Computing Curie temperature of two-dimensional ferromagnets in the presence of exchange anisotropy

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, and the renormalized spin-wave (RNSW) methods. We find that the Green's function method overestimates the Curie temperature in high-anisotropy regimes compared to the MC method, whereas the RNSW method underestimates the Curie temperature compared to the MC and the Green's function methods. Next, we propose a closed-form formula for calculating the Curie temperature of 2D FMs, which provides an estimate of the Curie temperature that is greatly improved 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 ${\mathrm{Fe}}_{2}{\mathrm{F}}_{2}$ and ${\mathrm{MoI}}_{2}$ emerging as the most promising 2D ferromagnets. Finally, by comparing to experimental results for ${\mathrm{CrI}}_{3}$, ${\mathrm{CrCl}}_{3}$, and ${\mathrm{CrBr}}_{3}$, 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.