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. The simple asymptotic expressions are given in various limiting cases. For instance, the conductivity for kv0 ≪ T ≪ ω is equal to σ(ω, k) = e 2 /4h 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.

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We analyze the features of the graphene mono-and multilayer reflectance in the far-infrared region as a function of frequency, temperature, and carrier density taking the intraband conductance and the interband electron absorbtion into account. The dispersion of plasmon mode of the multilayers is calculated using Maxwell's equations with the influence of retardation included. At low temperatures and high electron densities, the reflectance of multilayers as a function of frequency has the sharp downfall and the subsequent deep well due to the threshold of electron interband absorbtion.PACS numbers: 81.05. Uw, 78.67.Ch, Monolayer and bilayer graphenes 1,2,3 are gapless twodimensional (2D) semiconductors 4,5,6 whereas its 3D predecessor-graphite is a semimetal 7,8,9 . Hence the dimensionality effects for the unique substance can be studied 10 . Monolayer graphene has a very simple electron band structure. Near the energy ε = 0, the energy bands are cones ε 1,2 (p) = ±vp at the K points in the 2D Brillouin zone with the constant velocity parameter v = 10 8 cm/s. Such a degeneration is conditioned by symmetry because the small group C 3v of the K points has two-dimensional representation.While the carrier concentration is decreasing in the field gate experiment, the graphene conductivity at low temperatures goes to the finite minimal values 1,2 . Much theoretical efforts 11,12,13,14 have been devoted to evaluate the minimal conductivity in different approaches. Theoretical 15,16,17 and experimental researches show that the main mechanism of the carrier relaxation is provided by the charged impurities and gives the collision rate τ −1 ∼ 2π 2 e 4 n imp /hǫ 2 g ε, where ǫ g is the dielectric constant of graphene, ε is the characteristic electron energy (of the order of the Fermi energy or temperature), and n imp is the density of charged impurities per the unit surface. Plasmons in graphene are considered in Refs.18,19 . The optical visibility of both monolayer and bilayer graphene is studied in Ref.20 focusing on the role of the underlying substrate.In the present paper, we analyze the spectroscopy of the graphene monolayer and multilayers in the infra-red region. In order to calculate the reflection coefficient for the multilayers, we follow the method used in Ref.21 and determine the spectrum of electromagnetic excitationsplasmons. We use the appropriate boundary conditions at interfaces and the complex conductivity σ as a function of frequency ω, temperature T , and chemical potential µ. The chemical potential of ideal pure graphene equals to zero at any temperature. With the help of the gate voltage, one can control the density and type (n or p) of carriers varying their chemical potential.The general expression for the conductivity used here is obtained in our previous paper 18 and is valid under a restriction that the collision rate of carriers is less than the frequency and spatial dispersion of the electric ac field, τ −1 ≪ ω, kv. In limiting cases, our result coincides with the formulas of Ref.22,23 . For hig...