Spin waves in deterministic fractals

We calculate spin wave spectra of low-dimensional self-similar nanostructures, namely deterministic Sierpinski carpets, in the framework of short ranged ferromagnetic exchange. The integrated density of states of magnetic excitations are shown to exhibit singularities resulting in devil's staircase spectra which remain at all iteration steps; the spectra are found to be singular continuous functions of the frequency with numerous gaps and plateaux linked to symmetry and degeneracy of the associated eigen-modes. These spin wave spectra are shown to be sensitive not only to the fractal dimension, but to additional connectivity properties of the structures as already pointed out in the study of critical properties of fractals. This result is closely linked to the fractal subdimensions arising from the set of eigenvalues of the connectivity matrix describing the construction of the fractal structure; our set of results strongly suggests that the link between the integrated density of states and fractal subdimensions is a much more general feature of deterministic fractals; connectivity marks mode localization. Lastly, magnetic fractals are shown to be able to filter radiofrequencies.