Ferromagnetism in diluted magnetic semiconductors: A comparison between ab initio mean-field, RPA, and Monte Carlo treatments

We have determined from first principles the Curie temperature in the diluted magnetic semiconductor $\left({\mathrm{Ga}}_{1-x-y}{\mathrm{Mn}}_x{\mathrm{As}}_y\right)$ As alloys, where the compensation effect by As antisites is taken into account. The disorder due to random positions of Mn impurities and As antisites is taken into account in the framework of the coherent potential approximation. We demonstrate that a simple mean-field approximation (MFA) already gets $T_c$ accurately. In particular, it is shown that the calculated $T_c$ in the random-phase approximation and Monte Carlo simulations differ only weakly from the MFA values. The reason is that a proper treatment of the disorder beyond the virtual-crystal approximation leads, in this doped material, to a flat magnon dispersion in a large part of the Brillouin zone.