Tubular graphic carbon structures

Abstract

We define two classes of infinite graphitic carbon tubes (buckytubes) depending on the underlying morphology of their hexagonal ring structures. The principal axis of a cylindrical (1, 0) tube (type one, helical pitch zero) is parallel to two of the sides of each regular hexagon, and the orientation of hexagon sides in a (2, 0) cylinder (type two, pitch zero) lies at 90° to the tube axis. Helical graphitic tubes where the pitch is larger than zero can also be constructed. The tubes in which the helical pitch takes on the values zero and unity are considered in this work. Based on HMO level calculations, we obtain general formulae and/or recurrence relationships for characteristic polynomials of these tubes. From the formulae, it is found that (1,0) tubes with a circumference of 3R hexagonal rings (R is an integer) have zero HOMO-LUMO band gaps; other (1, 0) tubes always have nonzero band gaps. This pattern is slightly altered when the pitch equals one. In this case, the (1,1) tubes with a circumference of 3R-1 rings are found to have zero band gaps. In the type two systems all (2,0) tubes are predicted to have zero band gaps, whereas all (2,1) tubes have finite band gaps. All carbon structures treated in this paper may be qualitatively characterized as ‘graphitic.’ However, only the pitch zero tubes extrapolate exactly to graphite for infinite size. We speculate on the relationships of these structures and their calculated properties to experimentally observed structures and postulated growth mechanisms.