Band Diagram of Spin Waves in a Two-Dimensional Magnonic Crystal

Abstract

The dispersion curves of collective spin-wave excitations in a magnonic crystal consisting of a square array of interacting saturated nanodisks have been measured by Brillouin light scattering along the four principal directions of the first Brillouin zone. The experimental data are successfully compared to calculations of the band diagram and of the Brillouin light scattering cross section, performed through the dynamical matrix method extended to include the dipolar interaction between the disks. We found that the fourfold symmetry of the geometrical lattice is reduced by the application of the external field and therefore equivalent directions of the first Brillouin zone are characterized by different dispersion relations of collective spin waves. The dispersion relations are explained through the introduction of a bidimensional effective wave vector that characterizes each mode in this magnonic metamaterial.

Reconfigurable wave band structure of an artificial square ice

“…The term magnonics has been coined to describe this field of study [5,6]. One pathway to control magnon dispersions is to construct magnonic crystals [7,8] that are metamaterials with a spatial modulation of the magnetic properties on length scales comparable to relevant magnonic wavelengths [9][10][11]. Patterned thin magnetic films [12,13] or topographically modulated thin films have been used to manipulate the magnon spectra [14].…”

Asymmetry of spin wave dispersions in a hexagonal magnonic crystal

“…To interpret the dispersion of different modes, Tacchi et al 18 introduced a 2-D effective wave vector k eff , which includes and replaces both the band index and the Bloch wave vector. The effective wave vector represents the overall oscillation of the magnetization across the array, because it takes into account both the oscillation within individual disks due to the mode character (i.e., due to the number and orientation of nodal lines) and the variation between adjacent dots due to the Bloch factor, e ikÁr .…”

“…Conversely, k DE increases for the fundamental mode and for 1-DE (as well as for any n-DE with odd n) but decreases for any n-DE with even n. 18 Due to the misalignments of neighbor disks along y direction, the effective wave vector variation for m-BA modes depends also on m. We checked that if p is a nonzero integer, m-BA modes with m ¼ 4p-3 and m ¼ 4p-2 have a negative frequency dispersion, instead with m ¼ 4p-1 and m ¼ 4p have a positive frequency dispersion; of course, the bandwidth amplitude decreases as m increases.…”

“…[1][2][3][4][5][6][7][8] This tunability makes MCs promising candidates for creation of versatile devices such as adjustable filters and waveguides operating in the microwave frequency range. 3,5,9,10 Dense arrays of in-plane magnetized nano-disks have been extensively investigated by ferromagnetic resonance, [11][12][13][14][15] Brillouin light scattering (BLS), [16][17][18][19][20][21][22][23] and time resolved magneto-optical measurements. [24][25][26][27][28] However, arrays of disks arranged into a hexagonal mesh have been sparsely studied so far, and the experimental data have been presented for wave vector k ¼ 0 only.…”

“…A comparison of the calculated and measured mode frequencies and cross-sections was used to identify the modes, which were then labeled according to the scheme described in Ref. 18. The dynamic magnetization of each mode (Bloch wave) can be interpreted using the expression dmðrÞ ¼ dm k ðrÞe ikÁr , where r is the radius-vector in the direct space, k is the Bloch wave vector, and dm k is the cell function, which has the periodicity of the array.…”