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Magic-angle twisted bilayer graphene (MATBG) is notable as a highly tunableplatform for investigating strongly correlated phenomena such as high-$T_c$superconductivity and quantum spin liquids, due to easy control of doping levelthrough gating and sensitive dependence of the magic angle on hydrostaticpressure. Experimental observations of correlated insulating states,unconventional superconductivity and ferromagnetism in MATBG indicate that thissystem exhibits rich exotic phases. In this work, using density functionaltheory calculations in conjunction with the effective screening medium method,we find the MATBG under pressure at a twisting angle of $2.88\unicode{xb0}$ andsimulate how its electronic states evolve when doping level and out-of-planeelectric field are gate-tuned. Our calculations show that, at doping levelsbetween two electrons and four holes per moir\'{e} unit cell, a ferromagneticsolution with spin density localized at AA stacking sites is lower in energythan the nonmagnetic solution. The magnetic moment of this ferromagnetic statedecreases with both electron and hole doping and vanishes at fourelectrons/holes doped per moir\'{e} unit cell. Hybridization between the flatbands at the Fermi level and the surrounding dispersive bands can take place atfinite doping. Moreover, upon increasing the out-of-plane electric field atzero doping, a transition from the ferromagnetic state to the nonmagnetic oneis seen. We also analyze the interlayer bonding character due to the flat bandsvia Wannier functions. Finally, we report trivial band topology of the flatbands in the ferromagnetic state at a certain doping level.

First-principles calculation of gate-tunable ferromagnetism in magic-angle twisted bilayer graphene under pressure