Metamaterial properties of one-dimensional and two-dimensional magnonic crystals

In this paper some new and recent results on metamaterial properties of one-dimensional and two-dimensional magnonic crystals are presented. These results were obtained by means of a recently developed micromagnetic approach, the so called Hamiltonian-based Dynamical Matrix Method formulated for studying dynamical properties of of isolated magnetic particles and generalized to periodic magnetic systems. The method is essentially based on an eigensystem problem according to which the equation of motions are written in terms of the second derivates of the energy density evaluated at the equilibrium. The formalism of the differential scattering cross-section associated to each collective mode and generalized to periodic systems is also outlined. Applications of the micromagnetic formalism to one-dimensional and two-dimensional magnonic crystals are presented and the metamaterial properties found according to the method are illustrated and discussed. First, metamaterial properties of one-dimensional magnonic crystals represented by chains of rectangular dots are discussed by showing the calculated frequency dispersion for two scattering geometries, namely with the external magnetic field either perpendicular or parallel to the Bloch wave vector in the plane of the system. As a further example of metamaterial properties typical of chains of rectangular dots, calculated and measured band widths and band gaps between adjacent collective modes are shown and discussed. Second, application of the Hamiltonian-based Dynamical Matrix Method is reviewed for presenting the most important metamaterial properties in two prototypes of two-dimensional magnonic crystals: a) arrays of close-packed disks of Permalloy coupled via dipolar magnetic interaction and b) arrays of circular antidots embedded into a Permalloy ferromagnetic film. For the array of disks it is shown the calculation of the band structure and the band behaviour is discussed in terms of an effective wave vector. For the arrays of antidots frequency dispersion of extended and localised modes collective modes are presented and the opening of band gaps is explained by studying the inhomogeneity of the internal field according to a recently developed analytical model. A few examples of micromagnetic calculations performed on ferromagnetic arrays of antidots with different periodicities are presented.