Spin Wave Modes In Out-Of-Plane Magnetized Nanorings

We investigated the spin wave modes in flat circular permalloy rings with a canted external bias field using ferromagnetic resonance spectroscopy. The external magnetic field H was large enough to saturate the samples. For theta = 0 degrees (perpendicular geometry), three distinct resonance peaks were observed experimentally. In the case of the cylindrical symmetry violation due to H inclination from normal to the ring plane (the angle. of H inclination was varied in the 0 degrees-6 degrees range), the splitting of all initial peaks appeared. The distance between neighbor split peaks increased with the. increment. Unexpectedly, the biggest splitting was observed for the mode with the smallest radial wave vector. This special feature of splitting behavior is determined by the topology of the ring shape. Developed analytical theory revealed that in perpendicular geometry, each observed peak is a combination of signals from the set of radially quantized spin wave excitation with almost the same radial wave vectors, radial profiles, and frequencies, but with different azimuthal dependencies. This degeneracy is a consequence of circular symmetry of the system and can be removed by H inclination from the normal. Our findings were further supported by micromagnetic simulations.