Dynamical Conductivity of Dirac Materials

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- Gr\"{u}neisen ($\Theta_{BG}$) temperature and T linear in opposite case ($T>>\Theta_{BG}$). Going beyond the DC case, we investigate the dynamical conductivity in graphene using the powerful method of memory function formalism. In the DC (zero frequency regime) limit, we obtained the above mention behavior which was previously obtained using the Bloch-Boltzmann kinetic equation. In the finite frequency regime, we obtained several new results: (1) the generalized Drude scattering rate, in the zero temperature limit, shows $\omega^4 $ behavior at low frequencies ($\omega << k_B \Theta_{BG}/ \hbar$) 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, $\omega>>k_B \Theta_{BG}/ \hbar$, and higher temperatures $T>>\Theta_{BG}$, 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.