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.