Carrier-induced ferromagnetism in diluted magnetic semiconductors

We present a theory for carrier-induced ferromagnetism in diluted magnetic semi-conductor (DMS). Our approach treats on equal footing both quantum fluctuations within the random-phase approximation and disorder within the coherent potential approximation (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 ${\mathrm{Ga}}_{1-c}{\mathrm{Mn}}_c\mathrm{As}$ and compare our results with available experimental data.