  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.
