The density of states (DoS), $varrho(E)$, of graphene is investigated numerically and within the self-consistent T-matrix approximation (SCTMA) in the presence of vacancies within the tight binding model. The focus is on compensated disorder, where t
he concentration of vacancies, $n_text{A}$ and $n_text{B}$, in both sub-lattices is the same. Formally, this model belongs to the chiral symmetry class BDI. The prediction of the non-linear sigma-model for this class is a Gade-type singularity $varrho(E) sim |E|^{-1}exp(-|log(E)|^{-1/x})$. Our numerical data is compatible with this result in a preasymptotic regime that gives way, however, at even lower energies to $varrho(E)sim E^{-1}|log(E)|^{-mathfrak{x}}$, $1leq mathfrak{x} < 2$. We take this finding as an evidence that similar to the case of dirty d-wave superconductors, also generic bipartite random hopping models may exhibit unconventional (strong-coupling) fixed points for certain kinds of randomly placed scatterers if these are strong enough. Our research suggests that graphene with (effective) vacancy disorder is a physical representative of such systems.
Quantum size effects in armchair graphene nano-ribbons (AGNR) with hydrogen termination are investigated via density functional theory (DFT) in Kohn-Sham formulation. Selection rules will be formulated, that allow to extract (approximately) the elect
ronic structure of the AGNR bands starting from the four graphene dispersion sheets. In analogy with the case of carbon nanotubes, a threefold periodicity of the excitation gap with the ribbon width (N, number of carbon atoms per carbon slice) is predicted that is confirmed by ab initio results. While traditionally such a periodicity would be observed in electronic response experiments, the DFT analysis presented here shows that it can also be seen in the ribbon geometry: the length of a ribbon with L slices approaches the limiting value for a very large width 1 << N (keeping the aspect ratio small N << L) with 1/N-oscillations that display the electronic selection rules. The oscillation amplitude is so strong, that the asymptotic behavior is non-monotonous, i.e., wider ribbons exhibit a stronger elongation than more narrow ones.
