Extended Data Figure 8: Exchange interactions in the COII structure of LuFe₂O₄.

a, Schematic of the COII LuFe₂O₄ structure with intra-layer, inter-layer and in-plane interactions labelled. The Fe–O–Fe bond angles in the undistorted structure are indicated by the black arrows. The red arrows demonstrate the change to the bond angles as the distortions turn on. Lutetium, Fe³⁺, Fe²⁺ and oxygen are shown in turquoise, yellow, green and brown, respectively. b, Calculated exchange interactions as a function of the lutetium distortion Q. Circles, squares and diamonds denote the DFT-estimated value of the exchange interactions between two Fe²⁺ spins, two Fe³⁺ spins and Fe²⁺–Fe³⁺ spins, respectively. We considered in-plane interactions, intra-bilayer interactions and the interaction between two FeO₂ bilayers.