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We perform a detailed analysis of electronic polarizability of graphene withdifferent theoretical approaches. From Kubo's linear response formalism, wegive a general expression of frequency and wave-vector dependent polarizabilitywithin the random phase approximation. Four theoretical approaches have beenapplied to the single-layer graphene and their differences are on theband-overlap of wavefunctions. By comparing with the \textit{ab initio}calculation, we discuss the validity of methods used in literature. Our resultsshow that the tight-binding method is as good as the time-demanding \textit{abinitio} approach in calculating the polarizability of graphene. Moreover, dueto the special Dirac-cone band structure of graphene, the Dirac modelreproduces results of the tight-binding method for energy smaller than\SI{3}{\electronvolt}. For doped graphene, the intra-band transitions dominateat low energies and can be described by the Lindhard formula fortwo-dimensional electron gases. At zero temperature and long-wavelength limit,with the relaxation time approximation, all theoretical methods reduce to along-wave analytical formula and the intra-band contributions agree to theDrude polarizability of graphene. Effects of electrical doping and temperatureare also discussed. This work may provide a solid reference for researches andapplications of the screening effect of graphene.

Dynamical Polarizability of Graphene with Spatial Dispersion