Anisotropic electronic structure and transport properties of the $\mathcal{H}$-$0$ hyperhoneycomb lattice

Carbon, being one of the most versatile elements of the periodic table, forms solids and molecules with often unusual properties. Recently, a novel family of three-dimensional graphitic carbon structures, the so-called hyperhoneycomb lattices, has been proposed, with the possibility of being topological insulators [K. Mullen, B. Uchoa and D. T. Glatzhofer, Phys. Rev. Lett. 115, 026403 (2015)]. In this work, we present electronic structure calculations for one member ($\mathcal{H}$-0) of this family, using Density Functional Theory and non-equilibrium Green's functions transport calculations to show that the $\mathcal{H}$-0 structure should have strongly anisotropic electronic properties, being an insulator or a conductor depending on the crystalline orientation chosen for transport. Calculations in the framework of Extended H\"uckel Theory indicate that these properties can only be understood if one considers at least $2^{nd}$ nearest-neighbor interactions between carbon atoms, invalidating some of the conclusions of Ref. [K. Mullen, B. Uchoa and D. T. Glatzhofer, Phys. Rev. Lett. 115, 026403 (2015)], at least for this particular material.