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We consider the superconducting pairing induced by spin waves exchange in aferromagnet with both conduction and localized electrons, the latter beingdescribed as spins. We use the microscopic Eliashberg theory to describe thepairing of conducting electrons and the RPA approach to treat the localizedspins assuming an exchange coupling between the conducting electrons and spins.In the framework of non relativistic Hamiltonian twe found that he spin waveexchange results in equal spin electron pairing described by the two componentsof the order parameter, $\Delta^{\uparrow}$ (both spins up) and$\Delta^{\downarrow}$ (both spins down). Due to the conservation of total spinprojection on the axis of the spontaneous ferromagnetic moment, the spin waveexchange at low temperatures includes an emission of magnons and an absorptionof thermal magnons by the conduction electrons. The absorption and emissionprocesses depend differently on the temperature, with the absorption beingprogressively suppressed as the temperature drops. As a result, thesuperconducting pairing exists only if the electron-spin wave exchangeparameter $g$ exceeds some critical value $g_c$. At $g>g_c$ pairing vanishes ifthe temperature drops below the lowest point $T_{cl}$ or increases above theupper critical point $T_{ch} \approx T_m$ (the Curie temperature) where thespin waves cease to exist. This behavior inherent to the spin carrying glue isin an obvious disagreement with the results of conventional BCS approach whichassumes that the effective electron-electron attraction is simply proportionalto the static magnetic susceptibility.

Triplet superconductivity in ferromagnets due to magnon exchange