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Massless Dirac particles on the helicoid are theoretically investigated. Withits possible application being helical graphene, we explore how thepeculiarities of Dirac particles appear on the curved, screw-symmetric surface.The zweibein is used to derive the massless Dirac equation on the helicoid, aswell as general curved surfaces. Bound states of massless Dirac electrons areshown to be absent on the helicoid, and then the scattering probabilities andthe phase shifts on the surface are obtained from numerically calculated wavefunctions. We find the local density of states and the phase shiftsidiosyncratic especially around the axis of the helicoid. Bound states ofmassive Dirac electrons on the surface are also shown to be absent as anextension of the above result on massless Dirac electrons.

Electronic structure of helicoidal graphene: massless Dirac particles on a curved surface with a screw symmetry