Effect of Nonuniform Magnetization on the Spin-Wave Spectrum in Thin Ferromagnetic Films

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Abstract
It was first suggested by Portis that the linear positioning of the first few spin-wave modes observed in some ferromagnetic films could be due to a nonuniform static magnetization M and he examined a simple model which showed this effect. This model, where the magnetization falls parabolically from the film center, has been studied in greater detail. The modes found by Portis are incorrect in detail, since they all collapse onto the uniform mode as the distortion of M goes to zero. A more suitable choice of boundary conditions gives rise to a spectrum where the lower modes are linearly positioned whereas the higher modes tend to the more usual quadratic positioning. For a given relative distortion, the number of linear modes should be greater the thicker the film. This is in agreement with the measurements of Kooi, Wigen, and Nisenoff on Permalloy films, though the transition between linear and quadratic positioning observed by Nisenoff is much more abrupt than this model predicts.