Optimal traps in graphene

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We transform the two-dimensional Dirac-Weyl equation, which governs the charge carriers in graphene, into a non-linear first-order differential equation for scattering phase shift, using the so-called variable phase method. This allows us to utilize the Levinson Theorem to find zero-energy bound states created electrostatically in realistic structures. These confined states are formed at critical potential strengths, which leads to us posit the use of `optimal traps' to combat the chiral tunneling found in graphene, which could be explored experimentally with an artificial network of point charges held above the graphene layer. We also discuss scattering on these states and find the zero angular momentum states create a dominant peak in scattering cross-section as energy tends towards the Dirac point energy, suggesting a dominant contribution to resistivity.