Carrier induced ferromagnetism in diluted magnetic semi-conductors

We present a theory for carrier induced ferromagnetism in diluted magnetic semi-conductor (DMS). Our approach treats on equal footing quantum fluctuations within the RPA approximation and disorder within CPA. This method allows for the calculation of $T_c$, magnetization and magnon spectrum as a function of hole, impurity concentration and temperature. It is shown that, sufficiently close to $T_c$, and within our decoupling scheme (Tyablicov type) the CPA for the itinerant electron gas reduces to the Virtual Crystal Approximation. This allows, in the low impurity concentration and low density of carriers to provide analytical expression for $T_c$. For illustration, we consider the case of $Ga_{1-c}Mn_{c}As$ and compare our results with available experimental data.