Bumpy graphene membranes: a roadmap for flat-band engineering and topological transitions.*

Graphene’s high flexibility is a property that allows using strain engineering to control electronic properties. Wrinkled or rippled graphene -suspended or on substrates-develops inhomogeneous charge distributions due to underlying strain fields. STM measurements on locally deformed samples demonstrated electron confinement with broken sublattice symmetry and induced valley currents. Motivated by these ideas, we analyzed charge distributions in models of graphene with several Gaussian-shaped deformations arranged in different geometries. Our work revealed moiré-like patterns with pockets of localized charges like those reported for twisted bilayer graphene. Next, we modeled experimental settings of graphene on top of substrates with periodic arrays of deformations. Strains induced by the substrate geometry modify electron dynamics and suggest a practical method for band structure engineering. We identify geometries for the emergence of flat bands and optimal parameters for maximal gaps. Surprisingly, electronic states separate into ‘trivial’-bound and ‘percolating'-extended states that coexist in spatial regions with distinct lattice properties. For a wide range of geometries, and breaking of valley degeneracy, flat bands acquire topological properties with the corresponding emergence of valley chiral edge states.