Comparison between ab-initio and phenomenological modeling of the   exchange couplings in diluted magnetic semiconductors: the case of   $Zn_{1-x}Cr_{x}Te$
  Using a recently developed semi-analytical method (Self-Consistent Local RPA or SC-LRPA) we study the stability of the ferromagnetic phase in diluted magnetic systems where the exchange coupling between magnetic impurities are of RKKY form. A short discussion of the relevance of these calculations with respect to the ferromagnetism observed in diluted ferromagnetic materials is provided. Then, within a two step approach, we study ferromagnetism in $Zn_{1-x}Cr_{x}Te$. In the first step of our study, we calculate the magnetic couplings between Mn impurities within the LDA. In the second step, we diagonalize the resulting effective Heisenberg Hamiltonian using the SC-LRPA. We also compare, when available, our calculations with Monte Carlo simulations and experimental measurements. 
Georges ~Bouzerar, Richard Bouzerar, Josef ~Kudrnovsk\'y and Timothy   Ziman