We present exact analytical zero-energy solutions for a class of smooth
decaying potentials, showing that the full confinement of charge carriers in
electrostatic potentials in graphene quantum dots and rings is indeed possible
without recourse to magnetic fields. These exact solutions allow us to draw
conclusions on the general requirements for the potential to support fully
confined states, including a critical value of the potential strength and
spatial extent.Comment: 8 pages, 3 figures, references added, typos corrected, discussion
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