Graphene as a Tunable Nonradiative Bath for Moiré Excitons

A minimal theory for the nonradiative transfer of energy from a two-dimensional (2D) exciton -- especially a moir\'e-localized exciton -- to a nearby graphene layer is presented. Starting from Fermi's golden rule, the transfer rate is written as the overlap between the exciton near-field spectrum and the long-wavelength electronic loss function of graphene, weighted by an exciton form factor. In the point-dipole limit the framework reproduces the established $\GET\propto z^{-4}$ law for energy transfer to graphene. Including the finite spatial extent of a moir\'e exciton through a Gaussian form factor with localization length $\lX$, we show that high-momentum components of the near field are filtered out for $z\lesssim\lX$, so that the transfer rate -- and hence the photoluminescence (PL) quenching -- can serve as a probe of exciton localization. Treating graphene as a gate-tunable bath, a Pauli-blocking model predicts that interband electron-hole excitations are strongly suppressed once $2|\muF|$ approaches $\hbar\omega$, partially restoring PL intensity and lifetime. Benchmarking against the full random-phase-approximation loss function of doped graphene confirms the minimal model to within a few percent over the relevant distance range for representative near-infrared exciton parameters. We map the resulting PL observables over experimentally relevant ranges of spacer thickness, localization length, emission energy, and Fermi level, and identify when graphene-induced quenching dominates the optical response of transition-metal dichalcogenide/hexagonal boron nitride/graphene heterostructures. A graphene gate thus acts not as a passive electrostatic element but as a tunable 2D electronic reservoir whose long-wavelength response can be probed through exciton PL quenching.