Exactly solvable model for carrier-induced paramagnetic-ferromagnetic phase transition in diluted magnetic semiconductors

Abstract We present a model that permits to obtain and exactly describe the paramagnetic-ferromagnetic phase transition induced by carrier-ion exchange interaction and assess the different theoretical approaches to this problem. To modeling the ferromagnetic phase transition, we consider the cluster consisted of N m magnetic ions and N e electrons. Carrier-ion exchange interaction allows the transitions between two electronic levels. Exact energy spectrum of the model under consideration is found. Then thermodynamic parameters are calculated with straightforward calculations of partition function. The final expression for the magnetic susceptibility χ 0 we obtain in thermodynamic limit for cluster volume V →∞. The model describes two different contributions to the inverse magnetic susceptibility. One term T MF stems from the diagonal part of carrier-ion exchange interaction that corresponds to mean field approximation. Next term T LeL stems from the off-diagonal part of interaction and describes the indirect interaction between localized spins via free carriers. Thus, the critical temperature of ferromagnetic ordering is determined by two aforementioned contributions. A general approach is proposed and compared with different approximations to the problem under consideration.