Vortices in two-dimensional nanorings studied by means of the dynamical matrix method

This paper concerns an investigation of the spin wave excitations in magnetic nanoparticles. We provide a detailed derivation of the theoretical method for the determination of the normal modes of confined magnetic systems based on a discrete lattice of magnetic moments. The method is based on the damping-free Landau-Lifshitz equation and general enough to be utilized for the magnetic system of any dimensionality, magnetic structure, shape, and size. As an example we explore the influence of the competition between exchange and dipolar interactions on the spectrum of normal modes as well as on the stability of the vortex state in two-dimensional nanorings. We show the lowest-frequency mode to be indicative of the dipolar-to-exchange iterations ratio. We also study behavior of the fundamental mode and present the influence of both, the discreteness of the lattice and the dipolar-to-exchange iterations ratio, on its hybridization with azimuthal modes. We complete the paper with a selective review of the spin wave excitations in circular dots to compare with the results obtained for the rings. (C) 2015 AIP Publishing LLC.