Field evolution of the magnetic normal modes in elongated permalloy nanometric rings.

The eigenmode spectrum of elongated permalloy rings with relatively wide arms is investigated by combined Brillouin light scattering and ferromagnetic resonance measurements as a function of the applied field intensity, encompassing both vortex and onion ground states. To reproduce the frequencies and the spatial profiles of the measured modes we performed micromagnetic simulations which solve the discretized Landau-Lifshitz-Gilbert equation in the time domain and calculate locally the Fourier transform. This allowed us to correlate the field dependence of different modes to their localization inside different portions of the rings. With the rings in the vortex ground state, in addition to radial, fundamental, and azimuthal modes, a localized mode, existing in the element portions where the internal field assumes its minima, has been measured and identified. This latter mode, whose frequency decreases for increasing field intensity, becomes soft near the transition from vortex to onion state and determines the change in symmetry of the magnetic ground state. After the transition, it is replaced by two edge modes, localized on the internal and external boundary of the rings, respectively.