Disorder, spin-orbit, and interaction effects in dilute Ga₁₋ₓ Mnₓ As

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We derive an effective Hamiltonian for Ga₁₋ₓ Mnₓ As in the dilute limit, where Ga₁₋ₓ Mnₓ As can be described in terms of spin F=3/2 polarons 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 non-perturbative 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 non-collinear 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.