Many-body theory of radiative lifetimes of exciton-trion superposition states in doped two-dimensional materials

Optical absorption and emission spectra of doped two-dimensional (2D) materials exhibit sharp peaks that are often mistakenly identified with pure excitons and pure trions (or charged excitons), but both peaks have been recently attributed to superpositions of two-body exciton and four-body trion states and correspond to the approximate energy eigenstates in doped 2D materials. In this paper, we present the radiative lifetimes of these exciton-trion superposition energy eigenstates using a many-body formalism that is appropriate given the many-body nature of the strongly coupled exciton and trion states in doped 2D materials. Whereas the exciton component of these superposition eigenstates are optically coupled to the material ground state and can emit a photon and decay into the material ground state provided the momentum of the eigenstate is within the light cone, the trion component is optically coupled only to the excited states of the material and can emit a photon even when the momentum of the eigenstate is outside the light cone. In an electron-doped 2D material, when a four-body trion state with momentum outside the light cone recombines radiatively, and a photon is emitted with a momentum inside the light cone, the excess momentum is taken by an electron-hole pair left behind in the conduction band. The radiative lifetimes of the exciton-trion superposition states with momenta inside the light cone are found to be in the few hundred femtoseconds to a few picoseconds range and are strong functions of the doping density. The radiative lifetimes of exciton-trion superposition states with momenta outside the light cone are in the few hundred picoseconds to a few nanoseconds range and are again strongly dependent on the doping density. The doping density dependence of the radiative lifetimes of the two peaks in the optical emission spectra follows the doping density dependence of the spectral weights of the same two peaks observed in the optical absorption spectra, as both have their origins in the Coulomb coupling between the excitons and trions in doped 2D materials.