Research ArticlePHYSICS

Acoustic spin Hall effect in strong spin-orbit metals

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Science Advances  06 Jan 2021:
Vol. 7, no. 2, eabd9697
DOI: 10.1126/sciadv.abd9697

Abstract

We report on the observation of the acoustic spin Hall effect that facilitates lattice motion–induced spin current via spin-orbit interaction (SOI). Under excitation of surface acoustic wave (SAW), we find that a spin current flows orthogonal to the SAW propagation in nonmagnetic metals (NMs). The acoustic spin Hall effect manifests itself in a field-dependent acoustic voltage in NM/ferromagnetic metal bilayers. The acoustic voltage takes a maximum when the NM layer thickness is close to its spin diffusion length, vanishes for NM layers with weak SOI, and increases linearly with the SAW frequency. To account for these results, we find that the spin current must scale with the SOI and the time derivative of the lattice displacement. These results, which imply the strong coupling of electron spins with rotating lattices via the SOI, show the potential of lattice dynamics to supply spin current in strong spin-orbit metals.

INTRODUCTION

Spin current represents a flow of spin angular momentum carried by electrons. The spin Hall effect (1) allows electrical generation of spin current in materials with strong spin-orbit interaction (SOI) (2). The spin Hall angle, a material parameter that characterizes charge to spin conversion efficiency, scales with the longitudinal resistivity and the spin Hall conductivity (3). For the intrinsic spin Hall effect, the spin Hall conductivity is determined by the electron band structure (4, 5) (i.e., the Berry curvature of the bands near the Fermi level) and the SOI of the host material. As spin current can be used to control the direction of magnetization of a ferromagnetic layer placed adjacent to the spin source, developing materials and means to create it with high efficiency are the forefront of modern spintronics (68).

Recent studies have shown that not only electrons but also other degrees of freedom can generate spin current. Precessing magnetization pumps out spin current from magnetic materials, a mechanism known as spin pumping (911). In the spin Seebeck effect (12, 13), a temperature gradient applied to a magnetic material induces a magnon population gradient and the associated diffusion spin current. Spin current can also be produced from exchange of angular momentum between a rotating body and electrons, an effect referred to as spin-rotation coupling (14). The effect has been observed in liquid metals (15) and nonmagnetic light metals (e.g., Cu) (16). Generation of spin current via spin pumping, spin Seebeck effect, and spin-rotation coupling does not require large SOI of the host material.

Here, we show a profoundly different approach to generate spin current. We find a spin current directly emerges from the dynamics of lattice via SOI. Similar to the spin Hall effect where a spin current flows transverse to electrical current, a spin current develops orthogonal to the propagation direction of a surface acoustic wave (SAW) in nonmagnetic metals (NMs). The efficiency to generate spin current is proportional to the spin Hall angle and may be influenced by a factor that depends on the film texture. To account for the experimental results, we find that the spin current must scale with the SOI and the time derivative of the lattice displacement.

RESULTS

Experimental setup

Thin film heterostructures are grown on piezoelectric LiNbO3 substrates using radio frequency (rf) magnetron sputtering. The film structure is sub./X(d)/CoFeB(1)/MgO(2)/Ta(1) with X = W, Pt, Ta, and Cu (thickness in unit of nanometers). The heterostructures are referred to as X/CoFeB bilayers hereafter. Standard optical lithography is used to pattern Hall bars from the film and electrodes/interdigital transducers (IDTs) (17) from conducting metals (see Materials and Methods for the details of sample preparation).

The experimental setup and the coordinate system are schematically illustrated in Fig. 1A. The Hall bar is placed between the two IDTs. Figure 1B shows a representative optical microscope image of the device. A vector network analyzer (VNA) is connected to the IDTs to excite a Rayleigh-type SAW from one end and to detect its transmission at the other end. Figure 1C shows typical transmission spectra with a W/CoFeB bilayer placed between the IDTs. The transmission amplitude takes a maximum at ∼194 MHz, which corresponds to the fundamental excitation frequency of the SAW (fSAW) defined by the geometry of the IDTs and the sound velocity of the substrate.

The acoustoelectric properties of the films are studied as a function of magnetic field. A continuous rf signal with fixed frequency f and power P is fed from one of the VNA ports to the corresponding IDT, which launches a SAW along x that propagates to the film and induces lattice motion. The longitudinal (along x) and transverse (along y) voltages of the Hall bar, defined as Vxx and Vyx, respectively, are measured during the SAW excitation. Since Vxx and Vyx contain similar information, here, we focus on the results of Vxx; see section S1 for the characteristics of Vyx. To extract the voltage originating from the SAW, we subtract the average voltage measured under off-resonance conditions and obtain the acoustic voltage ΔVxxVxxVxxoff. Vxxoff is the average value of Vxx when f is set far from fSAW (see Materials and Methods for the details). We apply an in-plane magnetic field of magnitude H during the voltage measurements. The angle between the field and the x axis is defined as φH. As the magnetic easy axis of the CoFeB layer points along the film plane and the in-plane magnetic anisotropy is weak, we assume the orientation of the magnetization follows that of the magnetic field.

Fig. 1 Experimental setup to probe the acoustic spin Hall effect.

(A) Schematic illustration of the experimental setup including the substrate, the film, the IDTs, and the VNA. The bottom image illustrates lattice motion–induced spin current, i.e., the acoustic spin Hall effect. Spin current flows orthogonal to the SAW propagation. Note that the direction of spin current reverses when the SAW propagation direction is reversed. (B) Representative optical microscopy image of the device. The bright regions are the electrodes, and the gray square at the center is the Hall bar made of the film. (C) SAW transmission amplitude from IDT1 to IDT2 (IDT2 to IDT1) is plotted as a function of frequency (f) by the blue (red) line. A Hall bar made of the W(2.4)/CoFeB bilayer is placed between the IDTs.

The acoustic voltage

Figure 2 (A, C, and D) shows the field angle (φH) dependence of ΔVxx for W/CoFeB, Pt/CoFeB, and Cu/CoFeB bilayers when a rf signal of ffSAW and P∼10 dBm is applied to IDT2. For W/CoFeB and Pt/CoFeB bilayers, ΔVxx shows a sinusoidal variation with a period of 90°. Note that the sign (i.e., the phase) of the sinusoidal variation is the same for the two bilayers although the sign of the spin Hall angle is opposite between Pt and W (3). In contrast, no such variation is found for the Cu/CoFeB bilayer. (We also find that ΔVxx shows little dependence on φH for W single layer films.) Figure 2B shows ΔVxx versus φH of the W/CoFeB bilayer when the rf signal is applied to IDT1. The mean offset voltage and the sinusoidal variation change their signs as the SAW propagation direction is reversed. Similar features are observed for the Pt/CoFeB bilayers.

Fig. 2 Field angle dependence of the acoustic voltage.

(A to D) Magnetic field angle (φH) dependence of ΔVxx when a rf signal of ffSAW and P∼10 dBm is applied to IDT2 (A, C, and D) and IDT1 (B). Films placed between the IDTs are W(1.8)/CoFeB (A and B), Pt(2.0)/CoFeB (C), and Cu(1.8)/CoFeB (D) bilayers. The magnetic field magnitude is fixed to ∼55 mT. The error bars, which represent standard deviation of the repeated measurements, are smaller than the symbols. The black lines show fit to the data with Eq. 1.

We fit the φH dependence of ΔVxx with the following functionΔVxx=ΔVxx0+ΔVxx2φcos2φH+ΔVxx4φsin22φH(1)where ΔVxxnφ (n= 2, 4) represents the coefficient of the sinusoidal function with a period of 360°/n and ΔVxx0 is the φH-independent component. ΔVxx0 is proportional to what is known as the acoustic current, which originates from rectification of the localized electric field and charge density (18).

The f dependence of ΔVxx0 is plotted in Fig. 3A. ΔVxx0 takes a peak at f ∼194 MHz, which corresponds to fSAW (see Fig. 1C) and changes its sign as the SAW propagation direction is reversed (19). The f dependence of ΔVxx2φ and ΔVxx4φ is shown in Fig. 3 (B and C), respectively. ΔVxx4φ is significantly larger than ΔVxx2φ and shows a clear peak at ffSAW, suggesting that its appearance is associated with the excitation of SAW. (The origin of the small ΔVxx2φ remains unknown.) For all components (ΔVxx0, ΔVxx2φ, and ΔVxx4φ), the signal magnitude slightly varies depending on the SAW propagation direction. As the SAW transmission amplitude shows little dependence on the propagation direction (see Fig. 1C), nonreciprocity of the system (20, 21) may contribute to the asymmetry. The rf power (P) dependence of ΔVxx4φ is shown in Fig. 3D. ΔVxx4φ increases linearly with P.

Fig. 3 Resonant excitation of the acoustic voltage.

(A to C) rf frequency (f) dependence of ΔVxx0 (A), ΔVxx2φ (B), and ΔVxx4φ (C). The blue (red) triangles represent results when the rf signal is applied to IDT1 (IDT2). The rf power (P) is fixed to ∼10 dBm. (D) P dependence of ΔVxx4φ when f is varied. The solid lines show fit to the data with a linear function. Top and bottom panels show results when a rf signal is applied to IDT1 and IDT2, respectively. (A to D) The error bars show fitting errors of ΔVxx with Eq. 1. Data presented are obtained using W(2.4)/CoFeB bilayer.

To identify the origin of ΔVxx4φ, we have studied its dependence on the X layer thickness (d). Hereafter, we use ΔVxx0 and ΔVxx4φ to represent the corresponding value at ffSAW. As the transmittance of the SAW slightly varies from device to device due to subtle differences in the IDTs, we normalize ΔVxx4φ with ΔVxx0 and define vxx4φΔVxx4φ/ΔVxx0. Figure 4A shows the d dependence of vxx4φ for W/CoFeB bilayers. We find that vxx4φ takes a maximum at d ∼ 2 nm. Such d dependence of vxx4φ resembles that of the spin Hall magnetoresistance (SMR) (22, 23). The d dependence of the SMR ratio rxx2φΔRxx2φ/Rxx0 is plotted in Fig. 4B. ΔRxx2φ represents the resistance change when the magnetization of the CoFeB layer is rotated in the xy plane (24) and Rxx0 is the base resistance that does not vary with φH. The d dependence of vxx4φ and rxx2φ is similar. According to the drift-diffusion model of spin transport in NM/ferromagnetic metal (FM) bilayers (23, 24), the maximum of rxx2φ is proportional to the square of the NM layer spin Hall angle (θSH), and the NM layer thickness at the maximum is close to its spin diffusion length (λN). Using the model (see Materials and Methods), we obtain θSH ∼ 0.23 and λN ∼ 0.73 nm from the d dependence of rxx2φ for W, which are in good agreement with previous studies (24).

Fig. 4 X layer thickness, magnetic field, and resonance frequency dependence of the acoustic voltage.

(A) Normalized acoustic voltage vxx4φ=ΔVxx4φ/ΔVxx0 plotted against W layer thickness (d) for W/CoFeB bilayers. The rf frequency (f) and power (P) are set to ∼fSAW and ∼10 dBm, respectively. (B) d dependence of rxx2φ of the same system shown in (A). The black line is a fit to the data with Eq. 4. (C) The field angle (φH) dependence of the acoustic voltage ΔVxx obtained using various field magnitude (H). Purple, green, and orange lines are for H∼ 8, 14, and 55 mT, respectively. (D) ΔVxx4φ plotted as a function of H. (C and D) Data are obtained using rf signal of ffSAW and P∼10 dBm applied to IDT1. (E) H dependence of ΔRxx2φ. (F) The SAW resonance frequency (fSAW) dependence of ΔVxx4φ. The rf power (P) is fixed to 10 dBm. The solid lines show linear fits passing through the origin. (C to F) Data presented are obtained using W(2.4)/CoFeB bilayer. The blue (red) triangles in (A) and (F) represent results when the rf signal is applied to IDT1 (IDT2). The error bars in (A), (D), and (F) show fitting errors of ΔVxx with Eq. 1.

The similarity in the d dependence of vxx4φ and rxx2φ suggests that a spin current is generated in the X layer. The fact that vxx4φ is almost absent for Cu/CoFeB bilayers (see Fig. 2D) further supports this notion: The spin Hall angle of Cu is significantly smaller than that of Pt and W. Note, however, that there are a few differences between the acoustic voltage and the SMR. First, the field angle dependence of the two is different. Typically, the resistance due to SMR varies as cos 2φH [see, for example, (22)], whereas the dominant contribution to the acoustic voltage ΔVxx varies as sin22φH. Second, vxx4φ is more than one order of magnitude larger than rxx2φ. Third, we find a notable difference in the magnetic field magnitude (H) dependence between the two. In Fig. 4C, we show the H dependence of ΔVxx versus φH for W/CoFeB bilayer. As evident, the offset voltage (ΔVxx0) hardly changes with H. In contrast, the magnitude of ΔVxx4φ increases with decreasing H. The H dependence of ΔVxx4φ, plotted in Fig. 4D, shows that ΔVxx4φ scales with 1/H. As a reference, we show in Fig. 4E the H dependence of ΔRxx2φ. Contrary to ΔVxx4φ, ΔRxx2φ is nearly constant against H.

Model description

To account for these results, we modify the drift-diffusion model of spin transport that is used to describe SMR (23). First, we include SAW-induced straining of the FM layer and magnetoelastic coupling (25, 26), which cause changes in the magnetization direction with respect to the magnetic field (2729). Consequently, ΔVxx acquires an extra factor of 1Hsin 2φH compared to the resistance change that originates from SMR. Next, to generate a (rectified) dc current, the spin current must vary in time and space such that it couples to the motion of magnetic moments driven by the SAW-induced strain. We find that the following form of spin current js,zy (electron spin orientation along y and flow along z) produces a rectified dc current and accounts for the experimental resultsjs,zy=Auxt(2)where ux is the lattice displacement along the wave propagation direction (x). A is a prefactor that determines the spin current generation efficiency and is proportional to λso, the SOI.

The spin current js,zy generated in the NM layer, which varies in time and space in sync with the SAW, drifts to the NM/FM interface and causes spin accumulation. The amount of spin accumulation at the interface depends on the direction of the FM layer magnetization due to the action of spin transfer torque (22, 23). The spin accumulation generates a back flow spin current in the NM layer, which is converted to charge current via the inverse spin Hall effect (11). Since the magnetization direction varies in time and space via magnetoelastic coupling, the back flow spin current is locally rectified and causes a φH-dependent dc acoustic voltage. Thus, a FM layer with a sizable magnetoelastic coupling is required to observe the rectified dc acoustic voltage. [See section S1 where we show that ΔVxx4φ is absent for the W/Py bilayer due to the small magnetoelastic coupling of Py (Ni81Fe19)]. The resulting acoustic voltage reads (see section S2)ΔVxxcλso2K(d)fSAWP sgn(k)bHMSsin22φH(3)where c is a constant that depends on the material and the geometry of the device, K(d) characterizes the d dependence similar to that of the SMR (see Eq. 4), k is the wave vector of the Rayleigh-type SAW [sgn(x) takes the sign of x], and b and MS are, respectively, the magnetoelastic coupling constant and the saturation magnetization of the FM layer. The sgn(k) in the right-hand side of Eq. 3 originates from the form of the lattice-induced spin current (Eq. 2).

Equation 3 captures many features of the acoustic voltage found in the experiments. As evident, ΔVxx varies as sin22φH. The coefficient of sin22φH in Eq. 3, equivalent to ΔVxx4φ, changes its sign upon reversal of the wave propagation direction (defined by the sign of k), scales with 1H and P, and is proportional to the square of the spin-orbit coupling of the NM layer and thus independent of the sign of the NM layer spin Hall angle. The thickness dependence of ΔVxx4φ, coded in K(d), is in relatively good agreement with the experimental results. We have also studied the fSAW dependence of ΔVxx4φ for the W/CoFeB bilayer; the results are plotted in Fig. 4F. As evident, ΔVxx4φ scales with fSAW. We emphasize that Eq. 2 is the only form of spin current that can account for these results. Note that the linear dependence of ΔVxx4φ with fSAW allows us to exclude contributions from spin-dependent inertial force (30) and related effects in the presence of SOI (31), which are proportional to higher order of fSAW.

These results therefore demonstrate that the lattice motion induces a spin current. Recent studies have shown that spin-rotation coupling (14, 15) can induce spin accumulation in the NM layer, which results in generation of spin current if the NM layer thickness is larger than the SAW decay length (typically, of the order the SAW wavelength, which is a few micrometers here) (16). To clarify the role of spin-rotation coupling, we have studied ΔVxx4φ of inverted structures, CoFeB/W bilayers. In both W/CoFeB and CoFeB/W bilayers, spin-rotation coupling induces spin density in the W layer, which can cause a flow of spin current toward the CoFeB layer as the latter can act as a spin sink. If such spin current were to flow, the flow direction will be opposite for the normal (W/CoFeB) and inverted (CoFeB/W) structures and consequently results in ΔVxx4φ with opposite sign.

We find that the signs of ΔVxx4φ for W/CoFeB and CoFeB/W bilayers are the same, demonstrating that spin-rotation coupling is not the source of spin current (see sections S1 and S3). For the same reason, we can rule out SAW-induced spin pumping (27, 32) from the CoFeB layer and the inverse spin Hall effect of the W layer. This is also supported by the fact that the signs of ΔVxx4φ for W/CoFeB and Pt/CoFeB bilayers are the same (see Fig. 2) albeit the difference in the sign of θSH for W and Pt.

In Fig. 5A, we summarize the maximum value of vxx4φ and rxx2φ when d is varied, denoted as vxx,max4φ and rxx,max2φ, respectively, for each bilayer (X = Ta, W, Pt). Results from the CoFeB/W bilayers are included. Note that the structure of W depends on the growth condition: From the film resistivity (33, 34), we consider W forms a highly resistive β-phase in W/CoFeB bilayer, whereas it is a mixture of the β-phase and the low-resistivity crystalline α-phase in CoFeB/W bilayer. Consequently, the SMR ratio (rxx,max2φ) is smaller for the latter due to the smaller θSH (3436). In contrast, we find that vxx,max4φ takes nearly the same value for the two bilayers, indicating that there are factors other than θSH that set the magnitude of vxx,max4φ. In Fig. 5B, we plot the ratio γvxx,max4φrxx,max2φ to characterize such contribution. We find that γ is notably larger for bilayers with Pt and (β + α)-W (CoFeB/W) than that with β-W (W/CoFeB) and Ta. Since the former two layers are textured, whereas the latter two are highly disordered (i.e., amorphous-like), we consider that the texture of the films may influence γ. Little correlation is found between γ and the bulk modulus of the X layer.

Fig. 5 Efficiency to generate lattice motion–induced spin current.

(A and B) Maximum values of the normalized acoustic voltage vxx,max4φΔVxx4φ/ΔVxx0max (red bars) and the maximum SMR ratio rxx,max2φΔRxx2φ/Rxx0max (green bars) (A) and their ratio γvxx,max4φrxx,max2φ (B) obtained for X/CoFeB (X = Ta, W, and Pt) bilayers and CoFeB/W bilayers. W(β) and W(β + α) represent W/CoFeB and CoFeB/W bilayers, respectively.

DISCUSSION

Last, we discuss the source of spin current that scales with the time derivative of lattice displacement (Eq. 2). First, a conventional mechanism would be to consider internal electric field associated with the SAW and the resulting spin Hall effect of the NM layer. There are two major sources of internal electric field. One is the piezoelectric field (Ep) localized at the film/substrate interface. Spin current generated from Ep can only reach the NM/FM interface when the film thickness is smaller than λN. The thickness dependence of vxx4φ (Fig. 4A) rules out such contribution. The other source is the time varying electric field (Eb) caused by the motion of atoms (3739). Eb is uniform along the film normal as long as the film thickness is sufficiently smaller than the SAW decay length. In general, Eb is screened by the conduction electrons in metallic films: We infer that it generates negligible spin current. With the current understanding, we consider that it is difficult to quantitatively account for the experimental results with the combination of the SAW-induced electric field and the spin Hall effect. Second, Eq. 2 can be derived assuming the following interaction (40, 41): Hint = su · (p × σ), where s is a constant, u is the lattice displacement vector, and p and σ are electron momentum and spin orientation, respectively. This interaction derives from the SOI (40, 41) and the coefficient s is proportional to λso, similar to the relation between θSH and λso. Hint resembles the Rashba Hamiltonian (42) but can exist here since the inversion symmetry is broken by the dynamical lattice displacement u. Further studies are required, however, to justify the presence of such Hamiltonian. Third, the time derivative of the lattice displacement can cause changes in the Berry curvature of electron wave function. Theoretical studies have identified the right-hand side of Eq. 2 as the Berry electric field (43, 44). It remains to be seen whether spin current emerges from the Berry electric field under strong SOI. Last, the phonon angular momentum (4547) may contribute to the generation of spin current. Similar to the spin Seebeck effect (12), where the spin angular momentum of magnons is transferred to electrons, the angular momentum of phonons (i.e., sound waves) can be transferred to the electrons and induce spin current. The efficiency of such process must be addressed to assess its contribution. See section S3 for discussion on possible sources of voltage that may contribute to ΔVxx4φ.

In summary, we have shown that spin current is directly created from lattice motion associated with SAW. Such acoustic spin Hall effect is observed in NM/FM bilayers through a field-dependent dc acoustic voltage. The acoustic voltage roughly scales with the square of the spin Hall angle of the NM layer and is proportional to the SAW frequency. The NM layer thickness dependence of the acoustic voltage is similar to that of the SMR. Using a diffusive spin transport model, we show that such characteristics of the acoustic voltage can be accounted for when a spin current that scales with the time derivative of lattice displacement is generated in the NM layer. Possible sources of such spin current include a Berry electric field associated with time varying Berry curvature and/or an unconventional SOI-mediated spin-lattice interaction that resembles the form of Rashba Hamiltonian. The efficiency to generate spin current, represented by the maximum acoustic voltage, also seems to depend on a factor related to the film texture; the efficiency is nearly the same for amorphous-like β-W and textured Pt despite the difference in their spin Hall angle. The finding of the acoustic spin Hall effect thus implies a mechanism that facilitates an SOI-mediated coupling of electron spins and a rotating lattice. Further studies are required to unveil the microscopic mechanism to describe such coupling.

MATERIALS AND METHODS

Sample preparation

rf magnetron sputtering is used to deposit the films on piezoelectric Y+128°-cut LiNbO3 substrates. The film structure is sub./X(d)/CoFeB(1)/MgO(2)/Ta(1) with X = W, Pt, Ta, and Cu (thickness in unit of nanometers). The inverted structure is sub./MgO(2)/CoFeB(1)/X(d)/MgO(2)/Ta(1) with X = W. The MgO(2)/Ta(1) layers serve as a capping layer to prevent oxidation of the films. For bilayers with X = Pt and Cu, a 0.5-nm-thick Ta layer is inserted before deposition of X to promote their smooth growth. Hall bars are formed from the films using optical lithography and Ar ion etching. Subsequently, we use optical lithography and a liftoff process to form IDTs and electrodes made of Ta(5)/Cu(100)/Pt(5).

Schematic illustration of the SAW device and definition of the coordinate system are shown in Fig. 6. The distance of the two IDTs is ∼600 μm, and each IDT has 20 pairs of single-type fingers. The width and gap of the fingers are set to a: The corresponding SAW wavelength is ∼4a. The finger overlap, i.e., the SAW aperture (La), is fixed to ∼450 μm. A Hall bar made of the film is placed at the center of the two IDTs. The length and width of the Hall bar are set to ∼450 and ∼400 μm, respectively.

Fig. 6 Schematic illustration of the SAW device.

The orange structure represents the IDTs, and the dark gray area shows the film.

We vary a to change the SAW resonance frequency (fSAW). a is fixed to ∼5 μm for most of the results shown, which gives fSAW ∼ 194 MHz. In Fig. 4F, we vary a to change fSAW: a is set to ∼5, ∼4, ∼3, and ∼2 μm to obtain fSAW of ∼194, ∼242, ∼321, and ∼479 MHz, respectively. The input rf power P to the IDT is calibrated using the VNA.

Voltage measurements

The longitudinal (along x) and transverse (along y) voltages, defined as Vxx and Vyx, respectively, are measured during the SAW excitation. To extract the voltage originating from the SAW, we subtract the average voltage measured under off-resonance conditions, defined as Vxx(yx)off. Vxx(yx)off is obtained as follows. Under a fixed magnetic field and rf power, we study the frequency (f) dependence of Vxx(yx). Vxx(yx) takes a peak when ffSAW. We choose frequencies (foff) that are outside the peak structure of Vxx(yx), typically a few tens of megahertz away from fSAW (see Fig. 1C for a typical transmission spectra). Vxx(yx)off is the average value of Vxx(yx) measured at several foff. Vxx(yx)off is subtracted from the measured voltage Vxx(yx) at frequency f to obtain the acoustic voltage ΔVxx(yx)Vxx(yx)Vxx(yx)off. Vxx(yx)off is always measured before the measurement of Vxx(yx) at frequency f. Voltage measurements at each condition are repeated 5 to 100 times to improve the signal-to-noise ratio.

Spin Hall magnetoresistance

In the RESULTS section, we have used ΔRxx2φ, the resistance change when the magnetization of the CoFeB layer is rotated in the xy plane, to estimate SMR. ΔRxx2φ is equal to the sum of the SMR and the anisotropic magnetoresistance. Since the latter is significantly smaller than the former for the system under study (24), we assume that ΔRxx2φ represents the SMR. To obtain the SMR more accurately, it is customary to measure the resistance change when the magnetization of the CoFeB layer is rotated in the yz plane (22), defined as ΔRxxsmr. We have verified that ΔRxx2φ and ΔRxxsmr take similar value, justifying the assumption that ΔRxx2φ/Rxx0 represents the SMR.

The X layer thickness dependence of the SMR is fitted using the following equation (22, 23, 24)ΔRxx2φRxx0=θSH21+ζK(d),K(d)λNdtanh2d2λNtanh dλN(4)where ζρNtFρFd, ρF and tF are the resistivity and thickness of the FM (=CoFeB) layer, respectively, and ρN is the resistivity of the X layer. Here, we have assumed a transparent X/FM interface for spin transmission and neglected the effect of longitudinal spin absorption of the FM layer (24). The base longitudinal resistance Rxx0 is defined as the resistance when the magnetization of the FM layer points along the y axis. For fitting the data (Fig. 4B) with Eq. 4, we have used ρN ∼ 147 μΩ·cm and ρF ∼ 160 μΩ·cm.

SUPPLEMENTARY MATERIALS

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/7/2/eabd9697/DC1

https://creativecommons.org/licenses/by-nc/4.0/

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REFERENCES AND NOTES

Acknowledgments: Funding: This work was partly supported by JSPS Grant-in-Aid for Specially Promoted Research (15H05702) and the Center of Spintronics Research Network of Japan. T.K. acknowledges support from JSPS Research fellowship for young scientists and the Program of Excellence in Photon Science (XPS). Author contributions: T.K., M.K., and M.H. planned the study. T.K. grew the samples, performed the measurements, and carried out data analysis with input from M.K. and M.H. T.F. and H.K. developed microscopic models that describe lattice motion–induced spin current. T.K., M.K., and M.H. wrote the manuscript with input from T.F. and H.K. All authors contributed to the discussion of results. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the corresponding authors.
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