Experimental study of Coulomb corrections and single-particle energies for single-walled carbon nanotubes using cross-polarized photoluminescence

We report the observation of cross-polarized transitions of single-walled carbon nanotubes (SWCNTs) isolated with aromatic polymers. The use of photoluminescence excitation mapping allows the identification of the transitions of individual species of SWCNT. Comparing experimental observation with theory yields an estimate for the Coulomb interactions, which is dependent on the nanotubes’ diameter and environment. Values for the single-particle energies are deduced and found to be in good agreement with the predictions of tight-binding models.

Figure 1

(Color online) (a) PLE map of CoMoCAT nanotubes wrapped by PFO in o-xylene with only four nanotube species observed. Species are determined by using a semiempirical model (Ref. 27). The circles identify $E_{12}$ transitions associated with each species of nanotube. Clear resonant Raman features associated with the $G$ and $G^{\prime }$ band can also be observed. (b) PLE spectrum of (7,5) nanotube species. This spectrum is taken from a vertical slice of the PLE map, indicated by the black (dotted) line in Fig. 1a. The intensity is normalized to the main peak, which corresponds to the $E_{22}$ transition, while several other smaller features are also observed and identified. The red (dashed) line corresponds to parallel polarized PLE spectra of the same species with the intensity decreased in absolute terms by a factor of seven for clarity. The transverse peak feature is much weaker in this spectrum.

Figure 2

(Color online) (a) Plot of $E_{12}^{\text{obs}}$ against $\left(E_{11}^{\text{obs}}+E_{22}^{\text{obs}}\right)/2$. The red (solid) line indicates linear best fit and the black (dashed) line indicates $E_{12}^{\text{obs}}=\left(E_{11}^{\text{obs}}+E_{22}^{\text{obs}}\right)/2$. (b) Blueshift $\Delta E$ against the diameter for polymer wrapped species (hollow points) with $\Delta E=A{x}^{-B}$, with $A=0.18$ and $B=0.57$. Results for air suspended nanotubes from the work of Lefebvre et al. (Ref. 22) are shown as full squares.

Figure 3

(Color online) (a) Plot of Coulomb interaction energy against the diameter of SWCNTs. The strength of the Coulomb interactions is greater for nanotubes suspended in air than in solvent isolated by polymers. (b) Comparison of experimental (solid symbols) and theoretical (open symbols) single-particle $E_{ii}^{\text{gap}}$ as a function of nanotube diameter. The experimental values were deduced using measurements with both SDBS and polymer wrapped nanotubes. The theoretical results are from the tight-binding model developed by Popov et al. (Ref. 29).

Figure 4

(Color online) (a) A plot of the ratio $E_{22}^{\text{gap}}/E_{11}^{\text{gap}}$ for the experimental single-particle gaps shown in Fig. 3b as a function of diameter. The chiral index $q$ is given by $n-m=3p+q$, where $p$ is an integer. (b) $E_{11}^{\text{gap}}$ deduced from the SDBS wrapped nanotube data. The green (solid) line shows the band gap as predicted from a linear dispersion, $E_g=4\hslash c/3d$, with $c=0.88\times {10}^6{\text{ms}}^{-1}$.