Microscopic dipole-exchange theory for magnonic crystal arrays of interacting ferromagnetic nanorings

A microscopic theory is employed to study magnonic crystals represented by one-dimensional arrays of ferromagnetic nanorings. The Hamiltonian-based formalism includes the exchange and dipole-dipole coupling within each ring, as well as the long-range dipolar coupling between rings and an external applied magnetic field. Depending on the applied field, the size of the rings, and the inter-ring spacing, the array elements may be in a vortex (flux closure), onion (bidomain), or other inhomogeneous magnetization state. Numerical results are reported for the magnonic bands and gaps in Permalloy nanorings with outer diameter typically of order 100 nm. In a strong-coupling regime it is found typically that the bands are wide and the gaps small in the vortex state, and vice-versa in the onion state.