We present here a brief review on the remarkable consequences of the flat bands formed at the edges of the Zigzag graphene nanoribbon (ZGNR). The inclusion of the on-site Coulomb interaction is shown to induce the edge spin ferromagnetism, whose spin stiffness demonstrates a nonmonotonic dependence on the lateral electric field. The critical electric field strength corresponds to that of the insulator to half-metal transition. The inclusion of the spin–orbit coupling (SOC) has been believed to generate the quantum spin Hall effect (QSHE) guiding into the interesting new field of topological insulator. By carefully investigating the SOC near the edge, we have shown that the additional σ-edge band gives a marginal perturbation and hence the existence of the QSHE depends on the coupling strength between the π-edge bands and the σ-edge band. We demonstrate that for the charge neutral ZGNR, the QSHE does not occur in the pristine ZGNR, while the hydrogen passivation along the edge may recover the expected feature of the QSHE.
Electric field control of the indirect magnetic coupling through a short graphene nanoribbon
