Search for Toroidal Ground State and Magnetoelectric Effects in Molecular Spin Triangles with Antiferromagnetic Exchange

Using first-principles methods and spin models, we investigate the magnetic properties of transition-metal trimers Cr₃ and Cu₃. We calculate exchange coupling constants and zero-field splitting parameters using density functional theory and, with these parameters, determine the ground spin state as well as thermodynamic properties via spin models. Results for Cr₃ indicate uniaxial magnetic anisotropy with a magnetic easy axis aligned along the 3-fold rotational symmetry axis and a mostly isotropic exchange interaction. The Cu₃ molecule lacks rotational symmetry and our results show strong antisymmetric interactions for three distinct exchange couplings within the molecule. We are able to reproduce experimental findings on magnetic susceptibility and magnetization of Cr₃ with the first-principles spin-Hamiltonian parameters. Our results show no presence of a toroidal ordering of spins for Cr₃ and a finite toroidal moment for Cu₃ in the ground state. We apply an external electric field up to 0.08 V/Å to each system to reveal the field dependence of exchange coupling as magnetoelectric effects. Finally, we scan the parameter space of a spin Hamiltonian to gain insights into which parameters would lead to a sizable toroidal moment in such systems.