Electronic properties of twisted bilayer graphene

In this thesis we will focus on the study of a two-layer graphene system, where a small relative rotation is applied. This system is called twisted bilayer graphene, tBLG. In the last ten years many experimental works have shown that by twisting two layers of graphene by small angles, one can enhance the conducting properties of the system. Moreover at some specific angles, called “magic angles”, the lowest energy band turns to be very close to the Fermi level and becomes extremely flat. This is of particular interest in the presence of ordered phases of matter like superconductivity. The densities of charge carriers is orders of magnitude lower than the typical two-dimensional superconductors and the measured critical temperature is however relatively high. This makes tBLG a strong coupling superconductor. The great advantage in tBLG with respect to other systems is the simple and fine tunability of carrier densities, magnetic field and temperature, that enables a complete and fine investigation of the rich phase diagram of such a strongly correlated system. Also the interlayer interactions can be fine-tuned by the modulation of the twist angle and/or the application of perpendicular electric fields and of uniaxial strain induced by non hydrostatic pressure. In this work we study the tight binding model for the tBLG, which, in the continuum limit, correctly predicts the presence of angles at which the Fermi velocity vanishes, and, therefore, the presence of flat bands near these magic angles. In particular, we show how, in the low energy limit, one gets an analytical expression for the first magic angle, rederiving the effective model and providing all the details useful for the calculation skipped in literature. Moreover we perform the numerical calculation in order to obtain the full spectrum and the bandwidth for the lowest energy band. The latter result is useful for us when considering the interacting system. In the presence of electron-electron attractive interaction mediated by the phonons, the system seems to sustain a superconducting phase. By means of a path integral approach we finally derive the corresponding critical temperature.