Low-Energy Termination of Graphene Edges via the Formation of Narrow Nanotubes

We demonstrate that free graphene sheet edges can curl back on themselves, reconstructing as nanotubes. This results in lower formation energies than any other nonfunctionalized edge structure reported to date in the literature. We determine the critical tube size and formation barrier and compare with density functional simulations of other edge terminations including a new reconstructed Klein edge. Simulated high resolution electron microscopy images show why such rolled edges may be difficult to detect. Rolled zigzag edges serve as metallic conduction channels, separated from the neighboring bulk graphene by a chain of insulating $s{p}^3$-carbon atoms, and introduce van Hove singularities into the graphene density of states.

Figure 1

Formation energy ($\mathrm{eV}/\AA$) for different graphene edges (dashed lines indicated on the figure: Klein, reconstructed Klein, zigzag, armchair, and 5-7 reconstructed zigzag), freestanding zigzag and armchair tubes, armchair-, armchairlike-, and zigzag-tube terminated edges.

Figure 2

Upper panels: Structure of (4,4) armchair (a), (4,4) armchairlike (b), and (8,0) zigzag (c) nanotube-terminated graphene sheet. $s{p}^3$-like coordinated carbon atoms are marked in light gray (yellow). Bottom panels: Corresponding simulated HRTEM images.

Figure 3

Calculated density of states (DOS) for (a)  (8,8) armchair- and (b) (8,0) zigzag-nanotube terminated graphene edge. The van Hove singularities are marked by arrows.

Figure 4

(a) Band structure for (8,8) armchair-nanotube terminated edges and (b) distribution of highest occupied state at the $\Gamma$ point and $X$ point.