Magnonic minibands in antidot lattices with large spin-wave propagation velocities

Antidot lattices fabricated from permalloy thin films have been investigated by all-electrical spin-wave spectroscopy and Brillouin light scattering. Periodic arrays of 120-nm-diameter nanoholes have been prepared using focused ion beam etching. The periodicity of the square lattices was varied from 300 to 4000 nm. By applying an in-plane field of 40 mT, we discover surprisingly large spin-wave velocities of up to 6 km/s for a periodicity $<$400 nm. Using micromagnetic modeling and the further-developed plane wave method, we show that edge excitations at neighboring holes couple and form an allowed miniband supporting fast spin waves. By varying the orientation of the magnetic field we control the miniband characteristics. The coupling of edge modes opens interesting perspectives for magnonic crystals.

Figure 1

(a) and (b) Scanning electron microscope image of the antidot lattice in permalloy with integrated coplanar waveguides (CPWs). The holes have a diameter of $d=120\mathrm{nm}$ and are arranged on a square lattice of period $p=300$ nm, as highlighted by the magnified image. (c) Sketch of the device geometry displaying relevant parameters: magnetic field $\mathbf{H}$, angle $\eta$, and wave vector $\mathbf{k}$ addressed by the CPWs.

Figure 2

Experimental data taken at $\eta =1^{\circ }$ on an ADL with $p=0.3\mu$m: (a) AESWS amplitude $a_{11}$ in reflection configuration (gray-scale plot, where dark color denotes an SW resonance) and BLS data at $k=0$ (open circles). Thin solid lines are results of the PWM calculations (see text). (b) Micromagnetic simulation data for $k=0$ related to the reflection configuration. (c) AESWS real part $a_{12}$ in transmission configuration. Black and white contrast oscillations indicate a phase change due to SW propagation. (d) Single spectrum of the AESWS data (real part) in transmission configuration for ${\mu }_{0}H=40$ mT.

Figure 3

(a) Eigenfrequency $f$ (solid line, right axis) of the low-frequency mode of the ADL with $p=0.3\mu \mathrm{m}$. $f$ is shown as a function of angle $\eta$ for ${\mu }_{0}H=40\mathrm{mT}$. Symbols show the corresponding group velocities $v_{\mathrm{g}}$ (left axis) measured at $k_{\mathrm{CPW}}={10}^4$ rad/cm. The broken lines are to guide the eye. (b) Group velocities $v_{\mathrm{g}}$ at $k_{\mathrm{CPW}}={10}^4\mathrm{rad}/\mathrm{cm}$ of propagating modes in ADLs with different $p$. The values have been obtained for $2^{\circ }\le \eta \le 3^{\circ }$ by AESWS (circles) and micromagnetic simulations (squares) at ${\mu }_{0}H=40\mathrm{mT}$ and $k=k_{\mathrm{CPW}}={10}^4\mathrm{rad}/\mathrm{cm}$. The horizontal dashed line indicates the plain-film value. At $p=0.4\mu \mathrm{m}$ we find two modes of different eigenfrequency which both propagate with a characteristic but different $v_{\mathrm{g}}$. We explain this by a transition in the ADL behavior at this value of $p$: below (above) $p=0.4\mu \mathrm{m}$, the propagating mode is found to exhibit an eigenfrequency $f=2.9\left(6.0\right)\mathrm{GHz}$ indicated by the solid and open symbols (see text). At $p=0.4\mu \mathrm{m}$ these modes coexist in the experiment, but are not well separated. The group velocities obtained have a large error and are not considered for detailed analysis. Triangles indicate $v_{\mathrm{g}}$ predicted by PWM (given for $\eta =1^{\circ }$, see text). The gray shaded area represents group velocities calculated using the nanowire model for $25\le \delta \le 28$ nm (see text).

Figure 4

Spatial SW profiles for ${\mu }_{0}H=40\mathrm{mT}$, $\phi =1^{\circ }$, and $k=0$, simulated for (a) $p=0.3\mu \mathrm{m}$, $f=2.9\mathrm{GHz}$. (b) $p=0.8\mu \mathrm{m}$, $f=2.9\mathrm{GHz}$. (c) $p=0.8\mu \mathrm{m}$, $f=5.6\mathrm{GHz}$. Large (small) precession amplitude corresponds to large (small) $z$ elevation. (d) $x$ component of demagnetization field $H_{\mathrm{d}}$ for $p=0.3\mu \mathrm{m}$ (solid line) and $p=0.8\mu \mathrm{m}$ (dashed line). $H_{\mathrm{d}}$ is taken in the $y$ direction along the lines (A-B) in (a) and (c), respectively. (e) Spatial SW profile for $p=0.3\mu \mathrm{m}$, $f=4.5\mathrm{GHz}$, and $\eta ={10}^{\circ }$. The color coding for zero and maximum spin precession amplitude is shown.

Figure 5

Dispersions obtained from micromagnetic simulations (gray-scale plot) and BLS measurements (symbols) at ${\mu }_{0}H=60\mathrm{mT}$. A miniband behavior is resolved at low frequency for $p=0.3\mu$m. The solid line is the calculated plain-film dispersion for $M_{\mathrm{s}}=760\mathrm{kA}/\mathrm{m}$ and film thickness $\delta =25$ nm. The vertical dashed line indicates the Brillouin zone boundary at $\pi /0.3\mu$m. (b) Zoom into the SW dispersion at ${\mu }_{0}H=40$ mT near $k=0$, as obtained from PWM calculations (solid line) and micromagnetic simulations (gray-scale plot). The dashed lines indicate the $v_{\mathrm{g}}$ obtained for $k=0.5\times {10}^4\mathrm{rad}/\mu \mathrm{m}$ and $k=1.5\times {10}^4\mathrm{rad}/\mu \mathrm{m}$. They correspond to $v_{\mathrm{g}}=4.5$ and $v_{\mathrm{g}}=3.6\mathrm{km}/\mathrm{s}$, respectively. Such data are evaluated to obtain values $v_{\mathrm{g}}$ considered in Fig. 3b (see text for details). (c) Mode frequencies calculated by PWM for different $p$. The thick solid lines mark regions where, for $p<0.35\mu$m the edge mode i and for $p>0.35\mu$m the extended mode ii is predicted to be the propagating mode. Circles mark predicted mode crossings (see text).