Abstract
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.
Highlights
Thanks to the recent discovery of the two-dimensional (2D) magnets CrI3 [1], CrBr3 [2], and CrGeTe3 [3], research in the field of 2D magnets has garnered unprecedented attention in the past few years
We have investigated the impact of nearest-neighbor exchange anisotropy as well as the next-nearest-neighbor anisotropy on the Curie temperature calculated using the mentioned methods
We have shown that the Curie temperature calculated using the Green’s function and the Monte Carlo (MC) methods as a function of nearest-neighbor anisotropy results in three regions
Summary
Thanks to the recent discovery of the two-dimensional (2D) magnets CrI3 [1], CrBr3 [2], and CrGeTe3 [3], research in the field of 2D magnets has garnered unprecedented attention in the past few years. Monte Carlo simulations with anisotropy result in a rather accurate estimation of the Curie temperature for most of the experimentally verified 2D magnets yet discovered [24,27]. The recent application of methods that take into account the quantum mechanical fluctuations in the Heisenberg model to 2D magnets, e.g., the Green’s function [29,30] and the renormalized spin-wave [31] methods, raises further questions on how much the Curie temperature depends on the level of approximation used to solve the Heisenberg Hamiltonian. For 34 2D materials, three methods of calculating the Curie temperature from a Heisenberg Hamiltonian: the Monte Carlo (MC), Green’s function, and renormalized spin-wave (RNSW) methods. We provide an analytical formula to approximate the Curie temperature calculated using the three solution methods, as a function of nearest-neighbor exchange strength (J) and anisotropy ( NN). We show that the Curie temperature calculated using the three methods depends quantitatively on the long-range interactions; the qualitative trend remains the same and the analytical formula we develop provides a good estimation for a first-level theoretical screening
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