Figure 4

Evolution of magnetic couplings and competing ground states driven by Co-doping and pressure. (a) Change of all magnetic interactions with Co doping and external pressure (GPa) calculated by LSDA + U for the ground states at zero temperature. Points represent DFT results and the connecting lines are a guide for eye. \({J}_{{\rm{V}}-{\rm{V}}}^{{\rm{in}}}\) and \({J}_{{\rm{V}}-{\rm{V}}}^{{\rm{out}}}\) are expected to be degenerate at x = 0.8 (cubic) in Mn_1−x Co _x V₂O₄ from experiments^9,24. Bold (dotted) lines represent the exchange (J) and SIA (D) interactions. (b) The anisotropic J _V−V prefers the TI/TO state and the antiferromagnetic J _Mn−V also stabilizes TI/TO state in Mn-rich region. (c) The isotropic J _V−V prefers the AI/AO state but the antiferromagnetic J _Co−V tries to stabilizes the TI/TO state in Co-rich region. Thus, two states of the isosymmetric one-angle TI/TO state and the two-angle state that evolves from the AI/AO state compete with each other. (d) The energy landscape of CoV₂O₄ with the disappearance of SIA at high pressures. The massive degeneracies in the energy landscape of two angles (θ ₁ and θ ₂) may induce spin glass or liquid phases as explained in the text.