Submonolayer island growth with anomalous diffusion

Island nucleation and growth play an important role in the early stages of thin-film growth. Of particular interest is the exponent $\chi$ which describes the dependence of the peak island density on deposition flux, and which also depends sensitively on the critical island-size $i$. While the dependence of $\chi$ on $i$ is known for normal diffusion, the case of anomalous monomer diffusion is also of interest, since this appears to play a role in recent experiments. Here we derive general expressions for $\chi$ which are valid for arbitrary substrate dimension, island fractal dimension, critical island size, and monomer diffusion exponent $\mu$. Excellent agreement is obtained between our predictions and kinetic Monte Carlo simulations carried out for the case of irreversible growth ($i=1$), and monomer superdiffusion with $1<\mu\le 2$, although unusually large crossover effects are also observed. These results also confirm and generalize a previous prediction for the case of ballistic diffusion ($\mu=2$). We also consider the case of monomer subdiffusion corresponding to $0\le\mu < 1$. Good agreement with our predictions for $\chi(\mu)$ is also found in this case, although the general scaling behavior is more complex due to the presence of large fluctuations.