  Nucleation and growth mechanisms of short chains of carbon atoms on
single-layer, hexagonal boron nitride (h-BN), and short BN chains on graphene
are investigated using first-principles plane wave calculations. Our analysis
starts with the adsorption of a single carbon ad-atom and examines its
migrations. Once a C$_2$ nucleates on h-BN, the insertion of each additional
carbon at its close proximity causes a short segment of carbon atomic chain to
grow by one atom at at a time in a quaint way: The existing chain leaves its
initial position and subsequently is attached from its bottom end to the top of
the carbon ad-atom. The electronic, magnetic and structural properties of these
chains vertically adsorbed to h-BN depend on the number of carbon atoms in the
chain, such that they exhibit an even-odd disparity. An individual carbon chain
can also modify the electronic structure with localized states in the wide band
gap of h-BN. As a reverse situation we examined the growth of short BN atomic
chains on graphene, which attribute diverse properties depending on whether B
or N is the atom bound to the substrate. These results together with ab-initio
molecular dynamics simulations of the growth process reveal the interesting
self-assembly behavior of the grown chains. Furthermore, we find that these
atomic chains enhance the chemical activity of h-BN and graphene sheets by
creating active sites for the bonding of various ad-atoms and can act as
pillars between two and multiple sheets of these honeycomb structures leaving
wider spacing between them to achieve high capacity storage of specific
molecules.
