Spin-orbit torques (SOTs) and SOT-driven magnetization switchings in ferromagnet/heavy-metal bilayers have been extensively explored owing to the potential applications in energy-efficient and ultrafast spintronic devices1. Most of these SOT-devices employ the spin Hall effect2,3,4 and/or interfacial inverse spin-galvanic effect5,6,7, which are believed to originate in the spin-orbit coupling from the heavy-metals and the hetero-interfaces. However, such SOTs are not exclusively inherited from the bilayers whereas, after the discoveries of the inverse spin Hall effect8,9,10,11 and charge-to-spin conversion12,13,14,15,16,17,18,19 in a single-layer ferromagnet without the nominal spin-orbit source from the heavy-metal, it has recently been revealed that the SOT can also be generated within the ferromagnet solely, which even allows to switch the magnetization of the ferromagnet itself20,21,22,23,24,25. Although the mechanism of such SOTs remains elusive, its generation, detection, and exploitation have drawn significant attention, which enable the electrical control of the magnetization without any external polarizer and offer promising advantages in improving the device scalability and stability.

So far, this SOT has been observed in FeMn26, L10-FePt20,21,27, CoTb22,28,29, Py30,31,32, CoPt33,34,35,36, Gd-Fe-Co37, Mn3Sn24, FeTb25, and Co2MnGa23, all of which are metallic and exhibit a dominant damping-like (DL) SOT over the field-like (FL) SOT in most cases. Among them, the role played by the spin-orbit coupling is found to be nonnegligible, whereas some of them possess spin Hall angles even comparable to those in heavy-metals. It is known that topological insulators (TIs) can serve as a powerful reservoir of spin-orbit coupling that the spin-momentum locking of surface electrons can be utilized for high-efficient spin accumulation and SOT-exertion. Therefore, TIs have been widely used to replace the heavy-metal in the bilayer to couple with the ferromagnet, via which the magnetization switchings with higher efficiencies and lower threshold current densities have been achieved38,39,40,41,42,43,44,45. By incorporating with a small amount of magnetic dopants, the TI becomes ferromagnetic while the strong spin-orbit coupling can still be preserved, as exemplified by the archetypal material Cr-doped (Bi,Sb)2Te3.

Here, in a thin single-layer magnetic TI Cr-doped (Bi,Sb)2Te3 we discover a large DL-SOT accompanied with considerable unidirectional magnetoresistance (UMR) and nonlinear Hall effect. After demagnetizing the magnetic TI to zero field, the magnetization of the magnetic TI itself could be switched by applying a dc current solely. Based on the magnetoelectric transport and magneto-optical Kerr effect (MOKE) measurements, we found that such a switching, which was assisted by the magnetic history, behaved nonvolatile at zero field but otherwise volatile, which could be accomplished by either static or quasistatic current. We proposed that the SOT in the single-layer Cr-doped (Bi,Sb)2Te3 that probably originates in the compositional asymmetry of the Cr-profile may play a role in the observed switchings.


Second harmonic measurements

To probe SOT in the single-layer magnetic TI, second harmonic magnetoresistance (\({R}_{\parallel }^{2\omega }\)) and Hall resistance (\({R}_{\perp }^{2\omega }\)) were tracked when rotating the applied magnetic field \({{{{{\bf{H}}}}}}\) in the xy-, zx-, and zy-planes, as schematically shown in Fig. 1a, respectively. This is performed at 4.5 K by feeding an ac current \({I}_{{{{{{\rm{ac}}}}}}}\) = 10 μA into the Hall bar along the x-direction and measuring the voltages with lock-in technique. As \({R}_{\parallel }^{2\omega }\) of the zx-scan only stems from the anisotropic magnetoresistance \({R}_{{{{{{\rm{AMR}}}}}}}\) perturbed by DL-SOT and magneto-thermalelectric effect (MTE) via46:

$${R}_{\parallel {zx}}^{2\omega }=-\frac{\Delta {R}_{{{{{{\rm{AMR}}}}}}}}{2}\frac{{h}_{{{{{{\rm{DL}}}}}}}}{{H}_{{{{{{\rm{K}}}}}}}{{\cos }}\theta +H{{\cos }}\left({\theta }_{H}-\theta \right)}{{\sin }}\left(2\theta \right)+{C}_{{{{{{\rm{MTE}}}}}}}{{\sin }}\left(2\theta \right)\nabla {T}_{z}$$

where \({h}_{{{{{{\rm{DL}}}}}}}\) is the current-induced effective DL-field, \({H}_{{{{{{\rm{K}}}}}}}\) is the effective anisotropy field, \({C}_{{{{{{\rm{MTE}}}}}}}\) is the coefficient for MTE, and \(\nabla {T}_{z}\) represents the temperature gradient along the z-direction induced by Joule heating, the extraction of \({h}_{{{{{{\rm{DL}}}}}}}\) can be obtained by fitting \({R}_{\parallel {zx}}^{2\omega }\) data obtained above the saturation field of \({\mu }_{0}H\) ≥ 0.8T using Eq. (1). Here, \(\Delta {R}_{{{{{{\rm{AMR}}}}}}}\) = 3400 Ω is obtained from the first harmonic magnetoresistance (\({R}_{\parallel {xy}}^{\omega }\)) when rotating \({\mu }_{0}H\) = 3T in the xy-plane (Fig. 2b) and \({H}_{{{{{{\rm{K}}}}}}}\) = 0.48T from the first harmonic Hall resistance when rotating various \({\mu }_{0}H\) in the zx-plane, \({R}_{\perp {zx}}^{\omega }\) (Supplementary Note 1 and Supplementary Fig. 1). The first term of Eq. (1), which is related to the DL-SOT, has a magnetic field dependence of \(H{{\cos }}\left({\theta }_{H}-\theta \right)\), whereas the second term, which is related to MTE, is independent of the external magnetic field. Therefore, by fitting \({R}_{\parallel {zx}}^{2\omega }\) data under 1–3T (Fig. 1b and Supplementary Fig. 2), these two terms can be separated and \({h}_{{{{{{\rm{DL}}}}}}}\) is determined to be 0.65 mT. The corresponding SOT-efficiency \({\xi }_{{{{{{\rm{SOT}}}}}}}=\frac{2e{M}_{{{{{{\rm{s}}}}}}}t{h}_{{{{{{\rm{DL}}}}}}}}{\hslash {j}_{{{{{{\rm{c}}}}}}}}\) is about 12.4 as evaluated using parameters of the saturation magnetization \({M}_{{{{{{\rm{s}}}}}}}\) = 35 kA/m (Supplementary Fig. 3), the film thickness t = 6 nm, and the current density jc = 3.33 × 107 A/m2. Such a \({\xi }_{{{{{{\rm{SOT}}}}}}}\) is much higher than those in metallic ferromagnet/heavy-metal bilayers47,48,49 and comparable to those in TI/ferromagnet bilayers43,44. It is noted that the magnitude of \({R}_{\parallel {zx}}^{2\omega }\) is not very sensitive to the \({\mu }_{0}H\)-strength, so there should be a large contribution from MTE, whose magnitude is almost independent with \({\mu }_{0}H\).

Fig. 1: The damping-like spin-orbit torque, unidirectional magnetoresistance, and nonlinear Hall effect.
figure 1

a Geometry of the measurements. The Hall bar is made of a single-layer magnetic topological insulator. b, c Second harmonic magnetoresistance (\({R}_{\parallel }^{2\omega }\)) and Hall resistance (\({R}_{\perp }^{2\omega }\)) recorded when rotating the magnetic field of 3T at 4.5 K in three different planes with fits. df Extracted field-dependent \({R}_{\parallel }^{2\omega }\) and \({R}_{\perp }^{2\omega }\) in different planes with fits.

Fig. 2: Effects of the dc current.
figure 2

a Switchings of the out-of-plane magnetizations by the dc current under zero magnetic field. The geometry of the first harmonic magnetoresistance (\({R}_{\parallel }^{\omega }\)) and first harmonic Hall resistance (\({R}_{\perp }^{\omega }\)) measurements is also shown in the upper panel. (DL damping-like, M magnetization, \({{{{{{\boldsymbol{\tau }}}}}}}_{{{{{{\rm{DL}}}}}}}\) DL torque, \({{{{{{\bf{h}}}}}}}_{{{{{{\rm{DL}}}}}}}\) DL effective field.) The lower panel shows the optical image of the device. Scale bar: 100 μm. b \({R}_{\parallel }^{\omega }\) measured by applying different current combinations when rotating in the xy-plane. c \({\mu }_{0}{H}_{z}\)-dependent \({R}_{\perp }^{\omega }\) measured by applying different current combinations as plotted by different colors. Insets are enlarged views.

Likewise, the UMR can be evaluated from \({R}_{\parallel }^{2\omega }\) when rotating in the xy- and zy-planes.\(\,{R}_{\parallel }^{2\omega }\) in the xy-plane can be expressed as46:

$${R}_{\parallel {xy}}^{2\omega } = -\frac{\Delta {R}_{{{{{{\rm{AMR}}}}}}}}{4}\frac{{h}_{{{{{{\rm{FL}}}}}}}}{H}\left({{{{{\rm{sin }}}}}}\, 3\varphi +{{{{{\rm{sin }}}}}}\, \varphi \right)+\left({C}_{{{{{{\rm{ANE}}}}}}}+{C}_{{{{{{\rm{SSE}}}}}}}\right){{{{{\rm{sin }}}}}}\, \varphi \nabla {T}_{z}\\ +{N}_{{{{{{\rm{ONE}}}}}}}H{{{{{\rm{sin }}}}}}\, \varphi \nabla {T}_{z}+{R}_{\parallel {{{{{\rm{UMR}}}}}}}{{{{{\rm{sin }}}}}}\, \varphi$$

where \({h}_{{{{{{\rm{FL}}}}}}}\) is the current-induced effective FL-field, \({C}_{{{{{{\rm{ANE}}}}}}}\) and \({C}_{{{{{{\rm{SSE}}}}}}}\) are the coefficients for the anomalous Nernst effect (ANE) and spin Seebeck effect (SSE), \({N}_{{{{{{\rm{ONE}}}}}}}\) is the coefficient for the ordinary Nernst effect (ONE), and \({R}_{\parallel {{{{{\rm{UMR}}}}}}}\) is the UMR, respectively. Experimentally, these terms could be distinguished from each other by their distinct angle- and H-dependences: \({h}_{{{{{{\rm{FL}}}}}}}\) contributes a \({{\sin }}3\varphi\) component, ANE/SSE and their coefficients (\({C}_{{{{{{\rm{ANE}}}}}}}\) and \({C}_{{{{{{\rm{SSE}}}}}}}\)) are essentially H-independent, whereas ONE is H-linear; in TI-related systems, \({R}_{{{{{{\rm{UMR}}}}}}}\) usually arises from the asymmetric magnon scattering due to the spin-momentum locking which results in a \({H}^{-p}\) dependence with \(p\) close to 150,51,52. As the helicity of spin-momentum locking is opposite at the top and bottom surfaces, the corresponding scattering also has opposite sign. The UMR will be canceled out if the top and bottom surfaces are perfectly symmetric, but this ideal situation can never be achieved in any real sample. Hence, the presence of UMR suggests the structural asymmetry in the as-grown single-layer magnetic TI, which provides hints for the presence of SOT. Take the \({R}_{\parallel {xy}}^{2\omega }\) data under 3T in Fig. 1b as an example, it well follows a \({{\sin }}\varphi\)-behavior whereas the \({{\sin }}3\varphi\) component is vanishingly small and, hence, the contribution from \({h}_{{{{{{\rm{FL}}}}}}}\) is negligible. Similar fittings are carried out in other \({R}_{\parallel {xy}}^{2\omega }\) data under various \({\mu }_{0}H\), which are scatter-plotted in Fig. 1d. By fitting to these \({\mu }_{0}H\)-dependent \({{\sin }}\varphi\) component, the \(H\)-independent, \(H\)-linear, and \({H}^{-p}\)-dependent terms could be determined as 42.53, −3.52, and −10.90 Ω, respectively. This suggests that the \(H\)-independent ANE/SSE and \({H}^{-p}\)-dependent UMR dominate \({R}_{\parallel {xy}}^{2\omega }\), while the \(H\)-linear ONE only takes a minor role. The above evaluation is cross-checked using \({R}_{\parallel {zy}}^{2\omega }\). Rotating in this plane, only the ANE, ONE, and UMR contribute via:

$${R}_{\parallel {zy}}^{2\omega }={C}_{{{{{{\rm{ANE}}}}}}}\, {{{{{\rm{sin }}}}}}\, \theta \, \nabla {T}_{z}+{N}_{{{{{{\rm{ONE}}}}}}}\, H\, {{{{{\rm{sin }}}}}}\, \theta \nabla {T}_{z}+{R}_{\parallel {{{{{\rm{UMR}}}}}}}\, {{{{{\rm{sin }}}}}}\, \theta .$$

The extracted \({\mu }_{0}H\)-dependent \({R}_{\parallel {zy}}^{2\omega }\) is scatter-plotted in Fig. 1e, the fitting to which evaluates the H-independent, H-linear, and H-p-dependent contributions as 64, −3.95, and −14 Ω, respectively. These values essentially agree with those from \({R}_{\parallel {xy}}^{2\omega }\) except the H-independent term owing to the extra contribution from SSE in \({R}_{\parallel {xy}}^{2\omega }\) and/or a slight change of ANE after thermal cycling the sample. Therefore, UMR can be identified after excluding the thermal effects and SOT. To quantify UMR using the ratio of the second to first harmonic magnetoresistance over the applied ac current density jac, \(\alpha =(\frac{{R}_{||xy}^{2\omega }}{{R}_{||xy}^{\omega }})/{j}_{ac}\), we then acquire the φ-dependent \({R}_{\parallel {xy}}^{\omega }\) under 3T in Fig. 2b, which follows the \({{\cos }}\left(2\varphi \right)\) relation with a periodicity of π. We could then determine α = 6.51 × 10−6 cm2/A, which is several orders of magnitude higher than those in metallic bilayers53,54 and comparable to those in TI-multilayers52. To obtain the direct signature of UMR, dc currents of opposite polarities \(\pm {I}_{{{{{{\rm{dc}}}}}}}\) are respectively added onto \({I}_{{{{{{\rm{ac}}}}}}}\) while the first harmonic voltages are measured using lock-in technique (Fig. 2b). In this way, a pronounced antisymmetric \({R}_{\parallel {xy}}^{\omega }\) between φ = 90° and 270° arises that \({R}_{\parallel {xy}}^{\omega }\) is higher (lower) when \({I}_{{{{{{\rm{dc}}}}}}}\) is parallel (antiparallel) to the y-axis. Such an antisymmetry is regarded as the signature of the unidirectional and bilinear magnetoresistance, showing that the reversal of \({\mu }_{0}H\) is equivalent to the reversal of \({I}_{{{{{{\rm{dc}}}}}}}\)55,56. As shown in Supplementary Note 2, since \({R}_{\parallel {xy}}^{2\omega }\) inevitably comes with \({R}_{\parallel {xy}}^{\omega }\) in lock-in measurements, ONE in \({R}_{\parallel {xy}}^{2\omega }\) will also contribute to the resulted \({R}_{\parallel {xy}}^{\omega }\), which hence complicates the estimation of UMR from \({R}_{\parallel {xy}}^{\omega }\). Anyway, such \({R}_{\parallel {xy}}^{\omega }\) also supports the existence of UMR in the single-layer magnetic TI as those extracted from \({R}_{\parallel {xy}}^{2\omega }\), and provides another method beyond the second harmonic method to probe the UMR, which is particularly useful for cases with negligible thermal effect or small \({R}_{\parallel }^{2\omega }\) signal.

The above evaluations are cross-checked by the second harmonic Hall resistance \({R}_{\perp }^{2\omega }\) of the xy-plane, which can be expressed by46:

$${R}_{\perp {xy}}^{2\omega } = \frac{\Delta {R}_{{{{{{\rm{AMR}}}}}}}}{4}\frac{w}{l}\frac{{h}_{{{{{{\rm{FL}}}}}}}}{H}\left({{\cos }}\, 3\varphi -{{\cos }}\, \varphi \right)+\frac{{R}_{{{{{{\rm{AHE}}}}}}}}{2}\frac{{h}_{{{{{{\rm{DL}}}}}}}}{-{H}_{K}+H}{{\cos }}\, \varphi \\ -{C}_{{{{{{\rm{ANE}}}}}}}\frac{w}{l}{{\cos }}\, \varphi \nabla {T}_{z}-{N}_{{{{{{\rm{ONE}}}}}}}H\frac{w}{l}{{\cos }}\, \varphi \nabla {T}_{z}+{R}_{\perp {{{{{\rm{UMR}}}}}}}{{\cos }}\, \varphi$$

where w = 50 μm and l = 200 μm are the width and length of the Hall bar respectively, \({R}_{{{{{{\rm{AHE}}}}}}}\) is the resistance from the anomalous Hall effect (AHE), and \({R}_{\perp {{{{{\rm{UMR}}}}}}}\) is the nonlinear Hall component of UMR. Note that the FL-SOT term should be excluded according to the analysis on \({R}_{\parallel }^{2\omega }\). Therefore, \({R}_{\perp {xy}}^{2\omega }\) should only follow a \({{\cos }}\varphi\)-periodicity, which is captured by the φ-dependent \({R}_{\perp }^{2\omega }\) data with the cosinusoidal fit in Fig. 1c. Similar to \({R}_{\parallel {xy}}^{2\omega }\) and \({R}_{\parallel {zy}}^{2\omega }\), since the remaining terms have different H-dependences, it is possible to separate them by fitting to the \({\mu }_{0}H\)-dependent \({R}_{\perp {xy}}^{2\omega }\). Here, in addition to the H-independent, H-linear, and Hp-dependences discussed above, the \({\left(-{H}_{K}+H\right)}^{-1}\) dependence that corresponds to the contribution from DL-SOT needs to be extracted separately. From the fit shown in Fig. 1f these terms are determined as −9.7, 0.8, 2.7, and −0.17 Ω, respectively. As expected from Eqs. (3) and (4), thermal terms of ANE and ONE in \({R}_{\parallel {zy}}^{2\omega }\) and \({R}_{\perp {xy}}^{2\omega }\) should differed by \(-w/l\) = −0.25. After counting \(-w/l\), the former two terms (i.e. ANE and ONE terms) become 38.8 and −3.2 Ω, respectively. Compared with those extracted from \({R}_{\parallel {zy}}^{2\omega }\), i.e., 64 and −3.95 Ω, the ONE term can reach a quantitative consensus, whereas a large contrast exists between these ANE terms. Besides the measurement uncertainty and angular misalignment in experiments, such a contrast probably comes from a H-linear UMR, which has been observed in the intrinsic TIs57,58. Ideally, the \({R}_{\perp {xy}}^{2\omega }\)-measurement allows to quantify the DL-SOT but it may not be well compatible with a system with a perpendicular anisotropy (like the magnetic TI of this study) since a strong \({\mu }_{0}H\) is required in this case. However, such a strong \({\mu }_{0}H\) diminishes the contribution from DL-SOT and hence may cause larger measurement uncertainties due to the reduced magnetic susceptibility.

Magnetization switching by dc current

By applying \({I}_{{{{{{\rm{dc}}}}}}}\) to the single-layer magnetic TI, we detected a signature of magnetization switching particularly near zero magnetic field. As illustrated in Fig. 2a that the initial magnetizations are prepared to the out-of-plane and \({\mu }_{0}H\) returns to zero. As a sizable DL-SOT is observed in the single layer, when \({I}_{{{{{{\rm{dc}}}}}}}\) is applied along the x-axis, the DL-SOT \({{{{{{\boldsymbol{\tau }}}}}}}_{{{{{{\rm{DL}}}}}}}={{{{{\bf{M}}}}}}\times \left({{{{{\bf{M}}}}}}\times {{{{{\boldsymbol{y}}}}}}\right)\) will force the magnetization \({{{{{\bf{M}}}}}}\) to switch to align with the DL effective field \({{{{{{\bf{h}}}}}}}_{{{{{{\rm{DL}}}}}}}={h}_{{{{{{\rm{DL}}}}}}}{{{{{\bf{M}}}}}}\times {{{{{\boldsymbol{y}}}}}}\) and realize a magnetization switching. Generally, this scheme requires an in-plane \({\mu }_{0}H\) parallel to the applied current for a deterministic switching, and the result is shown in Supplementary Fig. 4. When applying different magnetic fields along the x-direction (\(\pm {\mu }_{0}{H}_{x}\)) in the setup of Fig. 1 and sweeping \({I}_{{{{{{\rm{dc}}}}}}}\) to drive the switching of \({R}_{\perp }^{\omega }\), along with the change of the \({I}_{{{{{{\rm{dc}}}}}}}\)-polarity, \({R}_{\perp }^{\omega }\) switches sign for each \(\pm {\mu }_{0}{H}_{x}\). This test suggests that the SOT is successfully induced in our single-layer magnetic TI. Surprisingly, under an out-of-plane field \({\mu }_{0}{H}_{z}\) in Fig. 2c, when the role of \({I}_{{{{{{\rm{dc}}}}}}}\) is gradually promoted by increasing the relative magnitude of \({I}_{{{{{{\rm{dc}}}}}}}\) over \({I}_{{{{{{\rm{ac}}}}}}}\) (\({I}_{{{{{{\rm{ac}}}}}}}\) is always fixed at 10 μA), the first harmonic Hall resistance \({R}_{\perp }^{\omega }\) at the saturation exhibits a successive decrease. When \({I}_{{{{{{\rm{dc}}}}}}}\) reaches 60 μA, a pair of small zero-Hall plateaus appear near the coercivity (left inset in Fig. 2c); above this value, the hysteresis loop even switches sign as evidenced by the reversed loop direction (right inset in Fig. 2c). Regarding the origin of the observed switching, we try to exclude the Joule heating from the possible main mechanism as follows. As shown in Supplementary Note 3 and Supplementary Fig. 5, while a dc-dominant mixed current \({I}_{{{{{{\rm{ac}}}}}}}+{I}_{{{{{{\rm{dc}}}}}}}\) = 10 μA + 100 μA realized the switching at 4.5 K, the ac current \({I}_{{{{{{\rm{ac}}}}}}}\) = 110 μA with the similar heating-power to the former only heats up the sample without showing any trace of switching. This heating effect is supported by the fact that \({R}_{\perp }^{\omega }\) measured by applying such \({I}_{{{{{{\rm{ac}}}}}}}\) at 4.5 K coincides with that by a much smaller \({I}_{{{{{{\rm{ac}}}}}}}\) = 10 μA at an elevated sample temperature, i.e. 8 K. The above results underline that the switching is caused by \({I}_{{{{{{\rm{dc}}}}}}}\) whereas \({I}_{{{{{{\rm{ac}}}}}}}\) only heats up the sample (otherwise the result using \({I}_{{{{{{\rm{ac}}}}}}}\) = 110 μA should have shown the switching). That is, the directionality of the applied current is the key for switching, while the consequent Joule heating cannot induce or explain the nonvolatile switching.

As discussed above, the resulted \({R}_{\perp }^{\omega }\) may include some \({R}_{\perp }^{2\omega }\) component like thermal effects more than the AHE. This leads to a question whether the observed switching in \({R}_{\perp }^{\omega }\) really comes from the magnetization switching. To address this, the switching process in transport measurement is performed using pulse currents (\({I}_{{{{{{\rm{p}}}}}}}\)) in order to minimize the Joule heating and also reexamined using MOKE simultaneously as follows.

Switching by quasistatic current

The switching using \({I}_{{{{{{\rm{p}}}}}}}\) is found to be equivalent with that using static current, further diminishing the role played by Joule heating. The experiment is performed as follows: fully magnetizing the magnetic TI under μ0 Hz = ±0.5 T, i.e. \(\pm {M}_{z}\), followed by sweeping to a specific magnetic field \({\mu }_{0}{H}_{z}^{* }\); then injecting one or multiple \({I}_{{{{{{\rm{p}}}}}}}\) of various amplitudes and 200-ms length to the Hall bar. During this process, Iac = 1 μA constantly flows through the device so that \({R}_{\perp }^{\omega }\) can be always recorded by lock-in amplifier. In Fig. 3a, the gray loops are the full hysteresis loops measured using \({I}_{{{{{{\rm{ac}}}}}}}\) which serve as references. The Hall bar is fully magnetized under 0.5T (left/right panel) before being demagnetized to zero field (i.e., \({\mu }_{0}{H}_{z}^{* }\) = 0) and fed by one \({I}_{{{{{{\rm{p}}}}}}}\) = 50 μA. Suddenly \({R}_{\perp }^{\omega }\) jumps from \(\mp {M}_{z}\) to \(\pm {M}_{z}\) during the application of \({I}_{{{{{{\rm{p}}}}}}}\), signifying the magnetization switching under zero field. After this switching, \({I}_{{{{{{\rm{p}}}}}}}\) is removed but \({I}_{{{{{{\rm{ac}}}}}}}\) still sustains to track \({R}_{\perp }^{\omega }\) which gradually converges onto the full hysteresis loop when sweeping toward ±0.5T. Results using Ip = 20 μA and 100 μA are shown in Supplementary Fig. 6 which show similar behaviors. The magnitude of the sudden jump is found to increase when ramping \({I}_{{{{{{\rm{p}}}}}}}\) to 50 μA, and saturate if ramping further. This signature is analogous to the SOT-induced magnetization switching in magnetic TI heterostructures39,40, in which only a portion of magnetization can be switched no matter the current is further increased. To examine the equivalence of static and pulse current in switchings, various \({\mu }_{0}{H}_{z}^{* }\) demagnetizing from +0.5T are adopted and Ip = 200 μA is used (Supplementary Fig. 7) to sketch the \({\mu }_{0}{H}_{z}\)-dependent \({R}_{\perp }^{\omega }\) switching loop, which is then used to compare with those obtained using \(\pm {I}_{{{{{{\rm{dc}}}}}}}\). These three curves are plotted in Fig. 3b, which show consistency after minimizing the Joule heating by \({I}_{{{{{{\rm{p}}}}}}}\). This again proves that the switching mechanism associates with the directionality of the applied current, which excludes the Joule heating from the dominant mechanism. Moreover, from the coincidence of the curves by \(\pm {I}_{{{{{{\rm{dc}}}}}}}\), one can find that both polarities function in the same way and the switching is polarity-independent. To further confirm this, multiple \({I}_{{{{{{\rm{p}}}}}}}\) of opposite polarities (\({I}_{{{{{{\rm{p}}}}}}}\) = ±100 μA) are successively injected. As shown in Fig. 3c, after demagnetizing from +0.5T to 0, both \({I}_{{{{{{\rm{p}}}}}}}\) can quickly reduce \({R}_{\perp }^{\omega }\) from 4 to −1.2 kΩ. Such a resulted \(-{M}_{z}\) state is very stable under the perturbation of \({I}_{{{{{{\rm{ac}}}}}}}\) = 1 μA until another \({I}_{{{{{{\rm{p}}}}}}}\) is followed, demonstrating its nonvolatile nature. Interestingly, \({I}_{{{{{{\rm{p}}}}}}}\) except the first one can only induce a sudden increase of \({R}_{\perp }^{\omega }\) from −1.2 to 0.2 kΩ during the application of \({I}_{{{{{{\rm{p}}}}}}}\), the removal of which immediately drives \({R}_{\perp }^{\omega }\) back to −1.2 kΩ. Such a volatile switching can be well repeated by applying several subsequent \({I}_{{{{{{\rm{p}}}}}}}\), in sharp contrast to the nonvolatile one induced by the first \({I}_{{{{{{\rm{p}}}}}}}\). This volatile switching is likely to originate in the DL-SOT demonstrated above because the lack of an in-plane field for a deterministic switching could cause the restoration to the initial magnetization after the disappearance of the \({I}_{{{{{{\rm{p}}}}}}}\)-induce SOT. However, there is no reason that the first \({I}_{{{{{{\rm{p}}}}}}}\) inducing the nonvolatile switching has a different effect when compared to the subsequent \({I}_{{{{{{\rm{p}}}}}}}\) that result in SOT-induced volatile switchings, except that the first \({I}_{{{{{{\rm{p}}}}}}}\) is applied to the special magnetic state that is preset by the magnetic history. To check this, we explore the \({I}_{{{{{{\rm{p}}}}}}}\)-magnitude dependence by varying from 20, 50, to 100 μA respectively, and the result is shown in Fig. 3d. One can find the threshold \({I}_{{{{{{\rm{p}}}}}}}\) ~ 50 μA, below which the change of \({R}_{\perp }^{\omega }\) by \({I}_{{{{{{\rm{p}}}}}}}\) is smaller than those above and in a step-by-step manner (as exemplified by the 20 μA curve), which is nonvolatile. When above the threshold, the switching is accomplished right after applying the first \({I}_{{{{{{\rm{p}}}}}}}\) (as exemplified by the coincidence of the 50 and 100 μA curves) while subsequent \({I}_{{{{{{\rm{p}}}}}}}\) cannot cause further change of \({R}_{\perp }^{\omega }\). Thus, the amount of the switched magnetization rises with the \({I}_{{{{{{\rm{p}}}}}}}\)-magnitude before reaching the threshold, consistent with the DL-SOT origin. Such a threshold is much lower than those in metallic bilayers.

Fig. 3: Zero-field magnetization switchings probed by transports.
figure 3

a The first harmonic Hall resistance (\({R}_{\perp }^{\omega }\)) measured by \({I}_{{{{{{\rm{ac}}}}}}}\) (10 μA) except a pulse current Ip (50 μA, 200 ms) is applied under zero field, which results in a sudden jump in \({R}_{\perp }^{\omega }\) (dashed arrow). The gray loops are the full hysteresis loops measured using \({I}_{{{{{{\rm{ac}}}}}}}\) as references. b \({\mu }_{0}{H}_{z}\)-dependent \({R}_{\perp }^{\omega }\) probed by different current configurations. c, d \({R}_{\perp }^{\omega }\) recorded when repeatedly applying \({I}_{{{{{{\rm{p}}}}}}}\) of different amplitudes under zero field. e Phenomenological explanation to the observed magnetization switching. The red circles represent the magnetic state, blue dashed lines indicate the possible transitions, and \(\triangle E\) is the energy difference between different magnetic states. Activation energies given by various \({I}_{{{{{{\rm{dc}}}}}}}\) or \({I}_{{{{{{\rm{p}}}}}}}\) can drive the magnetic state out from different local minima to others (navy dashed arrows).

Phenomenological model

The above observed switching can be described by a phenomenological model in Fig. 3e. On the one hand, the perpendicular magnetic anisotropy of the CBST film is not strong enough to maintain a single domain at zero-field. So the magnetization shows a multidomain state at zero-field with the majority of domains preset by the applied magnetic field history (see Figs. 2c and 4c). This state may thus host some metastable states and be easy to be switched. On the other hand, since the ferromagnetism in CBST is induced by Cr dopants, which could not be ideally homogeneous, it results in an amount of magnetic grains isolated by different energy barriers. The formation of these magnetic grains, and thus their energy landscape, is a combined effect from the native crystalline topography, specific Cr-distribution, and the magnetic history. Whereas the former two cannot be changed for the as-grown thin film, the magnetic history that controls the magnetization energy of the grains then determines the switching process of these grains. The above two reasons lead to the energy landscape as shown in the left panel of Fig. 3e, which exhibits a native asymmetry preset by the magnetic history. Probably because of this, the nonvolatile switching would therefore only depend on the duration and magnitude of \(I\) instead of its polarity. The object of this nonvolatile switching is likely to be the magnetic grains addressed above. \(\Delta E\) in this picture could be determined by the grain distribution (size, density, etc.), magnetic stiffness, magnetization, anisotropy, etc., which requires different activation energies (\({I}_{{{{{{\rm{dc}}}}}}}\) or \({I}_{{{{{{\rm{p}}}}}}}\) in this case) to drive the magnetic state out from various local minima. For the specific microscopic switching process, the magnetization switching in our single layer should be dominated by the domain wall propagation in the multidomain state under zero field. Furthermore, since our sample is in the micron-scale, the switching accomplished purely through a coherent rotation should be rather difficult. Usually, the switching process is initiated through a coherent rotation, i.e., the nucleation of a reversed domain via the applied torque/effective field. Then the domain wall propagates and further promotes the whole switching process. Such a process has also been observed through magnetic imaging59. According to the experimental results, the final state looks more energetically stable than the initial state, as marked by the energy difference \(\Delta E\) between them. Although the formation of this \(\Delta E\) remains unclear but it should associate with the magnetic history based on the fact that the nonvolatile switching direction solely depends on the magnetization procedure, i.e., from \(+{M}_{z}\) or \(-{M}_{z}\) to zero field. After switching, the energy landscape is smoothened and the switched state becomes the only stable state. Therefore, the native asymmetry is removed. At this stage, the application of \({I}_{{{{{{\rm{dc}}}}}}}\) or \({I}_{{{{{{\rm{p}}}}}}}\) only results in metastable magnetic dynamics, the removal of which immediately restores the switched state.

Fig. 4: Zero-field magnetization switchings probed by magneto-optical Kerr effect (MOKE).
figure 4

a \({\mu }_{0}{H}_{z}\)-dependent Kerr rotation angle (\({\theta }_{{{{{{\rm{K}}}}}}}\)). A pulse current \({I}_{{{{{{\rm{p}}}}}}}\) (50 μA, 200 ms) is applied under zero field, which results in a sudden change of \({\theta }_{{{{{{\rm{K}}}}}}}\) (dashed arrow). The gray loops are the full hysteresis loops measured using \({I}_{{{{{{\rm{ac}}}}}}}\) as references. b \({\theta }_{{{{{{\rm{K}}}}}}}\) when repeatedly applying \({I}_{{{{{{\rm{p}}}}}}}\) of different magnitudes under zero field. ce After sweeping from −0.5T to 0 followed by applying \({I}_{{{{{{\rm{p}}}}}}}\) of different magnitudes, MOKE images are taken by spatially scannings around the Hall bar, confirming the zero-field magnetization switching. Due to the large the laser spot size (~15 μm), the MOKE signal varies gradually around edges. The mapping areas near edges exhibit color gradients when compared with those in the bulks, which have length scales of ~ 12–14 μm, similar to the laser spot size. Due to this similar length scale and limited resolution, whether the switching process initiates from the edge remains unclear.

Switching probed by MOKE

The nonvolatile and volatile switchings are further confirmed using MOKE measurements, which were performed simultaneously with transports. The nonvolatile switchings in Fig. 3a and Supplementary Fig. 6 are reexamined by tracking the \({\mu }_{0}{H}_{z}\)-dependent Kerr rotation angle (\({\theta }_{{{{{{\rm{K}}}}}}}\)), and the results are shown in Fig. 4a and Supplementary Fig. 8, which show identical behaviors to those by transport. The transport results obtained using multiple-\({I}_{{{{{{\rm{p}}}}}}}\) of various magnitudes in Fig. 3d can also be reproduced by MOKE in Fig. 4b, in which the current threshold of the nonvolatile switching agrees with that in transport. The volatile switchings by subsequent multiple-\({I}_{{{{{{\rm{p}}}}}}}\) also recur consistently. The most direct evidence to the nonvolatile switching refers to the MOKE images which were obtained by spatially scannings around the Hall bar after demagnetizing from −0.5T to 0 followed by applying \({I}_{{{{{{\rm{p}}}}}}}\) = 0, 15, and 50 μA, as shown in Fig. 4c–e respectively. The image without applying any \({I}_{{{{{{\rm{p}}}}}}}\) (Fig. 4c) can serve as a reference of the \(-{M}_{z}\) state under zero field, which shows negative \({\theta }_{{{{{{\rm{K}}}}}}}\) throughout the Hall bar. After \({I}_{{{{{{\rm{p}}}}}}}\) = 15 μA is applied, the magnitude of \({\theta }_{{{{{{\rm{K}}}}}}}\) decreases but \({\theta }_{{{{{{\rm{K}}}}}}}\) still stays negative, which indicates a partial switching of \(-{M}_{z}\). With \({I}_{{{{{{\rm{p}}}}}}}\) = 50 μA above the threshold, \({\theta }_{{{{{{\rm{K}}}}}}}\) throughout the Hall bar becomes positive, which signifies the \(+{M}_{z}\) state resulted from magnetization switching. Such a process conforms with results in Fig. 4b using \({I}_{{{{{{\rm{p}}}}}}}\) smaller and greater than the threshold, strongly proving the nonvolatile zero-field switching by \({I}_{{{{{{\rm{p}}}}}}}\).

Trivial effect in metallic system

To gain insights about the mechanism of the switching, similar experiments have been performed on a ferromagnet/heavy-metal multilayer Ta(3)/CoFeB(1.3)/MgO(1)/Ta(3) (unit: nm) with a perpendicular anisotropy. This multilayer can realize a conventional SOT-induced magnetization switching when applying an in-plane μ0 H = 300 Oe and a switching current Idc = ±10 mA. By applying different current combinations to the multilayer, including Iac = 15 μA ~ 15 mA and Iac (15 μA) + Idc (100 μA ~ 15 mA), we found that \({I}_{{{{{{\rm{dc}}}}}}}\) functions no more than Joule heating when compared the results obtained by \({I}_{{{{{{\rm{ac}}}}}}}\) only (Supplementary Fig. 9), which is in contrast to those in the single-layer magnetic TI. This observation highlights the difference between the magnetic TI and the metallic multilayer in the present switching methodology. This difference may be ascribed to the difference in magnetic energy or anisotropy between the two systems.


The presence of SOT generally requires the breaking of structural inversion symmetry. In the ferromagnet/heavy-metal bilayer this can be achieved by the native vertical asymmetry. In our single-layer system, the matrix material, (Bi,Sb)2Te3, is a layered hexagonal material that has a trigonal axis (three-fold rotation symmetry), a binary axis (two-fold rotation symmetry), and a bisectrix axis (in the reflection plane)60. Thus, there is no apparent symmetry breaking in terms of the crystalline structure. As the single layer exhibits nonreciprocal transport, like UMR above, there must be some inversion symmetry breaking in this system besides the time-reversal symmetry broken by the magnetic order. However, according to the intensity oscillation of a reflection high-energy electron diffraction (RHEED) during the epitaxial growth of the Cr-doped (Bi,Sb)2Te3 layer (Supplementary Fig. 10), one can see that it takes ~118 s to grow the first layer, whereas it changes to ~75 s for each of the following layers. This originates in the lattice transition from the GaAs substrate (zinc-blende) to the Cr-doped (Bi,Sb)2Te3 film (rhombohedral), during which the high-vapor-pressure materials Bi/Sb/Te are less prone to be deposited than the low-vapor-pressure metal Cr. This contrast may lead to a higher Cr-concentration in the first layer than those in the followings. Mathematically, the Cr-concentration in the first layer is ~118 s/75 s = 1.57 times of the following, but this number is overestimated since the Cr-deposition rate is not ideally constant but also depends on the Bi/Sb/Te-deposition rates. Therefore, the epitaxial process is likely to result in a compositional asymmetry in the Cr-profile, which may be the origin of the observed UMR and SOT. Another symmetry-breaking candidate may come from the structural inversion asymmetry between the top and bottom surfaces of the single layer due to the existence of the substrate, which is a well-known effect in magnetic TI thin film61,62,63,64.

The above results have shown a large SOT extracted from the single-layer magnetic TI and volatile switchings induced by SOT, we thus intuitively regard that the SOT is likely to also play a role in nonvolatile switchings. When the Cr-concentration at the bottom surface of the single-layer magnetic TI is higher than that of the top, correspondingly, the gap opening at the bottom surface will be larger than that of the top. Due to the extremely small gap size (about the order of 10 meV) and the elevated temperature (like 4.5 K in the present study), surface electrons will be thermally excited to Fermi surface and contribute to transport. Because of the smaller gap on the top surface, more top surface electrons will be excited and contribute than those from the bottom, which will lead to a non-equilibrium spin accumulation. Since electrons from the top and bottom surfaces have helicities of opposite spin-momentum locking, this will lead to a non-zero SOT from the single layer. It could be plausible that the SOT kicks the initial magnetic state and induces the switching in assistance with the Joule heating.

The present study not only shows the existence of a large DL-SOT accompanied with UMR and nonlinear Hall effect in the single-layer magnetic TI but also provides a methodology to realize a zero-field magnetization switching using topological magnets and perspective on understanding SOT-induced magnetization dynamics. This work can motivate research toward ultralow-power and scalable topological spintronic devices.


Film growth and device fabrication

The magnetic TI film used in this study is the modulation-doped Cr-doped (Bi,Sb)2Te3 epitaxially grown on a semi-insulating GaAs (111)B substrate in a molecular beam epitaxy system. During the growth, the substrate was held at ~200 °C with high purity Bi (6N) evaporated from a dual-filament cell, Cr (4N5) from a high-temperature cell, while Sb (6N) and Te (7N) from thermal cracker cells. Similar to refs. 65,66, the top-one and bottom-one quintuple layer has a higher Cr concentration resulted from the Cr temperature of 1130 °C, while the middle four quintuple layers have lower Cr concentration resulted from the Cr temperature of 1110 °C, forming a magnetically-symmetric profile. The temperatures of remaining elements were fixed throughout the growth. The total thickness is six quintuple layers, corresponding to ~6 nm, which was controlled by the reflection high-energy electron diffraction in situ. The films were patterned into Hall bars with w = 50 μm and l = 200 μm using standard photolithography and reactive ion etching. Contacts were made by depositing Ti (15 nm)/Au (85 nm) layers and bonded by Al wires.

Magneto-electrical transport

The Hall bar was loaded into a cryostat with the magnetic field created by a superconducting magnet. The current was applied by a Keithley 6221 current source meter and the voltage was picked up using lock-in amplifiers (SR830). The Hall bar can be rotated by a rotator on the cryogenic insert. The resulted Cr-doped (Bi,Sb)2Te3 film is ferromagnetic with the Curie temperature of above 12 K as shown in Supplementary Fig. 11. In other transport measurements, the sample is held at 4.5 K so that all the currents in the present study can be applied without temperature increment higher than 0.2 K, while the signal-to-noise ratios of the measured quantities are still satisfactory.

MOKE measurement

The scanning magneto-optic Kerr effects measurement was carried out using an all-fiber zero-loop Sagnac interferometer similar to refs. 67,68 installed in a dilution refrigerator. A high sensitivity of 0.2 μrad is achieved with about 50 μW light incident onto the sample. A three-axis piezo scanner cooperating with a focusing gradient-index lens allows spatial mapping with resolution of 15 μm. The probing light and applied magnetic field were perpendicular to the sample. The single-layer magnetic TI is much thinner than the skin depth of the operating wavelength of 1550 nm and the Kerr rotation angle induced by the intrinsic GaAs substrate is negligible. The detected Kerr rotation angle should be proportional to the out-of-plane magnetization69.