Do We Really Understand Graphene Nanoribbons? A New Understanding of the 3n, 3n ± 1 Rule, Edge “Magnetism”, and Much More

Using the inherent shell structure of graphene and geometrical/topological constrains, we verify that there are only three families of armchair graphene nanoribbons (AGNR) with Z zigzag edge-rings, categorized by Z = 3n, 3n ± 1, n = 1, 2, ..., each with unique aromatic, electronic and topological properties. The Z = 3n + 1, 3n AGNR families are aromatic with large bandgaps, characteristic aromaticity patterns, and unique “active” frontier orbitals, in contrast to the ordinary ones. Such AGNRs due to sublattice/molecular-group symmetry-conflict develop 2n zigzag-edge-localized “gapless” frontier-states, which are “pseudospin-polarized” (not real-spin-polarized) with total pseudospin S = n, effectively optimizing sublattice “balance” and total energy. The “active” frontier orbitals, obtained after neglecting such gapless nonbonding states, have large “active” bandgaps, which are in very good agreement with experiment. The Z = 3n – 1 AGNRs have mixed aromatic, electronic, and topological character, with vanishingly small bandgaps. Zigzag GNRs, contrary to other reports, have no magnetic zigzag edges, unless magnetization due to polarization of atomic p_z orbital angular momentum is operative.