Curie temperature of emerging two-dimensional magnetic structures

Recent realizations of intrinsic, long-range magnetic orders in two-dimensional (2D) van der Waals materials have ignited tremendous research interest. In this work, we employ the XXZ Heisenberg model and Monte Carlo simulations to study a fundamental property of these emerging 2D magnetic materials, the Curie temperature ($T_c$). By including both on-site and neighbor couplings extracted from first-principles simulations, we have calculated $T_c$ of monolayer chromium trihalides and $\mathrm{C}{\mathrm{r}}_2\mathrm{G}{\mathrm{e}}_2\mathrm{T}{\mathrm{e}}_6$, which are of broad interest currently, and the simulation results agree with available measurements. We also clarify the roles played by anisotropic and isotropic interactions in deciding $T_c$ of magnetic orders. Particularly, we find a universal, linear dependence between $T_c$ and magnetic interactions within the parameter space of realistic materials. With this linear dependence, we can predict $T_c$ of general 2D lattice structures, omitting the Monte Carlo simulations. Compared with the widely used Ising model, mean-field theory, and spin-wave theory, this work provides a convenient and quantitative estimation of $T_c$, giving hope to speeding up the search for novel 2D materials with higher Curie temperatures.

Figure 1

(a) The top and side views of the atomic structure of monolayer $\mathrm{Cr}{\mathrm{I}}_3$. The magnetic Cr atoms form hexagonal lattices. (b) The XXZ model applies to hexagonal lattices. Each site has a localized magnetic moment, and the NN and NNN coupling interactions are indicated.

Figure 2

The MC-simulated magnetism vs temperature for monolayer $\mathrm{Cr}{\mathrm{X}}_3$ ($X=I$, Br, Cl) with the error bar.

Figure 3

The fitted linear relation between $T_c$ and corresponding parameters of the XXZ model of 2D hexagonal lattices. The MC simulation points are solid blue points. (a–d) are the linearly fitted planes for corresponding on-site anisotropic $A$ and NN isotropic, $J_1$ NN anisotropic coupling ${\lambda }_1$ and $J_1$, NNN anisotropic coupling ${\lambda }_2$ and $J_1$, and NNN isotropic coupling $J_2$ and $J_1$, respectively. The corresponding parameters of monolayer $\mathrm{Cr}{\mathrm{I}}_3$ are marked as white stars.

Figure 4

The fitted linear relation between $T_c$ and corresponding parameters of the XXZ model of 2D square lattices. The MC simulation points are solid blue points. (a–d) are the linearly fitted planes for corresponding on-site anisotropic $A$ and NN isotropic $J_1$, NN anisotropic coupling ${\lambda }_1$ and $J_1$, NNN anisotropic coupling ${\lambda }_2$ and $J_1$, and NNN isotropic coupling $J_2$ and $J_1$, respectively.

Figure 5

The MC-simulated magnetism vs temperature for different magnetic-moment systems. The products $\alpha S_i^{\left(z\right)}{S}_j^{\left(z\right)}\left(\alpha =A,\lambda ,J\right)$ are fixed at the same value for different magnetic moments.

Figure 6

(a1,b1) are phase diagrams generated from MC simulations of the Hamiltonian of Eqs. (10) and (11), respectively. The open circles are MC-simulated phase boundaries. (a2,b2) are amplified from the shaded area with more data points.