Time-domain propagating spin-wave spectroscopy for forward spin waves in a ferromagnetic metal

Figure 2(a) shows a typical waveform of a FVSW packet at μ₀H_ext = 1.16 T. At the propagation distance g = 5 µm, the SW packet emerged at t_center = 3.8 ns as indicated by the arrow, and SW showed the amplitude A = 110 µV. The FVSW packet exhibits a clear oscillation from 1.1 to 7 ns (Δt = 6 ns). Compared with the DE spin-wave packets with Δt = 2 ns,⁶⁾ the SW packet shows an elongated shape. In the case of g = 10 µm, a FVSW locates at t_center = 6.9 ns and diminishes its amplitude to A = 45 µV. In the case of g = 40 µm, the SW was too attenuated (A = 5 µV) to detect its packet shape (not shown).

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To understand the basic property of FVSW, the propagation distance dependences of A and t_center are plotted in Fig. 2(b). The amplitude A shows an exponential decay.⁶⁾ Using the formula A₀ exp(−g/Λ), the SW attenuation length Λ is estimated to be 6.5 µm which is the same those obtained in previous works. As shown by the linear dependence of t_center, the group velocity of SW is determined to be v_g = 1650 m/s, which is much smaller than those of BV^32,33) and DE^6,23) spin waves.

To confirm that the SW packets reflect the FVSW property, we measured the magnetic field dependence of resonant frequency. By analyzing the fast Fourier transform (FFT) of waveforms, the resonant frequency is deduced and shown in Fig. 3. As shown in Fig. 3(a), in stronger magnetic fields 1.1 < μ₀H_ext < 1.2 T, the peaks exhibit a clear frequency shift in response to the variation in magnetic field. The frequencies of the peaks are plotted in Fig. 3(b) and categorized by colors. As shown in solid circles in blue, the magnetic-field-dependent frequencies are fitted by the FVSW dispersion relation

$$\begin{equation} f = \frac{\gamma}{2\pi}\sqrt{H_{\text{eff}}^{2} + H_{\text{eff}} \cdot M_{\text{s}}\left(1 - \frac{1 - e^{-kd}}{kd}\right)}, \end{equation} \tag{ 2 }$$

where M_s is the saturation magnetization, H_eff (= H_ext − M_s) is the effective field, γ is the gyromagnetic ratio, k is the wavenumber, and d is the film thickness. The fitting parameters were M_s = 1.03 T, γ/2π = 28 GHz/T, and k = 0.4 rad/µm, which were in agreement with the previously reported values. The wavenumber is determined using our ACPS geometry. Simultaneously, the group velocity v_g = ∂ω/∂k was calculated and is shown in the inset of Fig. 3(b). The expected group velocity is v_g = 1526 m/s at μ₀H_ext = 1.16 T, which is the same as the experimental value within an 8% error. From this analysis, we confirmed the existence of propagating FVSWs in the field range of μ₀H_ext > 1.1 T.

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In weaker magnetic fields 0.9 < μ₀H_ext < 1.1 T, we can observe a double-peak structure having centers at f = 1.69 and 2.30 GHz. With increasing magnetic field from 0.9 T, the peaks at f = 1.69 GHz shift very slightly toward 1.63 GHz (Δf = −60 MHz) without a change in its magnitude. On the other hand, the peak position at f = 2.30 GHz shifts toward 1.94 GHz (Δf = −360 MHz) with decreasing peak magnitude. Two peaks appear to be a single peak at 1.05 T and may vanish at approximately 1.06 T. In weaker magnetic fields of 0.9 < μ₀H_ext < 1.05 T, the magnetization of the waveguide does not perfectly saturate toward the perpendicular direction (this is consistent with the parameter M_s = 1.03 T), and thus a simple picture of FVSW, DESW, and BVSW is not valid. Since the SW resonance under the unsaturated condition is not within the scope of this paper, we do not discuss its details but only comment as follows: owing to the in-plane components of magnetization, the peak at f = 1.69 GHz may originate from the BVSW nature and the peak at f = 2.30 GHz may originate from the DESW nature. For example, if we use a rough assumption of in-plane magnetization with a very small internal field of 3.2 mT, we can deduce the resonant frequencies f_DE = 2.8 GHz and f_BV = 1.6 GHz from the SW dispersion relations. This is not a rigorous picture owing to the tilting magnetization; however, both the resonance frequency and its tendency to depend on the magnetic field are well explained.

To investigate the electric-current effects on the propagating FVSWs, we examined the electric-current injection into the SW waveguide at μ₀H_ext = 1.16 T. The propagation distance is fixed at g = 5 µm. As proved in previous works,^6,23) the antenna excitation method causes an asymmetric excitation field distribution around the antenna, involving a SW nonreciprocity. The spin-wave packet can be excited by four different excitation configurations using two antennas: $(k,M)$, $(k, - M)$, $( - k,M)$, and $( - k, - M)$. Here, −M is set by applying the external magnetic field H_z along the −z-direction. The reversed SW wavevector −k is realized by switching the roles of excitation and detection antennas.

Figure 4 shows the electric-current modulation of SW packets in each configuration. The amplitude A and the center position t_center of the SW packets were deduced by fitting using the Gaussian-shaped packet. As mentioned above, the initial amplitude A(0) differs for each configuration owing to the excitation nonreciprocity: 117 µV for $(k,M)$, 68 µV for $( - k,M)$, 89 µV for $( - k, - M)$, and 56 µV for $(k, - M)$.

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As shown in Fig. 4(a), in the $( - k, - M)$ configuration, a positive current j = +6.2 × 10¹⁰ A/m² attenuated the SW amplitude by 32 µV but accelerated by −380 ps, while the negative current j = −5.7 × 10¹⁰ A/m² amplified the SW amplitude by 32 µV and decelerated by +140 ps. In the case that the direction of the SW wavevector was inverted to be $(k, - M)$ as shown in Fig. 4(b), the positive current j = +6.2 × 10¹⁰ A/m² attenuated the SW amplitude by 19 µV but accelerated by −310 ps, while the negative current j = −5.7 × 10¹⁰ A/m² amplified the SW amplitude by 7.7 µV and decelerated by +176 ps. In the case that the direction of magnetization was changed to M, the roles of positive and negative currents were inverted; the positive sense of current decelerated the SW packet, whereas the negative sense of current accelerated the SW packet [see $( - k,M)$ shown in Fig. 4(c) and $(k,M)$ in Fig. 4(d)]. The electric current effect strongly depends on k and M.

In Fig. 5(a), the detailed current dependences of t_center(j) for all configurations are shown. The time difference Δt_center = t_center(j) − t_center(0) was measured from the SW packet without current. Clearly, Δt_center shows a nonlinear dependence on j for each configuration. For the fixed magnetization direction, for instance, $(k, - M)$ and $( - k, - M)$, note that the total values of Δt_center are almost the same but slightly different. This point is recognized for the other configurations between $(k,M)$ and $( - k,M)$. A weak spin-transfer torque effect can be superimposed on a nonlinear background signal, which will be the Oersted field effect induced by electric current. This effect was pointed out and discussed in several papers.^23,25,33) In contrast, the SW amplitude shown in Fig. 5(b) shows a linear dependence on j. Here, note that we used a normalized SW amplitude given by $\tilde{A} = A(j \ne 0)/A(j = 0)$ owing to the nonreciprocity. As in Δt_center, we can detect small differences in the $\tilde{A}$ slopes, if we compare the pair of $(k, - M)$ and $( - k, - M)$ and the pair of $(k,M)$ and $( - k,M)$.

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To analyze the STT effect, we have deduced the changes in group velocity and amplitude using the following subtraction procedures:

$$\begin{align} \Delta v_{\text{STT}} & = [\Delta v_{\text{g}}(k) - \Delta v_{\text{g}}(-k)]/2, \end{align} \tag{ 3 }$$

$$\begin{align} \Delta\tilde{A}_{\text{STT}} & = [\Delta\tilde{A}(k) - \Delta\tilde{A}(-k)]/2. \end{align} \tag{ 4 }$$

For a fixed j-direction, only the intrinsic effect of STT changes its sense with respect to the inversion of the wavevector k, whereas the effects of the Oersted field and Joule heating, among others do not. According to Eq. (1), we can deduce the following equations as in a previous paper:²³⁾

$$\begin{align} \Delta v_{\text{STT}} & = u_{0}, \end{align} \tag{ 5 }$$

$$\begin{align} \Delta\tilde{A}_{\text{STT}} & = \frac{g\beta k\Delta u}{\pi\gamma M_{\text{s}}d}, \end{align} \tag{ 6 }$$

where u₀ (= −μ_BjP/eM_s) is the magnitude of adiabatic STT, and g is the propagation distance.

From the slope in Fig. 5(c), we can estimate the spin polarization P of electric current. In the case of the −M configuration, we have P = 0.67, which is in agreement with the reported value for DESW. The spin transfer torque on FVSW was successfully detected by this time-domain measurement; however, we should be careful about the evaluation of its magnitude. If P is evaluated from the slope of the M configuration, the spin polarization becomes P = 2.1 resulting in a nonphysical interpretation. This is caused by a small signal-to-noise (S/N) ratio in the raw data. Compared with the DESW amplitude, the SW amplitude itself was smaller than 1/20 of the DESW amplitude and easily affected by the current-induced Oersted field. The effect of the small S/N ratio was not perfectly eliminated. If we carefully check the data of M and analyze it only in the positive current region j > 0, we can find a reasonable value of P = 0.56. However, we cannot proceed with an accurate discussion about the magnitude of P. For further P evaluation from FVSW, we need a propagation distance smaller than 5 µm to obtain a much better S/N ratio. At the present stage, it is only possible to state that the spin transfer torque has an approximately 8% contribution to the total current-induced modulation.

Concerning the evaluation of the nonadiabatic torque β from the slope shown in Fig. 5(d), we were unable to detect a physically reasonable β owing to the small S/N ratio. The $\Delta \tilde{A}_{\text{STT}}$ values are almost unchanged and remained at the noise level. As shown in Fig. 5(d), ΔA_STT caused by both the Oersted field and spin transfer torque effects reaches 35% and is much higher than those of BV and DE spin waves.^23,33) This means that the effect of the Oersted field is much greater than that of the spin transfer torque, which affects the evaluation of $\Delta \tilde{A}_{\text{STT}}$. The effect of the current-induced Oersted field on the waveform and why the FVSW is strongly affected by Oersted field compared with the other SW modes will be the next subjects of our investigation.