Magnetostatic modes in ferromagnetic nanowires. II. A method for cross sections with very large aspect ratio

We present a method which allows one to calculate the frequencies of the magnetostatic spin waves and their eigenvectors, for ferromagnetic nanowires of arbitrary cross section. The approach is based on application of the extinction theorem, and has the virtue that one need not resort to expansions in special basis sets built around a specific geometry of interest. Instead we develop integral equations in the form of contour integrals around the periphery of the wire which allows us to obtain both eigenfrequencies and eigenvectors. Our earlier formulation of the problem allowed computation of the dispersion relation of each magnetostatic mode, i.e., their frequency as a function of the wave vector parallel to the wire. It is a numerical challenge to apply this method to rectangular wires whose cross section has a very large aspect ratio, though it works splendidly when the aspect ratio is modest. In the present paper, we focus our attention on modes of zero wave vector, and through use of conformal mapping cast the extinction theorem description of the spin wave modes into a form that allows exact calculations for rectangular wires whose cross sections have extremely large aspect ratios. Comparison of our exact calculations with approximation schemes found in the literature allow us to assess the accuracy of such schemes. We also compare our calculated frequencies to Brillouin light scattering data on ferromagnetic ribbons of very high aspect ratio.