Spin modes and layer magnetization for surface and impurity layer embedded in a ferromagnet

A theoretical study is made for the role of an impurity layer embedded within a semi-infinite ferromagnet in determining the spectra of (0 0 1) surface spin waves and the layer magnetization for the surface and impurity layer. The calculations are described using simultaneously a closed form of the spin-wave Green's function and the matching procedure in the random-phase approximation. Analytic expressions for the Green's functions are also derived in a low-temperature spin-wave approximation. The theoretical approach determines the bulk and evanescent spin fluctuation fields in the two-dimensional plane normal to the surface. The results are used to calculate the energies of localized modes associated with the impurity layer as well as with the surface. Numerical examples of the modes are given and they are found to exhibit various effects due to the interplay between the impurity layer and surface modes. The results derived from the dynamic correlation functions between a pair of spin operators at any two sites are employed to evaluate the spin deviation in the ferromagnet due to the localized modes associated with the surface and with the impurity layer obtained by means of the matching procedure. The correlation functions and the layer magnetization are then illustrated as function of the impurity layer distance from the surface for a given temperature.