Systematics of Toroidal, Helically-Coiled Carbon Nanotubes, High-genus Fullernens, and Other Exotic Graphitic Materials

Abstract

We systematically examine the tiling rules of graphene on various 3D surfaces including torus, helicoid, high-genus closed surface, doubly periodic quasi-minimal surface, and torus knot. Firstly, a thorough examination on the problem of finding possible isomers of highly symmetric toroidal carbon nanotubes (TCNT), which are isomorphic to torus, is presented. We then show that a particular class of TCNTs with n-fold rotational symmetry and well-defined latitude coordinates, uniquely characterized by a set of four indices, can be used as starting building blocks for constructing graphitic structures with nontrivial topology. Particularly, we discovered that complex graphitic structures are categorized into two major groups: (A) Tubu-lar CNTs as complex as torus knots can be obtained by altering the nonhexagon distribution on the molecular surface. In general, carbon nanotubes along any space curve such as HCCNT, can be approached with the geometric manipulation schemes (B) A large family of porous graphitic structures, either closed (0D) or extended (2D, 3D), can be classified by assembling suitable sets of neck-like structures. The neck structure is obtained by peeling the outer-rim of a TCNT off leaving the central hole unchanged. Depending on the occasion, necks with different geometric features are used and the resulting porous molecules can be high-genus fullerenes (0D), doubly periodic supergraphene (2D), or triply periodic quasi-minimal surfaces (3D).