A variety of nanocrystals, nanoparticles or quantum dots are fabricated using nucleation and growth processes. Therefore, a fundamental understanding of nucleation and growth is crucial to materials science and engineering on the nanoscale. In this dissertation, we explore the fundamental characteristics of nucleation and growth in multiple dimensional systems using several different methods. One method which has been found to be particularly useful is the Monte Carlo (MC) method. In particular, the kinetic Monte Carlo (KMC) method has made MC simulations of complicated many body systems very efficient. In this dissertation, we use KMC simulations to study nucleation, growth, and coarsening in a variety of different systems. In addition, we have carried out a theoretical analysis using rate equations. We have also carried out investigations of the fundamental characteristics of the coarsening process using parallel methods based on a newly developed parallel KMC algorithm.
This dissertation is organized in two parts, the first part is about fundamental characteristics of multiple dimensional systems, the second part is about parallel KMC calculation of coarsening process. In Part I, we first study the fundamental characteristics of nucleation and growth in 3 dimensional (3D) systems using a simplified model of nucleation and growth. One of the main goals of this work is to compare with previous work on 2D nucleation and growth in order to understand the effects of dimensionality.
The scaling of the average island-size, island density, monomer density, island-size distribution (ISD), capture number distribution (CND), and capture zone distribution (CZD) are studied as a function of the fraction of occupied sites (coverage) and the ratio D/F of the monomer hopping rate D to the (per site) monomer creation rate F. Our model may be viewed as a simple model of the early-stages of vacancy cluster nucleation and growth under irradiation. Good agreement is found between our mean-field (MF) rate-equation results for the average island and monomer densities and our simulation results. In addition, we find that due to the decreased influence of correlations and fluctuations in 3D as compared to 2D, 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, the scaled ISD obtained in kinetic Monte Carlo (KMC) simulations appears to diverge more slowly with increasing D/F than the MF prediction while the divergence occurs at a value of the scaled island-size which is somewhat beyond the MF prediction. These results are supported by an analysis of the asymptotic CND.
The final goal for understanding the mechanism of nucleation and growth is to develop a theory to concisely and precisely disclose the law underlying the nucleation and growth process. From the theoretical point view, dimension can be taken as a variable to develop theory. In order to obtain the upper critical dimension corresponding to MF theories, we compare the results of kinetic Monte Carlo (KMC) simulations of a point-island model of irreversible nucleation and growth in four-dimensions 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 D/F 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 the upper critical dimension for irreversible cluster nucleation and growth is equal to 4.
While most of this work focusses on the correlations between various quantities and the island-size, it is also interesting to study the dependence of global quantities, such as the global CZD on dimensionality. In order to compare with recent analytic theories for the global CZD we have also studied the dependence of the global CZD using KMC simulations as a function of dimensionality in, one, two, three, and four dimensions. In general, we find that while the CZD depends on the short-range interaction for finite D/F (corresponding to the ratio of monomer hopping rate D to deposition rate F) in the asymptotic limit of infinite D/F there is no significant dependence.
However, poor agreement is found between the asymptotic CZD and the predicted Wigner distributions. Our results also indicate that for d = 2 and 3 the asymptotic CZD is independent of model details and dimension. However, for d = 1 and d = 4 the resulting distribution is significantly more sharply peaked. We also find that in contrast to the island-size distribution, for which mean-field-like behavior is observed in d = 3 and above, the asymptotic CZD is significantly broadened by fluctuations even in d = 4.
In addition to nucleation and growth, coarsening also plays a significant role in nano-particle fabrication and thin film growth. Therefore, in Part II, we explore the coarsening process using KMC simulations. Exploring the fundamental characteristics of coarsening process requires large-scale simulations which cover a large time scale. In this study, the results of parallel kinetic Monte Carlo (KMC) simulations of island coarsening based on a bond-counting model are obtained. Our simulations were carried out both as a test of and as an application of the recently developed semi-rigorous synchronous sublattice (SL) algorithm. By carrying out simulations over long times and for large system sizes the asymptotic coarsening behavior and scaled island-size distribution (ISD) were determined. Our results indicate that while cluster diffusion and coalescence play a role at early and intermediate times, at late times the coarsening proceeds via Ostwald ripening. In addition, we find that due to fluctuations which become important in 2D, the asymptotic scaled ISD is significantly narrower and more sharply peaked than the mean-field theory prediction. The dependence of the scaled ISD on coverage is also studied. Our results demonstrate that parallel KMC simulations can be used to effectively extend the time-scale over which realistic coarsening simulations can be carried out. These results also suggest that the SL algorithm is likely to be useful in the future in parallel KMC simulations of more complicated models of coarsening.