Non-equilibrium electron relaxation in Graphene

We apply the powerful method of memory function formalism to investigate non-equilibrium electron relaxation in graphene. Within the premises of Two Temperature Model (TTM), explicit expressions of the imaginary part of the Memory Function or generalized Drude scattering rate ($1/\tau$) are obtained. In the DC limit and in equilibrium case where electron temperature ($T_e$) is equal to phonon temperature (T), we reproduce the known results (i.e. $1/\tau \propto T^4$ when $T<<\Theta_{BG}$ and $1/\tau \propto T$ when $T>>\Theta_{BG}$, where $\Theta_{BG}$ is the Bloch-Gr\"{u}neisen temperature). We report several new results for $1/ \tau$ where $T \neq T_e$ relevant in pump-probe spectroscopic experiments. In the finite frequency regime, we find that $1/\tau \propto \omega^2$ when $\omega<<\omega_{BG}$, and for $\omega>>\omega_{BG}$ it is $\omega$ independent and also electron temperature independent. These results can be verified in a typical pump-probe experimental setting for graphene.