Topologically-Induced Gapping in Graphene

Disclinated sp² carbon surfaces show a clear pattern in their density of states (DOS) with the separation of these disclinations. This arises from the effect of the topology of the π-bonding network on phase matching around these disclinations, and can produce a striking scaling behavior of the DOS to a level of fine detail within an energy window determined by the areal density of the disclinations. Motivated by the presumed existence of a low-energy continuum description of the system, which should give a scaling of DOS that varies inversely with the separation of disclinations, we look for this scaling and find it exists in families grouped by their separation in distance in the bond network, much like the (n-m) mod 3 rule for carbon nanotubes. These groupings (and their associated spacings mod 3) predict whether a specific structure has a gapped DOS at the Fermi energy, and the size of the gaps in the gapped structures are of the order predicted by a low-energy continuum model description. Notably, this behavior (including the scaling of the DOS) persists down to separations where the disclinations are not significantly distant from each other, and a continuum model would not be anticipated to work well.