Electrically Controllable Magnetism in Twisted Bilayer Graphene

Twisted graphene bilayers develop highly localized states around $AA$-stacked regions for small twist angles. We show that interaction effects may induce either an antiferromagnetic or a ferromagnetic (FM) polarization of said regions, depending on the electrical bias between layers. Remarkably, FM-polarized $AA$ regions under bias develop spiral magnetic ordering, with a relative 120° misalignment between neighboring regions due to a frustrated antiferromagnetic exchange. This remarkable spiral magnetism emerges naturally without the need of spin-orbit coupling, and competes with the more conventional lattice-antiferromagnetic instability, which interestingly develops at smaller bias under weaker interactions than in monolayer graphene, due to Fermi velocity suppression. This rich and electrically controllable magnetism could turn twisted bilayer graphene into an ideal system to study frustrated magnetism in two dimensions.

Figure 1

Zero-energy local density of states in real space (a),(b), band structure (c),(d), and density of states (e),(f) for a $\theta =1.5°$ twisted graphene bilayer. The left column has no interlayer bias, and the right column has a bias $V_b=300\mathrm{meV}$. This enhances the localization of the $AA$ quasibound states, red in (a) and (b). The said states arise from almost flat subbands at zero energy, which show up as large DOS peaks in (e) and (f). The solid (dashed) lines in (c) and (d) correspond to a scaled (unscaled) tight-binding model, see the main text.

Figure 2

Spatial distribution of the magnetic moment in the ground state of an interacting twisted bilayer with Hubbard $U=3.7\mathrm{eV}$. In the first row (a),(b) we show the ferromagnetic component of the two sublattices, $M_A+M_B$, in units of electrons per (monolayer) unit cell, both for zero interlayer bias $V_b=0$ (a) and $V_b=200\mathrm{meV}$ (b). Analogous plots of the lattice-AFM component $M_A-M_B$ are shown in (c) and (d). The scale in all color bars is expressed in units of one electron spins per supercell. Panels (e) and (f) show the variation of the total electronic energy per supercell as a function of the angle ${\alpha }_M$ between polarizations of adjacent $AA$ regions, indicating parallel alignment of the lattice-AFM order (e), and a spiral misalignment of 120° for the lattice-FM case (f).

Figure 3

(a) Critical value $U_c$ of the Hubbard $U$ beyond which the twisted bilayer develops lattice-AFM order at the mean-field level. The red dots show $U_c$ as a function of the twist angle $\theta$, and the dashed line shows the corresponding Fermi velocity at the Dirac point, normalized to the monolayer value $v_F^0$. At high twist angles both $U_c$ and $v_F$ converge to the monolayer values, while they become strongly suppressed at smaller angles. (b) Phase diagram for the ground state magnetic order in a $\theta =1.5°$ twisted bilayer as a function of Hubbard $U$ and interlayer bias $V_b$. The blue and red regions denote the spatial integral of the lattice-AFM and spiral-FM polarizations, respectively, while the yellow region is nonmagnetic.