Interaction of a conduction electron with critical fluctuations of the spin density in the $s$-$d$ exchange model of a magnetic semiconductor. Estimate of the free energy and static conductivity

Estimates of the free energy of conductivity electrons coupling with $T\gtrsim T_c$ critical spin density fluctuations are obtained by means of Feynman path integral technique and corresponding variational principle. Condition is found under which the fluctuations may be treated as gaussian ones; the opposite case of the Kac model for spin subsystem is also considered. The relations between the quantum-statistical approach proposed and the phenomenological theory [1, 13] is briefly discussed. In the gaussian case the expression for the static critical conductivity is obtained which is calculated in the considered regions of values of model parameters.