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We calculate the static polarization of AAA-stacked trilayer graphene (TLG)and study its screening properties within the random phase approximation (RPA)in all undoped, doped and biased regimes. We find that the static polarizationof undoped AAA-stacked TLG is a combination of the doped and undoped singlelayer graphene static polarization. This leads to an enhancement of thedielectric background constant along a Thomas-Fermi screening with theThomas-Fermi wave vector which is independent of carrier concentrations and a1/r^3 power law decay for the long-distance behavior of the screened coulombpotential. We show that effects of a bias voltage can be taken into account bya renormalization of the interlayer hopping energy to a newbias-voltage-dependent value, indicating screening properties of biasedAAA-stacked TLG can be tuned electrically. We also find that screeningproperties of doped AAA-stacked TLG, when $\mu$ exceeds $\sqrt{2}\gamma$, aresimilar to that of doped SLG only depending on doping. While for$\mu<\sqrt{2}\gamma$, its screening properties are a combination of SLG andAA-stacked screening properties and they are determined by doping and theinterlayer hopping energy.

Effects of doping and bias voltage on the screening in AAA-stacked trilayer graphene