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The electronic nonlinear transport through ultra narrow graphene nanoribbons(sub-$10nm$) is studied. A stable region of negative differential resistance(NDR) appears in the I-V characteristic curve of {\it odd} zigzag graphenenanoribbons (ZGNRs) in both positive and negative polarity. This NDR originatesfrom a transport gap inducing by a selection rule which blocks electrontransition between disconnecting energy bands of ZGNR. Based on this transitionrule, on/off ratio of the current increases exponentially with the ribbonlength up to $10^5$. In addition, charging effects and also spatialdistribution of bond currents was studied by using non-equilibrium Green'sfunction formalism in the presence of e-e interaction at a mean field level. Onthe other hand, we also performed an {\it ab initio} density functional theorycalculation of transmission through a passivated graphene nanoribbon todemonstrate robustness of the transport gap against hydrogen termination of thezigzag edges.

Nonlinear Transport through ultra-narrow Zigzag Graphene Nanoribbons: non-equilibrium charge and bond currents