  The two-dimensional van der Waals (vdW) materials MPS_3(M = Mn, Fe, Co, 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 parameterize 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 DFT+U(+ 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 MnPS_3 is found to be an almost isotropic antiferromagnet
with a tiny gap, the biquadratic interaction opens an increasingly wider gap
for FePS_3, CoPS_3 and NiPS_3. In line with this observation, Monte Carlo
simulations demonstrate that the biquadratic interactions contribute to a
systematic rise in the Neel temperature from FePS_3 to NiPS_3.
