Investigation of spin wave damping in three-dimensional magnonic crystals using the plane wave method

The Landau-Lifshitz equation with a scalar damping constant predicts that the damping of spin waves propagating in an infinite homogeneous magnetic medium does not depend on the direction of propagation. This is not the case in materials with a periodic arrangement of magnetic constituents (known as magnonic crystals). In this paper, the plane wave method is extended to include damping in the calculation of the dispersion and relaxation of spin waves in three-dimensional magnonic crystals. A model material system is introduced and calculations are then presented for magnonic crystals realized in the direct and inverted structure and for two different filling fractions. The ability of magnonic crystals to support the propagation of spin waves is characterized in terms of a figure of merit, defined as the ratio of the spin wave frequency to the decay constant. The calculations reveal that in magnonic crystals with a modulated value of the relaxation constant, the figure of merit depends strongly on the frequency and wave vector of the spin waves, with the dependence determined by the spatial distribution of the spin wave amplitude within the unit cell of the magnonic crystal. Bands and directions of exceptionally long spin wave propagation have been identified. The results are also discussed in terms of the use of magnonic crystals as metamaterials with designed magnetic permeability.