Upper critical dimension for irreversible cluster nucleation and growth in the point-island regime.

We compare the results of kinetic Monte Carlo (KMC) simulations of a point-island model of irreversible nucleation and growth in four dimensions (4D) with the corresponding mean-field (MF) rate-equation predictions for the monomer density, island density, island-size distribution (ISD), capture-number distribution (CND), and capture-zone distribution (CZD), in order to determine the critical dimension d(c) for mean-field behavior. The asymptotic behavior is studied as a function of the fraction of occupied sites (coverage) and the ratio DF of the monomer hopping rate D to the (per site) monomer creation rate F. Excellent agreement is found between our KMC simulation results and the MF rate equation results for the average island and monomer densities. For large D/F, the scaled CND and CZD do not depend on island size, in good agreement with the MF prediction, while the scaled ISD also agrees well with the MF prediction except for a slight difference at the peak values. Coupled with previous results obtained in d = 3 , these results indicate that for growth in the point-island regime, the upper critical dimension for irreversible cluster nucleation and growth is equal to 4.