Soft and anisotropic local moments in 4d and 5d mixed-valence M2O9 dimers

We investigate via exact diagonalization of finite clusters the electronic structure and magnetism of ${\mathrm{M}}₂{\mathrm{O}}₉$ dimers in the mixed-valence hexagonal perovskites ${\mathrm{A}}₃{\mathrm{B}}^{\ensuremath{'}}{\mathrm{M}}₂{\mathrm{O}}₉$ for various different fillings of $4d$ and $5d$ transition-metal M ions. We find that the magnetic moments of such dimers are determined by a subtle interplay of spin-orbit coupling, Hund's coupling, and Coulomb repulsion, as well as the electron filling of the M ions. Most importantly, the magnetic moments are anisotropic and temperature dependent. This behavior is a result of spin-orbit coupling, magnetic field effects, and the existence of several nearly degenerate electronic configurations whose proximity allows the occupation of excited states already at room temperature. This analysis is consistent with experimental susceptibility measurements for a variety of dimer-based materials. Furthermore, we perform a survey of ${\mathrm{A}}₃{\mathrm{B}}^{\ensuremath{'}}{\mathrm{M}}₂{\mathrm{O}}₉$ materials and propose ground-state phase diagrams for the experimentally relevant M fillings of ${d}^4.5$, ${d}^3.5$, and ${d}^2.5$. Finally, our results show that the usually applied Curie-Weiss law with a constant magnetic moment cannot be used in these spin-orbit-coupled materials.