Spin clustering and ferromagnetic couplings in a dilute magnetic semiconductor

Ferromagnetic couplings in spin clusters are shown to be strongly enhanced compared to those for an ordered impurity arrangement, even for the same spin separation and hole doping. The consequent energy-enhancement of the cluster-localized spin-wave modes indicates a potentially significant role of positional Mn disorder in enhancing $T_{\rm C}$. Within a simple model involving two spin-excitation energy scales corresponding to weakly and strongly coupled spins, the temperature dependence of magnetization is found to be in good agreement with the SQUID magnetization data for Ga$_{1-x}$Mn$_x$As.