Abstract
We demonstrate the existence of a new type of zero energy state associated to vacancies in multilayer graphene that has a finite amplitude over the layer with a vacancy and adjacent layers, and the peculiarity of being quasi-localized in the former and totally delocalized in the adjacent ones. In a bilayer, when a gap is induced in the system by applying a perpendicular electric field, these states become truly localized with a normalizable wavefunction. A transition from a localized to an extended state can be tuned by the external gate for experimentally accessible values of parameters.
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