Two-dimensional Heisenberg model with material-dependent superexchange interactions

The two-dimensional (2D) van der Waals ferromagnetic semiconductors, such as CrI$_3$ and Cr$_2$Ge$_2$Te$_6$, the 2D metals, Fe$_3$GeTe$_2$ MnSe$_2$, have been obtained in recent experiments attracted a lot of attentions. superexchange interaction has suggested to dominate magnetic interactions these systems. In usual theoretical studies, expression Heisenberg models were fixed by hand due experiences. Here, we propose method determine counting possible paths with density functional theory (DFT) Wannier function calculations. With this method, obtain model six different nearest-neighbor exchange coupling constants for metal Cr$_3$Te$_6$, which is very crystal structure Cr atoms Cr$_3$Te$_6$. calculated Curie temperature Tc = 328 K close 344 Cr$_3$Te$_6$ reported experiment. addition, predict two stable semiconductors Cr$_3$O$_6$ Mn$_3$O$_6$ sharing same similar are Mn$_3$O$_6$, where 218 208 K, respectively. Our offers general approach builds up solid basis further studies.