Single-Valley Engineering in Graphene Superlattices

The two inequivalent valleys in graphene preclude the protection against inter-valley scattering offered by an odd-number of Dirac cones characteristic of $Z_2$ topological insulator phases. Here we propose a way to engineer a chiral single-valley phase in a honeycomb lattice via folding K and K$'$ valleys onto $\Gamma$ point in tailored $\sqrt{3}N\times\sqrt{3}N$ or $3N\times 3N$ superlattices. The corresponding effective Hamiltonians for top-site adatom adsorption leads to inter-valley coupling and valley-orbit coupling mechanisms that resemble the conventional in-plane Zeeman fields and spin-orbit coupling of the electron spins that have important implications in valleytronics to control the valley polarization coherently. By folding the inequivalent K and K$'$ valleys together, the single valley phase can be formed and engineered by the the inter-valley coupling and staggered sublattice potentials from inversion symmetry breaking. A topological phase transition can take place from the quantum valley-Hall phase at large staggered sublattice potential to a chiral single-valley phase with quadratic band crossover resembling the electronic structure of one \textit{half} AB-stacked bilayer graphene for sufficiently strong inter-valley coupling.