Spin-Wave Spectral Analysis in Crescent-Shaped Ferromagnetic Nanorods

The research on the properties of spin waves (SWs) in three-dimensional nanosystems is an innovative idea in the field of magnonics. Mastering and understanding the nature of magnetization dynamics and binding of SWs at surfaces, edges, and in-volume parts of three-dimensional magnetic systems enables the discovery of alternative phenomena and suggests other possibilities for their use in magnonic and spintronic devices. In this work, we use numerical methods to study the effect of geometry and external magnetic field manipulations on the localization and dynamics of SWs in crescent-shaped (CS) waveguides. It is shown that changing the magnetic field direction in these waveguides breaks the symmetry and affects the localization of eigenmodes with respect to the static demagnetizing field. This, in turn, has a direct effect on their frequency. Furthermore, CS structures are found to be characterized by significant saturation at certain field orientations, resulting in a cylindrical magnetization distribution. Thus, we present chirality-based nonreciprocal dispersion relations for high-frequency SWs, which can be controlled by the field direction (shape symmetry) and its amplitude (saturation).

Figure 1

Model of the nanorod with a CS section. Spherical coordinates for the external magnetic field ${\mathbf{H}}_{\mathrm{ext}}$ and the main dimensions are marked.

Figure 2

Eigenanalysis of the CS nanowire as a function of the changing direction of the external magnetic field that saturates the magnetization, ${\mu }_0{H}_{\mathrm{ext}}=3$ T. Plot (a) shows the dependence of the eigenfrequencies on the azimuthal angle $\varphi$ in the range $0$–${90}^{\circ }$ for the polar angle $\theta ={90}^{\circ }$. The colors correspond to the intensities of the individual eigenmodes according to Eq. (6). In (b) we can see the distribution of the magnetization precession intensity, while in (c) the static demagnetizing fields, Eq. (4), are shown for the selected $\varphi$ values. The number insets in the plots refer to the eigenmode visualizations.

Figure 3

Edge-mode amplitude distribution for the first two eigenfrequencies and external magnetic field polar angle equal to (a) ${90}^{\circ }$ and (b) ${45}^{\circ }$. In both cases the azimuthal angle remains $0^{\circ }$, and the colors represent the sum of the dynamic components of the magnetization.

Figure 4

Frequencies of the SWs in the CS nanowire as a function of the polar angle $\theta$ for a constant azimuthal angle $\varphi ={90}^{\circ }$ of the 3-T external magnetic field (auxiliary diagram in the lower-left corner). The dark color of the dots represents the resulting intensity of the eigenmodes calculated according to Eq. (6). The images of the cross sections show the distribution of the magnetization precession intensity for successively marked values of $\theta$.

Figure 5

Scheme of the analyzed shape transformation—from ellipse to the crescent.

Figure 6

Eigenanalysis of a variably shaped nanowire going from a full ellipse to the CS cross section (according to Fig. 5). In (a), the magnetization precession intensity distributions for the lowest eigenfrequency branch are shown for the five selected steps of overlapping distance $|L|=350$, 270, 240, 90, and 0 nm. In (b) the distributions of the static demagnetizing field [see Eq. (4)] are presented.

Figure 7

Comparison of the localization of the SW modes and their frequencies (a),(c) depending on the different edge sharpness and comparison with the corresponding demagnetizing fields (b),(d). The color distributions for the modes represent the magnetization precession intensity from Eq. (6), while the demagnetizing field is from Eq. (4).

Figure 8

Comparison of the CS nanowire’s eigenanalysis results from comsol (gray dots) and time-domain simulation results from Mumax3 (color scale) as a function of the azimuthal angle $\varphi$ for the polar angle $\theta ={90}^{\circ }$ at external magnetic field 3 T. On the right, the static magnetization plots of the CS cross section from Mumax3 for three magnetic field configurations are shown. The color map visualizes the angle deviation of the static magnetization vector (arrows) from the direction of the external magnetic field.

Figure 9

Dispersion relations of CS nanorods for (a),(c) 1-T and (b),(d) 3-T external magnetic field directed along the $x$ axis (a),(b) and the $y$ axis (c),(d) for 1 T and (d) for 3 T. The wave vector is directed along the $z$ axis. Nonreciprocity is shown in insets as the frequency differences $\delta f$ between the most intense branches for positive (yellow dots) and negative (red dots) wave vectors. The plots also include visualizations of volumetric and edge modes (where they occur) for $k_z=0$.