Soft and anisotropic local moments in 4$d$ and 5$d$ mixed-valence M$_2$O$_9$ dimers

Abstract: We investigate via exact diagonalization of finite clusters the electronic structure and magnetism of M$_2$O$_9$ dimers in the mixed-valence hexagonal perovskites A$_3$B'M$_2$O$_9$ for various different fillings of 4$d$ and 5$d$ 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 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 A$_3$B'M$_2$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.