Vacancy-driven orbital and magnetic order in (K,Tl,Cs)_yFe _2-xSe₂

We investigate the effects of the √5×√5 Fe vacancy ordering on the orbital and magnetic order in (K,Tl,Cs)_yFe _2-xSe₂ using a three-orbital (t_2g) tight-binding Hamiltonian with generalized Hubbard interactions. We find that vacancy order enhances electron correlations, resulting in the onset of a block antiferromagnetic phase with large moments at smaller interaction strengths. In addition, vacancy ordering modulates the kinetic energy differently for the three t_2g orbitals. This results in a breaking of the degeneracy between the d_xz and d_yz orbitals on each Fe site, and the onset of orbital order. Consequently, we obtain a novel inverse relation between orbital polarization and the magnetic moment. We predict that a transition from high-spin to low-spin states accompanied by a crossover from orbitally-disordered to orbitally-ordered states will be driven by doping the parent compound with electrons, which can be verified by neutron scattering and soft x-ray measurements.