Account of helical and rotational symmetries in the linear augmented cylindrical wave method for calculating the electronic structure of nanotubes: Towards the ab initio determination of the band structure of a (100, 99) tubule

Every carbon single-walled nanotube (SWNT) can be generated by first mapping only two nearest-neighbor C atoms onto a surface of a cylinder and then using the rotational and helical symmetry operators to determine the remainder of the tubule [C. T. White et al., Phys. Rev. B 47, 5485 (1993)]. With account of these symmetries, we developed a symmetry-adapted version of a linear augmented cylindrical wave method. In this case, the cells contain only two carbon atoms, and the ab initio theory becomes applicable to any SWNT independent of the number of atoms in a translational unit cell. The approximations are made in the sense of muffin-tin (MT) potentials and local-density-functional theory only. An electronic potential is suggested to be spherically symmetrical in the regions of atoms and constant in an interspherical region up to the two essentially impenetrable cylinder-shaped potential barriers. To construct the basis wave functions, the solutions of the Schrödinger equation for the interspherical and MT regions of the tubule were sewn together using a theorem of addition for cylindrical functions, the resulting basis functions being continuous and differentiable anywhere in the system. With account of analytical equations for these functions, the overlap and Hamiltonian integrals are calculated, which permits determination of electronic structure of nanotube. We have calculated the total band structures and densities of states of the chiral and achiral, semiconducting, semimetallic, and metallic carbon SWNTs (13, 0), (12, 2), (11, 3), (10, 5), (9, 6), (8, 7), (7, 7), (12, 4), and (100, 99) containing up to the 118 804 atoms per translational unit cell. Even for the (100, 99) system with huge unit cell, the band structure can be easily calculated and the results can be presented in the standard form of four curves for the valence band plus one curve for the low-energy states of conduction band. About 150 functions produce convergence of the band structures better then $0.01\mathrm{eV}$ independent of the number of atoms in the translational unit cell.

Figure 1

Graphite layer segment showing indexed lattice points. Nanotubes designated $(n_1,n_2)$ are obtained by rolling the sheet from (0,0) to $(n_1,n_2)$ along a roll-up vector. The chiral angle (from 0° to 30°) is measured between that vector and the zigzag axis.

Figure 2

Atoms of SWNT inside two cylindrical potential barriers.

Figure 3

Dispersion diagram for an empty 1D lattice with screw axis for $\omega =2\pi ∕3$. The indices label the values of $P$ and ${\mathrm{k}}_{\Phi }$, e.g., 2,1 corresponds to dispersion curves with $P=2$ and ${\mathrm{k}}_{\Phi }=1$. Energy $E\left(\mathrm{k}\right)$ is in units of $\left(\frac{\pi }{h}\right)$.

Figure 4

Triply degenerate dispersion curves for the same system as in Fig. 3, but calculated taking into account only translational symmetry. Account of the screw symmetry results in a shift of some curves. The indices label the values of $P$. Energy $E\left(\mathrm{k}\right)$ is in units of $\left(\frac{\pi }{3h}\right)$.

Figure 5

General cylindrical and local spherical coordinate systems.

Figure 6

Illustration to the theorem of addition.

Figure 7

LACW basis set convergence test for the (11,3) SWNT with 652 atoms in the translational unit cell.

Figure 8

Band structure of the (11,3) SWNT.

Figure 9

Band structure of the (8,7) SWNT.

Figure 10

Band structure of the (12,2) SWNT.

Figure 11

Band structure of the (12,4) SWNT.

Figure 12

Band structure of the (10,5) SWNT.

Figure 13

Band structure of the (13,0) SWNT.

Figure 14

Total DOSs of the SWNTs with $d=10.15\pm 0.15\mathrm{\AA }$.

Figure 15

Partial DOSs of the (12,2) tubule.

Figure 16

Band structure of the (9,6) SWNT.

Figure 17

Band structure of the (7,7) SWNT.

Figure 18

LACW basis set convergence test for the (100,99) SWNT with 118 804 atoms in the translational unit cell.

Figure 19

Total band structure of the (100,99) SWNT.

Figure 20

Band structure of the (100,99) SWNT in the Fermi energy region.

Figure 21

Tight-binding $\pi$-band structures of semiconducting (11,3), (12,2), and (12,4) SWNTs plotted as the functions of two quantum numbers k and $L$.

Figure 22

Tight-binding $\pi$-band structures of two metallic (9,6) and (7,7) and one semiconducting (10,5) SWNTs plotted as functions of two quantum numbers k and $L$.