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We derive an effective Hamiltonian for ${\rm Ga}_{1-x}{\rm Mn}_x {\rm As}$ inthe dilute limit, where ${\rm Ga}_{1-x}{\rm Mn}_x {\rm As}$ can be described interms of spin $F=3/2$ polarons hopping between the {\rm Mn} sites and coupledto the local {\rm Mn} spins. We determine the parameters of our model frommicroscopic calculations using both a variational method and an exactdiagonalization within the so-called spherical approximation. Our approachtreats the extremely large Coulomb interaction in a non-perturbative way, andcaptures 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 finitetemperature. We study the resulting magnetic properties, such as themagnetization and spin disorder manifest in the generically non-collinearmagnetic state. We find a well formed impurity band fairly well separated fromthe valence band up to $x_{\rm active} \lesssim 0.015$ for which finite sizescaling studies of the participation ratios indicate a localization transition,even in the presence of strong on-site interactions, where $x_{\rmactive}<x_{\rm nom}$ is the fraction of magnetically active Mn. We study thelocalization transition as a function of hole concentration, Mn positionaldisorder, and interaction strength between the holes.Comment: 15 pages, 12 figure

physical review. b, condensed matter and materials physics

Disorder, spin-orbit, and interaction effects in dilute<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mi mathvariant="normal">Ga</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>x</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi mathvariant="normal">Mn</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mi mathvariant="normal">As</mml:mi></mml:mrow></mml:math>

Disorder, spin-orbit, and interaction effects in diluteGa1−xMnxAs