Onset of ferromagnetism in low-doped GaMnAs

We develop a quantitatively predictive theory for impurity-band ferromagnetism in the low-doping regime of GaMnAs and compare with experimental measurements of a series of samples whose compositions span the transition from paramagnetic insulating to ferromagnetic conducting behavior. The theoretical Curie temperatures depend sensitively on the local fluctuations in the Mn-hole binding energy, which originates from disorder in the Mn distribution as well as the presence of As antisite defects. The experimentally-determined hopping energy at the Curie temperature is roughly constant over a series of samples whose conductivities vary more than 10^4 and whose hole concentrations vary more than 10^2. Thus in this regime the hopping energy is an excellent predictor of the Curie temperature for a sample, in agreement with the theory.