analog of submonolayer growth and may also be viewed as a simplified model of the early stages of vacancy cluster nucleation and growth under irradiation. Good agreement is found between mean-field (MF) rate-equation results for the average island and monomer densities and our simulation results. In addition, due to the decreased influence of correlations and fluctuations in 3D, the scaled CND depends only weakly on the island-size. As a result, the scaled ISD is significantly sharper than obtained in 2D and diverges with increasing $D∕F$. However, for large $D∕F$ both the scaled ISD and the scaled CND differ from the MF prediction. In particular, the scaled ISD diverges more slowly than the MF prediction while the asymptotic divergence occurs at a value of the scaled island size which is larger than the MF prediction. These results are further supported by an analysis of the asymptotic scaled capture-number distribution.

Figure 1

Comparison between KMC results (symbols) and the corresponding RE results (solid lines) for the monomer density $N_1$ and island density $N$ as a function of coverage for $D_h∕F={10}^5$ (circles), ${10}^7$ (triangles), and ${10}^9$ (diamonds).

Figure 2

Scaled island-size distributions for $D_h∕F={10}^5$, ${10}^7$, and ${10}^9$. KMC simulation results (symbols), RE results (solid lines), and asymptotic MF limit (dashed curve).

Figure 3

Log-log plot of peak value of scaled ISD as function of $D_h∕F$.

Figure 4

KMC simulation results for scaled CND ${\sigma }_s∕{\sigma }_{av}$ versus scaled island size for $D_h∕F={10}^5–{10}^{10}$. Horizontal dashed line corresponds to MF CND; solid line corresponds to $(2∕3)(s∕S)$.

Figure 5

Plot of $u_{pk}(D∕F)$ and $u_{\mathit{cr}}(D∕F)$ as a function of ${(D_h∕F)}^{-\gamma }$ for $D_h∕F$ ranging from ${10}^5$ to ${10}^{10}$. Lines are fits as described in text while values of $\gamma$ correspond to best fits.

Figure 6

KMC simulation results for scaled capture zone $z_s∕Z$ versus scaled island size $s∕S$ for $D_h∕F={10}^5–{10}^9$.