Curvature-Induced Asymmetric Spin-Wave Dispersion

In magnonics, spin waves are conceived of as electron-charge-free information carriers. Their wave behavior has established them as the key elements to achieve low power consumption, fast operative rates, and good packaging in magnon-based computational technologies. Hence, knowing alternative ways that reveal certain properties of their undulatory motion is an important task. Here, we show using micromagnetic simulations and analytical calculations that spin-wave propagation in ferromagnetic nanotubes is fundamentally different than in thin films. The dispersion relation is asymmetric regarding the sign of the wave vector. It is a purely curvature-induced effect and its fundamental origin is identified to be the classical dipole-dipole interaction. The analytical expression of the dispersion relation has the same mathematical form as in thin films with the Dzyalonshiinsky-Moriya interaction. Therefore, this curvature-induced effect can be seen as a “dipole-induced Dzyalonshiinsky-Moriya-like” effect.

Figure 1

(a) Schematic illustration of a nanotube in a vortex state and the cylindrical coordinate system. SWs are excited in the middle with a radial rf field, as illustrated by the orange ring. The SWs travel toward the ends of the nanotube with a wave vector $k_{\mathit{z}}$ perpendicular to the magnetization. $+k_{\mathit{z}}$ and $-k_{\mathit{z}}$ indicate the right and left propagation directions, respectively. (b) A snapshot in time of the SW profiles (radial component of the magnetization color coded) for the three different excitation frequencies 8, 10 and 20 GHz for a circular field of 80 mT. The orange bar indicates the position and width of the rf field. ${\lambda }_L$(${\lambda }_R$) denotes the wavelength of the waves traveling to the left (right).

Figure 2

SW dispersion relation obtained by micromagnetic simulations (red and blue dots) and analytical calculations (solid line) for circular fields of 80 mT (a) and 1 T (b). The blue squares mark the frequencies for which the SW profile is shown in Fig. 1. A nearly perfect agreement between the results of the micromagnetic simulations and the analytical calculations is found.

Figure 3

The dispersion of SWs is summarized for several circular fields as a function of wave number for nanotubes with (a) 30 nm and (b) 150 nm outer radius and a 10 nm film thickness. The minima of the dispersion are shifted towards larger $k_z$ values with increasing circular field for both diameters. The open dots represent the minima for each circular field and the solid line connecting them is a guide to the eye only.

Figure 4

(a) The volume charge amplitude as a function of wave vector. (b) SW asymmetry as a function of wave vector $k_{\mathit{z}}$ for nanotubes with varying radius. (c) The wavelength ${\lambda }_{\mathrm{SW}}$ of the excited SWs for which the maximum asymmetry is reached versus the nanotube radius. (d) SW profile for waves with equal wavelength but opposite travel direction and the corresponding volume charges. The color scheme encodes the radial component of the dynamic magnetization. The dark yellow rectangles mark the excitation region.