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Recent realizations of intrinsic magnetic order in truly two-dimensionalmaterials have opened new avenues in the fundamental knowledge and spintronicapplications. Here we develop an anisotropic Heisenberg model with relativisticexchange interactions that are obtained from the first-principles calculations.We demonstrate the crucial importance of magnetic interactions beyond thefirst-neighbour to qualitatively and quantitatively reproduce the experimentalresults. Once we ascertain the predictive capacity of the model for chromiumtrihalides and CrGeTe$_3$, we investigate the feasibility of tuning themagnetic ordering by electrical means in these materials. A remarkablefive-fold increase in the ferromagnetic Curie temperature is predicted inmonolayer CrI$_3$ within experimentally obtainable hole density. The elusivemicroscopic mechanism behind the doping-dependent Curie temperature isillustrated. Further, in the present context, the effects of biaxial strain andchemical doping are also addressed. The results should trigger furtherexperimental attention to test the present conclusions.

Long-range anisotropic Heisenberg ferromagnets and electrically tunable ordering