Effects of shape distortions and imperfections on mode frequencies and collective linewidths in nanomagnets

Brillouin light scattering shows that shape distortions in ${\mathrm{Ni}}_{80}{\mathrm{Fe}}_{20}$ nanomagnets can have a dramatic effect on the measured collective linewidth of certain spin-wave modes. The intentional introduction of quantifiable asymmetric egglike shape distortion to an ideal elliptical structure lifts the degeneracy of end modes with concentrated amplitude at the nanomagnet edges. In contrast, modes with concentrated amplitude at the interior are significantly less affected by the distortion. The splitting of end modes by asymmetric distortions explains the large inhomogeneous linewidth broadening in end modes found in large ensembles of nanomagnets that contain a relatively small statistical variation in the degree of distortion.

Figure 1

SEM images of the four different arrays with nanomagnets exhibiting different amounts of eggcentricity. The yellow line along the contour of the individual ellipse is a fit of Eq. (1) to the actual shape of the nanomagnets. The resultant fitting parameter values for the long axis $A$, the short axis $B$, and the eggcentricity $c$ are given in Table .

Figure 2

Sketch of the BLS measurement geometry, where ϕ is the in-plane angle between the long axis of the ellipsoids and the applied magnetic field $H$. The incident laser beam is scattered by a spin wave with in-plane wave vector $k^{\mathrm{s}-\mathrm{w}}$. The wave vector of the incident and backscattered beams are $k^{\mathrm{in}}$ and $k^{\mathrm{bs}}$, respectively. The vector $n$ denotes the surface normal.

Figure 3

Dependence of nanomagnet spin-wave spectra on $c$ for different directions of the applied magnetic field. The red line is a fit to the data by use of a superposition of several Lorentians raised to the sixth power, corresponding to the number of peaks measured in the respective spectra. The colored arrows together with the symbols denote the different spin-wave modes.

Figure 4

Angular dependence of the strongest spin-wave modes measured with BLS for samples of varying eggcentricity. Three major spin-wave modes can be identified. The lowest frequency mode is denoted with filled black circles. This mode is identified as the edge mode. The next highest frequency mode is indicated with blue triangles. It is identified as the center mode. The highest frequency mode is denoted with filled green diamonds. For certain angles of the applied field these modes split into two modes.

Figure 5

Micromagnetic simulation results. The left panel shows the frequency and the spatial distributions for the spin-wave modes of the ellipse and the right panel for sample egg5 with the largest value of $c$. The rows show the simulations for the different angles between the long axis of the ellipse and the applied magnetic field. The static ground state of the magnetization of the ellipse and the egg5 sample are shown in each row next to the results of the frequency distribution.

Figure 6

Angular dependence of the spin-wave modes determined by micromagnetic simulations. The thickness of the lines is proportional to the amplitude of the respective spin-wave mode.

Figure 7

The frequency dependence of the FWHM linewidth for the center and the end modes measured by FR-MOKE in Ref. 8 for 5-nm-thick ${\mathrm{Ni}}_{80}{\mathrm{Fe}}_{20}$ 220 nm × 202 nm with the magnetic field applied parallel to the long axis of the ellipsoid. The dashed black and solid blue lines are calculated linewidths for the center and the end mode, respectively, where inhomogeneous broadening due to size variations alone are assumed. The green lines are predicted linewidths using a model where asymmetric shape distortions of egglike character are included in the model; see Eq. (7). The inset shows the resonance frequency versus applied field for the end mode (blue triangles) and the center mode (black circles).