  In this article we study the static polarization in ABC-stacked multilayer
graphene. Since the density of states diverges for these systems if the number
of layers exceeds three, screening effects are expected to be important. In the
random phase approximation, screening can be included through the polarization.
We derive an analytical integral expression for the polarization in both the
full-band model and an effective two-band model. Numerical evaluation of these
integrals are very time consuming in the full-band model. Hence, for
ABC-stacked trilayer graphene, we use the two-band model to calculate the low
momentum part of the polarization. The results for the two-band model are
universal, i.e. independent of doping. The high momentum part is linear and is
determined by calculating two points, such that we can determine the slope. For
ABC stacked trilayer graphene, the slope is given by three times the monolayer
value. We compare our results to previous ones in the literature and discuss
the similarities and discrepancies. Our results can be used to include
screening in ABC-stacked multilayer systems in a way that all the
characteristics of the polarization function are included. The numerical
results for the polarization of trilayer graphene are used to sketch the
screened potential.
