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Carbon, being one of the most versatile elements of the periodic table, formssolids and molecules with often unusual properties. Recently, a novel family ofthree-dimensional graphitic carbon structures, the so-called hyperhoneycomblattices, has been proposed, with the possibility of being topologicalinsulators [K. Mullen, B. Uchoa and D. T. Glatzhofer, Phys. Rev. Lett. 115,026403 (2015)]. In this work, we present electronic structure calculations forone member ($\mathcal{H}$-0) of this family, using Density Functional Theoryand non-equilibrium Green's functions transport calculations to show that the$\mathcal{H}$-0 structure should have strongly anisotropic electronicproperties, being an insulator or a conductor depending on the crystallineorientation chosen for transport. Calculations in the framework of ExtendedH\"uckel Theory indicate that these properties can only be understood if oneconsiders 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 particularmaterial.

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