Fig. 1: Total and interlayer exchange energy of bilayer CrBr₃ as predicted by first-principles calculations.

a Top view of monolayer CrBr₃, where the Cr atoms (orange balls) form a honeycomb lattice and lie inside edge-sharing octahedra formed by the Br atoms (pink balls; a and b are the two primitive lattice vectors of the unit cell). b Atomic arrangement and interlayer magnetic coupling for the three stacking configurations corresponding to local minima in total energy: AA stacking, where the Cr atoms of the top layer (orange) lie exactly on top of those of the bottom layer (blue); Monoclinic (M) stacking, where the Cr atoms of the top layer are shifted by [0,1/3] (in units of a and b) with respect to the bottom layer; AB stacking, where one of the Cr atoms of the top layer lies above the center of the hexagons in the bottom layer (i.e. with a relative shift by [1/3, 2/3]). DFT predicts that AB stacking favors ferromagnetic (FM) interlayer magnetic coupling, while AA and M stackings lead to antiferromagnetic (AFM) ordering. c Color plot of the total energy E_min (the minimum energy between FM and AFM configurations, with zero set at the AB FM stacking), as a function of interlayer shift along the two lattice vectors, showing three non-equivalent local minima (indicated by AA, M and AB). d Total energy along the gray dashed line in panel (c) (the three non-equivalent local minima are indicated by red arrows). e Color plot of the effective interlayer exchange energy, J_L = (E_FM -E_AFM)/2, as a function of interlayer shift. The orange regions correspond to AFM (J_L > 0) interlayer coupling while the blue regions correspond to FM (J_L < 0). f The interlayer exchange energy along the gray dashed line path in panel e.