Polymorphism of carbon forms: Polyhedral morphology and electronic structures

We consider the morphology of fullerene families which have negative Gaussian curvature, formed by the introduction of seven- or eight-membered rings in a graphite sheet. The existence of seven- or eight-membered rings makes it possible to form a sponge-shaped periodic graphite network of carbon atoms in three-dimensional space. We first propose a construction method for the structures, which we will call spongy graphite, based on polyhedral geometry. Infinite spatial networks of equilateral triangles and hexagons can be obtained by space filling or open packing of regular icosahedra, octahedra, tetrahedra, and truncated octahedra which have triangular or hexagonal faces. We demonstrate the formation of various types of spongy graphite by decorating each face with honeycomb patches, by which we mean triangular or hexagonal fragments of a graphite sheet. Further, we study their electronic structures using a tight-binding model for the network of π electrons, and find that such networks show a variety of electronic states including metal, zero-gap semiconductor, and insulator, depending on the geometrical parameters.