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La displacement driven Double-exchange like mediation in Titanium $d_{xy}$ ferromagnetism at the LaAlO$_3$/SrTiO$_3$

The epitaxial atomistic interfaces of two insulating oxides, LaAlO$_3$ (LAO)/SrTiO$_3$ (STO), have attracted great interest owing to rich emergent phenomena \cite{hwang12:nmat,coey13:mrs,ohtomo04:} such as interface metallicity \cite{ohtomo04:,nakagawa06:}, thickness dependent insulator-metal transition \cite{huijben06:nmat}, superconductivity \cite{N.Reyren08312007}, ferromagnetism \cite{brinkman.07:natmat}, and even their coexistence \cite{bert11:nphys,li11:nphys,dikin11:prl}. However, the physics origin of ferromagnetic ordering in the $n$-type LAO/STO interface is in debate. Here we propose that the polar distortion of La atom can ignite the ferromagnetism at the interface even without oxygen vacancy. The induced hybridization between La $d_{z^2}$ and O $p_{x,y}$ states can mediate double-exchange like interaction between Ti $d_{xy}$ electrons. We further suggest that the structural and electrical modification of the outermost surface of LAO or switching the polarization direction of ferroelectric overlayers on LAO/STO can promote such La displacement.