Critical behavior of the ferromagnets ${\mathrm{CrI}}_3, {\mathrm{CrBr}}_3$, and ${\mathrm{CrGeTe}}_3$ and the antiferromagnet ${\mathrm{FeCl}}_2$: A detailed first-principles study

We calculate the Curie temperature of layered ferromagnets, chromium tri-iodide (${\mathrm{CrI}}_3$), chromium tri-bromide ($\mathrm{Cr}{\mathrm{Br}}_3$), chromium germanium tri-telluride ($\mathrm{Cr}\mathrm{Ge}{\mathrm{Te}}_3$), and the Néel temperature of a layered antiferromagnet iron di-chloride ($\mathrm{Fe}{\mathrm{Cl}}_2$), using first-principles density functional theory calculations and Monte Carlo simulations. We develop a computational method to model the magnetic interactions in layered magnetic materials and calculate their critical temperature. We provide a unified method to obtain the magnetic exchange parameters ($J$) for an effective Heisenberg Hamiltonian from first principles, taking into account both the magnetic ansiotropy as well as the out-of-plane interactions. We obtain the magnetic phase change behavior, in particular the critical temperature, from the susceptibility and the specific-heat, calculated using the three-dimensional Monte Carlo (metropolis) algorithm. The calculated Curie temperatures for ferromagnetic materials (${\mathrm{CrI}}_3, \mathrm{Cr}{\mathrm{Br}}_3$, and $\mathrm{Cr}\mathrm{Ge}{\mathrm{Te}}_3$), match well with experimental values. We show that the interlayer interaction in bulk $\mathrm{Cr}{\mathrm{I}}_3$ with $R\overline{3}$ stacking is significantly stronger than the $C2/m$ stacking, in line with experimental observations. We show that the strong interlayer interaction in $R\overline{3}$ $\mathrm{Cr}{\mathrm{I}}_3$ results in a competition between the in-plane and the out-of-plane magnetic easy axes. Finally, we calculate the Néel temperature of $\mathrm{Fe}{\mathrm{Cl}}_2$ to be $47\pm 8\mathrm{K}$ and show that the magnetic phase transition in $\mathrm{Fe}{\mathrm{Cl}}_2$ occurs in two steps with a high-temperature intralayer ferromagnetic phase transition and a low-temperature interlayer antiferromagnetic phase transition.

Figure 1

Illustration of the elements of matrix $J_{ij}$ for various magnetic axis orientation in different directions for atoms $i$ and $j$ [(a), (b), and (c)] and the azimuthal angle ${\theta }_{ij}$ between atoms $i$ and $j$ in two different layers of a magnetic material and the magnetic anisotropic angle ${\mathrm{\Theta }}_j$ for the magnetic atom $j$ (d).

Figure 2

(a) Side view of $R\overline{3}$ $\mathrm{Cr}{\mathrm{I}}_3$. (b) Top view of the net of Cr atoms in $R\overline{3}$ ${\mathrm{CrI}}_3$. (c) Side view of the Cr atoms in $R\overline{3}$ $\mathrm{Cr}{\mathrm{I}}_3$ with A, B, and C stacks are shown explicitly. (d) Side view of $\mathrm{Fe}{\mathrm{Cl}}_2$ bulk. (e) Top view of the net of Fe atoms in $\mathrm{Fe}{\mathrm{Cl}}_2$ bulk. (f) Side view of $C2/m$ ${\mathrm{CrI}}_3$.

Figure 3

(a) Shows the $J$ parameter $\left[J^{zz}(r,\theta )\right]$ when the magnetic easy axis is out-of-plane for (a)$R\overline{3}$ $\mathrm{Cr}{\mathrm{I}}_3$ and (b)$C2/m$ $\mathrm{Cr}{\mathrm{I}}_3$. The circle shows the range of interaction ($r_{\mathrm{c}}=6.81$ Å for $R\overline{3}$ and $r_{\mathrm{c}}=7.01$ Å for $C2/m$) with respect to the atom at the center (in red). The atoms inside the interaction circle are the ones considered while obtaining the $J$ parameters.

Figure 4

Shows the magnetization vs temperature (left) and susceptibility vs temperature (right) for $\mathrm{Cr}{\mathrm{I}}_3$ with $R\overline{3}$ space group (solid lines) and $C2/m$ space group (dotted lines). Solid lines show the Curie-Weiss function $[M\propto {(T-T_{\mathrm{c}})}^{\beta }]$ fit, dots show the magnetization ($M$) obtained from the MC simulation.

Figure 5

The interplay between the magnetic and geometric anisotropy. The difference in the $J$ parameter [$\mathrm{\Delta }J=J^{zz}(r,\theta )-J^{\parallel }(r,\theta )$] for $\mathrm{Cr}{\mathrm{I}}_3$ bulk when the magnetic easy axis is out-of-plane (001) vs when the magnetic easy axis lies in-plane (010).

Figure 6

The magnetization vs temperature (blue) and the specific-heat vs temperature (red) of bulk $\mathrm{Cr}{\mathrm{Br}}_3,\mathrm{Cr}\mathrm{Ge}{\mathrm{Te}}_3,{\mathrm{CrI}}_3$. Solid lines show the interpolated data using a savgol filter while dots show the data obtained from the MC simulations. The saturation magnetization for $\mathrm{Cr}{\mathrm{Br}}_3,\mathrm{Cr}\mathrm{Ge}{\mathrm{Te}}_3$, and ${\mathrm{CrI}}_3$ are 2.7 ${\mu }_{\mathrm{B}}$, 2.7 ${\mu }_{\mathrm{B}}$, and 2.9 ${\mu }_{\mathrm{B}}$, respectively.

Figure 7

The magnetic orientation of a supercell of $R\overline{3}$ $\mathrm{Cr}{\mathrm{I}}_3$ at various temperatures. Magnetic orientations within the demarcated box are the orientations below the Curie temperature.

Figure 8

Solid lines show the interpolated magnetization vs temperature (blue) and the specific-heat vs temperature (red), of bulk $\mathrm{Fe}{\mathrm{Cl}}_2$ obtained from the MC simulations. Dots show the data obtained from the MC simulations. The dotted line shows the magnetization vs temperature for one of the layers of $\mathrm{Fe}{\mathrm{Cl}}_2$. The maximum magnetizations are $|M_{\max }|\approx 0.9{\mu }_B$ at 250 K and $|M_{\mathrm{sat}}|\approx 0.1{\mu }_B$ at 6 K, respectively.

Figure 9

Shows the magnetic ground state of a $5\times 5\times 2$ supercell of $\mathrm{Fe}{\mathrm{Cl}}_2$ at various temperatures obtained from the MC simulations. Magnetic orientations within the demarcated box are the orientations below the Néel temperature.

Figure 10

The $J$ parameters for the $\mathrm{Fe}{\mathrm{Cl}}_2$ bulk with magnetic easy axis in the out-of-plane direction $\left[J^{zz}(r,\theta )\right]$. The circle shows the range of the interaction ($r_{\mathrm{c}}=6.22$ Å) with respect to the atom at the center (in red). The atoms inside the interaction circle are the ones considered while obtaining the $J$ parameters.

Figure 11

The various blocks of our computational model.

Figure 12

Illustration of a ferrimagnetic configuration of $\mathrm{Cr}{\mathrm{I}}_3$ with (a) out-of-plane magnetic easy axis, and (b) in-plane magnetic easy axis. The circle shows the range up to which the exchange interaction is considered.

Figure 13

Illustration of the geometric ($\theta$) and the magnetic azhimuthal angle ($\mathrm{\Theta }$).