Electric detection of nonlinear effect upon spin-wave spin current

In Fig. 2, we showed typical waveforms of propagating SW in MSSW and MSBVW modes. To examine the linear SW transport properties, the RF amplifier was set to be ×1 amplification and the propagation distance was fixed to be g = 5 mm. In case of MSSW, the power and duration of excitation signal were set to be P_in = 3 mW and T₀ = 10 ns, respectively. In case of MSBVW, the power and duration of excitation signal were P_in = 20 mW and T₀ = 10 ns, respectively. Since we used fixed frequency excitation, a resonant condition was searched by detecting an external magnetic field at which the SW signals maximize their amplitude.

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SW packets were clearly observed and exhibited in Fig. 2. In case of MSSW configuration, at t = 0, an excitation pulse was applied and caused an initial magnetization oscillation, inducing small signal at t = 0 less than 3 mV. The initial oscillation was induced by the direct radiation of electromagnetic wave from input antenna, resulting no time delay of excitation. The SW shows a propagating time delay and reached at detection antenna later than 35 ns, exhibiting a relatively large amplitude up to 34 mV. As increasing the external magnetic field, the velocity of SW decreases [see Fig. 2(a)] ranging in 43 < v_g < 100 km s⁻¹. The power spectra calculated by FFT analysis of time domain waveform are shown in Fig. 2(b) and they exhibit sharp peaks.

To confirm the SW mode of SW packet, we depicted the magnetic field dependence of resonant frequency f in the inset of Fig. 2(b). As shown, the experimental data are well explained by using the theoretical MSSW dispersion relations:^13,32)

$$\begin{eqnarray}&&{\omega }_{{\rm{M}}{\rm{S}}{\rm{S}}{\rm{W}}}^{2}/{\gamma }^{2}={\left({H}_{y}+\displaystyle \frac{4\pi {M}_{s}}{2}\right)}^{2}-{\left(\displaystyle \frac{4\pi {M}_{s}}{2}\right)}^{2}\exp \left(-2kd\right)\end{eqnarray} \tag{ 1 }$$

with the gyromagnetic ratio γ = 17.6 MHz/Oe and the film thickness d = 5.1 μm. The fitting parameters are the saturation magnetization 4πM_s = 1750 G and wavenumber k = 14 600 ± 1730 m⁻¹. While the wavenumber of excited SW can be roughly estimated by the geometry of excitation antenna $1/2{w}_{{\rm{antenna}}}$ where w_antenna = 50 μm is a width of antenna, the wavenumber is estimated to be 10 000 m⁻¹ in the same order and it is plausible. The MSSW propagation was successfully demonstrated by our RF switch method.

In Fig. 2(c), we presented time domain waveforms in the MSBVW configuration. Compared to the MSSW, the amplitude of MSBVW becomes weak, approximately less than 4 mV. The MSBVW wave packets are indicated by arrows which overlapped with the tail of initial magnetization oscillation. The group velocity of MSBVW were ranging in 24 < v_g < 28 km s⁻¹. To check the SW mode, the FFT analysis was performed in Fig. 2(d), and f– H relation was shown in the inset.

The theoretical MSBVW dispersion relations^20,21,32)

$$\begin{eqnarray}&&{\omega }_{{\rm{MSBVW}}}^{2}/{\gamma }^{2}={H}_{x}\left[{H}_{x}+4\pi {M}_{s}\left(\displaystyle \frac{1-{e}^{-{kd}}}{{kd}}\right)\right]\end{eqnarray} \tag{ 2 }$$

explains the experimental data using the parameter 6700 < k < 14 600 m⁻¹ and 4πM_s = 1750 G. Since the frequency change was smaller than MSSW, the curve fitting has a smaller dependence on the magnitude of wavevector k and did not be perfectly converged. Basically, the wavevector k is determined by the antenna geometry, however, the coupling between excitation field and MSBVW could be different from MSSW. The wavevector range may corresponds to this fact.

The linear transport properties both in MSSW and MSBVW were well deduced by the experiments. Here, we should note that it is important to focus on the nonlinear parameter N defined by³¹⁾

$$\begin{eqnarray}&&N=\partial \omega /\partial {\left|u\right|}^{2},\end{eqnarray} \tag{ 3 }$$

where ω and u correspond to the frequency and amplitude of SW, respectively. To cause a nonlinear effect, the amplitude of SW must be tuned by experimental condition, i.e. input excitation power. According to the Lighthill criteria,³³⁾ we have to also select a SW mode which satisfies the condition

$$\begin{eqnarray}&&\omega ^{\prime\prime} N\lt 0,\end{eqnarray} \tag{ 4 }$$

where ${\omega }_{k}^{\prime\prime} ={\partial }^{2}\omega /\partial {k}^{2}$ is the second derivative of SW frequency with respect to wavenumber k. The MSBVW is the possible candidate according to the dispersion relation in Eq. (2), since the ${\omega }_{k}^{\prime\prime} \gt 0$ and N < 0.

Figure 3 exhibits an input excitation power dependence of MSBVWs at various propagation distance (g = 5, 6, and 7 mm). The external magnetic field was set to be 1410 Oe. Note that the SW packets were converted into power by calculating ${V}_{{\rm{SW}}}^{2}/{Z}_{0}$ with characteristic Z₀ = 50 Ω impedance.

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In each propagation distance g with fixed the excitation duration T₀ =  10 ns, the SW signals become strong as increasing the input excitation power from 0.17 to 0.90 W. When the input excitation power reached to 0.90 W, we can easily recognize the larger SW in Figs. 3(a)–3(c). To investigate the nonlinear effect, the SW power dependence on input power was shown in Figs. 3(d)–3(f). In case of g = 5 mm, the SW power dependence deviates from a linear relation shown by the broken line at 0.49 W. In case of g = 6 mm, it deviates at 0.73 W. The SWs were actually in a nonlinear regime, showing the formation of SW soliton. In case of g = 7 mm, the power dependence fluctuates at higher P_in, however, we could not recognize the clear deviation even at P_in = 0.90 W. This means that the relaxation and dispersion effects prevent the formation of soliton, since the long propagation distance means an enough time to lose the nonlinear power. The nonlinear effect competes with the SW relaxation and dispersion.

To discuss this point, we evaluated the full-width at half-maximum (FWHM) ΔT for Figs. 3(a)–3(b) and shown in the Table I. At P_in = 0.17 W, the ΔT simply increases and represents the fact that the dispersion is always stronger than the nonlinear effect. The SW stays in the linear regime. While at P_in = 0.50 W, the soliton is only possible at g = 5 mm. After further propagation, at g = 6 mm, the dispersion becomes stronger than the nonlinear effect and the ΔT starts to increase. At P_in = 0.90 W, the soliton formation is possible at g = 5 mm and 6 mm. At g = 7 mm, the dispersion may stronger or same as the nonlinear effect.

Table I.  The FWHM of the SW packets shown in Figs. 3(a), 3(a), and 3(c).

Pin	${\rm{\Delta }}{T}_{5\mathrm{mm}}$ (ns)	${\rm{\Delta }}{T}_{6\mathrm{mm}}$(ns)	${\rm{\Delta }}{T}_{7\mathrm{mm}}$ (ns)
0.17 W	11.6	13.6	15.4
0.50 W	11.2	15.4	16.0
0.90 W	12.2	16.0	17.0

Similarly, if the excitation duration was increased to be T₀ =  30 ns, we can recognize strong increase of SW power in each propagation distance g. As shown in Figs. 3(g) and 3(j) with g = 5 mm, the deviating threshold decreases to be 0.28 W and the magnitude of deviation from linear relation increases. In case of g = 6 mm, it is possible to see the second solitary-wave formation as shown by the arrow in Fig. 3(h). The second soliton was not separated from the first soliton. In Fig. 3(k), we plotted the power of the first soliton, and the SW power dependence appears to be linear relation. Here, the power contribution of second soliton was not considered. However, the relation starts to fluctuate at 0.32 W, representing 2-soliton formation. The input power is consuming to generate 2nd soliton.

In case of g = 7 mm, the 2-soliton formation becomes clear as shown in Fig. 3(i), which exhibits 2 independent SW solitons. The fluctuation in power dependence in Fig. 3(l) becomes larger at 0.4 W. The result represents that the longer duration T₀ of excitation brings about stronger nonlinear effect than short excitation signal, since the total power W_in = P_in × T₀ is larger.

As detected by the above experiments, the duration of excitation pulse plays a crucial role in the SW soliton formations. The time evolutions of SW solitons were depicted in Fig. 4.

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Figure 4(a) represents the formation of SW soliton in propagation distance g = 5 mm as a function of the excitation duration T₀, with fixed input power P_in = 0.60 W. As clearly seen, the formation of SW soliton shows complicated process as increasing excitation duration. It is possible to detect the evolutions of 4-soliton propagation as labelled 1, 2, 3, and 4. However, each soliton is not perfectly separated from the others in time. To understand the 4-soliton propagation, the cross section of Fig. 4(a) at T₀ = 150 ns is shown in the top panel of Fig. 4(b). By using a multi-peak analysis with Gaussian functions, we detected the position of each soliton shown in the bottom panel. The centers of each soliton are 231 ns, 253 ns, 308 ns, and 372 ns, respectively.

If the propagation distance is increased to be g = 6 mm [see Fig. 4(c)], we can see the separation of the 2nd and 3rd solitons from the others; 1st soliton at 250 ns and 4th at 390 ns soliton shows independent wave packet. From Fig. 4(d), only the 2nd and 3rd soliton shows an overlap. Increasing the propagation distance to be 7 mm, each soliton becomes independent showing the center positions at 280 ns, 319 ns, 355 ns, and 447 ns, respectively. Interestingly in case of g = 8 mm, the 1st and 3rd soliton are diminished but 2nd and 4th soliton keep their clear packet shapes.

To discuss the nonlinear effect, we analyzed the FWHM of solitons shown in Figs. 4(b), 4(d), 4(f), and 4(h). The magnitudes of FWHM are summarized in Table II.

Table II.  The FWHM of the SW packets shown in Figs. 4(b), 4(d), 4(f), and 4(h).

Pin	${\rm{\Delta }}{T}_{5\mathrm{mm}}$ (ns)	${\rm{\Delta }}{T}_{6\mathrm{mm}}$(ns)	${\rm{\Delta }}{T}_{7\mathrm{mm}}$ (ns)	${\rm{\Delta }}{T}_{8\mathrm{mm}}$ (ns)
1st	23.4	24.3	29.9	33.3
2nd	54.2	23.0	14.7	15.0
3rd	52.6	38.5	38.9	33.0
4th	27.5	50.1	49.9	42.6

We consider the 1st SW soliton. Since the P_in = 0.60 W is enough larger to cause nonlinear effect, we can consider the soliton propagation at g = 5 mm and g = 6 mm. However, at g = 7 mm, the FWHM (ΔT) slightly increases by 5.6 ns and starts to show the dispersion. For the 2nd soliton, it is possible to detect the clear decrease of ΔT at g = 6 mm. This corresponds to the separation from the other solitons. The FWHM (ΔT) keeps almost the same magnitude up to g = 8 mm. The dispersion and the nonlinear effect counterbalance with each other. In case of 3rd soliton, the decrease of ΔT is also observed at g = 6 mm. The ΔT is larger than 2nd soliton, however, it keeps the ΔT up to g = 8 mm. The 4th soliton shows a small ΔT only at g = 5 mm. At g = 6 mm, it increases to 50 ns. However, the ΔT keeps the same magnitude.

From the above analysis, there is the possibility that the nonlinear effects for 1st and 3rd solitons were consumed for the fusion and separation of wave packets (modulation instability). The 1st and 3rd solitons lose the nonlinear effects due to the overlap of waveforms and were diminished their amplitudes. While the 2nd soliton may absorb the nonlinear effects of 1st and 3rd solitons. Concerning to the 4th soliton, it exhibits a solitary shape already in g = 5 mm. The 4th soliton may not lose the nonlinear effect and could keep its shape up to g = 8 mm. This is the possible explanation why the 1st and 3rd solitons are diminished but 2nd and 4th soliton keep their clear packet shapes.

The competition between nonlinear effect and dispersion/relaxation shows a complicate behavior. As deeply discussed in the previous paper³¹⁾, if we use the NLSE, it could be possible to deduce the characteristic time scales of dispersion, relaxation, and nonlinear effects. According to the Chen's analysis, we analyzed the simplest case of Fig. 4(a). The important time scales to consider are a relaxation time T_r, a dispersion time T_d, a nonlinear response time T_n, and propagation time T_p given by

$$\begin{eqnarray*}\begin{array}{rcl}{T}_{r} & = & 1/\eta ,\\ {T}_{d} & = & {v}_{g}^{2}{T}_{0}^{2}/\left|\omega {{\prime\prime} }_{k}\right|,\\ {T}_{n} & = & \pi /\left|N\right|{\left|{u}_{0}\right|}^{2},\\ {T}_{p} & = & g/\left|{v}_{g}\right|,\end{array}\end{eqnarray*}$$

where η = 2.22 × 10⁶ rad s⁻¹ is the characteristic decay time defined by ${e}^{-\eta t},\left|{v}_{g}\right|=2.54\times {10}^{4}$ m s⁻¹ is the group velocity, $\omega {{\prime\prime} }_{k}=6.91\times {10}^{-2}$ m²/rad s, N = −7.43 × 10⁹ rad s⁻¹ is the nonlinear parameter, ${\left|{u}_{0}\right|}^{2}$ is the input power amplitude, in our experimental systems. The deduced values are listed in the Table III.

From the Table III, we can understand that the nonlinear response time is smaller than the propagation time T_n < T_p, which means the nonlinear effect was effectively contribute to the soliton formation within the propagation. Furthermore, the relaxation time T_r and dispersion time T_d is enough larger than T_n and T_p. This fact guarantees that the balance between nonlinear effect and dispersion effect to form soliton shape is possible during the propagation time.

To consider multi-soliton formation with the NLSE analysis, we should analyze data with a careful calculation, since we need to consider additional parameter of duration time T₀. The detail comparison between 5 parameters for each independent soliton is beyond the scope of present paper. M. Chen et al. predicted 4-soliton propagation by using the NLSE, however, their experiments showed 3-soliton propagation. Although our experiment detected the 4-soliton propagation, we must be careful to compare with their NLSE analysis since material parameters are different from our system. The detail is left to be elucidated at this state, however, we clearly detected that the enough excitation power characterized by T₀ = 150 ns is possible to cause 4 independent soliton propagation.