Nonequivalent Spin Exchanges of the Hexagonal Spin Lattice Affecting the Low-Temperature Magnetic Properties of RInO3 (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 3 (R = Gd, Tb, Dy) consist of spin-frustrated hexagonal spin lattices made up of rare-earth ions R 3+ , where R 3+ = Gd 3+ (f 7 , L = 0), Tb 3+ (f 8 , L = 3), and Dy 3+ (f 9 , L = 5). We carried out DFT calculations for RInO 3 , 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 3+ and Dy 3+ ions are smaller than their free-ion values by ∼2μ B while the Tb 3+ spins have an in-plane magnetic anisotropy, in agreement with the experiments. This suggests that the f orbitals of each R 3+ (R = Tb, Dy) ion are engaged, though weakly, in bonding with the surrounding ligand atoms. The magnetic properties of GdInO 3 with the zero orbital moment are adequately described by the spin exchanges extracted by DFT + U calculations. In describing the magnetic properties of TbInO 3 and DyInO 3 with nonzero orbital moments, however, the spin exchanges extracted by DFT + U + SOC calculations are necessary. The spin exchanges of RInO 3 (R = Gd, Tb, Dy) are dominated by the two NN spin exchanges J 1 and J 2 of their hexagonal spin lattice, in which the honeycomb lattice of J 2 forms spin-frustrated ( J 1 , J 1 , J 2 ) triangles. The J 2 / J 1 ratios are calculated to be ∼3, ∼1.7, and ∼1 for GdInO 3 , TbInO 3 , and DyInO 3 , respectively. This suggests that the antiferromagnetic (AFM) ordering of GdInO 3 below 1.8 K is most likely an AFM ordering of its honeycomb spin lattice and that TbInO 3 would exhibit low-temperature magnetic properties similar to those of GdInO 3 while DyInO 3 would not.