Spin motive force induced in Fe₃O₄ thin films with negative spin polarization

Spin motive force (SMF) is induced by a time and spatial derivative of magnetizations^1–3) as represented by

$$\begin{equation} E_{i} = -\frac{P\hbar}{2e}\boldsymbol{{m}}\cdot(\partial_{t}\boldsymbol{{m}}\times \partial_{i}\boldsymbol{{m}}), \end{equation} \tag{ 1 }$$

where P is the spin polarization, ħ is the Planck constant, e is the elementary charge, and m is a normalized magnetization vector. $\partial _{t}\boldsymbol{{m}}$ is a time derivative of the magnetization and $\partial _{i}\boldsymbol{{m}}$ is a spatial derivative of the magnetization. $\partial _{i}\boldsymbol{{m}}$ is finite when structures such as a domain wall and a vortex core possess spatially nonuniform magnetizations. The equation indicates that the dynamics of the spatially nonuniform magnetizations induce a motive force E_i. The SMF has been observed in magnetic dynamics in various systems such as a domain wall motion in an FeNi wire,^4,5) magnetization dynamics in MnAs nanoparticles in a magnetic tunnel junction,⁶⁾ a vortex core motion in an FeNi disc,⁷⁾ and a ferromagnetic resonance in asymmetrically patterned FeNi wires.⁸⁾ The SMF has a prospect for spintronic applications that can effectively convert the energy of magnetic dynamics into electric energy. Equation (1) indicates that the polarity of the SMF is dependent on the sign of the spin polarization of the ferromagnetic material. Although most of the SMFs measured so far are in FeNi having positive spin polarization, there are no reports on the SMF induced in materials with negative spin polarization.

In this report, we investigate the SMF in a magnetite (Fe₃O₄) having negative spin polarization and compare it with that in FeNi having positive spin polarization.⁹⁾ Fe₃O₄ is one of the good candidates for testing the spin polarization dependence of the SMF, which has a unique character of ferrimagnetism and a half-metallic character with negative spin polarization.^10–15)

We prepared fully epitaxial 50-nm-thick Fe₃O₄ films covered with 2-nm-thick Au on a MgO(001) single-crystal substrate. The quality of the epitaxy is described elsewhere.¹⁶⁾ We then patterned it into a wedged wire followed by the deposition of a coplanar waveguide as shown in Fig. 1(a). In order to detect the SMF, we measure the dc voltage longitudinal to the wire under the excitation of magnetic resonance. The measurement is performed by flowing an rf current through the wire while varying the magnitude of an in-plane magnetic field. The applied rf power is up to 16 dBm. θ_H is the in-plane magnetic field angle away from the x-axis [see Fig. 1(a) for the coordinate system]. The wedge shape is designed so that the rf current generates a nonuniform oscillatory Ampère field around the wire. This nonuniform field induces a spatially nonuniform magnetization dynamics in the longitudinal direction to the wire, as shown in blue arrows in Fig. 1(a). Since this configuration can involve parasitic dc voltages originating from the rectification effect due to the nonlinear coupling between the oscillation of anisotropic magnetoresistance and the rf current, the dc voltage spectrum is expected to be a superposition of a symmetric and antisymmetric Lorentzian^17–20) around the resonant frequency as shown in Fig. 1(b). We expect it to be composed of three different contributions:

$$\begin{equation} V_{\text{dc}}(\omega) = V_{\text{SHE}}(\omega) + V_{\text{Amp$\unicode{232}$re}}(\omega) + V_{\text{SMF}}(\omega). \end{equation} \tag{ 2 }$$

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The first term on the right-hand side originates from the spin Hall torque due to the spin Hall effect (SHE) of the Au layer.^21,22) The resistance oscillation upon the magnetization precession induced by the spin Hall torque by the Au layer couples with the rf current, leading to the rectified dc voltage. The phase of the resistance oscillation is the same as that of the rf current, resulting in a dc voltage spectrum with a symmetric Lorentzian shape at the resonant frequency,²⁰⁾

$$\begin{equation} V_{\text{SHE}}\propto S(\omega)\frac{\sin 2\theta_{H}}{2}. \end{equation} \tag{ 3 }$$

S(ω) is the normalized symmetric Lorentzian given by

$$\begin{equation*} S(\omega) = \cfrac{1}{\cfrac{(\omega - \omega_{0})^{2}}{\sigma^{2}} + 1}, \end{equation*}$$

where σ is the linewidth, ω₀ is the resonant frequency, and ω is the frequency of the rf current. It should be noted that this dc voltage is proportional to sin 2θ_H.

The second term originates from the Ampère field induced by the current flowing in the Au layer. The resistance oscillation upon the magnetization precession due to the Ampère field around the Au layer couples with the rf current, leading to the rectified dc voltage. The phase of the resistance oscillation differs from that of the rf current by π, resulting in a dc voltage spectrum of an antisymmetric Lorentzian at the resonant frequency,

$$\begin{equation} V_{\text{Amp$\unicode{232}$re}}\propto A(\omega)\frac{\sin 2\theta_{H}}{2}\sin\theta_{H}, \end{equation} \tag{ 4 }$$

where

$$\begin{equation*} A(\omega) = \cfrac{\cfrac{\omega - \omega_{0}}{\sigma}}{\cfrac{(\omega - \omega_{0})^{2}}{\sigma^{2}} + 1} \end{equation*}$$

is the normalized antisymmetric Lorentzian. It should be noted that this dc voltage is proportional to sin θ_H sin 2θ_H.

The third term is from the dc voltage induced by the SMF. Setting θ_H = 90°, we rewrite Eq. (1) as

$$\begin{equation} V_{\text{SMF}} = S(\omega)\frac{P\hbar\omega}{2e}(\cos\phi_{L} - \cos\phi_{0}), \end{equation} \tag{ 5 }$$

where the coordinate corresponds to that shown in Fig. 1(a). The coordinate originates from the long edge of the wire. Here, ϕ_L and ϕ₀ are the precessional angles of the magnetization at y = 0 and y = L, respectively. When ϕ is small, Eq. (5) is approximately

$$\begin{equation*} {-S}(\omega)\frac{P\hbar\omega}{4e}[m_{x}^{2}(L) - m_{x}^{2}(0)]. \end{equation*}$$

In our experiment, $[m_{x}^{2}(L) - m_{x}^{2}(0)]$ is nonzero because of the excitation of the spatially nonuniform magnetization dynamics. With P = 1, θ_H = 90°, and $m_{x}^{2}(L) - m_{x}^{2}(0) = 1$, we can obtain $V_{\text{SMF}} = 1.1 \times 10^{ - 6}$ V/GHz. We emphasize that the SMF is linearly proportional to the resonant frequency, whereas the two other parasitic dc voltages are not. Therefore, it is possible to separate V_SMF from V_SHE and V_Ampère by looking into the frequency dependence.

In the experiment, we mainly focus on the dc voltage under the magnetic field θ = 90° at which the parasitic voltage contributions are negligible [see Eqs. (3) and (4)]. FeNi(20 nm)/Au(3 nm) control samples are also investigated in the same manner for comparison.

We measure the dc voltage while varying the magnitude of the magnetic field ranging from 0 to about 1600 Oe at the field angle θ_H of 90°. Typical results of wedged samples are shown in Figs. 2(a) and 2(b) for Fe₃O₄ and FeNi samples, respectively. The peak amplitude in Fe₃O₄ is small but is clearly visible above the noise level. To specify the dynamics of the resonance, we fit it with Kittel's equation $\omega _{0} \approx \gamma \sqrt{(H + H_{k})(H + \mu _{0}M)}$, where γ = 1.76 × 10¹¹ s⁻¹ T⁻¹, M, and H_k are the gyromagnetic ratio, saturation magnetization, and anisotropy field, respectively. The fitting yields M = 573 mT and H_k = 26 mT for Fe₃O₄, and M = 1113 mT and H_k = 3.2 mT for FeNi, as shown in Figs. 2(c) and 2(d), respectively. These values are in good agreement with previous reports.^23–27) Since the resonant mode in the Fe₃O₄ sample is well explained by Kittel's equation, the observed resonance is ferromagnetic and we do not observe any ferrimagnetic resonances.

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In Figs. 3(a) and 3(b), we show the symmetric component of the dc voltage at the field angle θ_H of 90° as a function of resonant frequency. We observe that the symmetric dc component is linearly proportional to the resonant frequency. Although no gradient of the dc voltage as a function of frequency is observed in the rectangular wire, a finite gradient is observed in wedged wires. The sharper the sample, the larger the gradient observed. This is consistent with $[m_{x}^{2}(L) - m_{x}^{2}(0)]$ being larger in sharper samples because a larger precession is induced by a larger Ampère field at the sharp end. Therefore, these frequency-dependent dc voltages are SMFs. Furthermore, we found that the gradient of Fe₃O₄ is opposite to that of FeNi, suggesting that Fe₃O₄ induces the opposite SMF owing to the negative spin polarization. The results indicate that the measurement of the SMF can be a way of determining the sign and magnitude of the spin polarization. This measurement technique for determining the spin polarization is potentially advantageous over other methods since it does not require any adjoined materials for reference.¹²⁾ The nonzero extrapolation of the dc voltage at 0 GHz for the wedged samples implies a residual rectified voltage due to the magnetic inhomogeneity in the wire edge. In Figs. 3(c) and 3(d), we also show antisymmetric components of dc voltage for various shapes of samples. These components are almost 0 for all the samples and do not depend on the resonant frequency. The absence of antisymmetric components shows that there is no significant experimental error of the field angle θ_H of 90°.

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One may be still concerned with a few other possibilities of parasitic dc voltages. However, we will be able to rule those out because of the following reasons. One possibility is the temperature gradient of the wire that induces the Seebeck effect. Upon magnetic resonance, a magnon–phonon scattering can induce heat in the system. The cross section of the magnon–phonon scattering depends on the precessional angle. Since the precessional cone angle is different between the narrow and wide parts of the wire, a thermal gradient along the y-axis is possible. Taking into account the Seebeck constants of −20 µV/K for FeNi, −60 µV/K for Fe₃O₄, and 1.7 µV/K for Au,^28,29) we would expect the induced voltage to be in the same polarity for both FeNi and Fe₃O₄. However, in our results, the gradient of dc voltage against the resonant frequency of Fe₃O₄ is opposite to that of FeNi, which cannot be solely explained by the Seebeck effect. Another possibility is the Nernst effect as well as the anomalous Nernst effect. These can develop the dc voltage perpendicular to both the magnetic field and the temperature gradient. With both the temperature gradient and the magnetic field being in the y-axis, there should be no Nernst voltages in our voltage measurement direction. Lastly, the frequency dependence of the rf magnetic susceptibility could also induce the gradient of the dc voltage. A higher resonant frequency makes the precessional cone angle smaller, giving a smaller resistance oscillation and, therefore, inducing a smaller rectified dc voltage. However, in Fig. 3, the dc voltage of Fe₃O₄ increases when the resonant frequency increases. This result cannot be explained by the frequency dependence of the ac magnetic susceptibility.

In Fig. 4, we show SMF per GHz as a function of wedge angle. The definition of the wedge angle is shown in the inset. We obtain −0.67 and 0.08 µV/GHz at maximum for FeNi and Fe₃O₄, respectively. With the frequency vs field data shown in Figs. 2(c) and 2(d), these values translate to 5.5 and 0.53 nV/Oe for FeNi and Fe₃O₄, respectively. We suspect that the smaller SMF in Fe₃O₄ is due to a shunt resistance of the Au layer. The resistivity of Fe₃O₄ is $4.0 \times 10^{ - 5}$ Ω m, which is much higher than that of FeNi (2.2 × 10⁻⁷ Ω m). By assuming that the SMF is shunted by the surface Au layer, we can calculate $V_{\text{SMF}} = 73V'_{\text{SMF}}$ for Fe₃O₄ samples and $V_{\text{SMF}} = 2.0V'_{\text{SMF}}$ for FeNi samples. In the calculation, we used the resistivities ρ_Au = 2.2 × 10⁻⁸ Ω m, ρ_Py = 2.2 × 10⁻⁷ Ω m, and ρ_Fe3O4 = 4.0 × 10⁻⁵ Ω m for each material. Here, V_SMF is the corrected SMF and $V'_{\text{SMF}}$ is the measured SMF. Therefore, the Au shunt resistance is more effective in Fe₃O₄ samples, resulting in the underestimation of SMF in Fe₃O₄.

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In summary, we observed the SMFs induced in the nonuniform ferromagnetic resonance of Fe₃O₄ with negative spin polarization and in that of FeNi with positive spin polarization. By ruling out the parasitic dc voltage irrelevant to the SMF, Fe₃O₄ is found to induce the SMF that is opposite to that in FeNi. This originates from the opposite spin polarizations between FeNi and Fe₃O₄.

Acknowledgment