Microscopic origin of ferromagnetism in trihalides CrCl$_3$ and CrI$_3$

Microscopic origin of the ferromagnetic (FM) exchange coupling in CrCl$_3$ and CrI$_3$, their common aspects and differences, are investigated on the basis of density functional theory combined with realistic modeling approach for the analysis of interatomic exchange interactions. We perform a comparative study based on the pseudopotential and linear muffin-tin orbital methods by treating the effects of electron exchange and correlation in GGA and LSDA, respectively. The results of ordinary band structure calculations are used in order to construct the minimal tight-binding type models describing the behavior of the magnetic Cr $3d$ and ligand $p$ bands in the basis of localized Wannier functions, and evaluate the effective exchange coupling ($J_{rm eff}$) between two Cr sublattices employing four different technique: (i) Second-order Greens function perturbation theory for infinitesimal spin rotations of the LSDA (GGA) potential at the Cr sites; (ii) Enforcement of the magnetic force theorem in order to treat both Cr and ligand spins on a localized footing; (iii) Constrained total-energy calculations with an external field, treated in the framework of self-consistent linear response theory. We argue that the ligand states play crucial role in the ferromagnetism of Cr trihalides, though their contribution to $J_{rm eff}$ strongly depends on additional assumptions, which are traced back to fundamentals of adiabatic spin dynamics. Particularly, by neglecting ligand spins in the Greens function method, $J_{rm eff}$ can easily become antiferromagnetic, while by treating them as localized, one can severely overestimate the FM coupling. The best considered approach is based on the constraint method, where the ligand states are allowed to relax in response to each instantaneous reorientation of the Cr spins, controlled by the external field.