Tunable metamaterial response of a Ni${}_{80}$Fe${}_{20}$ antidot lattice for spin waves

All-electrical spin-wave spectroscopy and frequency-resolved magneto-optical Kerr-effect measurements are combined to study spin waves propagating through a magnetic antidot lattice nanopatterned from a Ni${}_{80}$Fe${}_{20}$ thin film. Spin waves are injected from a plain film into the antidot lattice and the transmission across the interface is explored in detail for different wavelengths. We find that spin waves with a wavelength much greater than the lattice periodicity are not described well by recently discussed approaches. Instead the spin-wave dispersion is consistent with an effective magnetization smaller than the saturation magnetization measured on the unstructured ferromagnetic material. Consistently, we find that the transmission coefficients are modeled well by assuming an effectively continuous metamaterial for spin waves characterized by the reduced magnetization. The experimental data and interpretation are substantiated theoretically using the plane-wave method and micromagnetic modeling. The results are interesting for the development of frequency-selective mirrors in magnonics through lateral nanopatterning.

Figure 1

(a) Optical micrograph of sample 3. Two collinear CPWs for AESWS are integrated on top of a hybrid device consisting of a plain film and an ADL. The inner conductors of the CPWs act as emitter and detector of spin waves. The separation in $y$ direction is $s=19.5$ $\mu$m. The additional metal stripes are ground lines. (b) Scanning electron microscopy image showing focused ion-beam-etched holes. The circular holes (diameter of 120 nm) are arranged on a square lattice with a period $p=800$ nm. (c) Sketch of the AESWS experiment. CPWs labeled emitter and detector are integrated with a sample partially structured into an antidot lattice (gray area) extending along the $x$ axis. SWs are excited in the plain film with a wave vector $k_1$ pointing in the $y$ direction. Behind the film-ADL boundary SWs are scattered to a wave vector $k_2$. The orientation of the field with the $x$ axis is given by the angle $\eta$.

Figure 2

(a) Maximum amplitude $\max \left(a_{21}\right)$ measured between two CPWs in transmission configuration for sample 1 (plain film, dotted line), sample 2 (full ADL, solid gray line), and sample 3 (hybrid device, black line). We use ${\mu }_{0}H=20$ mT. (b) Gray-scale plot of spectra $a_{21}\left(f\right)$ for different $\eta$ for sample 3. Dark (bright) represents large (small) SW amplitude. Open circles (crosses) mark the frequency of the main ADL (plain film) mode. (c) $a_{21}\left(f\right)$ for $\eta =0^{\circ }$, sample 3. Arrows highlight local minima in the resonance curve (see text).

Figure 3

(a) Measured and calculated SW dispersions for a plain film and an ADL at ${\mu }_{0}H=20$ mT and $\eta =0$. Open squares mark the MOKE data obtained on a plain film. The solid line reflects the calculated dispersion assuming $M_s=770$ kA/m.[43] Open circles mark the MOKE data measured on the ADL. The long-dashed line is a SW dispersion calculated for a film assuming a reduced saturation magnetization $M_s^{☆}=600$ kA/m. This remodels the measured dispersion far better than previously discussed nanowire SW dispersions[32, 44] (see text): the dotted and dashed-dotted lines represent calculated dispersions obtained from a model assuming $M_s=770$ kA/m and quantized wave vectors $k_{\parallel }=n\pi /w_{\mathrm{eff}}$ [Eq. (9)] with $w_{\mathrm{eff}}=600$ nm. The gray-scale plot is the SW dispersion as obtained from wave-vector-resolved micromagnetic simulations (see text). Dark (bright) corresponds to large (small) SW amplitude. Spatial spin-wave profiles as obtained from micromagnetic simulations at $k=0$ are shown for (b) $f=3.7$ GHz ($n=1$), (c) $f=4.2$ GHz ($n=2$), and (d) $f=4.4$ GHz ($n=3$). Dark (bright) colors correspond to small (large) spin-precession amplitude. The direction of $\mathbf{H}$ is indicated.

Figure 4

(a) $\eta =0^{\circ }$. SW amplitude between emitter and boundary of the ADL measured by MOKE at $f=4.48$ GHz (depicted as a continuous line). This shows an interference pattern due to partial back-reflection of SWs at the boundary. The CPW outer conductor edge (boundary) is at $y=8\left(25\right)$ $\mu$m. A fitted function according to Eq. (6) considers the incident and back-scattered SWs (long-dashed line). The short-dashed line indicates the nonoscillatory decay that would have been expected for a plain film without a boundary. It is arbitrarily shifted with respect to the other curve for clarity. (b) $t\left(\eta \right)$ from AESWS (open circles), MOKE measurements (solid square at $\eta =0^{\circ }$), and micromagnetic simulations (solid triangles). The solid line depicts the maximum transmission amplitude $\max \left({\mathrm{a}}_{21}\right)$ as obtained via AESWS [from Fig. 2a].

Figure 5

(a) Transmission coefficient $t$ for SWs propagating from the plain film into the ADL of sample 3. Solid circles represent data obtained by MOKE. The continuous function (solid line) follows from the dispersions fitted to the measured data of samples 1 and 2 in Fig. 2a. It is calculated using Eq. (2) and represents an independent consistency check of the data measured on sample 3. The data with $t<0$ at $f=4.05$ and 4.5 GHz are marked with arrows and explained in the text. Triangles depict simulated results obtained from the wave-vector-resolved micromagnetic simulations. (b) Solid lines represent SW dispersions of the ADL as obtained by the PWM. To aid the comparison the gray-scale plot and MOKE data (circles) of Fig. 3a are reproduced. The inset is a magnification of one of the three anticrossings observed in the PWM data. (c) Spectrum $a_{21}\left(f\right)$ measured by AESWS for $\eta =0^{\circ }$ [replotted from Fig. 2c]. We attribute local minima in $a_{21}\left(f\right)$ with avoided crossings predicted by the PWM (highlighted by arrows).

Figure 6

Modulus of spatial SW profiles as calculated by the PWM at ${\mu }_{0}H=20$ mT and $\eta =0^{\circ }$ for $k_{\perp }$ amounting to (a)–(d) 0 rad/$\mu$m, (e)–(h) 0.5 rad/$\mu$m, and (i)–(l) 1.0 rad/$\mu$m. The number of nodes $(n-1)$ in the direction of $\mathbf{H}$ is changing. In each row $n=1$ is highlighted by a thick border. It is found in different columns. This illustrates the avoided crossing predicted in the SW dispersions of the PWM. The specific parameters are as follows: (a) $f=3.5$ GHz, $n=1$; (b) $3.7$ GHz, $n=2$; (c) $f=4.0$ GHz, $n=3$; (d) $f=4.4$ GHz, $n=4$; (e) $f=3.7$ GHz, $n=2$; (f) $f=3.8$ GHz, $n=1$; (g) $f=4.0$ GHz, $n=3$; (h) $f=4.4$ GHz, $n=4$; (i) $f=3.8$ GHz, $n=2$; (j) $f=4.0$ GHz, $n=3$; (k) $f=4.2$ GHz, $n=1$; (l) $f=4.4$ GHz, $n=4$. Bright color indicates a large spin-precession amplitude.

Figure 7

(a) Spatiotemporal spin-wave precession for pulsed excitation in the plain film at time $T=0.1$ ns and $y_{\mathrm{ex}}=27$ $\mu$m. The film-ADL boundary is located at $y_0=33$ $\mu$m and marked by the vertical dashed line. Dark (light) colors denote large positive (negative) precession. (b) Maximum SW amplitude over all simulation times (light gray). Exponential fits to the data are depicted as dark solid and dotted lines. At the boundary between the ADL and the film, a sudden decrease in the maximum amplitude $\Delta$ due to partial reflection is observed. From this the transmission coefficient $t$ is evaluated (see text). (c) Spatiotemporal spin-wave precession for continuous-wave microwave excitation in the plain film where $y_{\mathrm{ex}}=17$ $\mu$m and $f=4.7$ GHz. For $T\gtrsim 5$ ns, a steady state is reached and minima and maxima of the amplitude are observed in between the excitation region and the boundary (positioned at $y_0=33$ $\mu$m). (d) Maximum SW amplitude for $T>5$ ns (gray solid line). The sinusoidal interference pattern is visible. The black dashed line is a fit to the data corresponding to Eq. (6) from which $t$ is evaluated.