Effective magnetic properties of arrays of interacting ferromagnetic wires exhibiting gyromagnetic anisotropy and retardation effects

We present a model for the effective magnetic properties of metamaterials composed of gyromagnetic inclusions in which electromagnetic retardation is accounted for. The formalism is used to derive the effective permeability tensor of axially magnetized ferromagnetic-metallic wires. The complex size-dependent gyromagnetic response of a single wire subjected to an external dynamic magnetic field is first obtained and expressed compactly in terms of renormalized permeability components. We then incorporate this response into an extended Maxwell-Garnett homogenization procedure to yield the effective permeability tensor of the array. An effective external susceptibility tensor, which includes dipolar interactions explicitly, is further introduced to describe the ferromagnetic resonance response of finite-size arrays placed in microwave resonant cavities. We examine the impact of electromagnetic retardation on the resonance, antiresonance, and linewidth and compare the magnetic response expected for conducting and insulating magnetic wires. The conditions under which arrays of ferromagnetic wires may exhibit a region of negative effective permeability are established and are found to be dependent on both the intrinsic magnetic and size-dependent eddy-current losses, thus providing useful guidelines for the design of magnetic metamaterials. The proposed formalism is sufficiently general to be applied or extended to various systems of interest.