This is a copy of the slides presented at the meeting but not formally written up for the volume.

Stripe domains in ferroelectric thin films form in order to minimize the total energy of the film. It has been known for some time that a stable configuration is reached when the decrease in elastic energy from domain formation is balanced by the energetic costs of domain wall formation, local elastic strains in the substrate, and internal electric field formation from domain polarizations. The size and strain of each domain is determined by the lattice mismatch and the energetic costs of interface formation. Recent piezoelectric force microscopy measurements have shown that BiFeO3 (BFO) films on SrRuO3/SrTiO3 (001) substrates form striped polarization domains. Since the details of the local structure and polarization cannot be measured at the same time with conventional techniques, we have used synchrotron x-ray microdiffraction to study these effects. Probing only a few domains at a time with the submicron x-ray spot resulted in a diffraction pattern near the substrate (103) reflection consisting of several BFO peaks. We have unambiguously assigned these peaks to individual structural variants. Based on these results, we propose a physical model that includes the striped domains. The structural variants within the stripes are similar to those predicted by striped patterns in rhombohedral films which minimize elastic energy. The local piezoelectric properties were measured using time-resolved microdiffraction in order to examine the role of the striped domains in the linear responses of the film. The out of plane piezoelectric coefficient d33 was approximately 50 pm/V and the piezoelectric strain was proportional to electric field was up to 0.55%, the maximum strain we have measured. The projection of the in-plane piezoelectric coefficients onto the reciprocal space maps for different structural variants had vastly different values due to the differences in orientation of the domains.