  In heterostructures consisting of different transition-metal dichalcogenide
monolayers, a staggered band alignment can occur, leading to rapid charge
separation of optically generated electron-hole pairs into opposite monolayers.
These spatially separated electron-hole pairs are Coulomb-coupled and form
interlayer excitons. Here, we study these interlayer excitons in a
heterostructure consisting of MoSe$_2$ and WSe$_2$ monolayers using
photoluminescence spectroscopy. We observe a non-trivial temperature dependence
of the linewidth and the peak energy of the interlayer exciton, including an
unusually strong initial redshift of the transition with temperature, as well
as a pronounced blueshift of the emission energy with increasing excitation
power. By combining these observations with time-resolved photoluminescence
measurements, we are able to explain the observed behavior as a combination of
interlayer exciton diffusion and dipolar, repulsive exciton-exciton
interaction.
