Flexoelectricity and polarization tilt near defects in improper ferroelectric hexagonal ferrites

Flexoelectricity originates from the electromechanical coupling between polar order and strain gradients. It been widely studied in proper ferroelectrics. However, direct atomic-scale observation of flexoelectricity in improper ferroelectrics and their phenomenological modeling are missing. In this work, based on epitaxial h-YbFeO₃ films and quantitative analysis of the position of atomic columns from HAADF-STEM images, we reveal that YbFe₂O₄-type stacking fault induced bending of atomic layers leads to significant changes in polarization in hexagonal ferrites. The strong strain gradient near defects reaches 2 ~ 6 *10⁶ m^-1, and the magnitude of primary order parameter Q is suppressed from ~ 30pm for bulk state to below 15 pm, corresponding to the giant flexoelectricity in h-YbFeO₃. A phenomenological model from Landau theory is proposed to reveal the general physical picture for flexoelectricity in improper ferroelectric hexagonal ferrites or manganites.