Local Moment Formation and Magnetism at LaAlO$_3$/SrTiO$_3$ Interfaces*

One of the most exciting observations at oxide interfaces relate to the observation of magnetism at the LaAlO$_3$/SrTiO$_3$ (LAO/STO) interface, since neither material is magnetic in the bulk even with doping. Experiments [1,2] give incontrovertible evidence for local moments at the LAO/STO interface, consistent with an areal density close to 0.5 per interfacial Ti atom. The particular splitting of the $t_{2g}$ orbitals at the interface, leads to a quarter-filled $d_{xy}$ band on the top band. Using a slave-rotor approach for the on-site Coulomb interaction U and Hartree-Fock for nearest neighbor V, we show that local moments form in a checkerboard charge-ordered insulating (COI) state, even for a very modest values of $U$. Phonons further stabilize the COI state as the breathing mode couples cooperatively to the charge order. To understand the magnetic interactions between moments, we examine both the small superexchange and the dominant kinetic exchange mediated by conduction electrons. We show that this leads to a ferromagnetic double exchange model with some very interesting twists arising from the Rashba SOC of the conduction electron due to broken inversion at the interface. Ref: [1] L. Li et al., Nature Phys. 7 762. [2] J. A. Bert et al., Nature Phys. 7 76.