Deviations from the usual $R^{-6}$ dependence of the rate of fluorescence resonance energy transfer (FRET) on the distance between the donor and the acceptor have been a common scenario in the recent times. In this paper, we present a critical analysis of the distance dependence of FRET, and try to illustrate the non-$R^{-6}$ type behaviour of the rate for the case of transfer from a localized electronic excitation on the donor, a dye molecule to three different energy acceptors with delocalized electronic excitations namely, graphene, a two-dimensional semiconducting sheet and the case of such a semiconducting sheet rolled to obtain a nanotube. We use simple analytic models to understand the distance dependence in each case.

Recently, we have reported theoretical studies on the rate of energy transfer from an electronically excited molecule to graphene. It was found that graphene is a very efficient quencher of the electronically excited states and that the rate $\alpha z^{−4}$. The process was found to be effective up to 30 $nm$ which is well beyond the traditional FRET limit. In this report, we study the transfer of an amount of energy $\hbar \Omega$ from a dye molecule to doped graphene. We find a crossover of the distance dependence of the rate from $z^{−4}$ to exponential as the Fermi level is increasingly shifted into the conduction band, with the crossover occurring at a shift of the Fermi level by an amount $\hbar \Omega/2$.