Dielectric screening dependence of excitonic transition energies in single-wall carbon nanotubes

The measured optical transition energies E$_{ii}$ of single-wall carbon nanotubes are compared with bright exciton energy calculations. The E$_{ii}$ differences between experiment and theory are minimized by considering first, a diameter/chiral angle-dependent dielectric constant and second, a diameter/exciton size-dependent dielectric constant (k). In our description, k is composed of the screening contributions from the tube, represented by k$_{tube,}$ and from the environment, represented by k$_{env}$. We discuss the main aspects of each approach and show that in the first case, different k dependencies are obtained for (E$^{S}_{11}$, E$^{S}_{22}$, E$^{M}_{11})$ relative to (E$^{S}_{33}$, E$^{S}_{44})$ which is understood as follows: A changing environment changes the k diameter dependence for (E$^{S}_{11}$, E$^{S}_{22}$, E$^{M}_{11})$, but for (E$^{S}_{33}$, E$^{S}_{44})$ the environmental effects are minimal. We show that in order to achieve a single dependence for all E$_{ii}$, the exciton's size should be taken into account, as considered in the second approach. The resulting calculated exciton energies reproduce experimental E$_{ii}$ values within $\vert $50$\vert $ meV for a diameter range (0.7$<$ dt $<$3.8 nm) and 1.2 $<$ Eii $<$2.7 eV, thus providing a theoretical justification for E$_{ii}$ and important insights into the dielectric screening in one-dimensional structures.