Spin waves parametrically excited via three-magnon scattering in narrow NiFe strips

Three-magnon scattering can be used to evaluate the lowest frequency of the magnon band in a ferromagnet. Both the ferromagnetic-resonant (FMR) frequency $f_{\mathrm{FMR}}$ and the lowest frequency of the magnon band $f_{\min }$ in narrow-shaped NiFe strips were electrically measured using the anisotropic-magnetoresistance effect. The comparison with a micromagnetic simulation shows that $f_{\min }$ of the magnon band can be controlled independent of $f_{\mathrm{FMR}}$ by varying the width $w$ and thickness $t$ of the NiFe strip while maintaining a constant $t/w$ ratio. In addition, we found that the frequency difference, $f_{\mathrm{FMR}}-f_{\min }$, can be greatly increased in thicker NiFe strips. Our results show that narrow-shaped ferromagnets allow us to tune the magnon-band structures by varying their $w$ and $t$. This ability is important for designing magnon circuits in integrated magnonic devices and for improving the quantitative study on the Bose-Einstein condensation of magnons.

Figure 1

Schematic illustration of the three-magnon scattering process in the magnon-band structure. (a) Magnon-band structures calculated for the 60-nm-thick NiFe film at (a1) ${\mu }_0{H}_{\mathrm{dc}}=20$ mT, (a2) 80 mT, and (a3) 120 mT. The blue (green) thick solid line shows the dispersion curve for the BVW mode with ${\theta }_k=0^{\circ }$ (MSSW mode, ${\theta }_k={90}^{\circ })$. The thin gray solid lines show the dispersion relationship of the magnons with oblique wave vectors at an interval of $5^{\circ }$; the solid magenta lines show the dispersion relationship for ${\theta }_k={15}^{\circ }$. The black open circles indicate the uniform magnons, which were forcibly excited by the microwave field; the magnons scattered via three-magnon scattering are indicated by red ellipses. (b) Schematic magnetic field dependence of ${\mu }_0{H}_{\mathrm{ac}}^{\mathrm{th}}$ (i.e., a butterfly curve). (c) Magnetic-field dependence of the critical propagation angle ${\theta }_k^{\mathrm{th}}$, which gives ${\mu }_0{H}_{\mathrm{ac}}^{\mathrm{th}}$.

Figure 2

(a) Scanning electron microscopy image of the sample and setup for the AMR experiment and (b) a schematic enlargement of the dashed square area in (a). A microwave synthesizer applies a continuous microwave to the CPW, which is embedded below the NiFe strip. The dc source and nanovoltmeter are connected to the four terminals, which are attached to the NiFe strip. (c) ${\mu }_0{H}_{\mathrm{dc}}$ dependence of the electric resistance of the NiFe strip for each sample. The red (blue) curve shows the AMR curve of the NiFe strip when ${\mu }_0{H}_{\mathrm{dc}}$ was swept parallel (perpendicular) to the electric current $J$.

Figure 3

The ${\mu }_0{H}_x$ dependence of $f_{\mathrm{FMR}}$ for each NiFe strip measured by the VNA-FMR technique. The inset shows an example of the $f_{\mathrm{ac}}$ dependence of $\mathrm{Re}\left[\mathrm{\Delta }{S}_{11}\right]$ at ${\mu }_0{H}_{\mathrm{dc}}=145$ mT for sample 2. A clear dip appears at $f_{\mathrm{ac}}=12$.2 GHz, which we denote as $f_{\mathrm{FMR}}$. The dashed lines are the lines of best fit calculated by Eq. (5).

Figure 4

AMR curves for each sample measured while applying ${\mu }_0{H}_{\mathrm{ac}}$ with a frequency in the range from 9 to 14 GHz when (a) ${\mu }_0{H}_{\mathrm{ac}}<{\mu }_0{H}_{\mathrm{ac}}^{\mathrm{th}}$ and (b) ${\mu }_0{H}_{\mathrm{ac}}^{\mathrm{th}}<{\mu }_0{H}_{\mathrm{ac}}$ for samples 1 to 3. When the magnetization precession was excited, the electric resistance of the NiFe strips decreased. The solid black triangles show the FMR modes, and the open red triangles show the parametric magnons excited via the three-magnon scattering process.

Figure 5

(a) Transition of the AMR curves under various ${\mu }_0{H}_{\mathrm{ac}}$ measured for sample 2. The dips indicated by the blue and red arrows correspond to the excitation of the FMR and three-magnon scattering, respectively. (b) Definition of $\mathrm{\Delta }R$ for the two resonant modes. (c) ${\mu }_0{H}_{\mathrm{ac}}$ dependence of $\mathrm{\Delta }R$ for the FMR and three-magnon scattering mode. ${\mu }_0{H}_{\mathrm{ac}}^{\mathrm{th}}$ is defined by the value of ${\mu }_0{H}_{\mathrm{ac}}$ at which $\mathrm{\Delta }R$ begins to increase from zero.

Figure 6

(a) Butterfly curves measured for sample 2 at frequencies of 9 to 12 GHz. The shaded area shows a certain magnetic field at which ${\mu }_0{H}_{\mathrm{ac}}^{\mathrm{th}}$ diverges, which yields the divergent field ${\mu }_0{H}_{\mathrm{dc}}^{\mathrm{div}}$. (b) ${\mu }_0{H}_x$ dependence of $f_{\min }$. Blue open circles represent the experimental results, and red open squares are the values obtained from the simulated dispersion relationship in Fig. 8(a2). The dashed line serves as a guide for the eyes.

Figure 7

Resonant frequencies of the FMR (open circles) and spin wave (open squares) modes. The solid line shows the FMR condition measured using the VNA-FMR technique. The dashed lines serve as a guide for the eyes for the spin wave modes excited via three-magnon scattering.

Figure 8

Color plot of $\langle n_{\mathrm{mag}}\rangle$ as a function of the wave number $k_x$ along the strip and the frequency calculated for the NiFe strips with $(w,t)$ of (a1) (400 nm, 30 nm), (a2) (800 nm, 60 nm), and (a3) (1600 nm, 120 nm). The aspect ratio $t/w$ of the NiFe strip is fixed at 0.075 for (a1), (a2), and (a3). Bright contrast shows the magnon excitation condition, which corresponds to the spin wave dispersion relationship. For comparison, the spin wave dispersion relationships obtained from the analytical solutions given by Eqs. (1, 2, 3), which are generally used for the thin-film limit, are shown in (b1), (b2), and (b3), where the thickness of the NiFe is the same as that in (a1), (a2), and (a3), respectively.

Figure 9

(a) Spin wave dispersion relationship for the BVW modes calculated for the NiFe strip (red curve) and NiFe film (blue curve) with $t=60$ nm at ${\mu }_0{H}_x=20$ mT. (b) and (c) Schematic magnetic configurations of the BVW mode propagating along the static magnetization direction $M_0$. Blue arrows show the dynamic component of magnetization $m\left(t\right)$ for the (b) strip and (c) thin film.

Figure 10

(a) $f_{\mathrm{FMR}}$ (open squares) and $f_{\min }$ (open circles) as a function of (a1) $t$ and (a2) $t/w$ for the NiFe strips with $w=200$, 400, and 800 nm. The solid curve in (a2) shows the $f_{\mathrm{FMR}}$ calculated using the Kittel relationship of Eq. (5). (b) $\mathrm{\Delta }f$ as a function of $t$. The dashed line shows $\mathrm{\Delta }f$ calculated from Eq. (1).

Figure 11

Results of the micromagnetic simulation. (a) Examples of the temporal development of $M_x/M_s$ over 50 ns for (a1) sample 1 to (a3) sample 3. ${\mu }_0{H}_{\mathrm{dc}}$ and $f_{\mathrm{ac}}$ dependence of the magnon number $n_{\mathrm{mag}}$ when (b) ${\mu }_0{H}_{\mathrm{ac}}<{\mu }_0{H}_{\mathrm{ac}}^{\mathrm{th}}$ and (c) ${\mu }_0{H}_{\mathrm{ac}}^{\mathrm{th}}<{\mu }_0{H}_{\mathrm{ac}}$. (d) Magnon frequency spectra obtained from the FFT analysis, which was applied to the temporal development of the $m_y$ component for the unit cell at the center of the NiFe strip. The FFT spectra were calculated at the condition marked by the open circles in (c).