arXiv2003
We introduce a Heisenberg Hamiltonian for describing the magnetic properties of GaMnAs. Electronic degrees of freedom are integrated out leading to a pairwise interaction between Mn spins. Monte Carlo simulations in large systems are then possible, and reliable values for the Curie temperatures of diluted magnetic semiconductors can be obtained. Comparison of mean field and Monte Carlo Curie temperatures shows that fluctuation effects are important for systems with a large hole density and/or increasing locality in the carriers-Mn coupling. We have also compared the results obtained by using a realistic ${\bf k} \cdot {\bf p}$ model with those of a simplified parabolic two band model. In the two band model, the existence of a spherical Fermi surface produces the expected sign oscillations in the coupling between Mn spins, magnifying the effect of fluctuations and leading to the eventual disappearance of ferromagnetism . In the more realistic ${\bf k} \cdot {\bf p}$ model, warping of the Fermi surface diminishes the sign oscillations in the effective coupling and, therefore, the effect of fluctuations on the critical temperature is severely reduced. Finally, by studying the collective magnetic excitations of the this model at zero temperature, we analyze the stability of the fully polarized ferromagnetic ground state.