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We present exact analytical zero-energy solutions for a class of smoothdecaying potentials, showing that the full confinement of charge carriers inelectrostatic potentials in graphene quantum dots and rings is indeed possiblewithout recourse to magnetic fields. These exact solutions allow us to drawconclusions on the general requirements for the potential to support fullyconfined states, including a critical value of the potential strength andspatial extent.

Zero-energy states in graphene quantum dots and rings