Detection of a Biexciton in Semiconducting Carbon Nanotubes Using Nonlinear Optical Spectroscopy

We report the observation of the biexciton in semiconducting single-wall carbon nanotubes by means of nonlinear optical spectroscopy. Our measurements reveal the universal asymmetric line shape of the Fano resonance intrinsic to the biexciton transition. For nanotubes of the (9,7) chirality, we find a biexciton binding energy of 106 meV. From the calculation of the ${\chi }^{(3)}$ nonlinear response, we provide a quantitative interpretation of our measurements, leading to an estimation of the characteristic Fano factor $q$ of $7\pm 3$. This value allows us to extract the first experimental information on the biexciton stability and we obtain a biexciton annihilation rate comparable to the exciton-exciton annihilation one.

Figure 1

Energy level diagram displaying the biexciton (a) and trion (b) transitions probed in our nonlinear optical experiments. $E_P$ and $E_{P2}$ stand for the energy of the first and second pump lasers, respectively, $E_{\mathrm{probe}}$ for the energy of the probe laser, and ${\gamma }_B$ is the biexciton annihilation rate characterizing the Coulomb coupling efficiency between the biexciton state $|XX⟩$ and the $S_{11}$ continuum of free electron-hole pair states $|\mathrm{eh}⟩$.

Figure 2

(a) Differential transmission $\Delta T/T$ of the probe laser at 10 K, as a function of the energy detuning $E_{\mathrm{probe}}-E_P$ between the probe and first pump lasers. $E_P=0.83\mathrm{eV}$ for the right part (blue dots), and $E_P=0.93\mathrm{eV}$ for the left one (red squares) [15]. The solid (black) line is a fit of the experimental data. Inset: expanded view of the biexciton line after subtracting the linear background and the induced absorption signal of the trion (described in the text and in Ref. 15). The dashed line is a fit of the experimental data in the Lorentzian limit ($q=+\infty$). (b) Temperature dependence of the differential transmission $\Delta T/T$ for $E_P=0.93\mathrm{eV}$, between 10 K and 50 K (the spectrum at 25 K is shifted by $+0.2$ for clarity). (c) Pump power dependence of the nonlinear signal amplitude for the $XX$ line (blue squares), and $X^{*}$ line (red triangles).

Figure 3

(a) Differential transmission $\Delta T/T$ of the probe laser at 10 K, as a function of the sum of the probe energy $E_{\mathrm{probe}}$ and $E_P=0.93\mathrm{eV}$. (Inset) Differential transmission $\Delta T/T$ spectrum in the biexciton spectral range, with (open blue circles) and without (full red squares) the additional pump laser at an energy $E_{P2}=1.76\mathrm{eV}$. (b) Deformation $\delta$ of the differential transmission $\Delta T/T$ spectrum as a function of the energy $E_{P2}$ of the additional pump laser. The dashed line is a guide for the eyes of Lorentzian form.