Triplet excitations in carbon nanostructures

We show that the energy differences between the lowest optical singlet exciton and the lowest triplet exciton in semiconducting single-walled carbon nanotubes with diameter $\sim 1$ nm and graphene nanoribbons with widths $\sim 2$ nm are an order of magnitude smaller than in the $\pi$-conjugated polymer poly(para-phenylenevinylene). Our calculated energy gaps between the singlet and triplet excitons are in excellent agreement with the measured values in three different nanotubes with diameters close to 1 nm. The spatial extent of the triplet exciton is nearly the same as that of the singlet exciton in wide nanotubes and nanoribbons, in contrast to that in $\pi$-conjugated polymers, in which the triplet exciton exhibits strong spatial confinement. Weakly confined behavior of the triplet state begins in nanoribbons with widths as narrow as 2.5 times the graphene unit lattice vector. We discuss possible consequences of the small singlet-triplet energy difference in the carbon nanostructures on device applications.