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We apply memory function formalism to investigate nonequilibrium 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/[Formula: see text]) are obtained. In the DC limit and in equilibrium case where electron temperature (Te) is equal to phonon temperature (T), we reproduce the known results (i.e., 1/[Formula: see text][Formula: see text]T4 when T[Formula: see text][Formula: see text] and 1/[Formula: see text][Formula: see text]T when T[Formula: see text][Formula: see text], where [Formula: see text] is the Bloch–Grüneisen temperature). We report several new results for 1/[Formula: see text] where T[Formula: see text][Formula: see text][Formula: see text]Te relevant in pump–probe spectroscopic experiments. In the finite-frequency regime we find that 1/[Formula: see text] when [Formula: see text], and for [Formula: see text] it is [Formula: see text]-independent. These results can be verified in a typical pump–probe experimental setting for graphene.

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For graphene (a Dirac material) it has been theoretically predicted and experimentally observed that DC resistivity is proportional to T when the temperature is much less than Bloch-Grüneisen temperature ([Formula: see text]) and T-linear in the opposite case ([Formula: see text]). Going beyond this case, we investigate the dynamical electrical conductivity in graphene using the powerful method of the memory function formalism. In the zero frequency regime, we obtain the above mentioned behavior which was previously obtained using the Bloch-Boltzmann kinetic equation. In the finite frequency regime, we obtain several new results: (1) the generalized Drude scattering rate, in the zero temperature limit, shows [Formula: see text] behavior at low frequencies ([Formula: see text]) and saturates at higher frequencies. We also observed the Holstein mechanism, however, with different power laws from that in the case of metals; (2) at higher frequencies, [Formula: see text], and higher temperatures [Formula: see text], we observed that the generalized Drude scattering rate is linear in temperature. In addition, several other results are also obtained. With the experimental advancement of this field, these results should be experimentally tested.

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