  Through an investigation of photoemission properties of highly-photoexcited
single-walled carbon nanotubes, we demonstrate that there is an upper limit on
the achievable excitonic density. As the intensity of optical excitation
increases, all photoluminescence emission peaks arising from different
chirality single-walled carbon nanotubes showed clear saturation in intensity.
Each peak exhibited a saturation value that was independent of the excitation
wavelength, indicating that there is an upper limit on the excitonic density
for each nanotube species. We propose that this saturation behavior is a result
of efficient exciton-exciton annihilation through which excitons decay
non-radiatively. In order to explain the experimental results and obtain
excitonic densities in the saturation regime, we have developed a model, taking
into account the generation, diffusion-limited exciton-exciton annihilation,
and spontaneous decays of one-dimensional excitons. Using the model, we were
able to reproduce the experimentally obtained saturation curves under certain
approximations, from which the excitonic densities were estimated. The validity
of the model was confirmed through comparison with Monte Carlo simulations.
Finally, we show that the conventional rate equation for exciton-exciton
annihilation without taking into account exciton diffusion fails to fit the
experimentally observed saturation behavior, especially at high excitonic
densities.
