We compute the DC and the optical conductivity of graphene for finite valuesof the chemical potential by taking into account the effect of disorder, due tomid-gap states (unitary scatterers) and charged impurities, and the effect ofboth optical and acoustic phonons. The disorder due to mid-gap states istreated in the coherent potential approximation (CPA, a self-consistentapproach based on the Dyson equation), whereas that due to charged impuritiesis also treated via the Dyson equation, with the self-energy computed usingsecond order perturbation theory. The effect of the phonons is also includedvia the Dyson equation, with the self energy computed using first orderperturbation theory. The self-energy due to phonons is computed both using thebare electronic Green's function and the full electronic Green's function,although we show that the effect of disorder on the phonon-propagator isnegligible. Our results are in qualitative agreement with recent experiments.Quantitative agreement could be obtained if one assumes water molelcules underthe graphene substrate. We also comment on the electron-hole asymmetry observedin the DC conductivity of suspended graphene.
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