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We explore the use of applied electrical potentials to induce band flatteningin graphene for bands near zero energy. We consider various families of doublyperiodic potentials and simulate their effect on the electronic band structureusing a tight-binding and a continuum approach. From these families, we findthat an applied potential with symmetries of wallpaper group 17, in particulara Kagome potential, works best for inducing a high degree of band flatteningfor a range of realistic potential amplitudes and periods. Our work indicatesthat it should be possible to engineer the band structure of graphene usingelectrostatic gating, thus enabling a new approach to the development ofgraphene-based metamaterials.

Theory of engineering flat bands in graphene using doubly-periodic electrostatic gating