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

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 Dh /F of the monomer hopping rate Dh 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 Dh /F. In addition, while the peak height of the scaled ISD for fractal islands appears to diverge with increasing Dh /F, the dependence on Dh /F is much weaker than for point-islands in d=3. The results of a self-consistent rate-equation calculation for the coverage and Dh /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 describing the Dh /F dependence of the island density at fixed coverage, e.g., Nsat Dh /F − , is in good agreement with the value =1 /3 expected for irreversible growth. However, for both compact and fractal islands in d=3, our results indicate that the value of 0.42 is significantly larger. In order to explain this behavior, an analytical expression e.g., =df / 3df −2 for the dependence of on island fractal dimension df in d=3 is derived and found to give reasonable agreement with our simulation and rate-equation results for the case of point-islands df = , compact islands df =3 , and fractal islands df 2.5 . A general expression for the exponent , valid for d 2, as a function of the critical island size i and df is also derived.