Rate-equation approach to irreversible island growth with cluster diffusion

Abstract

A self-consistent rate-equation (RE) approach to irreversible island growth and nucleation is presented which takes into account cluster mobility. As a first application, we consider the irreversible growth of compact islands on a two-dimensional surface in the presence of monomer deposition (with rate F) and monomer diffusion (with rate D(1)) while the mobility of an island of size s is assumed to satisfy D(s)=D(1)s(-μ) where μ>0. Results are obtained for the dependence of the island-density and island-size distribution (ISD) on the parameters D(1)/F, μ, and coverage θ. For all values of μ, we find excellent agreement between our self-consistent RE results and simulation results for the island and monomer densities, up to and even somewhat beyond the coverage corresponding to the peak island density. We also find good agreement between our self-consistent RE and simulation results for the portion of the ISD corresponding to island sizes less than the average island-size S. However, for larger island sizes the effects of correlations become important and as a result the agreement is not as good. Using our self-consistent RE approach we also demonstrate that the discrepancies between simulations and recent mean-field predictions for the exponent τ(μ) describing the power-law size dependence of the ISD for μ<1 can be explained almost entirely by geometric effects. Our results are also compared with those obtained using a simpler mean-field Smoluchowski approach. In general, we find that, except for the case μ=1/2 (for which the island and monomer densities are reasonably well predicted), such an approach leads to results which are in poor agreement with the simulations.

Critical island size, scaling, and ordering in colloidal nanoparticle self-assembly

“…One possible explanation for the relatively broad ISD is that, due to the relatively small NP size (which leads to less dipole repulsion between islands) there is significantly more island coalescence. The increased island coalescence in the nucleation regime tends to smear out the peak near A/ A = 1 in the island-size distribution, in a manner similar to that which was previously found in simulations of irreversible island growth (i = 1) with cluster diffusion [18,19]. In addition, due to the weaker binding in this case, there is significant NP detachment from large islands, and as a result we expect that coarsening effects [75] may also play a role [54].…”

“…In addition, we present results for the scaled island-size distribution (ISD) and capture-zone distribution (CZD), and their dependence on NP size. Surprisingly, we find that, despite the existence of significant cluster diffusion and coalescence, which has been previously shown [16][17][18][19] to dramatically broaden the island-size distribution for the case of a critical island size of 1, in our experiments the scaled island-size distribution is sharply peaked as in epitaxial growth. In particular, for large NPs, we find good agreement between the scaled ISD and epitaxial growth models as well as good agreement between the scaled CZD and scaled ISD.…”

Area-Selective Deposition of Ruthenium by Area-Dependent Surface Diffusion

“…Modeling island-growth on SiO2. To quantify the relative importance of adsorption and surface diffusion during Ru CVD on SiO2 and the effect of surface termination, we linked experimental observables to atomistic mechanisms through a rate-equation mean-field modelling approach that builds on the theory of nucleation and growth of thin films 28, 41,[59][60][61][62] . This model is a variant of the ones developed in our previous works [31][32]34 for island-growth in ALD of noble metals.…”

“…terms are calculated using the self-consistent method proposed by Hubartt et al 64 for irreversible island growth with cluster diffusion as = + , where and are the diffusion coefficient and the so-called capture number of a cluster of size , respectively. The capture numbers are slow-varying correction factors that account for the fact that, by capturing adatoms and other clusters in their vicinity, clusters introduce local gradients in , thus effectively creating depletion zones, which depend on their size as compared to the average diffusion length (or capture length), at each given time.…”

“…The capture numbers therefore allow one to reconcile the diffusion problem at the nanoparticle scale with the global mean-field balance defined by Equations 1 and 2. As prescribed by Hubartt et al 64 , we calculate the capture numbers at each time step by solving the coupled equations: −2 = ∑ and = 2 1 ( ) / 0 ( ). where 0 and 1 are the modified Bessel functions of order zero and one, respectively.…”