  We show that graphene-dielectric multilayers give rise to an unusual
tunability of the Casimir-Lifshitz forces, and allow to easily realize
completely different regimes within the same structure. Concerning thermal
effects, graphene-dielectric multilayers take advantage from the anomalous
features predicted for isolated suspended graphene sheets, even though they are
considerably affected by the presence of the dielectric substrate. They can
also archive the anomalous non-monotonic thermal metallic behavior by
increasing the graphene sheets density and their Fermi energy. In addition to a
strong thermal modulation occurring at short separations, in a region where the
force is orders of magnitude larger than the one occurring at large distances,
the force can be also adjusted by varying the number of graphene layers as well
as their Fermi energy levels, allowing for relevant force amplifications which
can be tuned, very rapidly and in-situ, by simply applying an electric
potential. Our predictions can be relevant for both Casimir experiments and
micro/nano electromechanical systems and in new devices for technological
applications.
