Diameter and chirality dependence of exciton properties in carbon nanotubes

We calculate the diameter and chirality dependences of the binding energies, sizes, and bright-dark splittings of excitons in semiconducting single-wall carbon nanotubes. Using results and insights from ab initio calculations, we employ a symmetry-based variational method within the effective-mass and envelope-function approximations using tight-binding wave functions. Binding energies and spatial extents show a leading dependence on diameter as $1∕d$ and $d$, respectively, with chirality corrections providing a spread of roughly 20% with a strong family behavior. Bright-dark exciton splittings show a $1∕d^2$ leading dependence. We provide analytical expressions for the binding energies, sizes, and splittings that should be useful to guide future experiments.

Figure 1

(Color online) Lowest-energy singlet exciton wave function of the (11,0) tube. Black thin line: Ab initio ${\mid \psi \mid }^2$ after integrating out on the coordinates perpendicular to the tube. Red thick dashed line: Envelope fit using a Gaussian. Blue thick solid line: Envelope fit using a Whittaker function. Notice that the two fits are almost indistinguishable.

Figure 2

Binding energies for the lowest-energy bright excitons in 38 SWNTs with varying diameter and chirality. The dots are our model results and the lines represent the analytical fit using Eq. (7). The labels indicate the $(2n+m)$ families.

Figure 3

Sizes of the bright lowest-energy excitons for 38 SWNTs with varying diameter and chirality. The dots are our model results and the lines represent the analytical fit using Eq. (7). The labels indicate the $(2n+m)$ families.