LaAlO$_3$ thickness window for electronically controlled magnetism in LaAlO$_3$/SrTiO$_3$ heterostructures*

Complex oxide heterostructures, especially LaAlO$_3$/SrTiO$_3$ (LAO/STO) exhibit emergent conductivity, superconductivity, and magnetism. A recent investigation\footnote{F. Bi, \textit{et. al}, Nature Communications \textbf{5}, 5019 (2014).} demonstrates that in-plane magnetism at the LAO/STO interface can be controlled electronically at room temperature. Here we employ the magnetic force microscopy to investigate the emergence of magnetism as the LAO thickness is varied between 4 u.c. and 40 u.c. Magnetism is observed only within a certain thickness window 8 u.c.-25 u.c; outside of this window the magnetic signal becomes negligible. Simultaneous capacitance measurements for different LAO thickness show that the devices with negligible magnetic signatures all have capacitance values far below the geometric capacitance. The existence of the observed window is attributed to the existence of direct or Zener tunneling, which prevents the insulating phase from being reached.