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 $\hbar \Omega/2$.