Bright and dark excitons in semiconductor carbon nanotubes: insights from electronic structure calculations

Abstract

We review electronic structure calculations of finite-length semiconducting carbon nanotubes using time-dependent density functional theory (TD-DFT) and the time dependent Hartree-Fock (TD-HF) approach coupled with semi-empirical AM1 and ZINDO Hamiltonians. We specifically focus on the energy splitting, relative ordering, and localization properties of 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 AM1 calculations overbind excitons, inaccurately predicting the lowest energy state as a bright exciton. Changing the AM1 with the 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 functionals 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 ( approximately 0.1 eV), the dense excitonic manifold below the bright exciton allows for fast non-radiative relaxation leading to the rapid population of the lowest dark exciton. This rationalizes the low luminescence efficiency in nanotubes.

Progress and Challenges in the Calculation of Electronic Excited States

“…This method is commonly employed because of its simplicity and apparent black-box behavior, in particular for large systems out of reach from the more accurate ab initio methods. [75,76] Based, as the propagator methods, on the solution of the frequency-dependent polarizability equation, [77,78] the procedure avoids the calculation of the explicit state just by obtaining the excitation energy and the transition dipole moment. As for any other single-reference approach, the adequate application of the method is typically restricted to structures which have closed-shell ground-state references.…”

Electronic Structure of Ligated CdSe Clusters: Dependence on DFT Methodology

“…70,73 Namely, the pure XC functionals (LSDA and GGA) underestimate the band gaps and do not include relevant excitonic effects in the optical transitions. Hybrid functionals, such as B3LYP and PBE1, provide very reasonable values of the energy gap and transition energies (∼3 eV), which correlate well with experimental data.…”

DFT studies on the β-glycosidase catalytic mechanism: The deglycosylation step

“…In the last years, the computational power and the development of sophisticated density functionals has increased. Therefore, many reviews have been published in order to provide an outline of the several types of density functionals currently available and the practical application of DFT to solve chemical problems [14][15][16][30][31][32][33][34][35][36]. The performance of each functional is evaluated taking into account its capacity to reproduce a wide range of chemical properties, namely barrier heights, atomization energies, binding energies, ionization potentials, electron affinities, heats of formation, hydrogen bonding energies, molecular geometries and vibrational properties.…”