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Acoustic spin Hall effect in strong spin-orbit metals

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 a spin current flows orthogonal to the propagation direction of a surface acoustic wave (SAW) in non-magnetic metals. The acoustic spin Hall effect manifests itself in a field-dependent acoustic voltage in non-magnetic metal (NM)/ferromagnetic metal (FM) 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 the spin current must scale with the SOI and the time derivative of the lattice displacement. Such form of spin current can be derived from a Berry electric field associated with time varying Berry curvature and/or an unconventional spin-lattice interaction mediated by SOI. 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.

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 [6][7][8].
Recent studies have shown that not only electrons but other degrees of freedom can generate spin current. Precessing magnetization pumps out spin current from magnetic materials, a mechanism known as spin pumping [9][10][11]. 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 spinrotation coupling [14]. The effect has been observed in liquid metals [15] and non-magnetic light metals (e.g., Cu) [16]. Generation of spin current via spin pumping, spin Seebeck effect and spin-rotation coupling do 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 non-magnetic metals. 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 structure. To account for the experimental results, we find the spin current must scale with the SOI and the time derivative of the lattice displacement.
Thin film heterostructures are grown on piezoelectric LiNbO 3 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 The experimental setup and the coordinate system are schematically illustrated in Fig. 1(a). The Hall bar is placed between the two IDTs. Figure 1(b) 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 1 The experimental setup and the coordinate system are schematically illustrated in Fig. 1(a). The Hall bar is placed between the two IDTs. Figure 1(b) 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 1(c) 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 (f SAW ) 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 V xx and V yx , respectively, are measured during the SAW excitation.
Since V xx and V yx contain similar information, here we focus on the results of V xx ; see supplementary material section I for the characteristics of V yx . In order to extract the voltage originating from the SAW, we subtract the average voltage measured under offresonance conditions (f = f SAW ) and obtain the acoustic voltage xx is the average value of V xx when f is set far from f SAW (see 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 ϕ H . 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.
continuous rf signal with fixed frequency and power is fed from one of the VNA ports to the corresponding IDT, which launches a SAW along that propagates to the film and induces lattice motion. The longitudinal (along ) and transverse (along ) voltages of the Hall bar, defined as xx and yx , respectively, are measured during the SAW excitation.
Since xx and yx contain similar information, here we focus on the results of xx ; see supplementary material section I for the characteristics of yx . In order to extract the voltage originating from the SAW, we subtract the average voltage measured under offresonance conditions ( SAW ) and obtain the acoustic voltage ∆ xx xx − off xx off xx is the average value of xx when is set far from SAW (see Methods for the details). We apply an in-plane magnetic field of magnitude during the voltage measurements. The angle between the field and the -axis is defined as . 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.  For W/CoFeB and Pt/CoFeB bilayers, ∆V xx 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. Figure 2(b) shows ∆V xx vs.
ϕ H of the W/CoFeB bilayer when the rf signal is applied to IDT1. Clearly, 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.
We fit the ϕ H dependence of ∆V xx with the following function: where ∆V nϕ xx (n =2,4) represents the coefficient of the sinusoidal function with a period of 360 • /n and ∆V 0 xx is the ϕ H -independent component. ∆V 0 xx 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 ∆V 0 xx is plotted in Fig. 3(a). ∆V 0 xx takes a peak at f ∼194 MHz, which corresponds to f SAW (see Fig. 1(c)), and changes its sign as the SAW propagation direction is reversed [19]. The f dependence of ∆V 2ϕ xx and ∆V 4ϕ xx are shown in Figs. 3(b) and 3(c), respectively. ∆V 4ϕ xx is significantly larger than ∆V 2ϕ xx and shows a clear peak at f ∼ f SAW , suggesting that its appearance is associated with the excitation of SAW. The rf power (P ) dependence of ∆V 4ϕ xx is shown in Fig. 3(d). ∆V 4ϕ xx increases linearly with P . To identify the origin of ∆V 4ϕ xx , we have studied its dependence on the X layer thickness (d). Hereafter, we use ∆V 0 xx and ∆V 4ϕ xx to represent the corresponding value at f ∼ f SAW . As the transmittance of the SAW slightly varies from device to device due to subtle differences in the IDTs, we normalize ∆V 4ϕ xx with ∆V 0 xx and define v 4ϕ Figure 4(a) shows the d-dependence of v 4ϕ xx for W/CoFeB bilayers. We find v 4ϕ xx takes a maximum at d ∼ 2 nm. Interestingly, such d-dependence of v 4ϕ xx resembles that of the spin Hall magnetoresistance (SMR) [20,21]. The d-dependence of the SMR ratio, r 2ϕ xx ≡ |∆R 2ϕ xx /R 0 xx | is plotted in Fig. 4(b). ∆R 2ϕ xx represents the resistance change when the magnetization of the CoFeB layer is rotated in the xy plane [22] and R 0 xx is the base resistance that does not vary with ϕ H . Clearly, the d-dependence of v 4ϕ xx and r 2ϕ xx are similar. According to the drift-diffusion model of spin transport in non-magnetic metal (NM)/ferromagnetic metal (FM) bilayers [21,22], the maximum of xx is proportional to the square of the NM layer spin Hall angle ( SH ), and the NM layer thickness at the maximum is close to its spin diffusion length ( ). Using the model (see Methods), we obtain SH 23 and 73 nm from the -dependence of xx for W, which are in good agreement with previous studies [22].
The similarity in the -dependence of xx and xx suggests that a spin current is generated in the X layer. The fact that xx is almost absent for Cu/CoFeB bilayers (see Fig. 2 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 of spin transport in non-magnetic metal (NM)/ferromagnetic metal (FM) bilayers [21,22], the maximum of r 2ϕ xx is proportional to the square of the NM layer spin Hall angle (θ SH ), and the NM layer thickness at the maximum is close to its spin diffusion length (λ N ). Using the model (see Methods), we obtain θ SH ∼ 0.23 and λ N ∼ 0.73 nm from the d-dependence of r 2ϕ xx for W, which are in good agreement with previous studies [22]. The similarity in the d-dependence of v 4ϕ xx and r 2ϕ xx suggests that a spin current is generated in the X layer. The fact that v 4ϕ xx is almost absent for Cu/CoFeB bilayers (see Fig. 2(d)) 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 2ϕ H (see for example, Ref. [20]), whereas the dominant contribution to the acoustic voltage ∆V xx varies as sin 2 2ϕ H . Second, v 4ϕ xx is more than one order of magnitude larger than r 2ϕ xx . Third, we find a striking difference in the magnetic field magnitude (H) dependence between the two. In Fig. 4(c) Fig. 4(e) the H dependence of |∆R 2ϕ xx |. Contrary to ∆V 4ϕ xx , |∆R 2ϕ xx | is nearly constant against H. To account for these results, we modify the drift-diffusion model of spin transport that is used to describe SMR [21]. First, we include SAW-induced straining of the FM layer and magnetoelastic coupling [23,24], which cause changes in the magnetization direction with respect to the magnetic field [25,26]. Consequently, ∆V xx acquires an extra factor of 1 H sin 2ϕ H compared to the resistance change that originates from SMR. (See supplementary material section I where we show that ∆V 4ϕ xx is absent for W/NiFe bilayer due to the small magnetoelastic coupling of NiFe.) 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 the following form of spin current j y s,z (electron spin orientation along y and flow along z) produces a rectified dc current and accounts for the experimental results: where u x 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 λ so , the SOI.
The spin current j y s,z generated in the NM layer drifts to the NM/FM interface and causes spin accumulation. The accumulated spin at the interface causes a back flow of spin current within the NM layer, which is converted to electrical current via the inverse spin Hall effect [11]. 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 [20,21], thus causing the ϕ H dependent acoustic voltage. The resulting acoustic voltage reads (see supplementary material section II) 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 M S are, respectively, the magnetoelastic coupling constant and the saturation magnetization of the FM layer.
Equation (3) captures many features of the acoustic voltage found in the experiments.
As evident, ∆V xx varies as sin 2 2ϕ H . The coefficient of sin 2 2ϕ H in Eq. (3), equivalent to ∆V 4ϕ xx , changes its sign upon reversal of the wave propagation direction (defined by the sign of k), scales with 1 H 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 ∆V 4ϕ xx , coded in K(d), is in relatively good agreement with the experimental results. We have also studied the f SAW dependence of ∆V 4ϕ xx for W/CoFeB bilayer; the results are plotted in Fig. 4

(f). As evident, ∆V 4ϕ
xx scales with f SAW . We emphasize that Eq. (2) is the only form of spin current that can account for these results. Note that the linear dependence of ∆V 4ϕ xx with f SAW excludes contributions from spin-dependent inertial force [27] and related effects in the presence of SOI [28], which are proportional to higher order of f SAW .
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 where is a constant that depends on the material and the geometry of the device, characterizes the -dependence similar to that of the SMR (see Eq. (4)), is the wave vector of the Rayleigh-type SAW (sgn( ) takes the sign of ), and and are, respectively, the magnetoelastic coupling constant and the saturation magnetization of the FM layer.
Equation (3) captures many features of the acoustic voltage found in the experiments.
As evident, ∆ xx varies as sin . The coefficient of sin in Eq. (3), equivalent to xx , changes its sign upon reversal of the wave propagation direction (defined by the sign of ), scales with and , 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 ∆ xx , coded in ), is in relatively good agreement with the experimental results. We have also studied the SAW dependence of ∆ xx for W/CoFeB bilayer; the results are plotted in Fig. 4(f). As evident, ∆ xx scales with SAW . We emphasize that Eq. (2) is the only form of spin current that can account for these results. Note that the linear dependence of ∆ xx with SAW excludes contributions from spin-dependent inertial force [27] and related effects in the presence of SOI [28], which are proportional to higher order of SAW These results therefore demonstrate that the lattice motion induces a spin current. Recent studies have shown that spin-rotation coupling [14,15]  than the SAW decay length (typically, of the order the SAW wavelength, which is a few µm here) [16]. To clarify the role of spin-rotation coupling, we have studied ∆V 4ϕ xx 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 ∆V 4ϕ xx with opposite sign. We find that the signs of ∆V 4ϕ xx for W/CoFeB and CoFeB/W bilayers are the same, demonstrating that spin-rotation coupling is not the source of spin current (see supplementary material sections I and III).
For the same reason, we can rule out SAW-induced spin pumping [25,29] 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 ∆V 4ϕ xx for W/CoFeB and Pt/CoFeB bilayers are the same (see Fig. 2) albeit the difference in the sign of θ SH for W and Pt.
In Fig. 5(a), we summarize the maximum value of v 4ϕ xx and r 2ϕ xx when d is varied, denoted as v 4ϕ xx,max and r 2ϕ xx,max , 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 [30,31], 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 (r 2ϕ xx,max ) is smaller for the latter due to the smaller θ SH [31][32][33]. Interestingly, we find that v 4ϕ xx,max takes nearly the same value for the two bilayers, indicating that there are factors other than θ SH that sets the magnitude of v 4ϕ xx,max . In Fig. 5(b), we plot the ratio γ ≡ v 4ϕ xx,max r 2ϕ xx,max to characterize such contribution. We find γ is significantly 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 the texture of the films may influence γ. Little correlation is found between γ and the bulk modulus of the X layer.
Finally, 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 (E p ) localized at the film/substrate interface. Spin current generated from E p can only reach the NM/FM interface when the film thickness is smaller than λ N . The thickness dependence of v 4ϕ xx (Fig. 4(a)) rules out such contribution. The other source is the time varying electric field (E b ) caused by the motion of ions [34][35][36]. E b is uniform along the film normal as long as the film thickness is significantly smaller than the SAW decay length. In general, E b is screened by the conduction electrons in metallic films: we infer it generates negligible spin current. With the current understanding, we consider 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 [37,38]: , 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 [37,38] and the coefficient s is proportional to λ so , similar to the relation between θ SH and λ so . H int resembles the Rashba Hamiltonian [39] 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.
Indeed, theoretical studies have identified the right hand side of Eq. (2) as the Berry electric field [40,41]. It remains to be seen whether spin current emerges from the Berry electric field under strong SOI. Finally, the phonon angular momentum [42][43][44] may contribute to the generation of spin current. Similar to the spin Seebeck effect [12], where the spin angular momentum of magnons are 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.
In summary, we have shown that spin current is directly created from lattice motion associated with surface acoustic wave (SAW). Such acoustic spin Hall effect is observed in non-magnetic metal (NM)/ferromagnetic metal (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 spin Hall magnetoresistance. 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.

A. Sample preparation
Radio frequency magnetron sputtering is used to deposit the films on piezoelectric 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 (L a ), 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 µm and ∼ 400 µm, respectively.
We vary a to change the SAW resonance frequency (f SAW ). a is fixed to ∼ 5 µm for most of the results shown, which gives f SAW ∼ 194 MHz. In Fig. 4(f), we vary a to change f SAW : a is set to ∼5, ∼4, ∼3, ∼2 µm to obtain f SAW of ∼194, ∼242, ∼321, ∼479 MHz, respectively. is obtained as follows. Under a fixed magnetic field and rf power, we study the frequency ( ) dependence of xx yx xx yx takes a peak when SAW . We choose frequencies ( off ) that are outside the peak structure of xx yx , typically a few tens of MHz away from SAW (see Fig. 1(c) for a typical transmission spectra).

B. Voltage measurements
The longitudinal (along x) and transverse (along y) voltages, defined as V xx and V yx , 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 V off xx(yx) . V off xx(yx) is obtained as follows. Under a fixed magnetic field and rf power, we study the frequency (f ) dependence of V xx(yx) . V xx(yx) takes a peak when f ∼ f SAW . We choose frequencies (f off ) that are outside the peak structure of V xx(yx) , typically a few tens of MHz away from f SAW (see Fig. 1(c) for a typical transmission spectra). V off xx(yx) is the average value of V xx(yx) measured at several f off . V off xx(yx) is subtracted from the measured voltage V xx(yx) at frequency f to obtain the acoustic voltage ∆V xx(yx) ≡ V xx(yx) − V off xx(yx) . V off xx(yx) is always measured prior to the measurement of V xx(yx) at frequency f . Voltage measurements at each condition are repeated 5-100 times to improve the signal to noise ratio.

C. Spin Hall magnetoresistance
In the main text, we have used ∆R 2ϕ xx , the resistance change when the magnetization of the CoFeB layer is rotated in the xy plane, to estimate SMR. ∆R 2ϕ xx is equal to the sum of the SMR and the anisotropic magnetoresistance (AMR). Since the latter is significantly smaller than the former for the system under study [22], we assume ∆R 2ϕ xx 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 [20], defined as ∆R smr xx . We have verified that ∆R 2ϕ xx and ∆R smr xx take similar value, justifying the assumption that ∆R 2ϕ xx /R 0 xx represents the SMR. The X layer thickness dependence of the spin Hall magnetoresistance is fitted using the following equation [20,21]: where ζ ≡ ρ N t F ρ F d , ρ F and t F are the resistivity and thickness of the FM (=CoFeB) layer, respectively and ρ N 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 [22]. The base longitudinal resistance R 0 xx is defined as the resistance when the magnetization of the FM layer points along the y-axis. For fitting the data (Fig. 4(b)) with Eq. (4), we have used ρ N ∼ 147 µΩ·cm and ρ F ∼ 160 µΩ·cm.