Delving into the anisotropic interlayer exchange in bilayer CrI₃

Abstract Bilayer CrI₃ attracted much attention due to stacking-induced switching between the layered ferromagnetic and antiferromagnetic order. This discovery brought under the spotlight the interlayer Cr–Cr exchange interaction, which despite being much weaker than the intralayer exchange, plays an important role in shaping the magnetic properties of bilayer CrI₃. In this work we delve into the anisotropic part of the interlayer exchange with the aim to separate the contributions from the Dzyaloshinskii–Moriya (DMI) and the Kitaev interactions (KI). We leverage the density functional theory calculations with spin Hamiltonian modeling and develop an energy mapping procedure to assess these anisotropic interactions with μeV accuracy. After inspecting the rhombohedral and monoclinic stacking sequences of bilayer CrI₃, we reveal a considerable DMI and a weak interlayer KI between the sublattices of a monoclinic bilayer. We explain the dependence of DMI and KI on the interlayer distance, stacking sequence, and the spin–orbit coupling strength, and we suggest the dominant superexchange processes at play. In addition, we demonstrate that the single-ion anisotropy in bilayer CrI₃ is highly stacking-dependent, increasing by 50% from monoclinic to rhombohedral bilayer. Remarkably, our findings prove that iodines are highly efficient in mediating the DMI across the van der Waals gap, much owing to spatially extended 5p orbitals which feature strong spin–orbit coupling. Our study gives promise that the interlayer chiral control of spin textures, demonstrated in thin metallic films where the DMI is with a much longer range, can be achieved with similar efficiency in semiconducting two-dimensional van der Waals magnets.