Discontinuity in the family pattern of single-wall carbon nanotubes

The higher lying bright exciton energies $(E_{11}^M,E_{33}^S,E_{44}^S,E_{22}^M,E_{55}^S,E_{66}^S,E_{33}^M)$ of single-wall carbon nanotubes are calculated by solving the Bethe-Salpeter equation within an extended tight binding method. For smaller diameter nanotubes, some higher $E_{ii}$ excitonic states are missing. In particular, some $E_{ii}$’s on the one-dimensional Brillouin zone (cutting line) are no longer relevant to the formation of excitons and are skipped in listing the order of the $E_{ii}$ values. Thus the family patterns show some discontinuities in $k$ space and this effect should be observable in Raman $G^{\prime }$ band spectroscopy. The higher exciton energies $E_{33}^S$ and $E_{44}^S$ have a large chirality dependence due to many body effects, since the self-energy becomes larger than the binding energy. Thus the chirality dependence of the higher $E_{ii}$ comes not only from a single particle energy but also from many-body effects.

Figure 1

(Color online) Wave vectors of a (6, 1) nanotube around the $K$ point are shown. The wave vector lines must cross the $M_l{M}_r$ line segment in order to have an $E_{ii}$ within the $3M$ region.

Figure 2

(Color online) The bright exciton energy Kataura plot as a function of inverse diameter. $E_{ii}$ is up to $E_{66}^S$ and $E_{33}^M$. Red, blue and black symbols denote type I, type II, and metallic nanotubes, respectively. The eight diamond symbols are experimental data by Sauvajol et al. (Ref. 32).

Figure 3

(Color online) The bright exciton energy Kataura plot for $E_{33}^S$, $E_{44}^S$. Open (solid) triangles and squares denote type I (type II) $E_{33}^S$ and $E_{44}^S$, respectively. Circles denote type II $E_{55}^S$. Red symbols denote the exchange of the order of cutting lines.

Figure 4

(Color online) The bright exciton energy Kataura plot for $E_{22}^M$, $E_{55}^S$, $E_{66}^S$, $E_{33}^M$. Red, blue and black symbols denote type I, type II, and metallic nanotubes, respectively.

Figure 5

The $E_{11}^M$ exciton energy Kataura plot. Open and solid symbols denote $E_{11L}^M$ and $E_{11H}^M$, respectively. The symbols $A$ and $Z$ denote armchair and zigzag SWNTs, respectively.

Figure 6

The self-energy $\Sigma$, the binding energy $E_{\mathrm{bd}}$, and the difference energy $\Sigma -E_{\mathrm{bd}}$ for $E_{11}^M$ are plotted as a function of $d_t^{-1}$. The dashed line is the scaling law with $p=3$. Open and solid symbols denote $E_{11L}^M$ and $E_{11H}^M$, respectively.

Figure 7

The polarization function $ϵ(0,q)$ of metallic zigzag nanotubes for (6,0), (12,0) and (18,0). The inset shows $ϵ(0,q)$ of semiconducting zigzag nanotubes (10,0), (13,0), (16,0), and (19,0) from the bottom as a function of $q$.

Figure 8

(Color online) $\Sigma -E_{\mathrm{bd}}$ of $E_{ii}^S(i=1,2,3,4)$ is plotted as a function of $d_t^{-1}$. The red (blue) symbols denote semiconducting type I (type II) SWNTs, in which circles, pluses, triangles and squares are for $E_{11}^S$ (circles), $E_{22}^S$ (pluses), $E_{33}^S$ (triangles), and $E_{44}^S$ (squares), respectively. The family numbers $2n+m$ are shown for some cases.

Figure 9

(Color online) Effective mass of (a) an electron and (b) a hole as a function of $d_t^{-1}$. The red (blue) symbols denote semiconducting type I (type II), SWNTs, in which circles, pluses, triangles, and squares are for $E_{11}^S$ (circles), $E_{22}^S$ (pluses), $E_{33}^S$ (triangles), and $E_{44}^S$ (squares), respectively.

Figure 10

(Color online) Quasiparticle energy $E$, exciton energy $\Omega$, single particle energy $\epsilon$, self-energy $\Sigma$, and binding energy $E_{\mathrm{bd}}$ of zigzag nanotubes are plotted as a function of wave vector $k$ along the $\Gamma KM$ line as we move along from the $K$ point. Each of the following symbols denote $E_{11}$ (circle), $E_{22}$ (plus), $E_{33}$ (triangle), and $E_{44}$ (square). Red, blue, and black symbols denote semiconducting type I, type II, and metallic nanotubes, respectively. The arrows show that $E_{ii}$ exist in the indicated regions.