Calculations of (1 1 0) ferromagnetic surface modes: A method of solution

We present a theoretical method for the calculations of (1 1 0) ferromagnetic surface modes in the context of the Heisenberg model with only nearest neighbors interactions. It does not include either relaxation or reconstruction at the surface. The propagating and evanescent spin waves fields in the two-dimensional plane normal to the surface are investigated in this theoretical approach. Emphasis has been laid upon the advantage of handling each displacement field, characterized by its symmetry in particular. Analytic expressions for the bulk and surface equations of motion are also derived in a low-temperature spin wave approximation which enable us to analyze qualitatively the nature of the modes, by observing the response of the energy branches to the variation of structure factors and exchange parameters occurring on the surface. The results are used to calculate the localized modes of spin waves associated with the surface. Numerical examples of the localized modes are given and they are found to exhibit various effects due to the interplay between the bulk and surface modes. It is shown, that under certain circumstances, either one or two branches of surface spin waves may exist. Also the bulk spin fluctuation field as well as the magnetic surface waves, are in qualitative overall agreement with those obtained in the literature by using a Green’s function formalism.