Hidden direct exchange and alternative description of magnetic interactions in two-dimensional Fe$_3$GeTe$_2$

We propose an alternative description of the magnetic properties in monolayer Fe$_3$GeTe$_2$ based on the state-of-the-art first-principles calculations combined with model simulations. We show that due to structural peculiarities of the system, short distance between the neighboring iron atoms gives rise to a strong direct exchange coupling. The direct exchange is important for an accurate description of the magnon spectrum at high energies, but remains insignificant for moderate-energy spin excitations, as well as for thermodynamic averages. A strong coupling between the iron atoms allows us to consider them as an effective cluster with a magnetic moment $\sim$5 $\mu_B$, giving rise to a simplified spin model on a bipartite honeycomb lattice with the reduced number of long-range interactions. The simplified model perfectly reproduces the results of the conventional spin model, but allows for a more tractable description of the magnetic properties of Fe$_3$GeTe$_2$, which is important, e.g., for large-scale simulations. Also, we discuss the role of biaxial strain in the stabilization of ferromagnetic ordering in Fe$_3$GeTe$_2$.