Island density scaling in reversible submonolayer growth with attachment barriers

Island nucleation and growth play an important role in thin-film growth. One quantity of particular interest is the exponent χ, which describes the dependence of the saturation island density Nₛₐₜ~(Dₕ/F)^-χ on the ratio Dₕ/F of the monomer hopping rate Dₕ to the deposition rate F. While standard rate equation (RE) theory predicts that χ=i/(i+2) (where i is the critical island size), more recently it has been predicted that in the presence of a strong barrier to the attachment of monomers to islands, a significantly larger value χ=2i/(i+3) may be observed. While this prediction has recently been tested using kinetic Monte Carlo simulations for the case of irreversible growth corresponding to $i=1$, it has not been tested for the case of reversible island growth corresponding to $i>1$. Here we present a mean-field self-consistent RE method which we have used to study the dependence of the effective value of χ on Dₕ/F and barrier-strength for $i=1,2,3$, and 6. Both the no-nucleation-barrier case in which there exists a barrier for monomers to attach to islands larger than the critical island size (but not to smaller islands) and the nucleation-barrier case in which there is a barrier for monomers to attach to islands of all sizes are studied. In all cases, we find that the existence of attachment barriers significantly increases the effective value of χ for a given barrier strength. In addition, for $i=1$ we find good agreement between our extrapolated asymptotic value of χ and the theoretical strong-barrier prediction both with and without a nucleation barrier. In contrast, for $i>1$ the value of χ is significantly larger in the presence of a nucleation barrier than in its absence. In particular, while an asymptotic analysis of our results for $i>1$ also leads to excellent agreement with the strong barrier prediction in the presence of a nucleation barrier, in the absence of a nucleation barrier the asymptotic values are significantly lower.