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# Suppression of the field-like torque for efficient magnetization switching in a spin–orbit ferromagnet

## Abstract

Spin–orbit torque magnetization switching is an efficient method to control magnetization. In perpendicularly magnetized films, two types of spin–orbit torque are induced by driving a current: a damping-like torque and a field-like torque. The damping-like torque assists magnetization switching, but a large field-like torque pushes the magnetization towards the in-plane direction, resulting in a larger critical switching current density and making deterministic switching challenging. Control of the field-like torque strength is difficult because it is intrinsic to the material system used. Here, we show that the field-like term can be suppressed in a spin–orbit ferromagnetic single layer of (Ga,Mn)As by a current-induced Oersted field due to its non-uniform current distribution, making the damping-like torque term (the result of strong Dresselhaus spin–orbit coupling) dominant. The Oersted field can be controlled by the film thickness, resulting in an extremely low switching current density of 4.6 × 104 A cm–2. This strategy can thus provide an efficient approach to spin–orbit torque magnetization switching.

## Main

Current-induced spin–orbit torque (SOT) magnetization switching has been studied in bilayer systems consisting of a perpendicularly magnetized ferromagnetic layer and a non-magnetic metal layer using the spin Hall effect1. In damping-like torque (DLT)-induced SOT switching, the large field-like torque (FLT) hinders the magnetization reversal process by tilting the magnetic moment towards the in-plane direction. Though the presence of a small FLT can reduce the magnetization switching barrier between the up and down states, a large FLT makes deterministic magnetization switching in perpendicular ferromagnets more difficult, because it leads to an increase in the critical switching current density (Jc). The interfacial Rashba effect plays an important role in magnetization switching in conventional metal bilayer systems and induces a large FLT contribution. In such systems, due to the large FLT, a Jc value of approximately 107 A cm–2 is necessary1,2,3,4,5,6,7. Because the FLT is intrinsic to materials, control of the FLT is nearly impossible so long as the same material systems are used.

Spin–orbit ferromagnets, such as the ferromagnetic semiconductor (Ga,Mn)As, have recently been found to be promising for resolving the aforementioned problem, because they have high spin polarization and large spin–orbit interaction8,9,10,11,12,13,14,15,16,17,18,19,20,21. Magnetization rotation occurs by driving a current in only a single layer without a non-magnetic layer8,13. In this case, the current that flows inside the film rather than that at the surface plays the main role in the magnetization reversal. (Ga,Mn)As is a unique ferromagnetic semiconductor because the local manganese concentration has a gradient in the direction perpendicular to the film due to the segregation of gallium and manganese atoms at the surface22,23,24, resulting in a non-uniform current distribution (the current mainly flows near the top surface; Fig. 1b) and generating a current-induced Oersted field (HOe).

In this Article, we show that the current-induced Oersted field in a layer of (Ga,Mn)As can be used to suppress the FLT contribution induced at the interface by controlling the current direction and layer thickness. By suppressing the FLT contribution, Jc can be decreased to 4.6 × 104 A cm–2, three orders of magnitude smaller than that observed in typical metal bilayers. Our device structure is shown in Fig. 1a. When the current flows in the $$[\bar 110]$$ direction, an Oersted torque ($$\widehat {\mathbf{\tau }}_{{\mathrm{Oe}}}$$) with the same direction as the FLT ($$\widehat {\mathbf{\tau }}_{{\mathrm{FL}}}$$) is induced by HOe, enhancing the field-like term contribution (Fig. 1c). In contrast, when applying current in the $$[\bar 1\bar 10]$$ direction, $$\widehat {\mathbf{\tau }}_{{\mathrm{Oe}}}$$ points in the opposite direction of $$\widehat {\mathbf{\tau }}_{{\mathrm{FL}}}$$ and suppresses the field-like term contribution, decreasing Jc (Fig. 1d).

## Thickness dependence on magnetization switching

The heterostructure examined in our study is (Ga0.94,Mn0.06)As (t nm)/In0.3Ga0.7As (500 nm)/GaAs (50 nm) grown on a GaAs(001) substrate by molecular beam epitaxy (Fig. 1a and Supplementary Note 1), where the (Ga,Mn)As thickness, t, is 5, 7, 10, 13, 15 and 20 nm. The 500-nm-thick lattice-relaxed insulating In0.3Ga0.7As layer applies a tensile strain to the (Ga0.94,Mn0.06)As thin film, inducing perpendicular magnetic anisotropy. The film was patterned into a crossbar with a channel width and length of 5 and 20 μm, respectively, for the transport measurements. Here, using electron beam evaporation, the electrodes were capped with Au (100 nm)/Cr (5 nm) layers, which also work as heat sinks.

First we show that HOe increases with increasing t. This behaviour can be clearly observed when applying a current along the $$[\bar 110]$$ axis. In Fig. 2a,b, when t = 5 and 7 nm, we clearly see full SOT magnetization switching in the single (Ga,Mn)As thin films. Here, to detect the magnetization direction, we measured the anomalous Hall resistance (RH), which is proportional to the vertical component of the magnetization (Mz) of (Ga,Mn)As (ref. 25). Before beginning the measurement, we applied a large magnetic field of 10 kOe along the –z direction to align the magnetization along this direction, where RH takes the largest negative value. After decreasing this large field to zero, we applied a small constant in-plane magnetic field (Hext) of 500 Oe along the $$[\bar 110]$$ direction (the current direction) to realize deterministic magnetization reversal. In this case, Mz is reversed from –z to +z with increasing J in the $$[{\bar 1}10]$$ direction (black curves in Fig. 2a,b). When the sign of Hext is reversed, the polarity of the SOT magnetization switching is also inverted; Mz is reversed from –z to +z with increasing J in the $$[1\bar10]$$ direction (red curves in Fig. 2a,b). As shown in Fig. 2d, for J $$[\bar 110]$$, a spin component along the –x direction ($$\widehat {\mathbf{\sigma }}_x$$) is induced by the generated dominant effective field HD due to the strong Dresselhaus spin–orbit coupling in the (Ga,Mn)As layer. This $$\widehat {\mathbf{\sigma }}_x$$ exerts DLT $$\widehat {\mathbf{\tau }}_{{\mathrm{DL}}}$$, which is proportional to $$\left( {{\widehat {\mathbf{m}}} \times {\widehat {\mathbf{\sigma }}}_x \times {\widehat {\mathbf{m}}}} \right)$$ and whose direction is the same as that of $$\widehat {\mathbf{\sigma }}_x$$, on the magnetization (the torque system in Fig. 2e). Here $$\widehat {\mathbf{m}}$$ is the unit vector along the magnetization. Magnetization reversal occurs when $$\widehat {\mathbf{\tau }}_{{\mathrm{DL}}}$$ overcomes the torque $$\widehat {\mathbf{\tau }}_{{\mathrm{an}}}$$ induced by the anisotropy field $$\widehat {\mathbf{H}}_{{\mathrm{an}}}$$, that is, $$\widehat {\mathbf{\tau }}_{{\mathrm{an}}} = - \widehat {\mathbf{m}} \times \widehat {\mathbf{H}}_{{\mathrm{an}}}$$, with the assistance of the torque $$\widehat {\mathbf{\tau }}_{{\mathrm{ext}}}$$ induced by the external field $$\widehat {\mathbf{H}}_{{\mathrm{ext}}}$$, that is, $$\widehat {\mathbf{\tau }}_{{\mathrm{ext}}} = - \widehat {\mathbf{m}} \times \widehat {\mathbf{H}}_{{\mathrm{ext}}}$$. As clearly shown in Fig. 2a,b, full (180°) magnetization switching occurs when t = 5 and 7 nm. Note that Jc increases from 1.6 × 105 A cm–2 at t = 5 nm to 8.5 × 105 A cm–2 at t = 7 nm. When t is increased to 10 nm, the magnetization cannot be reversed by 180°, as shown in Fig. 2c, where the maximum value of RH is decreased to approximately half the saturated value RH_sat (right axis in Fig. 2c), as shown in Supplementary Note 2. As described below, this result is caused by the increase in HOe due to the enhancement in the non-uniform current distribution. This HOe exerts an additional Oersted torque, $$\widehat {\mathbf{\tau }}_{{\mathrm{Oe}}}$$, on the magnetization, that is, $$\widehat {\mathbf{\tau }}_{{\mathrm{Oe}}} = - \widehat {\mathbf{m}} \times \widehat {\mathbf{H}}_{{\mathrm{Oe}}}$$, with the same direction as $$\widehat {\mathbf{\tau }}_{{\mathrm{FL}}}$$ on the magnetization, enhancing the field-like term contribution, as shown by the following equation of the total torque $$\widehat {\mathbf{\tau }}$$:

$$\widehat {\mathbf{\tau }} = \tau _{{\mathrm{DL}}}\left( {\widehat {\mathbf{m}} \times \widehat {\mathbf{\sigma }}_x \times \widehat {\mathbf{m}}} \right) - (\tau _{{\mathrm{FL}}} + \tau _{{\mathrm{Oe}}})\left( {\widehat {\mathbf{m}} \times \widehat {\mathbf{\sigma }}_x} \right).$$
(1)

Here the field-like term consisting of $$\widehat {\mathbf{\tau }}_{{\mathrm{FL}}}$$ and $$\widehat {\mathbf{\tau }}_{{\mathrm{Oe}}}$$ hinders the magnetization switching process when $$\widehat {\mathbf{\sigma }}_x$$ exerts $$\widehat {\mathbf{\tau }}_{{\mathrm{DL}}}$$ on the magnetization, which reverses it. The obtained SOT switching curve is well reproduced by the Landau–Lifshitz–Gilbert (LLG) equation, and the switching performance depends on the ratio, r, of the DLT strength to the total SOT strength. By fitting the theoretical curves to the experimental results shown in Fig. 2 (Supplementary Note 3), r is estimated to be approximately 1.00, 0.95 and 0.90 for t = 5, 7 and 10 nm, respectively. Our results indicate that the field-like term contribution is enhanced by HOe with increasing t, which results in an increase in Jc, as shown in Fig. 2f.

With a further increase in t (t ≥ 13 nm), the field-like term becomes dominant, and the magnetization cannot be fully switched but can only be switched by an angle smaller than 90° towards the in-plane direction (that is, RH = 0), as shown in Fig. 3. The switching polarity is determined by the relative strength between the DLT and the field-like term. In the thinner samples (t < 10 nm), the DLT is dominant, where a positive current reverses the magnetization from the –z direction to the +z direction under Hext = 500 Oe, as shown in Fig. 2a,b. However, in the thicker samples (t ≥ 13 nm), a positive current switches the magnetization from the +z direction to the in-plane direction, where the field-like term starts to be dominant. Although the t dependence of the coercive field Hc is similar to that of Jc (Supplementary Fig. 5f), the Hc value at t = 13 nm is higher than that at t = 20 nm and the Jc value at t = 13 nm is lower than that at t = 20 nm, which indicates that Hc is not a crucial factor for the SOT switching behaviour. In addition, Han linearly decreases with the increase in t (Supplementary Note 4), but Jc does not show a linear relationship with t at all, which means that Han is also not a decisive factor in our study. Here the enhancement in the field-like term can be attributed to the increase in HOe due to the non-uniform charge current in the thick (Ga,Mn)As single layers (Supplementary Note 5), where HOe can effectively modulate the field-like term because the contribution of Han to the switching barrier is small. This HOe exerts a torque $${\widehat{\mathbf{\tau}}}_{{\mathrm{Oe}}}$$, whose direction is the same as that of $$\widehat {\mathbf{\tau }}_{{\mathrm{FL}}}$$, on the magnetization. Meanwhile, the polarity of SOT switching can be changed by the application direction of Hext. When Hext is positive along the +y direction, the magnetization prefers to stay along the +z direction with positive J. Hence, when J is swept back from the positive side to the negative side, the magnetization starts to be released towards the +z direction at a positive J (see black curves in Fig. 3). On the contrary, the negative Hext makes the magnetization more stable when it stays along the −z direction with positive J and the magnetization starts to be released towards the −z direction at a positive J when J is swept back (see red curves in Fig. 3). Therefore, the sign of RH depends on the sign of Hext, where RH is mainly positive when Hext is positive, but it is negative when Hext is negative.

## Suppression of the FLT contribution

In contrast, by changing the directions of J and Hext from $$[\bar 110]$$ to $$[\bar 1\bar 10]$$ (or [110]) for the thick (Ga,Mn)As thin films (t ≥ 13 nm), HOe can be used to suppress the field-like term, and the DLT becomes dominant again. As shown in Fig. 4a–c, the magnetization is switched by an angle larger than 90°. In this case, as shown in Fig. 4d, HD is generated along $$[\bar 110]$$, inducing a spin component along this direction (+y direction). Thus, as shown in Fig. 4e, the hole spin has a +y component ($$\widehat {\mathbf{\sigma }}_y$$), which exerts a torque $$\widehat {\mathbf{\tau }}_{{\mathrm{DL}}}$$ whose direction is the same as that of $$\widehat {\mathbf{\sigma }}_y$$ on the magnetization. When Hext is applied along the [110] direction, $$\widehat {\mathbf{\tau }}_{{\mathrm{DL}}}$$ is parallel to $$\widehat {\mathbf{\tau }}_{{\mathrm{ext}}}$$, whose direction is opposite to that of $$\widehat {\mathbf{\tau }}_{{\mathrm{an}}}$$. Hence, with increasing J, $$\widehat {\mathbf{\tau }}_{{\mathrm{DL}}}$$ reverses the magnetization when it overcomes $$\widehat {\mathbf{\tau }}_{{\mathrm{an}}}$$ with the assistance of $$\widehat {\mathbf{\tau }}_{{\mathrm{ext}}}$$ (black curves in Fig. 4a–c). This is a typical DLT-dominant switching process in thin films prepared using perpendicular magnetic anisotropy. Here the DLT becomes dominant because $$\widehat {\mathbf{\tau }}_{{\mathrm{FL}}}$$ is suppressed by the $$\widehat {\mathbf{\tau }}_{{\mathrm{Oe}}}$$ induced by HOe. As shown in Fig. 4d, the direction of HOe induced by the non-uniform current distribution is along $$[1\bar 10]$$ (−y direction), which is opposite to the direction of the induced $$\widehat {\mathbf{\sigma }}_y$$ (+y direction). Hence, the direction of $$\widehat {\mathbf{\tau }}_{{\mathrm{Oe}}}$$ is opposite to that of $$\widehat {\mathbf{\tau }}_{{\mathrm{FL}}}$$ (Fig. 4e), and the total torque $$\widehat {\mathbf{\tau }}$$ can be expressed as

$$\widehat {\mathbf{\tau }} = \tau _{{\mathrm{DL}}}\left( {\widehat {\mathbf{m}} \times \widehat {\mathbf{\sigma }}_y \times \widehat {\mathbf{m}}} \right) - (\tau _{{\mathrm{FL}}} - \tau _{{\mathrm{Oe}}})\left( {\widehat {\mathbf{m}} \times \widehat {\mathbf{\sigma }}_y} \right).$$
(2)

From equation (2), we see that $$\widehat {\mathbf{\tau }}_{{\mathrm{Oe}}}$$ suppresses the field-like term contribution, which makes the DLT dominant again and switches the magnetization between the ±z axes. Thus, by properly controlling HOe, we can achieve full SOT magnetization switching in the (Ga,Mn)As single layers with t ≥ 13 nm. As shown in Fig. 4f (orange curve), when t = 15 nm, SOT magnetization switching occurs at Jc = 4.6 × 104 A cm–2, which is one of the lowest values so far among those reported for SOT magnetization switching26. When t < 15 nm, the suppression of FLT is not as effective as the case for t = 15 nm, and thus, Jc decreases with increasing t. Additionally, when t > 15 nm, HOe increases with increasing t, which makes the suppression of the FLT contribution excessively effective so that the magnetization switching barrier becomes larger, thus increasing Jc. Furthermore, the switching power P = Ic2R shows the lowest value in the (Ga,Mn)As thin film with t = 15 nm, as shown in Fig. 4f (blue curve), further eliminating the heating effect and reducing the energy consumption. Here Ic is the critical switching current and R is the direct current resistance of (Ga,Mn)As. Therefore, by carefully adjusting HOe, we can obtain ultra-efficient SOT magnetization reversal with ultralow energy consumption.

## LLG simulation

The enhancement and suppression of the field-like term contribution in the magnetization switching can be well reproduced by the LLG equation8

$$\frac{{\mathop {{{\hat{\mathbf m}}}}\limits^ \cdot }}{{\gamma ^\prime }} = - {\hat{\mathbf m}} \times \left( {{\hat{\mathbf H}} + \beta {\hat{\mathbf S}}^ \ast } \right) - \alpha {\hat{\mathbf m}} \times {\hat{\mathbf m}} \times \left( {{\hat{\mathbf H}} + \frac{1}{\alpha }{\hat{\mathbf S}}^ \ast } \right),$$
(3)

where

$$\gamma ^\prime = \frac{\gamma }{{1 + \alpha ^2}},\beta = \frac{{1 - r({\mathrm{1}} + \alpha ^2)}}{\alpha }.$$

Here $$\mathop {{{\hat{\mathbf m}}}}\limits^ \cdot$$ is the derivative of $${\hat{\mathbf m}}$$ with respect to time; $${\hat{\mathbf H}}$$ is the effective field consisting of the total equivalent field of the Oersted torque and the FLT (−H*), Hext and HD; γ is the gyromagnetic ratio; α is the damping constant; and $$\widehat {\mathbf{S}}^ \ast$$ is the SOT effective magnetic field in the in-plane direction. Further, r denotes the strength of the DLT relative to the total SOT: when r is 0, only the FLT is present, and when r is 1, only the DLT is present. When J is along the $$\left[ {\bar 1{\mathrm{10}}} \right]$$ direction, HOe points in the same direction as the FLT equivalent field, HFLT; −H* = HFLT + HOe. When J is along the [110] direction, HOe is opposite to the direction of HFLT, where −H* = HFLT − HOe. In our LLG simulation, $$\widehat {\mathbf{S}}^ \ast$$ points in the same direction as that of the spin component along the in-plane direction, $$\widehat {\mathbf{\sigma }}^ \ast$$. Since both S* and H* are current-induced fields and show similar current dependence, we conducted the LLG simulation with H* = −nS*, where we assumed n = 0, 0.1, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45 and 0.5. Other parameters used here are Hext = 500 Oe, Han = 1.67 kOe, α = 0.05 and r = 0.95. The simulation results are shown in Fig. 5a and the H* dependence of the critical switching equivalent field, $${{S}}_{\mathrm{C}}^{\ast}$$, is shown in Fig. 5b, where $${{S}}_{\mathrm{C}}^{\ast}$$ is the average of the S* values at the magnetization switching with the positive and negative sweep directions: S* = (|S*()| + |S*()|)/2, where the subscript arrow represents the sweeping direction of S*. When |H*| ≤ 0.2|S*|, the field-like term can assist the SOT switching by reducing the switching barrier, which makes $${{S}}_{\mathrm{C}}^{\ast}$$ decrease with increasing |H*|. When 0.2|S*| < |H*| ≤ 0.35|S*|, $${{S}}_{\mathrm{C}}^{\ast}$$ increases with increasing |H*|, because the field-like term hinders the magnetization reversal by pulling the magnetization towards the in-plane direction. When |H*| is increased to be larger than 0.35|S*|, the magnetization cannot be fully switched with an angle of 180° (Fig. 5a). This is because the magnetization rearrangement is mainly induced by the application of the in-plane Oersted magnetic field. The larger the field-like term is, the more easily the magnetization can be aligned along the in-plane direction.

By comparing the experimental results of the SOT switching in the 15-nm-thick thin film shown in Fig. 5c with the simulation results using H* = −0.35S* and H* = −0.2S* shown in Fig. 5d, we found that the experimental results can be well reproduced. When J flows along the $$\left[ {\bar 1{\mathrm{10}}} \right]$$ direction (light blue curve in Fig. 5c), $$\widehat {\mathbf{\tau }}_{{\mathrm{Oe}}}$$ points in the same direction as that of $$\widehat {\mathbf{\tau }}_{{\mathrm{FL}}}$$ and the magnetization is switched by an angle smaller than 90°. When J flows along the [110] direction (orange curve in Fig. 5c), $$\widehat {\mathbf{\tau }}_{{\mathrm{Oe}}}$$ is opposite to the direction of $$\widehat {\mathbf{\tau }}_{{\mathrm{FL}}}$$ and the magnetization can be fully reversed with a switching angle of 180°. The switching behaviour with J sweeping along the $$\left[ {\bar 1{\mathrm{10}}} \right]$$ and [110] direction can be well reproduced by the simulation with H* = −0.35S* (light blue curve in Fig. 5d, where H* and S* point in the ±x direction) and H* = −0.2S* (orange curve in Fig. 5d, where H* and S* point in the ±y direction). Therefore, we can conclude that the field-like term is suppressed by HOe when J [110], which contributes to more efficient magnetization switching.

## Conclusions

We have shown that the field-like term contribution in a spin–orbit ferromagnetic single layer of perpendicularly magnetized (Ga,Mn)As can be controlled by the introduction of an Oersted field induced by a bias current with a possible non-uniform current distribution. When applying a current along the [110] direction in the (Ga,Mn)As single layer with a thickness of 15 nm, $$\widehat {\mathbf{\tau }}_{{\mathrm{Oe}}}$$ assists in reducing the switching current density to 4.6 × 104 A cm–2. These results will help in understanding the mechanism of magnetization switching from the viewpoint of torque contribution and advance the development of memory technology with higher efficiency and lower energy consumption.

## Methods

### Sample preparation

(Ga0.94,Mn0.06)As thin films with various thicknesses (t) were grown on semi-insulating GaAs(001) substrates in an ultrahigh-vacuum molecular beam epitaxy system. After removing the surface oxide layer of the GaAs substrate at 580 °C, a 50-nm-thick GaAs buffer layer was grown to obtain an atomically smooth surface. After that, the substrate was cooled to 450 °C for the growth of In0.3Ga0.7As with a thickness of 500 nm to induce a tensile strain in the (Ga0.94,Mn0.06)As layer, giving rise to perpendicular magnetic anisotropy. Then, the sample was cooled to approximately 290 °C for the growth of the t-nm-thick (Ga0.94,Mn0.06)As layer. The growth process was monitored in situ by means of reflection high-energy electron diffraction. The Curie temperatures of the (Ga0.94,Mn0.06)As layers were estimated to be 67, 88, 66 and 65 K for t = 5, 7, 10 and 15 nm, respectively (Supplementary Note 1).

### Device preparation and transport measurements

The sample was patterned into a crossbar with a width of 5 μm and a length of 20 μm using photolithography and argon ion milling. Then, Au (100 nm)/Cr (5 nm) layers were deposited on the electrodes as heat sinks by electron beam evaporation. For the SOT measurements, a Keithley 2636A instrument was used as the current source for applying a direct current. The Hall voltage was measured with a Keithley 2400 instrument. The measurements were carried out at 40 K (20 K for t = 5 nm).

## Data availability

The data that support the plots within this paper and other findings of this study are available at https://doi.org/10.6084/m9.figshare.12961430.v1. Source data are provided with this paper.

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## Acknowledgements

This work was partly supported by Grants-in-Aid for Scientific Research (no. 16H02095 and no. 18H03860), the CREST program of the Japan Science and Technology Agency (JPMJCR1777), the Spintronics Research Network of Japan (Spin-RNJ) and the China Scholarship Council (no. 201706210086).

## Author information

Authors

### Contributions

Sample preparation: M.J. and H.A.; measurements: M.J.; data analysis: M.J. and S.S.; writing and project planning: M.J., S.O. and M.T.

### Corresponding authors

Correspondence to Miao Jiang, Shinobu Ohya or Masaaki Tanaka.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Supplementary information

### Supplementary Information

Supplementary Figs. 1–8, Discussion and Table 1.

### Supplementary Data

Statistical source data for Supplementary Fig. 2.

### Supplementary Data

Statistical source data for Supplementary Fig. 3.

### Supplementary Data

Statistical source data for Supplementary Fig. 4.

### Supplementary Data

Statistical source data for Supplementary Fig. 5.

### Supplementary Data

Statistical source data for Supplementary Fig. 6.

### Supplementary Data

Statistical source data for Supplementary Fig. 7.

### Supplementary Data

Statistical source data for Supplementary Fig. 8.

## Source data

### Source Data Fig. 2

Statistical source data.

### Source Data Fig. 3

Statistical source data.

### Source Data Fig. 4

Statistical source data.

### Source Data Fig. 5

Statistical source data.

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Jiang, M., Asahara, H., Sato, S. et al. Suppression of the field-like torque for efficient magnetization switching in a spin–orbit ferromagnet. Nat Electron 3, 751–756 (2020). https://doi.org/10.1038/s41928-020-00500-w