The infrared conductivity of graphene

We study the infrared conductivity of graphene at finite chemical potential and temperature taking into account the effect of phonons and disorder due to charged impurities and unitary scatterers. The screening of the long-range Coulomb potential is treated using the random phase approximation coupled to the coherent potential approximation. The effect of the electron-phonon coupling is studied in second-order perturbation theory. The theory has essentially one free parameter, namely, the number of charge impurities per carbon, n^{{\rm C}}_i. We find an anomalous enhancement of the conductivity in a frequency region that is blocked by Pauli exclusion and an impurity broadening of the conductivity threshold. We also find that phonons induce Stokes and anti-Stokes lines that produce an excess conductivity, when compared to the far infrared value of \sigma_0 = (\pi/2) e^2/h.