Separated spin evolution quantum hydrodynamics of degenerated electrons with spin-orbit interaction and extraordinary wave spectrum

To consider a contribution of the spin-orbit interaction in the extraordinary wave spectrum we derive a generalization of the separate spin evolution quantum hydrodynamics. Applying corresponding nonlinear Pauli equation we include Fermi spin current contribution in the spin evolution. We find that the spectrum of extraordinary waves consists of three branches: two of them are well-known extraordinary waves and the third one is the spin-electron acoustic wave (SEAW). Earlier SEAWs have been considered in the electrostatic limit. Here we include the electromagnetic effects in their spectrum at the propagation perpendicular to the external magnetic field. We find that the SEAW spectrum considerably changes at the account of transverse part of electric field. We obtain that the separate spin evolution modifies spectrum of the well-known extraordinary waves either. A change of the extraordinary wave spectrum due to the spin-orbit interaction is obtained as well.