Chirality dependence of exciton effects in single-wall carbon nanotubes: Tight-binding model

We have studied the exciton properties of single-wall carbon nanotubes by solving the Bethe-Salpeter equation within tight-binding models. The screening effect of the $\pi$ electrons in carbon nanotubes is treated within the random phase and static screened approximations. The exciton wave functions along the tube axis and circumference are discussed as a function of $(n,m)$. A $2n+m=\mathrm{const}$ family behavior is found in the exciton wave function length, excitation energy, binding energy, and environmental shift. This family behavior is understood in terms of the trigonal warping effect around the $K$ point of a graphene layer and curvature effects. The large family spread in the excitation energy of the Kataura plot is found to come from the single-particle energy.

Figure 1

The three inequivalent regions in the 2D BZ of graphene. The cutting lines for a (6,5) SWNT are shown. The electron-hole pairs and the corresponding center-of-mass momentum $2\overline{\mathbf{K}}={\mathbf{k}}_c-{\mathbf{k}}_v$ for an $E_{12}\left(E_1\right)$ exciton of the (6,5) SWNT are indicated. The electron-hole pair with the electron and hole lying on the second and first cutting lines to the $K$ point and the electron-hole pair with the electron and hole lying on the first and second cutting lines to the $K^{\prime }$ point correspond to an $E_{12}\left(E_1\right)$ exciton with the center-of-mass momentum $2\overline{\mathbf{K}}$ on the first cutting line to the $\Gamma$ point.

Figure 2

The excitation energy dispersions for (a) $E_{11}\left(A_2\right)$ $(S=0)$ and $E_{22}\left(A_2\right)$ $(S=0)$, (b) $E_{11}\left(A_1\right)$ $(S=0,1)$ and $E_{22}\left(A_1\right)$ $(S=0,1)$, and (c) $E_{11}\left(A_2\right)$ $(S=1)$ and $E_{22}\left(A_2\right)$ $(S=1)$ excitons for (6,5) SWNTs. The excitation energy levels for $\overline{K}=0$ excitons are also shown in (d).

Figure 3

The exciton wave functions along a line passing through an $A$ carbon site parallel to the tube axis of an (8,0) SWNT. (a) $E_{22}\left(A_2^0\right)$, (b) $E_{22}\left(A_2^1\right)$, and (c) $E_{22}\left(A_2^2\right)$ are the exciton wavefunctions for the three lowest $E_{22}\left(A_2^{\nu }\right)$ excitations in the order of increasing energy. The hole is put on an $A$ atom.

Figure 4

The half width $l_k$ of the wave functions in 1D $k$ space for $E_{11}\left(A_2^0\right)$ and $E_{22}\left(A_2^0\right)$ states for many SWNTs. The cutting line spacing $2∕d_t$ is shown by the solid line for comparison. Open and filled circles are for SI and SII SWNTs. Integers denote $2n+m$ family values.

Figure 5

The wave function ${\Psi }_0$ of the $E_{22}\left(A_2^0\right)$ state along a line on the circumference of an (8,0) SWNT. (a) ${\Psi }_0$ on the unfolded line and (b) $\mid {\Psi }_0\mid$ on the folded line. The small open circles in (a) and (b) denote the hole position.

Figure 6

The wave function ${\Psi }_{-1}$ for $E_{12}\left(E_1^0\right)$ or $E_{21}\left(E_1^0\right)$ along a line on the circumference of an (8,0) SWNT. The hole indicated by a small circle is put at an $A$ atom with $\theta \approx -11°$. (a) ${\Psi }_{-1}$ on the unfolded line and (b) $\mid {\Psi }_{-1}\mid$ on the folded line.

Figure 7

The wave function ${\Psi }_{-1}$ for $E_{12}\left(E_1^0\right)$ or $E_{21}\left(E_1^0\right)$ along a line on the circumference of an (8,0) SWNT. The hole indicated by a small circle is put at an $A$ atom with $\theta \approx 126°$. (a) ${\Psi }_{-1}$ on the unfolded line and (b) $\mid {\Psi }_{-1}\mid$ on the folded line.

Figure 8

The excitation and binding energy Kataura plots using the simple TB model for $E_{11}^S\left(A_2^0\right)$ and $E_{22}^S\left(A_2^0\right)$ states in $S$ SWNTs and $E_{11L}^M\left(A_2^0\right)$ and $E_{11H}^M\left(A_2^0\right)$ states in $M$ SWNTs. (a) Excitation energy Kataura plot. Open, filled, and crossed circles are for SI, SII, and $M$ SWNTs. (b) Binding energy Kataura plots for $S$ SWNTs. Open and filled circles are for SI and SII SWNTs. (c) Binding energy Kataura plots for $M$ SWNTs.

Figure 9

The dielectric function $ϵ(0,q)$ for (a) $S$ SWNTs and (b) $M$ SWNTs. The integers denote $(2n+m)$ values (a) 13, 14, 25, 26, 37, and 38 and (b) 15, 27, and 39. The dots indicate the position with $q=1∕d_t$. In (a) solid and dashed lines are for SI and SII SWNTs.

Figure 10

The exciton binding energy for $E_{33}\left(A_2^0\right)$ and $E_{44}\left(A_2^0\right)$ states with $1.0\mathrm{nm}<d_t<1.6\mathrm{nm}$. Open and filled circles are for SI and SII SWNTs for $E_{33}\left(A_2^0\right)$ states, and open and filled squares are for SI and SII SWNTs for $E_{44}\left(A_2^0\right)$ states. For SI SWNTs, $(n,m)=(9,7)$ and (10, 8) SWNTs with families $2n+m=25$ and 28, respectively, have the same binding energies for $E_{33}\left(A_2^0\right)$ and $E_{44}\left(A_2^0\right)$ states.

Figure 11

The excitation energy differences $E_{11}\left(A_2^1\right)-E_{11}\left(A_2^0\right)$ and $E_{22}\left(A_2^1\right)-E_{22}\left(A_2^0\right)$ (upper). Open and filled circles are calculated data for SI and SII SWNTs. Open and filled squares are experimental data by Dukovic et al. (Ref. 42) for SI and SII SWNTs. The dashed line is a fitted relationship between $E_{11}\left(A_2^1\right)-E_{11}\left(A_2^0\right)$ and $1∕d_t$ for the experimental data. The theoretical data around the dashed line are calculated only for $E_{11}\left(A_2^1\right)-E_{11}\left(A_2^0\right)$ with $\kappa =2.22$.

Figure 12

The $E_{11}\left(A_2^0\right)$ and $E_{22}\left(A_2^0\right)$ excitation energy shifts by $\kappa$ changing from 3 to 2, $\Delta E_{ii}={\phantom{\mid }{E}_{ii}\mid }_{\kappa =2}-{\phantom{\mid }{E}_{ii}\mid }_{\kappa =3}$. Open and filled circles are for SI and SII SWNTs.

Figure 13

The excitation energy dependence on $\kappa$ for states $E_{11}\left(A_2^0\right)$ and $E_{22}\left(A_2^0\right)$ for (6,5) SWNTs. Solid and dashed lines are with and without considering the $\pi$ electron screening effect, respectively. (a) Excitation energy vs $1∕\kappa$. The three curves below $E_{11}$ are exciton binding energies for $E_{22}\left(A_2^0\right)$ and $E_{11}\left(A_2^0\right)$, and the function $E={(1∕\kappa )}^{1.4}$ from top to bottom, respectively. (b) Excitation energy vs $\kappa$.

Figure 14

The excitation energy Kataura plot based on the ETB model for $E_{11}^S\left(A_2^0\right)$ and $E_{22}^S\left(A_2^0\right)$ for $S$ SWNTs and $E_{11}^M\left(A_2^0\right)$ for $M$ SWNTs. Open and filled circles are for SI and SII SWNTs, respectively, and crossed circles are for $M$ SWNTs.

Figure 15

The excitation energy ratio $E_{22}^S∕E_{11}^S$ for $A_2^0$ states based on the ETB model for $S$ SWNTs with $0.5\mathrm{nm}<d_t<1.6\mathrm{nm}$. Open and filled circles are for SI and SII SWNTs, respectively. (a) and (b) are the results with and without Coulomb interaction. The dashed lines are used to indicate the values for the average ratio.

Figure 16

The excitation energy $E_{11}$, self-energy $\Sigma$, binding energy $E_{\mathrm{bd}}$, and energy corrections $\Sigma -E_{\mathrm{bd}}$ based on ETB for $E_{11}\left(A_2^0\right)$ states. Open and filled circles are for SI and SII SWNTs. The dashed line is calculated by Eq. (28) with $p=1$.

Figure 17

The effective mass for the first and second transitions for $S$ SWNTs based on the ETB model. Open and filled circles are for SI and SII SWNTs.