Bright and dark excitons in semiconductor carbon nanotubes

We report electronic structure calculations of finite-length semiconducting carbon nanotubes using the time dependent density functional theory (TD-DFT) and the time dependent Hartree Fock (TD-HF) approach coupled with semiempirical AM1 and ZINDO Hamiltonians. We specifically focus on the energy splitting, relative ordering, and localization properties of the optically active (bright) and optically forbidden (dark) states from the lowest excitonic band of the nanotubes. These excitonic states are very important in competing radiative and non-radiative processes in these systems. Our analysis of excitonic transition density matrices demonstrates that pure DFT functionals overdelocalize excitons making an electron-hole pair unbound; consequently, excitonic features are not presented in this method. In contrast, the pure HF and A111 calculations overbind excitons inaccurately predicting the lowest energy state as a bright exciton. Changing AM1 with ZINDO Hamiltonian in TD-HF calculations, predicts the bright exciton as the second state after the dark one. However, in contrast to AM1 calculations, the diameter dependence of the excitation energies obtained by ZINDO does not follow the experimental trends. Finally, the TD-DFT approach incorporating hybrid functions with a moderate portion of the long-range HF exchange, such as B3LYP, has the most generality and predictive capacity providing a sufficiently accurate description of excitonic structure in finite-size nanotubes. These methods characterize four important lower exciton bands. The lowest state is dark, the upper band is bright, and the two other dark and nearly degenerate excitons lie in-between. Although the calculated energy splittings between the lowest dark and the bright excitons are relatively large (∼0.1 eV), the dense excitonic manifold below the bright exciton allows for fast non-radiative relaxation leasing to the fast population of the lowest dark exciton. This rationalizes the low luminescence efficiency in nanotubes.