We investigate the optical properties of spin-triplet interlayer excitons in heterobilayer transition metal dichalcogenides in comparison with the spin-singlet ones. Surprisingly, the optical transition dipole of the spin-triplet exciton is found to be in the same order of magnitude to that of the spin-singlet exciton, in sharp contrast to the monolayer excitons where the spin triplet species is considered as dark compared to the singlet. Unlike the monolayer excitons whose spin-conserved (spin-flip) transition dipole can only couple to light of in-plane (out-of-plane) polarization, such restriction is removed for the interlayer excitons due to the breaking of the out-of-plane mirror symmetry. We find that as the interlayer atomic registry changes, the optical transition dipole of interlayer exciton crosses between in-plane ones of opposite circular polarization and the out-of-plane one for both the spin-triplet and spin-singlet species. As a result, excitons of both species have non-negligible coupling into photon modes of both in-plane and out-of-plane propagations, another sharp difference from the monolayers where the exciton couples predominantly into the out-of-plane propagation channel. At given atomic registry, the spin-triplet and spin-singlet excitons have distinct valley polarization selection rules, allowing the selective optical addressing of both the valley configuration and the spin singlet/triplet configuration of interlayer excitons.
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