  We compare three first-principles methods of calculating the Curie
temperature in two-dimensional (2D) ferromagnetic materials (FM), modeled using
the Heisenberg model, and propose a simple formula for estimating the Curie
temperature with high accuracy that works for all common 2D lattice types.
First, we study the effect of exchange anisotropy on the Curie temperature
calculated using the Monte-Carlo (MC), the Green's function method, and the
renormalized spin-wave (RNSW). We find that the Green's function overestimates
the Curie temperature in high-anisotropy regimes compared to MC, whereas RNSW
underestimates the Curie temperature compared to the MC and the Green's
function. Next, we propose a closed-form formula for calculating the Curie
temperature of 2D FMs, which provides an estimate of the Curie temperature
greatly improving over the mean-field expression for magnetic material
screening. We apply the closed-form formula to predict the Curie temperature 2D
magnets screened from the C2DB database and discover several high Curie
temperature FMs with Fe2F2 and MoI2 emerging as the most promising 2D
ferromagnets. Finally, comparing to experimental results for CrI3, CrCl3, and
CrBr3, we conclude that for small effective anisotropies, the Green's
function-based equations are preferable, while, for larger anisotropies
MC-based results are more predictive.
