  We develop an analytical approach for studying the FMR frequency shift due to
dipolar interactions and surface effects in two-dimensional arrays of
nanomagnets with (effective) uniaxial anisotropy along the magnetic field. For
this we build a general formalism on the basis of perturbation theory that
applies to dilute assemblies but which goes beyond the point-dipole
approximation as 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 reached an agreement that improves as
the size of the array increases. We find that the red-shift of the
ferromagnetic resonance due to dipolar interactions decreases for smaller
arrays. Surface effects may induce either a blue-shift or a red-shift of the
FMR frequency, depending on the crystal and magnetic properties of the
nano-elements themselves. In particular, some configurations of the
nano-elements assemblies may lead to a full compensation between surface
effects and dipole interactions.
