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1 Magnetization switching through giant spin-orbit torque in a magnetically doped topological insulator heterostructure Yabin Fan, 1,,* Pramey Upadhyaya, 1, Xufeng Kou, 1, Murong Lang, 1 So Takei, 2 Zhenxing Wang, 1 Jianshi Tang, 1 Liang He, 1,* Li-Te Chang, 1 Mohammad Montazeri, 1 Guoqiang Yu, 1 Wanjun Jiang, 1 Tianxiao Nie, 1 Robert N. Schwartz, 1 Yaroslav Tserkovnyak, 2 and Kang L. Wang 1,* 1 Electrical Engineering Department, University of California, Los Angeles, California 90095, USA 2 Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA These authors contributed equally to this work. *To whom correspondence should be addressed. yabin@seas.ucla.edu; liang.heliang@gmail.com; wang@seas.ucla.edu SUPPLEMENTARY INFORMATION Contents: 1. Materials, Electric Properties and Methods 2. Second Harmonic Measurement for Rotating the External Magnetic Field 3. Second Harmonic Measurement for Sweeping the In-plane External Magnetic Field: the Dependence 4. Determining the Effective Spin-orbit Field from the Field-like Spin-orbit Torque 5. Anisotropic Magneto-resistance (AMR) Effect and Planar Hall Effect (PHE) 6. Determining the Curie temperature in the TI/Cr-doped TI Bilayer Heterostructure 7. Temperature Dependence of the Switching Current Density and the Anisotropy Field 8. Thermal Effect when Passing an AC or DC Current in the Device 9. Comparison of the Switching Phase Diagrams revealed in the Experiments and the Single Domain Simulations NATURE MATERIALS 1

2 1. Materials, Electric Properties and Methods To reveal the crystalline structure of our grown TI/Cr-doped TI bilayer films, high-resolution scanning transmission electron microscopy (HRSTEM) 1 is performed on one of the bilayer films with composition (Bi 0.5 Sb 0.5 ) 2 Te 3 /(Cr 0.04 Bi 0.57 Sb 0.39 ) 2 Te 3. Fig. S1a reveals the atomically sharp TI/Cr-doped TI GaAs (111) interface and highly-ordered tetradymite-type quintuple-layered (QL) structure. Such epitaxiallyordered lattice configuration also confirms that there is no second phase segregation inside the TI/Crdoped TI bilayer thin film within the resolution of HRSTEM. In addition, the Bi/Sb composition ratio as well as the Cr-doping level is determined by an energy-dispersive X-ray (EDX) spectroscopy as shown in Fig. S1b. While the top TI layer shows no obvious Cr peak in the EDX spectrum, there is a clear Cr peak in the bottom Cr-doped TI layer EDX spectrum, indicating that the top TI layer is free of magnetic impurities and the diffusion of Cr dopants from the bottom Cr-doped TI layer into the top TI layer is negligible. Impedance-match for the top (Bi 0.5 Sb 0.5 ) 2 Te 3 layer and the bottom (Cr 0.08 Bi 0.54 Sb 0.38 ) 2 Te 3 layer in the (Bi 0.5 Sb 0.5 ) 2 Te 3 /(Cr 0.08 Bi 0.54 Sb 0.38 ) 2 Te 3 bilayer heterostructure thin film---the essential requirement for our magnetization switching experiments is to generate a uniform lateral current distribution inside the bilayer thin film. As we show the bilayer-based Hall bar device structure in Fig. 1b in the main text, the electric current is applied laterally from the source contact (S) to the drain contact (D) during the transport measurements. This in turn results in the parallel conduction configuration between the top undoped TI layer and the bottom Cr-doped TI layer. Consequently, we have to match the conductivities (S/cm) between these two channels in order to make sure we have a uniform current density (A/cm 2 ) flowing from S to D inside the whole bilayer film. By taking this concern into consideration, we deliberately adjust the Bi/Sb ratio in both the TI layer and the Cr-doped TI layer during the modulation growth so that the conductivities of them are nearly the same. The requirement is finally achieved in the (Bi 0.5 Sb 0.5 ) 2 Te 3 /(Cr 0.08 Bi 0.54 Sb 0.38 ) 2 Te 3 configuration where the conductivities in these two layers are 2 NATURE MATERIALS

3 SUPPLEMENTARY INFORMATION almost the same at 1.9 K. To demonstrate the impedance-match for these two layers, we separately grow a 6QL (Bi 0.5 Sb 0.5 ) 2 Te 3 thin film and a 6QL (Cr 0.08 Bi 0.54 Sb 0.38 ) 2 Te 3 thin film, and the magneto-transport experiments are carried out to probe their magneto-conductivities. In Fig. S2 we show the obtained data which demonstrate that the two films have nearly the same magneto-conductivities within a certain magnetic field region. Besides, the measured 2D carrier densities of these two films, as shown in Table S1, are low enough to ensure the Fermi level is inside the bulk band gap 2, as comparable with other reported data 3-5. The related electric properties of these two different films are summarized in Table S1. a b (Bi 0.5 Sb 0.5 ) 2 Te 3 Te Sb (Cr 0.04 Bi 0.57 Sb 0.39 ) 2 Te 3 Intensity (a.u) Bi Bi Sb Te Sb Te Te Sb GaAs (111) 1 nm Intensity (a.u) 2 Bi Bi 3 Sb Te Sb Te Sb Te Te Sb 4 Energy (kev) 5 Cr 6 Figure S1 HRSTEM and EDX spectrum of the (Bi 0.5 Sb 0.5 ) 2 Te 3 /(Cr 0.04 Bi 0.57 Sb 0.39 ) 2 Te 3 bilayer thin film. a, Highresolution cross-section HRSTEM image of the bilayer thin film. Typical quintuple-layered crystalline structure and sharp TI-GaAs interface can be clearly observed. No Cr segregations are detected, which indicates a uniform Cr dopant distribution in the magnetic TI layer matrix. b, EDX spectrum of the (Bi 0.5 Sb 0.5 ) 2 Te 3 and (Cr 0.04 Bi 0.57 Sb 0.39 ) 2 Te 3 layers. Adapted with permission from ref.1. Copyright (2013) American Chemical Society. NATURE MATERIALS 3

4 a 222 b 230 σ xx (S/cm) σ xx (S/cm) B (Oe) B (Oe) Figure S2 Magneto-conductivity data showing similar conductivities for the (a) 6 QL (Bi 0.5 Sb 0.5 ) 2 Te 3 and (b) 6 QL (Cr 0.08 Bi 0.54 Sb 0.38 ) 2 Te 3 thin films on GaAs (111) substrate at 1.9 K. The external magnetic field is applied perpendicularly to the samples, and the arrows indicate the sweeping directions of the field. Adapted with permission from ref.1. Copyright (2013) American Chemical Society. Table S1 Electric properties of both the 6QL TI thin film and the 6QL Cr-doped TI thin film at 1.9 K Sample σ xx (0 T) (S/cm) R sheet (0 T) (kω ) n 2D (p-type) ( cm -2 ) µ (cm 2 /V s) (Bi 0.5 Sb 0.5 ) 2 Te (Cr 0.08 Bi 0.54 Sb 0.38 ) 2 Te Adapted with permission from ref.1. Copyright (2013) American Chemical Society. 2. Second Harmonic Measurement for Rotating the External Magnetic Field The expression for extracting the effective spin-orbit field from the second harmonic signal is derived in refs 6,7 for the case of varying the magnitude of the external magnetic field while keeping its orientation fixed. However, for the rotation experiments presented in this paper we apply a large external magnetic field of constant magnitude,, to ensure single domain behavior, and extract the effective spin-orbit field by varying the orientation of the external magnetic field. In this supplementary 4 NATURE MATERIALS

5 SUPPLEMENTARY INFORMATION section, following ref. 6, we derive the modified expression used to extract the effective spin-orbit field, i.e., equation (2) of the main text. For the geometry used in this paper, the external magnetic field is applied in the -plane making a polar angle with the -axis (see Fig. 1a in the main text). For the harmonic measurements the device is excited by a small AC current,, flowing along the -axis, while simultaneously measuring the Hall voltage,, at first and second harmonics. To this end, taking, the Hall resistance can be Taylor expanded up to linear order in, 0, giving rise to the Hall voltage at different harmonics: 0. (S2.1) and can be directly measured by the standard lock-in technique 6,7. In the -plane or -plane, as shown in Fig. 1a and Fig. 1b in the main text, the first harmonic Hall resistance is given by, 0, where is the ordinary Hall coefficient and is the anomalous Hall effect (AHE) coefficient; is the polar angle between the magnetization and the axis. For clarity, we define the latter part as the first harmonic AHE resistance,, since it is frequently used in the main text. The second harmonic Hall resistance is given by,. Since the ordinary Hall coefficient experimentally does not show any obvious dependence on the current around 0, i.e., 0, the second harmonic Hall resistance is therefore also referred to as the second harmonic AHE resistance,. At this point we note that a full quantitative characterization of the Hall resistance would require inclusion of the planar Hall effect (PHE) resistance. However, in the plane or -plane the PHE does not contribute to the first harmonic Hall resistance since 0 or NATURE MATERIALS 5

6 . Furthermore, in section 5 we will show that in the -plane the PHE contribution to the second harmonic Hall resistance is actually negligible for the device studied here. As a result, in the following we will focus only on the first and second harmonic signals from the AHE resistance. In the rotation experiment the external magnetic field,, can rotate continuously in the -plane, which allows us to probe the effective spin-orbit field,, which is also lying in the -plane as shown in Fig. 2a in the main text. arises from the spin transfer-like spin-orbit torque (SOT) term 6. We want to mention that an equally important field-like SOT term is shown to be an order of magnitude smaller (section 4), and its contribution to the second harmonic signal is negligible (section 5) when we do the rotation experiment in the -plane. With such a consideration in mind, using the form of the AHE resistance we get:. S2.2 Here, represents the current-induced effective spin-orbit field value, expanded up to first order in I, with its orientation as shown in Fig. S3b. Next, similar to ref. 6, noting that the change in caused by the current-induced effective spin-orbit field can be replaced by that due to change in the component of the external magnetic field transverse to the magnetization, we write: 1. S2. Here, the essential difference from ref. 6 is that instead of the magnitude,, the orientation,, is varied. Using equations (S2.1), (S2.2) and (S2.3), we immediately arrive at the formula used in the main text for extracting : 2 S2.4 6 NATURE MATERIALS

7 SUPPLEMENTARY INFORMATION In order to extract by varying, is found by fitting the first harmonic AHE resistance within a single domain model as described in the following. The equilibrium orientation of the magnetization in the presence of a large external magnetic field can be obtained by balancing the torques due to the external magnetic field and the perpendicular anisotropy field, i.e., the total torque, which leads to the following equation (see Fig. S3a): where is the anisotropy coefficient. Noting that the first harmonic AHE resistance reads,, the numerical solution of equation (S2.6) is then fitted to the experimental data to obtain, as shown in Fig. S4. Consequently, the relation between and is also established. a z b z I B K M B K B θ M θ B θ M B ext θ B θ M B ext 0 y 0 y Figure S3 a, The equilibrium orientation of the magnetization in the presence of an external magnetic field with a constant magnitude of 2T. is the perpendicular anisotropy field. and are the polar NATURE MATERIALS 7

8 angles of and from the -axis, respectively. b, The orientation of the magnetization M in the presence of current-induced effective spin-orbit field which is transverse to the magnetization m z experiment fitting θ B (radian) Figure S4 The normalized -component of the magnetization, (black squares), obtained from the rotation experiment ( ) and from the fitting by solving equation (S2.6) (red circles), respectively, as a function of the field angle. In the fitting, we used. 8 NATURE MATERIALS

9 SUPPLEMENTARY INFORMATION 3. Second Harmonic Measurement for Sweeping the In-plane External Magnetic Field: the 1 Dependence In this section, we derive, within the single domain model, the expression for the second harmonic AHE resistance for the case of varying magnitude of the external magnetic field applied along the -axis. We are interested in the regime where the magnitude of the in-plane external magnetic field,, is larger than. In this case, the equilibrium condition of the magnetization, equation (S2.6), gives 2, i.e., the equilibrium magnetization in the absence of current-induced effective spin-orbit field is pointing along the -axis. However, when an AC current,, is applied, the corresponding effective spin-orbit field,, transverse to the magnetization, will cause the orientation of the magnetization to oscillate with an amplitude, whose magnitude depends on the strength of the external magnetic field. This gives rise to a second harmonic AHE resistance as explained in ref. 6. The amplitude, for the given and, can be obtained by balancing the torques due to the external magnetic field, the anisotropy field and the current-induced effective spin-orbit field (see Fig. S5), resulting in the following equation: cos..1 We are interested in the case when the transverse fluctuation 1. To this end, expanding equation (S3.1) up to first order in gives the solution,. Using this solution in conjunction with equations (S2.1) and (S2.2) gives the following solution for the second Harmonic AHE resistance: NATURE MATERIALS 9

10 As seen in the experiments (Fig. 3 in the main text), within the current-induced spin-orbit torque model we obtained and verified the dependence of the second harmonic AHE signal. z B K I 0 0 θ M M B SO B y δθ M y Figure S5 The orientation of the magnetization in the presence of a large in-plane external magnetic field along the -axis and an AC current,. is the current-induced effective spin-orbit field; is the small transverse deflection caused by. is the perpendicular anisotropy field.. 4. Determining the Effective Spin-orbit Field from the Field-like Spin-orbit Torque In addition to the spin transfer-like SOT term that has been explored in the main text, usually there is an equally important field-like SOT term, which can be described as,, where the effective spin-orbit field is,, and is a coefficient. This effective field is pointing along (or ) direction (i.e., transverse to the Hall bar structure). In order to measure this effective field by the second harmonic method, the external magnetic field needs to rotate in the -plane (detailed explanation can be found in ref. 6). We managed to reconstruct our measurement setup such that can rotate 10 NATURE MATERIALS

11 SUPPLEMENTARY INFORMATION locally in the -plane near the 0 and points, as shown in the insets of Fig. S6a-b. Similar to the rotation experiment we already introduced in the main text, now we rotate near the 0 and points in the -plane while keeping its magnitude fixed at 2T. Meanwhile an in-plane AC current is sent in the Hall bar device along the -direction. The first and second harmonic AHE resistances, and, are measured simultaneously. The first harmonic AHE resistance is similar to the one as shown in Fig. 4a in the main text. The second harmonic AHE resistance,, which is caused by the alternating effective field arising from the field-like SOT, is displayed in Fig. S6a-b, for near the 0 and regions, respectively. Since at the 0 and points, and all approach zero, using the L Hopital s rule the formula employed to extract the effective spin-orbit field can be modified as, cos. The obtained results are plotted in Fig. S6c. It can be seen that the effective field,, is negative at 0 and changes to positive at, which means the effective field is pointing along the direction for both cases. From Fig. S6c we can get the effective field versus current density ratio is mt/(a/cm 2 ), which is 30 times smaller than the one revealed from the spin transfer-like SOT ( mt/(a/cm 2 )) at 0 in the main text. The above analysis is valid if we assume the PHE coefficient is zero. If we don t neglect the PHE contribution (the PHE coefficient is measured to be around ), by combining the second harmonic measurements in the -plane and -plane and using the iteration method as introduced in ref. 6, we can get the corrected effective field vs. current density ratio is mt/(a/cm 2 ) for the field-like SOT and mt/(a/cm 2 ) for the spin transfer-like SOT at 0. The field-like SOT is thus still much smaller than the spin transfer-like SOT. When we carry out the second harmonic measurement by rotating the field in the -plane as shown in the main text, the effective spin-orbit field from the field-like SOT mainly affects the second harmonic signal through modulation of the PHE resistance 6. Since the PHE resistance is very small compared with NATURE MATERIALS 11

12 the AHE resistance (section 5), the total effect from the field-like SOT to the second harmonic signal is almost negligible when we do the rotation experiment in the -plane in the main text. a R 2ω AHE (Ohm) I ac =2μA I ac =4μA I ac =6μA I ac =8μA I ac =10μA z B ext M θ B - + x b R 2ω AHE (Ohm) I ac =2μA I ac =4μA I ac =6μA I ac =8μA I ac =10μA z θ B M - + B ext x θ B (radian) [θ B -π] (radian) c J ac (10 4 A/cm 2 ) Effective Field B θ (mt) θ B =0 θ B =π AC current amplitude (μa) Figure S6 a-b, Second harmonic AHE resistance,, as a function of the field angle for the applied AC current with different amplitudes ranging from 2 µa to 10 µa, when varies locally near 0 and in the -plane, respectively. c, The transverse effective spin-orbit field as a function of the AC current amplitude for two different angles, 0 and, respectively. In the experiments, the field magnitude is fixed at 2 T and the temperature is kept at 1.9 K. Straight lines in the figures are linear fittings. 12 NATURE MATERIALS

13 SUPPLEMENTARY INFORMATION 5. Anisotropic Magneto-resistance (AMR) Effect and Planar Hall Effect (PHE) The AMR effect depends on the angle between the magnetization and the electric current direction ( direction, see Fig. 1b in the main text). In order to measure the AMR effect and the associated PHE in our Hall bar device, we reconstruct our experiment setup such that an external magnetic field with constant magnitude (2T) can rotate in the -plane, as illustrated in the insets of Fig. S7a-b. A small probing AC current (1μA) is applied along the -direction. The external magnetic field is large enough to ensure single domain state of the magnetization and when we rotate, the magnetization follows, as shown in the insets of Fig. S7a-b. Both the longitudinal sheet resistance and the transverse Hall resistance are measured simultaneously. The results are shown in Fig. S7a and S7b, respectively. In Fig. S7a, we plot the percentage change in the sheet resistance, %, as a function of the azimuthal angle. % is defined as % /, where is the sheet resistance when is perpendicular to the current direction. From Fig. S7a we can observe that the sheet resistance has a maximum value when is parallel to the current direction and approaches the minimum value when is perpendicular to the current direction. This agrees with the general formula for the AMR effect, /2, where /2 is the angle between and the current direction ( -axis) and is the sheet resistance in the special case when is parallel to the current direction. In Fig. S7a we can see that % is very small in the whole range of, indicating does not differ from too much. The transverse PHE resistance,, is plotted in Fig. S7b as a function of the azimuthal angle. The PHE resistance depends on the product where and are the magnetization components along the and axes 6, respectively. As a result, the PHE resistance is simply zero in the -plane or plane. In the -plane, the PHE resistance has a simple formula 6, 2, where is the PHE coefficient. In Fig. S7b we can clearly see the double periods for from 0 to 2, and the coefficient NATURE MATERIALS 13

14 can be obtained as Ohm, which is much smaller than the anomalous Hall effect (AHE) coefficient, Ohm. Next, we will estimate how much the PHE resistance affects the determination of the effective spin-orbit field when we perform the second harmonic measurements in the main text. In the -plane, the general formula for the second harmonic Hall resistance is (see ref. 6), where is the effective field along the polar angle direction and is the effective field along the azimuthal angle direction. When we perform the rotation experiments in the -plane, stands for the effective field arising from the spin transfer-like SOT and the effective field from the field-like SOT. Now let us show that in the -plane, the PHE contribution to the second harmonic signal is indeed negligible. Suppose we use an AC current with amplitude of 1µA, so m and m (see section 4). In the region, the latter part in equation (S5.1) stands for of the total signal; when, e.g., at, the latter part in equation (S5.1) can be estimated to be Ohm, while the total measured second harmonic resistance is Ohm at (see Fig. 4b in the main text). In other words, the latter part in equation (S5.1) only accounts for 1.8% of the total signal. In conclusion, the field-like SOT induced second harmonic resistance is negligible when we perform the rotation experiments in the -plane, and to a good approximation, we have attributed all of the measured second harmonic signals to the spin transfer-like SOT induced second harmonic AHE resistance in the main text. 14 NATURE MATERIALS

15 SUPPLEMENTARY INFORMATION a 0.6 b ΔR sheet % 0.2 x φ B ext M y R PHE (Ohm) 0-2 x φ B ext M y ϕ (radian) ϕ (radian) Figure S7 a, The percentage change in the sheet resistance, %, as a function of the field azimuthal angle when the external magnetic field rotates in the -plane, as shown in the inset figure. b, The transverse PHE resistance,, as a function of the field azimuthal angle when the external magnetic field rotates in the -plane, as shown in the inset figure. In the experiments, the field magnitude is fixed at 2 T and the temperature is kept at 1.9 K. 6. Determining the Curie temperature in the TI/Cr-doped TI Bilayer Heterostructure In order to determine the Curie temperature in our TI/Cr-doped TI bilayer heterostructure, we carried out the out-of-plane Hall measurement at different temperatures to determine the trend in the coercivity field,, as a function of the temperature. The result is shown in Fig. S8. We observe that the changes dramatically as the temperature increases, indicating a decrease in the magnetization magnitude. The approaches zero when the temperature is around 8.5K, and consequently we can estimate the Curie temperature is about 8.5K. From the trend in we may underestimate the Curie temperature and a more accurate estimation can be provided by the Arrott-Noakes method 8. Nevertheless, our estimation here is NATURE MATERIALS 15

16 good enough to demonstrate the pronounced ferromagnetic ordering in the TI/Cr-doped TI bilayer heterostructure in the experiments mentioned in the main text B C (mt) R H (Ohm) T=2K T=5K Field (mt) T (K) Figure S8 The coercivity field as a function of the temperature. approaches zero at around 8.5K. Inset: the out-of-plane Hall resistance hysteresis loops for two different temperatures, and. 7. Temperature Dependence of the Switching Current Density and the Anisotropy Field Experiments on the in-plane DC current induced magnetization switching in the presence of an in-plane external magnetic field are carried out at different temperatures and the corresponding phase diagrams for different temperatures are obtained, similar to Fig. 2d in the main text. The critical switching current 16 NATURE MATERIALS

17 SUPPLEMENTARY INFORMATION density, defined as the current density needed to switch the magnetization when the in-plane external magnetic field approaches zero, can be obtained from the phase diagram. The anisotropy field, which is also a function of temperature, can be obtained using the method introduced in section 2. In Fig. S9, we plot both the critical switching current density and the anisotropy field as a function of temperature. We can observe that both and the anisotropy field decrease as the temperature increases. Interestingly, the critical switching current density approaches zero at around 8K, while the anisotropy field still displays some remnant value even at 9K, possibly due to the fact that the in-plane DC current induced magnetization switching is likely multi-domain switching while the anisotropy field obtained here is for the single domain case (see section 2). J switch (10 4 A/cm 2 ) Switching Current Density Anisotropy Temperature (K) Anisotropy (T) Figure S9 The critical switching current density and the anisotropy field as a function of temperature. NATURE MATERIALS 17

18 8. Thermal Effect when Passing an AC or DC Current in the Device In this section, we estimate the rise in temperature on the device due to Joule heating. This is done by measuring the longitudinal resistance, as a function of temperature ( ) and comparing it with the measurement of as a function of current ( ). First, the longitudinal resistance was measured as a function of temperature in the presence of a constant out-of-plane external magnetic field of 2T. The corresponding percentage change in, defined as 1 1.9K 1.9K, is plotted in Fig. S10 (black solid circles). Next, was measured at 1.9K (the temperature provided surrounding the device) as a function of the current in the presence of the same out-of-plane external magnetic field of 2T (this field is large enough so that the current induced spin-orbit torque cannot tilt the magnetization much). The measured percentage change in for this case, defined as 1, is also plotted in Fig. S10 (blue open circles). Comparing the two cases we can see, from Fig. S10, that for a current of 1, the temperature rises by ~ 2.5 K. In the current-induced magnetization switching experiments (Fig. 2b and 2c in the main text), this is in agreement with the fact that for this range of currents the sample is still below the Curie temperature. K and shows hysteretic switching (see Fig. 2b and 2c in the main text). However, this Joule heating might assist the current-induced magnetization switching by lowering the anisotropy field (section 7). On the other hand, for the maximum AC current amplitude of (root mean square value is. ) in the second harmonic experiments (Fig.4 in the main text), the corresponding rise in temperature is ~ 1 K and thus is not expected to affect the strength of the extracted effective spinorbit field. We also note that the modulation of the Hall resistance due to a modulating temperature via Joule heating will lead to the generation of a third harmonic Hall voltage signal 6 and thus will not affect the second harmonic measurements. 18 NATURE MATERIALS

19 SUPPLEMENTARY INFORMATION Current (μa) Temperature induced Current induced ΔR xx % Temperature (K) Figure S10 The percentage change in as a function of the temperature (bottom abscissa) and the current (upper abscissa), respectively. The corresponding rise in temperature due to current-induced Joule heating can be obtained by comparisons. For example, when the current is, the corresponding % is ~ -2.2, and the corresponding temperature is ~4.4 K, meaning the rise in temperature is around 2.5 K. A constant out-of-plane external magnetic field of 2T is applied in the measurements. NATURE MATERIALS 19

20 9. Comparison of the Switching Phase Diagrams revealed in the Experiments and the Single Domain Simulations In this section we compare the switching phase diagram revealed in the experiments (Fig. 2d in the main text) with the one obtained from the single domain simulations. We find that the switching current density exhibited in the experiment is nearly one order of magnitude smaller than the one required for single domain switching. As a result, the current-induced magnetization switching in our experiment occurs most likely via a multi-domain switching process (Fig. 2c in the main text). The single domain simulations were performed by solving the following Landau-Lifshitz-Gilbert equation, augmented with a current-induced spin-orbit torque (SOT) term:.1 Here is the magnetization vector, is the gyromagnetic ratio, is the saturation magnetization magnitude and is the Gilbert damping parameter; stands for the external magnetic field and the anisotropy field, with coefficients given already in the main text; the last term in equation (S9.1) models the spin transfer-like SOT induced by the in-plane current and the corresponding effective spin-orbit field is, as described in detail in the main text. For certain in-plane current density, can be obtained via the second harmonic measurements (Fig. 4d in the main text), from which the / ratio is known to be between mt/(a/cm 2 ) and mt/(a/cm 2 ). It is to be noted that the field-like SOT term has not been included in equation (S9.1) because it is much smaller than the spin transfer-like SOT term in our current device, as argued in section 4. For simplicity, using the parameters, rad s T, 0.1 (it turns out that the choices of and do not affect the simulation results of the phase diagram), 1 emu/cm and / = mt/(a/cm 2 ), the phase diagram can be readily obtained by initializing the 20 NATURE MATERIALS

21 SUPPLEMENTARY INFORMATION magnetization to point both up ( direction) and down ( direction) and allowing it to relax to the equilibrium positions according to equation (S9.1). The resulting phase diagram for the case of positive in-plane current and positive in-plane external magnetic field in the -direction is shown in Fig. S11. For comparison with the experiments, corresponding phase diagram measured experimentally, from Fig. 2d of the main text, is superimposed in Fig. S11. As can be seen, the critical current for magnetization switching within the single domain model is approximately one order of magnitude larger than the one we measured experimentally. This one order of magnitude difference could be attributed to multi-domain switching, possibly in conjunction with current-induced heating effect, as has also been seen for heavy metal/ferromagnetic heterostructures 9. In-plane field B y (T) J dc (10 4 A/cm 2 ) Single Domain Simulation Experimental Data M z = M z = / I dc (μa) Figure S11 Comparison of the phase diagrams obtained from the experiments (blue triangles) and from the single domain simulations (red squares), for the case of positive in-plane current and positive in-plane external magnetic field in the -direction. The symbol means and means, not. NATURE MATERIALS 21

22 In addition, it is worth mentioning that in the perpendicular external magnetic field induced switching in the absence of the in-plane DC current, as shown in Fig. 1c in the main text, the measured coercivity field (around 0.1T) is about one order of magnitude smaller than the perpendicular anisotropy field (0.9T as measured for the single domain case, see section 2), which suggests that the magnetization switching in our micrometer scale Hall bar device most likely favors multi-domain switching in the low field region. References 1 Kou, X. et al., Manipulating Surface-related Ferromagnetism in Modulation-doped Topological Insulators. Nano Lett. 13, (2013). 2 Kou, X. et al., Interplay between Different Magnetisms in Cr-Doped Topological Insulators. ACS Nano 7, (2013). 3 Kong, D. et al., Ambipolar field effect in the ternary topological insulator (Bi x Sb 1-x ) 2 Te 3 by composition tuning. Nature Nanotech. 6, (2011). 4 Hong, S.S., Cha, J.J., Kong, D.S., & Cui, Y., Ultra-low carrier concentration and surface-dominant transport in antimony-doped Bi 2 Se 3 topological insulator nanoribbons. Nat. Commun. 3, 757 (2012). 5 Arakane, T. et al., Tunable Dirac cone in the topological insulator Bi 2-x Sb x Te 3-y Se y. Nat. Commun. 3, 636 (2012). 6 Garello, K. et al., Symmetry and magnitude of spin-orbit torques in ferromagnetic heterostructures. Nature Nanotech. 8, (2013). 7 Kim, J. et al., Layer thickness dependence of the current-induced effective field vector in Ta CoFeB MgO. Nature Mater. 12, (2013). 8 Kou, X.F. et al., Magnetically doped semiconducting topological insulators. J. Appl. Phys. 112, (2012). 9 Liu, L., Lee, O.J., Gudmundsen, T.J., Ralph, D.C., & Buhrman, R.A., Current-Induced Switching of Perpendicularly Magnetized Magnetic Layers Using Spin Torque from the Spin Hall Effect. Phys. Rev. Lett. 109, (2012). 22 NATURE MATERIALS