Figure 1

ZGNRs with defect lines composed of (a) consecutive octagons and (b) octagons with an adjacent pair of pentagons. The ZGNRs extend horizontally. Rectangles mark the unit cells of ribbons including the defects. Klein nodes (see the text) in the unit cells are highlighted in red.Reuse & Permissions

Figure 2

(a) Energy bands of a ZGNR with a defect line composed of octagons, corresponding to two 8-ZGNRs joined by an extra row of Klein-like atoms, as depicted in Fig. 1a, calculated within the TB approximation without considering spin. The wave vector $k$ is given in units of 1/$T$, where $T$ is the length of the unit cell. The degeneracies of $E=0$ bands at $k=0$ and $k=\pi$ without considering spin are indicated in the figure. (b) Energy spectrum for the same ribbon calculated including electron-electron interactions by means of a Hubbard model. Blue solid lines denote spin-up bands; red dashed lines show the spin-down bands. The Fermi energy is at 0 eV in both cases. (c) TB wave functions corresponding to the four zero-energy bands at $k=0$. Blue (dark gray) and white colors represent positive and negative signs of the wave function, respectively.Reuse & Permissions

Figure 3

(a) Energy bands of the armchair (8,8) carbon nanotube with a defect line composed of octagons obtained by rolling up a ZGNR with octagonal defects, calculated with a TB model. Degeneracies are indicated without considering spin. (b) Energy spectrum for the same tube calculated with a Hubbard model. Blue solid lines, spin-up bands; red dashed lines, spin-down bands. (c) TB wave functions corresponding to two zero-energy bands at $k=0$. Blue (dark gray) and white colors represent positive and negative signs of the wave function, respectively.Reuse & Permissions

Figure 4

(a) Band structure of a ZGNR with a defect line composed of octagons and pentagon pairs, as depicted in Fig. 1b, calculated within the TB model. (b) Band structure for the same system calculated with the Hubbard model. Spin-down and spin-up bands are degenerate in energy. (c) TB wave functions corresponding to the three zero-energy bands at $k=0$ found in the TB calculation. Blue (dark gray) and white colors represent positive and negative signs of the wave function, respectively.Reuse & Permissions

Figure 5

(a) Band structure of an (8,8) armchair nanotube (rolled-up ZGNR) with a defect line composed of alternating octagons and pentagon pairs calculated with the TB approximation. (b) Band structure calculated within the Hubbard model. (c) TB wave function at $k=0$ of the flatband localized at the octagonal ring.Reuse & Permissions

Figure 6

(a) LDOS at the junction between two zigzag graphene half planes with an octagon–pentagon-pair defect line. Interface bands appear in the gap as dark lines, while the shaded area is the graphene band continuum. Vertical long dashed line marks $k=\pi /2$. (b) LDOS at the junction between two $(8,0)$ nanotubes with an octagon–pentagon-pair junction. Horizontal dotted lines mark the correspondence between the energies of the interface bands of the graphene junction at the allowed wave vectors given by the Born–von Kármán boundary condition $k=0,\frac{\pi }{2},\pi$ and the LDOS peaks for the (8,0) nanotube junction. The LDOSs in (a) and (b) are calculated at the eight nodes of the octagon of the graphene and nanotube junctions, respectively.Reuse & Permissions

Figure 7

Schematic band structures close to the Fermi energy for the systems investigated in Sec. 3. Shaded regions stand for the band continua in case of very wide ribbons. $Z_e$ marks states localized at the outer zigzag edges, $Z_i$ marks states localized at the internal zigzag edges forming the octagon, and $K$ represents Klein edge-localized states. (a) Two independent ZGNRs, one of them with Klein edges at one side (right). (b) Two ZGNRs connected to form a ZGNR with an octagon-only line of defects. (c) ZGNR with an octagon defect line, but the band structure is folded from the previous case (b), because the real-space unit cell has twice the length. (dl) Rolled-up ZGNR with octagons only. (dr) ZGNR with octagon–double-pentagon defects (i.e., with Klein nodes connected). (e) Armchair nanotube with a line of defects composed of octagons and pentagon pairs.Reuse & Permissions