Band-gap formation in $(n,0)$ single-walled carbon nanotubes $(n=9,12,15,18)$: A first-principles study

We study the electronic structure of the carbon nanotube theoretically by the first-principles techniques using the local-density approximation (LDA) with the many-body correction in the GW approximation. We find that the (9,0) tube is gapful irrespective of naive expectation from the graphene band structure. All of the $\pi \text{−}\sigma$ hybridization effect, lattice relaxation effect, and many-body effect due to electron interaction enhance the band gap, and the value is as large as $0.17\mathrm{eV}$ when taking into account all effects. For the $(n,0)$ nanotubes with $n=9$, 12, 15, and 18, the LDA gap is found to range from $0.08\text{to}0.02\mathrm{eV}$. These sizable gap values obtained by the most reliable methods to date shed light on the classification of carbon nanotubes by their electronic transport properties.

Figure 1

(Color online) Electronic structure of the (9,0) nanotube obtained by (a) $\pi$ tight-binding method, (b) $\pi \text{−}\sigma$ tight method, and (c) LDA. Energy is measured from the top of the valence band, which is shown by a dotted line.

Figure 2

(Color online) (a) LDA band structure of the (9,0) nanotube for the relaxed geometry. In (b), the band structure for the relaxed geometry (filled circles) is compared to the one for the fixed geometry (open circles) in an enlarged scale.

Figure 3

(Color online) (a) Quasiparticle band structure of the (9,0) nanotube by LDA (solid line) and GW (circles), and (b) those in an enlarged scale. Dashed lines are guides for the eyes.

Figure 4

(Color online) Density of states of the (9,0) nanotube taking account of the GW self-energy correction. The energy is measured from the top of the valence band.