Fluorescence dynamics and fine structure of dark excitons in semiconducting single-wall carbon nanotubes

Exact diagonalization results are reported for the bright and dark exciton structure of semiconducting single-wall carbon nanotubes in the framework of the Hubbard model combined with a small crystal approach for several values of the correlation coupling strength U/t. Our findings, in the low-intermediate correlation regime (1.5 < U/t < 2.1), show the presence of dark states above and below the first bright exciton |B> and can account for reported experimental values of deep triplet states below |B> and of a K-momentum singlet dark exciton above this state. In order to fit the temporal profile of the photoluminescence (PL) decay, a bottleneck mechanism is considered involving a few dark states, with the respective energy gaps correspondingly obtained in the above-mentioned correlation range. We find that a kinetic model with one dark state above and two below |B> is able to recover the observed biexponential features of the PL behaviour with a reasonable set of parameters. Within this model we attribute the long tail of the PL to a delayed luminescence process of the bright state caused by the nearby calculated dark states.