Same effect of biquadratic exchange interaction and Heisenberg linear interaction in a spin spiral

Monolayer (ML) NiCl₂ exhibits a strong biquadratic exchange interaction between the first neighboring magnetic atoms (B₁), as demonstrated by the spin spiral model in J. Ni et al., Phys. Rev. Lett., 2021, 127, 247204. This interaction is crucial for stabilizing the ferromagnetic collinear order within the ML NiCl₂. However, they neither point out the role of B₁ nor discuss the dispersion relation from spin orbit coupling (SOC) in the spin spiral. As we have done in this work, these parameters might theoretically potentially be derived directly by fitting the calculated spin spiral dispersion relation. Here, we draw attention to the fact that B₁ is equivalent to half of J₃ in Heisenberg linear interactions and that the positive B₁ partially counteracts the negative J₃'s impact on the spin spiral to make the ML NiCl₂ ferromagnetic. The comparatively small J₃ + 1/2B₁ from the spin spiral led us to believe that J₃ could be substituted by B₁, yet it still exists and plays a crucial function in magnetic semiconductors or insulators. The dispersion relation, which we also obtain from SOC, displays weak antiferromagnetic behavior in the spin spiral.