Effects of deformation on the electronic structure of a single-walled carbon nanotube bundle

We have studied the effects of uniaxial pressure on the geometric structure and the electronic structure of single-walled carbon nanotube bundles theoretically. The local-density approximation in the density-functional theory has been applied to three types of carbon nanotube bundles, made up of the (8, 0), (10, 0), and (11, 0) tubes under uniaxial pressure perpendicular to the tubule axis. In all these types of bundles, an abrupt change is observed in the deformation of the tubes and their configuration at a certain pressure. It is also found that, despite a similar change of the lattice constants of the bundle, the deformation and the configuration of the tubes depend strongly on their types: While the (8, 0) tube bundle has a dense structure at high pressures, larger tube bundles prefer a loose one. All types of bundles, which are calculated to be semiconducting, exhibit a semiconductor-metal transition before or at the beginning of the abrupt change of the lattice constants when they are deformed by the uniaxial pressure. The pressure effect on the energy gap, however, is not monotonous: a decrease and an upturn followed by its disappearance. By analyzing the atomic arrangement, the band structure, and the wave functions of the three types of carbon nanotube bundles, the relationship between them is also established.

Figure 1

Uniaxial pressure with the cross section of the bundle.

Figure 2

(Color online) Variations of (a) lattice constants and (b) angle $\gamma$ with pressure in the (8, 0) carbon nanotube bundle.

Figure 3

Variation of the energy gap with pressure in the (8, 0) carbon nanotube bundle.

Figure 4

(Color online) Schematic view of the cross section of the (8, 0) bundle at (a) $0\mathrm{GPa}$ ($a=17.6\text{bohr}$, $b=18.3\text{bohr}$, $e^{\prime }=0.23$), (b) $3.0\mathrm{GPa}$ ($a=20.9\text{bohr}$, $b=17.2\text{bohr}$, $e^{\prime }=0.34$), and (c) $5.0\mathrm{GPa}$ ($a=33.4\text{bohr}$, $b=18.4\text{bohr}$, $e^{\prime }=0.67$). Contour plot of the electron density distribution is also illustrated in (c). All contour lines are in the range of $0–0.05e∕{\text{bohr}}^3$. The gray spheres (yellow online) represent carbon atoms.

Figure 5

(Color online) Variations of (a) lattice constants and (b) angle $\gamma$ with pressure in the (11, 0) carbon nanotube bundle.

Figure 6

Variation of energy gap with pressure in the (11, 0) carbon nanotube bundle.

Figure 7

(Color online) Schematic view of the cross section of the (11, 0) bundle at (a) $0\mathrm{GPa}$ ($a=22.0\text{bohr}$, $b=22.8\text{bohr}$, $e^{\prime }=0.24$), (b) $5.5\mathrm{GPa}$ ($a=28.2\text{bohr}$, $b=19.3\text{bohr}$, $e^{\prime }=0.86$), and (c) $6.7\mathrm{GPa}$ ($a=48.8\text{bohr}$, $b=23.6\text{bohr}$, $e^{\prime }=0.93$). Contour plot of electron density distribution is also illustrated in (c). All contour lines are in the range of $0–0.03e∕{\text{bohr}}^3$. The gray spheres (yellow online) represent carbon atoms.

Figure 8

(Color online) Variations of (a) lattice constants and (b) angle $\gamma$ with pressure in the (10, 0) carbon nanotube bundle.

Figure 9

Variation of energy gap with pressure for the (10, 0) bundle.

Figure 10

(Color online) Symmetry points and lines in hexagonal Brillouin zone.

Figure 11

(Color online) Band structures of the (8, 0) bundle along several symmetry lines under (a) $0\mathrm{GPa}$, (b) $1.0\mathrm{GPa}$, and (c) $1.125\mathrm{GPa}$. The top of the valence band on the Fermi level $E_F$ is set to zero.

Figure 12

Calculated pressure dependence of energy bands for the lowest conduction band (LCB) and the highest valence band (HVB) at the M3 and $\Gamma$ points of the (10, 0) bundle. The top of the valence band on the Fermi level $E_F$ is set to zero.

Figure 13

(Color online) Equivalue plots of the wave function amplitude of LCB at M3 of the (10, 0) bundle under pressure at (a) $0\mathrm{GPa}$ $(e^{\prime }=0.26)$, (b) $1.0\mathrm{GPa}$ $(e^{\prime }=0.47)$, (c) $2.25\mathrm{GPa}$ $(e=0.52)$, and (d) $2.5\mathrm{GPa}$ $(e^{\prime }=0.62)$. The gray surface (yellow online) is plotted for the absolute value 0.15 of the wave function normalized in the unit cell. The dark spheres (red online) represent the carbon atoms.

Figure 14

(Color online) (a) Definition of the atomic orientational angle $\Phi$ for a tube in the (10, 0) bundle, (b) schematic view of atomic positions in one of the planes of the unit cell, and (c) variations of the angle difference $\Delta \Phi$ for different atoms as a function of pressure. The gray spheres (yellow online) in (a) and (b) represent carbon atoms.