Approximate quantum mechanical method for describing excitations and related properties of finite single-walled carbon nanotubes

Optical properties of three kinds of zigzag (5,0), (13,0), and (9,0) single-walled carbon nanotubes (SWCNTs) are studied using an approximate quantum mechanical method named complete neglect of differential overlap, which distinguishes basis atomic orbitals with different azimuthal $l$ quantum numbers (CNDOL). This method models the electron energy transitions and excited state charge distributions through a configuration interaction of singly (CIS) excited determinants allowing the direct understanding of properties related with the total electronic wave function of nanoscopic systems, projecting a reliable quantum mechanical understanding to real life objects. The finite SWCNT’s structures were obtained by replicating the unit cells of periodic SWCNTs and saturating the edge dangling bonds with hydrogens. The unit cell was previously relaxed using standard density functional theory methods. The behavior of these SWCNTs were interpreted in the framework of the CNDOL scheme by increasing the lengths of the tubes above 3 nm. As the nanotubes grow in length, the position of excited states for each SWCNT evolve differently: in contrast with (9,0) SWCNT, which exhibits favorable conditions for photoexcitation, the (13,0) and (5,0) SWCNTs do not show a lowering of the lowest excited states. This behavior is discussed by taking into account electron—electron interactions as considered in the framework of the CIS procedure. Furthermore, the (13,0) and (5,0) SWCNTs present forbidden transitions for the lowest excitations and its first dipole-allowed transitions are at 0.9–1.0 and 1.4–1.6 eV, respectively. In contrast, (9,0) SWCNT allows excitations by photon at less than 0.4 eV as the length of the nanotube tends to infinite. Excitons appear more bounded, energetically and spatially, in the (13,0) than in the (9,0) and (5,0) SWCNTs.

Figure 1

(Color online) Schematic representation of a (9,0,2) finite SWCNT. A table with the corresponding length in nanometres for each $L$ value is shown in the box.

Figure 2

Single particle DOS of $\left(5,0,L\right)$ SWCNT.

Figure 3

Lowest energy excited states of $\left(5,0,L\right)$ SWCNT; (a) the density of states of CIS states and (b) the predicted absorption spectra between 0 and 4 eV. The absorption spectrum of $L=20$ has been multiplied by two times.

Figure 4

Middle, experimental optical absorption spectra of SWCNT with diameter near $4\text{Å}$ located in the pores of a zeolite. Top, calculated imaginary part of the dielectric function, obtained with the Bethe-Salpeter equation. Bottom, CNDOL absorption spectrum of (5,0,20) SWCNT. The electric field of light is polarized along the nanotube axis.

Figure 5

(Color online) (5,0) SWCNT singlet state energies as a function of the inverse length of the system. Letters $f$, $s$, $m$, and $w$ indicate forbidden, strong, medium and weak transitions, respectively.

Figure 6

Single particle densities of states of $\left(9,0,L\right)$ (on the left) and $\left(13,0,L\right)$ SWCNTs (on the right).

Figure 7

Lowest energy excited states of $\left(9,0,L\right)$ (in a and b) and $\left(13,0,L\right)$ (in c and d) SWCNTs; density of states of CIS states (a and c), and predicted spectra (b and d) between 0 and 4 eV. In both SWCNTs, the absorption spectra of $L=20$ has been multiplied by two times.

Figure 8

(Color online) (9,0) and (13,0) SWCNTs singlet state energies as a function of the inverse length. Letters $f$, $s$, and $w$ indicate forbidden, strong, and weak transitions, respectively.

Figure 9

Size dependence $\Delta E_{\text{HL}}$ and the lowest excited states.