Geometrical perturbation of graphene electronic structure

Abstract

We discuss low-energy electronic properties of distorted graphene sheets from a local geometric viewpoint, treating curvature and strain as perturbations of a smooth surface. This allows a unified description of the variety of deformations to which carbon nanotubes are susceptible. By using a general symmetry analysis in conjunction with a four-orbital, nonorthogonal tight-binding model, we calculate accurate values of the relevant couplings. Carbon nanotubes 1 have a high Young’s modulus, but are not otherwise mechanically strong, being easily bent, or deformed by van der Waals interactions with a substrate. 2 Hence, the electronic repercussions of mechanical deformations are of interest for both fundamental and device-oriented reasons. Kane and Mele 3 have studied this issue with the aid of a tight-binding model incorporating only p electrons. We use a model-independent group-theoretical analysis to reveal the relevant couplings, which are then evaluated with a fourorbital non-orthogonal tight-binding model that takes into account the strong effects of rehybridization. By expressing curvature and strain as a modification of the Hamiltonian and overlap, we avoid the use of the large unit cells implied by small deformations. The response of electronic structure to curvature and strain is essentially a local coupling, largely independent of boundary conditions or global topology. We therefore isolate the effects of geometric deformation by first studying a single graphene sheet. Although our perturbative approach is in principle limited to gentle distortions, the results are in good agreement with full tight-binding calculations for a radius of curvature as small as two graphene lattice spacings @i.e., a ~12,0! tube#. The conformation of a surface in space ~e.g., a single graphene sheet! is described by a vector-valued embedding function X(x m ), specifying the location of the point with coordinates x m ,m51,2. The pair of vectors tm5]X/]x m ,(m 51,2) provide a basis for the tangent space and determine the metric g mn 5t m it n , which converts coordinate differences into physical distances. If the coordinates are determined relative to a flat unstretched state, the metric carries information about strain. The inverse of this matrix is denoted g ls , so that g lm gmn5d l n . Curvature is detected by second derivatives 4,5 of the embedding function X,