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.

Figure 1

Three different type of lattices for 2D materials: (a) honeycomb, (b) hexagonal, and (c) square lattices. For the (b) hexagonal and the (c) square lattices, the unit cell contains only one atom, whereas for the (a) honeycomb lattice, the unit cell has two atoms, referred to as sublattice A (pink) and sublattice B (black). (d) The in-plane ($J_{ij}^{xx}$) and out-of-plane ($J_{ij}^{zz}$) exchange interaction between spins $i$ and $j$.

Figure 2

(a) Comparison of Curie temperature calculated using the Green's function, the Monte Carlo (MC), and the renormalized spin-wave (RNSW) methods as a function of nearest-neighbor exchange anisotropy (${\mathrm{\Delta }}_{\mathrm{NN}}$) for a honeycomb lattice. For the MC method, the solid line shows the median and the shading shows the 25th to 75th percentile of the calculated Curie temperature. The dashed lines show a fit using function ${[{\alpha }_1-{\alpha }_{2}ln\left({\mathrm{\Delta }}_{\mathrm{NN}}\right)]}^{-1}J(S^2+\theta S)/k_{\mathrm{B}}$ for the respective methods. The horizontal tick shows the Ising limit ($1.52S^2$, with $S=3/2$) and the quantum mean field ($1/3\left[S(S+1)N_{\mathrm{NN}}\right]$, $N_{\mathrm{NN}}=3$ for honeycomb lattice). The yellow shaded area shows the region where the MC and the Green's function methods have a difference of less than 10%. (b) Comparison of Curie temperature sensitivity calculated using the Green's function, the MC, and the RNSW methods as a function of nearest-neighbor exchange anisotropy (${\mathrm{\Delta }}_{\mathrm{NN}}$) for a honeycomb lattice.

Figure 3

(a) Materials obtained from the C2DB database with positive out-of-plane anisotropy and positive exchange interaction. The Curie temperature of the screened materials using the analytical formula for the MC, the Green's function, and the RNSW (top abscissa) methods sorted as a function of (b) exchange interaction ($J_{ij}$) and (c) exchange anisotropy (${\mathrm{\Delta }}_{\mathrm{NN}}$).

Figure 4

Comparison of Curie temperature calculated using the Green's function, Monte Carlo, and the renormalized spin-wave methods as a function of next-nearest-neighbor exchange anisotropy (${\mathrm{\Delta }}_{\mathrm{NNN}}$) for a honeycomb lattice. The vertical dashed line shows the ${\mathrm{\Delta }}_{\mathrm{NNN}}=0$ at which Fig. 2 is calculated. For the MC method, the solid line shows the median and the shading shows the 25th to 75th percentile of the calculated Curie temperature.