Graphene ribbons 

Graphene ribbons are one dimensional stripes of graphene. They can be considered as unfolded carbon nanotubes. Graphene ribbons were proposed by Nakada et al in 96 (URL). Using the same single orbital tight-binding model that succesfully portraits two dimensional graphene as a semimetal,  graphene ribbons are either metallic or semiconducting depending on their crytallographic orientation and on their width.  More realistic calculations using the Hubbard model in a mean field approximation and Density functional calculations, show that zigzag ribbons are insulating due to the magnetization of their edges with opposite spin orientation in each edge.  I have found that this anfiterromagnetic insulator phase has a hidden underlying ferroelectric order which can be described as excitonic insulator whose order parameter is the spin-resolved dipole operator, the analog of the spin current operator (see paper 4 in the list below).

Ferrimagnetic zigzag ribbon (see paper 4) Figure created with xcrysden	Graphene nanoconstriction.  (see paper 1) (Figure created with molden, courtesy of F. Muñoz-Rojas)

Field effect transistor based on graphene armchair ribbon constriction  (see papers 2 and 3) (Figure  courtesy of F. Muñoz-Rojas)

Our first papers focused on the transport and electrical properties of graphene ribbons. In (1) we have studied coherent quantum transport accross a narrow constriction in graphene zigzag ribbons.  In (2) we calculated the relation between gate voltage and induced density both in graphene and armchair graphene ribbons. In the case of 2D graphene we found an anomalous quantum contribution to the capacitance  related to the linear bands in graphene.   transport.