Comparison of electronic and geometric structures of nanotubes with subnanometer diameters: A density functional theory study

All-electron calculations based on density functional theory have been carried out to study the electronic structures of single-walled nanotubes with subnanometer diameters. Present studies suggest the need for performing all-electron calculations, specifically for the nanotubes with very small diameters. Complete geometry optimization is found to be very crucial for predicting the electronic properties. We report here the electronic properties of two of the smallest single-walled carbon nanotubes (SWCNTs)—an armchair (2,2) and a zigzag (4,0) SWCNT—with comparable diameters of about $3\mathrm{\AA }$. It is observed that, both geometrically and electronically, they are quite different. The discussion on the properties of zigzag $(n,0)$ SWCNTs, with diameters in the subnanometer regime, as a function of $n$, involves the elucidation of their geometric and band structures. In the band structures, systematic variations of a nearly-free-electron-like state and a nearly-dispersion-free state are shown as a function of $n$ and their implications have been discussed. The Fermi surfaces calculated for the metallic SWCNTs show signature of a collection of quasi-Fermi-points. This shows the quasi-one-dimensional nature of these nanotubes (NTs). From the present calculations, it is expected that the number of conduction channels is restricted between 2 and 3 for the NTs studied here.

Figure 1

Two types of bond lengths (d1 and d2) and bond angles (a1 and a2) are shown in both armchair and zigzag tubes. Axis of tube is marked with an arrow. a1 is the angle between two d2 bond lengths, and a2 is between d1 and d2 in the case of both the armchair and zigzag tubes.

Figure 2

(Color online) A part of the armchair tube SWCNT(2,2) along with its valence charge density (in left panel) is shown in the top row. Below, part of zigzag tubes with $n=4$, 7, and 9 (from second to last row, respectively) are also shown along with their valence charge densities (in left panel).

Figure 3

Band structures of (a) SWCNT(6,0), (b) SWCNT(8,0), (c) SWCNT(7,0), and (d) SWCNT(9,0). The Fermi energy is shown by a dashed line. The band energies are normalized with respect to the Fermi level.

Figure 4

Band structures of (a) SWCNT(2,2) without geometry optimization and (b) SWCNT(2,2) with geometry optimization. The Fermi energy is shown by a dashed line. The band energies are normalized with respect to the Fermi level.

Figure 5

Band structures of (a) SWCNT(4,0) without geometry optimization and (b) SWCNT(4,0) with geometry optimization. The Fermi energy is shown by a dashed line. The band energies are normalized with respect to the Fermi level.

Figure 6

Band structures of (a) SWCNT(5,0) without geometry optimization and (b) SWCNT(5,0) with geometry optimization. The Fermi energy is shown by a dashed line. The band energies are normalized with respect to the Fermi level.

Figure 7

Band gap of SWCNT(5,0) as a function of the residual force, averaged over different carbon atoms. The dashed line is a guide for the eye.

Figure 8

Energies of the (a) nearly-dispersion-free states and (b) nearly-free-electron-like states as a function of $n$, the wrapping index. In (a), the solid line and the dashed line correspond to the energies, normalized with respect to $E_F$, at the $Z$ point and $\Gamma$ point, respectively. The dot-dashed line corresponds to the difference between these two energies. In (b), the energy of the nearly-free-electron-like state for graphene is indicated by a star.