Island-size distribution and capture numbers in three-dimensional nucleation: Dependence on island morphology

John Royston and Jacques G. Amar

Phys. Rev. E 80, 041602 – Published 13 October 2009

Abstract

The scaling of the monomer and island densities, island-size distribution (ISD), and capture-number distribution (CND) as a function of the fraction of occupied sites (coverage) and ratio $D_{h} / F$ of the monomer hopping rate $D_{h}$ to the (per site) monomer creation rate $F$ are studied for the case of irreversible nucleation and growth of fractal islands in three dimensions $(d = 3)$ . We note that our model is a three-dimensional analog of submonolayer growth in the absence of island relaxation and may also be viewed as a simplified model of the early stages of vacancy cluster nucleation and growth under irradiation. In contrast to results previously obtained for point-islands in $d = 3$ , for which mean-field behavior corresponding to a CND which is independent of island size was observed, our results indicate that for fractal islands the scaled CND increases approximately linearly with island size in the asymptotic limit of large $D_{h} / F$ . In addition, while the peak height of the scaled ISD for fractal islands appears to diverge with increasing $D_{h} / F$ , the dependence on $D_{h} / F$ is much weaker than for point-islands in $d = 3$ . The results of a self-consistent rate-equation calculation for the coverage and $D_{h} / F$ dependence of the average island and monomer densities are also presented and good agreement with simulation results is obtained. For the case of point-islands, the value of the exponent $χ$ describing the $D_{h} / F$ dependence of the island density at fixed coverage, e.g., $N_{s a t} \sim {(D_{h} / F)}^{- χ}$ , is in good agreement with the value $(χ = 1 / 3)$ expected for irreversible growth. However, for both compact and fractal islands in $d = 3$ , our results indicate that the value of $χ$ $(χ ≃ 0.42)$ is significantly larger. In order to explain this behavior, an analytical expression [e.g., $χ = d_{f} / (3 d_{f} - 2)$ ] for the dependence of $χ$ on island fractal dimension $d_{f}$ in $d = 3$ is derived and found to give reasonable agreement with our simulation and rate-equation results for the case of point-islands $(d_{f} = \infty)$ , compact islands $(d_{f} = 3)$ , and fractal islands $(d_{f} ≃ 2.5)$ . A general expression for the exponent $χ$ , valid for $d \geq 2$ , as a function of the critical island size $i$ and $d_{f}$ is also derived.