Symmetry and curvature effects on spin waves in vortex-state hexagonal nanotubes

Analytic and numerical studies on curved magnetic nano-objects predict numerous exciting effects that can be referred to as magneto-chiral effects, which do not originate from intrinsic Dzyaloshinskii-Moriya interaction or interface-induced anisotropies. In contrast, these chiral effects stem from isotropic exchange or dipole-dipole interaction, present in all magnetic materials, which acquire asymmetric contributions in case of curved geometry of the specimen. As a result, for example, the spin-wave dispersion in round magnetic nanotubes becomes asymmetric; namely, spin waves of the same frequency propagating in opposite directions along the nanotube exhibit different wavelenghts. Here, using time-resolved scanning transmission x-ray microscopy experiments, standard micromagnetic simulations, and a dynamic-matrix approach, we show that the spin-wave spectrum undergoes additional drastic changes when transitioning from a continuous to a discrete rotational symmetry, i.e., from round to hexagonal nanotubes, which are much easier to fabricate. The polygonal shape introduces localization of the modes to both the sharp, highly curved corners and flat edges. Moreover, due to the discrete rotational symmetry, the degenerate nature of the modes with azimuthal wave vectors known from round tubes is partly lifted, resulting in singlet and duplet modes. For comparison with our experiments, we calculate the microwave absorption from the numerically obtained mode profiles, which shows that a dedicated antenna design is paramount for magnonic applications in 3D nanostructures. To our knowledge these are the first experiments directly showing real space spin-wave propagation in 3D nano-objects.

Figure 1

(a) Scanning-electron-transmission-microscopy image of a hexagonal permalloy nanotube with $250\mathrm{n}\mathrm{m}$ outer diameter, $30\mathrm{n}\mathrm{m}$ thickness, and $12\mu \mathrm{m}$ length on a GaAs wire. The gold stripline antenna (here colored for visual purposes) was patterned on a SiN membrane, and the nanotube was placed on the top using a focused ion beam (FIB) tool and a micromanipulator. The Oersted field of an rf current is used to excite SWs. (b) Cross-sectional sketch of the hexagonal nanotube showing the layer structure. The permalloy layer is directly evaporated on the GaAs wire in situ and capped with Al to avoid oxidation. A relative angle of ${60}^{\circ }$ was used between the x-ray beam and the symmetry axis of the nanotube. This configuration allows for being sensitive to the in-plane dynamic magnetization of the top and bottom surfaces of the tube, which is expected to be larger than the out-of-plane dynamic magnetization component.

Figure 2

Numerical (red-blue) and experimental (gray scale) snapshots of the dynamic magnetization at fixed times for modes propagating in the corners of the hexagonal tube (a) at $5.571\mathrm{G}\mathrm{Hz}$ and (b) at $4.571\mathrm{G}\mathrm{Hz}$, and for modes mostly propagating in the top and bottom facets (c) at $8.571\mathrm{G}\mathrm{Hz}$ and (d) at $9.571\mathrm{G}\mathrm{Hz}$. For the numerical profiles, a side view is shown next to the projection of the upper half the nanotube. For better illustration, they have been stretched in the width direction. Below we show all frames of the respective STXM movies. In addition to each last frame, we show an average line scan along the tube, together with sinusoidal fits used to obtain the wave lengths. The position of the antenna is in all cases marked with a translucent gold-colored patch. The dashed lines in each STXM frame mark approximately the contour of the hexagonal tube.

Figure 3

Dispersion relation and modes profiles directly resulting from the dynamic-matrix approach for propagating spin waves. (a) Full dispersion for all modes below $12\mathrm{G}\mathrm{Hz}$. The modes can be categorized based on their localization position as corner c, facet f, and corner facet cf modes, shown in (b). A second categorization can be made based on their singlet and duplet nature by looking at the phase of the wave along the azimuthal direction in (c); singlets are standing spin-wave solutions, while duplets are two degenerate solutions of counterpropagating waves along the azimuthal direction. In panels (d) and (e) the mode magnitude as well as its phase is shown for several examples of the singlet and duplet solutions. Below the modes, we annotate the corresponding irreducible representations (irreps) to which they belong.

Figure 4

Generators of the symmetry group 6/$m^{\prime }mm$ shown for the hexagonal tube containing a vortex state, where 6 is the sif-fold rotational symmetry, $m$ indicate mirror planes and ' is the time reversal operation.

Figure 5

Predicted microwave absorption calculated using the mode profiles from the dynamic-matrix approach for two different antenna geometries. (a) The zeroth-order peak of the excitation efficiency at the surface of the antennas is shown on the top of the full dispersion relation including all modes up to $16\mathrm{G}\mathrm{Hz}$. The dispersion relation obtained when exciting spin-wave modes with a current loop antenna is shown in (b). Only singlets belonging to the fully symmetric $A_1$ representation are susceptible to the spatial distribution of such a field. The dispersion relation for a stripline antenna used in the experiments is summarized in panel (c). The usual asymmetry of the excitation efficiency known for Damon-Eshbach modes is recovered. This antenna will excite simultaneously a multiple of modes with the same frequency. The white dashed lines indicate the excitation frequencies for which the experimental spatial spin-wave profiles are shown in Fig. 2.