Calculations of surface spin dynamics associated with biquadratic exchange and uniaxial anisotropy field in (S > 1) heisenberg ferromagnet by means of matching procedure.

Abstract

Calculations within framework of both non-interacting spin-wave theory and random phase approximation have been carried out to investigate elementary magnetic excitations by scanning the occurrence of localized modes associated with surface spin dynamics. Mean field approximation and single spin-wave function theory in the context of semi-infinite Heisenberg ferromagnet are exploited. We exploit the mathematical formalism of matching method which starting from secular equation of spin motion established for the bulk of semi-infinite structure will make it possible to describe the symmetry properties of propagating (evanescent) precessional fields in parallel (perpendicular) direction to surface terrace. The localized states associated with surface are obtained by matching the properties of quantized propagating modes with evanescent modes occurring in vicinity of surface region. This mathematical procedure allows treating the continuity equations for spin precessional field amplitude operating in the border bulk media-surface region from their special suitable symmetry properties related to those induced by the broken translational symmetry arising from the existence of surface. The solutions are described considering the presence of uniaxial anisotropy field and biquadratic exchange as well as magnetic anisotropy field. The focus is on the existence, the nature as well as on the form and stability of localized surface modes. Also, obtained solutions are discussed qualitatively by observing the response of energy branches to the variation of strength of biquadratic exchange term and surface anisotropy field as well as magnetic exchange parameters occurring on the surface domain. Numerical examples of the localized modes energies associated with the surface are shown and they are found to exhibit various effects due to the strength of biquadratic exchange and magnetic anisotropy field under certain circumstances. The analytical formulation exposed in this work and its methodology based solely on the symmetry properties as well as the qualitative agreement describing our results with those obtained in the literature provide a considerable contribution to the surface spin dynamics calculations under the influence of biquadratic coupling and uniaxial anisotropy term. Moreover, the results obtained through matching procedure make it possible to remove an ambiguity concerning the existing disparities in analysis and interpretation of the main results already reported in literature.