Figure 1

(Color online) Diagrams for the electronic bands and symmetries for (a) chiral $(n,m)$, (b) zigzag $(n,0)$, and (c) armchair $(n,n)$ nanotubes and for their respective excitonic bands (d), (e), and (f). The electron hole and exciton states at the band edges are indicated by a solid circle and labeled according to their irreducible representation. Different line types and colors in this figure are related to bands with different symmetries. Thick (black) solid lines correspond to the $E_{{\tilde{\mu }}^{\prime }}$ representation, the blue (thin) solid lines correspond to $A_1$ excitons while the cyan (thin) dashed lines correspond to the $A_2$ excitonic states. In the case of achiral nanotubes, we also have inversion and mirror plane symmetries. For a better visualization, the bands with different parities under the inversion and mirror planes were grouped together and appear with the same line color and pattern. In the case of armchair nanotubes, the bands that transform as the $B^{\prime }$ and $B^{″}$ representations are shown using a red dot-dash pattern. The electronic and excitonic band structures shown here are only pictorial. Group theory does not order the values for the eigenenergies and energy dispersions.Reuse & Permissions