Red-shift of the ferromagnetic resonance of an interacting $2D$ array of nanomagnets

Through analytical developments we study the FMR frequency shift due to dipolar interactions in two-dimensional arrays of nanomagnets with effective uniaxial anisotropy along the magnetic field. For this we develop a general formalism on the basis of perturbation theory that applies to dilute assemblies but which goes beyond the point-dipole approximation. Indeed, it takes account of the size and shape of the nano-elements, in addition to their separation and spatial arrangement. The contribution to the frequency shift due to the shape and size of the nano-elements has been obtained in terms of their aspect ratio, their separation and the lattice geometry. We have also varied the size of the array itself and compared the results with a semi-analytical model and have seen that the agreement improves as the size of the array increases. On the other hand, it turns out that the red-shift of the ferromagnetic resonance decreases for smaller arrays.