Dispersion of collective magnonic modes in stacks of nanoscale magnetic elements

We report a numerical study of the dispersion of collective magnonic modes in magnonic crystals formed by stacks of magnetostatically coupled magnetic nanoelements. The calculations reveal that the sign of the magnonic dispersion is determined by the spatial character and ellipticity of precession for the eigenmodes of the isolated elements that give rise to the magnonic bands. We identify a critical value of the ellipticity at which the dispersion of the collective magnonic modes changes sign. The critical value is independent of the magnetic parameters and shape of the elements but is a characteristic of their arrangement (superstructure).

Figure 1

The geometry of the problem is shown schematically. The inset shows the coordinate system used in the calculations, together with the orientations of the bias and excitation magnetic fields.

Figure 2

The frequencies of the modes of the isolated elements of different sizes are shown as a function of their ellipticity. In the inset, the mode spectrum of the isolated 100 × 50 × 10 nm${}^3$ element is shown. For each mode, the top and bottom images represent the spatial distributions of the DFT amplitude and phase, respectively, while the numbers are the frequencies in GHz.

Figure 3

The magnonic dispersion is shown for stacks of 240 (left-hand side) and 60 (right-hand side) elements with dimensions of 100 × 50 × 10 nm${}^3$. The insets show the spatial profiles of the modes of the isolated element, while the horizontal dashed lines represent their frequencies. The arrows point to the magnonic bands formed from the modes of the isolated element. The solid vertical short dashed lines show the boundaries of the Brillouin zone.

Figure 4

The widths of the collective magnonic bands are shown as a function of the ellipticity of the corresponding modes of the isolated elements. The inset shows the same dependence, but for the edge, bulk, and high-order bulk modes, only.