Scaling relation of domain competition on（2＋1）-dimensional ballistic deposition model with surface diffusion

During heteroepitaxial overlayer growth multiple crystal domains nucleated on a substrate surface compete with each other in such a manner that a domain covered by neighboring ones stops growing.The number density of active domains ρ decreases as the height h increases.A simple scaling argument leads to a scaling law of ρ~ h~（-γ） with a coarsening exponent γ=d/z,where d is the dimension of the substrate surface and z the dynamic exponent of a growth front.This scaling relation is confirmed by performing kinetic Monte Carlo simulations of the ballistic deposition model on a two-dimensional（d=2） surface,even when an isolated deposited particle diffuses on a crystal surface.