Universal nonresonant absorption in carbon nanotubes

Photoluminescence excitation measurements in semiconducting carbon nanotubes show a systematic nonresonant contribution between the well-known excitonic resonances. Using a global analysis method, we were able to delineate the contribution of each chiral species, including its tiny nonresonant component. By comparison with the recently reported excitonic absorption cross section on the $S_{22}$ resonance, we found a universal nonresonant absorbance which turns out to be of the order of one-half of that of an equivalent graphene sheet. This value, as well as the absorption line shape in the nonresonant window, is in excellent agreement with microscopic calculations based on the density-matrix formalism. This nonresonant absorption of semiconducting nanotubes is essentially frequency independent over 0.5-eV-wide windows and reaches approximately the same value between the $S_{11}$ and $S_{22}$ resonances and between the $S_{22}$ and $S_{33}$ resonances. In addition, the nonresonant absorption cross section turns out to be the same for all the chiral species we measured in this study. From a practical point of view, this study provides a solid framework for sample content analysis based on photoluminescence studies by targeting specific excitation wavelengths that lead to almost uniform excitation of all the chiral species of a sample within a given diameter range.

Figure 1

Photoluminescence map of a micelle suspension of HiPCO nanotubes in ${\mathrm{D}}_2\mathrm{O}$. The $(n,m)$ indices are assigned to each PL resonance according to the scheme of Ref. [1]. The spots at the bottom of the map correspond to the excitation on the $S_{22}$ resonances of the nanotubes, whereas the spots in the upper part of the map correspond to the excitation on the $S_{33}$ levels.

Figure 2

Linear absorption spectrum (black) and PLE spectrum (red) of a $(6,5)$ enriched suspension. Both spectra are normalized to the maximum of the $S_{22}$ resonance to show their respective background contributions.

Figure 3

Deconvoluted PLE spectra for the (a) $(6,5)$ and (b) $(10,2)$ species [HiPCO (black) and CoMoCat (red)] extracted from the PL map of Fig. 1. The absolute value of $\sigma$ is retrieved from the knowledge of the resonant $S_{22}$ absorption cross section [14] and assuming a flat internal conversion rate. The values are expressed per unit area, allowing a direct comparison with the graphene case. The blue dashed line represents the equivalent absorption of a graphene layer. The black dashed lines show the nonresonant absorption plateaus that make up the input of Fig. 5. The solid blue line is the microscopic calculation of the absorption spectrum of the corresponding species after applying a global energy shift and a global convolution with a (a) 150 meV or (b) 130 meV broad Lorentzian. The inset in (a) shows the calculated spectra for a set of broadenings of 70, 100, 150, and 200 meV, showing that the magnitude and the shape of the nonresonant absorption are hardly altered.

Figure 4

Calculated absorption of nanotubes for selected diameters. For larger diameters, the absorption clearly becomes diameter independent and reaches that of graphene. The calculations are normalized to set the value of this plateau to $\pi \alpha$.

Figure 5

Diameter dependence of the nonresonant absorption contribution extracted from deconvoluted PLE spectra for ten species (a) between the $S_{22}$ and $S_{33}$ (${\sigma }_{\text{NR}23}$) transition and (b) between the $S_{11}$ and $S_{22}$ transition (${\sigma }_{\text{NR}12}$). The black symbols stand for the HiPCO material, and the red ones are for the CoMoCat material. The blue dashed line represents the graphene absorption, while the magenta dashed line shows nonresonant absorption of nanotubes estimated within the zone-folding approximation.

Figure 6

Absorption spectrum (black line) of a sodium cholate suspension of HiPCO nanotubes (offset corrected). PL spectrum (red line) of the same sample for an off-excitation wavelength at 3 eV.

Figure 7

Transient bleaching of the $S_{11}$ transition of $(6,5)$ nanotubes in a micellar suspension subsequent to femtosecond excitation either at their $S_{22}$ transition (black line) or at several nonresonant energies indicated in the legend. In all cases, the product of the pump-power density and the absorbance was kept constant, resulting in similar excitation densities. The inset shows the same data plotted on an extended time window.

Figure 8

Normalized PLE spectra of the $(6,5)$ species measured at several temperatures.

Figure 9

Absorption calculated from the joint density of states in the conical zone-folding approximation (red line) as a function of the reduced energy ${\omega }^{\prime }=3\omega d_t/\left(4v_F\right)$, where $v_F\simeq 1\times {10}^6$ m ${\mathrm{s}}^{-1}$ stands for the graphene Fermi velocity. The amplitude is rescaled globally so that the value computed for a graphene layer matches the universal value of 0.023 (blue solid line). An effective broadening of 0.01 eV was used.

Figure 10

Comparison of regular PLE spectra (black solid line) with deconvoluted spectra (red line) from the global analysis method for selected chiral species with associated error bars.

Figure 11

Nonresonant absorption of several chiral species measured at the foot of their (a) $S_{33}$ and (b) $S_{22}$ transitions as a function of the trigonal parameter $qcos3\theta$, where $q=n-m$ mod 3.