LAUSR

In addition to the well-known anti-ferromagnetic and ferromagnetic edge states in zigzag graphene nanoribbons (GNR), we find that there also exist some excited spin density wave (ESDW) states, the energies of which are close to the anti-ferromagnetic state (ground state). We thus argue that these ESDW states may coexist in experiment. Our numerical results from the self-consistent mean-field method as well as the first-principles calculations indicate that the allowed ESDWs are commensurate; and their dispersion curves are linear in the long wave limit. The coupling of the two edge portions in an ESDW becomes very weak in a wide GNR, in which case these ESDW portions may have different phases. Finally our calculations in the non-equilibrium Green's function theory combined with the Hubbard model show that these ESDW states can also exist in some localized (middle or terminal) region of GNR.