Influences of Exciton Diffusion and Exciton-Exciton Annihilation on Photon Emission Statistics of Carbon Nanotubes

Pump-dependent photoluminescence imaging and second-order photon correlation studies have been performed on individual single-walled carbon nanotubes (SWCNTs) at room temperature. These studies enable the extraction of both the exciton diffusion constant and the Auger recombination coefficient. A linear correlation between these parameters is attributed to the effect of environmental disorder in setting the exciton mean free path and capture-limited Auger recombination at this length scale. A suppression of photon antibunching is attributed to the creation of multiple spatially nonoverlapping excitons in SWCNTs, whose diffusion length is shorter than the laser spot size. We conclude that complete antibunching at room temperature requires an enhancement of the exciton-exciton annihilation rate that may become realizable in SWCNTs allowing for strong exciton localization.

Figure 1

(a),(g) Wide-field PL images of two SWCNTs with lengths of $\sim 1.2\mu \mathrm{m}$ (a) and $2.6\mu \mathrm{m}$ (g), respectively. (b)–(e), (h)–(k) PL images of the same SWCNTs confocally excited at the center of the tubes. The excitation intensity was increased from $6.6\mathrm{W}/{\mathrm{cm}}^2$ (b) to $535.0\mathrm{W}/{\mathrm{cm}}^2$ (e), and $41.2\mathrm{W}/{\mathrm{cm}}^2$ (h) to $223.1\mathrm{W}/{\mathrm{cm}}^2$ (k), respectively. f, l, Axial intensity profiles of SWCNT PL images [black circles in (b),(h); red circles in (e),(k)] together with the corresponding fitting curves (solid lines). Insets, FWHM of measured intensity profiles (black triangles), uncertainty (error-bars) determined from Gaussian fit of the intensity profiles (see the Supplemental Material [25], Sec. S3) and FWHM of calculated (red triangles) PL intensity profiles are plotted as the function of pump fluence given in terms of average number of absorbed photons per laser pulse $⟨N⟩$.

Figure 2

Dots show values of Auger coefficient $C_A$ and diffusion constant $D$, obtained from global fitting of the intensity profiles of individual SWCNTs using our generalized diffusion model [Eqs. (1) and (2)]. Straight line $C_A\sim D/d_0$ with parameter $d_0=6.5\mathrm{nm}$ is a linear fit of the experimental points.

Figure 3

(a)–(c), Representative $g^{(2)}$ ($\tau$) of three individual SWCNTs with diffusion lengths of 188 nm (a), 336 nm (b), and 525 nm (c) measured at increasing pump fluences (from bottom to top, black dots). Each peak is well fitted by a two-sided exponential function (red curves). The degree of photon-antibunching, $R$, is indicated for each curve. $\mathbf{d}$, $R$ of the three individual SWCNTs in (a)–(c) plotted as a function of $⟨N⟩$ [black in (a), red in (b), and blue in (c)].

Figure 4

Measured (green circles) and calculated (solid line) $R_0$ values plotted along with calculated $p_2{Q}_{2X}/(m{Q}_{1X})$ (dotted line) and $(m-1)/m$ (dashed line) factors as a function of the ratio ${\sigma }_0/L_D$ and number of independent emitters $m={\sigma }_0/1.6L_D$.