Femtosecond Spectroscopy of Optical Excitations in Single-Walled Carbon Nanotubes: Evidence for Exciton-Exciton Annihilation

Frequency-resolved femtosecond transient absorption spectra and kinetics measured by optical excitation of the second and first electronic transitions of the $(8,3)$ single-walled carbon nanotube species reveal a unique mutual response between these transitions. Based on the analysis of the spectra, kinetics, and their distinct amplitude dependence on the pump intensity observed at these transitions, we conclude that these observations originate from both the excitonic origin of the spectrum and nonlinear exciton annihilation.

Figure 1

Schematic model describing the exciton dynamics in a semiconducting SWNT. $E_g$ and $E_i(k)$ represent the ground state and the $i$th exciton band, respectively, $k$ is the exciton center-of-mass momentum, and arrows depict the transitions between the exciton bands caused by relaxation. $k_{ij}$ describes the relaxation rate from the $i$th exciton state to the $j$th exciton state, and $\gamma n_1^2/2$ is the rate of populating a higher exciton state as the result of exciton-exciton annihilation.

Figure 2

Transient absorption spectra measured in the visible (a) and NIR (b) regions upon excitation at 660 (filled circles) and 953 nm (open squares). The spectra are recorded at delay times of 50 fs and 1 ps, respectively, with a pump fluence of $\sim {10}^{14}\mathrm{\text{photons}}/\mathrm{\text{pulse}}{\mathrm{cm}}^2$. Those excitation conditions correspond to transitions into the $E_2(k)$ and $E_1(k)$ exciton bands as indicated in the scheme shown in Fig. 1, respectively. The solid line is the linear absorption spectrum in the relevant region (scaled for ease of visualization), and the dotted lines are the spectra of the pump pulses. The transient absorption spectra are vertically offset for clarity, and the thin dashed lines are the baselines ($\Delta \mathrm{OD}=0$).

Figure 3

Plot of the maximum amplitude of the transient absorption kinetics probed at 660 nm (filled circles) and 953 nm (open squares) versus the intensity of the pump pulses at 953 nm. The dashed line is the linear fit of the data obtained by probing at 660 nm, and the dotted line is drawn to guide the eye for the data obtained by probing at 953 nm. The inset shows the same data measured at 953 nm, but plotted on the scale of the square root of pump intensity. The solid line represents the linear fit.

Figure 4

(a) Comparison of the squared profile at 953 nm (solid line) and the kinetics at 660 nm; both data are recorded with the $E_1(k)$ excitation. (b) Kinetics probed at 953 nm under the lowest (open squares) and highest (solid line) pump intensities at 953 nm. (c) Kinetics probed at 660 nm upon resonant excitation of the $E_1(k)$ (open squares) and $E_2(k)$ (solid line) exciton bands of the $(8,3)$ nanotubes. All kinetics shown in (a)–(c) are normalized at the signal maxima.