Correlations between magnetic and piezoelectric response at gated LaAlO$_3$/SrTiO$_3$ interfaces*

The interface between perovskite oxide semiconductors LaAlO$_3$ and SrTiO$_3$ exhibits remarkable conducting, superconducting, magnetic, and spintronic properties that are strongly influenced by electron density. Scanning probe methods have the ability to probe local properties of interest. For example, magnetic force microscopy (MFM) has be used to measure magnetism at the LaAlO$_3$/SrTiO$_3$ interface\footnote{F. Bi, \textit{et al.}, Nat. commun. \textbf{5}, 5019 (2014); F. Bi, \textit{et al.}, Appl. Phys. Lett.. \textbf{107}, 082402 (2015). }, while piezoelectric force microscopy has been used to measure the local electron density\footnote{M. Huang, \textit{et al.}, APL Mater. \textbf{1}, 052110 (2013).}. Here we directly compare these two methods to provide further insight into the relationship between electron density and magnetic properties.