On the Formulation of the Exchange Field in the Landau-Lifshitz Equation for Spin-Wave Calculation in Magnonic Crystals

The calculation of the magnonic spectra using the plane-wave method has limitations, the origin of which lies in the formulation of the effective magnetic field term in the equation of motion (the Landau-Lifshitz equation) for composite media. According to ideas of the plane-wave method the system dynamics is described in terms of plane waves (a superposition of a number of plane waves), which are continuous functions and propagate throughout the medium. Since in magnonic crystals the sought-for superposition of plane waves represents the dynamic magnetization, the magnetic boundary conditions on the interfaces between constituent materials should be inherent in the Landau-Lifshitz equations. In this paper we present the derivation of the two expressions for the exchange field known from the literature. We start from the Heisenberg model and use a linear approximation and take into account the spacial dependence of saturation magnetization and exchange constant present in magnetic composites. We discuss the magnetic boundary conditions included in the presented formulations of the exchange field and elucidate their effect on spin-wave modes and their spectra in one- and two-dimensional planar magnonic crystals from plane-wave calculations.