Stochastic models of carrier dynamics in single-walled carbon nanotubes

Recent time-resolved measurements in optically excited individual semiconducting single-walled carbon nanotubes have revealed rapid electron-hole pair annihilation. A stochastic model of the carrier dynamics has been proposed that includes Auger interactions and discrete occupancy of the nanotubes by the carriers [F. Wang et al., Phys. Rev. B 70, 241403(R) (2004)]. Here we solve this model analytically and also analyze another physically reasonable stochastic model which involves Auger ionization.

Figure 1

Decay in the number of excitons per nanotube, $n\left(t\right)$, from the initial $n_0=1,2,3,5$ (from bottom to top) for $\gamma ∕{\gamma }_A=0.1$, according to Eq. (10). Dots represent direct numerical solution of the master equation (3) for $n_0=2$. Log-linear plot.

Figure 2

Decay in the average number of excitons per nanotube, $\overline{n}\left(t\right)$, for the Poissonian initial distribution (13) with ${\overline{n}}_0=2,3,5$ (from bottom to top) and $\gamma ∕{\gamma }_A=0.1$, according to Eq. (15). Dots represent direct numerical solution of the master equation (3) for ${\overline{n}}_0=2$. Dashed line shows the long-time decay with the rate $\gamma$ and the amplitude ${\overline{n}}_{\infty }$ given by Eq. (16), for ${\overline{n}}_0=2$. Log-linear plot.

Figure 3

Normalized amplitude of the long-time decay, ${\overline{n}}_{\infty }∕(1+z)$, as a function of the initial average number of excitons per nanotube, ${\overline{n}}_0$, for Poissonian initial condition (13) and $\gamma ∕{\gamma }_A=0.1$. Solid line illustrates Eq. (16); dashed line corresponds to Eq. (17).

Figure 4

Decay in the number of excitons per nanotube, $n\left(t\right)$, from the initial $n_0=2,3,4$ for $\gamma ∕{\gamma }_A=0.1$, according to Eq. (22). Dots represent direct numerical solution of the master equation (19) for $n_0=2$. Dashed line shows the long-time decay with the rate $\gamma$ and the amplitude $n_{\infty }=2\gamma ∕(\gamma +{\gamma }_A)$ for $n_0=2$. Here $n_{\infty }$ is the fraction of excitons that escaped Auger recombination due to trapping. Log-linear plot.