Reconfigurable spintronic logic gate utilizing precessional magnetization switching

Abstract

In traditional von Neumann computing architecture, the efficiency of the system is often hindered by the data transmission bottleneck between the processor and memory. A prevalent approach to mitigate this limitation is the use of non-volatile memory for in-memory computing, with spin–orbit torque (SOT) magnetic random-access memory (MRAM) being a leading area of research. In this study, we numerically demonstrate that a precise combination of damping-like and field-like spin–orbit torques can facilitate precessional magnetization switching. This mechanism enables the binary memristivity of magnetic tunnel junctions (MTJs) through the modulation of the amplitude and width of input current pulses. Building on this foundation, we have developed a scheme for a reconfigurable spintronic logic gate capable of directly implementing Boolean functions such as AND, OR, and XOR. This work is anticipated to leverage the sub-nanosecond dynamics of SOT-MRAM cells, potentially catalyzing further experimental developments in spintronic devices for in-memory computing.

Introduction

With the rapid development of data-driven applications, the energy consumption and delay caused by data communications between memory and processor have become the major challenge for the computing system with von Neumann architecture. The concept of In-memory computing (IMC)^1,2,3,4,5 has been proposed to address this bottleneck by employing the same non-volatile memory cell for both data processing and storage. Among all the solutions, spintronic devices manipulate data by controlling the spin of electrons, showing advantages in access speed, energy consumption and durability. Representative examples of such devices include spin-transfer torque (STT)^6,7,8,9 magnetic memory and SOT^{10,11,12,13,14,15,16} magnetic memory. Despite the benefits in reading performance and power consumption, the SOT memory cell requires the assistance of a small in-plane external magnetic field to break the switching symmetry, which negatively impacts the integration density and stability of devices. On the other hand, although the logic gate functionality AND and OR have been proposed by adjusting the threshold switching current density of the MTJ, a straightforward solution for XOR function is still absent. Particularly, the realization of XOR function requires a pre-processing of the input signal, or otherwise the cascading of multiple memory cells, leading to a substantial increase in power consumption and the circuitry complexity^{17,18,19,20,21,22}. In general, the roadmap in developing the SOT-MRAM cell for the IMC application is largely elusive up to now.

Here we propose a simple but effective scheme based on the well-understood precessional switching^23,24,25,26 of the MTJ to realize the reconfigurable Boolean operations. Through the controlling of either the amplitude or the width of the current pulses, the 180°, 360°, and 540° magnetization switching of the free layer can be selected in a deterministic manner without the assistance of external field. This typical process of the precessional magnetization switching is further utilized to implement AND, OR, and XOR operations with a single MTJ, at a time scale of sub-nanosecond. The simulation results combining macro spin modeling and micromagnetic modeling demonstrate the predominant role of field-like torque^{27,28,29,30,31,32} in realizing the above-mentioned operations. The aim of this work is to exploit the magnetization dynamics driven by spin–orbit torque within nano-second, which has been long-neglected since the switching at this time scale is usually considered nondeterministic, and potentially catalyzing further experiments in developing spintronic devices for in-memory computation.

Results

Figure 1 illustrates the schematic of the proposed IMC unit, which is developed from a general SOT-MRAM cell consisting of the heavy metal layer (HM), the free layer (FL), the oxide barrier (OB), the reference layer (RL), and the top electrode. Both the FL and RL have perpendicular magnetic anisotropy (PMA). We assume the diameter of the MTJ is approximately the domain wall width, in this case the dynamics of the free layer is quasi-uniform, and the macro spin modeling is valid.

Figure 1

Schematic of the simulated SOT based IMC cell. Current flows into the HM along the x direction, and the total current density sums the Input1, Input2 and Configuration terminals. The red arrows indicate the directions of the charge current I_c, the spin polarization σ, and the effective field of the interfacial Rashba-Edelstein effect H_FL. The spin orbit torques drive the free layer magnetization into precession, enabling the multi-cycle switching.

When the charge current flows through the HM in the -x direction, the spin of the electrons will be polarized in the -y direction due to the spin–orbit coupling, and exert spin torques on the adjacent free layer. The spin torques can be attributed to both the bulk Spin Hall effect, giving rise to the damping-like component, and the interfacial Rashba-Edelstein effect, giving rise to the field-like component. The magnetization dynamics of the free layer can be described by the Landau-Lifshitz-Gilbert (LLG) Eq.³³ including the spin–orbit torques as:

$$\frac{{\partial {\mathbf{m}}}}{\partial t} = - \gamma {\varvec{m}} \times \left( {{\varvec{H}}_{{{\mathbf{anis}}}} + {\varvec{H}}_{{{\mathbf{thm}}}} } \right) + \alpha {\varvec{m}} \times \frac{{\partial {\varvec{m}}}}{\partial t} + \frac{{\gamma \hbar J_{{\text{c}}} }}{{2eM_{{\text{s}}} \mu_{0} t_{{\text{F}}} }}\left[ {\xi_{{{\text{DL}}}} {\varvec{m}} \times \left( {{\varvec{\sigma}} \times {\varvec{m}}} \right) + \xi_{{{\text{FL}}}} {\varvec{\sigma}} \times {\varvec{m}}} \right]$$

(1)

where $\gamma$ is the gyromagnetic ratio, $\alpha$ the Gilbert damping constant, $\hbar$ the reduced Planck constant, J_c the charge current density injected into the HM, e the electron charge, $\mu_{0}$ the permeability of vacuum, M_s the saturation magnetization, $t_{{\text{F}}}$ the thickness of the free layer, $\xi_{{{\text{DL}}}}$ and $\xi_{{{\text{FL}}}}$ are the efficiency constants damping-like torque and field-like torque, ${\varvec{m}}$ the magnetization unit vector of the free layer, ${\varvec{\sigma}}$ is the spin polarization vector. The origin of the damping-like SOT is the spin Hall effect, and the origin of the field-like spin torque is commonly considered a combination of the interfacial Rashba–Edelstein effect and the oersted field from the charge current³⁴. We note that by the definition in Eq. (1), the efficiency constant of damping-like SOT, $\xi_{{{\text{DL}}}}$, and the efficiency constant of field-like SOT, $\xi_{{{\text{FL}}}}$, should have opposite sign to trigger the precessional switching dynamics. In this case, the two SOT components compete to align the magnetization in opposite directions. Previous experiments demonstrated the ratio $\xi_{{{\text{FL}}}} /\xi_{{{\text{DL}}}}$ = -0.7 to -1 in the Co/Pt/Al₂O₃ heterostructure ^35,36. While the Ta/CoFeB/MgO heterostructure can provide a larger ratio up to $\xi_{{{\text{FL}}}} /\xi_{{{\text{DL}}}}$ = -4 ^26,37,38. We assume the HM is Ta, the FL is CoFeB, and the corresponding parameters we used for simulation are listed in Table 1.

Table 1 Simulation parameters used for the macro spin modeling.

We first used a single shot current pulse with varied current density J_c and pulse width T_c to excite the magnetic dynamics of the free layer at room temperature, and calculated the magnetization switching probability p_sw averaged over 200 trails. For the case where the damping-like and the field-like components of the SOT having opposite sign and equal strength ($\xi_{{{\text{DL}}}} = - 0.1$ and $\xi_{{{\text{FL}}}} = 0.1$), the deterministic switching (p_sw > 0.95) only occurs when T_c < 1 ns, as shown in Fig. 2a. The increasing of T_c will first prohibit the magnetization switching, and create a gap where p_sw = 0, and then set stochastic switching (p_sw = 0.5). The switching probability for the case $\xi_{{{\text{DL}}}} = - 0.1,\xi_{{{\text{FL}}}} = 0.2$ is shown in Fig. 2b. The increased field-like SOT effectively reduces the threshold current density from 1.2 × 10¹² A/m² to about 0.8 × 10¹² A/m². More importantly, we observed a second branch in the current parameter space where the switching is deterministic. The result indicates the dominance of the field-like SOT will lead to magnetization precession around the direction of the spin polarization, agrees with previous experimental findings^23,26. We also note that this phenomenon is a representation of binary memristivity, since the resistance of the cell alternates between two states as the current density, or the pulse width increasing.

Figure 2

Magnetization switching probability as a function of the current density and pulse width. (a) ${\xi }_{\text{FL}}$ = 0.1 and (b) ${\xi }_{\text{FL}}$ = 0.2. The magnetization is fully relaxed to reach ± z direction when the current is turned off. And the switching probability p_sw is obtained by counting the switching times from 200 trials with the parameters listed in Table 1.

We further fixed the pulse width T_c = 0.7 ns, and investigated the detailed magnetization switching processes at zero temperature with increasing J_c. As shown in Figs. 3a and d, when J_c = 0.94 × 10¹² A/m² the SOT overcomes the perpendicular magnetic anisotropy, and drives the magnetization over the energy barrier. When the excitation stops, the magnetization precesses to the -z-direction and realizes the 180° switching, representing the general dynamics for the current configuration located in the first branch of deterministic switching in Fig. 2b. When the current density is increased to 1.24 × 10¹² A/m², as shown in Fig. 3b and e, the field-like component of SOT drives the magnetization into in-plane precession, and completes the 360° switching. In this case, the magnetization direction of the free layer does not change. When the current amplitude is further increased to 1.7 × 10¹² A/m², as shown in Fig. 3c and f, the in-plane precession completes the 540° switching, representing the dynamics for current configuration in the second branch of deterministic switching in Fig. 2b. Besides, we have verified the switching stability by including 5 ns thermal fluctuation before the charge current is applied (see Supplementary Material Fig. S1). Here we denote the critical current density that enables the magnetization switching of 180°, 360°, 540° as J_π, J_2π, and J_3π, respectively.

Figure 3

Precessional magnetization switching processes driven by the combined damping-like and field-like SOT. (a)–(c) Evolution of the magnetization components with time. The duration of the current pulse is highlighted by the yellow part. The current pulse width T_c = 0.7 ns, and the current density J_c = (a) 0.94 $\times$ 10¹² A/m², (b) 1.24 $\times$ 10¹² A/m² and (c) 1.7 $\times$ 10¹² A/m². (d)–(f) Magnetization trajectories corresponding to (a)–(c). We highlight the dynamics excited by current pulses in solid, and the relaxation dynamics in transparent.

Both the field-like SOT and the current pulse width have pronounced impacts on the precessional magnetization switching dynamics. We further investigated the critical current density as a function of the pulse width with varying $\xi_{{{\text{FL}}}}$, as shown in Fig. 4. For both the cases of 180° switching and 360° switching, the increasing $\xi_{{{\text{FL}}}}$ effectively reduces the critical current densities. On the other hand, the dependences of J_π and J_2π on T_c are quite different. As shown in Fig. 4a, J_π changes sightly when T_c > 400 ps. However, J_2π nonlinearly decreases with the pulse width, and approaches J_π when T_c > 1 ns, as shown in Fig. 4b. The results indicate that the precessional switching is actually a dynamics process, and can only be excited by short current pulses, since the current window for deterministic switching quickly closes and leads to stochastic switching as the pulse width increases.

Figure 4

Critical current density vs. pulse width with varied field-like torque. The relationship between the critical current density of (a) 180° and (b) 360° switching with the pulse width T_c.

We used the open-source micromagnetic simulator Mumax3³⁹ to confirm the precessional magnetization switching dynamics demonstrated by the macro spin modeling. The parameters we adopted for the micromagnetic simulation are the same to those listed in Table 1. We assume the diameter of the FL is 50 nm, and the mesh size is 1 nm × 1 nm × 1 nm. The exchange constant A_ex = 1 × 10^–11 J/m, the PMA energy density K_u = 2.36 × 10⁶ J/m³, and the corresponding effective PMA energy density K_eff = 7.3 × 10⁴ J/m³. The strength of damping-like and field-like spin–orbit torques are adjusted by tuning the secondary Slonczewski term. In particular, we use Slonczewski spin transfer torque to replace spin orbit torque in Mumax3 by setting ${\Lambda }$ = 1, P = $\xi_{{{\text{DL}}}}$, $\epsilon {\prime } = \eta \epsilon$, ϵ defined as:

$$\epsilon = \frac{{P\left( {{\text{r}},{\text{t}}} \right){\Lambda }^{2} }}{{\left( {{\Lambda }^{2} + 1} \right) + \left( {{\Lambda }^{2} - 1} \right)\left( {{\varvec{m}} \times {\varvec{\sigma}}} \right)}}$$

(2)

m is the FL magnetization, ${\varvec{\sigma}}$ the spin polarization vector, P the spin polarization, $\epsilon {\prime }$ the secondary Slonczewski STT term, ${\Lambda }$ the barrier layer thickness, $\xi_{{{\text{FL}}}} = \eta \xi_{{{\text{DL}}}}$. We excited the FL by a current pulse of 0.7 ns, and then relax the magnetization for 10 ns. In this case, the threshold current densities for 180°, 360° and 540° switching are J_π = 0.88 × 10¹² A/m², J_2π = 1.21 × 10¹² A/m², J_3π = 1.97 × 10¹² A/m², respectively. They are increased compared to the those obtained by macro spin modeling, and could be attributed to the slightly non-coherent switching of the free layer. Figure 5 shows the free layer magnetization evolution of the 180°, 360° and 540° switching, which also captures the non-coherent magnetization distribution during the switching process. We observed that the domain first nucleates in the middle and then propagates outward. This may leverage the exchange energy and the threshold current density for multicycle magnetization switching as well. However, for the MTJ with a diameter smaller than the domain wall width, the precessional switching dynamics will not be qualitatively affected by demagnetization and exchange.

Figure 5

Snapshots of the magnetization distribution at different time during the switching process. From the top to the bottom, 180°, 360°, and 540° switching are excited by the current pulses with a density of 1.0 $\times$ 10¹² A/m², 1.5 $\times$ 10¹² A/m², 2.0 $\times$ 10¹² A/m², respectively.

Utilizing the precessional switching mechanism, a spintronic logic gate integrating reconfigurable AND, OR and XOR operations can be implemented. The logic operation to be performed can be selected by a configuring current pulse with density of J_config. The two binary logic inputs are single current pulse with the current density of J_in1 and J_in2, and the pulse width T_c. Here we assume these current pulses are additive, which can be realized by parallel connection of the inputs, and in this case the total current density in the heavy metal layer is J_c = J_config + J_in1 + J_in2. The logic output can be read from the resistance of the MTJ. Noting that no external field is introduced to break the switching symmetry, thus the magnetic switching of the free layer is nonpolar, and is independent of the initial magnetization (Please see Supplementary Material Fig. S2). For simplicity, we assume the output being “0” if the resistance state of the MTJ is not changed, while the output being “1” if the high resistance state changes to low resistance state, or vice versa. In previous researches, the logic operation based on a single MTJ requires a reset process to initialize the magnetization state^20,40. This is also realizable in our proposed scheme by introducing magnetic field to break the switching symmetry. We also highlight that the implementation of the logic operation XOR directly utilizes the 360° switching, which is actually the precessional dynamics introduced by the field-like SOT. For the example hereafter, we calculated that when T_c = 0.7 ns, the zero-temperature critical current density J_π = 0.88 × 10¹² A m^-2, J_2π = 1.04 × 10¹² A/m², J_3π = 1.48 × 10¹² A/m². The required current settings for achieving different logic operations are listed in Table 2.

Table 2 Current configurations for logic gates implementation.

As shown by the left part of Fig. 6, the logic AND function can be realized by setting J_config = 0 A/m², and the operation is as follows: when J_in1 = J_in2 = 0 A/m², corresponding to the logic input '0,0', J_c is less than the 180° switching critical current. The magnetization of the FL does not switch, and the logic output is ‘0’; When current pulse passes through either the input, i.e. J_in1 = 0.5 × 10¹² A/m² or J_in2 = 0.5 × 10¹² A/m², corresponding to the logic input '1,0' or '0,1'. In this case J_c = 0.5 × 10¹² A/m² < J_π, the magnetization does not switch, and the logic output is ‘0’; When J_in1 = J_in2 = 0.5 × 10¹² A/m², corresponding to the logic input '1,1', J_π < J_c = 1.0 × 10¹² A/m² < J_2π, the magnetization of the free layer deterministically switching, and the logic output is ‘1’.

Figure 6

The implementation of AND, OR and XOR logic operations. The total current density in the heavy metal layer is J_config + J_in1 + J_in2. The bottom panel shows the magnetization response to the current pulses, and the logic outputs '0' and '1' are based on whether the magnetization switches. The current densities in this figure are normalized by 0.5 × 10¹² A/m².

The logic operation OR can be realized by setting J_config = 0.5 × 10¹² A/m², as shown in the middle part of Fig. 6. When there is no logic input current, J_c = J_config + J_in1 + J_in2 = 0.5 × 10¹² A/m² < J_π, the logic output is ‘0’; When the current pulse passes through either the logic input, corresponding to the logic input '1,0' or '0,1', J_π < J_c = 1.0 × 10¹² A/m² < J_2π, the magnetization switches and the logic output is ‘1’; When J_in1 = J_in2 = 0.5 × 10¹² A/m², the logic input is '1,1', and J_c = 1.5 × 10¹² A/m² > J_3π. In this case, the magnetization switches 540°, the resistance is changed, and the logic output is ‘1’.

Similarly, the logic operation XOR can be realized by setting J_config = 0.4 × 10¹² A/m², as shown in the right part of Fig. 6. When there is no logic input current, J_c = 0.4 × 10¹² A/m² < J_π, the logic output is ‘0’; When the logic inputs are '1,0' or '0,1', J_π < J_c = J_config + J_in1 + J_in2 = 0.9 × 10¹² A/m² < J_2π, the magnetization switches 180°, the logic output is ‘1’; When J_in1 = J_in2 = 0.5 × 10¹² A/m², the logic input is '1,1', J_2π < J_c = 1.4 × 10¹² A/m² < J_3π, the magnetization switches 360°, in this case the resistance does not change, and the logic output is ‘0’.

We further denote the configuration current densities for realizing logic gate AND, OR and XOR by J_and, J_or, and J_xor, respectively. For the implementation of the AND gate, J_and = 0 A/m², and the current density should satisfy the following relationship:

$$\left\{ {\begin{array}{*{20}l} {0 \le J_{{{\text{and}}}} < J_{{\uppi }} } \hfill \\ {0 \le J_{{{\text{and}}}} + J_{{{\text{in}}}} < J_{{\uppi }} } \hfill \\ {J_{{\uppi }} \le J_{{{\text{and}}}} + 2J_{{{\text{in}}}} < J_{2\pi } } \hfill \\ \end{array} } \right.$$

(3)

For the implementation of OR gate:

$$\left\{ {\begin{array}{*{20}l} {0 \le J_{{{\text{or}}}} < J_{{\uppi }} } \hfill \\ {J_{{\uppi }} \le J_{{{\text{or}}}} + J_{{{\text{in}}}} < J_{2\pi } } \hfill \\ {J_{{3{\uppi }}} \le J_{{{\text{or}}}} + 2J_{{{\text{in}}}} } \hfill \\ \end{array} } \right.$$

(4)

For the implementation of XOR gate:

$$\left\{ {\begin{array}{*{20}l} {0 \le J_{{{\text{xor}}}} < J_{{\uppi }} } \hfill \\ {J_{{\uppi }} \le J_{{{\text{xor}}}} + J_{{{\text{in}}}} < J_{2\pi } } \hfill \\ {J_{{2{\uppi }}} \le J_{{{\text{xor}}}} + 2J_{{{\text{in}}}} < J_{3\pi } } \hfill \\ \end{array} } \right.$$

(5)

Solving the above inequalities, we have 0.49 × 10¹² A/m² ≤ J_in ≤ 0.50 × 10¹² A/m², 0.50 × 10¹² A/m² ≤ J_or < 0.54 × 10¹² A/m², 0.39 × 10¹² A/m² ≤ J_xor < 0.48 × 10¹² A/m². We performed further investigation and find that by leveraging J_and, utilizing 360° and 540° switching to realize the AND gate functionality, the critical current variability can be effectively improved. In this way, Eq. (3) will be replaced by:

$$\left\{ {\begin{array}{*{20}l} {0 \le J_{{{\text{and}}}} < J_{{\uppi }} } \hfill \\ {J_{{2{\uppi }}} \le J_{{{\text{and}}}} + J_{{{\text{in}}}} < J_{{3{\uppi }}} } \hfill \\ {J_{{3{\uppi }}} \le J_{{{\text{and}}}} + 2J_{{{\text{in}}}} } \hfill \\ \end{array} } \right.$$

(6)

The calculated range of the current densities are sensitive to the variation of the pulse width T_c, as shown in Fig. 7. The results indicate that the logic input current density J_in should be precisely controlled, and the optimized variability is about 6% when the current pulse width T_c = 0.58 ns. Correspondingly, the variability of J_and, J_or and J_xor are 52%, 46% and 40%.

Figure 7

The current variability when the pulse width from 0.5 ns to 0.7 ns. The orange, red, green and blue regions represent the variability of J_in, J_and, J_or, and J_xor, respectively.

Conclusions

In this study, we conducted numerical simulations to investigate the magnetization switching of the Ta/CoFeB/MgO/CoFeB heterostructure, driven by spin–orbit torque. By controlling either the amplitude or the width of the current pulses, we achieved 180°, 360°, and 540° magnetization switching without the need for an external field. A critical element in this mechanism is the field-like torque efficiency, which serves two key functions: it reduces the critical switching current density and induces an oscillatory switching behavior. Building on these findings, we propose a reconfigurable multi-functional logic gate. This gate can efficiently perform Boolean operations, including AND, OR, and XOR, without the necessity for cascading.

Method

Macro spin modeling

$$\frac{{\partial {\varvec{m}}}}{\partial t} = - \gamma \mu_{0} {\varvec{m}} \times {\varvec{H}}_{{{\mathbf{eff}}}} + \alpha {\varvec{m}} \times \frac{{\partial {\varvec{m}}}}{\partial t} + {\varvec{\tau}}_{{{\mathbf{SOT}}}}$$

(7)

$${\varvec{\tau}}_{{{\mathbf{SOT}}}} = \user2{ }\frac{{\gamma \hbar J_{{\text{c}}} }}{{2eM_{{\text{s}}} t_{{\text{F}}} }}\left[ {\xi_{{{\text{DL}}}} {\varvec{m}} \times \left( {{\varvec{\sigma}} \times {\varvec{m}}} \right) + \xi_{{{\text{FL}}}} {\varvec{\sigma}} \times {\varvec{m}}} \right]$$

(8)

Here, ${\varvec{\tau}}_{{{\mathbf{SOT}}}}$ is the SOT torque including the damping-like component and the field-like component, H_eff the effective magnetic field, m the magnetization unit vector of the free layer, ${\varvec{\sigma}}$ the spin polarization vector, $\gamma$ the gyromagnetic ratio, $\alpha$ the Gilbert damping constant, $\hbar$ the reduced Planck constant, $J_{{\text{c}}}$ the charge current density injected into the HM, e the electron charge, $\mu_{0}$ the permeability of vacuum, $M_{{\text{s}}}$ the saturation magnetization, $t_{{\text{F}}}$ the thickness of the free layer, $\xi_{{{\text{DL}}}}$ and $\xi_{{{\text{FL}}}}$ are the efficiency constants damping-like torque and field-like torque.

Equation (7) is transformed to the integrable form:

$$\frac{{\partial {\varvec{m}}}}{\partial t} = \frac{\gamma }{{1 + \alpha^{2} }}\left[ { - {\varvec{m}} \times {\varvec{H}}_{{{\mathbf{eff}}}} - \alpha {\varvec{m}} \times \left( {{\varvec{m}} \times {\varvec{H}}_{{{\mathbf{eff}}}} } \right) + \left( {a_{{\text{J}}} + \alpha b_{{\text{J}}} } \right){\varvec{m}} \times \left( {{\varvec{m}} \times {\varvec{\sigma}}} \right) + \left( {b_{{\text{J}}} - \alpha a_{{\text{J}}} } \right){\varvec{m}} \times {\varvec{\sigma}}} \right]\user2{ }$$

(9)

with $a_{J} = \frac{{\xi_{DL} \hbar J_{{\text{c}}} }}{{2eM_{{\text{s}}} \mu_{0} t_{{\text{F}}} }}$ and $b_{J} = \frac{{\xi_{FL} \hbar J_{{\text{c}}} }}{{2eM_{{\text{s}}} \mu_{0} t_{{\text{F}}} }}$. H_eff is the sum of here, ${\varvec{H}}_{{{\mathbf{anis}}}} = \frac{{2K_{eff} }}{{\mu_{0} M_{s} }}$ is the effective field of the PMA, and ${\varvec{H}}_{{{\mathbf{thm}}}} = \left( {\frac{{2\alpha k_{b} T}}{{\gamma M_{s} V_{FL} }}} \right)^{\frac{1}{2}} I_{{{\text{ran}}}}$ is a Gaussian random field ⁴¹ representing the influence from the temperature, here T is the temperature, $k_{{\text{b}}}$ is the Boltzmann constant, $V_{{{\text{FL}}}}$ is the volume of free layer, $I_{{{\text{ran}}}}$ is a random Gaussian variable with mean of 0 and standard deviation of 1. Equation (7) is numerically solved using the second order Huen’s method to include the thermal fluctuations, with a time step of 1 $\times$ 10^–13 ns.

Micromagnetic simulation

Our simulation is performed by the MuMax3 program. Based on the LLG equation, the magnetization under the current-induced SOT is expressed as:

Setting $a_{{\text{J}}} = \frac{{\xi_{DL} \hbar J_{{\text{c}}} }}{{2eM_{{\text{s}}} t_{{\text{F}}} }}$, $b_{{\text{J}}}$ = $\eta a_{{\text{J}}}$, $\xi_{{{\text{FL}}}} = \eta \xi_{{{\text{DL}}}}$. Equation (7) can be transformed as:

$$\frac{{\partial {\varvec{m}}}}{\partial t} = - \gamma \mu_{0} {\varvec{m}} \times {\varvec{H}}_{{{\mathbf{eff}}}} - a_{{\text{J}}} \gamma {\varvec{m}} \times \left( {{\varvec{m}} \times {\varvec{\sigma}}} \right) - b_{{\text{J}}} \gamma {\varvec{m}} \times {\varvec{\sigma}} + \alpha {\varvec{m}} \times \frac{{\partial {\varvec{m}}}}{\partial t}$$

(10)

$${\varvec{m}} \times \frac{{\partial {\varvec{m}}}}{\partial t} = - \gamma \mu_{0} {\varvec{m}} \times \left( {{\varvec{m}} \times {\varvec{H}}_{{{\mathbf{eff}}}} } \right) + a_{{\text{J}}} \gamma {\varvec{m}} \times {\varvec{\sigma}} - b_{{\text{J}}} \gamma {\varvec{m}} \times \left( {{\varvec{m}} \times {\varvec{\sigma}}} \right) - \alpha \frac{{\partial {\varvec{m}}}}{\partial t}$$

(11)

Combining Eq. (10) and Eq. (11):

$${\varvec{\tau}}_{{{\mathbf{SOT}}}} = \user2{ } - \frac{{\upgamma }}{{\left( {1 + {\upalpha }^{2} } \right)}}a_{{\text{J}}} \left[ {\left( {1 + \eta \alpha } \right){\varvec{m}} \times \left( {{\varvec{m}} \times {\varvec{\sigma}}} \right) + \left( {\eta - \alpha } \right)\left( {{\varvec{m}} \times {\varvec{\sigma}}} \right)} \right]$$

(12)

In MuMax3, the spin torques is expressed as:

$${\varvec{\tau}}_{{{\mathbf{SL}}}} = \gamma \frac{{\hbar J_{c} }}{{eM_{s} t_{F} }}\frac{{ \epsilon+ \alpha \epsilon^{\prime}}}{{\left( {1 + \alpha^{2} } \right)}}\left( {{\varvec{m}} \times \left( {{\varvec{m}}_{{\mathbf{P}}} \times {\varvec{m}}} \right)} \right) + \gamma \frac{{\hbar J_{c} }}{{eM_{s} t_{F} }}\frac{{\epsilon^{\prime} - \alpha \epsilon }}{{\left( {1 + \alpha^{2} } \right)}}\left( {{\varvec{m}}_{{\mathbf{P}}} \times {\varvec{m}}} \right)$$

(13)

$$\epsilon = \frac{{P\left( {{\text{r}},{\text{t}}} \right){\Lambda }^{2} }}{{\left( {{\Lambda }^{2} + 1} \right) + \left( {{\Lambda }^{2} - 1} \right)\left( {{\varvec{m}} \times {\varvec{\sigma}}} \right)}}$$

(14)

${\varvec{m}}_{{\mathbf{P}}}$ is the reference layer magnetization, P the spin polarization, $\epsilon^{\prime}$ the secondary Slonczewski STT term, ${\Lambda }$ the barrier layer thickness. We use Slonczewski spin transfer torque to replace spin orbit torque in Mumax3 by setting ${\Lambda }$ = 1, P = $\xi_{{{\text{DL}}}}$, $\epsilon {\prime } = \eta \epsilon$. And we take the exchange constant A_ex = 1 × 10^–11 J/m, the PMA energy density K_u = 2.36 × 10⁶ J/m³.