Saturation of Photoluminescence from Carbon Nanotubes at High Laser Intensities: Exciton-Exciton Annihilation near the Mott Density

We have carried out a nonlinear photoluminescence excitation (PLE) spectroscopy study of carbon nanotube ensembles using intense, femtosecond, and wavelength-tunable optical pulses. For all PL features we examined, their intensities were seen to saturate at high laser fluence, irrespective of whether the excitation was resonant or non-resonant with the $E_{22}$ transition. As the fluence was increased from the linear regime to the saturation regime, excitation resonances at $E_{22}$ energies gradually broadened and eventually became completely flat at the highest fluence, indicating that the PL intensity became independent of the excitation wavelength. However, the energies and lineshape of PL \textit{emission peaks} did not show any changes throughout the entire range of fluence used. Through absorption spectroscopy at high laser intensities, we also demonstrated that $E_{22}$ \textit{absorption peaks} do not show any shift or broadening even at high laser fluence, indicating that state-filling or scattering is not the cause of the observed ``flattening'' of the excitation spectra. We developed a model to explain these observations by carefully taking into account the spatial overlap of excitons when the average inter-exciton distance approaches the Bohr radius in the exciton-exciton annihilation process.