Non-local piezoresponse in 3 u.c. LaAlO$_3$/SrTiO$_3$

Nanoscale control of the metal-insulator transition in 3 unit cell (u.c.) LaAlO$_3$/SrTiO$_3$ heterostructures can be achieved by the conducting AFM lithography,\footnote{C. Cen, S. Thiel, G. Hammerl, C. W. Schneider, K. E. Andersen, C. S. Hellberg, J. Mannhart, and J. Levy, Nat. Mater. \textbf{7}, 2136 (2008)} however the mechanism behind this transition is still not well understood. One proposed mechanism invokes ionic transport through the LaAlO$_3$ layer.\footnote{A. Kumar, S. Jesse, A. Gruverman, C. B. Eom, S.V.Kalinin, unpublished} We have performed a variety of local and non-local piezoforce measurements on 3 u.c. LaAlO$_3$/SrTiO$_3$ heterostructures. The existence and nature of the non-local piezoelectric effect places strong constraints on the origin of the piezoelectric response. This work is supported by NSF DMR-1104191.