Effective spin-orbital Hamiltonian for the double perovskite Sr₂FeWO₆: Derivation of the phase diagram

We formulate a superexchange theory of insulating double-perovskite compounds such as Sr₂FeWO₆. An effective spin-orbital Hamiltonian is derived in the strong-coupling limit of Hubbard model for d electrons on Fe and W ions. The relevant degrees of freedom are the spins S=2 and the threefold orbital degeneracy of Fe²⁺ ions. W sites are integrated out by means of a fourth-order perturbative expansion. The magnetically and orbitally ordered ground states of the effective Hamiltonian are discussed as a function of the model parameters. We show that for realistic values of such parameters the ground state is antiferromagnetic, as experimentally observed. The order found is of type II, consisting of {111} ferromagnetic planes stacked antiferromagnetically. The orbital order energy scale found is one order of magnitude less than the spin one.