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