Disorder, spin-orbit, and interaction effects in dilute Ga_1−xMn_xAs

We derive an effective Hamiltonian for Ga_1−xMn_xAs in the dilute limit, where Ga_1−xMn_xAs can be described in terms of spin F=3∕2 holes hopping between the Mn sites and coupled to the local Mn spins. We determine the parameters of our model from microscopic calculations using both a variational method and an exact diagonalization within the so-called spherical approximation. Our approach treats the extremely large Coulomb interaction in a nonperturbative way and captures the effects of strong spin-orbit coupling and Mn positional disorder. We study the effective Hamiltonian in a mean-field and variational calculation, including the effects of interactions between the holes at both zero and finite temperature. We study the resulting magnetic properties, such as the magnetization and spin-disorder manifest in the generically noncollinear magnetic state. We find a well-formed impurity band fairly well separated from the valence band up to x_active≲0.015 for which finite-size scaling studies of the participation ratios indicate a localization transition, even in the presence of strong on-site interactions, where x_active<x_nom is the fraction of magnetically active Mn. We study the localization transition as a function of hole concentration, Mn positional disorder, and interaction strength between the holes.