Curvature-Induced Asymmetry of Spin-Wave Dispersion in Nanotubes

Due the their stable magnetisation states and small sizes magnetic nanotubes are perfect candidates for magnonic waveguides. Such novel structures can nowadays be very well produced [1,2], motivated by the broad range of applications for magnetoresistive devices, optical metamaterials, cell-DNA separators, and drug delivery vectors [3,4].
The high stability of their equilibrium state [5,6] against external perturbations and their robust domain walls propagating with velocities faster than the SW phase velocity [7] promote MNTs as appealing candidates for racetrack memory devices [8,9] and information processing [7].
We show using micromagnetic simulations and analytical calculations that spin-wave propagation in ferromagnetic nanotubes is fundamentally different than in flat thin films. The dispersion relation is asymmetric regarding the sign of the wave vector.
As shown in figures 1 and 2, the spin-wave dispersion is asymmetric for both the zeroth and first order azimuthal modes. This is a purely curvature induced effect and its origin is identified to be the classical dipole-dipole interaction.
In certain cases the Damon-Eshbach modes in nanotubes behave as the volume-charge-free backward volume modes in flat thin films. Such non-reciprocal spin-wave propagation [10] is known for flat thin films with Dzyalonshiinsky-Moriya interaction (DMI), an antisymmetric exchange due to spin-orbit coupling. The analytical expression of the dispersion relation has the same mathematical form as in flat thin films with DMI. The influence of curvature on spin waves is thus equivalent to an effective dipole-induced Dzyalonshiinsky-Moriya interaction [11].
We also derive the dispersion relation for the limiting cases k=0 and k much larger than 1/R, where k is the wave vector of the spin wave and R the nanotube radius. For the first case, the mathematical formula of the dispersion relation resembles the well-known Kittel formula for the ferromagnetic resonance of a thin film with the in-plane magnetization parallel to the applied field, and both oriented perpendicularly to the in-plane easy axis of the shape anisotropy field. In the latter case the expression is identical to the exchange-dominated dispersion relation of a planar thin film in the Damon-Esbach configuration with the in-plane magnetization oriented perpendicularly to the in-plane easy axis.