Figure 3: Polar forces leading to the improper ferroelectric polarization.
From: Improper electric polarization in simple perovskite oxides with two magnetic sublattices
Average polar forces acting on the Gd and Fe atom sublattices, calculated as a function of various distortions of a reference (cubic) perovskite structure, in the scalar-magnetism approximation and assuming that both spin sublattices are G-AFM ordered (see text for more details). Results as a function of O6 rotations (a), anti-polar distortion patterns (b) and shear strain (c). Circles and squares correspond to the results for Gd and Fe, respectively. In a, solid and open symbols correspond to antiphase and in-phase rotations, respectively. In b, solid and open symbols correspond to R-point and X-point modulated anti-polar patterns, respectively. For reference, note that our relaxed (the experimental) orthorhombic GdFeO3 structure is characterized by antiphase and in-phase tilting angles of 11.99° (11.38°) and 11.91° (11.95°), respectively; anti-polar distortions of 0.092 Å (0.083 Å) and 0.369 Å (0.353 Å), respectively, for the 
and 
modes; and a shear strain of about 5.1% (4.8%).
