Spin wave localization and softening in rod-shaped magnonic crystals with different terminations

The spin dynamics of simple cubic arrays of magnetic dipoles with the shape of elongated prisms is investigated in dependence of their terminations (flat or cusp) and of the applied field. We used two different calculation approaches: in the first, we solve the Landau-Lisfshits equation of motion of planar arrangements of magnetic dipoles; the static magnetization of the array is supposed to be uniform along the direction of the applied field, and the calculated modes have nodal planes perpendicular to the magnetization. In the second approach, we use the dynamical matrix method, which is a micromagnetic method, considers the exact (non-uniform) 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 studied the internal field profile as a function of the termination details, the localization of spin modes, in particular of the lowest frequency mode, and the space resolved density of states. Finally, we address the soft modes of these systems, showing their frequency vs. applied field behavior in relation to the discontinuity of the magnetization curve, and investigating the symmetry transfer from the soft mode profile to the static magnetization, with possible applications