Unusual nature of confined modes in a chiral system: Directional transport in standing waves

Nonreciprocity of spin-wave propagation is a well-known consequence of antisymmetric exchange contributions in magnetic spin systems that lack inversion symmetry. In this case, the energy of a state depends on the sign of its momentum as $ħ\omega \left(k\right)\ne ħ\omega \left(-k\right)$. We discuss here the consequences of this nonreciprocity on counterpropagating traveling spin-wave states. In a confined geometry we find states with well-defined nodes which are inherently phase modulated such that space-inversion symmetry of the mode profile is lost. This entails that additional spectral features become visible in ferromagnetic resonance studies of microelements with Dzyaloshinskii-Moriya interaction (DMI), allowing a quantification of the amplitude and direction of the DMI. Moreover, this interference between nonreciprocal modes forms the basis for a generalized concept of mode confinement.

Figure 1

(a) Schematic of nonreciprocal dispersion relation when DMI is present. (b) The direction of propagation leads to different wavelengths. (c) Profiles of the spin excitations, calculated with the one-dimensional model, at different times for the mode at 2.58 GHz. We see a node position that is constant, but the profile of the mode slides to the left with time.

Figure 2

One-dimensional Heisenberg spin chain simulation results for excitations caused by a small uniform canting of the spins from equilibrium. Comparison of frequencies calculated (a) without DMI (top) and (b) with DMI (bottom). The numbers indicate the mode frequencies in GHz. Three modes are visible with DMI while the original mode has decreased in spectral amplitude and shifted from 2.89 to 2.33 GHz. (c) (Right panel) Spatial amplitude profiles for the modes at the frequencies seen in (b).

Figure 3

Results from micromagnetic simulations of spin waves confined in an elliptical disk. The FMR spectra without DMI (left) and with DMI (right) are shown at the top. They are represented by Fourier transformed of the time dependent z-component of the magnetization. The mode profiles, along the short (y-) axis of the ellipse, which correspond to the highest amplitudes in the spectra (each marked with a red vertical line) are shown in the lower section together with the phase angle. The outline of the mode profiles is indicated by dashed lines. Time snapshots of the wave are shown as green and blue curves. The green curve is plotted a quarter period later then the blue curve in the time evolution.

Figure 4

Micromagnetic calculations showing how mode intensities and profiles depend upon interfacial DMI for an elliptical ferromagnet. The upper plot summarizes the frequencies and intensities of modes as a function of the DMI strength. Shown in the lower left are mode spectra calculated by taking the Fourier transform in time after excitation by a shaped pulse. The pulse is shaped in time such that a broad spectral regime is excited with uniform spectral amplitude (i.e. its Fourier transformed approximates a step-function) and is uniform in space. The corresponding mode intensity profiles are shown on the right and identified by labels A0, etc. As an example, one can see that for $D<1\mathrm{mJ}/{\mathrm{m}}^2$, the shift in frequency is little and additional modes faintly appear. For values $D>1\mathrm{mJ}/{\mathrm{m}}^2$ the shift of the eigenfrequencies has become much stronger and appears to assume linear behaviour. The spectral amplitudes have also increased and the chiral envelope that selects which modes are excited is very broad and appears to broaden linearly. These features can be detected in an experiment as a clear fingerprint of DMI.