Hybrid method for modeling epitaxial growth: Kinetic Monte Carlo plus molecular dynamics

We propose a concurrently coupled hybrid molecular dynamics (MD) and kinetic Monte Carlo (KMC) algorithm to simulate the motion of grain boundaries between fcc and hcp islands during epitaxial growth on a fcc (111) surface. The method combines MD and KMC in an adaptive spatial domain decomposition, so that near the grain boundary, atoms are treated using MD but away from the boundary atoms are simulated by KMC. The method allows the grain boundary to interact with structures that form on spatial scales significantly larger than that of the MD domain but with a negligible increase in computational cost.

Figure 1

(Color online) (Top) Snapshots showing the (instantaneous) decomposition of the system into KMC domains (far left and far right) and a MD domain (center) of width $w$, centered on a grain boundary. The atoms on the right occupy fcc sites, while the atoms on the left occupy hcp sites. (Bottom) A snapshot and a schematic representation of a typical grain boundary configuration showing a mixture of $A$- and $B$-type boundaries and a kink site where they meet.

Figure 2

(Color online) (Top) Evolution of the grain boundary position (defined as the average position of the boundary across the short direction of the slab) during typical simulations using the full MD method, the hybrid method with $w=30\sigma$, and the hybrid method with $w=5\sigma$. The mobility of the boundary in the $w=5\sigma$ MD domain is clearly suppressed. (Bottom) The mean square displacement $\overline{\Delta x^2}$ versus $\Delta t$ for the full MD simulation and for the hybrid method for various MD domain sizes.

Figure 3

(Color online) The dependence of the logarithm of the grain boundary diffusion coefficient $D$ on temperature, calculated using the hybrid method. The slope of the plot gives an energy barrier $\Delta E=0.70ϵ$. As expected, this is very close to the barriers for kink migration (shown in the inset are the important hcp to fcc kink migration steps), calculated using the NEB method, which limits the diffusion of the boundaries.

Figure 4

The figure illustrates the change in energy barrier as calculated by the NEB method for a kink migration step as a function of the distance between kink site and the rigid KMC boundary.

Figure 5

(Color online) (Top) Time series showing the movement of grain boundaries at different adatom coverages (0%, 2.5%, and 30%, respectively). At 30% coverage, the boundaries are generally pinned by adatoms and adatom islands, although motion can occur when the boundaries pass under islands (two such events are noted by arrows). At 2.5%, the pinning is generally not observed, although mobility is reduced as the boundaries encounter single adatoms. (Bottom) The mean square displacement $\overline{\Delta x^2}$ of the boundary versus $\Delta t$ for a range of adatom coverages from 0% to 60%. The inset shows one of the main mechanisms for pinning of the boundaries: a kink site decorated by an adatom in an off-lattice position.