Third and Fourth Optical Transitions in Semiconducting Carbon Nanotubes

We have studied the optical transition energies of single-wall carbon nanotubes over broad diameter (0.7–2.3 nm) and energy (1.26–2.71 eV) ranges, using their radial breathing mode Raman spectra. We establish the diameter and chiral angle dependence of the poorly studied third and fourth optical transitions in semiconducting tubes. Comparative analysis between the higher lying transitions and the first and second transitions show two different diameter scalings. Quantum mechanical calculations explain the result showing strongly bound excitons in the first and second transitions and a delocalized electron wave function in the third transition.

Figure 1

(a) 2D color map showing the SWNT RBM spectral evolution as a function of excitation laser energy. In (b), the intensity of each spectrum is normalized to the strongest peak, and we plot the inverse Raman shift. (c) The 378 optical transition energies (dots) of all SWNTs in the experimental range [20, 21]. Grey lines are a guide to the eyes for SWNTs with $2n+m=\mathrm{const}$. Open, filled, and dotted circles stand for $E_{ii}$ for semiconducting type 1 [$(2n+m)\mathrm{\text{mod}}3=1$], type 2 [$(2n+m)\mathrm{\text{mod}}3=2$], and metallic SWNTs [$(2n+m)\mathrm{\text{mod}}3=0$], respectively). (d) The ${\omega }_{\mathrm{RBM}}$ versus tube diameter $d_t=0.142\sqrt{3(n^2+mn+m^2)}/\pi$. Circles are experimental data, and the solid line is given by ${\omega }_{\mathrm{RBM}}=217.8/d_t+15.7$. Inset: transmission electron microscopy image of the SWNT samples used in our experiment [17]. (e) Experimental optical transition energies as a function of $p/d_t$, after correcting for the chiral angle dependence ($E_{ii}^{\mathrm{EXP}}-{\beta }_p\cos 3\theta /d_t^2$). The chirality dependence corrected $E_{11}^S$ (black and white diamonds from Ref. 8), $E_{22}^S$ (filled stars online), and $E_{11}^M$ (open stars) are fitted with Eq. (1). Inset: the experimental ${\beta }_p$ values for the lower (upper) $E_{ii}$ branches are $-0.07(0.05)$, $-0.19(0.14)$, $-0.19(?)$, $-0.42(0.42)$, and $-0.4(0.4)$ for $p=1$, 2, 3, 4, and 5, respectively. (f) Deviation ($\Delta E$) of the ($E_{ii}^{\mathrm{EXP}}-{\beta }_p\cos 3\theta /d_t^2$) data from the fitting curve in (e), versus $1/d_t$. The solid line ($\Delta E=0.305/d_t$) fits the $\Delta E_{33}^S$ (circles) and $\Delta E_{44}^S$ (squares). Inset: exciton binding energy $\Delta E_b$ obtained from Ref. 13 (circles) and Ref. 26 (diamonds), on top of the $\Delta E=0.305/d_t$.

Figure 2

Traces from top to bottom plot the electronic probability distribution for excited electrons on $E_{11}^S$, $E_{22}^S$, and $E_{33}^S$, respectively, along the (7,6) SWNT shown on top, considering the hole fixed in the middle.