  We calculate the inelastic scattering rates and the hot electron inelastic
mean free paths for both monolayer and bilayer graphene on a polar substrate.
We study the quasiparticle self-energy by taking into account both
electron-electron and electron-surface optical (SO) phonon interactions. In
this calculation the leading order dynamic screening approximation (G$_0$W
approximation) is used to obtain the quasiparticle self-energy by treating
electrons and phonons on an equal footing. We find that the strong coupling
between the SO phonon and plasmon leads to a new decay channel for the
quasiparticle through the emission of the coupled mode, and gives rise to an
abrupt increase in the scattering rate, which is absent in the uncoupled
system. In monolayer graphene a single jump in the scattering rate occurs,
arising from the emission of the low energy branch of the coupled
plasmon-phonon modes. In bilayer graphene the emission of both low and high
energy branches of the coupled modes contributes to the scattering rate and
gives rise to two abrupt changes in the scattering rate. The jumps in the
scattering rate can be potentially used in the hot electron device such as
switching devices and oscillators.
