Theory of collective spin waves and microwave response of ferromagnetic nanowire arrays

We formulate the theory of the collective spin wave modes and the microwave response of arrays of ferromagnetic nanowires of cylindrical cross section, each magnetized parallel to the axis of symmetry. The theory is based on a multiple scattering approach, and may be applied to the regime where the diameter of the constituents is small enough that exchange enters the response importantly. The formalism can be applied to ordered arrays, or disordered arrays of nanowires. We present explicit results for the spin wave normal mode frequencies of a pair of nanowires, and for a linear array. The dispersion curves for the collective modes in the latter case show a dispersive collective mode which crosses and hybridizes with resonances in the individual cylinders. For a nanowire pair, and in the magnetostatic limit, we obtain a closed form expression for the collective spin wave modes in the limit as the wavevector parallel to the symmetry axis vanishes. Interestingly, the resonance does not split as the wires are brought together, but just shifts downward in frequency, from the value $\gamma {(H}_0+2\mathrm{}\pi M_s)$ appropriate to the isolated wire at large separations, to $\gamma [H_0{(H}_0+4\mathrm{}\pi M_s){]}^{1/2}$ when the wires just touch.