Effective quantities and effective rules in two-dimensional ferromagnetic antidot lattices

In this paper the metamaterial properties of two-dimensional arrays of circular antidots (holes) embedded into a ferromagnetic medium of Permalloy are studied according to both micromagnetic and analytical calculations. The periodicity of the arrays and the diameters of the antidots are in the nanometric range. The collective mode dynamics is described by means of effective physical quantities for the scattering geometry with the external magnetic field applied perpendicularly to the Bloch wave vector in the antidot plane. As an example, the definition of an effective field, incorporating the demagnetizing effects due to the holes, permits to describe the dynamical properties of collective modes in terms of effective properties in the travelling regime. An effective wavelength and a small wave vector are introduced both for extended and localized magnonic modes. By means of these effective quantities it is shown that holes play the role of point defects affecting the spin dynamics in the microwave range. Relations between the effective wavelength and the Bloch wavelength and between the corresponding small wave vector and the Bloch wave vector are found. Some effective rules on the dynamic magnetization, based upon the effective wavelength and the corresponding small wave vector, are derived. An application that exploits the definition of the small wave vector is proposed and an experiment based upon the notion of effective wavelength and small wave vector is suggested.