Analytical analysis of the spin wave dispersion in the cycloidal spin structures under the influence of magneto-electric coupling

Spin waves and coupling of the spin waves with electromagnetic waves are considered in the multiferroic materials with the electric dipole moment proportional to the scalar product of spins. Nature of this interaction is discussed within the spin current model. Dispersion dependence for the spin waves propagating as the perturbation of the equilibrium state described by spin cycloid is found analytically. Contribution of the wave vector of the equilibrium cycloid is traced and it is found that it decreases the contribution of the anisotropy constant, which can lead to the instability. Limit regime of the spin waves in systems of collinear spins is described to show the role of the magnetoelectric coupling for two regimes of the electric dipole moment: it proportional to the scalar product of spins or to the vector product of spins. The dielectric permeability as the response on the electromagnetic perturbations associated with the magneto-electric coupling for the same equilibrium state is calculated.