Hippopede curves for modeling radial spin waves in an azimuthally graded magnonic landscape

We propose a mathematical model for describing radially propagating spin waves emitted from the core region in a magnetic patch with $n$ vertices in a magnetic vortex state. The azimuthal anisotropic propagation of surface spin waves (SSWs) into the domain, and confined spin waves (or Winter's magnons) in domain walls, increases the complexity of the magnonic landscape. In order to understand the spin wave propagation in these systems, we first use an approach based on geometrical curves called “hippopedes”; however, it provides no insight into the underlying physics. Analytical models rely on generalized expressions from the dispersion relation of SSWs with an arbitrary angle between magnetization M and wave number k. The derived algebraic expression for the azimuthal dispersion is found to be equivalent to that of the hippopede curves. The fitting curves from the model yield a spin wave wavelength for any given azimuthal direction, number of patch vertices, and excitation frequency, showing a connection with fundamental physics of exchange-dominated surface spin waves. Analytical results show good agreement with micromagnetic simulations and can be easily extrapolated to any $n$-corner patch geometry.

Figure 1

Dispersion relations for the laterally emitted spin wave from a first higher order dynamical core, see Ref. [12], and a Winter's magnon (WM), for the material parameters indicated in Sec. 2, as limiting cases. Dashed horizontal line indicates an excitation frequency of 8.8 GHz. Gray area highlights the expected magnitudes for intermediate wave vectors (see bottom-right inset) between the two limiting cases, when being simultaneously excited. Insets: Color arrows show the wave vectors for the laterally emitted spin wave (in a SSW configuration) and the Winter's magnon, which are assumed $k_{\text{WM}}\approx 2k_{\text{SSW}}$ for the excitation frequency of 8.8 GHz (length of the vectors represent their magnitudes). Black arrows show the orientation of magnetization in the domains. Bottom inset shows a simplified schematic of the angular distribution of wave vectors at the top-right corner of a square patch ($n=4$). Gray arrow is an intermediate case for the spin wave wave vector between the limiting cases, from which an angular dependence can be inferred.

Figure 2

(a) Schematic of various magnetic patches depending of the number of vertices $n$ with magnetization around the core (solid black line). Domain walls bisect the sharp corners. (b) Polar representation of Eq. (2) showing the azimuthal variation of wavelength within the shape, with the appropriate rotation of the patch (${\varphi }_0$) so the equation describes correctly the shape of the patch. Minimum and maximum radial amplitudes are assumed ${\lambda }_{\text{SSW}}=2{\lambda }_{\text{WM}}$ (length of the vectors represent their magnitudes). In terms of the hippopedes parameters $a$ and $b$, this means a ratio of $b/a=0.75$.

Figure 3

Collection of curves from Eq. (5) for $n=0,1,2$, and 4. Curves for $n=3$ and higher orders $(n>4)$ can be easily inferred. Physically nonrealizable zeros are placed at every $\pi /n$ angle.

Figure 4

Ratio (black solid curve) between wavelengths from the dispersion relation from [11] (${\lambda }_{\text{DE}}$) and the extended model linked to the hippopedes (${\lambda }_{\text{Hp}}$) for $p\approx 1.75$ showing good agreement, and practically no deviation from unity at any angle apart from the domain wall region for $p\approx 10.75\gg 1$ (red solid curve). Left inset: Normalized results from Eq. (7) adding azimuthal periodicity for a number of corners $n=0$ (blue), $n=1$ (red), $n=2$ (yellow), and $n=4$ (purple). Right inset: Comparison of the normalized results from Eq. (7) (solid purple curve) and Eq. (5) (dashed purple curve) for $n=4$ and $p\approx 1.75$.

Figure 5

(a) Left: Snapshots of numerical results for a double-teardrop shape ($n=2$) and a square ($n=4$). Red arrows indicate the direction of propagation of the two main modes (surface spin wave and Winter's magnon). Right: k space of the snapshots where the wave number profile (inverse of the wavelength profile) is shown. Results are interpolated to 5 extra points between data points for clarity. White arrows show the direction of propagation of the main modes. (b) Comparison of the maximum values of $\lambda =2\pi /k$ found in (a) for the double-teardrop shape (blue) and the square (orange) with the results from Eq. (6) where ${\lambda }_{\text{SSW}}=135$ nm and ${\lambda }_{\text{WM}}=89$ nm found from numerical results at ${\omega }_0/2\pi =8.8\mathrm{GHz}$ and $p\approx 1.75$. Error bars are found after the interpolation process, which introduces a measured error of approximately 8 nm.