Space-time dispersion of graphene conductivity

We present an analytic calculation of the conductivity of pure graphene as a function of frequency ω, wave-vector k, and temperature for the range where the energies related to all these parameters are small in comparison with the band parameter γ≃3 eV, but much larger than the collision rate τ^-1. The simple asymptotic expressions are given in various limiting cases. For instance, the conductivity for kv₀≪ T≪ω is equal to σ(ω,k)=e²/4ħ and independent of the band structure parameters γ and v₀. Our results are also used to explain the known dependence of the graphite conductivity on temperature and pressure.