Ultralow-frequency shear modes of 2-4 layer graphene observed in scroll structures at edges

The in-plane shear (C) modes between neighbor layers of 2-4 layer graphenes (2-4LGs) and the corresponding 2-4 layer graphene scrolls (2-4LGSs) rolled up by 2-4LGs at edges are investigated by Raman scattering. In contrast to the result that only one C mode is observed in 3-4LGs, all the C modes of 3-4LGs are observed in 3-4LGSs, whose frequencies agree with the theoretical prediction by the force-constant and linear chain models. The results indicate that the C mode intensity of 2-4LGSs is resonantly enhanced by the electronic transition gaps of band structures of 2-4LGS structures at edges, which makes it possible to observe all the C modes. Indeed, for a simple assumption, the calculated band structures of twisted ($n+n$)LGs ($n=2$,3,4) show parallel conduction and valence bands and the corresponding Van Hove singularities in the joint density of states along Γ-M, Γ-K and/or K-M directions. The intensity resonance of the C modes provides direct evidence to explain how the band structure of few layer graphenes can be sensitive to local stacking configurations. This result can be extended to n layer graphene ($n>4$) for understanding the basic phonon and electronic properties of multilayer graphenes. This observation of all the C modes in graphene scrolls can be foreseen in other two-dimensional materials with similar scroll structures.