  Grain boundary roughness can affect electronic and mechanical properties of
two-dimensional materials. This roughness depends crucially on the growth
process by which the two-dimensional material is formed. To investigate the key
mechanisms that govern the roughness, we have performed kinetic Monte Carlo
simulations of a simple model that includes particle attachment, detachment,
and diffusion. We have studied the closure of the gap between two flakes during
growth, and the subsequent formation of the grain boundary (GB) for a broad
range of model parameters. The well known near-equilibrium (attachment-limited)
and unstable (diffusion-limited) growth regimes are identified, but we also
observe a third regime when the precursor flux is sufficiently high to fully
cover the gap between the edges. This high coverage regime forms GBs with
spatially uncorrelated roughness, which quickly relax to smoother
configurations. Extrapolating the numerical results (with support from a
theoretical approach) to edge lengths and gap widths of some micrometers, we
confirm the advantage of this regime to produce GBs with minimal roughness
faster than near-equilibrium conditions. This suggests an unexpected route
towards efficient growth of two-dimensional materials with smooth GBs.
