Disorder and the effective Mn-Mn exchange interaction in Ga$_{1-x}$Mn$_x$As diluted magnetic semiconductors

We perform a theoretical study, using {it ab initio} total energy density-functional calculations, of the effects of disorder on the $Mn-Mn$ exchange interactions for $Ga_{1-x}Mn_xAs$ diluted semiconductors. For a 128 atoms supercell, we consider a variety of configurations with 2, 3 and 4 Mn atoms, which correspond to concentrations of 3.1%, 4.7%, and 6.3%, respectively. In this way, the disorder is intrinsically considered in the calculations. Using a Heisenberg Hamiltonian to map the magnetic excitations, and {it ab initio} total energy calculations, we obtain the effective $JMn$, from first ($n=1$) all the way up to sixth ($n=6$) neighbors. Calculated results show a clear dependence in the magnitudes of the $JMn$ with the Mn concentration $x$. Also, configurational disorder and/or clustering effects lead to large dispersions in the Mn-Mn exchange interactions, in the case of fixed Mn concentration. Moreover, theoretical results for the ground-state total energies for several configurations indicate the importance of a proper consideration of disorder in treating temperature and annealing effects.