Influence of Surface Anisotropy and Next-Nearest-Neighbor Coupling on Surface Spin Waves

The spin-wave spectrum of a Heisenberg ferromagnet is derived in the framework of the method of De Wames and Wolfram, with the next-nearest-neighbor coupling, the surface and bulk exchange, and the surface anisotropy taken into account. The calculations are performed for a {100} surface in a simple-cubic structure. When the next-nearest-neighbor coupling is strong enough, the character of the surface modes is shown to change from acoustical to optical (or vice versa) as the components (kₓ, ky) of the wave vector parallel to the surface vary. This transition from acoustical to optical character occurs for a definite value λₖ⁰ of the parameter λₖ=1-1/2(coskₓa+coskya), where a is the lattice parameter. For this value λₖ⁰, the effective coupling between the spin deviation in different bulk layers is shown to vanish, so that each of the bulk modes corresponds to vibrations of the spins of one single layer and to a surface spin wave strictly localized on the first layer. Moreover, the equilibrium spin configuration near the surface could be modified by the existence of the surface anisotropy.