Mode localization transition in ferromagnetic microscopic rings

We present a combined experimental, computational, and semianalytical study of the magnetization dynamics of permalloy disks and nanostructured rings with a systematically varied ring width. We investigate the dynamics of the quasisaturated state. In the case of wide rings the spin wave mode spectrum is similar to that of a disk. The small inner hole can be viewed as a weak perturbation. When the central hole increases, its influence becomes more important and two characteristic modes form gradually. They become localized at different positions in the ring. We explain the localization by the increasing inhomogeneity of the internal magnetic field and the formation of domain wall regions as the rings grow more narrow. In narrow rings one of the modes is clearly confined to the two segments where the internal field $H_{\mathrm{int}}$ is at a maximum and the other to the domain wall region where $H_{\mathrm{int}}$ is small. Dynamic magnetic simulations agree well with the measured spectra and confirm this interpretation. We also applied a semianalytical model, which confirms that the mode localization is driven by the internal field inhomogeneity.

Figure 1

(a) Schematics of the broadband spin wave spectrometer based on a vector network analyzer and a coplanar wave guide (CPW). A magnetic field $H_{\mathrm{ext}}$ may be generated by two orthogonal electromagnets at any in-plane direction. (b) Atomic force micrograph of a CPW with an integrated ring array on the central conductor having a width of $15\mu \mathrm{m}$. The coordinate system used throughout the paper is shown. The ellipse illustrates the approximate shape of the excitation field.

Figure 2

Magnetic excitation fields $100\mathrm{nm}$ above the center conductor of the CPW calculated according to Biot-Savart’s law (Ref. 41) with a realistic current density. The black and grey lines are the in-plane and out-of-plane fields, respectively. The sketched inset shows the cross section of the center conductor (grey rectangle) and the ring rows (black rectangles) to scale. The dashed line indicates the level of the magnetic field for a fictitious constant current density.

Figure 3

Computational magnetization configuration for the disk and rings at ${\mu }_0{H}_{\mathrm{ext}}=90\mathrm{mT}$. The calculation was performed with geometrical parameters from AFM measurements and magnetic parameters from measurements on a witness sample. For all ring widths we find the quasisaturated state with the magnetization in the ring arms in parallel.

Figure 4

Measured absorption spectra of disks and rings at ${\mu }_0{H}_{\mathrm{ext}}=90\mathrm{mT}$. The rings had different widths (listed in the panels). The insets show scanning electron micrographs of an individual ring from each array.

Figure 5

Calculated absorption spectra for a disk and rings with geometrical parameters corresponding to the experimental ones (for details see text). Widths are given in each panel. The damping constant $\alpha$ is chosen to be 0.01.

Figure 6

Comparison of the (a) experimental and computed relative absorption strength ${\eta }_{AB}$ and (b) the frequency splitting $\Delta f_{AB}$. Black filled symbols: Experiment, empty symbols: Computation.

Figure 7

(Color) Calculated mode profiles for the $2\text{−}\mu \mathrm{m}$-wide disk and the different ring widths as indicated at the column bottom (for details see text). In the simulation an external field of $90\mathrm{mT}$ lies along the horizontal axis. We show the mode profiles of the main modes $A$ and $B$ in the two bottom rows. Panel (i) on the top depicts the phase of mode $A$ for the ring with $w=930\mathrm{nm}$ and panel (ii) the satellite mode at $12.0\mathrm{GHz}$ below mode $A$ for $w=300\mathrm{nm}$. For color coding compare the colorbar.

Figure 8

(a) Micromagnetically calculated internal field profile for the $300\mathrm{nm}$ ring. White corresponds to high values grey to low values. The external field is ${\mu }_0{H}_{\mathrm{ext}}=90\mathrm{mT}$, applied in the horizontal direction. (b) Azimuthal cut of the internal field profile along the dashed line defined in (a) by $\overline{R}$ with the ring center at ($1\mu \mathrm{m}$, $1\mu \mathrm{m}$). The angle $\phi$ is defined in the inset.

Figure 9

Simulated internal field profiles at $\overline{R}=D_0∕2-w∕2$ along the azimuthal direction for the four different widths $w$ of 930, 830, 480, and $300\mathrm{nm}$ (from top to bottom). The ring geometries are sketched in insets. Note that the $H_{\mathrm{int}}$ scales differ for the four rings.

Figure 10

(a) Micromagnetically calculated internal field profile for the ring with $w=300\mathrm{nm}$. The symbols 엯, ▿, ▵, ◻, ◇ mark the positions at which the dispersion curves in panel (b) are calculated for mode $A(0,0)$. The grey line in (a) is the azimuthal wave vector $k_{\phi }(\phi ,H_{\mathrm{int}})$ extracted from the intersections of the calculated dispersion curves at different $H_{\mathrm{int}}$ in (b) with the experimental frequency. The dashed line (엯) in panel (b) is the lowest-lying dispersion curve which has an intersection with at $k_{\phi }⩾0$. This curve defines the localization length ${\Phi }_0$ along the azimuthal direction (measured in degrees).

Figure 11

(Color) Comparison of mode profiles of mode $A$ for the $w=300\mathrm{nm}$ ring from the micromagnetic computation (left column) and the approximate analytic mode profiles with wave vectors calculated as described in the text. For a color bar see Fig. 7. All panels are scaled such that the maximum amplitude equals 1.