Possibility for exciton Bose-Einstein condensation in carbon nanotubes

We demonstrate a possibility for exciton Bose-Einstein condensation in individual small-diameter ($\sim$$1\text{--}2$ nm) semiconducting carbon nanotubes. The effect occurs under the exciton-interband-plasmon coupling controlled by an external electrostatic field applied perpendicular to the nanotube axis. It requires fields $\sim$1 V/nm and temperatures below 100 K that are experimentally accessible. The effect offers a testing ground for fundamentals of condensed matter physics in one dimension and opens up perspectives to develop tunable coherent polarized light source with carbon nanotubes.

Figure 1

Calculated energy dependence of the dimensionless (normalized by $e^2/2\pi \hslash$) axial surface conductivity ${\sigma }_{zz}$ for the four zigzag nanotubes, (13,0), (16,0), (20,0) and (26,0), of increasing diameters. Peaks of $\text{Re}{\sigma }_{zz}$ represent excitons ($E_{11}$, $E_{22}$); peaks of $\text{Re}(1/{\sigma }_{zz})$ represent interband plasmons ($P_{11}$, $P_{22}$). Dimensionless energy is defined as $\left[\text{Energy}\right]/2{\gamma }_0$, where ${\gamma }_0=2.7$ eV is the C-C overlap integral.

Figure 2

(a) The geometry of the problem. (b) Exciton-plasmon dispersion as a function of the perpendicular electrostatic field and longitudinal momentum for the lowest bright exciton coupled to the nearest interband plasmon in the (20,0) nanotube ($E_{11}^{(20,0)}$ and $P_{11}^{(20,0)}$ in Fig. 1). See text for dimensionless momentum.

Figure 3

Critical temperatures (a) as functions of the perpendicular electrostatic field applied and longitudinal momentum, and mean upper-branch BEC population fractions (b) as given by Eq. (7), for the four CNs under consideration. See text for dimensionless momentum.

Figure 4

Function $ln[1+{\langle n_{10}\rangle }_{{}_{E/P}}(0,T,F)]$ calculated using Eqs. (7) and (8) for the CNs under consideration.