Observation of spin-current striction in a magnet

The interplay among magnetization and deformation of solids has long been an important issue in magnetism, the elucidation of which has made great progress in material physics. Controlling volume and shapes of matter is now indispensable to realizing various actuators for precision machinery and nanotechnology. Here, we show that the volume of a solid can be manipulated by injecting a spin current: a spin current volume effect (SVE). By using a magnet Tb0.3Dy0.7Fe2 exhibiting strong spin-lattice coupling, we demonstrate that the sample volume changes in response to a spin current injected by spin Hall effects. Theoretical calculation reflecting spin-current induced modulation of magnetization fluctuation well reproduces the experimental results. The SVE expands the scope of spintronics into making mechanical drivers.

T he magneto-volume effect [1][2][3] (MVE), one of the magnetostriction effects, has been a central issue in the physics of magnetism in itinerant electron systems for a long time. One notable example is an invar alloy Fe 64 Ni 36 [4][5][6] , where the magneto-volume change compensates for thermal expansion; the volume change is attributed to spin fluctuation in the alloy (see Fig. 1a). Since the discovery of the effect, extensive studies on MVE have made remarkable progress in the physics of spin fluctuation and electronic correlation in ferromagnetic metals 1 .
Recently, a powerful tool for controlling spin fluctuation emerged in the field of spintronics: a spin current [7][8][9][10][11][12] , a flow of spin angular momentum in a solid. By injecting a spin current into ferromagnets, magnetization fluctuation can be modulated via the angular momentum transfer between magnetization and a spin current 13,14 ; as shown in Fig. 1b, when the injected spin current carries spins along the field direction (the z-direction), the spin current turns the magnetization M toward the z-direction via the spin-transfer torque 15 , and the magnetization fluctuation is suppressed. By combining the effect with MVE, a fascinating hypothesis is made: volume of matter could be manipulated by applying a spin current: a spin current volume effect (SVE).
Here, we report an observation of SVEs in a Tb 0.3 Dy 0.7 Fe 2 film. Tb 0.3 Dy 0.7 Fe 2 is a typical ferromagnet exhibiting strong magnetostriction due to spin-lattice coupling [16][17][18][19] . We demonstrate the thickness modulation of the Tb 0.3 Dy 0.7 Fe 2 film by the spin current injection, a spin-current induced magnetostriction effect, which offers a way for magneto-mechanical control of mechanical actuators based on spintronics.

Results
Sample characterization and measurement setup. Figure 1d shows a schematic illustration of the sample system used in the present study. To inject a spin current into a Tb 0.3 Dy 0.7 Fe 2 film, we used the spin Hall effect 20 (SHE) in a paramagnetic metal Pt. When a charge current, j c , is applied to a Pt film, it is converted into a spin current, j s , via the SHE, as shown in Fig. 1c. By putting a Tb 0.3 Dy 0.7 Fe 2 film onto a Pt film on a Si substrate by an electroplating method 21 (see Methods for details), the spin current with the spin polarization σ / j c n is injected into the Tb 0.3 Dy 0.7 Fe 2 film, where n is a normal vector to the interfacial plane. The magnetostriction coefficient of the Tb 0.3 Dy 0.7 Fe 2 film fabricated in the present study was found to be ∼550 × 10 −6 around 1400 Oe 21 (see Supplementary Note 5 for details).
The thickness change of the film is measured by means of laser Doppler vibrometry 22 (LDV). In the measurement, as shown in Fig. 1d, an a.c. spin current is injected into the Tb 0.3 Dy 0.7 Fe 2 film by applying an a.c. electric current j c to the Pt film, where the spin-transfer torque / j c can increase and decrease the M fluctuation alternatively. The M fluctuation decrease (increase) causes the volume expansion (shrinkage) of the Tb 0.3 Dy 0.7 Fe 2 film. Due to the in-plane constraints of the film fabricated on the Si substrate, the volume change should accompany a thickness change of the film, as shown in Fig. 1c. The LDV detects the outof-plane displacement of the film surface in response to the a.c. spin current injection in terms of the Doppler shift of light reflected at the surface of the film. To extract the displacement synchronized with the input a.c. spin current, we performed Fourier transform and obtained its frequency spectra (see Methods for details). All the measurements were performed at room temperature.
Observation of spin current volume effect. Figure 2a shows the obtained frequency f spectrum of the vibrational amplitude A and phase ϕ for the Pt/Tb 0.3 Dy 0.7 Fe 2 sample. We applied j c at the frequency of f AC = 10 kHz and the external magnetic field H = 1115 Oe along the z-axis. Importantly, as shown by the red arrow in Fig. 2a, a clear peak appears at f = 10 kHz (= f AC ) in the A spectrum. The result means that the surface of the Tb 0.3 Dy 0.7 Fe 2 film mechanically vibrates at the same frequency as that of the a.c. current. The j c dependence of A at f = f AC , A peak , also indicates that the amplitude of the mechanical vibration is proportional to j c (see the inset to Fig. 2b). The results exclude the thermal expansion due to the Joule heating (/ j 2 c ), whose frequency is 2f AC (see Supplementary Note 2). We also confirmed that the observed peak exhibits the frequency shift in response to the change in the f AC value (see Supplementary Note 3).
To clarify the origin of the observed mechanical vibration, we carried out control experiments by replacing Pt with paramagnetic metals W and Cu, where the sign of the spin-Hall angle in W is opposite to that in Pt 23,24 , while Cu exhibits minute SHEs 23 . As shown in Fig. 2b, the signed amplitude of the vibration d Acosϕ for a W/Tb 0.3 Dy 0.7 Fe 2 sample exhibits an opposite sign to that for the Pt/Tb 0.3 Dy 0.7 Fe 2 sample. This sign reversal is consistent with the signs of the spin-transfer torque created by Pt and W on Tb 0.3 Dy 0.7 Fe 2 . We also performed a similar experiment for a Cu/Tb 0.3 Dy 0.7 Fe 2 sample. In the sample, the peak of d at f = f AC is much smaller than that in the Pt/Tb 0.3 Dy 0.7 Fe 2 sample (a gray solid curve in Fig. 2b). The results indicate that the observed mechanical vibration is due to the spin current injection via the SHE. Figure 3 shows the magnetic field H dependence of the mechanical vibration. In the inset of Fig. 3b, we show the magnetization curve of the Pt/Tb 0.3 Dy 0.7 Fe 2 sample measured with changing H from −1700 Oe to 1700 Oe (the maximum field we can apply in our LDV system) along the z-direction. In this H range (much less than the magnetization saturation field of the Tb 0.3 Dy 0.7 Fe 2 film ∼7000 Oe), M is not fully saturated along the field direction. To compare the mechanical vibration signal with the magnetization curve, we show the A and ϕ spectra at each H for the Pt/Tb 0.3 Dy 0.7 Fe 2 sample (see Fig. 3c). Here, ϕ at f = f AC , ϕ peak , is shifted by 180°between H = 1410 Oe and H = −1400 Oe, showing that the sign of the vibration displacement is reversed by reversing the field direction. In the entire field range, A peak increases with |H|, while ϕ peak changes by 180°at around H = 0 (blue plots in Fig. 3a). In Fig. 3b, we plot the H dependence of the singed amplitude of the vibration d peak ¼ A peak cosϕ peak . d peak increases with the increase of H, which aligns M along with the field direction. Furthermore, as shown in Fig. 3a, b, d, e, the W/Tb 0.3 Dy 0.7 Fe 2 sample exhibits a clear sign reversal of the displacement (green plots and curves) before the possible oxidization of the W film and the Cu/ Tb 0.3 Dy 0.7 Fe 2 sample shows suppression of the displacement (gray plots and curves), being consistent with the characteristics of the spin current injection via the SHE. The results imply that the observed mechanical vibration originates from the interaction between M and spin currents.
Comparison between experiments and theoretical model for spin current volume effect. Now we discuss the origin of the observed mechanical displacement due to the spin current injection. The injected spin current interacts with m, partial magnetization responsible for the volume effect, and exerts the spintransfer torque 15 τ stt / m ðm σÞ on m. The frequency of the current 10 kHz is much less than that of the magnetization dynamics ∼GHz, and the effective damping and fluctuation of m are modulated by τ stt via the anti-damping spin torque (Slonczewski spin torque 15 ) mechanism; τ stt suppresses (enhances) the m thermal fluctuation and increases (decreases) the thermally averaged m intensity when σ is antiparallel (parallel) to m. The m intensity increase (decrease) induces the expansion (shrinkage) of the Tb 0.3 Dy 0.7 Fe 2 film via the spin-lattice coupling (see Fig. 4b), resulting in the out-of-plane mechanical displacement. In contrast, when σ is perpendicular to m, the m fluctuation remains unchanged due to the cancellation of τ stt , and the volume change does not occur. We note that the effect of the anti-damping torque is maximized when m || σ because it is exerted on the m fluctuation component 7 . Based on the above scenario, we constructed a theoretical model for the SVE. The magnetization dynamics under thermal fluctuation is calculated from the stochastic LLG equation 25 where γ, α, and m s are the gyromagnetic ratio, the magnetic damping coefficient, and the saturation magnetization, respectively. The thermal fluctuation of m is taken into account with the random magnetic field h(t). By combining the above equation, the fluctuation-dissipation theorem, and phenomenological magnetoelastic theory 26 , we derived the mechanical displacement d SVE due to the SVE in the linear response to j s (see Supplementary where k B , T, V, a, and θ are the Boltzmann constant, temperature, the magnetic coherence volume 25 of the Tb 0.3 Dy 0.7 Fe 2 film, a known constant parameter, and the relative angle between H and j c (see Fig. 4a), respectively. Here, the result gives us the θ dependence of the SVE: d SVE is proportional to sinθ. When the external field is much weaker than the magnetization saturation field, d SVE / sinθ, where H in Eq. (1) is replaced with the internal magnetic field in each magnetic domain and θ represents the relative angle between local magnetization in each domain and j c . The sinθ averaged over the magnetic domains increases with the external magnetic field application, consistent with the observed H dependence of d peak (see also Supplementary Note 8). Figure 4c shows the θ dependence of the measured d peak . We found that d peak exhibits a clear sinθ dependence for both the Pt/ The agreement between the experimental results and the theoretical calculation supports our interpretation that the observed mechanical vibration is attributed to the volume change due to the SVE.

Discussion
Here, we discuss the influence of other effects on the observed mechanical displacements in the (Pt, W, and Cu)/Tb 0.3 Dy 0.7 Fe 2 samples. The first one is Lorentz force due to the a.c. current under the magnetic fields. By carrying out control experiments using (Pt, W, and Cu)/Si samples without the Tb 0.3 Dy 0.7 Fe 2 layer, we found that the mechanical peak signal disappears in the absence of the Tb 0.3 Dy 0.7 Fe 2 layer (see Supplementary Note 1). The result implies that the Lorentz force is irrelevant to the observed mechanical effect. We also examined the Oersted field effect due to the a.c. current flowing in the paramagnetic metals which might induce magnetostriction of the Tb 0.3 Dy 0.7 Fe 2 film, but we found that this cannot be responsible for the observed paramagnetic metal dependence (sign reversal between Pt/ Tb 0.3 Dy 0.7 Fe 2 and W/Tb 0.3 Dy 0.7 Fe 2 ), although the small mechanical signal in the Cu/Tb 0.3 Dy 0.7 Fe 2 sample might be attributed to such Oersted field effects or a small finite SHE in the Cu film [27][28][29] .
In summary, we found spin current volume effects (SVEs), volume modulation by spin current injection, in Tb 0.3 Dy 0.7 Fe 2 films. The SVE observed here enables the direct mechanical actuation of a magnetostrictive thin film by using a spin current, which can be applied to making mechanical actuators driven by spin currents free from electricity. The high controllability of the SVE in terms of magnetic fields will present great advantages in designing spintronics-based mechanical devices.

Methods
Sample preparation. We used an electroplating method 21 to grow the Tb 0.3 Dy 0.7 Fe 2 film on the paramagnetic metals (see Supplementary Note 5 for details). The polycrystalline Tb 0.3 Dy 0.7 Fe 2 film with the thickness of ∼100 nm was fabricated on the Pt film with the thickness of 140 nm (the W and Cu films with the thickness of 100 nm), which was sputtered as a seed electrode for electroplating on a Si substrate with the size of 20 mm × 20 mm. The obtained samples were cut into 2 mm wide and 10 mm long pieces. The resistance of the Tb 0.3 Dy 0.7 Fe 2 film is in the order of 0.1 MΩ while the resistance of the paramagnetic films is less than 10 Ω. Therefore, when an electric current flows in the (Pt, W, and Cu)/Tb 0.3 Dy 0.7 Fe 2 samples, the portion of the electric current in the paramagnetic metal films is much greater than that in the Tb 0.3 Dy 0.7 Fe 2 film. The spin current is injected into the Tb 0.3 Dy 0.7 Fe 2 film via the SHE in the paramagnetic films and it modulates magnetization fluctuation (a type of the reverse processes of the dynamic spin pumping 7,13 ). In electroplating methods, it is not possible to fabricate a single layer of Tb 0.3 Dy 0.7 Fe 2 without the seed electrode films, such as Pt.
Mechanical vibration measurement setup. The samples were fixed with varnish on a stage located between the magnetic poles of an electromagnet. An a.c. charge current was applied to the samples to induce the SVE. The mechanical vibration of the sample surface was measured by means of LDV, where a laser light with the wavelength λ = 532 nm was split into a reference beam and an incident beam. The incident laser beam was focused on the Tb 0.3 Dy 0.7 Fe 2 film surface. The reflected light from the sample surface was analyzed with an LDV system (MSA-100-3D, Polytec, Inc.) to obtain the displacement and velocity of the surface along the laserbeam direction as a function of time. The data were Fourier transformed into f spectra of A and ϕ. All the measurements were performed at room temperature and in a high vacuum of ∼10 −4 Pa.

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.

Code availability
The codes used in theoretical calculations are available from the corresponding author upon reasonable request.