Multiscale modeling of island nucleation and growth during $\mathrm{Cu}\left(100\right)$ homoepitaxy

The long-time scale dynamics of small $\mathrm{Cu}∕\mathrm{Cu}\left(100\right)$ islands are studied. Atomistic simulations using embedded atom method (EAM) potentials and the dimer method saddle point searches provide pathways and their temperature-dependent rates to lattice-based kinetic Monte Carlo (KMC) simulations. The KMC utilizes translational symmetry to identify previously visited sites and re-use the atomistic rates. As a result very long time scales are accessible to the simulation which reveals the dissociation as well as the diffusion mechanisms of the small islands in an unbiased manner. Our results for island diffusion reproduce well the activation energies calculated in previous work, and provide in addition the associated frequency prefactors. The island dissociation pathways are rationalized in terms of previously anticipated mechanisms. We also utilize our results in mean field rate equations to predict “kinetic phase diagrams” for the critical island size as a function of temperature and vapor deposition rate during $\mathrm{Cu}\left(100\right)$ homoepitaxy. We predict that the higher critical island sizes $(i>2)$ should be observable at higher temperatures (above $\sim 500\mathrm{K}$) at experimentally accessible deposition rates.

Figure 1

Diagrammatic representation of the self-learning KMC simulation. The KMC uses lattice “animals” [column (a)] to represent the island configuration. New configurations are passed to the atomistic simulation illustrated in column (b). The atomistic simulation performs dimer method saddle point searches to find possible pathways out of the current configuration, along with their associated activation energies and frequency prefactors. This “event list” is returned to the KMC which selects the next configuration and increments the simulation clock as usual. If the new configuration has been visited before, then no new searches need be initiated in the atomistic simulation.

Figure 2

Results for the center-of-mass movement of a tetramer island at $500\mathrm{K}$. In (a) and (b) only 50 dimer method searches were used for each configuration visited; (a) shows 5600 steps whereas (b) shows several million steps. In (c) several million steps are shown using 250 saddle point searches for each configuration.

Figure 3

Arrhenius plots for the dynamics of islands of various size: (a) Shows center-of-mass hopping rates, and (b) shows island dissociation rates.

Figure 4

Island mobility results from this work. In (a) we show the temperature at which the island will remain stationary for $1\mathrm{s}$ (on average). In (b) and (c) we show the frequency prefactors and the activation energies, respectively, for island hopping.

Figure 5

The main dissociation pathways for a trimer island discovered in our self-learning KMC simulations.

Figure 6

Some dissociation pathways for a tetramer island discovered in our self-learning KMC simulations.

Figure 7

Results for the island dissociation mechanisms discovered in the self-learning KMC simulation. In (a) we show the temperature at which islands are stable for $1\mathrm{s}$ (on average). In (b) and (c) we show the frequency prefactors and the activation energies, respectively, for island dissociation.

Figure 8

Results from the mean field rate equations for the island densities vs temperature, taken at 10% substrate coverage with $F=0.005\mathrm{ML}∕\mathrm{s}$. In (a) only monomers are mobile and all islands are stable; in (b) islands sizes 2–8 can also diffuse; and in (c) islands sizes 2–8 can dissociate and diffuse.

Figure 9

Results from the mean field rate equations for the total island density vs deposition rate, taken at 10% substrate coverage. In (a) only monomers are mobile and all islands are stable; in (b) island sizes 2–8 can dissociate and diffuse.

Figure 10

Kinetic phase diagrams showing regimes of critical island size predicted by this work. In (a) only monomers are mobile and all islands are stable; in (b) islands sizes 2–8 can also diffuse; and in (c) islands sizes 2–8 can dissociate and diffuse.