Introduction

Spin-orbit torque (SOT) is a spin torque arising from spin–orbit coupling in heavy metal (HM)/ferromagnet (FM)/oxide structures, in which a spin current generated by the spin Hall effect1,2,3 in the HM and the Rashba effect4,5,6 at the HM/FM and/or FM/oxide interfaces exerts a torque on the FM and manipulates the magnetization direction. Providing fast and energy-efficient magnetization switching7,8,9 and domain wall motion10,11, SOT is being intensively investigated12,13 as an alternative technique to manipulate the magnetization for various spintronic devices including magnetic random access memories8,9,14, spintronic logics15,16, and oscillators17,18. While the spin Hall effect is of bulk origin from the spin–orbit coupling in HM19, the Rashba effect is of interface origin from the broken inversion symmetry at interfaces in HM/FM/oxide heterostructures20. Electrical control of the Rashba effect was demonstrated at semiconductor interfaces;21,22,23,24 however, it has not been clearly reported in metallic structures so far for the following reasons. First, the application of an electric field in the metallic structures is limited to one or two atomic layers due to the Coulomb screening effect. Second, it is difficult to distinguish the changes in the Rashba effect from the concomitant modification of magnetic properties owing to the applied gate voltage. Note that the voltage control of the magnetism has been actively studied25,26,27,28,29 since it offers the reduction of power consumption for the magnetization switching without degrading the thermal stability. Recent studies demonstrated that the SOT in HM/FM/oxide structures is controlled using a gate voltage30,31,32 through modulation of oxygen concentration at the interface by inducing oxygen ion migration. Note that the voltage effect based on oxygen ion migration is large but slow compared to that caused by the charge distribution mechanism33,34,35.

In this article, we report that the lateral modulation of Rashba effect in Pt/Co/AlOx structures by using two side gates generates out-of-plane SOT, allowing electrical control of field-free switching of perpendicular magnetization. In the device configuration illustrated in Fig. 1a, applying different gate voltages to the two side gates induces an additional lateral symmetry-breaking along the y-direction. In the presence of a charge current along the x-direction, the symmetry analysis36 shows that the lateral symmetry-breaking along the y-direction results in additional SOTs in directions of \({{{{{\bf{m}}}}}}\times {{{{{\bf{z}}}}}}\) and \({{{{{\bf{m}}}}}}\times ({{{{{\bf{m}}}}}}\times {{{{{\bf{z}}}}}})\), where \({{{{{\bf{m}}}}}}\) and z are unit vectors along the magnetization and the thickness direction, respectively. This out-of-plane SOT or z-SOT can switch perpendicular magnetization without an external magnetic field36,37,38 and a theoretical study suggests that the z-SOT can greatly reduce the switching current39.

Fig. 1: Electrical control of field-free SOT switching in Pt/Co/AlOx/TiO2 samples.
figure 1

a Schematic of the gate voltage-induced lateral symmetry breaking. Gate voltage difference (ΔVG) induces an electric-field modulation along the y-direction, creating additional lateral symmetry-breaking. With a charge current along x-direction, this lateral symmetry-breaking generates out-of-plane spin–orbit fields (red arrows); field-like effective field (\({B}_{{{{{{\rm{FLT}}}}}}}^{z}\)), and damping-like effective field (\({B}_{{{{{{\rm{DLT}}}}}}}^{z}\)). Those result in additional SOTs (z-SOT) in direction of \({{{{{\bf{m}}}}}}\times {{{{{\bf{z}}}}}}\) by \({B}_{{{{{{\rm{FLT}}}}}}}^{z}\) and \({{{{{\bf{m}}}}}}\times ({{{{{\bf{m}}}}}}\times {{{{{\bf{z}}}}}})\) by \({B}_{{{{{{\rm{DLT}}}}}}}^{z}\), where \({{{{{\bf{m}}}}}}\) is located in y-z plane. The blue arrows indicate the in-plane spin–orbit fields (\({B}_{{{{{{\rm{FLT}}}}}}}^{y}\) and \({B}_{{{{{{\rm{DLT}}}}}}}^{y}\)) induced by symmetry-breaking along the z-direction. b Schematic illustration of the Hall-bar device with two side gates and the sample structure of Pt/Co/AlOx/TiO2, where the inset shows the optical microscopic image. c Current-induced SOT switching for ΔVG = 0 (VG,L = VG,R = 0 V). Bx = 20 mT. d, e Field-free spin–orbit torque switching for ΔVG > 0 (VG,L = +8 V, VG,R = 0 V) (d) and the ΔVG < 0 (VG,L = 0 V, VG,R = +8 V) (e). 8 V corresponds to the electric field of 2.5 MV/cm. Here, the blue (or red) dot arrows indicate from up-to-down (or down-to-up) switching direction.

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In this work, we show that the z-SOT and associated switching polarity are controllable by gate voltage in a reversible and non-volatile manner, offering programmable logic operations in spintronic login-in-memory devices as demonstrated below. Moreover, we provide a microscopic origin of the z-SOT induced by asymmetric gate voltages. It is noted that the above symmetry argument is valid regardless of the source of lateral symmetry-breaking along the y-direction. However, it is of crucial importance to identify the microscopic origin of the electric-field-induced lateral symmetry breaking for further improvement of device performance. To this end, we show that the z-SOT depends on gate oxide materials; the direction of the z-SOT in the identical Pt/Co/AlOx structures is reversed by changing the gate oxide from TiO2 to ZrO2. The two gate oxides exhibit opposite electric-field effects on the potential barrier; for TiO2 (ZrO2), a positive voltage decreases (increases) the potential barrier height, resulting in the modification of the built-in electric field and associated Rashba effect at the Co/oxide interface. This, together with the same sign of voltage-controlled magnetic anisotropy effect regardless of the gate oxide, suggests that the z-SOT is mainly due to the lateral modulation of the Rashba effect at the Co/AlOx interface, which is further supported by the measurements of the gate-voltage dependence of field-like SOTs.

Results

Electric-field control of deterministic spin–orbit torque switching

To demonstrate the z-SOT generated by asymmetric gate voltages, we fabricate Pt (5 nm)/Co (1.4 nm)/AlOx (2 nm) Hall-bar devices, in which two side gates consisting of a gate oxide of TiO2 (40 nm) and a gate electrode of Ru (50 nm) are integrated (Fig. 1b and Methods). Figure 1c–e show SOT-induced switching measurements of the sample depending on the polarity of ΔVG. Here, ΔVG is the difference in voltages applied to the left (VG,L) and right (VG,R) gate electrodes with respect to the ground connected to the Pt bottom layer. Without applying a gate voltage, i.e., ΔVG = 0 (VG,L = VG,R = 0 V), the sample shows a typical SOT switching behaviour that occurs only when an in-plane magnetic field (Bx) is applied (Fig. 1c);4,12,13 a positive current favours up-to-down switching under a positive Bx. This switching polarity corresponds to a positive spin Hall angle of Pt. Remarkably, when nonzero ΔVG is applied, the sample shows a deterministic SOT switching even in the absence of Bx (Fig. 1d, e). Furthermore, the switching polarity is determined by the sign of ΔVG; a positive current favours up-to-down switching for ΔVG > 0 (VG,L = +8 V, VG,R = 0 V), and it is opposite for ΔVG < 0 (VG,L = 0 V, VG,R = +8 V). This field-free deterministic SOT switching evidences the z-SOT due to ΔVG that breaks lateral symmetry. Scanning transmission electron microscopy and electron-energy loss spectroscopy measurements reveal that the ΔVG induces the oxygen ion redistribution in oxides depending on its polarity; a larger oxygen ion concentration on the side where a positive bias is applied, compared to the other side (Supplementary Note 1). This might be responsible for the voltage-induced lateral asymmetry. Note that field-free switching is achieved when the ΔVG is greater than 8 V, which is a critical ΔVG to create a net lateral asymmetry.

We point out that our device has a distinctive advantage that the deterministic SOT switching polarity is electrically controllable. This electrical controllability cannot be achieved with previously reported ones demonstrating z-SOT using a hybrid FM/ferroelectric structure30, a wedged structure36, a tilted magnetic anisotropy40,41, a chirally coupled nanomagnets42, a structural asymmetry38,43, and a low crystal symmetry material44,45. Moreover, this electrical controllability offers programmable logic operations by utilizing the reversible and non-volatile characteristics (Supplementary Note 2). As an example, we demonstrate that XOR and AND logic gate operations are realized in a single device using the gate voltage and input current as two input parameters and the magnetization direction as digital output (See details in Supplementary Note 3). This allows for multifunctional memories or programmable spin logic devices, offering the way for the realization of spin-based logic-in-memory devices.

We next systematically test the z-SOT in a Pt/Co/AlOx/TiO2 structure (TiO2 sample) by performing in-plane harmonic measurements46, in which the 1st and 2nd harmonic Hall resistances (\({R}_{{xy}}^{1\omega }{,{R}}_{{xy}}^{2\omega }\)) are measured with an a.c. current Iac by rotating the sample (azimuthal angle \(\varphi\)) under a fixed in-plane magnetic field Bext (Fig. 2a and Methods). The \({R}_{{xy}}^{2\omega }\) is given as

$${R}_{{xy}}^{2\omega }\left(\varphi \right)= \,\left({R}_{{{{{{\rm{AHE}}}}}}}\frac{{B}_{{{{{{\rm{DLT}}}}}}}^{y}}{{B}_{{{{{{\rm{eff}}}}}}}}+{R}_{\nabla T}^{2{{\omega }}}\right){{\cos }}\varphi +2{R}_{{{{{{\rm{PHE}}}}}}}\frac{{B}_{{{{{{\rm{FLT}}}}}}}^{y}+{B}_{{{{{{\rm{Oe}}}}}}}}{{B}_{{{{{{\rm{ext}}}}}}}}\left(2{{{\cos }}}^{3}\varphi -{{\cos }}\varphi \right)\\ -2{R}_{{{{{{\rm{PHE}}}}}}}\frac{{B}_{{{{{{\rm{DLT}}}}}}}^{z}}{{B}_{{{{{{\rm{ext}}}}}}}}{{\cos }}2\varphi +{R}_{{{{{{\rm{AHE}}}}}}}\frac{{B}_{{{{{{\rm{FLT}}}}}}}^{z}}{{B}_{{{{{{\rm{eff}}}}}}}},$$
(1)

where RAHE and RPHE are the anomalous Hall and planar Hall resistances, respectively; \({B}_{{{{{{\rm{DLT}}}}}}}^{y}\) (\({B}_{{{{{{{\mathrm{FLT}}}}}}}}^{y}\)) is the damping-like (field-like) effective field originating from conventional y-spin accumulation (y-SOT), and \({B}_{{{{{{\rm{DLT}}}}}}}^{z}\) (\({B}_{{{{{{\rm{FLT}}}}}}}^{z}\)) is the damping-like (field-like) effective field additionally generated by z-SOT, and BOe is the Oersted field; Beff is the effective magnetic field, defined as Beff = Bext + Bdem − Bani, where Bdem and Bani are the demagnetization field and the anisotropy field of FM, respectively; \({R}_{\nabla T}^{2\omega }\) is the anomalous Nernst contribution. Notably, the only \({B}_{{{{{{\rm{DLT}}}}}}}^{z}\) has a cos2φ dependence (Supplementary Note 4), which allows us to unambiguously demonstrate z-SOT if generated in the sample. Note that \({B}_{{{{{{\rm{FLT}}}}}}}^{z}\) creates angle-independent offset, which is uneasy to identify from harmonic Hall signals.

Fig. 2: Harmonic spin–orbit torque measurements in Pt/Co/AlOx/TiO2 samples.
figure 2

a Schematic for the measurement configuration. The second harmonic Hall resistances \(({R}_{{{{{{\rm{xy}}}}}}}^{2{{{{{\rm{\omega }}}}}}})\) for an a.c. current Iac are measured while rotating the sample in the plane (azimuthal angle \(\varphi\)) under an external field Bext. b The \({R}_{{{{{{\rm{xy}}}}}}}^{2{{{{{\rm{\omega }}}}}}}\) versus \(\varphi\) curves for the TiO2 samples with four different VG combinations and Bex = 3 T, where the single standard deviation uncertainty of the harmonic Hall voltage measurements is ±0.15 μV, which is included as error bars in the figures. Here, (+,+), (−,−), (+,−), and (−,+) denote (VG,L = VG,R = +8 V), (VG,L = VG,R = −8 V), (VG,L = +8 V and VG,R = −8 V), and (VG,L = −8 V and VG,R = +8 V), respectively. c–e The extracted \(\varphi\)-dependent components of \({R}_{{{{{{\rm{xy}}}}}}}^{2{{\omega }}}\); \({{{{{\rm{c}}}}}}{{{{{\rm{os}}}}}}\varphi\) component (c), \((2{{{\cos }}}^{3}\varphi -{{\cos }}\varphi )\) component (d), and \({{\cos }}2\varphi\) component (e). f–h Each \(\varphi\)-dependent component plotted as a function of 1/Beff (or 1/Bext), where the error bars are due to the uncertainty of the fitting of the \({R}_{{xy}}^{2\omega }\) versus \(\varphi\) curves to Eq. (1); \({{\cos }}\varphi\) component versus 1/Beff (f), \((2{{{\cos }}}^{3}\varphi -{{\cos }}\varphi )\) component versus 1/Bext (g), and \({{\cos }}2\varphi\) component versus 1/Bext (h).

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We examine the sample with four different gate-voltage conditions: (VG,L = VG,R = +8 V), (VG,L = VG,R = −8 V), (VG,L = +8 V and VG,R = −8 V), and (VG,L = −8 V and VG,R = +8 V), which are denoted as VG(+,+), VG(−,−), VG(+,−), and VG(−,+), respectively. We note that ΔVG = 0 for both VG(+,+) and VG(−,−) while ΔVG > 0 (ΔVG < 0) for VG(+,−) (VG(−,+)). Figure 2b shows the representative measurement data of the \({R}_{{xy}}^{2\omega }\) measured at Bext = 3 T for those voltage conditions, from which we separate cosφ, (2cos3φ–cosφ), and cos2φ components (Fig. 2c–e). In particular, Fig. 2e shows that the cos2φ component appears only in the sample under asymmetric gate voltages, i.e., VG(+,−) and VG(−,+). We repeat the measurement with a different Bext and plot each component as a function of the Beff (or Bext) in Fig. 2f–h. From the slope of the graphs, we extract the asymmetric gate voltage ( = ΔVG) dependence of the \({B}_{{{{{{\rm{DLT}}}}}}}^{y}\), \({B}_{{{{{{\rm{FLT}}}}}}}^{y}\), and \({B}_{{{{{{\rm{DLT}}}}}}}^{z}\) for a current density of 1 × 107A/cm2 [Table 1]. Two interesting observations are worth noting; firstly, the \({B}_{{{{{{\rm{DLT}}}}}}}^{y}\) and \({B}_{{{{{{\rm{FLT}}}}}}}^{y}\) due to y-SOT are enhanced (reduced) when a positive (negative) voltage is applied to both gates: larger \({B}_{{{{{{\rm{DLT}}}}}}}^{y}\) and \({B}_{{{{{{\rm{FLT}}}}}}}^{y}\) for VG(+,+) than for VG(−,−), whereas they are not significantly different for the asymmetric voltage application of VG(+,−) or VG(−,+). This result demonstrates that the y-SOT can be quantitatively modulated by a symmetric gate voltage [VG(+,+) and VG(−,−)]. Secondly, and more importantly, a sizable \({B}_{{{{{{\rm{DLT}}}}}}}^{z}\) due to z-SOT occurs when applying asymmetric voltages (VG(+,−), VG(−,+); i.e., nonzero ΔVG). The magnitude of \({B}_{{{{{{\rm{DLT}}}}}}}^{z}\) is about 10% of that of \({B}_{{{{{{\rm{DLT}}}}}}}^{y}\). We further confirm the z-SOT by performing the hysteresis loop shift experiments of the sample with various ΔVG’s (Supplementary Note 5), of which results are consistent with those of the SOT switching (Fig. 1) and in-plane harmonic measurements (Fig. 2). We note that the amount of the hysteresis loop shift increases gradually with increasing in-plane current without showing a threshold current that requires to overcome the intrinsic damping37, indicating the presence of \({B}_{{{{{{\rm{FLT}}}}}}}^{z}\), which cannot be clearly identified by the harmonic measurement.

Table 1 Gate voltage dependence of SOT effective fields of the TiO2 sample.
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To verify whether the ΔVG-induced z-SOT is general, we investigate another sample of a Pt/Co/AlOx/ZrO2 structure (ZrO2 sample), in which the gate oxide is replaced with ZrO2 while maintaining the same remaining structure. Note that the TiO2 and ZrO2 are widely exploited dielectric materials in resistive memory devices47 due to their unique material properties; TiO2 has a high oxygen mobility48 while ZrO2 has a high dielectric constant49. Figure 3a–c show SOT switching results of the sample; without gate voltages (ΔVG = 0), the up-to-down switching occurs for a positive current and a positive Bx (Fig. 3a). The switching polarity is the same with that of the TiO2 sample shown in Fig. 1c. When applying asymmetric electric voltages (ΔVG ≠ 0), the ZrO2 sample also shows deterministic switching similar to the TiO2 sample. It is found that the field-free switching current density (JSW) gradually reduces when increasing ΔVG (Supplementary Note 6). Furthermore, we examine the magnetization reversal characteristics during field-free SOT switching using magneto-optical Kerr effect (MOKE) measurement. This indicates that the domain reversal starts at the center of the FM island (or the edge of the gate electrode), where the ΔVG-induced z-SOT is maximum (Supplementary Note 7). These results unambiguously demonstrate that the generation of z-SOT is due to ΔVG. However, unexpectedly, the switching polarity of the ZrO2 sample is opposite from that of the TiO2 sample; a positive (negative) current favours the down-to-up switching for ΔVG > 0 (ΔVG < 0). This indicates that the direction of z-SOT generated in the ZrO2 sample is opposite to that of the TiO2 sample, which is supported by the additional measurements of z-SOT of the ZrO2 samples using in-plane harmonic Hall (Supplementary Note 8) and hysteresis loop shift (Supplementary Note 5) experiments. A possible reason of the opposite polarity depending on the gate oxide will be discussed later.

Fig. 3: Electric-field control of the field-free switching in Pt/Co/AlOx/ZrO2 samples.
figure 3

a Current-induced SOT switching for ΔVG = 0 (Bx = 20 mT). b, c Field-free SOT switching for ΔVG > 0 (b) and ΔVG < 0 (c). Here, the blue (or red) dot arrows indicate from up-to-down (or down-to-up) switching direction.

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Modulation of Rashba effect in ferromagnet/oxide interface

We now discuss the physical origin of the electric-field-induced z-SOT. The first possible cause is the lateral modulation of PMA through voltage-controlled magnetic anisotropy (VCMA) effect36,40,41. The asymmetric gate voltages give rise to a gradient of PMA along the y-direction of which sign depends on the sign of ΔVG, resulting in a field-free switching. To test this possibility, we measure the VCMA effect of the TiO2 and ZrO2 samples with four different gate-voltage conditions of VG(+,+), VG(−,−), VG(+,−), and VG(−,+) (Supplementary Note 9). We find an enhancement (reduction) in PMA for both samples with a gate voltage of VG(−,−) (VG(+,+)) and no significant variation in PMA for the VG(+,−) and VG(−,+) conditions. Notably, the polarity of the VCMA effect is the same for both the TiO2 and ZrO2 samples and thus ΔVG–induced PMA gradient along the y-direction cannot explain the opposite signs of the z-SOT. As a result, the VCMA effect is excluded as a cause of z-SOT. We also perform COMSOL simulations to check the gate voltage dependence of the current distribution (Supplementary Note 10), which allows us to rule out the possible contributions of the lateral current distribution.

Another possibility is the lateral variation of the Rashba effect at the Co/oxide interface, which can also be induced by asymmetric gate voltages. The magnitude of Rashba effect is proportional to the built-in electric-field originating from the band structure at the Co/oxide interface, which we estimate by measuring the potential barrier height50 in Co (10 nm)/AlOx (2 nm)/gate oxide (5 nm)/Ru (20 nm) tunnel junctions depending on gate voltage. Note that we use a 5 nm gate oxide that is thin enough for electrons to tunnel through. Figure 4a, b show the I–V characteristics of the tunnel junctions, demonstrating that the change in the potential barrier by a gate voltage (VG) depends on the gate oxide. For the junction with a TiO2, after applying a positive (negative) VG, the tunnelling current increases (decreases) (Fig. 4a), indicating that the potential barrier diminishes (increases). In contrast, the junction with a ZrO2 shows the reverse electric-field effect; a positive VG decreases the tunnel current and increases the potential barrier (Fig. 4b). The inset of Fig. 4a, b schematically illustrates the lateral variation of the barrier height (ϕ) for the gate oxide. The opposite electric-field effect may be due to the different transport mechanisms of the oxides; oxygen ion migration (charge trap) is the dominant mechanism in TiO2 (ZrO2)47,48,49. However, further investigation is required to clarify the gate oxide dependence of the electric-field effect. Nonetheless, the above results support the hypothesis that the z-SOT results from the lateral modulation of the built-in electric-field and associated Rashba effect at the Co/oxide interface. The asymmetric gate voltage (ΔVG) induces the variation of the potential barrier along the y-direction, leading to the generation of z-SOT. This result explains the different polarity of z-SOT between the TiO2 and ZrO2 samples.

Fig. 4: Voltage-induced variation of the potential barrier and field-like torque.
figure 4

a, b I–V characteristics in the Ta (10 nm)/Co (10 nm)/AlOx (2 nm)/TiO2 (5 nm)/Ru (20 nm) (a) and Ta (10 nm)/Co (10 nm)/AlOx (2 nm)/ZrO2 (5 nm)/Ru (20 nm) tunnel junctions (b) depending on pre-biased gate voltages VG corresponding to ±2.5 MV/cm. The inset corresponds to schematic drawing of the lateral variation of barrier height (ϕ) of the gate oxide, where the black and red line indicate the potential barrier height of oxide layer. c, d Field-like SOT component of depending on pre-biased gate voltage VG corresponding to ±2.5 MV/cm for Pt (0.5 nm)/Co (2 nm)/AlOx (2 nm)/TiO2 (40 nm) (c) and Pt (0.5 nm)/Co (2 nm)/AlOx (2 nm)/ZrO2 (40 nm) (d) samples. The error bars are due to the uncertainty of the fitting of the \({R}_{{xy}}^{2\omega }\) versus \(\varphi\) curves to Eq. (1).

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This scenario is further supported by in-plane harmonic Hall measurement of a Pt (0.5 nm)/Co (2 nm)/AlOx (2 nm)/gate oxide (40 nm) samples, in which FM Co is fully covered by a single gate. Figure 4c, d show the \((2{{{\cos }}}^{3}\varphi -{{\cos }}\varphi )\) component of \({R}_{{xy}}^{2\omega }\) as a function 1/Bext depending on VG, which demonstrates that the change in \({B}_{{{{{{\rm{FLT}}}}}}}^{y}\) with VG depends on gate oxide; for the sample with TiO2 (ZrO2) gate oxide, the \({B}_{{{{{{\rm{FLT}}}}}}}^{y}\) increases (decreases) by a positive VG. Since the SHE is negligible in a 0.5 nm Pt (Supplementary Note 11), the \({B}_{{{{{{\rm{FLT}}}}}}}^{y}\) primarily originates from the Pt/Co and/or Co/AlOx interfaces. Furthermore, the opposite electric-field effect of \({B}_{{{{{{\rm{FLT}}}}}}}^{y}\) between the TiO2 and ZrO2 samples suggests that the variation of the \({B}_{{{{{{\rm{FLT}}}}}}}^{y}\) can be attributed to the electric-field-controlled Rashba effect of the Co/AlOx interface. This is consistent with the trend of the electric-field-controlled potential barrier height, supporting that the lateral variation of the Rashba effect is the key element for the generation of the electric-field-induced z-SOT. It is reported that spin scattering at the FM/oxide interface can contribute to field-like torque51. This interfacial spin scattering is consistent with the Rashba effect at the FM/oxide interface, which causes the interfacial spin–orbit precession37,52 and associated field-like torques53.

Theoretical and numerical demonstration of electric-field-induced z-SOT

We now present that the lateral modulation of Rashba effect, which is induced by the lateral oxygen gradient, is the key element for the z-SOT by means of a symmetry argument and first-principles calculations. First, we present a general symmetry argument. The z-SOT \({\left.\frac{d{{{{{\bf{m}}}}}}}{{dt}}\right|}_{{{\perp }}}\) can be described as

$${\left.\frac{d{{{{{\bf{m}}}}}}}{{dt}}\right|}_{{{\perp }}}={aJ}{{{{{\bf{m}}}}}}\times {{{{{\bf{z}}}}}}{{{{{\boldsymbol{+}}}}}}{bJ}{{{{{\bf{m}}}}}}\times \left({{{{{\bf{m}}}}}}\times {{{{{\bf{z}}}}}}\right),$$
(2)

where \(J\) is the current density applied along the \(x\)-direction, \({{{{{\bf{m}}}}}}[=({m}_{x},{m}_{y},{m}_{z})]\) is the direction of magnetization, \({{{{{\bf{z}}}}}}\) is the direction normal to the film plane, and \(a\) (\(b\)) represents the magnitude of field-like (damping-like) component of the z-SOT. The mirror reflection \({{{{{{\mathscr{M}}}}}}}_{y}\) of Eq. (2) with respect to the \({xz}\) plane corresponds to the following operations: \({m}_{x}\to -{m}_{x}\), \({m}_{y}\to {m}_{y}\), \({m}_{z}\to -{m}_{z}\), \(\frac{d{m}_{x}}{{dt}}\to -\frac{d{m}_{x}}{{dt}}\), \(\frac{d{m}_{y}}{{dt}}\to \frac{d{m}_{y}}{{dt}}\), \(\frac{d{m}_{z}}{{dt}}\to -\frac{d{m}_{z}}{{dt}}\), and \(J\to J\). This mirror reflection results in \({\left.\frac{d{{{{{\bf{m}}}}}}}{{dt}}\right|}_{\perp }={-}{\left.\frac{d{{{{{\bf{m}}}}}}}{{dt}}\right|}_{{{\perp }}}\), meaning that the z-SOT is absent when the \({{{{{{\mathscr{M}}}}}}}_{y}\) symmetry is preserved. However, the z-SOT is allowed when the \({{{{{{\mathscr{M}}}}}}}_{y}\) symmetry is broken34. In our experiment, the \({{{{{{\mathscr{M}}}}}}}_{y}\) symmetry is broken by the lateral gradient of Rashba effect.

We next present our first-principles results. We numerically demonstrate that the lateral oxygen gradient (\(\nabla {V}_{{{{{{\rm{o}}}}}}}\)) results in the z-component non-equilibrium spin density, corresponding to the z-SOT. To this end, we compute a non-equilibrium spin density in Pt/Co/O structures with and without \(\nabla {V}_{{{{{{\rm{o}}}}}}}\), based on the linear response theory within density functional theory (Supplementary Note 12). Calculated non-equilibrium spin density \(\delta {{{{{\bf{s}}}}}}[{=}(\delta {s}_{x},\delta {s}_{y},\delta {s}_{z})]\) for Pt/Co/O structures are shown in Fig. 5. In our calculation, the electric field is applied along the \(x\)-axis and the magnetization \({{{{{\bf{m}}}}}}\) of Co is aligned in the \(y\)-axis. The results show that the magnitudes of \(\delta {s}_{y}\) are similar for both structures with and without \(\nabla {V}_{0}\) (Fig. 5d). In contrast, the magnitudes of \(\delta {s}_{x}\) and \(\delta {s}_{z}\) are nonzero only in the structure with \(\nabla {V}_{0}\) (Fig. 5c, e). Since the SOT is proportional to \({{{{{\bf{m}}}}}}{{{{{\boldsymbol{\times }}}}}}\delta {{{{{\bf{s}}}}}}\) and \({{{{{\bf{m}}}}}}\) is aligned in the y-direction, \(\delta {s}_{x}\) corresponds to the \(\left({{{{{\bf{m}}}}}}{{{{{\boldsymbol{\times }}}}}}{{{{{\bf{z}}}}}}\right)\)-component of \(\delta {{{{{\bf{s}}}}}}\) whereas \(\delta {s}_{z}\) corresponds to the \({{{{{\bf{z}}}}}}\)-component of \(\delta {{{{{\bf{s}}}}}}\). Comparing to Eq. (2), one finds that \(\delta {s}_{x}\) and \(\delta {s}_{z}\) correspond to \(b\) and \(a\), respectively. These results confirm that the lateral oxygen gradient, which induces the lateral modulation of Rashba effect (Supplementary Note 12), makes both damping-like and field-like z-SOT. Furthermore, it is also found that \(\delta {s}_{z}\) is linearly proportional to the \(\nabla {V}_{{{{{{\rm{o}}}}}}}\), indicating \({{{{{\rm{\delta }}}}}}{s}_{z}\) is always nonzero once (even small) oxygen gradient is present (Supplementary Note 13), and that \(\delta {s}_{z}\) is the same for the same oxygen gradient regardless of the system size, demonstrating that devices properties would be maintained even in the nanometer scale devices (Supplementary Note 14).

Fig. 5: Non-equilibrium spin density for the structures with and without the lateral oxygen gradient.
figure 5

a, b Schematics of the two different Pt/Co/O structures in yz-plane (side view) (a) and xy-plane (top view) (b). In b, the left panel shows the structure without the lateral oxygen gradient (\(\nabla {V}_{{{{{{\rm{o}}}}}}}\)), and the right panel shows the structure with the lateral oxygen gradient. The direction of lateral symmetry breaking is indicated by yellow arrow. In both a and b, the directions of magnetization (\({{{{{\bf{m}}}}}}\)) and electric field (\({{{{{\bf{E}}}}}}\)) are denoted by red arrows. c–e Atomic layer resolved non-equilibrium spin densities \(\delta {s}_{x}\) (c), \(\delta {s}_{y}\) (d), and \(\delta {s}_{z}\) (e) for the Pt/Co/O structure with and without lateral oxygen gradient.

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Discussion

We demonstrate the generation of z-SOT in Pt/Co/AlOx structures by inducing a lateral voltage gradient using two side gates, which enables field-free switching of perpendicular magnetization and electrical control of the switching polarity. We also show that the magnitude and direction of SOT is electrically controllable in such structures. Furthermore, we find that the z-SOT depends on gate oxides that exhibit an opposite electric-field effect on the potential barrier height while maintaining the same voltage-controlled magnetic anisotropy effect, indicating that the lateral modulation of Rashba effect at the Co/oxide interface by an electric voltage is a key ingredient of the generation of z-SOT.

We finally discuss the feasibility of our lateral gate SOT device. Our device performances such as speed and endurance depend on voltage-induced oxygen migration. In this study, we used a 40 nm thick gate oxide for which a gate voltage (VG) needs to be applied for 5 min at 100 °C to observe a noticeable electric-field effect. This is not a fundamental limitation of the operation speed of our device, but it can be improved by materials engineering. By introducing a double gate oxide of TiO2(2 nm)/ZrO2(5 nm) structure, we show that the gate operation works at room temperature with a short voltage pulse of 20 μs (Supplementary Note 15) and high endurance (Supplementary Note 16). Furthermore, we perform SPICE circuit simulations under the assumption that our technique, a field-free SOT switching device with a side gate, can be applicable to nanosized magnetic tunnel junctions (Supplementary Note 17), which demonstrate the potential of implementing an XOR gate based on our SOT device with reduced device area and comparable energy consumption compared to CMOS-based XOR gates.

Methods

Sample preparation

The samples of Ta (2 nm)/Pt (5 nm)/Co (1.4 nm)/AlOx (2 nm) structures were deposited on Si/Si3N4 wafer using d.c. magnetron sputtering with a base pressure of less than 4.0 × 10−6 Pa at room temperature. All metallic layers were grown with a working pressure of 0.4 Pa and a power of 30 W, while the AlOx layer was fabricated by deposition of an Al layer and subsequent plasma oxidation with an O2 pressure of 4.0 Pa and a power of 30 W for 75 s. The samples were then covered by a gate oxide of TiO2 or ZrO2 (40 nm) that is grown at 125 °C by plasma enhanced atomic layer deposition (PE-ALD) using TDMAT [Tetrakis (dimethylamido) titanium] or TEMAZ [Tetrakis(ethylmethylamido)zirconium] and O2 precursors. The oxygen plasma was formed with a rf power of 60 W and a flow of 500 sccm O2. The Hall-bar-structure devices with a 10 × 10 μm cross that includes a square-shaped ferromagnetic island with 8 × 8 μm were fabricated using photolithography and Ar ion-milling. Two gate electrodes on both sides of the ferromagnetic island with a spacing of 2 μm were fabricated by deposition of Ru (50 nm) and subsequent lift-off technique.

Gate voltage application

Prior to the electrical measurements, a gate voltage equivalent to ±2.5 MV/cm was applied to the Ru gate electrode for 5 min at 100 °C with respect to the ground connected to the bottom Pt layer unless otherwise specified. Thereafter, the spin–orbit torque measurements were conducted at room temperature with the gate electrode floating.

Spin-orbit torque measurements

Spin-orbit torque was characterized using an in-plane harmonic lock-in technique. The \({R}_{{xy}}^{1\omega }\) and \({R}_{{xy}}^{2\omega }\) for an a.c. current Iac of 11 Hz were simultaneously measured while rotating the sample in the plane (azimuthal angle \(\varphi\)) under an external field Bext. The magnitude of Bext is larger than the anisotropy field Bani of the sample, so the magnetization is aligned parallel to the direction of Bext. The single standard deviation uncertainty of the harmonic Hall voltage measurements is ±0.15 μV, which is included as error bars in the figures. The SOT-induced switching experiments were performed by measuring the \({R}_{{{{{{\rm{AHE}}}}}}}\) using a d.c. current of 100 μA after applying a current pulse of 70 μs with (or without) an in-plane magnetic field Bx. The single standard uncertainty of the Hall resistance measurements is ±1 mΩ. Corresponding error bars are included in the figures. In most case, they are much smaller than the size of symbols.

All measurements were conducted at room temperature. More than three samples are measured for each type of sample; data are qualitatively reproducible.

I–V curve measurements

The I–V characteristics of the sample were measured while sweeping a dc voltage between ±1.5 V at room temperature using a Ta (10 nm)/Co (10 nm)/AlOx (2 nm)/gate oxide (5 nm)/Ru (20 nm) tunnel junction with a junction area of 8 × 8 μm. The top Ru electrode was biased and the bottom Ta was grounded, so that a positive (negative) voltage corresponds to an electric-field pointing down (up). Prior to the I–V measurement, a gate voltage equivalent to 2.5 MV/cm was applied to the top Ru gate electrode for 5 min at 100 °C.