We investigate the electronic band structure of an undoped graphene armchair nanoribbon. We demonstrate that such nanoribbon always has a gap in its electronic spectrum. Indeed, even in the situations where simple single-electron calculations predict a metallic dispersion, the system is unstable with respect to the deformation of the carbon-carbon bonds dangling at the edges of the armchair nanoribbon. The edge bonds deformation couples electron and hole states with equal momentum. This coupling opens a gap at the Fermi level. In a realistic sample, however, it is unlikely that this instability could be observed in its pure form. Namely, since chemical properties of the dangling carbon atoms are different from chemical properties of the atoms inside the sample (for example, the atoms at the edge have only two neighbours, besides additional non-carbon atoms might be attached to passivate unpaired covalent carbon bonds), it is very probable that the bonds at the edge are deformed due to chemical interactions. This chemically-induced modification of the nanoribbons edges can be viewed as an effective field biasing our predicted instability in a particular direction. Yet by disordering this field (e.g., through random substitution of the radicals attached to the edges) we may tune the system back to the critical regime and vary the electronic properties of the system. For example, we show that electrical transport through a nanoribbon is strongly affected by such disorder.
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