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.
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