Orbital Ordering in Cs$_{2}$AgF$_{4}$ - an electronic structure study

The argentate Cs$_{2}$AgF$_{4}$, first synthesized in 1974 has many similarities to the high-T$_{c}$ cuprates, featuring AgF$_{2}$ sheets in place of CuO$_{2}$ sheets. While the undoped cuprates are antiferromagnetic, this argentate is ferromagnetic with a T$_{Curie}$ of about 15 K. Density functional calculations in the proposed tetragonal structure produce an itenerant half-metallic ferromagnet. Recent inelastic neutron scattering experiments have suggested an orthorhombic structure that allows an orbitally ordered ferromagnetic ground state. We performed electronic structure calculations using the LDA+U method on this system and were able to obtain an orbitally ordered ground state, not only for the newly proposed orthorhombic lattice but also for the orginal tetragonal lattice by constraining the bravais lattice and allowing the change of the atomic basis. In specific, very small changes in the position of the in-plane fluorine atoms already trigger an orbital ordering. Our calculated energy scale shows that this orbitally ordered state should be stable for all temperatures, consistent with the experiment.