Scaling of the island density and island-size distribution in irreversible submonolayer growth of three-dimensional islands

Abstract

Motivated by observations of three-dimensional (3D) island formation in a variety of experiments, we have carried out kinetic Monte Carlo simulations of a simplified model of 3D island growth with critical island size $i=1$ (irreversible growth). Of particular interest are the exponents ${y}_{1}$ and ${y}_{1}^{\ensuremath{'}}$ describing the dependence of the submonolayer island density in the precoalescence regime on dose and coverage respectively, as well as the exponent ${\ensuremath{\chi}}_{1}^{\ensuremath{'}}$ describing the dependence of the peak island density on the ratio $D/F$, where $D$ is the monomer hopping rate and $F$ is the (per site) deposition rate. We find that the values of ${y}_{1}$ and ${y}_{1}^{\ensuremath{'}}$ (e.g., ${y}_{1}\ensuremath{\simeq}0.24,\text{ }{y}_{1}^{\ensuremath{'}}\ensuremath{\simeq}0.32$) are significantly lower than the standard rate-equation (RE) predictions $({y}_{1}=1/3,\text{ }{y}_{1}^{\ensuremath{'}}=3/7\ensuremath{\simeq}0.43)$. This may be explained by the fact that, in contrast to the standard RE assumption of size-independent capture numbers, for 3D islands the island radius increases with the number of atoms in an island. Accordingly, the average capture number increases with coverage and as a result, the coverage dependence of the island density for 3D islands is intermediate between that for two-dimensional (2D) islands $({y}_{1}={y}_{1}^{\ensuremath{'}}=0)$ and the standard RE prediction. As a result, the measured value of ${\ensuremath{\chi}}_{1}^{\ensuremath{'}}\text{ }({\ensuremath{\chi}}_{1}^{\ensuremath{'}}\ensuremath{\simeq}0.30)$ is slightly larger than the standard RE prediction $({\ensuremath{\chi}}_{1}^{\ensuremath{'}}=2/7\ensuremath{\simeq}0.29)$ but still lower than the value $({\ensuremath{\chi}}_{1}=1/3)$ for irreversible 2D island growth. For comparison with our simulations we also present self-consistent RE results for the island and monomer densities as a function of coverage and excellent agreement is obtained without any adjustable parameters. Results for the scaled island-size distribution (ISD) for 2D and 3D irreversible island growth are also presented. Somewhat surprisingly, we find that there is very little difference between the scaled ISD for 3D islands and that for 2D islands. In addition to our simulation results, the scaling behavior of the 3D island density for general critical island size $i$ is also discussed. Our results suggest that while the exponent ${\ensuremath{\chi}}_{i}^{\ensuremath{'}}$ describing the flux dependence of the peak island density is the most accurate indicator, the exponent ${y}_{i}$ may also be useful to estimate the critical island size $i$ in submonolayer 3D growth.