Topology and electronic structure of nanotube junctions of tetrapod shape

We propose a series of carbon nanostructures in the shape of tetrapod as a kind of three-dimensional junction for carbon nanotubes. The tetrapod junctions are such open networks that are made of sp2 carbon atoms only, have negative Gaussian curvature, and connect four nanotubes together. We define the structure of standard tetrapod junctions, the simplest one, that have 12 heptagons other than hexagons and have the Td symmetry.Our tight-binding energy-band calculations for the standard tetrapod junctions of smaller sizes show that their electronic property mainly depends on one particular topological factor: the junctions having a carbon atom in the center of each triangular face of tetrahedron exhibit metallic band structure while the junctions having a benzene ring in the center of the faces are semiconductors. We also find that tetrapod junctions connecting (6,0) nanotubes exhibit a flat band near the Fermi energy in a particular momentum region. The origin of the flat band states can be figured out from the wavefunction distribution. We also show the possibility to extend the standard tetrapod junctions to some non-standard ones that can connect nanotubes of different kinds and/or radii.