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Tomohiro Taniguchi, [Shinji Isogami](https://orcid.org/0000-0001-7230-6090), Shuji Okame, Katsuyuki Nakada, Tomoyuki Sasaki, [Seiji Mitani](https://orcid.org/0000-0002-1348-0774), [Masamitsu Hayashi](https://orcid.org/0000-0003-2134-2563)

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[Probability maximization of magnetization switching by spin-orbit torques generated by a ferromagnet](https://mdr.nims.go.jp/datasets/744eaaf2-7d07-4f5f-9666-f36c16873773)

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manuscript.dviProbability maximization of magnetization switching by spin-orbit torques generatedby a ferromagnetTomohiro Taniguchi1,∗ Shinji Isogami2, Shuji Okame3, KatsuyukiNakada3, Tomoyuki Sasaki3, Seiji Mitani2, and Masamitsu Hayashi2,41National Institute of Advanced Industrial Science and Technology (AIST),Research Center for Emerging Computing Technologies, Tsukuba, Ibaraki 305-8568, Japan,2National Institute for Materials Science, Tsukuba 305-0047, Japan,3Advanced Products Development Center, Technology & Intellectual Property HQ,TDK Corporation, Ichikawa, Chiba 272-8558, Japan,4Department of Physics, The University of Tokyo, Tokyo 113-8654, Japan.(Dated: February 22, 2025)Spin-orbit torques (SOTs) caused by spin currents generated in a ferromagnetic electrode enablea fast and deterministic magnetization switching. One SOT (y-SOT), polarized orthogonal to bothelectric current flowing in the electrode and easy axis of a ferromagnetic free layer, causes fast mag-netization instability. The other SOT (z-SOT) points to the easy-axis direction and leads to thedeterministic switching. Here, we evaluate the magnetization switching probability by these SOTsfor various values of electric current density and the ratio of two SOTs from numerical simulation ofthe Landau-Lifshitz-Gilbert (LLG) equation. It is found that the switching probability is maximizedwhen the electric current density is close to a critical value for the magnetization destabilizationsolely by the y-SOT. The origin of such a current dependence is investigated by analyzing temporaldynamics, spectra of the magnetization distribution, and a steady-state solution of the LLG equa-tion. We reveal that the maximization of the switching probability originates due to two differentswitching behaviors. In the low current region, the magnetization in some trials remains near theinitial state because of the weak y-SOT, and thus, the switching error occurs. The number of sucha trial, as well as the error, decreases as the electric current density increases because both y andz-SOTs prompt the switching. In the large current region, the large y-SOT immediately tilts themagnetization toward the switched direction. However, since the y-SOT prefers an in-plane magne-tized state, the switching error due to a probabilistic return to the initial state after turning off theelectric current density occurs. As a result, the switching probability is optimized when the electriccurrent density is close to the critical value.I. INTRODUCTIONSpin-orbit torque (SOT) driven magnetization dynam-ics [1–3] has attracted significant attention to manipulatethe magnetization in nanostructured ferromagnets. SOToriginates from pure spin current generated by spin-orbitcoupling in nonmagnetic electrode [4–6]. The spin polar-ization of SOT points to a direction orthogonal to bothelectric and spin currents. (In this paper, we use a Carte-sian coordinate where the electric and spin currents flowalong x- and z-directions, respectively, and therefore, theSOT points to the y direction). One of the advantagesof SOT, compared to spin-transfer torque (STT) [7, 8]in the conventional two-terminal structures [9–15], is thefast magnetization destabilization in a perpendicularlymagnetized ferromagnet. SOT directly moves the magne-tization from the stable (z) direction to an in-plane (‖ y)direction without accompanying magnetization preces-sion. This is in contrast to the conventional STT switch-ing [16] that costs a much longer time. However, SOTfixes the magnetization to the y direction, and there-fore, the magnetization switching becomes probabilistic[17]. Although the switching becomes deterministic by∗ tomohiro-taniguchi@aist.go.jpapplying an external magnetic field along the x direc-tion, this method is not preferable from a practical view-point. Therefore, various approaches for magnetic field-free switching have been proposed, such as utilizing lat-eral structure asymmetry [18], tilted magnetic anisotropy[19, 20], exchange bias from the antiferromagnet [21], in-terlayer exchange coupling [22], and STT assistance [23–27].Another solution for magnetic field-free switching is touse SOTs caused by spin currents generated in ferromag-netic electrode, which have been studied both theoreti-cally [28–37] and experimentally [38–45]. An advantageutilizing such SOTs is a possibility to manipulate thetorque direction. For example, the SOT due to the spinanomalous Hall effect [28, 38, 41, 43] is parallel to themagnetization direction in the ferromagnetic electrode.SOTs due to the spin-orbit filtering and spin-orbit pre-cession effects near the ferromagnetic/nonmagnetic in-terface [29, 30, 39, 42] and bulk spin Hall effect in theferromagnet [44] generate two kinds of SOTs polarizedin y and ey × p directions, where ek (k = x, y, z) and pare the unit vectors in the k direction and in the magne-tization direction of the ferromagnetic electrode. There-fore, we can excite both the y-polarized and z-polarizedSOTs, called y and z-SOTs for simplicity in this paper,acting on the top free layer simultaneously when a fer-romagnetic electrode magnetized along the x direction2(p = ex). The y-SOT induces a fast magnetization in-stability from the easy axis while the z-SOT results ina deterministic switching. Accordingly, SOTs generatedfrom a ferromagnetic electrode are expected to be usefulfor realizing fast and deterministic magnetization manip-ulation.In this work, we evaluate the magnetization switchingprobability by solving the Landau-Lifshitz-Gilbert (LLG)equation with both y and z-SOTs at finite temperature.It is found that the switching probability is maximizedwhen the current density is close to a critical value for themagnetization destabilization solely by the y-SOT, andfurther increase of the electric current density leads toa reduction of the switching probability (or equivalently,the error rate increases). This is contrary to the intuitionthat the switching probability increases as the electriccurrent density increases, as found in the previous worksfor both STT and SOT switchings [15, 17]. The origin ofsuch a current dependence of the switching probabilityis investigated from temporal dynamics, spectra of themagnetization distribution, and an analytical solution ofthe LLG equation in a steady state. These analyses re-veal that the maximization of the switching probability isa result of the competition between two switching behav-iors. In the low current region, the z-SOT is necessaryfor the switching while the y-SOT contributes to reduc-ing the time necessary to escape from the initial state.When the current magnitude is small, the magnetizationremains near the initial state for some trials and causesthe switching error. Thus, the switching probability in-creases as the electric current density increases. Whenthe electric current density exceeds the critical value, onthe other hand, the magnetization immediately changesits direction and reach a steady state. The magnetizationdirection in the steady state is determined by the balancebetween two SOTs: it changes from the switched state toa state along the film plane as the electric current den-sity is increased. This change of the steady state causesa probabilistic return of the magnetization to the ini-tial state due to thermal fluctuation after turning off thecurrent and becomes an origin of an error. The switch-ing probability is therefore optimized when the electriccurrent density is close to the critical value. The resultindicates that an application of a large electric currentdensity does not guarantee a reduction of the switchingerror and implies a condition for maximizing the switch-ing probability.This paper is organized as follows. In Sec. II, wedescribe the system under study and show the switchingprobability evaluated by numerical simulation of the LLGequation. In Sec. III, the origin of the switching erroris studied by analyzing the temporal dynamics, evaluat-ing the magnetization distribution, and solving the LLGequation analytically. Section IV is devoted to the con-clusion.(a)(b)j Ferromagnetic electrodemSOTy-SOTz-SOTx yzTime (ns)mxmymzmx, my, mz0 1 2 4 5 6 7 8 9 10 1130-1.01.0j≠0 j=0j/jc,SOT=1.0θz/θy=0.4FIG. 1. (a) Schematic illustration of SOT devices. Electriccurrent density j flowing in a bottom ferromagnetic electrodegenerates spin current, whose spin polarization can be de-composed into y and z components. Thus, SOT acting onmagnetization m in a ferromagnetic free layer can also be de-composed into the y and z-SOTs. (b) An example of temporalmagnetization dynamics for ϑz/ϑy = 0.4 and j/jc,SOT = 1.0.The electric current density is finite from t = 0 to t = 1.0 nswhile it is turned off from t = 1.0 ns. The magnetization isregarded to be switched when mz after 10 ns from turning offj satisfies mz < 0.II. NUMERICAL SIMULATIONIn this section, we first introduce the system understudy and then discuss the dependence of the switchingprobability on various system parameters.A. Landau-Lifshitz-Gilbert equationFigure 1(a) shows a schematic illustration of the sys-tem. A top ferromagnetic free layer and the bottom fer-romagnetic electrode are separated by a thin nonmag-netic spacer. By applying electric current density j flow-ing in the x direction to the bottom ferromagnetic elec-trode, spin currents polarized in the y and z directionsare generated at the interface [29, 30, 39, 42] and bulk[44]. These spin currents are injected into the top ferro-3magnetic free layer and exert y and z-SOTs. The mag-netization dynamics excited in a perpendicularly magne-tized free layer by such SOTs is described by the LLGequation,dmdt=− γm×H− γHSOT,ym× (ey ×m)− γHSOT,zm× (ez ×m) + αm× dmdt,(1)where m is the unit vector pointing in the magneti-zation direction in the free layer. The magnetic fieldH = HKmzez corresponds to the perpendicular magneticanisotropy field (HK > 0): other magnetic fields, e.g., theexternal magnetic field, are neglected in this study. Theparameters γ and α are the gyromagnetic ratio and theGilbert damping constant, respectively. The second andthird terms on the right-hand side of Eq. (1) describethe y and z-SOTs. The k component (k = y, z) of theSOT strength isHSOT,k =~ϑkj2eMd, (2)where the parameters M and d are the saturation mag-netization and the thickness of the free layer. The kcomponent of the spin Hall angle is denoted as ϑk. Thepast experiments indicate that |ϑz/ϑy| < 1 [42, 44].In addition, if the z-SOT is much stronger than they-SOT, the switching becomes similar to the conven-tional STT switching, which has been studied previ-ously. Therefore, we consider the parameter region of|ϑz/ϑy| ≤ 1 in this work. A recent experiment showedthat ϑz/ϑy in Co-Ni ferromagnets changes as the mate-rial concentrations change [44]. Therefore, the materialinvestigation showing a large ϑz, which originates fromthe spin-orbit precession effect at the interface [29, 30]and spin Hall precession effect inside the bulk [44], willbe one research direction for achieving high switchingprobability. The values of the parameters are summa-rized in Table I [46]. Here HK is set such that the ther-mal stability ∆0 = MHKV/(2kBT ), with the volume Vand temperature T at room temperature (T = 300 K), isequal to 60.For later discussion, it is convenient to introduce criti-cal current densities. The definition of the critical currentdensity is that, when the current density becomes largerthan the critical value, the magnetization in the pres-ence of torque is no longer stable near the initial stateat zero temperature. In the presence of the y-SOT only,the critical current density is given by [47]jc,SOT =eMd~ϑyHK. (3)Using the values in Table I, the critical current density bythe y-SOT, given by Eq. (3), is 89 MA/cm2. We informthat the formula similar to Eq. (3) was already derivedin Ref. [48] before for a different purpose. Recall alsothat Ref. [47] assumes that the trajectory of the mag-netization switching satisfies my = 0 to derive Eq. (3).TABLE I. Parameters used in the numerical calculations:M , saturation magnetization; HK, perpendicular magneticanisotropy field; γ, gyromagnetic ratio; α, Gilbert dampingconstant; d, thickness of the free layer; V , volume of the freelayer; ϑy, spin Hall angle of the y-SOTQuantity ValueM 1500 emu/cm3HK 1.172 kOeγ 1.764 × 107 rad/(Oe s)α 0.030d 1 nmV d× π × 302 nm3ϑy 0.30This assumption is valid when the current pulse has a fi-nite rise time, while it is not applicable to, for example, astep-function-like input [49]; however, Eq. (3) works wellto estimate the instability by the y-SOT in many cases.In the presence of the z-SOT only, the critical currentdensity is the same with that for two-terminal MRAMdriven by spin-transfer torque [16, 50]jc,STT =2eαMd~ϑzHK. (4)Usually, the value of jc,SOT given by Eq. (3) is muchlarger than jc,STT given by Eq. (4) due to the small quan-tity α(≪ 1), which only appears in Eq. (4) [51]. Thisis because the y-SOT induced magnetization destabiliza-tion is a result of the competition between the y-SOTand the precessional torque [49], while the conventionalSTT switching is a result of the competition betweenSTT and the damping torque [16]. Strictly speaking, thecondition jc,SOT > jc,STT is satisfied when ϑz/ϑy > 2αholds. In the following calculations, we change the valueof ϑz/ϑy from 0 to 1 with an increment 0.05. Therefore,jc,SOT > jc,STT is satisfied in general, except for a nar-row parameter region ϑz/ϑy ∼ 0 to 0.1. In the following,we use ϑz/ϑy = 0.4 as a representative case. Under suchparameter set, jc,STT = 13 MA/cm2. We therefore havej > jc,STT when j > jc,SOT.B. Definition of switching probabilityWe add a random torque −γm × h to the right-handside of Eq. (1) to describe the thermally activated mag-netization dynamics. Component hk (k = x, y, z) of therandom field h satisfies the fluctuation-dissipation theo-rem [52],〈hk(t)hℓ(t′)〉 = 2αkBTγMVδ(t− t′), (5)where T and V are the temperature and the volume ofthe free layer. A numerical method to solve Eq. (1) usingEq. (5) was described in our previous work [46, 53].410.80.60.40.20Switching probabilityθz/θy0.40.20 0.6 0.8 1.0j/jc,SOT=0.810.80.60.40.20Switching probabilityθz/θy0.40.20 0.6 0.8 1.0j/jc,SOT=1.010.80.60.40.20Switching probabilityθz/θy0.40.20 0.6 0.8 1.0j/jc,SOT=1.510.80.60.40.20Switching probabilityθz/θy0.40.20 0.6 0.8 1.0j/jc,SOT=2.0(a) (b)(c) (d)FIG. 2. Dependence of the magnetization switching proba-bility on ϑz/ϑy for the electric current densities with respectto the critical value, j/jc,SOT, of (a) 0.8, (b) 1.0, (c) 1.5, and(d) 2.0.In the present study, we solve the LLG equation withdifferent random torque 107 times. In each trial, weinitially solve the LLG equation without SOTs during10 ns with the initial condition of m = +ez to obtaindistributed initial states induced by thermal fluctuation.Then we solve the LLG equation with a finite currentdensity j, where its pulse width is 1 ns. After that, wesolve the LLG equation without SOTs for a duration of 10ns. The trial is regarded as to be magnetization switch-ing when mz < 0 at this moment. Figure 1(b) showsan example of the magnetization dynamics in the pres-ence of thermal fluctuation, where j/jc,SOT = 1.0 andϑz/ϑy = 0.4. According to the definition, this case isclassified as magnetization switching. By repeating suchtrials 107 times, the switching probability is obtained.C. Switching probabilityFigures 2(a)-2(d) show the dependence of the switchingprobability on ϑz/ϑy for j/jc,SOT = 0.8, 1.0, 1.5, and2.0, respectively. In all cases, the switching probabilityincreases monotonically with ϑz/ϑy increases. This isreasonable because the z-SOT moves the magnetizationto the switched direction. We, however, notice that thedetails of the switching probability are different.For small ϑz/ϑy, the switching probability is smallerthan 0.5 when the current is small [j/jc,SOT < 1.0, Fig.2(a)] while it is approximately 0.5 or larger for the othercases [j/jc,SOT ≥ 1.0, Figs. 2(b)-2(d)]. Recall that the y-SOT induces a fast destabilization of the magnetizationand moves it to the in-plane direction (parallel to the yaxis) when the electric current density exceeds jc,SOT. Itis therefore reasonable that the switching probability is0.5 for j/jc,SOT ≥ 1.0 and ϑz = 0. When the electric cur-rent density is smaller than jc,SOT, however, the desta-bilization of the magnetization by the y-SOT becomes j/jc,SOT=0.8 j/jc,SOT=1.0 j/jc,SOT=1.5 j/jc,SOT=2.0j/jc,SOT=0.8j/jc,SOT=1.0, 1.5, 2.0Switching probability1.00.970.980.99θz/θy0.2 0.3 0.4FIG. 3. An enlarged view of the magnetization switchingprobability in Fig. 2 near 0.2 ≤ ϑz/ϑy ≤ 0.4, where the elec-tric current density with respect to the critical value, j/jc,SOT,of (a) 0.8 (red), (b) 1.0 (green), (c) 1.5 (blue), and (d) 2.0(purple).TABLE II. Switching probabilities for various j/jc,SOT andϑz/ϑy obtained from 107 trials of the LLG equation.jjc,SOTϑzϑy= 0.20 0.25 0.30 0.35 0.400.8 0.9708238 0.9956011 0.9994168 0.9999065 0.99998611.0 0.9876308 0.9983098 0.9999070 0.9999991 1.00000001.5 0.9846845 0.9967647 0.9995187 0.9999465 0.99999682.0 0.9837671 0.9961923 0.9993042 0.9998977 0.9999879weaker, and thus, the magnetization does not reach thein-plane direction in many cases. As a result, the switch-ing probability becomes much smaller than 0.5 for a smallϑz/ϑy when j/jc,SOT < 1, as shown in Fig. 2(a).For j/jc,SOT ≥ 1.0 [Figs. 2(b)-2(d)], one may con-sider the switching probability is approximately indepen-dent of the electric current density. However, the detailsshow a non-trivial behavior. In Fig. 3, we show an en-larged view of the switching probabilities of Fig. 2 for0.2 ≤ ϑz/ϑy ≤ 0.4. We also rearrange the probabil-ities as a function of the electric current density withrespect to jc,SOT in Fig. 4, while their numerical valuesare listed in Table II. It shows that the switching prob-ability is maximized near j/jc,SOT = 1.0. One mightconsider that the difference of the switching probabilityfor the different electric current density is negligible inthese cases. However, their differences are non-negligiblefor practical purposes such as magnetoresistive randomaccess memory. For example, an error rate less than10−3 for storage memory or 10−9 for working memoryis required [15, 54, 55]. In such applications, it will berequired to clarify the origin of the current dependenceof the switching probability. Note that the switchingprobability for a large current region (j/jc,SOT & 1.0)decreases as the electric current density increases. Theresult is against the intuition that the switching proba-5j/jc,SOT(a)0.9710.8 1.0 1.5 2.0switching probabilityj/jc,SOT(b)0.99510.8 1.0 1.5 2.0switching probabilityj/jc,SOT(c)0.99910.8 1.0 1.5 2.0switching probabilityj/jc,SOT(d)0.9998510.8 1.0 1.5 2.0switching probabilityj/jc,SOT(e)0.9999810.8 1.0 1.5 2.0switching probabilityθz/θy=0.20 θz/θy=0.25 θz/θy=0.30 θz/θy=0.35 θz/θy=0.40FIG. 4. Dependence of the magnetization switching probabilities on the electric current density, j/jc,SOT for ϑz/ϑy of (a) 0.20,(b) 0.25, (c) 0.30, (d) 0.35, and (e) 0.40.Time (ns)mz0 1.00.80.60.40.20-1.01.0j/jc,SOT=0.8, 1.0, 1.5, 2.0θz/θy=0.4FIG. 5. Examples of temporal dynamics of magnetization(mz) in the presence of electric current densities, j/jSOT = 0.8(red), 1.0 (green), 1.5 (blue), and 2.0 (purple). The value ofϑz/ϑy = 0.4.bility increases as the electric current density increases,which has been found in both STT and SOT switchings[17, 53]. Therefore, we develop theoretical analyses toclarify the current dependence of the switching probabil-ity in the present system.III. THEORETICAL ANALYSISIn this section, we investigate the origin of the switch-ing error found in Sec. II. We analyze the temporaldynamics in details, evaluate the spectra of the magneti-zation distribution, and derive an analytical solution ofthe LLG equation in a steady state.A. Temporal dynamicsWe firstly study the temporal dynamics from the nu-merical solution of the LLG equation. Figure 5 showexamples of the time development of mz in the presenceof the electric current density (0 ≤ t ≤ 1.0 ns), wherethe electric current density is j/jc,SOT = 0.8 (red), 1.0(green), 1.5 (blue), and 2.0 (purple) while ϑz/ϑy = 0.4.First, the time scale of the temporal change of mz be-comes shorter as the electric current density increases.This is because strong SOTs drive fast magnetizationdynamics. In particular, mz reaches a steady state forrelatively large electric current densities (j/jc,SOT = 1.5and 2.0) while it seems to be on a way to a steady state forrelatively small electric current densities (j/jc,SOT = 0.8and 1.0); see also Appendix A. Second, when the electriccurrent density is turned off (t = 1.0 ns), mz becomescloser to the switched state (mz = −1) when the electriccurrent density is small, or in other words, the magne-tization stays close to the in-plane direction when theelectric current is large. This is because a large electriccurrent density leads to a strong y-SOT, which prefers tofix the magnetization to the y direction. In Sec. III C,we verify this finding from an analytical solution of theLLG equation.These results partially explain the current dependenceof the switching probability in Figs. 3 and 4, in par-ticular for the large current cases (j/jc,SOT = 1.5 and2.0). As mentioned, a strong y-SOT originated from alarge electric current density forces the magnetization tothe in-plane (y) direction. When the magnetization lo-cates close to the in-plane direction, even if mz < 0,thermal fluctuation causes a probabilistic return to theinitial state (mz = 1). It results in the reduction of theswitching probability, or equivalently, in an increase ofthe switching error. Accordingly, the switching proba-bility decreases as the electric current density increaseswhen j/jc,SOT ≥ 1.0. This is consistent with the resultfor j/jc,SOT = 1.0, 1.5, and 2.0 shown in Figs. 3 and 4.We should, however, recall that, for the case of small6electric current density, the switching probability in-creases as the electric current density increases; see theswitching probabilities for j/jc,SOT = 0.8 and 1.0 in Figs.3 and 4. This result indicates that the current depen-dence of the switching probability cannot be explainedsolely by the y-SOT. To clarify the switching mechanismin this small current region, an analysis on one temporaldynamics here is not enough. Therefore, in the followingsections, we perform statistical analysis and analyticalcalculation.B. Magnetization distributionTo clarify the dependence of the switching probabilityon the electric current density in small current region,we evaluate the distribution of mz when the electric cur-rent density is turned off. Figure 6 shows the spectra ofthe distribution of mz as a function of the tilt angle ofthe magnetization from the z axis, cos−1 mz, for variousϑz/ϑy. The initial and switched states correspond to 0◦(mz = +1.0) and 180◦ (mz = −1) while the in-planemagnetized state corresponds to 90◦. First, we noticethat the peaks of the distribution move from close to theswitched state to the in-plane magnetized state as theelectric current density increases, which is consistent withmz found in Fig. 5. Second, we notice that the distribu-tions for large electric current densities (j/jc,SOT = 1.5and 2.0) are relatively sharp. This is because a strong y-SOT makes the magnetization dynamics approximatelydeterministic against thermal fluctuation. Third and im-portantly, the distribution for the small electric currentdensity (j/jc,SOT = 0.8) shows a wide width, and mz insome trials remains in the non-switched (mz > 0) state.The presence of such non-switched trials results in reduc-ing switching probability, even though the peak positionof the distribution is closer to the switched state thanthose of the larger current densities.The presence of the non-switched state for the smallelectric current density (j/jc,SOT = 0.8) is explained asfollows. Recall that jc,SOT is the critical current den-sity for the magnetization destabilization solely by the y-SOT. Therefore, y-SOT is insufficient to move the magne-tization to the negative mz region. Both the y-SOT andthe z-SOTs are necessary for the magnetization switch-ing. The z-SOT is necessary to move the magnetizationto the region of mz < 0. However, the switching bythe z-SOT is similar to the conventional STT switchingand thus, costs a long switching time because the STTswitching accompanies the magnetization precession [16].In particular, the time necessary to escape from the ini-tial state dominates the switching time. The y-SOT con-tributes to reducing this time, even though its magnitudeis smaller than the critical value, by moving the mag-netization from the easy (z) axis immediately; see alsoAppendix A, where we compare the magnetization dy-namics in the presence of both torques and solely by thez-SOT. We should also recall that the switching speed isrelatively slow for small current; see Appendix A again.Accordingly, the magnetization sometimes remains nearthe initial state for this case and leads to an increase ofthe switching error.To summarize the analyses up to this point, the currentdependence of the switching probability in Figs. 3 and 4is explained from two viewpoints, as follows. In the smallcurrent region (j/jc,SOT . 1), the switching probabilityincreases as the electric current increases. This is becausethe number of the trials remained near the initial statedecreases as the electric current density increases. Inthe large current region (j/jc,SOT & 1), in contrast, they-SOT forces the magnetization to lie in the xy-planeand causes a probabilistic return after turning off thecurrent. Therefore, the switching probability decreasesas the electric current density increases. Accordingly,the switching probability is maximized when the electriccurrent density is close to the critical value, jc,SOT.C. Steady state solutionThe above analyses for large current region are alsoverified by deriving a steady state solution of the LLGequation. By introducing the zenith and azimuth angles,θ and ϕ, as m = (sin θ cosϕ, sin θ sinϕ, cos θ), the steadystate solutions of the LLG equation are determined byp sin θ − cos θ sinϕ = 0, (6)sin θ cos θ − r2cosϕ = 0. (7)For simplicity, we introduce dimensionless quantities rand p given byr =jjc,SOT. (8)p =ϑzϑy. (9)A solution of mz = cos θ obtained from Eqs. (6) and (7)is given bymz = −√4− 3(1 + p2)r2 + 2A1/3 +A2/36A1/3, (10)where A isA =3[48p2r2 − 3(1 + 20p2 − 8p4)r4 + 3(1 + p2)3r6]1/2+ 8− 9(1− 2p2)r2.(11)The steady state is determined as a result of the com-petition between two SOTs and the precessional torquedue to the magnetic anisotropy field. The magneticanisotropy field appears through the definition of jc,SOT71001011021031041051071060 30 60 90 120 150 180Magnetization angle (deg)Countj/jc,SOT=0.8j/jc,SOT=1.0j/jc,SOT=1.5j/jc,SOT=2.0θz/θy=0.41001011021031041051071060 30 60 90 120 150 180Magnetization angle (deg)Countj/jc,SOT=0.8j/jc,SOT=1.0j/jc,SOT=1.5j/jc,SOT=2.0θz/θy=0.21001011021031041051071060 30 60 90 120 150 180Magnetization angle (deg)Countj/jc,SOT=0.8j/jc,SOT=1.0j/jc,SOT=1.5j/jc,SOT=2.0θz/θy=0.3(a) (b) (c)FIG. 6. Spectra of mz when the electric current density is turned off, where j/jSOT are 0.8 (red), 1.0 (green), 1.5 (blue), and2.0 (purple), while ϑz/ϑy is (a) 0.2, (b) 0.3, and (c) 0.4. . Note that the magnetization direction near θ = 0◦ is the initial state,θ = 90◦ is the in-plane magnetized state, and θ = 180◦ are the switched state.in r, while two SOTs appears in p. The damping torqueis proportional to dm/dt and thus, jc,STT ∝ α does notappear to determine the steady state (dm/dt = 0) solu-tion. In the low current region, z-SOT intends to movethe magnetization near mz = −1 while the y-SOT iscompensated to the precessional torque and is insuffi-cient to induce the switching [49]. Thus, the steady stateis mainly determined by the z-SOT and locates nearmz = −1. When the electric current density exceedsjc,SOT, the y-SOT overcomes the precessional torque.Then, the contribution of the y-SOT to determine thesteady state becomes relatively large. Since the y-SOTprefers to fix the magnetization to the y-direction, thesteady state moves toward mz = 0. In the large currentlimit (r → ∞), the steady state solution is determinedby the competition between two SOTs, and Eq. (10) sat-urates to −1/√1 + (1/p2). This value becomes 0 (−1)in the limit of p → 0 (∞), in which only the y (z) SOTacts on the magnetization and moves it to the in-plane(switched) direction.Figure 7(a) shows the dependence of Eq. (10) onj/jc,SOT for ϑz/ϑy = 0.4. The result is partially consis-tent with the temporal dynamics shown in Fig. 5. For ex-ample, mz deviates from the switched state (mz = −1.0)as the electric current density increases. The steady statesolution shows a sudden change near j/jc,SOT = 1.0. Thevalues of mz for j/jc,SOT = 1.5 and 2.0 in Fig. 7 areclose to those in Fig. 5. Therefore, we can concludethat the magnetization for j/jc,SOT = 1.5 and 2.0 inFig. 5 reaches the steady state (the value of mz satu-rates to −0.37 when j → ∞ in this case). We should,however, notice that the values of mz between the tem-poral dynamics in Fig. 5 and the steady state solu-tion in Fig. 7 are different when the electric currentdensity is small (j/jc,SOT = 0.8 and 1.0). This is be-cause the y-SOT in Fig. 5 is small and the switchingspeed is slow; therefore, mz in Fig. 5 does not reach thesteady state before turning off the electric current den-sity. This point is verified by evaluating the values ofmz at the pulse width (t = 1.0 ns) and the magnetiza-tion reaches the steady state (t → ∞) from the numeri-cal simulation of the LLG equation at zero temperature;-1.0-0.500.8 0.8 1.0 1.2 1.4 1.6 1.8 2.01.0 1.2 1.4 1.81.6 2.0j/jc,SOTmzθz/θy=0.4θz/θyj/jc,SOT0.81.00.60.40.201.0-1.00mz0.8 1.0 1.2 1.4 1.6 1.8 2.0θz/θyj/jc,SOT0.81.00.60.40.201.0-1.0 -1.00mz(a) (b)(c)0.8 1.0 1.2 1.4 1.6 1.8 2.0θz/θyj/jc,SOT0.81.00.60.40.201.00mz (t=20ns) - mz (t=1ns)(d)FIG. 7. (a) Dependence of the steady state solution of mz,given by Eq. (10), on the electric current density with respectto the critical value, j/jc,SOT. The value of ϑz/ϑy is 0.4. De-pendence of mz at (b) t = 1.0 ns and (c) t → 20 ns estimatedby solving the LLG equation at zero temperature on j/jc,SOTand ϑz/ϑy is also shown. Remind that the electric currentdensity is kept being constant until t = 20 ns in the simula-tion to clarify whether the pulse width of 1 ns is enough toreach a steady state, while the electric current is turned offat t = 1 ns in other simulations. Their difference of (b) and(c), mz(t = 20 ns)−mz(t = 1 ns), is shown in (d).see Figs. 7(b) and 7(c), which show mz at t = 1.0 and20.0 ns. Note that Figs.7(b) and 7(c) are obtained bythe numerical simulation with the constant electric cur-rent density, i.e., j is kept being finite until t = 20.0ns, while the electric current is turned off at t = 1.0 nsin other simulations such as Fig. 1(b). This is becausethe purpose showing these figures is to clarify whetherthe pulse width of 1.0 ns is enough or not to reach themagnetization to a steady state (see also Appendix A,where it is confirmed from the temporal dynamics thatt = 20.0 ns is enough to estimate the saturated valueof mz). We confirm that the values of mz betweenFigs. 7(b) and 7(c) are the same for large current region8(j/jc,SOT & 1.0) while they are different for the smallcurrent region (j/jc,SOT . 1.0); see also Fig: 7(d), whichshows their difference, mz(t = 20 ns)−mz(t = 1 ns). Itindicates that the magnetization reaches (does not reach)the steady state when the current is turned off for thelarge (small) current region. In Fig. 7, the lowest cur-rent is limited to j/jc,SOT = 0.8 because we are interestedin a steady state satisfying mz < 0 here. For one havingan interest in the phase diagram over a wide range of theelectric current density, see Appendix A.These results explain the current dependence ofthe switching probability in the large current region(j/jc,SOT & 1) again. When the electric current densityis large, mz reaches the steady state during the appli-cation of the current pulse. The magnetization movesfrom close to the switched state to the in-plane magne-tized state as the electric current density increases due tothe large y-SOT. It is consistent with the result in Fig. 6,where the peak of the spectra moves to the in-plane direc-tion (90◦) when j/jc,SOT changes from 1.5 to 2.0. As a re-sult, the rate of the probabilistic return to the initial stateincreases. Therefore, the switching probability decreasesas the electric current density increases. On the otherhand, the steady state solution in Fig. 7 cannot be com-pared to the temporal dynamics in Fig. 5 directly whenthe current density is relatively small (j/jc,SOT . 1).The existence of such a difference, however, still signi-fies that the switching behavior must be separated intotwo regions, j/jc,SOT & 1 and j/jc,SOT . 1, in orderto understand the current dependence of the switchingprobability.At the end of this subsection, we provide comments forreaders who are interested in the steady state solutionfrom the mathematical viewpoint. Note that Eq. (10) isnot a unique solution of the LLG equation in a steadystate. There are other steady state solutions of the LLGequation, and the magnetization reaches one of these pos-sible states, depending on the values of the parametersand the initial condition. For example, mz = ±1 are thesteady state solutions in the absence of SOTs (j → 0).Equation (10) represents one of the steady state solutionsnear the switched state. Therefore, it might differ to thesolution of numerical simulation in some cases; for exam-ple, while Fig. 7(a) shows that mz → −1 in the limit ofj → 0, the solution of numerically solved LLG equationdoes not reach this switched state because of the absenceof SOTs. In the present work, however, since the cur-rent density is always finite, Eq. (10) works for the mostcases. Note also that Eq. (10) is applicable to some val-ues of r and p, however, it becomes complex number andprovides unphysical solution for the other values of r andp. Some of the other steady state solutions are tediousand also give unphysical solutions, depending on the val-ues of r and p. For Eq. (10), for example, the quantityA in Eq. (11) is a real number for an arbitrary r onlywhen p > 1/(2√2) ≃ 0.35; however, when p < 1/(2√2),the discriminant of the square root in Eq. (11) becomesnegative, and thus, A becomes a complex number forsome values of r (especially near r ≃ 1). Therefore, weshow Eq. (10) only for p = ϑz/ϑy = 0.40 in Fig. 7(a).We, however, notice that, in reality, Eq. (10) is still areal number and well describes the steady state solutioneven for a small p[< 1/(2√2)] for a wide range of r.Inaddition, without such a strict mathematical treatmentof the steady state solution, the origin of the switchingerror can be well analyzed by Eq. (10), as done above.Therefore, we believe that Eq. (10) is enough for thepresent work.D. Applicability of present analysisOne might consider that the decrease of the switch-ing probability as the electric current density increasesin the large current region also appears for the conven-tional SOT switching, where only the y-SOT exists andan in-plane external magnetic field is applied to devices.Although a similar role of the y-SOT may exist in theconventional SOT switching, it is difficult to comparethe present work to it due to the following reason. Anissue in the conventional SOT switching is a sensitivityof the switching condition on the electric current den-sity even at zero temperature [47]. The meaning of thesensitivity is as follow. For certain current magnitudelarger than a critical value, the magnetization moves closeto the switched state by the SOT, and relaxes to theswitched state when the current is turned off. Whenthe current magnitude is slightly changed, however, themagnetization returns to the initial state after turningoff the current. This is in contrast to the conventionalSTT switching, where the magnetization switching al-ways occurs when the current magnitude exceeds thecritical value [16]. This sensitivity in the conventionalSOT switching originates from precessional dynamics ofthe magnetization around the external magnetic field af-ter turning off the current and has a deterministic na-ture, not a probabilistic one due the thermal fluctuation.Such precessional dynamics, and thus the sensitivity ofthe switching condition on the current, are unavoidableas the electric current density increases [49]. The exis-tence of this sensitivity provides an additional difficultyto analyze the switching probability by the conventionalSOT switching. Although the previous work observed amonotonic increase of the switching probability as theelectric current density increases, its current value is lim-ited to a relatively low current region [17]; thus, it re-mains unclear how these factors (the probabilistic returndue to thermal fluctuation and the precessional dynamicsaround the external magnetic field) affect the switchingprobability in a large current region. We keep this issueas a future work.Let us also comment on the role of the pulse width onthe switching probability. Regarding the above results,one might consider that the result depends on the pulsewidth of the electric current. When the pulse width is suf-ficiently long, high switching probability will be achieved9even if the electric current density is small (when thecondition j/jc,STT > 1 is satisfied). Such a switching,however, is basically similar to the conventional STTswitching and has been already studied extensively [16].Therefore, such a switching is out of interest in this work.When the pulse width is sufficiently short, on the otherhand, large current is necessary to make the switchingtime short. Therefore, the switching probability increasesas the electric current density increases. In such a case, acareful evaluation of the pulse shape from a circuit sim-ulator might be necessary, as in the case of a differentswitching scheme [56], which is beyond the scope of thiswork. In our opinion, the value of the pulse width usedin this work (1.0 ns, or more generally on the order of afew nanoseconds) is reasonable for developing SOT-basedspintronics applications. Therefore, we believe that thepresent results will be applicable to a wide range of spin-tronics research even in the presence of such an applicablerange.IV. CONCLUSIONIn conclusion, the magnetization switching probabilitydue to SOTs caused by spin currents generated in a fer-romagnetic electrode was studied theoretically. The nu-merical simulation of the LLG equation indicates that theprobability of the magnetization switching is maximizedwhen the electric current density is close to the criticalvalue for the magnetization destabilization solely by they-SOT. This is contrary to the intuition, as well as theprevious reports, that the switching probability increasesas the electric current density increases. The origin of theincrease of the switching error was investigated by ana-lyzing the temporal dynamics, evaluating the spectra ofthe magnetization distribution, and deriving an analyt-ical solution of the steady state LLG equation. Theseanalyses reveal that the maximization of the switchingprobability is a result of the current dependence of theswitching probability in two different current regions. Inthe low current region, the time necessary to escape fromthe initial state becomes relatively long. Then, the mag-netization in some trials remains near the initial statewhen the current is turned off and results in the switch-ing error. Therefore, a large current is necessary to in-crease the switching speed. As a result, the switchingprobability increases as the electric current density in-creases. When the electric current density exceeds thecritical value, on the other hand, the magnetization im-mediately saturates to the steady state due to a strongy-SOT. Since the y-SOT moves the magnetization to thein-plane direction, the magnetization in a steady statechanges from close to the switched state to the in-planemagnetized state. The change of the steady state causesa probabilistic return of the magnetization due to ther-mal fluctuation and results in an increase of the switch-ing error. Therefore, the switching probability decreasesas the electric current density increases. In summary,the switching probability is maximized when the electriccurrent density is close to the critical value, and an ap-plication of large electric current does not guarantee ahigh switching probability.DATA AVAILABILITY STATEMENTSThe data are available upon reasonable request.ACKNOWLEDGEMENTSThis work was supported by funding from the TDKCorporation.AUTHOR CONTRIBUTIONST. T. performed numerical simulation and developedtheoretical analyses. T. T. and M. H. conceptualized thework and wrote the paper with review by S. I., S. O., K.N., T. S.,and S. M.Appendix A: Zero temperature dynamics to steadystateIn Sec. III A, we mention that the magnetization forj/jc,SOT = 0.8 and 1.0 does not reach a steady state,while the magnetization for j/jc,SOT = 1.5 and 2.0 satu-rates its direction. This argument was confirmed in Sec.III C by deriving an analytical solution of the steadystate. Here, we confirm the argument again from thetemporal solution of the LLG equation.In Figs. 8(a)-8(d), we show the temporal dynamics ofmz for j/jc,SOT of 0.8, 1.0, 1.5, and 2.0, respectively, byblack lines. Here, the temperature is set to be zero, andthe electric current density is constant to estimate thesteady state. These results confirm that the magnetiza-tion for j/jc,SOT = 0.8 and 1.0 is not saturated to thesteady state solution when t = 1.0 ns, while the magneti-zation for j/jc,SOT = 1.5 and 2.0 immediately reach thesteady state during the application of the electric currentdensity.In Sec. III B, we described that the switching speed bythe z-SOT becomes fast as the electric current density in-creases. This argument can be verified both numericallyand analytically. First, we show the temporal dynamicsof mz solely by the z-SOT (or in other words, we ne-glect the y-SOT) in Fig. 8 by blue lines. Since ϑy = 0.3and ϑz/ϑy = 0.4 were mainly used in the main text, weuse ϑz = 0.12 in HSOT,z in the following. On the otherhand, we set HSOT,y = 0 because we are interested inthe magnetization dynamics solely by the z-SOT. Notethat the initial state of the magnetization is set to bem(0) = (sin θ0, 0, cos θ0) with θ0 = 1/√∆0 [50], in con-trast to m(0) = +ez before thermalization in the main10Time (ns)mz1.0-1.000 1 2 3 4 5j/jc,SOT=0.8(a) j/jc,SOT=1.0(b)Time (ns)mz1.0-1.000 1 2 3 4 5Time (ns)mz1.0-1.000 1 2 3 4 5j/jc,SOT=1.5(c) j/jc,SOT=2.0(d)Time (ns)mz1.0-1.000 1 2 3 4 5FIG. 8. Temporal dynamics of mz at zero temperature forj/jc,SOT of (a) 0.8, (b) 1.0, (c) 1.5, and (d) 2.0. A con-stant electric current is assumed in this calculation. Blackand blue lines correspond to the dynamics for HSOT,y 6= 0and HSOT,y = 0.text. This is because a finite tilt angle of the magnetiza-tion is necessary to make the z-SOT finite at the initialstate. Figures 8(a)-8(d) indicate that the switching speedbecomes faster as the electric current density increases.This result can also be confirmed by deriving an analyt-ical solution of the LLG equation asαγHKt =− ss2 − 1(tanh−1 cos θ − tanh−1 cos θ0)+12(s2 − 1)ln1− cos2 θ1− cos2 θ0− 1s2 − 1lns− cos θs− cos θ0, (A1)where s = j/jc,STT. Equation (A1) provides the mag-netization tilt angle θ = cos−1 mz at time t. We notethat this equation is applicable for an arbitrary value ofs. When the z-SOT is larger than the damping torqueand thus, s > 1, the magnetization moves from the initialstate θ0 to the switched direction, θ = π, i.e., θ(t) > θ0.On the other hand, when the z-SOT is smaller than thedamping torque (s < 1), the magnetization relaxes fromθ0 to θ = 0, i.e., θ(t) < θ0. Therefore, the time t be-comes positive value by substituting θ > (<)θ0 into Eq.(A1) when s > (<)1. If we substitute the opposite value,i.e., θ < (>)θ0 for s > (<)1 into Eq. (A1), t becomesnegative because it describes a process going backwardto the past. In both cases, t is the real value. In thepresent study, jc,STT = 13 MA/cm2 by using the parame-ters in the mentioned above, while jc,SOT = 89 MA/cm2,as already mentioned in the main text [note that jc,SOTdoes not appear in Eq. (A1) because only the z-SOTis taken into account here, although we use the valueof jc,SOT to give the value of j quantitatively]. Accord-ingly, j/jc,STT > 1 is satisfied in all cases in Fig. 8;therefore, θ increases as time increases, and the magne-tization switching is achieved. Equation (A1) indicatesthat the switching speed by the z-SOT becomes shorteras the electric current density increases. Equation (A1)also implies that the time necessary to tilt from the zaxis becomes long when the initial state (θ0) is closeto the z axis. Therefore, we concluded that both they and z-SOTs prompt the switching in the low current-1.0-0.500 0.5 1.0 1.5 2.0j/jc,SOTmzθz/θy=0.4θz/θy0.81.00.60.40.201.0-1.00mz0 1.00.5 1.5 2.0θz/θyj/jc,SOT0 1.00.5 1.5 2.0j/jc,SOT0 1.00.5 1.5 2.0j/jc,SOT0.81.00.60.40.201.0-1.0 -1.00mz(a) (b)(c)θz/θy0.81.00.60.40.201.00mz (t=20ns) - mz (t=1ns)(d)FIG. 9. Extensions of Fig. 7 to lower current region. A blacksolid line in (c) is derived from Eq. (4).region. The y-SOT contributes to moving the magnetiza-tion from the z axis and reduce the switching time, eventhough the current density is below the critical value andthus, the y-SOT cannot induce the switching solely by it-self. The z-SOT contributes to moving the magnetizationclose to the switched state.In Figs. 7(a)-7(d) in Sec. III C, we show the analyticalsolution of the steady state, the values ofmz at t = 1.0 ns11and 20.0 ns and their difference obtained from the numer-ical simulation, respectively. The lowest electric currentdensity is limited to j/jc = 0.8 because we are interestedin the magnetization switching, and thus, mz should be-come at least negative for given parameters. One mightbe, however, interested in extending the results for thelower current region. Figure 9 summarizes the extensionof Fig. 7 to j/jc = 0. Recall that the magnetization doesnot move to the region of negative mz when the currentis small, while Eq. (10) is derived by assuming mz < 0;therefore, there are differences between the analytical so-lution in Fig. 9(a) [or Eq. (10)] and the numerical solu-tion in Fig. 9(c) for the steady state. Comparing Figs.9(b)-9(d), we notice that the region of mz ≃ 1.0 in Fig.9(b) is greatly suppressed in Fig. 9(c), i.e., the mag-netization does not reach the steady state within 1.0 ns(recall that the electric current is kept being constant inthese calculations, and thus, the magnetization can moveto the switched direction by the SOTs even after t = 1.0ns). We notice that the boundary between mz > 0 andmz < 0 in the low current region of Fig. 9(c) is wellfitted by Eq. (4), which is shown by a black solid line inthe figure. This result supports the above argument thatthe z-SOT moves the magnetization close to the switchedstate when the electric current density is small.[1] L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, andR. A. Buhrman, Spin-torque switching with the giantspin Hall effect of tantalum, Science 336, 555 (2012).[2] L. Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ralph, andR. A. Buhrman, Current-induced switching of perpendic-ularly magnetized magnetic layers using spin torque fromthe spin Hall effect, Phys. Rev. Lett. 109, 096602 (2012).[3] C.-F. Pai, L. Liu, Y. Li, H. W. Tseng, D. C. Ralph, andR. A. Buhrman, Spin transfer torque devices utilizing thegiant spin Hall effect of tungsten, Appl. Phys. Lett. 101,122404 (2012).[4] M. I. Dyakonov and V. I. Perel, Current-induced spinorientation of electrons in semiconductors, Phys. Lett. A35, 459 (1971).[5] J. E. Hirsch, Spin Hall effect, Phys. Rev. Lett. 83, 1834(1999).[6] S. Zhang, Spin hall effect in the presence of spin diffusion,Phys. Rev. Lett. 85, 393 (2000).[7] J. C. Slonczewski, Current-driven excitation of magneticmultilayers, J. Magn. Magn. Mater. 159, L1 (1996).[8] L. Berger, Emission of spin waves by a magnetic mul-tilayer traversed by a current, Phys. Rev. B 54, 9353(1996).[9] J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers,and D. C. Ralph, Current-driven magnetization rever-sal and spin-wave excitations in Co/Cu/Co pillars, Phys.Rev. Lett. 84, 3149 (2000).[10] E. B. Myers, F. J. Albert, J. C. Sankey, E. Bonet, R. A.Buhrman, and D. C. Ralph, Thermally activated mag-netic reversal induced by a spin-polarized current, Phys.Rev. Lett. 89, 196801 (2002).[11] S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Em-ley, R. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph,Microwave oscillations of a nanomagnet driven by a spin-polarized current, Nature 425, 380 (2003).[12] H. Kubota, A. Fukushima, Y. Ootani, S. Yuasa, K. Ando,H. Maehara, K. Tsunekawa, D. D. Djayaprawira,N. Watanabe, and Y. Suzuki, Evaluation of spin-transferswitching in CoFeB/MgO/CoFeB magnetic tunnel junc-tions, Jpn. J. Appl. Phys. 44, L1237 (2005).[13] I. N. Krivorotov, N. C. Emley, J. C. Sankey, S. I. Kiselev,D. C. Ralph, and R. A. Buhrman, Time-domain measure-ments of nanomagnet dynamics driven by spin-transfertorques, Science 307, 228 (2005).[14] Z. Diao, D. Apalkov, M. Pakala, Y. Ding, A. Panchula,and Y. Huai, Spin transfer switching and spin polariza-tion in magnetic tunnel junctions with MgO and AlOxbarriers, Appl. Phys. Lett. 87, 232502 (2005).[15] B. Dieny, R. B. Goldfarb, and K.-J. Lee, eds., Introduc-tion to Magnetic Random-Access Memory (Wiley-IEEEPress, Hoboken, 2016).[16] J. Z. Sun, Spin-current interaction with a monodomainmagnetic body: A model study, Phys. Rev. B 62, 570(2000).[17] K.-S. Lee, S.-W. Lee, B.-C. Min, and K.-J. Lee, Ther-mally activated switching of perpendicular magnet byspin-orbit spin torque, Appl. Phys. Lett. 104, 072413(2014).[18] G. Yu, P. Upadhyaya, Y. Fan, J. G. Alzate, W. Jiang,K. L. Wong, S. Takei, S. A. Bender, L.-T. Chang,Y. Jiang, M. Lang, J. Tang, Y. Wang, Y. Tserkovnyak,P. K. Amiri, and K. L. Wang, Switching of a perpendicu-lar magnetization by spin-orbit torques in the absence ofexternal magnetic fields, Nat. Nanotechnol. 9, 548 (2014).[19] L. You, O. Lee, D. Bhowmik, D. Labanowski, J. Hong,J. Bokor, and S. Salahuddin, Switching of perpendicu-larly polarized nanomagnets with spin orbit torque with-out an external magnetic field by engineering a tiltedanisotropy, Proc. Natl. Acad. Sci. 112, 10310 (2015).[20] J. Torrejon, F. Garcia-Sanchez, T. Taniguchi, J. Sinha,S. Mitani, J.-V. Kim, and M. Hayashi, Current-drivenasymmetric magnetization switching in perpendicularlymagnetized CoFeB/MgO heterostructures, Phys. Rev. B91, 214434 (2015).[21] S. Fukami, T. Anekawa, C. Zhang, and H. Ohno,Magnetization switching by spin-orbit torque in anantiferromagnet-ferromagnet bilayer system, Nat. Mater.15, 535 (2016).[22] Y.-C. Lau, D. Betto, K. Rode, J. M. D. Coey, and P. Sta-menov, Spin-orbit torque switching without an externalfield using interlayer exchange coupling, Nat. Nanotech-nol. 11, 758 (2016).[23] M. Wang, W. Cai, D. Zhu, Z. Wang, J. Kan, Z. Zhao,K. Cao, Z. Wang, Y. Zhang, T. Zhang, C. Park, J.-P.Wang, A. Fert, and W. Zhao, Field-free switching of aperpendicular magnetic tunnel junction through the in-terplay of spin-orbit and spin-transfer torques, Nat. Elec-tron. 1, 582 (2018).[24] E. Grimaldi, V. Krizakova, G. Sala, F. Yasin, S. Couet,G. S. Kar, K. Garello, and P. Gambardella, Single-shot12dynamics of spin-orbit torque and spin transfer torqueswitching in three-terminal magnetic tunnel junctions,Nat. Nanotechnol. 15, 111 (2020).[25] S. Pathak, C. Youm, and J. Hong, Impact of spin-orbittorque on spin-transfer torque switching in magnetic tun-nel junctions, Sci. Rep. 10, 2799 (2020).[26] C. Zhang, Y. Takeuchi, S. Fukami, and H. Ohno, Field-free and sub-ns magnetization switching of magnetic tun-nel junctions by combining spin-transfer torque and spin-orbit torque, Appl. Phys. Lett. 118, 092406 (2021).[27] D. H. Kang and M. Shin, Critical switching current den-sity of magnetic tunnel junction with shape perpendic-ular magnetic anisotropy through the combination ofspin-transfer and spin-orbit torques, Sci. Rep. 12, 22842(2021).[28] T. Taniguchi, J. Grollier, and M. D. Stiles, Spin-transfertorques generated by the anomalous Hall effect andanisotropic magnetoresistance, Phys. Rev. Applied 3,044001 (2015).[29] V. P. Amin and M. D. Stiles, Spin transport at interfaceswith spin-orbit coupling: Formalism, Phys. Rev. B 94,104419 (2016).[30] V. P. Amin and M. D. Stiles, Spin transport at interfaceswith spin-orbit coupling: Phenomenology, Phys. Rev. B94, 104420 (2016).[31] V. P. Amin, J. Zemen, and M. D. Stiles, Interface-generated spin currents, Phys. Rev. Lett. 121, 136805(2018).[32] V. P. Amin, J. Li, M. D. Stiles, and P. M. Haney, In-trinsic spin currents in ferromagnets, Phys. Rev. B 99,220405(R) (2019).[33] A. Davidson, V. P. Amin, W. S. Aljuaid, P. M. Haney,and X. Fan, Perspectives of electrically generated spincurrents in ferromagnetic materials, Phys. Lett. A 384,126228 (2020).[34] G. Qu, K. Nakamura, and M. Hayashi, Magnetizationdirection dependent spin Hall effect in 3d ferromagnets,Phys. Rev. B 102, 144440 (2020).[35] G. G. B. Flores, A. A. Kovalev, M. van Schilfgaarde, andK. D. Belashchenko, Generalized magnetoelectronic cir-cuit theory and spin relaxation at interfaces in magneticmultilayers, Phys. Rev. B 101, 224405 (2020).[36] K.-W. Kim and K.-J. Lee, Generalized spin drift-diffusionformalism in the presence of spin-orbit interaction of fer-romagnets, Phys. Rev. Lett. 125, 207205 (2020).[37] Y. Miura and K. Masuda, First-principles calculations onthe spin anomalous Hall effect of ferromagnetic alloys,Phys. Rev. Mater. 5, L101402 (2021).[38] S. Iihama, T. Taniguchi, K. Yakushiji, A. Fukushima,Y. Shiota, S.Tsunegi, R. Hiramatsu, S. Yuasa, Y. Suzuki,and H. Kubota, Spin-transfer torque induced by the spinanomalous Hall effect, Nat. Electron. 1, 120 (2018).[39] S. h. C. Baek, V. P. Amin, Y.-W. Oh, G. Go, S.-J. Lee,G.-H. Lee, K.-J. Kim, M. D. Stiles, B.-G. Park, and K.-J. Lee, Spin currents and spin-orbit torques in ferromag-netic trilayers, Nat. Mater. 17, 509 (2018).[40] Y.-W. Oh, J. Ryu, J. Kang, and B.-G. Park, Material andthickness investigation in ferromagnet/Ta/CoFeB trilay-ers for enhancement of spin-orbit torque and field-freeswitching, Adv. Func. Mater. 5, 1900598 (2019).[41] T. Seki, S. Iihama, T. Taniguchi, and K. Takanashi,Large spin anomalous Hall effect in L10-FePt: Symmetryand magnetization switching, Phys. Rev. B 100, 144427(2019).[42] Y. Hibino, K. Hasegawa, T. Koyama, and D. Chiba, Spin-orbit torque generated by spin-orbit precession effectin Py/Pt/Co tri-layer structure, APL Mater. 8, 041110(2020).[43] Y. Koike, S. Iihama, and S. Mizukami, Composition de-pendence of the spin-anomalous Hall effect in a ferromag-netic Fe-Co alloy, Jpn. J. Appl. Phys. 59, 090907 (2020).[44] Y. Hibino, T. Taniguchi, K. Yakushiji, A. Fukushima,H. Kubota, and S. Yuasa, Giant charge-to-spin conver-sion in ferromagnet via spin-orbit coupling, Nat. Com-mun. 12, 6254 (2021).[45] G. Choi, J. Ryu, S. Lee, J. Kang, N. Noh, J. M. Yuk, andB.-G. Park, Thickness dependence of interface-generatedspin currents in ferromagnet/Ti/CoFeB trilayers, Adv.Mater. Interface 9, 2201317 (2021).[46] T. Taniguchi, S. Isogami, S. Okame, K. Nakada, E. Ko-mura, T. Sasaki, S. Mitani, and M. Hayashi, Probabilityof spin-orbit torque driven magnetization switching as-sisted by spin-transfer torque, Phys. Rev. B 108, 134431(2023).[47] K.-S. Lee, S.-W. Lee, B.-C. Min, and K.-J. Lee, Theo-retical current for switching of a perpendicular magneticlayer induced by spin Hall effect, Appl. Phys. Lett. 102,112410 (2013).[48] H. Morise and S. Nakamura, Relaxing-precessional mag-netization switching, J. Magn. Magn. Mater. 306, 260(2006).[49] T. Taniguchi, Theoretical condition for switching themagnetization in a perpendicularly magnetized ferromag-net via the spin Hall effect, Phys. Rev. B 100, 174419(2019).[50] T. Taniguchi, K. Yamada, and Y. Nakatani, Criticalcurrent formula of perpendicularly magnetized magneticrandom access memory revisited, Jpn. J. Appl. Phys. 58,058001 (2019).[51] M. Oogane, T. Wakitani, S. Yakata, R. Yilgin, Y. Ando,A. Sakuma, and T. Miyazaki, Magnetic damping in ferro-magnetic thin films, Jpn. J. Appl. Phys. 45, 3889 (2006).[52] W. F. Brown Jr, Thermal fluctuations of a single-domainparticle, Phys. Rev. 130, 1677 (1963).[53] T. Taniguchi, S. Isogami, Y. Shiokawa, Y. Ishitani,E. Komura, T. Sasaki, S. Mitani, and M. Hayashi, Mag-netization switching probability in the dynamical switch-ing regime driven by spin-transfer torque, Phys. Rev. B106, 104431 (2022).[54] A. V. Khvalkovskiy, D. Apalkov, S. Watts, R. Chepulskii,R. S. Beach, A. Ong, X. Tang, A. Driskill-Smith, W. H.Butler, P. B. Visscher, D. Lottis, E. Chen, V. Nikitin,and M. Krounbi, Basic principles of STT-MRAM cell op-eration in memory arrays, J. Phys. D. Appl. Phys. 46,074001 (2013).[55] D. Apalkov, B. Dieny, and J. M. Slaughter, Magnetore-sistive random access memory, Proc. IEEE 104, 1796(2016).[56] H. Lee, A. Lee, S. Wang, F. Ebrahimi, P. Gupta, P. K.Amiri, and K. L. Wang, A word line pulse circuit tech-nique for reliable magnetoelectric random access mem-ory, IEEE Trans. Very Large Scle Integr. (VLSI) Syst. 7,3302 (2017).