Effect of thermodynamic fluctuations of magnetization on the bound magnetic polaron in dilute magnetic semiconductors

We consider the magnetic field and temperature dependences of the spin splitting for an electron localized on a shallow donor, taking into account the $s-d$ coupling with the surrounding magnetic ions. We calculate the probability distribution of the spin splitting $\Delta$ with the thermodynamic fluctuations of magnetization included. They are responsible for persistence of the spin splitting at ambient temperature in the absence of the field. The probability distribution of the spin splitting is the main factor determining the energy and the shape of the optical transition line between the spin-split states. We compare our results with recent experimental results on spin-flip Raman scattering in ${\mathrm{Cd}}_{1-x}{\mathrm{Mn}}_x\mathrm{Se}$. Our theory provides a satisfactory quantitative description of the donor electron accompanied by a nonuniform cloud of magnetization in that material.