Magnetic and orbital ordering ofRuO2planes inRuSr2(Eu,Gd)Cu2O8

Abstract

We start from an effective Hamiltonian for $\mathrm{Ru}$ ions in a square lattice, which includes the on-site interactions between ${t}_{2g}$ orbitals derived from Coulomb repulsion, and a tetragonal crystal-field splitting. Using perturbation theory in the hopping terms, we derive effective Hamiltonians to describe the ${\mathrm{RuO}}_{2}$ planes of ${\mathrm{RuSr}}_{2}(\mathrm{Eu},\mathrm{Gd}){\mathrm{Cu}}_{2}{\mathrm{O}}_{8}$. For undoped planes (formal valence ${\mathrm{Ru}}^{+5}$), depending on the parameters we find three possible orderings of spin and orbitals, and construct a phase diagram. This allows us to put constraints on the parameters based on experimental data. When electron doping consistent with the hole doping of the superconducting ${\mathrm{RuO}}_{2}$ planes is included, we obtain (for reasonable parameters) a double-exchange model with infinite antiferromagnetic coupling between itinerant electrons and localized spins. This model is equivalent to one used before [H. Aliaga and A. A. Aligia, Physica B 320, 34 (2002)], which consistently explains the seemingly contradictory magnetic properties of ${\mathrm{RuSr}}_{2}(\mathrm{Eu},\mathrm{Gd}){\mathrm{Cu}}_{2}{\mathrm{O}}_{8}$.