Effect of biquadratic magnetic exchange interaction in the 2D antiferromagnets MPS3 ( M=Mn,Fe,Co,Ni )

The two-dimensional van der Waals (vdW) materials $M{\mathrm{PS}}_{3}\phantom{\rule{0.16em}{0ex}}(M=\mathrm{Mn},\mathrm{Fe},\mathrm{Co},\mathrm{Ni})$ display antiferromagnetic ordering of the magnetic moments at the transition metal ions. The possibility to exfoliate thin layers that preserve the magnetic order makes these materials interesting for numerous applications in devices that require integration of flexible patches of magnetic materials, e.g., in antiferromagnetic spintronics. Hence, an improved understanding of their magnetic properties is desirable. Here, we parametrize spin Hamiltonians for a monolayer of all four materials of this class using density functional theory plus Hubbard $U$ calculations. We provide a step-by-step guide for calculating the magnetic exchange interactions and magnetic anisotropy energy using the (non)collinear $\mathrm{DFT}+U(+\mathrm{SOC})$ approach with a suitably chosen $U$ for each material. It is found that the biquadratic interactions gain in importance while moving through the $3d$ series. Retaining the leading terms of a Holstein-Primakoff-transformed spin Hamiltonian, the magnon spectra are calculated. While ${\mathrm{MnPS}}_{3}$ is found to be an almost isotropic antiferromagnet with a tiny gap, the biquadratic interaction opens an increasingly wider gap for ${\mathrm{FePS}}_{3}, {\mathrm{CoPS}}_{3}$, and ${\mathrm{NiPS}}_{3}$. In line with this observation, Monte Carlo simulations demonstrate that the biquadratic interactions contribute to a systematic rise in the N\'eel temperature from ${\mathrm{FePS}}_{3}$ to ${\mathrm{NiPS}}_{3}$.