Braiding of edge states in narrow zigzag graphene nanoribbons: effect of the third neighbors hopping

We study narrow zigzag graphene nanoribbons (ZGNRs), employing density functional theory (DFT) simulations and the tight-binding (TB) method. The main result of these calculations is the braiding of the conduction and valence bands, generating Dirac cones for non-commensurate wave vectors $\vec{k}$. Employing a TB Hamiltonian, we show that the braiding is generated by the third-neighbor hopping (N3). We calculate the band structure, the density of states and the conductance, new conductance channels are opened, and the conductance at the Fermi energy assumes integer multiples of the quantum conductance unit $G_{o} = 2e^{2}/h$. We also investigate the satisfaction of the Stoner criterion by these ZGNRs. We calculate the magnetic properties of the fundamental state employing LSDA (spin-unrestricted DFT) and we confirm that ZGNRs with $N=(2,3)$ do not satisfy the Stoner criterion and as such the magnetic order could not be developed at their edges. These results are confirmed by both tight-binding and LSDA calculations.