Distance dependence of the energy transfer mechanism in WS$_2$-graphene heterostructures

We report on the mechanism of energy transfer in van der Waals heterostructures of the two-dimensional semiconductor WS$_2$ and graphene with varying interlayer distances, achieved through spacer layers of hexagonal boron nitride (hBN). We record photoluminescence and reflection spectra at interlayer distances between 0.5 nm and 5.8 nm (0-16 hBN layers). We find that the energy transfer is dominated by states outside the light cone, indicative of a F\"orster transfer process, with an additional contribution from a Dexter process at 0.5 nm interlayer distance. We find that the measured dependence of the luminescence intensity on interlayer distances above 1 nm can be quantitatively described using recently reported values of the F\"orster transfer rates of thermalized charge carriers. At smaller interlayer distances, the experimentally observed transfer rates exceed the predictions and furthermore depend on excess energy as well as on excitation density. Since the transfer probability of the F\"orster mechanism depends on the momentum of electron-hole pairs, we conclude that at these distances, the transfer is driven by non-thermalized charge carrier distributions.