Dynamical electrical conductivity of graphene

For graphene (a Dirac material) it has been theoretically predicted and experimentally observed that DC resistivity is proportional to T-4 when the temperature is much less than Bloch-Gruneisen temperature (circle minus(BG)) and T-linear in the opposite case (T >> circle minus(BG)). 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 omega(4) behavior at low frequencies (omega << k(B)circle minus(BG)/h) 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, omega >> k(B)circle minus(BG)/lh, and higher temperatures T >> circle minus(BG), 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.