Flat bands, electron interactions, and magnetic order in magic-angle mono-trilayer graphene

Starting with twisted bilayer graphene, graphene-based moir\'e materials have recently been established as a new platform for studying strong electron correlations. In this paper, we study graphene monolayers on trilayer and demonstrate that system can host flat bands when the twist angle is close to magic-angle of 1.16$^\circ$. When monolayer ABA are not isolated, but intersected by Dirac cone large Fermi velocity. contrast, ABC (denoted AtABC) exhibits gap between remote bands. Since known broken-symmetry phases, further investigate ostensibly similar magic AtABC system. We effect electron-electron interactions in using both Hartree theory an atomic Hubbard calculate magnetic phase diagram function doping, angle, perpendicular electric field. Our analysis reveals rich variety orderings, including ferromagnetism ferrimagnetism, demonstrates field makes more susceptible ordering.