Atomic-scale visualization of three-dimensional magnetization vectors with competing spin Hamiltonian interactions

The spin Hamiltonian describes the energy of a magnetic system, which originates from interactions between electron spins that have a three-dimensional (3D) vectorial nature and atomic-scale arrangement. It is crucial to know the energy contributions associated with spin behaviour, such as superexchange coupling, Dzyaloshinskii-Moriya interaction, Zeeman energy in external fields and single-ion anisotropy (SIA), for a fundamental understanding of the ground state and dynamics of magnetic materials. An experimental method, capable of resolving local 3D spin vectors on the atomic scale, is needed to provide direct correlation between the competing energy terms and corresponding atomic configurations. Here we demonstrate a beam-shift-based four-dimensional electron magnetic circular dichroism technique, which achieves quantitative 3D vector magnetometry and infers the energy contributions to the spin Hamiltonian on an individual atomic plane basis. The spatial resolution of 2.25 Å enables evaluation of spin Hamiltonian energy terms within individual atomic planes. In a canted antiferromagnet, YFeO₃, we confirm an asymmetric spin reorientation of the magnetic spin moments on the two antiferromagnetic spin sublattices. The inferred superexchange coupling, Dzyaloshinskii-Moriya interaction, SIA and Zeeman energies provide atomic-level insight into the interplay between these spin interactions during a field-induced spin reorientation transition. The observed asymmetry in spin reorientation angle arises primarily from the SIA and opposite Zeeman energies of the adjacent antiferromagnetically coupled atomic planes, resulting in both magnetic spin moments being nearly aligned with the external magnetic field. By bridging the gap in imaging resolution for 3D spin vectors and atomic positions, our method opens an avenue for the atomic-scale characterization and engineering of the energetics described by the spin Hamiltonian.