Nonreciprocal spin-wave channeling along textures driven by the Dzyaloshinskii-Moriya interaction

Ultrathin metallic ferromagnets on substrates with strong spin-orbit coupling can exhibit induced chiral interactions of the Dzyaloshinskii-Moriya (DM) form. For systems with perpendicular anisotropy, the presence of DM interactions has important consequences for current-driven domain-wall motion and underpins possible spintronic applications involving skyrmions. We show theoretically how spin textures driven by the DM interaction allow nonreciprocal channeling of spin waves, leading to measurable features in magnetic wires, dots, and domain walls. Our results provide methods for detecting induced DM interactions in metallic multilayers and controlling spin-wave propagation in ultrathin nanostructures.

Figure 1

(a) The transverse magnetization component $m_y$ at the boundary edges (located at $y=\pm 256$ nm) of a 512-nm-wide rectangular wire. (b) Illustration of the magnetization tilts for $D>0$, with the yellow shaded regions representing the tilts shown in panels (a) and (c). The partial wall (blue curve) is shown schematically, with $y_c$ denoting the wall center and $w$ the wire width. (c) The perpendicular component $m_z$ at the boundary edges, where the solid lines correspond to fits to a partial Néel wall profile. (d) Partial wall center $y_c$ as a function of $D$.

Figure 2

Néel wall eigenfrequencies calculated using perturbation theory for weak DMI. (a), (b) The DMI splits frequencies into two sheets that are otherwise degenerate and distorts the sheets such that propagation is nonreciprocal with respect to $k_x[\omega \left(k_x\right)\ne \omega (-k_x)]$. Dispersion relations for states propagating along the domain wall ($k_y=0$) for (c) $D=1.5$ mJ/m${}^2$ and (d) $D=-1.5$ mJ/m${}^2$.

Figure 3

Nonreciprocal propagation in a thin rectangular wire. Spatial profiles of $m_x$ resulting from a rf field excitation, ${\mathbf{h}}_{\mathrm{rf}}\left(t\right)=h_{0}sin\left(2\pi f_{\mathrm{rf}}t\right)\widehat{\mathbf{x}}$, where ${\mu }_0{h}_0=5$ mT, at (a) $f_{\mathrm{rf}}=50$ GHz and (b) $f_{\mathrm{rf}}=16$ GHz. The different wave-vector components considered are illustrated. In panel (a), $f_{\mathrm{rf}}$ is in the spin-wave band and nonreciprocal propagation occurs for $k_{\mathrm{top}}$ and $k_{\mathrm{bot}}$, while $k_{\mathrm{cen}}$ propagation is symmetric. In panel (b), $f_{\mathrm{rf}}$ is in the gap of the bulk modes and only edge modes are excited. (c) Dispersion relations computed from simulations for $D_{\mathrm{ex}}=4.5$ mJ/m${}^2$, with $f_{\mathrm{rf}}$ used in panels (a) and (b) indicated. Dots represent simulation results. The solid black curve (and gray shaded area) represents the theoretical dispersion relation for exchange modes. The solid red curve represents the fit given by Eq. (4). (d) Dispersion relation for $D_{\mathrm{ex}}=2.5$ mJ/m${}^2$.

Figure 4

Map of the eigenmode power spectral density (PSD) as a function of $D$ for (a) 100-nm-diameter circular dots and (b) 100-nm-wide square dots. Selected profiles of the four lowest modes for different strengths of the DMI for the (c) circular and (d) square dots.