Asymptotic exchange coupling of quasi-one-dimensional excitons in carbon nanotubes

An analytical expression is obtained for the biexciton binding energy as a function of the interexciton distance and binding energy of constituent quasi-one-dimensional excitons in carbon nanotubes. This allows one to trace biexciton energy variation and relevant nonlinear absorption under external conditions whereby the exciton binding energy varies. The nonlinear absorption line shapes calculated exhibit characteristic asymmetric (Rabi) splitting as the exciton energy is tuned to the nearest interband plasmon resonance. These results are useful for tunable optoelectronic device applications of optically excited semiconducting carbon nanotubes, including the strong excitation regime with optical nonlinearities.

Figure 1

(a) Schematic of the exchange coupling of two ground-state 1D excitons to form a biexcitonic state (arbitrary units). Two collinear axes, $z_1$ and $z_2$, which represent independent relative electron-hole motions in the first and second exciton, have their origins shifted by $\Delta Z$, the interexciton center-of-mass separation. (b) The coupling occurs in the configuration space of the two independent longitudinal relative electron-hole motion coordinates, $z_1$ and $z_2$, of each of the excitons, due to the tunneling of the system through the potential barriers formed by the two single-exciton cusp-type potentials [bottom, also in (a)], between equivalent states represented by the isolated two-exciton wave functions shown on the top.

Figure 2

Difference $E_g(\Delta Z)-2E_X$ for the coupled pair of the first bright excitons in the $(11,0)$ CN as a function of the center-of-mass-to-center-of-mass interexciton separation $\Delta Z$ and perpendicular electrostatic field applied. Inset: Biexciton binding energy $E_{XX}$ and equilibrium interexciton separation $\Delta Z_0$ ($y$ and $x$ coordinates, respectively, of the minima in the main figure) as functions of the field.

Figure 3

[(a), (b), and (c)] Linear (top) and nonlinear (bottom) response functions as given by Eq. (8) and by the imaginary part of Eq. (9), respectively, for the first bright exciton in the $(11,0)$ CN as the exciton energy is tuned to the nearest interband plasmon resonance (vertical dashed line). Vertical lines marked as $X$ and $\mathit{XX}$ show the exciton energy and biexciton binding energy, respectively. Dimensionless energy is defined as $[\text{Energy}]/2{\gamma }_0$, where ${\gamma }_0=2.7$ eV is the C-C overlap integral.