J. Phys. Soc. Jpn. 90, 081011 (2021) [10 Pages]

SPECIAL TOPICS: Renewed Interest in the Physics of Ferrimagnets for Spintronics

Spin-Motive Force in Ferromagnetic and Ferrimagnetic Materials

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¹Department of Advanced Science and Technology, Toyota Technological Institute, Nagoya 468-8511, Japan²Department of Physics, Toho University, Funabashi, Chiba 274-8510, Japan

Received January 5, 2021; Revised April 12, 2021; Accepted April 26, 2021; Published June 17, 2021

The magnetization dynamics of nonuniform magnetic structures induce the spin-dependent force acting on the conduction electrons via the s–d coupling. This emergent electric field, the so-called spin-motive force (SMF), is observed in a variety of magnetic structures such as magnetic domain walls, magnetic vortices, and staggered ferrimagnetic structures. The magnitude and direction of the SMF depend on the spatiotemporal derivative of magnetic structures, which is determined by the magnetic interaction and geometry of the sample. In this article, we review recent studies on the SMF using ferromagnetic and ferrimagnetic materials. We introduce the theoretical aspect of the SMF and show the relationship between the SMF and the topology of the magnetic structure. We present the experiment on the SMF induced by various magnetic samples in the presence of ferromagnetic coupling, the dipole–dipole interaction, and the antiferromagnetic coupling.

https://doi.org/10.7566/JPSJ.90.081011

1. Introduction

Recently, controllable nonuniform magnetic structures such as magnetic domain walls,¹^–⁶⁾ magnetic vortices,⁷^–¹²⁾ anti-vortices,¹³^–¹⁷⁾ skyrmions,¹⁸^–²⁵⁾ and anti-skyrmion²⁶^–²⁸⁾ have attracted much attention owing to the possibility of spintronics nonvolatile memory. Magnetic domain walls and skyrmion structures can be regarded as information carriers in memory devices³^,⁶^,²⁰⁾ and are successfully moved by a charge current. This effect, called the spin transfer torque (STT),²⁹^–³¹⁾ is naturally extended to the inverse effect, i.e., the dynamics of the nonuniform magnetic structure induce a charge current. A simple theoretical argument has pointed out that the effective electric field acting on the conduction electron spin emerges via the s–d coupling.³²^–⁴⁶⁾ The spin-dependent force results in a net charge current in the magnetic material. This spin-motive force (SMF) is proposed in many types of magnetic structure.⁴⁷^–⁶⁶⁾ Because the SMF depends on the magnetization dynamics, the electric signal of the SMF strongly depends on the sample geometry and the magnetic interaction inside the sample. Furthermore, it is also known that the spin–orbit interaction modulates the SMF both in its direction and amplitude. The SMF is the universal phenomenon in the magnetic metal and can be understood practically on the basis of the gauge field theory, the equation of motion, and the spin Berry phase.

In this paper, we review the SMF realized in ferromagnetic and ferrimagnetic systems. This paper is organized as follows. The theoretical aspects of the SMF are reviewed in Sect. 2. We also present the relationship between the SMF and the topology of the magnetic structure in this section. In Sect. 3, the experimental aspects of the SMF using the ferromagnetic material are reviewed. We introduce the SMF induced by the magnetic domain wall motion, gyrating magnetic vortex, and ferromagnetic resonance (FMR) in an asymmetric structure. In Sect. 4, the SMF induced in ferrimagnetic materials (GdFeCo alloys) is reviewed. We discuss experimental issues on the SMF in Sect. 5.

2. Theoretical Aspect of SMF

2.1 SMF for adiabatic approximation

The SMF has been proposed theoretically³²^–³⁴⁾ from the viewpoint of the Josephson effect,³²⁾ the gauge field,³³⁾ and the Berry phase.³⁴⁾ In 2007, Barnes and Maekawa reclaimed the SMF by generalizing Faraday's law³⁵⁾ in the solid state with the spin degree of freedom. They have pointed out the promising experimental setup for observing the SMF using the magnetic domain wall. The magnetic domain wall is formed between two ferromagnetic domains, and it can be moved by a charge current, the so-called STT effect.²⁹^–³¹⁾ The spin angular momentum of the conduction electrons transfers to the local magnetic moment via the s–d coupling. The SMF is the counterpart of the STT, and the electric voltage would appear across the domain wall when the magnetization dynamics occur. The theoretical aspect of the SMF is commonly based on the gauge field theory that applies the spatiotemporal derivative of the magnetization structure. The time scale of the magnetization dynamics is determined by the magnetic interaction inside the sample. Therefore, the SMF signal depends on the sample geometry and material parameters. Before introducing each experimental result, we first introduce the SMF theory.

The Hamiltonian of the conduction electrons with the s–d coupling is represented as \begin{equation} H = \frac{\boldsymbol{p}^{2}}{2m_{e}} - J_{\text{sd}}{\boldsymbol{\sigma}}\cdot \boldsymbol{M}(\boldsymbol{r},t), \end{equation} (1) where \(m_{e}\) is the effective mass of the conduction electrons, \({\boldsymbol{\sigma}}\) represents the Pauli matrices, \(\boldsymbol{M}(\boldsymbol{r},t) = M_{s}(\cos\phi (\boldsymbol{r},t)\sin\theta(\boldsymbol{r},t),\sin\phi (\boldsymbol{r},t)\sin\theta (\boldsymbol{r},t),\cos\theta(\boldsymbol{r},t))\) is the magnetization, \(M_{s}\) is the saturation magnetization and \(J_{\text{sd}}\) is the s–d coupling energy.⁴⁰^,⁴¹⁾ To calculate the effective field acting on conduction electrons, one performs the local gauge transformation, \begin{equation} \hat{U}^{\dagger} \hat{H}\hat{U} = \frac{[\boldsymbol{p}- e\hat{\boldsymbol{A}}^{\boldsymbol{s}}]^{2}}{2m_{e}} - J_{\text{sd}}\sigma_{z}M_{s} + \hat{V}^{s}, \end{equation} (2) where \(\hat{U}\) is the unitary matrix represented by \(\hat{U}(\boldsymbol{r},t) =\exp(- i\sigma_{y}\theta (\boldsymbol{r},t)/2)\exp(- i\sigma_{z}\phi (\boldsymbol{r},t)/2)\). The effective spin scalar and vector potentials are derived by solving \begin{align} \hat{V}^{s} &= - \left(\frac{i\hbar}{e} \right)\hat{U}^{\dagger} \frac{\partial}{\partial t}\hat{U}, \end{align} (3) \begin{align} \hat{\boldsymbol{A}}^{s} &= \left(\frac{i\hbar}{e} \right)\hat{U}^{\dagger} {\boldsymbol{\nabla}} \hat{U}. \end{align} (4) As the conventional electromagnetism, the effective electric field is obtained by calculating \begin{equation} \hat{E}_{\mu} = \frac{\partial}{\partial t}\hat{A}^{s} - \frac{\partial}{\partial x_{\mu}}\hat{V}^{s}. \end{equation} (5) This effective field is represented by a \(2\times 2\) matrix in the spin space. The second term in Eq. (2) gives the strong s–d coupling by which one can assume that the spin direction is parallel to that of the local magnetic moment. This adiabatic approximation leads one to calculate the net charge current as \begin{equation} E_{\mu} = p\mathop{\text{Tr}}\nolimits(\sigma_{z}\hat{E}_{\mu}), \end{equation} (6) where p is the spin polarization of conduction electrons. By using the normalized vector \(\boldsymbol{n}=\boldsymbol{M}/M_{s}\), we can express the effective electric field as \begin{equation} E_{\mu} = - \frac{p\hbar}{2e}\boldsymbol{n}\cdot \left(\frac{\partial \boldsymbol{n}}{\partial t} \times \frac{\partial \boldsymbol{n}}{\partial x_{\mu}} \right). \end{equation} (7) As shown in Eq. (7), the conduction electrons feel the effective gauge fields such as the Peierls phase of the magnetic field. We note that the staggered gauge field⁵⁵⁾ is introduced for the case of the antiferromagnetic spin configuration similar to that in a ferrimagnetic material as described in Sect. 4.

The SMF is also understood on the basis of the geometrical phase, called the spin Berry phase acting on the conduction electrons.⁶⁷⁾ Let us consider the one-dimensional metal ring under a magnetic field as shown Fig. 1(a). According to Faraday's law of induction, the time derivative of the magnetic flux penetrating the ring induces the electromotive force, \begin{equation} V = - \frac{d\Phi}{dt}. \end{equation} (8) When conduction electrons move around along the ring, the charge Berry phase, which corresponds to the Aharonov–Bohm phase,⁶⁸^–⁷¹⁾ is added to the phase component of the wave function of the conduction electrons. The charge Berry phase is proportional to the magnetic flux and is represented as \begin{equation} \gamma_{e} = \left(- \frac{e}{\hbar} \right)\Phi. \end{equation} (9) By substituting Eqs. (9) to (8), we can obtain the relationship between the electromotive force and the charge Berry phase as \begin{equation} E = - \frac{\hbar}{(-e)}\frac{d\gamma_{e}}{dt}. \end{equation} (10) Barnes and Maekawa derived the generalized Faraday's law in condensed matter by rewriting the charge Berry phase \(\gamma_{e}\) to the spin Berry phase.³⁵⁾ For the magnetic case shown in the Fig. 1, one can consider the one-dimensional ferromagnetic metal ring as shown in Fig. 1(b). The spin direction of conduction electrons is assumed to be in the local magnetization direction owing to the s–d exchange interaction in transition metals. When the conduction electrons move around along the ring, the spin Berry phase, which is proportional to the steradian generated by the local magnetization, is added to the phase component of the spin wave function. From the analogy with the charge Berry phase, a change in the spin Berry phase is expected to induce the spin-dependent motive force: \begin{equation} E^{\pm} = - \frac{\hbar}{(- e)}\frac{d\gamma_{s}^{\pm}}{dt}, \end{equation} (11) where \(\gamma_{s}^{\pm}\) is the spin-dependent Berry phase of majority and minority spins. \(\gamma_{s}^{\pm}\) is defined as \(\gamma_{s}^{\pm} =\frac{1}{2}\int A_{s}^{\pm}\cdot dr\), and \(A_{s}^{\pm} = 2i\langle \psi^{\pm}|\nabla_{r}\psi^{\pm}\rangle\), where \(\psi^{\pm}\) is the wave function of the majority and minority spins that are decoupled by the strong s–d interaction. The net current \(\boldsymbol{j}_{\text{net}}\) consists of majority and minority spin currents: \begin{equation} j_{\text{net}} = \sigma^{+} E^{+} + \sigma^{-}E^{-}. \end{equation} (12) Since the SMF gives the opposite sign for the different spins, the total (charge) force is represented as \begin{equation} E^{\text{smf}} = \frac{\sigma^{+} - \sigma^{-}}{\sigma^{+} + \sigma^{-}}E^{+} = pE^{+}. \end{equation} (13)

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Figure 1. (Color online) (a) Schematic diagram of Faraday's law of induction. (b) Schematic diagram of SMF. The green and blue arrows indicate the magnetization and spin of conduction, respectively.

2.2 Relationship between spin-motive force and topology

The SMF is firstly proposed for the case of the moving domain wall. Clearly, it would appear in the magnetization dynamics of many types of magnetization structure. From Eq. (7), the topological feature of the magnetization structure is related to the SMF. When the magnetization structure is rigid during the dynamics, Eq. (7) can be rewritten as \begin{equation} E_{\mu} = - \frac{p\hbar}{2e}v_{\nu} \boldsymbol{n}\cdot \left(\frac{\partial \boldsymbol{n}}{\partial x_{\nu}} \times \frac{\partial \boldsymbol{n}}{\partial x_{\mu}} \right) = - \frac{p\hbar}{2e}v_{\nu}G. \end{equation} (14)

The quantity G is called the scalar spin chirality and related to the topology of the magnetization structure. For example, this quantity is obtained by summarizing the skyrmion structure and called the topological charge (skyrmion number): \begin{equation} Q = \frac{1}{4\pi} \int \boldsymbol{n}\cdot \left(\frac{\partial \boldsymbol{n}}{\partial x_{\nu}} \times \frac{\partial \boldsymbol{n}}{\partial x_{\mu}} \right)\,dx_{\nu}\,dx_{\mu}. \end{equation} (15) \(v_{\nu}\) is the velocity of the topological charge. From Eq. (14), the direction of the SMF is perpendicular to the velocity of the skyrmion structure. This scalar spin chirality induces the topological Hall effect.⁷²^–⁷⁴⁾ The conduction electrons are bent by the spin-dependent effective magnetic field that comes from the spin Berry phase, as described in the previous section.

2.3 SMF with spin–orbit coupling

When the system includes the spin–orbit interaction, the SMF is modified owing to the spin flip process. Similarly to the spin Berry phase in Eq. (11), the spin–orbit interaction induces the Aharonov–Casher (AC) phase \(\gamma_{\textit{AC}}\),⁷⁵^,⁷⁶⁾ which is generated by a cross product of a spin and an electric field \(\gamma_{\textit{AC}}^{\pm} =\frac{1}{2}\int (\boldsymbol{s}\times \boldsymbol{E})\cdot d\boldsymbol{r}\), where \(\boldsymbol{s}\) (\(=\pm 1\)) is the conduction spin. By the analogy of the SMF, we expect that a change in the AC phase induces an electromotive force,⁷⁷^–⁸⁵⁾ which is represented by \begin{equation} V = - \frac{\hbar}{(- e)}\frac{d\gamma_{\textit{AC}}}{dt}. \end{equation} (16) The appearance of the electromotive force requires modulation of an electric field that is applied to a spin, \(\boldsymbol{s}\times \dot{\boldsymbol{E}}\),⁸⁰^,⁸¹^,⁸⁴^,⁸⁵⁾ or change of the spin direction under an electric field, \(\dot{\boldsymbol{s}}\times \boldsymbol{E}\).⁷⁹^,⁸²^,⁸³⁾ The former was studied by Ho et al.⁸⁰^,⁸⁴^,⁸⁵⁾ and Yamane et al.⁸¹⁾ They proposed the ferromagnetic film device having a gate electrode to modulate an electric field [Fig. 2(a)]. The motive force is induced in the direction perpendicular to both the spin and the time-derivative of the electric field by controlling the gate voltage. Its amplitude is expected to increase with increasing speed of change in the gate voltage. In addition, since the requirements for the motive force do not include nonuniform magnetic structures having the spin chirality, it is easy to enlarge the device. It is, however, difficult to rapidly control the gate voltage owing to a large time constant of \(\tau =\mathit{RC}\) in an RC circuit, which is a disadvantage. The latter was studied by Kim et al.⁷⁹⁾ and Tatara et al.⁸²^,⁸³⁾ We consider asymmetric multilayer structures such as AlO_x/Co/Pt, which have a large Rashba electric field in the Co layer [Fig. 2(b)]. When the microwave having the resonant frequency of the ferromagnetic layer is applied to the multilayer, the FMR is excited and the ac electromotive force is expected to be induced in the direction perpendicular to both the electric field and the time derivative of magnetization. According to the calculation by Kim et al.,⁷⁹⁾ the electromotive force is represented by \begin{equation} \boldsymbol{E}^{R} = \alpha_{R}\frac{m_{e}}{e\hbar} \left(\boldsymbol{e}_{z} \times \frac{\partial \boldsymbol{n}}{\partial t} \right), \end{equation} (17) where \(\alpha_{R}\boldsymbol{e}_{z}\) is the Rashba field. The electromotive force induced by the domain wall motion is expected to be enhanced approximately 100 times compared with the conventional SMF.

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Figure 2. (Color online) Schematic illustrations of measurement setups. (a) The SMF is induced by modulating the electric field applied between the gate electrode and the ferromagnetic layer. (b) The SMF is induced by magnetization precession excited by an ac magnetic field.

3. Experimental Aspect of SMF in Ferromagnetic Materials

There are few experimental reports on the SMF owing to the difficulty to detect it, compared with the spin pumping effect, spin-torque FMR, and other rectification effects. Both nonuniform magnetic structures and their dynamics are needed to induce the SMF. The Kittel mode, which is the most famous magnetization dynamics in the fields of magnetic engineering and spintronics, has a uniform magnetic structure and it does not induce the SMF. Dynamics of exotic magnetic structures such as a magnetic domain wall and a skyrmion are desired. In addition, we cannot apply a conventional dc circuit model the so-called lumped constant circuit, in the nanosecond region, corresponding to a time scale of typical magnetization dynamics. A distributed constant circuit, which considers the size of the devices, should be applied to a device structure. Moreover, although we often use a magnetic field generated by an electromagnet to control magnetization dynamics, a larger electromagnet has a higher inductance, and it is difficult to control an electric current flowing through a large electromagnet on nanosecond order. A few research groups have succeeded in observating the SMF to resolve or avoid these problems. Yang et al. prepared three electromagnets having different roles to control the magnetic domain wall motion in the ferromagnetic wire.⁴⁷^,⁴⁸⁾ They generated short pulse currents using a small electromagnet with a low inductance. Hayashi et al. also used multiple magnetic fields to control the magnetic domain wall motion.⁵¹⁾ They fabricated a nanowire in the device and controlled the domain wall motion using the short-pulse current flowing through the nanowire. Tanabe et al. focused on a gyration mode of a magnetic vortex, which is a continuous resonant mode.⁵⁰⁾ The mode is controlled by low-rf magnetic fields. Yamane et al. showed that a dc voltage is induced at the boundary between a fixed magnetization and a precessing magnetization.⁴⁹⁾ The method allows us to easily detect the SMF as long as we use an asymmetric structure. Their excellent idea has affected several studies by Nagata et al.⁵⁷⁾ and Zhou et al.⁶³⁾ Details of the experiments are shown in the section below.

3.1 SMF induced by magnetic domain wall motion

The first experimental report is about the SMF induced by the magnetic domain wall motion in nanowires.⁴⁷⁾ Because the SMF directly depends on the magnetization dynamics, the dynamics of the magnetic moments that the magnetic domain wall consists of is very important. In in-plane magnetic films such as permalloy, there are typically two magnetic domain wall structures: transverse wall and vortex wall. In addition, the field-driven magnetic domain wall motion has two modes as shown in Figs. 3(a) and 3(b). One is the steady motion, which indicates that the domain wall structure does not change, under a low magnetic field. The other is the precession mode under a magnetic field higher than the critical field, the so-called Walker breakdown field.⁸⁶^–⁸⁸⁾ In general, the velocity of the domain wall motion rapidly decreases when the magnetic field exceeds the Walker breakdown field and, after that, the velocity increases with increasing magnetic field. From Eq. (7), the temporal and spatial derivatives of the magnetic moment must be orthogonal to induce the SMF. Hence, since the temporal derivative is approximately equal to the spatial derivative in the steady motion, the steady motion does not induce the SMF. On the other hand, the magnetic moment in the domain wall rotates along the magnetic field direction in the precession mode following the Landau–Lifshitz–Gilbert (LLG) equation. Since the time derivative becomes perpendicular to the spatial derivative in the precession mode, the precession mode is expected to induce the SMF. Since the total amount of the SMF voltage during a period, \(\tau = 1/f\), is topologically protected and the frequency of the precession is expressed by \(f = (1/2\pi)\gamma H\), the electromotive force is proportional to the external magnetic field, \begin{equation} V = \frac{h\gamma}{2\pi e\mu_{0}}\mu_{0}H, \end{equation} (18) where \(\mu_{0}\) is the vacuum permeability. The proportionality constant is about \(h\gamma/2\pi e\mu_{0}\sim 116\) nV/mT.

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Figure 3. (Color online) Schematic diagrams of steady motion (a) and precession mode (b) in domain wall motions. Gray arrows indicate the magnetization. The domain walls, which are labeled as blue bars, are driven by the external magnetic field from left to right.

Yang et al. demonstrated the observation of the SMF induced by the magnetic domain wall motion in a ferromagnetic permalloy wire 500 nm wide, 20 nm thick, and 35 µm long using the special modulation technique [Fig. 4(a)].⁴⁷^,⁴⁸⁾ The important point is that the sign of the SMF is independent of domain wall structures such as the head-to-head and tail-to-tail walls but depends on the direction of the domain wall motion. Hence, when the domain wall repeatedly moves in one direction regardless of the structures, the constant electromotive force is averagely expected to be induced by the moving domain walls. They controlled the domain wall motion using three external magnetic fields. The first one is the magnetic field to nucleate a domain wall at the left edge of the magnetic wire, and both the head-to-head and tail-to-tail walls are repeatedly generated by the positive and negative magnetic fields, respectively. The second one is the magnetic field to drive the domain wall from the left to the right edge in the wire, and its amplitude is larger than the depinning field in the wire. The last one is the modulation field for the homodyne detection. They obtained the SMF that is proportional to the driving magnetic field under the magnetic field and over the Walker breakdown field [Fig. 4(b)]. They estimated the spin polarization \(P = 0.85\) from the slope.

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Figure 4. (Color online) (a) Schematic illustration of experimental setup. The inset indicates the image of the permalloy wire. (b) Wall velocity and wall-induced voltage as functions of drive magnetic field. The electromotive force of hundreds of nanovolts has been detected under the magnetic field of more than 10 Oe, which corresponds to the Walker breakdown field. Reprinted with permission from Ref. 47. © 2009 American Physical Society.

Later, Hayashi et al. succeeded in the real-time observation of the SMF induced by the magnetic domain wall motion in the ferromagnetic permalloy wires 300 and 600 nm wide, 20 nm thick, and 1, 2, 4, and 8 µm long.⁵¹⁾ Their setup is shown in Fig. 5(a). Electrodes A and D are to nucleate domain walls and electrodes B and C are used to detect the SMF. When the domain wall nucleated under electrode A (or D) enters the region between B and C, the SMF is detected by an oscilloscope. They repeated the real-time measurements about 16,000 times and obtained the average signals. They have studied the dependence on the head-to-head and tail-to-tail walls and removed the conventional electromotive force from the difference in dependences of between the SMF and the conventional electromotive force [Figs. 5(b) and 5(d)]. The SMF detected by Hayashi et al. is also proportional to the external magnetic field as shown in Fig. 5(e), and the spin polarization is estimated to be 0.69 from the slope.

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Figure 5. (Color online) (a) Schematic illustration of experimental setup. The gray and brown strips indicate the permalloy wire and the electrodes to create domain walls and to detect the electromotive force, respectively. (b) Real-time observation of electromotive force including conventional electromotive force and SMF. (c, d) Conventional electromotive force (c) and SMF (d) extracted from signals of the head-to-head and tail-to-tail walls. (e) Field dependence of the SMF. Reprinted with permission from Ref. 51. © 2012 American Physical Society.

3.2 SMF induced by gyrating magnetic vortex

The magnetic vortex, which is a swirling in-plane magnetic configuration in a magnetic disc, is one of the most fundamental nonuniform magnetic states. There is an area with the perpendicular component of the magnetization at the center of the vortex state, which is termed a vortex core. Progress on the topological aspect on the skyrmion allows the topological classification of the magnetic vortex. Although the skyrmion has the skyrmion number of \(Q = +1\), the magnetic vortex has the skyrmion number of \(Q = +1/2\). The skyrmion number Q is independent of the continuous deformation of the spin arrangement in a magnetic structure and it is a topologically protected number. In addition, Q is closely related to an effective magnetic flux. The effective magnetic flux is a surface integral of the magnetic field and is derived as \begin{equation} \Phi = \frac{h}{e}\left\{\frac{1}{4\pi} \int \boldsymbol{n}\cdot \left(\frac{\partial \boldsymbol{n}}{\partial x} \times \frac{\partial \boldsymbol{n}}{\partial y} \right)\,dx\,dy \right\}. \end{equation} (19) The flux is proportional to the skyrmion number and its proportionality constant is the quantum flux, \(h/e\sim 4\times 10^{-15}\) Wb. In addition, the skyrmion density, which is deeply related to an effective magnetic field, is also defined as \begin{equation} q = \frac{1}{4\pi} \boldsymbol{n}\cdot \left(\frac{\partial \boldsymbol{n}}{\partial x} \times \frac{\partial \boldsymbol{n}}{\partial y} \right). \end{equation} (20) The effective magnetic field is proportional to the skyrmion density: \(B = (h/e)q\). Since the magnetic vortex has a nonzero skyrmion number, it has the effective magnetic flux. When the size of the vortex core is assumed to be 10 nm,⁸⁾ the magnitude of the effective magnetic field is roughly estimated to be \((h/e)(Q/A)\sim 20\) T, which is inversely proportional to the core area. Here, A is the core area. The effective magnetic flux is estimated as \((h/e)Q\sim 2\times 10^{-15}\) Wb, which is half that of the skyrmion. When this effective magnetic flux moves, the electromotive force is expected to be induced by the magnetic flux, which is the SMF in the magnetic vortex. The magnetic vortex has the resonant mode of in-plane rotation in the disc, and its frequency is determined by the aspect ratio of the diameter and the thickness of the magnetic disc. Tanabe et al. reported the observation of the SMF of 1 µV in the magnetic vortex having the resonant frequency of \(f\sim 80\) MHz [Figs. 6(a) and 6(e)].⁵⁰⁾ Although the value of the SMF is larger than the theoretical value, \((h/e)Qf\sim 2\times 10^{-7}\) V, the reason is that the time scale of the measurement is shorter than \(1/f\) owing to the local-space measurement. The value is consistent with the results of the numerical calculation of the micromagnetics simulation shown in detail in Fig. 6(f).

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Figure 6. (Color online) Schematic illustrations of perpendicular components (a) and the in-plane components (b) of magnetic vortex, inducing SMF (c), and experimental setup (d). (e) Real-time observation of the electromotive force. (f) Simulated SMF. Reprinted with permission from Ref. 50. © 2012 Nature Publishing Group.

Recent theoretical studies have shown that an SMF is induced by the dynamics of the skyrmion, which has a spin structure similar to the magnetic vortex but a different skyrmion number. The skyrmion has three resonant modes: clockwise, counterclockwise, and breathing modes. The clockwise and counterclockwise modes are quite similar to the resonant dynamics in the magnetic vortex. Compared with the magnetic vortex, a skyrmion has higher resonance frequencies and double the skyrmion number. Hence, the SMF induced by the skyrmion motion is expected to become larger [\((h/e)Qf\sim 4\) µV as \(f\sim 1\) GHz].

3.3 SMF induced by ferromagnetic resonance

FMR is one of the most fundamental magnetization dynamics. The whole magnetic moment synchronously precesses around the direction of a magnetic field. The conventional FMR is not expected to induce the SMF because the spin structure having the spin chirality does not appear. Yamane et al. focused on an asymmetrically shaped thin film, which consists of a large pad and an array of wires [Figs. 7(a) and 7(c)]. The resonant frequency depends on the shape of the sample. The LLG equation is represented as \begin{align} \frac{d\boldsymbol{M}}{dt} &= - \gamma (\boldsymbol{M}\times \boldsymbol{H}^{\text{eff}}) + \frac{\alpha}{M_{s}}\left(\boldsymbol{M}\times \frac{d\boldsymbol{M}}{dt} \right), \end{align} (21) \begin{align} \boldsymbol{H}^{\text{eff}} &= \boldsymbol{H}_{0} + \boldsymbol{H}_{a} - \tilde{N}\boldsymbol{M}, \end{align} (22) where \(\boldsymbol{H}^{\text{eff}}\) is an effective magnetic field, \(\boldsymbol{H}_{0}\) is the external magnetic field, \(\boldsymbol{H}_{a}\) is the magnetic anisotropic field, and \(\tilde{N}\) is the demagnetizing tensor. The resonant frequency is derived from the LLG equation: \begin{equation} f = \frac{\gamma}{2\pi} \sqrt{\{H_{0} + (N_{y} - N_{z})M_{s}\}\{H_{0} + (N_{x} - N_{z})M_{s}\}}, \end{equation} (23) where \(N_{i}\) is a demagnetizing coefficient. Here, we neglect the magnetic anisotropy for simplicity. Since the demagnetizing coefficient is characterized by the shape of the sample, the resonant frequency depends on the shape. Hence, the application of an rf magnetic field can generate the condition under which only the magnetization in the pad precesses steadily while the magnetization in the array of the wires is static with the fixed axis and vice versa. In this case, the SMF appears between the pad and the wires. This SMF is a continuous dc voltage unlike the SMF in the experiments on the domain wall and the magnetic vortex.

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Figure 7. (Color online) (a) Schematic illustration of experimental setup. (b) Microwave absorption and (b) SMF signals due to FMR. (d) Microwave power dependence of SMF. Experiment and calculation correspond to open and closed symbols, respectively. The magnitude is roughly proportional to the microwave power. Reprinted with permission from Ref. 49. © 2011 American Physical Society.

The sample is measured using a modulation method in a cavity at room temperature. The frequency of the microwave is 9.43 GHz. A static magnetic field is applied perpendicular to the wires as shown in Fig. 7(c). The static magnetic field modulation at 100 kHz is 2 mT, which is sufficiently smaller than the FMR linewidth of 8 mT. Figures 7(b) and 7(c) show the microwave absorption and voltage derivative signals for a microwave power of 200 mW, respectively. Two peaks appear both in the absorption and the electromotive force around 120 and 230 mT, corresponding to the resonances of the pad and the wires, respectively. Whereas similarly shaped signals appear in the FMR absorption, the shape of the signal is reversed in the voltage, indicating the sign change of the electromotive force. The reversal stems from the reversal of the relationship between the excited and static magnetizations. The precession angle increases with increasing power. By rough estimation, we can represent the induced voltage as \begin{equation} V = \frac{p\hbar \omega}{2e}(\cos \theta_{W} - \cos \theta_{P}), \end{equation} (24) where \(\theta_{W}\) and \(\theta_{P}\) are the precession angles in the wire and the pad, respectively. Here, we assume that the precession trajectory is circular. Figure 7(d) shows the microwave-power dependence of the electromotive force. The detected voltage is proportional to the power, which is consistent with the theory because the precession angle is proportional to the square root of the power when the power is sufficiently small.

3.4 Other important experiments

Hai et al. succeeded in detecting a huge SMF in the magnetic tunneling junction that contains zinc-blende-structured MnAs ferromagnetic nanoparticles.⁸⁹⁾ The voltage of 3–7 mV is observed, which is quite larger than those in other experiments, and they claimed that the reason for this is the large spin angular momentum in MnAs particles. The huge electromotive force, however, continues to be detected over a period of 10 min and Ralph claimed a violation of the energy conservation law.⁹⁰⁾ The discrepancy is still controversial and more research is necessary.

Zhou et al. reported the observation of the SMF in the out-of-plane direction induced by the spinwave excitation in bilayer devices.⁶³⁾ The vertically structured devices are an area of research interest in multilayer thin films, which have become a building block for recent spintronic devices. In addition, their study is the first report on the observation of the SMF generated in the heterojunction between two different ferromagnetic layers, suggesting that complicated magnetic states in multiple junctions may become a good tool to enhance the SMF.

In 2019, Nagaosa proposed a novel method of inducing inductance in a helical magnet using the SMF,⁶²⁾ and in 2020, Yokouchi et al. succeeded in detecting the SMF as the inductance in the helical magnet Gd₃Ru₄Al₁₂ at low temperatures.⁶⁴⁾ The inductance is inversely proportional to the cross section of the sample, which enables the reduction in the size of the device, unlike the conventional solenoid-type inductor. Actually, they realized the inductor in a volume about a million times smaller than a commercial inductor.

4. SMF in Ferrimagnetic Materials

The amplitude of the SMF is characterized by both the temporal derivative of the magnetization and the deformation of the magnetic structure. Materials and (/or) structures having a high resonant frequency and large deformation of magnetic structures are desired to enhance the SMF. Since the ferrimagnetic materials satisfy such conditions, they are suitable materials and the studies on the SMF in ferrimagnetic materials may lead to new developments in this field.

4.1 Ferrimagnetic rare-earth/transition metal alloys

Ferrimagnetic materials have been attracting much attention owing to their much higher resonant frequencies than ferromagnetic materials and the easy control of the magnetization, compared with antiferromagnetic materials. In particular, magnetic parameters such as saturation magnetization, coercive force, and Gilbert damping constant are easily controllable in rare-earth (RE)/transition metal (TM) alloys by changing the compositions of the RE and the TM alloys,⁹¹^–⁹⁸⁾ and various types of the alloys were previously used in magnetooptical drives. The magnetic moment of the transition metal is antiferromagnetically coupled with that of a RE metal. Hence, the saturation magnetization decreases with increasing RE metal concentration in RE_xTM\(_{1 - x}\) and reaches zero, which is termed the magnetization compensation point (MCP, \(x = x_{m}\)). When \(x > x_{m}\), the saturation magnetization increases with increasing RE metal concentration in RE_xTM\(_{1 - x}\). When the net magnetization is parallel to the magnetic moment of the RE (TM) metals, the alloy is termed RE-rich (TM-rich). RE/TM alloys such as GdFe and TbCo tend to have a large perpendicular magnetic anisotropy near the MCP. Since the RE and TM metals have different g factors, which depend on the spin–orbit interaction, the gyromagnetic ratio is different between these metals. The gyromagnetic ratio is a significantly crucial parameter that combines an angular momentum in mechanics and a magnetic moment in electromagnetism, which is famous for being studied by Barnet, Einstein, and de Haas studied over 100 years ago.⁹⁹^,¹⁰⁰⁾ The difference in the gyromagnetic ratio generates a difference between the compensation points in angular momentum and a magnetization. In the case of the GdFeCo alloys, the angular-momentum compensation point (ACP), \(x_{a}\), is a few percent smaller than the MCP, \(x_{m}\), at room temperature. Moreover, the Gilbert damping constant in the RE/TM alloys strongly depends on the composition. The constant increases towards infinity near the ACP and this feature has been used for ultrafast magnetization reversal.⁹²⁾ Recently, Imai et al. have succeeded in the direct observation of the ACP, where the sign of the gyromagnetic ratio is changed, in a ferrimagnet insulator (HoDy)₃Fe₅O₁₂ by using the Barnett effect measurement technique.¹⁰¹^,¹⁰²⁾

Here, we consider the dynamics of the sublattice magnetization. The LLG equation in ferrimagnetic materials is represented as \begin{equation} \frac{d\boldsymbol{M}_{i}}{dt} = - \gamma_{i}(\boldsymbol{M}_{i} \times \boldsymbol{H}_{i}^{\text{eff}}) + \frac{\alpha_{i}}{\boldsymbol{M}_{i}}\left(\boldsymbol{M}_{i} \times \frac{d\boldsymbol{M}_{i}}{dt} \right)\quad (i = 1,2), \end{equation} (25) where \(\boldsymbol{M}_{i}\) is the magnetization vector, \(\gamma_{i}\) is the gyromagnetic ratio, and \(\alpha_{i}\) is the Gilbert damping constant. \(\boldsymbol{H}_{i}^{\text{eff}}\) is the effective magnetic field and is represented as \begin{equation} \left\{ \begin{array}{l} \boldsymbol{H}_{1}^{\text{eff}} = \boldsymbol{H}_{0} + \boldsymbol{H}_{A1} + \lambda \boldsymbol{M}_{2} - \tilde{N}(\boldsymbol{M}_{1} + \boldsymbol{M}_{2})\\ \boldsymbol{H}_{2}^{\text{eff}} = \boldsymbol{H}_{0} + \boldsymbol{H}_{A2} + \lambda \boldsymbol{M}_{1} - \tilde{N}(\boldsymbol{M}_{1} + \boldsymbol{M}_{2}) \end{array}\right., \end{equation} (26) where \(\boldsymbol{H}_{Ai}\) is the anisotropic magnetic field, \(\lambda\boldsymbol{M}_{i}\) is the exchange magnetic field, and \(\tilde{N}\) is the demagnetizing tensor. For simplicity, we neglect the magnetic anisotropy and assume the sphere sample. Two resonance modes are derived from the LLG equation. One is similar to the conventional FMR. The frequency is represented as \begin{align} \omega_{A} &= \gamma^{\text{eff}}H_{0}, \end{align} (27) \begin{align} \gamma^{\text{eff}} &= \frac{M_{1} - M_{2}}{M_{1}/\gamma_{1} - M_{2}/\gamma_{2}}. \end{align} (28) When the composition is far from the compensation points or \(\gamma_{1}\) is close to \(\gamma_{2}\), \(\gamma^{\text{eff}}\) is approximately \(\gamma_{i}\). Even if near the compensation points, the mode is almost the same as the FMR by using \(\gamma^{\text{eff}}\) instead of \(\gamma_{i}\). The other is derived as \begin{equation} \omega_{B} \sim \lambda (\gamma_{1}M_{2} - \gamma_{2}M_{1}), \end{equation} (29) whose resonant frequency, which depends on materials, is often in the THz region owing to the large exchange interaction. Kim et al. demonstrated ultrafast domain wall motion induced by the external magnetic field by using high-speed magnetization dynamics near the ACP in Gd₂₃Fe_67.4Co_9.6 alloys.⁹⁵⁾ The ferrimagnetic materials have many material properties such as controllable saturation magnetization, sublattice magnetization, Gilbert damping constant, controllable magnetic anisotropy, the MCP, ACP, and two different resonant modes, and are good tools for studying the relationship between the SMF and the material properties.

4.2 SMF in ferrimagnetic materials

Fukuda et al. reported the detection of the SMF in Gd_x(Fe₈₂Co₂₈)\(_{1-x}\)/Pt strips.⁶⁵⁾ The Pt layer under the GdFeCo layer is used as an electrode for rf currents that induce spin torques and (/or) rf Ampère magnetic fields, which is quite similar to an experimental method of the spin-torque FMR.¹⁰³⁾ When an rf current is injected into a bilayer of GdFeCo/Pt, a magnetic resonance is excited by the rf current in the Pt layer. Since the rf current density \(j_{e}\) is inversely proportional to the width w of the strip, the angle of the magnetization precession in the trapezoidal strip is changed in the longitudinal direction of the strip [Fig. 8(a)]. Hence, the excited magnetic resonance becomes nonuniform dynamics and is expected to induce the SMF. Figure 8(b) shows the Gd-composition dependence of the SMF \(V_{1}\). Except for \(x = 0.12\), the detected voltage increases with increasing Gd composition as \(x < 0.17\), which indicates that the SMF can become larger than that in alloys that consist of only transition metals such as the FeCo alloy via the Gd doping.

»View larger version

Figure 8. (Color online) (a) Schematic illustration of experimental setup. (b) Detected electromotive force as a function of Gd composition. The deposited GdFeCo film has the magnetization compensation point at about \(x_{m} = 0.23\). The GdFeCo film is in-plane-magnetized from \(x = 0.0\) to 0.17 and in \(x > 0.27\) owing to the large demagnetizing fields. (c) \(\alpha^{2}V_{1}\) as a function of Gd composition. The blue curve is a guide to the eye, which is obtained using Eq. (32). Reprinted with permission from Ref. 65. © 2020 American Institute of Physics.

To understand the enhancement of the SMF via the Gd doping as \(x < 0.17\), the experimental data were analyzed by modeling the motion of the magnetic moment in GdFeCo by assuming that the net magnetic moment that consists of Gd and FeCo is a single magnetic moment. The LLG equation used is represented as \begin{equation} \frac{d\boldsymbol{n}}{dt} = - \gamma \boldsymbol{n}\times \boldsymbol{H}^{\text{eff}} + \alpha \boldsymbol{n}\times \frac{d\boldsymbol{n}}{dt} + \frac{\hbar}{2e\mu_{0}M_{s}d}J_{s}(\boldsymbol{n}\times \boldsymbol{s}) \times \boldsymbol{n}. \end{equation} (30) Here, \(\boldsymbol{n}\) is a normalized net magnetization vector of GdFeCo, \(M_{s}\) is the net saturation magnetization of GdFeCo, \(\boldsymbol{s}\) is the spin magnetic moment, and d is the thickness of the GdFeCo layer. The xyz-coordinate direction is defined as the xyz arrows in Fig. 8(a). \(\boldsymbol{H}^{\text{eff}} = (h_{x}\ -(M_{s}-H_{y}^{\textit{ani}})n_{y}\ H_{0})^{T}\), where \(h_{x}\) is the Ampère field induced by the rf current, \(-M_{s}n_{y}\) is the demagnetizing field, \(H_{y}^{\textit{ani}}\) is the anisotropic field, and \(H_{0}\) is the external magnetic field. When \(\alpha\ll 1\), \(| n_{x} |,| n_{y} |\ll 1\), \(| h_{x} |\ll | H_{0} |\), \(h_{x} = he^{i\omega t}\), and \(J_{s} = je^{i\omega t}\), \begin{align} &\left\{ \begin{array}{l} n_{x} = \dfrac{\sqrt{S_{H}} \sqrt{(\omega \gamma h)^{2} + (\omega_{R}\beta j)^{2}}}{\alpha \gamma (2H_{R} + M_{s} - H_{y}^{\textit{ani}})}\dfrac{1}{\omega_{R}}\cos (\omega t + \delta)\\ n_{y} = \dfrac{\sqrt{S_{H}} \sqrt{(\omega \gamma h)^{2} + (\omega_{R}\beta j)^{2}}}{\alpha \gamma (2H_{R} + M_{s} - H_{y}^{\textit{ani}})}\dfrac{1}{\omega} \sin (\omega t + \delta) \end{array}\right., \end{align} (31) \begin{align} &\tan \delta = \frac{A_{H}\omega_{R}\beta j - S_{H}\omega \gamma h}{S_{H}\omega_{R}\beta j + A_{H}\omega \gamma h},\notag \end{align} where \(\omega =\gamma \sqrt{H_{R}(H_{R} + M_{s} - H_{y}^{\textit{ani}})}\) is the angular frequency of the rf current, \(\omega_{R} =\gamma H_{R}\), \(\beta =\hbar \|\boldsymbol{s}\|/2e\mu_{0}M_{s}d\), \(S_{H} =\Delta H^{2}/((H_{0} - H_{R})^{2} +\Delta H^{2})\), and \(A_{H} = (H_{0} - H_{R})\Delta H/((H_{0} - H_{R})^{2} +\Delta H^{2})\). \(\Delta H\) is the full width at half maximum and \(\Delta H = (2\pi\alpha/\gamma)f\). The results reveal that the magnetization trajectory is not a circular orbit, but an elliptical orbit when \(4\pi M_{s} - H_{y}^{\textit{ani}}\neq 0\). In particular, its ellipticity mainly depends on the magnetic anisotropy and the demagnetizing field. By substituting Eq. (32) into Eq. (7), we can roughly calculate \(V_{\text{SMF}}\) as \begin{equation} V_{\text{SMF}} \propto \frac{1}{\alpha^{2}}\{\gamma^{2}(M_{s} - H_{y}^{\textit{ani}})^{2} + 4\omega^{2}\}^{- \frac{1}{2}}. \end{equation} (32) Here, it is assumed that \(\omega\gamma h\cong \omega_{R}\beta j\). Since the SMF is inversely proportional to \(\alpha^{2}\), the SMF should be evaluated as the product of the SMF and the square of the damping constant, \(\alpha^{2}V_{\text{SMF}}\). Equation (32) means that the SMF decreases with increasing frequency and increases with decreasing \(4\pi M_{s} - H_{y}^{\textit{ani}}\). Fukuda et al. claimed that the enhancement of the SMF via the Gd doping originates from the modulation of the magnetization trajectory. Indeed, \(\alpha^{2}V_{1}\) increases with increasing Gd composition up to \(x = 0.16\), which is close to the boundary composition, and decreases with increasing Gd composition after that [Fig. 8(c)]. The curve in Fig. 8(c), which is a guide to the eye, is approximately drawn by using values obtained by solving Eq. (32). Since the curve overlaps with the experimental data, the enhancement by Gd doping can be roughly expressed by the competition between the demagnetizing field and the magnetic anisotropy field. In other words, when the trajectory of the magnetization precession becomes circular, the output of the SMF is maximized.

We now briefly comment on the comparison between the results of the ferrimagnetic materials with the transition metal alloys such as NiFe and FeCo, which are ferromagnetic materials. In the ferromagnetic resonance, the trajectory of the precession is an elliptical orbit with a large eccentricity owing to the large demagnetizing field originating from the saturation magnetization. Hence, the magnitude of the SMF is suppressed, compared with a circular orbit when the frequency is fixed. On the other hand, the magnetic moment compensates and the saturation magnetization is suppressed in ferrimagnetic materials. Moreover, a perpendicular magnetic anisotropy sometimes appears in the GdFeCo alloys. Hence, the trajectory is close to a circular orbit and the magnitude of the SMF is enhanced.

5. Concluding Remarks

Although many theorists have discussed multi directionally, there are only ten experimental reports on the SMF. In contrast, the topological Hall effect, which is a twin of the SMF, has been studied from both theoretical and experimental aspects. In this last section, we summarize three experimental issues on the SMF.

The first issue is the magnitude of the SMF. The magnitudes of the detected SMF are approximately 0.1–1 µV except for that in Ref. 89. To apply spintronic devices, the SMF should be enhanced. The simplest idea is to use more rapid magnetization dynamics because the SMF is proportional to the time derivative of the magnetization. Antiferromagnetic and ferrimagnetic materials having high resonant frequencies and spinwave resonances in ferromagnetic materials may become good subjects of the study on the SMF.⁵⁸^,⁶²⁾ Yamane studied the SMF induced by the domain wall motion in magnetic wires having the Dzyaloshinskii–Moriya exchange interaction.⁵⁹⁾ This result showed about tenfold enhancement of the SMF in the spiral domain wall. Moreover, it may be important to start to study the Rashba SMF. As mentioned earlier, Kim et al. showed that the Rashba SMF may become 100 times larger than the conventional SMF in the magnetic domain wall.⁷⁹⁾

The second issue is that the materials used in the experiments are extremely limited, and the relationship between the SMF and material parameters is still unclear. In almost all the experimental studies, transition metal alloys such as permalloy⁴⁷^–⁵¹⁾ and related oxides such as the magnetite Fe₃O₄⁵⁷⁾ were used, and the experiments were performed at room temperature. To find new guidelines to enhance the SMF, it is necessary to understand the material and temperature dependences of the SMF.

The last is studies on the application of the SMF. The SMF has been little studied from the aspect of applied physics. It is important to propose a clear application so that many researchers in a wide range of fields will become interested in the SMF. Nagaosa⁶²⁾ and Yokouchi et al.⁶⁴⁾ opened the possibility of the development of devices, i.e., inductance, using the SMF. Studies on several applications including inductance are desired.

Acknowledgment

The authors are grateful to T. Ono, K. Kobayashi, D. Chiba, S. Maekawa, H. Kohno, S. E. Barnes, H. Awano, S. Fukuda, and N. Nakamura for fruitful discussions. This work was partly supported by Grants-in-Aid for Scientific Research (C) (No. 20K05307) and (B) (No. 21H01799) from JSPS and by the Tokai Foundation of Technology.

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Author Biographies

Kenji Tanabe was born in Osaka, Japan in 1985. He obtained his B.Sc. (2008), M.Sc. (2010), and Ph.D. (2013) degrees from Kyoto University. He was a specially appointed assistant professor at Institute for Academic Initiatives, Osaka University (2013–2014) and an assistant professor at Department of Physics, Nagoya University (2015–2017). Since 2018, he has been an associate professor at Toyota Technological Institute. He has worked on various fields of materials science such as spintronics, nano-magnetics, and thermoelectrics.

Jun-ichiro Ohe was born in Tochigi, Japan in 1974. He obtained Ph.D. (2002) from Hokkaido University, and has worked as a postdoctoral researcher at Sophia University, Hamburg University, Tohoku University and Japan Atomic Energy Agency. Since 2011, he has been a Lecturer (2011–2013) and an Associate professor (2014–2019) and a Professor (2020–) at the Department of Physics in Toho University. His research interests include spintronics as well as mesoscopic physics and its application.

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Spin-Motive Force in Ferromagnetic and Ferrimagnetic Materials

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