  Theoretical 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 the intrinsic Dzyaloshinskii-Moriya interaction or
surface-induced anisotropies. The origin of these chiral effects is the
isotropic exchange or the dipole-dipole interaction present in all magnetic
materials but renormalized by the curvature. Here, we demonstrate
experimentally that curvature induced effects originating from the
dipole-dipole interaction are directly observable by measuring spin-wave
propagation in magnetic nanotubes with hexagonal cross section using time
resolved scanning transmission X-ray microscopy. We show that the dispersion
relation is asymmetric upon reversal of the wave vector when the propagation
direction is perpendicular to the static magnetization. Therefore
counter-propagating spin waves of the same frequency exhibit different
wavelenghts. Hexagonal nanotubes have a complex dispersion, resulting from
spin-wave modes localised to the flat facets or to the extremely curved regions
between the facets. The dispersion relations obtained experimentally and from
micromagnetic simulations are in good agreement. %The asymmetric spin-wave
transport is present for all modes, promoting hexagonal nanotubes for magnonic
applications. These results show that spin-wave transport is possible in 3D,
and that the dipole-dipole induced magneto-chiral effects are significant.

Mesoscale and Nanoscale Physics;Materials Science