Spin wave modes in 3D arrays of magnetic particles

In this paper we investigate the spin modes of 3-D arrays of magnetic dipoles, in order to simulate the collective magnetic response of the corresponding magnonic metamaterial. We consider simple cubic arrays with the shape of elongated prisms with different terminations (flat, pyramidal) and at different applied fields. Our investigation is carried out by applying two different calculation approaches and comparing the results. In the first method we solve the Landau-Lisfshits equation of motion of planar arrangements of magnetic dipoles with uniform precession in each plane. In the second approach we use the dynamical matrix method, which is based on micromagnetism, considers the exact magnetic equilibrium configuration, and returns the complete set of magnetic eigenvalues/eigenmodes. Calculations show the existence of modes with different localization: low frequency modes, localized at the prism ends, and high frequency bulk modes, including the fundamental or quasi-uniform mode. We aolso study the dependence of the surface modes on the termination of the prisms, and correlated it to the behavior of the internal field