  Based on first-principles calculations, an antiferromagnetic Ising model on a
triangular lattice has been proposed to interpret the order of Fe$(1)$-Ge pairs
and the formation of $\sqrt{3}$ $\times $ $\sqrt{3}$ superstructures in the
Fe$_5$GeTe$_2$ (F5GT), as well as to predict the existence of similar
superstructures in Ni doped F5GT (Ni-F5GT). Our study suggests that F5GT
systems may be considered as a structural realization of the well known
antiferromagnetic Ising model on a triangular lattice. Based on the
superstructures, a Heisenberg-Landau Hamiltonian, taking into account both
Heisenberg interactions and longitudinal spin fluctuations, is implemented to
describe magnetism in both F5GT and Ni-F5GT. We unveil that frustrated magnetic
interactions associated with Fe(1), tuned by a tiny Ni doping ($x \leq 0.2 $),
is responsible for the experimentally observed enhancement of the $T_c$ to 478
K in Ni-F5GT. Our calculations show that at low doping levels, monolayer
Ni-F5GT has almost the same magnetic phase diagram as that of the bulk, which
indicates a pervasive beyond room temperature ferromagnetism in this Ni doped
two-dimensional system.
