## 6 QD Emitters Near a Flat Interface
### Energy Transfer to a 2D Material  
Graphene, an atomic-thick monolayer of carbon, is a semimetal (or gapless semiconductor) with unusual electronic properties first demonstrated by [PERSON] and [PERSON] [205]. Low-energy excitations in graphene are massless, chiral, Dirac fermions. In neutral graphene, the chemical potential (hereafter called Fermi level) crosses exactly the Dirac point [206]. Pristine graphene is transparent in a broad spectral range from the infrared (IR) to the ultraviolet (UV), with the residual absorption of \(\approx\)2.3% originating from interband transitions [207]. The Fermi level, \(E_{F}\), can be shifted by up to \(\approx\)1 eV with respect to the Dirac point by applying an appropriate gate voltage. The usual doping via adsorbed impurities is also possible [206]. One can say that graphene is a transparent conductor with a tunable conductivity, both static and frequency-dependent (for the frequencies up to the mid-IR) [208]. It supports propagating surface plasmons whose dispersion relation can be controlled via gate voltage [209]. Interestingly, a specific type of quantum dots can form in graphene, namely, mass profile QDs, a system within current experimental reach [210], for which FRET can also be important [211]. Figure 7: Spectral dependence of different contributions to the emission decay rate for a vertically oriented emitter located in a dielectric with \(\epsilon_{1}=2\) at two different distances (indicated on the plots) above a flat interface with gold. Note that the contribution of the Ohmic losses (the main cause of the emission quenching near metal surfaces) is negligible in the upper plot, while it is completely dominating in the lower panel. For the calculation of the radiative losses and coupling to surface plasmons, the damping parameter, \(\Gamma\) of the Drude model was set equal to zero. Numerous experiments have shown that the nanoparticle or dye molecule emission is quenched in a broad spectral range by a single graphene sheet [39; 212; 213; 214; 215]. It is understandable for the lower-energy part of the spectrum, as graphene is a conductor, although with a relatively low free carrier concentrations (which can reach \(10^{13}\) cm\({}^{-2}\)), with the plasma frequency in the THz-to-IR range; it also is supported by calculations. However, the situation is more complex than in the case of a point emitter in the vicinity of a usual metal. In addition to the Ohmic losses and energy transfer to surface plasmons, there is an additional mechanism that may be named exciton transfer (ET) to graphene. It is similar to FRET and consists in energy transfer to electron-hole pairs generated by inter-band transitions, symmetric with respect to the Dirac point (see Figure 8). Doped graphene absorbs EM radiation due to inter-band transitions only for energies above \(2E_{F}\), therefore, it also applies to the ET. The main contribution of this mechanism scales with distance as \(h^{-4}\). It can be demonstrated in a rather simple way that elucidates its connection to FRET. Let us assume that acceptor dipoles are distributed within a plane (representing the graphene sheet), then the ET rate can be obtained by integrating (13) over this plane and written as follows,  
\[\frac{k_{ET}(h)}{\gamma_{0}^{(QD)}}=\rho_{A}\int_{0}^{2\pi}d\phi\int_{0}^{ \infty}\frac{rdrb_{0}^{6}}{(r^{2}+h^{2})^{3}}\,,\]  
where \(\rho_{A}\) is the surface density of acceptor dipoles. The emitter lineshape may be taken as a \(\delta\)-function. As \(\rho_{A}Q_{A}I_{A}=\beta(\omega)\) is the absorbance of the graphene, we obtain  
\[\frac{k_{ET}(h,\omega)}{\gamma_{0}^{(QD)}}=\frac{3c^{4}}{8\epsilon_{1}^{2} \omega^{4}h^{4}}\beta(\omega)\,. \tag{22}\]  
This formula can be derived in a more rigorous way [216], with just slightly different numerical coefficients. The \(h^{-4}\) scaling law was verified in several works and it seems to be obeyed for molecular emitters (down to \(h\approx 5\,\)nm) [212] but not so much for QDs where deviations were found for small distances [213].