Island nucleation and growth play an important role in thin-film growth. One quantity of particular interest is the exponent $\chi$, which describes the dependence of the saturation island density $N_{\text{sat}}\sim {(D_h/F)}^{-\chi }$ on the ratio $D_h/F$ of the monomer hopping rate $D_h$ to the deposition rate $F$. While standard rate equation (RE) theory predicts that $\chi =i/(i+2)$ (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 $\chi =2i/(i+3)$ 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 $\chi$ on $D_h/F$ 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 $\chi$ for a given barrier strength. In addition, for $i=1$ we find good agreement between our extrapolated asymptotic value of $\chi$ and the theoretical strong-barrier prediction both with and without a nucleation barrier. In contrast, for $i>1$ the value of $\chi$ 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.

• Received 6 November 2023
• Accepted 19 February 2024

DOI:https://doi.org/10.1103/PhysRevE.109.034803