Dirac electrons on three-dimensional graphitic zeolites: A scalable mass gap

A class of graphene wound into three-dimensional periodic curved surfaces ("graphitic zeolites") is proposed and their electronic structures are obtained to explore how the massless Dirac fermions behave on periodic surfaces. We find in the tight-binding model that the low-energy band structure around the charge neutrality point is dominated by the topology (cubic or gyroid) of the periodic surface as well as by the spatial period L in modulo 3 in units of the lattice constant. In both cubic and gyroid cases the Dirac electrons become massive around the charge neutrality point, where the band gap is shown to scale as 1/L within each mod-3 class. Wave functions around the gap are found to have amplitudes sharply peaked around the topological defects that are required to deform the graphene sheet into a three-dimensional periodic surface, and this is shown to originate from nontrivial Bloch phases at K and K' points of the original graphene.