Results 4.1. Localization at Octagonal DefectsWe begin by considering a diagonal junction between the (8,0) and (14,0) nanotubes, as shown in Figure 1(a). As worldwide distributors discussed in the previous section, when joining the (14,0) tube to a diagonally cut (8,0) tube, an 8R octagonal defect appears at the back of the knee. At the kneecap we may have either (i) an 8N octagon or (ii) a pair of pentagons (2 �� 5). This is visualized in the upper inset of Figure 1(a) and in Figure 1(b). In the latter case (ii) the pentagons mix the graphene sublattices, which causes the breaking of electron-hole symmetry.Here we consider the case (i), that is, with two octagons, 8R and 8N, present at each junction. Since there are no pentagons, the system can be still considered as a bipartite lattice.

Figure 2(a) shows the LDOS of an (8,0)/(14,0) single junction. A strong peak appears at E = 0 in the gap. In order to elucidate its origin, we perform calculations for two related superlattices M(8,0)/M(14,0) with M = 12 and M = 5 (M is the number of hexagons along the tube axis). The corresponding band structures are presented in Figures 2(b) and 2(c). Four bands, almost flat, appear near the Fermi level. They form two bonding and antibonding pairs because there are two junctions within a unit cell (recall Figure 1(c)). The wavefunctions of one pair are localized at the octagons 8R, whereas the wavefunctions of the other pair of bands are localized at the 8N octagons. The localization at the octagon 8N takes place at the sublattice to which the pair of nodes having only two neighbors belong.

Consequently, localization at the octagon 8R occurs in the complementary sublattice. The wavefunctions centered at each octagon defect extend in a decaying way into the region of the complementary octagon but are always confined into their own sublattice. In the limit of M �� �� we end up with the single junction case having a doubly degenerate E = 0 state, like the one shown in Figure 2(a).Figure 2(a) LDOS at the wedge for the single junction (8,0)/(14,0). (b) Band structure of the 12(8,0)/12(14,0) SL. (c) Band structure of the 5(8,0)/5(14,0) SL.Below we show that the appearance of states localized at the junctions is connected with the octagonal structure of the defects.

To do this we divide the diagonal junction between the (8,0) and (14,0) tubes into three parts: (T1) a regular semi-infinite (8,0) tube, (W) the 62-atom wedge of the (8,0) tube, and (T2) a regular semi-infinite (14,0) tube, as shown in Figure 1.According to the rules presented in [14], the semi-infinite (8,0) tube cut perpendicularly to Dacomitinib its axis has three zero-energy edge states localized at the sublattice to which the zigzag-edge atoms belong. Similarly, the semi-infinite (14,0) tube has five E = 0 edge states localized at the same sublattice.