Electrical control over perpendicular magnetization switching driven by spin-orbit torques
X. Zhang, C. H. Wan, Z. H. Yuan, Q. T. Zhang, H. Wu, L. Huang, W. J. Kong, C. Fang, U. Khan, X. F. Han
aa r X i v : . [ c ond - m a t . m t r l - s c i ] M a y Electrical control over perpendicular magnetization switching driven by spin-orbit torques
X. Zhang, C. H. Wan, ∗ Z. H. Yuan, Q. T. Zhang, H. Wu, L. Huang, W. J. Kong, C. Fang, U. Khan, and X. F. Han † Institute of Physics, Chinese Academy of Sciences, Beijing National Laboratory for Condense Matter Physics, Beijing, 100190, China (Dated: September 27, 2018)Flexible control of magnetization switching by electrical manners is crucial for applications of spin-orbitronics. Besides of a switching current that is parallel to an applied field, a bias current that is normal to theswitching current is introduced to tune the magnitude of effective damping-like and field-like torques and furtherto electrically control magnetization switching. Symmetrical and asymmetrical control over the critical switch-ing current by the bias current with opposite polarities is both realized in Pt/Co/MgO and α -Ta/CoFeB/MgOsystems, respectively. This research not only identifies the influences of field-like and damping-like torques onswitching process but also demonstrates an electrical method to control it. I. INTRODUCTION
Spin-orbitronics , aiming at current or voltage controlof magnetization ( M ) via spin-orbit coupling (SOC) ef-fect, has gradually manifested itself charming prospect innonvolatile magnetic storage and programmable spin-logicapplications. Spin Hall effect (SHE) in heavy metals or topologic insulators and Rashba effect at the heavymetal/ferromagnetic metal interfaces are two broadly utilizedeffects to realize spin-orbitronics due to their large SOCstrength. With the aid of magnetic field, SHE induced mag-netization switching has already been realized in many sys-tems comprising a magnetic layer (Co, CoFeB, NiFe) sand-wiched by an oxide layer(AlO x , MgO) and a heavy metallayer (Pt, β -Ta, W) with not only in-plane anisotropy butperpendicular anisotropy . Recently, field-free magnetiza-tion switching via current has been also achieved in a wedgedTa/CoFeB/TaO x or antiferromagnetic/ferromagnetic cou-pled perpendicular systems .In those perpendicular systems, current can generate viaSHE effect a damping-like torque which balances effectivetorques from perpendicular anisotropy and in-plane bias fieldand consequently switches magnetization as it becomes largeenough. In these previous researches, mainly spin Hall torque(along x axis) induced by one current (namely, switching cur-rent I along y axis) applied along the direction of an appliedor effective magnetic field is taken into account while the in-fluence of field-like torque (along y axis) on magnetic reversalprocess is rarely experimentally testified. Definition of coor-dinates is shown in Fig. 1(a).Here, we introduced another current (namely, bias current I B along x axis) to electrically control magnetization switch-ing process (Fig. 1(a)). The damping-like and field-liketorques of the bias current have the same symmetry with thefield-like and damping-like torque of the switching current,respectively. Therefore, as shown below, the influences ofboth field-like torque and damping-like torque of the switch-ing current on magnetization switching process become vis-ible with tuning the magnitude of the bias current. Further-more, the main features of aforementioned results can be wellreproduced by a macrospin model which provides further un-derstanding. This work can not only help to distill the in-fluences of different kinds of torques on the switching pro-cess but also demonstrate a practical manner of controlling SHE-driven magnetization switching process by electricallytuning the magnitude of effective damping-like and field-liketorques. II. EXPERIMENTAIL METHOD
SiO //Ta(5)/Co Fe B (1)/MgO(2)/Pt(3) andSiO //Pt(5)/Co(0.8)/MgO(2)/Pt(3) (thickness in nanome-ter) stacks were provided by Singulus GmbH. They weremagnetron-sputtered at room temperature. They have intrin-sically in-plane anisotropy. After annealing at 400 ◦ C and10 − Pa for 1 h in a perpendicular field of 0.7 T could thestacks exhibit strong PMA. Raw films were then patterned byultraviolet lithography and the following two-step argon ionetching into Hall bars with the size of the center squares being20 µ m (Fig. 1(a)). Cu(10 nm)/Au(30 nm) electrodes werefinally deposited to make contacts with four legs of Hall bars.After device microfabrication, the Hall bars were measuredwith two Keithley 2400 sourcemeters and Keithley 2182voltmeter sourcing devices and measuring Hall voltages,respectively. Meanwhile, PPMS-9T (Quantum Design)provided magnetic fields with proper directions. The twoKeithley 2400 sourcemeters first provided the current pulsesto the Hall bar. One applied switching current along the y axis and the other applied bias current along the x axis to thesample with a duration time of 50 ms. Then the two Keithley2400 stopped sourcing after the duration time. After waitingfor 100 ms, one Keithley 2400 applied another current pulseof 1 mA along the y axis to the sample for 100 ms. At the endof this pulse, Keithley 2182 picked up the Hall voltage alongthe x axis. Then the Keithley 2400 was switched off. After100 ms, the next round of destabilizing-measuring processwas performed. III. RESSULTS AND DISCUSSIONA. Experiment
Two typical perpendicular systemsSub//Pt(5)/Co(0.8)/MgO(2)/Pt(3) (PCM for short) andSub//Ta(5)/Co Fe B (1.0)/MgO(2)/Pt(3) (TFM) are usedfor comparison. Thickness is in nanometers. Here Ta is in FIG. 1. (color online). (a) Sample structure of a Hall bar. (b) Glanc-ing XRD pattern of Ta/Co Fe B /MgO/Pt stacks. (c) and (d)show H -dependence of Hall resistance of the PCM and TFM, respec-tively, as field is along x / y / z axis. The Hall resistance R xy ≡ V x /I y in (c) and (d) is obtained as I y =1 mA and I B x =0. α -phase instead of β -phase (Fig. 1(b)). The strong peakat 2 θ =55.6 ◦ can be only ascribed to (200) plane of α -Ta.Absence of the two main peaks at 2 θ =63.6 ◦ and 64.7 ◦ corre-sponding to (631) and (413) planes of β -phase, respectively,indicates nonexistence of β -phase. The wide peak at 2 θ =39 ◦ can be attributed to the merge of (110) plane of α -Ta and(111) plane of Pt. The other wide peak at 2 θ =68 ◦ can be dueto the merge of (211) plane of α -Ta and (220) plane of Pt. M t of PCM and TFM measured by Vibration SampleMagnetometry is 125 µ emu/cm and 145 µ emu/cm , respec-tively. M and t is saturated magnetization and thicknessof magnetic layer, respectively. Hall measurement demon-strates perpendicular magnetic anisotropy (PMA) of both sys-tems. PCM shows higher PMA energy than TFM. Anisotropyfield ( H an ) of PCM and TFM is about 13.6 kOe and 5.8kOe, respectively (Figs. 1(c) and 1(d)). Sophisticated har-monic lock-in technique is applied here to characterizespin-orbit torques of the above systems induced by appliedcurrent. The effective longitudinal field ∆ H L and effective transverse field ∆ H T corresponding to damping-like torqueand field-like torque, respectively, are shown in Fig. 1. Sam-ple structure is also shown (Fig. 1(a)).During measurement, current density ( j y = j y sin ωt ) is ap-plied along + y axis. Magnetic field ( H ) is applied along x or y axis. Direction of H determines which torque canbe detected. H x and H y are respectively used to measurecurrent-induced field-like torque (or effective transverse field ∆ H T corresponding to the field-like torque) and damping-liketorque (or effective longitudinal field ∆ H L corresponding tothe damping-like torque). First and second harmonic Hallvoltages along x axis ( V ω x = V ω x sin ωt and V ω x = V ω x cos 2 ωt )are picked up to indirectly show direction of magnetization( M ) respective to the + z axis and j y -tuned M change, accord-ingly. V ω x and V ω x exhibit parabolic and linear field depen-dence as M around ± z , respectively. Especially, the V ω vs. H curves (Fig. 2(b)) exhibit the same slopes at ± m z as H isalong y while they exhibit opposite slopes as H is along x (Fig.2(d)). From the slopes as well as ∂ V ω /∂H (Figs. 2(a) and2(c)) can we obtain ∆ H L along y axis and ∆ H T along x axisvia ∆ H L/T =-2( ∂V ω /∂H y / x ) / ( ∂ V ω /∂H y / x ). Here ∆ H L par-allel to σ × M originates from spin Hall effect. ∆ H T parallelto σ originates from Rashba field as well as Ostered field. The σ is the spin current density induced by the j y via σ ∝ j y × z .The ∆ H L/T shows linear dependence on applied currentdensity j y with zero intercepts as expected. Parameter β L/T de-fined as d ∆ H L/T /dj y characterizes conversion efficiency fromcharge current to effective field. Here, j y = I/ ( wh HM ) . I is theswitching current, w is the width of Hall bar (20 µ m) and h HM is the thickness of the heavy metal (5 nm). 1 mA of I thus corresponds to 1 MA/cm of j y . The shielding effect ofthe ferromagnetic layer and anti-oxidation layer is ignored.Thus, j y and β L/T should be deemed as an upper and lowerbound, respectively. The β L is about -40 nm and +4 nm forPCM and TFM, respectively (Fig. 2(e)). Meanwhile, the β T is about +1.2 nm and -4 nm for PCM and TFM, respectively(Fig. 2(f)). Especially, β L/T of α -Ta and Pt has opposite signs.The β L/T of Pt is reported in the order of 1 µ m-1 nm in different systems. Our value is closer to that of Liu andFan . Besides, | β L, Pt | ≫ | β T, Pt | , consistent with the resultsof Liu . The β L/T of Ta in Ta/CoFeB/MgO system is thor-oughly researched by Kim . It is in the order of 2-20 nm, de-pending on thickness of Ta and CoFeB. Besides, their resultsshow β T, Ta can be comparable and even larger than β L, Ta . Ourmeasured values are within their range and | β L, Ta | is equal to | β T, Ta | . However, the β L of α -Ta is smaller than that of β -Ta .Ratio of β T / β L for Pt and α -Ta is -0.03 and -1, respectively.Field-like torque can be nearly neglected in the PCM while itcannot be ignored in the TFM, which provides us a couple ofideal systems to research the influence of field-like torque anddamping-like torque on switching behavior of perpendicularfilms. The reason why field-like torque is insignificant andsignificant in PCM and TFM system respectively, we think, isthat the two systems may have different interfacial potentialsdue to different work functions of Pt (5.3 eV), Co (4.4 eV), Fe(4.3 eV), and Ta (4.1 eV) as elaborated in Ref. [25].In the following, we will use the PCM with β T / β L =-0.03and the TFM with β T / β L =-1 to study the influence of I B on FIG. 2. (color online). The H y dependence of (a) V ω and (b) V ω and the H x dependence of (c) V ω and (d) V ω in TFM. (e) and (f)shows, respectively, the current dependence of ∆ H T and ∆ H L inboth TFM and PCM films. Their linear fittings with zero interceptare also shown. As measuring the effective fields induced by theswitching current, we applied no bias current. FM and HM denoteferromagnetic and heavy metal, respectively. switching behaviors and introduce the underneath mechanismbased on a macrospin model. I and H y are applied along y . I B is applied along x . As I B =0, M can be switched back and forthbetween spin-up state and spin-down state (Fig. 3) by scan-ning I under nonzero H y . Due to opposite spin Hall angle,switching direction is opposite for PCM and TFM with thesame measurement setup. For example, switching directionfor TFM and PCM is clockwise and anticlockwise, respec-tively, at positive H y . Sign reversal of H y leads to reversal ofthe switching direction. Fig. 3 also shows nearly a full mag-netization switching can be realized as H y =0.3 kOe for TFM.In this condition, critical switching current ( I C ) is 63.5 mA.Meanwhile, the I C for PCM is about 80 mA as H y =0.7 kOe.These results manifest α -Ta can also function as a high effi-cient converter from charge current to spin current besides ofPt and β -Ta.As shown in Figs. 4(a) and 4(b) elevated I B can signifi-cantly reduce the I C in the PCM system. For example, I C =88 mA as I B =0 mA while I C =73 mA as I B =50 mA. I C decreasesby 17%. Meanwhile, positive and negative I B leads to nearlythe same amount of reduction, no matter the sign of H y asshown by the parabolic fitting lines in Figs. 4(c) and 4(d). This I B -induced decrease in I C can be ascribed to the damping-like torque from I B as shown in the theoretical part below. Itis worthy of accentuating that the damping-like torque of I B shares the similar symmetry with the field-like torque of I andthus a large field-like torque of I could also in principle reducethe I C .Certainly, I B will heat magnetic films as well and in prin-ciple reduce the effective H an , which could also reduce I C .However, our experiment shows that I C varies little as chang-ing duration time of I B from 50 ms to 1 s, which indicates thatthermal effect is at least not dominating factor in determining I C here.On the other hand, TFM system manifests a different re-sponse to I B with different symmetry in comparison withthe PCM counterpart. As shown in Figs. 5(a) and 5(c) for H y =+100 Oe and the transition from down-state to up-state, I C is reduced by about 67% under I B =40 mA while it is onlyreduced by 20% under I B =-40 mA. In contrast, for H y =-100Oe and the transition from up-state to down-state (Figs. 5(b)and 5(d)), besides of the opposite switching direction, the ef-fect of I B on I C is also reversed, i.e. I C decreased only byabout 5% under I B =40 mA while it decreased remarkably by53% under I B =-40 mA. Here, the asymmetric response of I C to positive and negative I B cannot be interpreted by damping-like torque induced by I B or heating effect as shown in thecase of PCM. Instead, field-like torque of I B is a key contrib-utor to the asymmetry as shown below. B. Macrospin model
In order to interpret the different response of PCM and TFMto I B , we have turned to a macrospin model (more details inAppendix). The magnetic energy includes uniaxial anisotropyenergy K sin 2 θ and Zeeman energy − H y M sin θ sin ϕ where θ and ϕ is the polar angle between M and the + z axisand the azimuthal angle between in-plane projection of M andthe + x axis, respectively (Fig. 1(a)). I and I B provide both adamping-like torque and a field-like torque on M with effi-ciency characterized by β T / β L . We use parameter a in unit of H an ≡ K/µ M to denote the damping-like torque providedby I , parameter c to denote the ratio of I B /I and parameter b to denote the ratio of β T / β L . Actually, c reflects the angle oftotal current density with respect to the direction of magneticfield. As I and I B are both applied, torque equilibrium condi-tion requires satisfaction of Eq. (1). ~m × ~H eff + a ~m × ( − ˆ e x ) × ~m + ab ~m × ( − ˆ e x )+ ac ~m × ˆ e y × ~m + abc ~m × ˆ e y (1)Here H eff = −∇ M E , m ≡ M / M , E ≡ K sin θ − µ M H y sin θ cos ϕ , e x and e y is a unit vector along the x and y axes, respectively. The 2 nd and 3 rd term in the RHS FIG. 3. (color online). The dependence of R xy on switching current ( I ) in (a) TFM and (b) PCM systems under different H y .FIG. 4. (color online). The switching current dependence of R xy of the PCM under different bias current as (a) H y =100 Oe and (b) -100 Oeand the dependence of I C on I B as (c) H y =100 Oe and (d) -100 Oe. Red and blue dots in (c) and (d) show, respectively, the I C of transitionsfrom down-state to up-state and from up-state to down-state. The dependence of I C on I B in (c) and (d) could be well reproduced by parabolicfittings. of Equation (1) is damping-like and field-like torque from I while the 4 th and 5 th term is damping-like and field-like torque from I B , respectively. Equation (1) can be furtherreduced as scalar equations. Equation (2) is one of them. sin θ cos θ − [ a ( b + 1)( c + 1) + h y − abch y ] h y − a cos θ − abc cos θ sin ϕ + ah y (1 + cos θ ) h y − a cos θ − abc sin ϕ = 0 (2)If I B =0 and b =0, sin θ cos θ − h y cos θ sin ϕ + a sin ϕ = 0 ,which shares the similar form as derived by Liu and Yan .Here h y ≡ H y /H an . Comparing Eq. (2) with the simplified one as I B =0 and b =0, we can see that introduction of I B leadsto an effective h eff y and an effective damping-like torque a eff FIG. 5. (color online). The switching current dependence of R xy of the TFM under different bias current as (a) H y =+100 Oe and (b) -100 Oeand the dependence of I C on I B as (c) H y =+100 Oe and (d) -100 Oe. as expressed in Equation (3). h effy = [ a ( b + 1)( c + 1) + h y − abch y ] h y − a cos θ − abc (3a) a eff = ah y (1 + cos θ ) h y − a cos θ − abc (3b)Simulated results according to Eq. (1) are shown in Fig.6 where τ C ∝ I C is the critical damping-like torque of I . As c =0, a nonzero b can significantly reduce critical switchingcurrent ( I C ), regardless of its sign (Fig. 6(g)). I C decreases by5.8% and 42% as b = ± b = ± , respectively, comparedwith the I C as b =0. This trend is consistent with the result inthe PCM sample in which the damping-like torque of I B canmimic the influence of the field-like torque of I . Though itcannot reverse M directly, large Rashba effect can still help toeffectively reduce I C .As b =0, bias current ( c =
0) can notably decrease I C and theamount of the reduction in I C does not depend on polarity of c (Figs. 6(a) and 6(b)), which manifests similar characteristicswith the switching behaviors of the PCM sample. As b =-1 and h y =0.4 (Fig. 6(c)), c =0.3 and c =-0.3 will result in asymmetricdecrease in I C . Here c =-0.3 is more effective in reducing I C .However, as h y =-0.4 (Fig. 6(d)), I C reduces more in the caseof c =+0.3. These characteristics (Figs. 6(c) and 6(d)) well re-produce the results of the TFM sample in Figs. 5(a) and 5(b).Figs. 6(e) and 6(f) shows the I B dependence of I C as b =0 and b =-1, respectively. The former indeed predicts a parabolic de-pendence as observed in Figs. 4(c) and 4(d) while the latter FIG. 6. (color online). Dependence of m z on damping-like torqueof switching current a (in unit of H an ) for different c , as (a) h y =0.4,(b) h y =-0.4 with b =0 and (c) h y =0.4, (d) h y =-0.4 with b =-1. (e)and (f) τ c as a function of damping-like torque of I B under h y = ± b =0 and b =-1, respectively. Here τ c is obtained by the transitionfrom spin-down to spin-up state. (g) τ c as a function of b as c =0 and h y =0.4. also predicts a linear dependence besides of the parabolic one,which qualitatively reproduces the results in Figs. 5(c) and5(d). It is worth bearing that field-like torque and damping-like torque are both indispensable to realize the asymmetryreduction of I C under opposite I B . Fig. 5 also indirectly man-ifests that the two types of torque both play important roles inmagnetization switching process of the TFM system.Other Pt/Co/MgO and Ta/CoFeB/MgO samples have ex-hibited similar switching symmetries. Noteworthy, though wedemonstrate the switching behaviors with aid of an appliedfield, the switching performance controlled by I B will be stillachievable in principle if the applied field is replaced by aneffective field from exchange coupling. IV. SUMMARY
Current induced torques of Pt and α -Ta, includingdamping-like torque and field-like torque, have been char-acterized by second-harmonic technique as β L, Pt =-40 nm, β L, Ta =+4 nm, β T, Pt =+1.2 nm and β T, Ta =-4 nm. Current cangenerate much larger field-like torque in α -Ta than in Pt.Current-induced magnetization switching has also been re-alized in the α -Ta system, indicating its high enough spin-orbit coupling strength and shedding light on its potential usein spin-orbitronics. Field-like torque, though incapable ofswitching M directly in our case, plays crucial role in reducing I C . I B results in different influences on switching behaviors forthe TFM and PCM systems. Opposite I B equally decreases I C in PCM while it asymmetrically influences the I C in TFMsystem. Furthermore this asymmetry originates from the field-like torque of I B and can be adjusted by polarity of H y . Ourwork not only brings to light the influence of damping-likeand field-like torques of switching current and bias current onswitching but also experimentally demonstrates an electricalmanner (via bias current) to symmetrically or asymmetricallycontrol the switching, which could advance the developmentof spin-logic applications in which control of the switchingprocess via electrical methods is crucial and beneficial. ACKNOWLEDGMENTS
This work was supported by the 863 Plan Project ofMinistry of Science and Technology (MOST) (Grant No.2014AA032904), the MOST National Key Scientific In-strument and Equipment Development Projects [Grant No.2011YQ120053], the National Natural Science Founda-tion of China (NSFC) [Grant No. 11434014, 51229101,11404382] and the Strategic Priority Research Program (B)of the Chinese Academy of Sciences (CAS) [Grant No.XDB07030200].
Appendix: DETAILS OF MACROSPIN MODEL
The schematic structure of the Pt/Co/MgO orTa/CoFeB/MgO is shown in Fig. 1(a). An applied field H y and the switching current ( I ) are along the + y axis.The bias current ( I B ) is along the + x axis. The ratio of I B / I is defined as a parameter c which actually reflects theangle between the direction of total current density withthat of the applied field. Easy axis of the perpendicularsystems (PCM or TFM) is along the z axis. Therefore thetotal energy ( E ) is K sin θ − µ M H y sin θ sin ϕ with K anisotropy energy, M saturation magnetization and µ permeability of vacuum. This energy drives an effective field H eff = −∇ M E . Here we use a macrospin model for simplicityand therefore only θ and ϕ are variable with the M beinga constant. H θ ,eff = − H an sin θ cos θ + H y cos θ sin ϕ and H ϕ ,eff = H y cos ϕ . H an ≡ K/µ M . H θ ,eff and H ϕ ,eff aretwo orthogonal components of H eff . As the currents I and I B are both applied, magnetization direction will be modulateddue to the damping-like and field-like torques originated fromthe I and I B . The damping-like torque of a unit of M inducedby the I via spin Hall effect is defined as a parameter a whichis proportional to the spin Hall angle and along the x axis.Then the damping-like torque induced by the I B is ac whichis however along the y axis. As shown in the maintext, β L(T) is defined as the effective field correspond to the damping(field)-like torque induced by an unit of I . Here we furtherdefine b as β T / β L . Thus field-like torque induced by the I viaRashba effect as well as Ostered mechanism is ab and alongthe y axis. In contrast, the field-like torque induced by the I B is abc and along the x axis. It is very important that thedirection of the field-like torque induced by the I is the sameas that of the damping-like torque induced by the I B . Theyare both along the y axis. The final state of the system isdetermined by the following LLG equation (A.1). − γ d ~MM dt = − α ~M × d ~MM dt + ~MM × ~H eff + a ~MM × ( − ˆ e x ) × ~MM + ab ~MM × ( − ˆ e x )+ ac ~MM × ˆ e y × ~MM + abc ~MM × ˆ e y (A.1)In the first line of Equation (A.1), γ and α are gyromagneticratio and damping constant, respectively. The quantity e x and e y is unit vector along the x and y axis, respectively. The 1 st and 2 nd term in the second line is damping-like and field-liketorque induced by the switching current ( I ), respectively. The1 st and 2 nd term in the third line is damping-like and field-liketorque induced by the bias current ( I B ), respectively. At thesteady state, d M /M dt = . Thus we arrive at Equation (A.2). ~m × ~H eff + a ~m × ( − ˆ e x ) × ~m + ab ~m × ( − ˆ e x )+ ac ~m × ˆ e y × ~m + abc ~m × ˆ e y (A.2)Here we have replaced M / M with m . Equation (A.2) givesthe scalar equations (A.3) which is also shown in the main text. H y cos ϕ − a cos θ cos ϕ − ab sin ϕ + ac cos θ sin ϕ − abc cos ϕ = 0 (A.3a) H y cos θ sin ϕ − H an sin θ cos θ sin ϕ − a sin ϕ + ab cos θ sin ϕ cos ϕ − acsinϕ cos ϕ − abc cos θ sin ϕ = 0 (A.3b)As c = b =0, Equation (A.3) is reduced as Equation (A.4) ( H y − a cos θ ) cos ϕ = 0 (A.4a) sinϕ ( H y cos θ sin ϕ − H an sin θ cos θ − a sin ϕ ) = 0 (A.4b)One possible solution as well as the final physically meaning- ful solution of Equation (A.4) is further reduced as Equation(A.5) cos ϕ = 0 (A.5a) H an sin θ cos θ − H y cos θ sin ϕ + a sin ϕ = 0 (A.5b)This solution shares the similar form with that derived inRef.[13], which demonstrates the rationality of our deriva- tions.In general case, Equation (A.3) can be transformed asEquation (A.6). cos ϕ = ( ab − ac cos θ ) sin ϕH y − a cos θ − abc (A.6a) H an sin θ cos θ − [( H y − abc ) + a ( b + c + 1)] H y − a cos θ − abc cos θ sin ϕ + aH y (1 + cos θ ) H y − a cos θ − abc sin ϕ = 0 (A.6b)Comparing Equation (A.5b) and (A.6b), we find that the intro-duction of I B actually updates the H y with an effective field of [( H y − abc ) + a ( b + c + 1)] / ( H y − a cos θ − abc ) and up-dates the a with an effective torque of aH y (1 + cos θ ) / ( H y − a cos θ − abc ) .As c =0, the effective field becomes [ H y + a ( b +1)] / ( H y − a cos θ ) . A nonzero b can make the effective field larger,which is very beneficial for higher efficient switching. As b =0, the effective field becomes [ H y + a ( c + 1)] / ( H y − a cos θ ) . Therefore, the introduction of the bias current (or nonzero c regardless of its polarity) can also increase the ef-fective field. Besides, the field-like torque of the switchingcurrent ( ab ) in the former case functions a similar role withthe damping-like torque of the bias current ( ac ) in the lattercase. Only as b = c with opposite sign asymmetricallyinfluence the effective field. It is also worth noting that the H y is still indispensable for magnetization switching becausea zero H y will also lead to a zero effective torque. The nu-merical results regarding the solutions of Equation (A.3) areshown in the Fig. 6 in the main text and not shown here. ∗ Email: [email protected] † Email: [email protected] T. Kuschel and G. Reiss, Nature Nanotech , 22 (2014). A. Manchon, Nat Phys , 340 (2014). M. I. 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