Nonequivalent Spin Exchanges of the Hexagonal Spin Lattice Affecting the Low-Temperature Magnetic Properties of RInO (R = Gd, Tb, Dy): Importance of Spin-Orbit Coupling for Spin Exchanges between Rare-Earth Cations with Nonzero Orbital Moments.

Rare-earth indium oxides RInO (R = Gd, Tb, Dy) consist of spin-frustrated hexagonal spin lattices made up of rare-earth ions R, where R = Gd (f, L = 0), Tb (f, L = 3), and Dy (f, L = 5). We carried out DFT calculations for RInO, including on-site repulsion U with/without spin-orbit coupling (SOC), to explore if their low-temperature magnetic properties are related to the two nonequivalent nearest-neighbor (NN) spin exchanges of their hexagonal spin lattices. Our DFT + U + SOC calculations predict that the orbital moments of the Tb and Dy ions are smaller than their free-ion values by ∼2μ while the Tb spins have an in-plane magnetic anisotropy, in agreement with the experiments. This suggests that the f orbitals of each R (R = Tb, Dy) ion are engaged, though weakly, in bonding with the surrounding ligand atoms. The magnetic properties of GdInO with the zero orbital moment are adequately described by the spin exchanges extracted by DFT + U calculations. In describing the magnetic properties of TbInO and DyInO with nonzero orbital moments, however, the spin exchanges extracted by DFT + U + SOC calculations are necessary. The spin exchanges of RInO (R = Gd, Tb, Dy) are dominated by the two NN spin exchanges J and J of their hexagonal spin lattice, in which the honeycomb lattice of J forms spin-frustrated ( J, J, J) triangles. The J/ J ratios are calculated to be ∼3, ∼1.7, and ∼1 for GdInO, TbInO, and DyInO, respectively. This suggests that the antiferromagnetic (AFM) ordering of GdInO below 1.8 K is most likely an AFM ordering of its honeycomb spin lattice and that TbInO would exhibit low-temperature magnetic properties similar to those of GdInO while DyInO would not.