Localization, confinement, and field-controlled propagation of spin waves in ${\text{Ni}}_{80}{\text{Fe}}_{20}$ antidot lattices

Spin-wave modes in ${\text{Ni}}_{80}{\text{Fe}}_{20}$ thin-film antidot lattices are investigated using micromagnetic simulations and a semianalytical theoretical approach. The simulations reveal a rich eigenmode spectrum consisting of edge and center modes. We find both spatially localized and spin waves extending over many unit cells. To classify the different types of modes and to analyze the microscopic properties, we adapt a semianalytical approach. We show how to reduce the two-dimensional problem of the antidot lattice to a one-dimensional problem if certain high-symmetry axes are considered. For lattices of unit-cell lengths ranging from 200 to 1100 nm, we find that the characteristic mode eigenfrequencies can be correlated with both local inhomogeneities of the demagnetization field and specific wave vectors caused by geometry-imposed mode quantization conditions. We compare our results with recently published experimental data and discuss the crossover from dipolar to exchange-dominated spin waves. Moreover, we simulate propagation of spin waves and find a preferred axis of propagation perpendicular to the external magnetic field.

Figure 1

(Color online) (a) Sketch of an antidot lattice where we define lateral parameters. Dark gray represents holes. We investigate holes of both square shape (full line) and circular shape (dashed line). (b) The square shown as a dotted line depicts the unit cell for the simulations. Relevant lattice directions $A$ to $D$ and specific areas $\alpha$, $\beta$, and $\gamma$ are indicated. (c)–(h) show the static behavior of lattice 3, with a lateral unit cell size of 272 nm. In particular: (c) Demagnetization field $H_{\text{Dem}}$ in the butterfly state [cf. (d)] at 100 mT (color code: see legend). Magnetization configuration: (d) the butterfly state at 100 mT $\left(\eta =0°\right)$ and (e) waterfall state at 30 mT $\left(\eta =0°\right)$. Small arrows indicate local orientation of spins. (f) Internal field profile through areas $\alpha$ and $\beta$ in the $x$ direction at $y=a/2$. For smaller fields the butterfly state gets more pronounced. (g) Demagnetization field $H_{\text{Dem}}$ and (h) magnetization configuration for $\eta =45°$ and ${\mu }_{0}H=100\text{mT}$. Dynamic demagnetization field changes as simulated (see Sec. 3A) are less than 0.1 mT.

Figure 2

Simulated magnetic field dispersions for (a) lattice 1 at $\eta =0°$, (b) lattice 3 at $\eta =0°$, and (c) lattice 3 at $\eta =45°$.

Figure 3

(Color online) Color-coded mode profiles for lattice 3 (a)–(f) and lattice 1 (g)–(j). Red (bright) means high-precession amplitude, dark blue reflects zero amplitude. The orientation of $H$ is indicated by the white arrow. Broken lines illustrate high-symmetry axes along which mode analysis is performed (see text). Lattice 3: eigenfrequencies at 60 mT are (a) 5.68 GHz (mode 0Deg-2), (b) 13.1 GHz (mode 0Deg-4), (c) 8.5 GHz (mode 45Deg-2), (d) 10.65 GHz (mode 45Deg-3), (e) 2.58 GHz (mode 0Deg-1, edge mode), (f) 8.95 GHz (mode 0Deg-3). Lattice 1: (g) 8.05 GHz (mode 0Deg-2 at 40 mT), (h) 12.2 GHz (mode 0Deg-4 at 40 mT), (i) 8.0 GHz (mode 0Deg-2 at 110 mT), and (j) 13.8 GHz (mode 0Deg-4 at 110 mT).

Figure 4

(Color online) Analysis of excitation 0Deg-2 at 400 mT for lattice 1: FFT of the resonance spectrum around 20 GHz. Insets show simulated spatial mode profiles for different frequencies. For frequencies at the peak and below areas $\alpha$ participate more strongly. This is consistent with experimental observations in Ref. 16.

Figure 5

(a) Mode 0Deg-4 analysis for lattice 1 at 40 mT. The straight line depicts the calculated mode profile using Eq. (3). The dashed line depicts the calculated wave vector $k_{\perp }$ using $k_{\parallel }=\pi /b=\pi /85\text{nm}$, where $b$ is the dot diameter. (b) Mode 45Deg-2 analysis (short dashes) and Mode 45Deg-3 analysis (long dashes) for lattice 3 at 100 mT. Only the mode profiles are shown for sake of simplicity.

Figure 6

Eigenfrequency of mode 0Deg-2 at 400 mT as a function of antidot-to-antidot distance (see text). Continuous line is an analytical result using the demagnetization field of a thin wire. Dots are simulated results.

Figure 7

(Color online) Spin-wave propagation within an antidot lattice: simulated patterns show the time evolution after a pulsed excitation for ${\mu }_{0}H=130\text{mT}$ with a spatial inhomogeneous magnetic field $h\left(t\right)$ at (a) 70 ps, (b) 180 ps, (c) 330 ps, and (d) 2200 ps after the field pulse. In the $x$ and $y$ direction eleven unit cells are depicted. Red (bright) represents a high spin-wave amplitude. For long times, i.e., (d), the damping has reduced the overall amplitude. The propagation is predominantly perpendicular to $\vec{H}$, which is applied along the $y$ direction.