## 6 QD Emitters Near a Flat Interface
### FRET between QDs Near Interface  
In the vicinity of a polarisable material, such as graphene, the dipole-dipole interaction responsible for the FRET between two QDs becomes renormalized because of the additional interaction between the induced charges. In the electrostatic limit, these additional interactions can be modelled with image dipoles (see Figure 9). This physical mechanism, of course, is not unique to graphene and can take place in the vicinity of plasmonic nanoparticles [36; 37; 227]. The effect of plasmonic enhancement of FRET has been detected experimentally by observing a decrease of the emission decay rate of donor QDs and a corresponding increase of the luminescence intensity for acceptor dots when they are placed in the vicinity of gold nanoparticles [37]. Of course, if the concentration of the nanoparticles becomes too high, the emission is quenched for both donor and acceptor QDs because of the energy transfer to lossy modes in the metal (the Ohmic losses, Section 6.2). Graphene is a conductor with electrically controllable conductivity and, in principle, it promises a possibility of controlling the inter-dot FRET rate. The exciton energy transfer between two QDs located near a graphene-covered interface occurs not just through the direct dipole-dipole interaction but also through polarization charges that they induce on the interface. One can think that the image dipoles of Figure 9 have magnitudes that depend on the optical conductivity of graphene and, therefore, on its Fermi energy [196]. This interaction can be written in the form  
\[V_{FRET}=\frac{1}{\epsilon_{1}}\mathbf{d}_{\mathbf{D}}\cdot\mathbf{\hat{T}} \cdot\mathbf{d}_{\mathbf{A}}\]  
with \(\mathbf{\hat{T}}=\mathbf{\hat{T}}^{(0)}+\mathbf{\hat{T}}^{(1)}\), where  
\[\mathbf{\hat{T}}^{(0)}(\mathbf{b})=b^{-3}(3\mathbf{n}\otimes\mathbf{n}-\mathbf{ \hat{I}})\]  
is the usual dipole-dipole interaction tensor (\(\mathbf{n}=\mathbf{b}/b\)) and  
\[\mathbf{\hat{T}}^{(1)}(\mathbf{b})=\int\!\frac{\mathrm{d}^{2}q}{(2\pi)^{2}}\left(- i\mathbf{e}_{q}+\mathbf{e}_{z}\right)\otimes(i\mathbf{e}_{q}+\mathbf{e}_{z})qA(q)e^{-2 qh+ iq\cdot b} \tag{23}\]  
is the part due to the image dipoles. In Equation (23), \(\mathbf{e}_{q}=q/q\), \(\mathbf{e}_{z}\) is the unit vector along \(z\)-axis,  
\[A(q)=\frac{\epsilon_{2}-\epsilon_{1}+f(q)}{\epsilon_{2}+\epsilon_{1}+f(q)}\,, \qquad f(q)=i\frac{4\pi\sigma_{g}(\omega)}{\omega}q\,,\]  
and \(\sigma_{g}(\omega)\) is the optical conductivity of graphene. The components of \(\mathbf{\hat{T}}^{(1)}\) have an oscillatory dependence upon the inter-dot distance [228], which arises from the Bessel functions appearing as the result of the angular integration in Equation (23). It means that instead of the usual monotonic \(b^{-6}\) dependence of the FRET rate one may have an oscillatory behaviour, controllable by the gate voltage through the graphene Fermi energy (which determines \(\sigma_{g}(\omega)\)). Beyond the non-monotonic dependence upon the inter-dot distance, the emission/transfer frequency also should affect it in a rather complex way. If this interaction is strong enough, the emitters (QDs in our case) can be coupled to create a collective radiative mode, a phenomenon called superradiance [229], which has been observed for epitaxial quantum dots [230]. As the coupling between the emitters can be strongly enhanced or suppressed in the vicinity of graphene, depending on the distance between them, the frequency and the graphene Fermi energy, both superradiance and subradiance regimes can be expected [228]. So far, we are not aware of an experimental demonstration of such effects. Losses associated with the real part of graphene's optical conductivity can be one possible reason for this. The influence of a polarisable surface (such as graphene) on the irreversible FRET should be an easier-to-observe effect than the superradiance.