Internal spin-wave confinement in magnetic nanowires due to zig-zag shaped magnetization

We perform broadband spin-wave spectroscopy on thin submicrometer-wide ${\text{Ni}}_{80}{\text{Fe}}_{20}$ magnetic wires. Intentionally, we apply the in-plane magnetic field under an angle that is a few degrees off with respect to the in-plane hard-axis direction. In an intermediate field regime we find dipole-exchange modes that, as substantiated by micromagnetic simulations, reflect spin waves in intrinsically formed nanometer-wide channels along the wire. Interestingly, this phenomenon is not ruled by the outer boundary conditions at the geometrical edges but by a deterministic zig-zag shaped magnetization configuration that varies on the nanometer scale in the inner part of the wire. In our case the individual channel width is only 65 nm, about a factor of five smaller than the geometrical width of 300 nm. Internal spin-wave guiding becomes possible, resembling the graded-index approach for optical wave guiding in fibers. This opens new perspectives for spin-wave propagation in magnonic waveguides.

Figure 1

(Color online) (a) Sketch of the experiment: Au leads form ground (g) and signal (s) lines on top of the Permalloy nanowires. Parameters are defined in the figure. (b) Transmission spectrum $T\left(f\right)$ obtained on an array with 185 Permalloy wires with $w=300\text{nm}$ at $\alpha =12°$ and ${\mu }_{0}H=-46\text{mT}$. This spectrum belongs to regime II in the experiment since ${\mu }_0{H}_{b,\text{exp}}=-44\text{mT}$ at $\alpha =12°$. Color-coded spectra for (c) $\alpha =5°$, (d) $7°$, and (e) $12°$ (bright means strong absorption). Labels are discussed in the text.

Figure 2

(Top) Magnetization $\vec{M}\left(x,y\right)$ in two different fields $H$ with $\alpha =12°$. The wire was initially saturated along the $y$ direction: (a) ${\mu }_{0}H=-44\text{mT}$ and (b) $-48\text{mT}$. Small arrows indicate local magnetic moments. (Bottom) Angle $\theta \left(x\right)$, i.e., the orientation of spins relative to the $x$ axis across the wire. The broken line is at $\left(180°+12°\right)$, i.e., the orientation of $\vec{H}$.

Figure 3

[(a)–(d)] Spatially resolved internal (solid dots), demagnetization (open squares), and exchange fields (open triangles) at $\alpha =12°$ for $H$. In (b) $H_{\text{int}}$ reenters large negative values in localized regions of the wire. They promote spin-wave confinement in $x$ direction that is, in particular, offset from the edges. Propagation in $y$ direction is possible.

Figure 4

(Color online) (a) Simulated magnetic-field dispersion for a single 300-nm-wide wire with $H$ tilted by $\alpha =12°$ from the $x$ direction. (b) Spin-precession intensity across the nanowire for modes A1–A3 as a function of $x$ (vertical axis) at the same fields $H$ shown in (a) (white represents a large value). The spin-wave confinement in regime II, originating from the zig-zag shaped $\vec{M}\left(x,y\right)$, is clearly seen. For each $H$, the maximum amplitude is normalized to one for illustration purposes. Bottom: Fourier-transform images (linescans) illustrating the spin-precession intensity along $x$ direction (horizontal axis) at specific fields. Here, we also show higher order modes. The labels refer to the different modes discussed in the text.