Dislocations and defects in graphene

Graphene is a two dimensional material whose surprising properties have arisen promising expectatives. Mathematically, it is described by an hexagonal lattice. This lattice exhibits geometrical defects: heptagon-pentagon pairs, Stone-Wales defects, vacancies, divacancies. These defects can be described as cores of edge dislocations and dislocation groupings. Their long time stability is then understood analyzing the response of isolated defects to applied forces. In simpler two dimensional lattice models, we argue that the onsets of both dislocation motion and dislocation nucleation correspond to different types of bifurcations. Dislocation motion is associated to global bifurcations in a branch of stationary dislocation solutions to give raise to traveling wave solutions. Homogeneous dislocation nucleation may be described by pitchfork bifurcations in simple geometries. As the way forces are applied changes, the bifurcation diagram varies. In real graphene lattices, curvature effects are important, specially at the core of defects. These effects are captured by discrete versions of Foppl-Von Karman equations in hexagonal lattices