Electron states in the s-f exchange model of a ferromagnetic semiconductor in the spin wave region

One-electron Green functions are obtained and investigated in the framework of the s-f exchange model of a ferromagnetic semiconductor, at temperatures T much lower than the Curie temperature T_c. An exact procedure is developed for working out the series expansions of the Green functions in powers of the magnon occupation numbers. Using these expansions the spectrum shift at T<<T_c and the damping of the conduction electrons due to the magnons are calculated, the ratio of the s-f exchange parameter to the band width and the value of the magnetic ion spin S being arbitrary quantities. The vanishing of the effective electron-magnon interaction as the magnon wavevector tends to zero is proved rigorously. In addition, a perturbation theory involving the formal parameter ¹/₂S that supports the idea of the Hamiltonian involving an effective electron-magnon interactions is developed.