Bandgaps of atomically precise graphene nanoribbons and Occam's razor

Rationalization of the "bulk" (ΔE_ac) or "zigzag-end" (ΔE_zz) energy gaps of atomically precise armchair graphene nanoribbons (AGNRs), which are directly related to fundamental applications in nanoelectronics, could be challenging and largely controversial with respect to their magnitude, origin, substrate influence (ΔE_sb), and spin-polarization, among others. Hereby a simple self-consistent and "economical" interpretation is presented, in full accordance with Occam's simplicity principle, which is highly successful (within less than 1%) in predicting all energy gaps of the 5-, 7-, and 9-AGNRs, in contrast to other complicated and/or contradicting prevailing views in the literature for ΔE_ac, ΔE_zz, and ΔE_sb. The present approach is based on "appropriate" DFT (TDDFT) calculations, general symmetry principles, and plausibility arguments. The excellent agreement with experiments and the new insight gained is achieved by invoking the approximate equivalence of Coulomb correlation energy with the staggered sublattice potential. Breaking established stereotypes, we suggest that the measured STS gaps are virtually independent of the substrate, essentially equal to their free-standing values, and that the "true" lowest energy state is a closed singlet with no conventional magnetism. The primary source of discrepancies is the finite length of AGNRs together with inversion/reflection symmetry conflict and the resulting topological end/edge states. Such states invariably mix with other "bulk" states making their unambiguous detection/distinction difficult. This can be further tested by eliminating end-states (and ΔE_zz), by eliminating "empty" zigzag rings.