Fig. 3

Evolution of antiferromagnetic-spin and polarization with strain. a Three-dimensional representation of the evolution of the magnetic energy landscape under various strain states. The color in the representation indicates the total energy when the antiferromagnetic-spin axis L points in that direction wherein cool and warm colors represent low and high energy, respectively, and thus easy and hard axes, respectively. b Summary of the evolution of antiferromagnetic-spin axis L (red arrow) and polarization P (blue arrow) with strain from first-principles calculations. c The three components of the polarization as a function of misfit strain from the first-principles calculations. d The angle θ between the polarization vector and the antiferromagnetic-easy axis as a function of misfit strain. e High-resolution high-angle annular dark field-scanning transmission electron microscopy (HAADF-STEM) image of a 70 nm BiFeO₃/GdScO₃ (010)_O heterostructure. f Bright field-scanning transmission electron microscopy (BF-STEM) image of the same heterostructure showing the Fe cation and O anion displacement details. Direct imaging of all species in the BF-STEM image enables direct extraction of the polar distortions. g Fe-cation displacement vector maps relative to its two neighboring Bi cations; the films are found to exhibit uniform structural distortion as noted, indicating that the polar direction of the film is uniform and close to the expected 〈111〉