The role of La displacement in Titanium $d_{xy}$ ferromagnetism at the LaAlO$_3$/SrTiO$_3$-- double-exchange like mediation

In this talk, we argue that La site polar distortion is the key for magnetism without oxygen vacancy, whereby coupling between La $d_{z^2}$ and O $2p_{x,y}$ offer a pathway for double-exchange like interaction to support Ti $d_{xy}$ ferromagnetism. While ferromagnetism found in the $n$-type LAO/STO has been mainly attributed to oxygen vacancy near the interface, the mechanism we propose a possibility of ferromagnetism without any defect. We show that there is a phase separation between charge-order and half-metallicity that can be characterized by La displacement - this can be tailored through adsorption on the outer AlO$_2$ surface or electrostatic bias.