Electronic band structure of carbon nanotube superlattices from first-principles calculations

We report on first-principles calculations for metallic carbon nanotube superlattices $N(12,0)∕N(6,6)$ with $N=1–4$. Although the calculated band structures show a good overall agreement with the results of the simpler tight-binding $\pi$-electron approximation, electron interaction and correlation effects strongly modify some peculiar flatbands, previously found within a tight-binding approach [W. Jaskólski and L. Chico, Phys. Rev. B 71, 155405 (2005)]. In the ab initio approach, these bands are no longer dispersionless, much closer to the Fermi level, and are always nondegenerate, in contrast to former tight-binding results.

Figure 1

(Color online). Schematic representation of a $(12,0)∕(6,6)$ junction. Ovals represent the bean-shaped orbitals that give rise to the dispersionless bands in the (12,0) section.

Figure 2

Band structure for the $N(12,0)∕N(6,6)$ superlattices around and below $E_F$ for (a) $N=1$, (b) $N=2$, (c) $N=3$, and (d) $N=4$. Ab initio calculations, left panel; tight-binding results, right panel. The Fermi levels for ab initio (tight-binding) calculations are $-4.34(-0.44)\mathrm{eV}$, $-4.13(-0.32)\mathrm{eV}$, $-4.12(-0.20)\mathrm{eV}$, and $-4.12(-0.24)\mathrm{eV}$ following the (a)–(d) order. Dispersionless and the corresponding weakly dispersive bands are indicated by arrows. The wave vector $k$ is given in reciprocal space units. The superlattice unit cell lengths $a_{\mathit{SL}}$ are $6.25\mathrm{\AA }$ for $N=1$, $13.54\mathrm{\AA }$ for $N=2$, $20.29\mathrm{\AA }$ for $N=3$, and $27.04\mathrm{\AA }$ for $N=4$.

Figure 3

(Color online) Orbitals of the WD (left) and DL (right) bands at $k=0$ for the (12, 0) CNT around $-2.66\mathrm{eV}$ below the Fermi energy (see caption of Fig. 4 for details).

Figure 4

(Color online) Some examples of WD (left panels) and dispersionless orbitals (right panels) around the Fermi energy, for superlattices with (a) $N=1$ and (b) $N=2$. WD orbitals are given in order of decreasing energies from top to bottom. The ab initio orbitals are cut at 10% of their maximum value. Positive lobes are shown in white (light gray), and negative ones in blue (dark). Likewise, each tight-binding eigenvector is represented by a sphere of radius proportional to its modulus; positive values are in white and negative values in blue (dark). Orbitals are labeled by their band energies in eV at $k=0$.

Figure 5

(Color online) Some examples of WD (left panels) and dispersionless orbitals (right panels) around the Fermi energy for the $3(6,6)∕3(12,0)$ superlattice. WD orbitals are given in order of decreasing energies from top to bottom (see caption of Fig. 4 for details).

Figure 6

(Color online) Some ab initio WD orbitals around the Fermi energy for the $4(6,6)∕4(12,0)$ superlattice, shown in order of decreasing energies from top to bottom. The ab initio orbitals are cut at 10% of their maximum value. Positive lobes are in white (light gray), and negative ones in blue (dark).