Interaction effects in decoupled graphene layers

Aim of the thesis was to study the effect of interactions on decoupled graphene layers, the main objective being to extract quantitative results for several quantities that are object of study in the ongoing experiments. The experimental realization of multi-layers of graphene has been reached only recently spurring a lot of theoretical and experimental interest. Already analyzed theoretically for the first time in the '40, graphene was discovered only in 2004. Band calculations predict a linear energy dispersion, which makes it an ideal candidate to observe relativistic-like phenomena at low energy. Moreover, the non-conventional band dispersion implies other peculiar features. Although most of these facts can be theoretically described in a non-interacting picture, the effect of interactions is relevant, as several experiments have demonstrated. As extensively shown on single layer graphene, interactions may play a relevant role in determining several of its properties. The same considerations hold for graphene multilayers. Particularly important are Coulomb interactions, whose strength in graphene is given by the dimensionless coupling constant $\alpha_{\rm ee}$, defined as the ratio between the Coulomb and kinetic energies, which is approximately $\approx 2.2$. It follows that graphene and multilayer systems are strongly interacting, the main difference being the presence in the latter ones of inter-layer Coulomb interactions. One prominent question is: what makes multilayers different from graphene because of such inter-layer interactions? In the context of linear response theory, we have introduced a multi-component random-phase approximation, supported from the G0W approximation which has allowed to calculate the electron self-energy. The quasiparticle properties have been obtained from the derivatives of the self-energy. We have shown numerical results for the renormalized Fermi velocity as a function of the electron densities. Furthermore, we have used the Hellman-Feynman theorem to calculate the ground-state energy, and shown numerical results. The knowledge of the fundamental energy at various values of electron densities has allowed us to study the electro-chemical equilibrium problem. After having introduced the system of the electro-chemical equilibrium equations, we have numerically solved it and furnished results for the equilibrium electronic densities on the layers as a function of an additional parameter related to the substrate. The collective modes have been found as the zeroes of the dielectric function. In the limit of large wavelength, we have obtained the exact dispersion of the modes by a series expansion in the wavevector. We have also numerically computed them at a generic wavevector, and made a comparison with the analytical results which are valid at large wavelenght. We have finally shown and discussed results for the spectral function, which is directly related to the experimentally found spectra. We have numerically computed it for various choices of parameters of the system, thus demonstrating the appearence of the acoustic plasmaron which could then become the subject of experimental investigation.