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Electrical switching of altermagnetism

Yiyuan Chen† ,1 Xiaoxiong Liu† ,1 Hai-Zhou Lu,1 , 2, ∗ and X. C. Xie3 , 4, 5 1 State Key Laboratory of Quantum Functional Materials, Department of Physics,

arXiv:2412.20938v2 [cond-mat.mes-hall]

4

Jun 2025

and Guangdong Basic Research Center of Excellence for Quantum Science, Southern University of Science and Technology (SUSTech), Shenzhen 518055, China 2 Quantum Science Center of Guangdong-Hong Kong-Macao Greater Bay Area (Guangdong), Shenzhen 518045, China

3 International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China 4 Institute for Nanoelectronic Devices and Quantum Computing, Fudan University, Shanghai 200433, China

5 Hefei National Laboratory, Hefei 230088, China

Switching magnetism using only electricity is of great significance for industrial applications but remains challenging. We find that, altermagnetism, as a newly discovered unconventional mag- netism, may open an avenue along this effort. Specifically, to have deterministic switching, i.e., reversing current direction must reverse magnetic structure, parity symmetry has to be broken. We discover that, due to their symmetry which depends on chemical environments, altermagnet devices may naturally carry the parity symmetry breaking required for deterministic electrical switching of magnetism. More importantly, we identify MnTe bilayers (Te-Mn-Te-Mn-Te) as candidate devices, with the help of symmetry analysis, first-principles calculations, and magnetic dynamics simulations. This scheme will inspire further explorations on unconventional magnetism. Introduction - Magnetic materials have been a focal point in condensed matter physics and materials science, brewing diverse applications in technology, energy stor- age, and sensing. While ferromagnetic materials are well applied, antiferromagnetic materials offer advantages like quicker dynamics and resilience to external magnetic fields [1–5]. Recently, there has been increasing inter- est in altermagnetism [6–14], a new category of collinear antiferromagnetic phase, offering unique properties and potential applications [13–16]. The symmetry of alter- magnets is determined not only by the magnetic moments but also the chemical environments (Fig. 1), dictated by spin space groups [17, 18]. Their rotation-related sublat- tices with opposite magnetic moments breaks time rever- sal symmetry, holding the potential to exhibit other fer- romagnetic features such as an anomalous Hall effect [19– 27], which offers a better readout signal in information storage, compared to the previous antiferromagnets. On the other hand, using pure electrical methods to switch magnetism is a long-standing goal for facilitating applica- tions, but switching altermagnetism still relies on mag- netic fields [28], which is highly unfavorable. However, the mechanism of pure electrical switching of altermag- netism remains an important and open question. In this Letter, we propose a scheme to switch alter- magnetism deterministically using a pure electric current. We discover that, due to their symmetry which depends on chemical environments, altermagnet devices may nat- urally carry the parity symmetry breaking required for deterministic electrical switching of magnetism. We iden- tify MnTe and FeS bilayers (Te-Mn-Te-Mn-Te and S-Fe- S-Fe-S) as candidate devices, based on a search of many promising material candidates, with the help of rigorous symmetry analysis, solid first-principles calculations, and magnetic dynamics simulations. More importantly, our scheme can be generalized to other altermagnets, to in- (a) (b) (c) chemical environment x y z x y z MnB Te Parity inversion on magnetic moments on MnA MnTe //[1120]- / /[1100]- Chemical environment Current V FIG. 1. (a) Crystal and magnetic structures of an altermag- net candidate MnTe. Te atoms around Mn atoms form dif- ferent chemical environments (cyan and pink shadows). (b) Schematic of the setup for deterministic switching of alter- magnetism by a pure electric current (black arrow) along the [11¯20] direction. The N´eel order can be read out through an anomalous Hall voltage V measured in the [1¯100] direction. (c) Parity symmetry has to be broken in any schemes of all- electric deterministic switching of magnetism. When apply- ing parity inversion to a MnTe bilayer, the magnetic moments (blue and red arrows) do not change, but the inversion of the chemical environments leads to the broken parity symmetry required for all-electric deterministic switching of altermag- netism. The bilayer here is more about the two Mn layers that solely contribute to the magnetic dynamics, although three Te layers are needed to maintain the altermagnetism. The effect of substrate can be balanced by capping the device. spire more future explorations. Symmetry analysis of deterministic switching - Parity symmetry breaking is always required for a magnet to be deterministically switched by a pure electric current, i.e., the magnetic order has to be reversed when the electric current is reversed. Reversing the current, which is a ra-

2 m x T T (a) (b) (c) (d) (c) (b) (h) Final states (f) 3 0 - 3 ×106 o n s [ ( e V · s / m 3 ) / ( V / m ) ]

Initial state Tm Final state

x

y (d)

φ

xCurrent (g) 3

0

-3 0 Magnetic moments (e)

Spin accumulations (a)

Spin accumulation 0.5 1

φ/π 1.5 2

FIG. 2. Deterministic switching of altermagnetism in a MnTe bilayer by a pure electric current. (a) Assume an electric current (big black arrow) in the +x direction and an initial N´eel order (measured by the angle φ) of Mn magnetic moments (blue and red solid arrows). The light yellow shadow indicates six stable positions of the N´eel order with a minimal energy, which happen to be aligned with the Te atoms (gray balls). The current can induce φ-dependent spin accumulations (with polarizations indicated by the dashed arrows) on the Mn atoms. The magnetic moments tend to chase the polarizations of spin accumulations (with the tendency indicated by the curved arrows), resulting the final N´eel order in (e). [(b)-(d)] Other possible initial N´eel orders all have the same tendency towards the final N´eel order in (e), because they can be related to the initial N´eel order in (a) by the symmetry operations mx , T , and T mx , where mx is the mirror reflection with respect to the y-z plane and T is time reversal. Therefore, the +x-direction current can deterministically switch the N´eel order to the +y direction in (e). [(f) and (g)] The spin accumulations as functions of φ, calculated by using the Kubo formula and first-principles calculations [see from Eqs. (A1) through (A4)]. sAx means the x-direction component of spin accumulations on Mn atom A, and so on. The dashed lines mark the initial N´eel order in panels (a), (b), (c), and (d), respectively. (h) The final N´eel orders in the MnTe bilayer tend to be perpendicular to the driving current, as shown for currents in −x and ±y directions. After turning off the current, the N´eel order tends to relax to the nearby stable positions, as shown by the light-color solid arrows. dial vector, corresponds to applying a parity inversion, under which however a magnetic order remains invariant if the magnet has parity symmetry. In other words, if the magnet has parity symmetry, opposite currents may yield the same magnetic order, which is against the de- terministic requirement. Parity symmetry can be broken in altermagnet devices due to their symmetry that depends on chemical environ- ments. We will focus on a promising candidate of alter- magnet, MnTe, though our scheme can be generalized to more altermagnets. MnTe has a large spin splitting (∼370 meV), high Curie temperature (∼267 K) [27], and clear anomalous Hall signal [29]. Te atoms in MnTe form octahedra around Mn atoms, creating different chemi- cal environments around the A, B Mn atoms, as shown by the cyan and pink shadows in Fig. 1(a). The bulk crystal of MnTe has parity symmetry due to the periodic boundary condition in the [0001] crystallographic direc- tion (defined as the z direction here). To break parity symmetry for deterministic switching as well as to facil- itate device applications, we consider a bilayer structure grown along the z direction. of Mn atoms, the magnetic structure remains invariant, but the inversion of the chemical environments gives rise to the broken parity symmetry required for determinis- tic switching. This is a significant property, showing the advantage and potential of the MnTe bilayer as well as other altermagnets in pure electrical switching. Deterministic switching driven by current-induced spin accumulations - Microscopically, the deterministic switching is guaranteed by the symmetry of the spin ac- cumulations induced by the current. The spin accumu- lations exert torques to rotate the magnetic moments of the Mn atom. Deterministic switching means that no matter the initial N´eel order, there must be a one-to-one correspondence between the direction of electric current and final N´eel order (defined by mA −mB , where mA and mB are the magnetic moments on the Mn atoms, as indi- cated by the blue and red arrows in Fig. 2). In addition, the final N´eel order has to be aligned with its correspond- ing spin accumulations, so that the torque vanishes and the rotation of the Mn moments stops. MnTe has six stable positions for the N´eel order, where the energy is minimal [30], as indicated by the shadows

Parity symmetry can be broken in altermagnet devices due to their symmetry that depends on chemical environ- ments. We will focus on a promising candidate of alter- magnet, MnTe, though our scheme can be generalized to more altermagnets. MnTe has a large spin splitting (∼370 meV), high Curie temperature (∼267 K) [27], and clear anomalous Hall signal [29]. Te atoms in MnTe form octahedra around Mn atoms, creating different chemi- cal environments around the A, B Mn atoms, as shown by the cyan and pink shadows in Fig. 1(a). The bulk crystal of MnTe has parity symmetry due to the periodic boundary condition in the [0001] crystallographic direc- tion (defined as the z direction here). To break parity symmetry for deterministic switching as well as to facil- itate device applications, we consider a bilayer structure grown along the z direction. More importantly, as shown in Fig. 1(c), if the par- ity inversion is applied only to the magnetic moments Deterministic switching driven by current-induced spin accumulations - Microscopically, the deterministic switching is guaranteed by the symmetry of the spin ac- cumulations induced by the current. The spin accumu- lations exert torques to rotate the magnetic moments of the Mn atom. Deterministic switching means that no matter the initial N´eel order, there must be a one-to-one correspondence between the direction of electric current and final N´eel order (defined by mA −mB , where mA and mB are the magnetic moments on the Mn atoms, as indi- cated by the blue and red arrows in Fig. 2). In addition, the final N´eel order has to be aligned with its correspond- ing spin accumulations, so that the torque vanishes and the rotation of the Mn moments stops. MnTe has six stable positions for the N´eel order, where the energy is minimal [30], as indicated by the shadows in Fig. 2(a). Without loss of generality, we assume a cur- rent in the +x direction and a N´eel order at an angle φ

3 +x current on off off -x current on off 2.5 Below threshold 1.5 Above threshold 0.5 -0.5 0 40 80 120

Time (Picosecond)

160 200

FIG. 3. Simulation of the switching dynamics in the MnTe bilayer. When applying a pure electric current (black arrow) above the threshold current density jc , the N´eel order between ±y directions (φ = ±π/2, 3π/2 = −π/2) can be deterministically switched. The insets show the switching tendency (curved arrows) and status (red and blue solid arrows) of the N´eel order. Here jc ≈ 2.5 × 105 A/cm2 for the parameters J = -1.756 eV, λ = 0.01 meV, K = 0.1 meV, and Jm = 4.46 meV. measured from the x direction. It is known that the Mn moments in MnTe are always polarized in the x-y plane [29], so the two Mn moments in the bilayer can always be transformed to each other by a mirror reflection mz with respect to the x-y plane so the N´eel order remains invariant, while the two Mn moments always have oppo- site current-induced spin accumulations, mz : φ → φ, sA ,B x (φ) = −sB ,A x (φ), sA ,B y (φ) = −sB ,A y (φ), (1) which guarantees the same rotation tendency (as indi- cated by the curved arrows in Fig. 2) for the two Mn moments. We numerically calculate the spin accumulations as functions of φ that describes the N´eel order, by using the Kubo formula and first-principles calculations (see Appendix A). Figures. 2(f) and 2(g) show the x and y components of the spin accumulations on the A and B Mn atoms, driven by the +x-direction current. In the present case, the moments are polarized in the x- y plane because of the magnetic anisotropy, so we only need to consider the spin accumulations polarized in the x-y plane [29]. As shown in Fig. 2(a), for the initial N´eel order (solid arrows), the calculated spin accumulations are polarized along the dashed arrows, which leads to a rotation tendency towards the y direction as indicated by the curved arrows. Roughly speaking, the moments ma tend to chase the polarization of the spin accumula- tions sa , until ma is aligned with sa to cease the rotation, approximately governed by the spin-orbit torque [31–33] Ta ∼ ma × sa . (2) Our calculated spin accumulations in Figs. 2(f) and 2(g) show that, for arbitrary φ in the first quadrant (φ ∈ [0, π/2]), the +x-direction current always switch the N´eel order anticlockwisely to the +y direction. More importantly, all possible initial N´eel orders can be classified into the four quadrants in Figs. 2(a)-2(d), which all show the same rotation tendency towards the +y direction for the +x-direction current, because the initial N´eel order in Fig. 2(a) can be related to those in the other three quadrants by three different symmetry operations that transform the N´eel order while keeping the crystal structure unchanged and Mn atoms A and B unexchanged (see details in Appendix B). Therefore, we conclude that the electric current in the +x direction will always deterministically switch the N´eel order to the +y direction. Moreover, we can analyze the switching results for cur- rents applied along other directions, as shown in Fig. 2(h) for currents in the −x and ±y directions. The currents in the ±y direction switches the N´eel order to the middle of two nearby stable positions (Sec. S1 of [43]). After turning off the current, the N´eel order will randomly re- lax to one of the two nearby stable positions. The two nearby stable positions, however, have opposite anoma- lous Hall signals [29], leading to failure of detectable 180◦ switching if the current is in the ±y directions (details in Sec. S2 of [43]). Therefore, to achieve a deterministic 180◦ switching of altermagnetism that can be detected by the anomalous Hall effect in the MnTe bilayer, the current should be injected along the x direction (or three directions equivalent to the [11¯20] crystallographic direc- tion). Magnetic dynamics simulation - To verify the switch- ing of the ±y-direction N´eel orders driven by the ±x-

4 [1120] RuF 4 , FeSb 2 [010] [010] ...

Rb 1-δ V 2 Te 2 O, KV 2 Se 2 O .

.

. .

.

.

[0001] - RuO 2 , MnF 2 CrSb ...

MnTe, FeS [0001]

FIG. 4. Symmetry analysis of whether the material candidates for altermagnets can be deterministically switched by pure elec- tric currents. We consider FeS [34], CrSb [25, 35], RuO2 [18, 36–38], MnF2 [9], Rb1 −δ V2 T e2 O [39], KV2 S e2 O [40], RuF4 [41] and FeSb2 [42], because they have large spin splitting and high N´eel temperatures. The arrowed balls indicate the magnetic atoms and the cyan and pink shadows show the chemical environments created by surrounding non-magnetic atoms. Unfortunately, by far most predicted altermagnets have parity symmetry so that they cannot be switched by pure electric currents. The FeS bilayer, on the other hand, has the same symmetry as that of MnTe bilayer, so it may also be switchable, even at room temperature because of its high N´eel temperature [34]. The CrSb bilayer, although has the broken parity symmetry, does not possesses anomalous Hall signal, due to its symmetries with a N´eel order along the [0001] direction. direction current in the MnTe bilayer, we simulate the dynamics of the switching, by using the Landau-Lifshitz- Gilbert equation [44, 45] (1 + α2 ) ˙ m a = αma × |γ| MS ma × δHm δma + Ta + |γ| MS ma × δHm δma + Ta , (3) where α = 0.1 is the Gilbert damping, γ(< 0) is the gyromagnetic ratio of the electron, MS = 3.89µB is the saturation moment of a Mn atom with µB the Bohr mag- neton, Hm is the magnetic Hamiltonian that describes the magnetic structure of the MnTe bilayer Hm = Jm m A · mB + K (mA · ˆz)2 + (mB · ˆz)2 + λ 2 [cos(6φA ) + cos(6φB )] . (4) The first term of Eq. (4) describes the antiferromagnetic exchange coupling between mA and mB . The exchange energy Jm ≈ 4.46 meV is calculated by using the Green’s function method implemented in the TB2J package [46]. The tight-binding Hamiltonian in the atomic basis, as delineated in Eq. (A2), is employed for the calculation. The second term with K = 0.1 meV [30] describes the anisotropy that keeps the magnetic structure within x-y plane. The third term with λ = 0.01 meV [30] describes the six-fold symmetric anisotropy of magnetic structure within x-y plane. Ta in Eq. (3) is the spin-orbit torques induced by the electric current Ta ≡ |γ| MS ma × δEc δma , (5) where the energy generated by the current-induced spin accumulations per unit cell can be found as Ec ( φ) = (2JVU / ℏ) [mA · sA ( φ) + mB · sB (φ)] , (6) where VU is the volume of the unit cell, and the exchange energy J between the Mn magnetic moments and spins of itinerary electrons is extracted from Eq. (A2). Specif- ically, the Fermi surface of MnTe is mainly composed of five 3d orbitals, so J is evaluated by the coupling between the spin-up and spin-down states on the 3d orbitals, when the Mn atoms are polarized along the x direction. It ranges from -1.830 eV to -1.691 eV and its average value is -1.756 eV. In this case the spin-orbit torques takes the form Ta = (2JVU | γ|/MS ℏ)ma × sa [31–33]. Figure 3 shows that the N´eel order can be switched from φ = 3π/2 (equivalent to −π/2) to π/2 by a +x direction current applied from 40 to 80 ps (cyan area) and from φ = π/2 to −π/2 by a −x direction current from 120 to 160 ps (pink area). Moreover, near 50 ps and 130 ps, the N´eel order oscillates to the stable positions, consistent with the symmetry analysis of the switching tendency in Fig. 2(a)-(d). The switching can be achieved only when the current density is above a threshold. For J ranges from -1.830 eV to -1.691 eV, the threshold current has a range jc ≈ 2.4 − 2.6 × 105 A/cm2 , (7) which can be understood by an energy analysis in Ap- pendix C. Here, the threshold is lower than those in other antiferromagnetic devices [1, 28, 47–49], probably because we consider only single crystals with single mag- netic domains. For realistic samples with polycrystals and multidomains, the threshold will be higher. Other altermagnet candidates - Recent studies have been suggesting an increasing number of candidate ma- terials for altermagnets [23–27, 35, 39–42, 50–53], includ- ing FeS [34], CrSb [25, 35], RuO2 [18, 36–38], MnF2 [9], Rb1 −δ V2 T e2 O [39], KV2 S e2 O [40], RuF4 [41] and FeSb2 [42]. Most of the altermagnet candidates are parity sym-

5 metric and thus can not be deterministically switched by a pure current, as shown in Fig. 4. To discover the al- termagnetic materials that can be electrically switched, we should aim for the materials with broken parity sym- metry, e.g., a MnTe bilayer as we study in this work. Nevertheless, parity symmetry breaking is merely a nec- essary requirement. For example, a CrSb bilayer has no parity symmetry, but the easy axis of its N´eel order is in the crystallographic [0001] direction, as shown in Fig. 4, leading to the absence of the anomalous Hall signal (de- tails in Sec. S5 of [43]). However, the Cr magnetic mo- ments can be deviated from the [0001] direction, i.e., by the Dzyaloshinskii-Moriya interactions, giving a readable anomalous Hall signal [16]. We thank fruitful discussions with Qihang Liu, Chang Liu, Xiangang Wan, and Xiaoqun Wang. 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End Matter Appendix A: Calculation of spin accumulations - We calculate the current-induced spin accumulations on a ∈ {A, B} Mn atom, by applying the first-principles calcu- lations to the Kubo formula [54, 55] (sax , say , saz ) ≡ sa = eℏ 2V τ ν,k ∂fν ∂ϵν (E · vν )σaν , (A1) where e is the elementary charge, ℏ is the reduced Planck constant, V is the volume of the system, τ is the re- laxation time, fν = 1/{exp[(ϵν − EF )/kB T ] + 1} is the Fermi-Dirac distribution function, ϵν is the eigen energy of eigen state |ν⟩, the electric field E can be converted from the switching current density j = σE, σ is the con- ductivity, v is the group velocity operator, vν = ⟨ν|v|ν⟩, σa = (σax , σay , σaz ) are the Pauli matrices for the local spin density on Mn atom a, and σaν = ⟨ν|σa |ν⟩. To evaluate the spin accumulations in the MnTe bi- layer accurately, we employ the Wannier interpolation scheme [56, 57] based on the ab initio disentangled Wan- nier functions (details in Sec. S3 of [43]). The Wannier Hamiltonian reads HWa α,bβ (R) = ⟨0aα|ˆH|Rbβ⟩, (A2) describing the hoppings from localized orthonormal Wan- nier function |Rbβ⟩ (β orbital of atom b at lattice R) to |0aα⟩ (α orbital of atom a at home unit cell 0). In mo- mentum space, the eigen energies ϵν and eigen vectors |ν⟩ can be found from the eigen equation H(k)|ν⟩ = ϵν |ν⟩, where the Fourier transform of the Wannier Hamiltonian H(k) = R ei k·R HWa α,bβ (R). (A3) The electronic structures of MnTe slab with various directions of the magnetic moments are calculated via density-functional theory in the pseudopotential frame- work and incorporating spin-orbit coupling. In these cal- culations, we use the projected augmented wave method, implemented in the Vienna ab initio simulation pack- age [58–60], adopting the Perdew-Burke-Ernzerhof pa- rameterization of the generalized gradient approxima- tion exchange-correlation functional [61]. To account for electron-electron correlation on Mn sites, the Dudarev method is applied with a Hubbard effective parameter term U = 3 eV is applied on d orbitals of Mn atoms. The electron wave function is expanded on a 6×6×1 k- mesh, and the self-consistent computation is carried out until the energy difference is less than 10− 6 eV with a cut-off energy of 400 eV. To freeze the magnetic moment of MnTe in a random direction, we first calculate the collinear spin-polarized scalar-relativistic electronic structure of the antiferro- magnetic MnTe bilayer. The charge density in real space ρ0 (r) is a scalar on a sampling grid. Then, we project the scalar charge density ρ0 (r) to a vector charge density, following the direction of the magnetic moment, ρx( r) = ρ0( r) sin θ cos ϕ, ρy (r) = ρ0( r) sin θ sin ϕ, ρz( r) = ρ0( r) cos θ, (A4) where θ and ϕ are the polar and azimuthal angles of the magnetic moment, respectively. The ρx (r), ρy (r), and ρz (r) are well initial guess for the local spin density approx- imation (LSDA) method to calculate the non-collinear spin-polarized full-relativistic electronic structure of the MnTe bilayer with a random direction of the magnetic moment. The Wannier functions are disentangled from the Kohn-Sham wave functions, using s, d orbitals of Mn and p orbitals of Te as projections, by employing Wan- nier90 code package [62]. To achieve accurate mapping of spin accumulations on the Fermi surface in the MnTe slab, we evaluate Eq. (A1) using Wannier interpolation method implemented in WannierBerrri [63]. In the cal- culation, a 200×200×1 k-point grid and 10 refinement iterations are used. The tetrahedron method [64, 65] is further applied to accurately describe the distribution function ∂fν /∂ϵν at zero temperature. The scheme in Fig. 2 are from the diagonal spin accumulations. We

8
show that the off-diagonal spin accumulations are three to π + φ in the third quadrant [Fig. 2(c)] and the spin
orders smaller and cannot support deterministic switch- accumulations transform as
ing in Sec. S3 and Sec. S4 of [43].

Tendency sA ,B x (φ) = sA ,B x (π + φ), sA ,B y (φ) = sA ,B y (π + φ),

Above threshold = 2.5×105 A/cm2

T mx : φ → π − φ,

φ → π − φ, sA ,B x (φ) = −sA ,B x (π − φ), sA ,B y (φ) = sA ,B y (π − φ),

sx (φ) = sx (π + φ), sA ,B y (φ) = sA ,B y (π + φ), (B2) as verified by the calculated results at the dashed line (c) in Figs. 2(f) and 2(g). Therefore, the N´eel order in the third quadrant also rotates clockwisely towards the +y direction. (iii) By applying T mx , the N´eel order can be trans- formed from φ in the first quadrant [Fig. 2(a)] to π − φ in the second quadrant [Fig. 2(b)] and the spin accumu- lations transform as

ec. S3 and Sec. S4 of [43]. 0.6 (a) T : φ → π + φ, sA ,B x (φ) = sA ,B x (π + φ),

2 1.5 1 0.5 0 0.1 0 -0.1 -0.2 [ m e V ] at = 1.25×105 A/cm2 FIG. 5. (a) Total energy Ec + Ea of the magnetic structure of MnTe in the presence of the switching current as a function of φ for different current density, where Ec is the energy of current-induced spin accumulations and Ea is the anisotropy of the magnetic structure. The curved arrow indicates the switching tendency from the energy maximum at φ = 3π/2. Inset: the local energy barriers near φ = 3π/2 when the cur- rent density is below the threshold jc ≈ 2.5 × 105 A/cm2 . (b) Ec at j =1.25×105 A/cm2 and Ea , which gives rise to the energy barrier. Appendix B: Symmetry operations mx , T , and T mx in Fig. 2 - (i) By applying the mirror reflection mx with respect to the y-z plane, the N´eel order can be trans- formed from φ in the first quadrant [Fig. 2(a)] to −φ in the fourth quadrant [Fig. 2(b)] and the spin accumula- tions transform as mx : φ → −φ, sA ,B x (φ) = −sA ,B x (−φ), sA ,B y (φ) = sA ,B y (−φ), (B1) as verified by the calculated results at the dashed line (b) in Figs. 2(f) and 2(g). Therefore, the N´eel order in the fourth quadrant also rotates anticlockwisely towards the +y direction. (ii) By applying the time reversal T , the N´eel order can be transformed from φ in the first quadrant [Fig. 2(a)] sx (φ) = −sx (π − φ), sA ,B y (φ) = sA ,B y (π − φ), (B3) as verified by the calculated results at the dashed line (d) in Figs. 2(f) and 2(g). Therefore, the N´eel order in the second quadrant also rotates clockwisely towards the +y direction. Appendix C: Estimate of threshold current density - To understand the threshold current density in the magnetic dynamics simulation, we perform an energy analysis. The current-induced spin accumulations can give rise to an energy. When the energy of spin accumulations over- comes the local energy barrier produced by the in-plane anisotropy of the magnetic structure, the switching can be achieved. The total energy of magnetic structure per unit cell in the presence of the switching current can be found as Ea ( φ) + Ec ( φ), (C1) where the anisotropy of the magnetic structure shows a six-fold rotational symmetry in the (0001) plane and can be described by Ea ( φ) = λ cos(6φ), (C2) where λ = 0.01 meV [30]. The energy of the current- induced spin accumulations has been given in Eq. (6) and J = −1.756 eV. For a +x direction current, Ec ( φ) + Ea ( φ) in Fig. 5(a) shows a minimum at φ = π/2 and a maximum at φ = 3π/2, so the switching tendency is from the maxi- mum at φ = 3π/2 to the minimum at φ = π/2. However, below a threshold current density, there is a local energy barrier near φ = 3π/2, as shown by the inset of Fig. 5(a). To flatten the barrier, the current density has to be above a threshold, then the switching can be achieved. Follow- ing this strategy, we numerically find that the threshold current density is about jc ≈ 2.5 × 105 A/cm2 . When J ∈ [-1.830,-1.691] eV, jc ∈ [2.4, 2.6] × 105 A/cm2 .