## 6 QD Emitters Near a Flat Interface
### Energy Transfer to a 2D Material
The above theory does not take into account the finite  
Figure 8: Schematics of the exciton energy transfer from a QD to graphene. The left part represents an excited QD with an electron–hole pair. It can be transferred as a whole to the graphene represented by its conduction and valence bands’ cones touching in the Dirac point. For the first-order vertical interband transition in graphene, at low temperatures the transfer is possible only if the exciton energy exceeds \(2E_{F}\). size of the QDs. As a first approximation, one can assume that the point dipole is located at the centre of the QD. In this case, the distance of the dipole from the graphene sheet can be approximated as \(h^{*}=(h+R_{QD}+s)\) where \(R_{QD}\) is the radius of the nanocrystal core of the QD and \(s\) is thickness of the organic shell for a colloidal QD. The latter can be estimated as the chain length of the capping agent, \(s\sim 1.5\,\)nm [12]. Using this correction, \(h\to h^{*}\), indeed improves the agreement between the theory and experiment [213]. However, one needs to bear in mind also the possibility of charge transfer from an excited QD to graphene if the distance is sufficiently small for tunnelling. Indeed, the demonstration of hybrid graphene-QD phototransistors [217] and solar cells [218] where the light is absorbed in the QDs and changes the electric current in graphene implies that such processes can take place in specially designed structures. Calculations including both intra-band (i.e., Drude plasmons) and inter-band transitions in graphene show that there can be a no-quenching spectral window for an emitter over a strongly doped graphene sheet [71]. Indeed, the graphene absorbance, \(\beta(\omega)\), due to Drude plasmons decreases with the frequency, while the inter-band transitions contribute to \(\beta\) (and, therefore, to the quenching) only for \(\omega>2E_{F}/\hbar\)[209]. As the Fermi level in graphene can be tuned electrically, it opens an interesting possibility of electrical switching of the QD emission by controlling the quenching rate; it has been demonstrated experimentally in [219; 220]. Energy transfer (ET) from QDs to other 2D materials, namely, few-atomic-layer-thick semiconductors (transition metal dichalcogenides (TMDs)) has also been investigated in the recent years [221; 222; 223; 224]. These materials support robust excitons, which determine their optical properties in the visible range (along with the inter-band transitions) and they depend strongly on the number of monolayers in the material [225]. The ET from a QD exciton to the continuum of states above the band gap in the 2D material should work similar to the case of graphene (Equation (22)). However, it was found that the donor QD emission is quenched most strongly near monolayer MoS\({}_{2}\) and then the effect decreases with the number of monolayers (presumably keeping the QD distance to the semiconductor surface fixed), in contrast with the case of few-layer graphene [222; 215]. Meanwhile, the tendency similar to graphene (quenching increases with the number of monolayers) was found for the less studied TMD SnS\({}_{2}\)[223]. On the theoretical side, it has been suggested that the total spontaneous emission rate of a quantum emitter can be enhanced several orders of magnitude due to the excitation of surface exciton-polariton modes supported by the 2D semiconductor [224]. In the vicinity of an excitonic transition, the real part of the dielectric function of the semiconductor material is negative, like in a metal, so one can indeed expect an enhancement of the local photonic DOS and, consequently, the Purcell effect [226]. However, it would require matching of the energies of the QD and 2D excitons.