Three-dimensional massless Dirac fermions in carbon schwarzites

The discovery of gapless linear energy dispersion in low-dimensional carbon-based nanostructures had a tremendous impact on conventional condensed-matter physics by imposing relativistic physics in the electronic properties of these nanosystems. Indeed, the electrons in graphene [two-dimensional (2D)] and carbon nanotubes [one-dimensional (1D)] behave like pseudorelativistic massless particles (Dirac fermions) as described by the crossing of linearly dispersive electronic bands, also called the Dirac cone. The presence of Dirac cones is not restricted to 1D or 2D nanostructures and has recently been observed in several more complex 3D materials. Here, using density functional theory and tight-binding approaches, we predict the presence of a Dirac cone in a 3D carbon-based material. Indeed, our simulations reveal a linear band crossing merging in a point forming a Dirac (hyper)cone for a large gyroidal schwarzite structure. Such a specific linear dispersion relation as reported in this 3D negatively curved sp2-bonded carbon allotrope is believed to be a direct consequence of the Dirac cone present in 2D graphene. The corresponding charge carriers are thus expected to behave as 3D massless Dirac fermions. Therefore, we expect this prediction to stimulate the experimental synthesis of such fascinating 3D bulk carbon allotropes, which are a remarkable playground to investigate relativistic physics of these exotic fermions