The Moiré pattern of the magic-angle twisted bilayers graphene and twisted bilayer MoS₂ leads to localization of the low energy electrons in the AA-stacking regions, reflected by very flat bands. This strong reduction of the kinetic energy enhances the importance of interactions and thus renders the bilayer systems much more susceptible to correlation effects, as show experimentally by the discovery of correlated insulators and superconductivity. Despite numerous theoretical and experimental studies, the understanding of this new electronic localization is still incomplete. The rotation angle is of course a key parameter, but it has also been shown that slight expansion or contraction of one layer relative to the other (“heterostrain”) can strongly modify the flat bands.

We will present theoretical study of the electronic structure and quantum transport properties of electronic flat bands, considering as well as possible the structural parameters that condition them (rotation angle, bias voltage, heterostrain, local defects or atomic relaxation). We also investigate the magnetic instabilities in twisted bilayer graphene using a real-space Hartree-Fock mean-field theories. Starting from a tight-binding description of the non-interacting bilayer systems we add a local Coulomb interaction U in order to model the Coulomb repulsion between electron. At half filling, localized magnetic states emerge for U

values well below those required to render an insulated layer antiferromagnetic.