Critical behavior of the ferromagnets CrI3, CrBr3 , and CrGeTe3 and the antiferromagnet

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 N\'eel 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 N\'eel temperature of $\mathrm{Fe}{\mathrm{Cl}}_{2}$ to be $47\ifmmode\pm\else\textpm\fi{}8\phantom{\rule{0.16em}{0ex}}\mathrm{K}$ 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.