Chiral-induced switching of antiferromagnet spins in a confined nanowire

In the development of spin-based electronic devices, a particular challenge is the manipulation of the magnetic state with high speed and low power consumption. Although research has focused on the current-induced spin-orbit torque based on strong spin-orbit coupling, the charge-based and the torque-driven devices have fundamental limitations: Joule heating, phase mismatching and overshooting. In this work, we investigate numerically and theoretically alternative switching scenario of antiferromagnetic insulator in one-dimensional confined nanowire sandwiched with two electrodes. As the electric field could break inversion symmetry and induce Dzyaloshinskii-Moriya interaction and pseudo-dipole anisotropy, the resulting spiral texture takes symmetric or antisymmetric configuration due to additional coupling with the crystalline anisotropy. Therefore, by competing two spiral states, we show that the magnetization reversal of antiferromagnets is realized, which is valid in ferromagnetic counterpart. Our finding provides promising opportunities to realize the rapid and energy-efficient electrical manipulation of magnetization for future spin-based electronic devices.


Introduction
In the development of highly efficient spintronic devices, one emerging issue is to discover and exploit novel phenomena with strong spin-orbit coupling (SOC) [1][2][3][4]. Due to scientific and technological interest, intensive research has focused on current-driven spin-orbit torques (SOT) for manipulation of magnetization. Most of experimental and theoretical works on SOT switching have been performed in a magnetic multilayer consisting of an ultrathin ferromagnets (FMs) or antiferromagnets (AFMs) and heavy metal layers [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Because SOT devices use a current, charge scattering and corresponding Joule heating inevitably occur [21]. This intrinsic property is an obstacle in reducing the switching power, although SOT efficiency is significantly improved in nanoscale devices [7,10,[22][23][24][25][26]. In heterostructures, especially with structural inversion asymmetry, Dzyaloshinskii-Moriya (DM) interaction, which is also induced by spin-orbit coupling, has received attention in spin dynamics research. In the presence of DM interaction, the competition between exchange and DM interaction allows for a nontrivial topological spin configuration to exist as a ground state [27][28][29][30], i.e., spiral configuration and skymion in a confined geometry. Topological robustness has been exploited to enhance the performance of SOT devices, such as DM interaction-stabilized Néel domain wall motion [31][32][33][34] and DM interaction-assisted current-driven switching [35]. The DM interaction plays a secondary role in current-driven dynamics. However, it is rarely studied as a driving source to replace a current to initiate spin motion. Actually, a few studies performed on electric field-induced DM interactions found that the conversion efficiency is proportional to the spin-orbit coupling strength as in SOT [36][37][38].
Here, we report an electric field-induced magnetization switching scenario through potential barrier modulation in a nanowire, instead of the spin current. This switching is realized by changing the ground spiral state and relaxing it into a switched configuration by controlling the DM interactions. This switching scenario is different from the precessional switching mechanism driven by external torques, efficiency of which relies upon the timing of the torque and magnetization precession. Figure 1 shows the spiral structure of antiferromagnetic insulator (AFI). Here, AFMs are aligned along the z axis and sandwiched by two electrodes of heavy and normal metal. We use two order parameters: the Néel order l = (si-sj)/2 and the ferromagnetic order parameter m = (si+sj)/2, where each spin is normalized by its magnitude si = Si/S0 with S0 = |Si|. Therefore, the wire length is defined as lw in Néel space. In heavy metal layer with strong SOC such as Pt, Ta and W, spin Hall current is typically generated when a charge current is applied in those materials [1,2]. The magnetic crystalline anisotropy has an easy axis along the z axis where anisotropy constant Kz is positive. The geometric inversion asymmetry induces DM interaction along the y axis according to

Results
where the x axis is normal to the interface and eij is the unit vector connecting neighbor spins si and sj [27,28]. We ignore this geometric DM interaction in the calculation and discuss it later. When the electric field along the x axis breaks inversion symmetry, the DM vector D, becomes effectively toward the y axis due to [27,28,[36][37][38]. Also, we introduce an electric-field-induced pseudo-dipolar anisotropy energy KE with easy plane, and it is induced from SOC that gives rise to the DM interaction [27,28,[36][37][38]. An electric-field-induced anisotropy is an effect of order of E 2 , but cannot be ignored in our switching scenario. In other indirect exchange interactions, known as double-exchange and Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction in metal, it has been reported that same SOC induces the DM interaction and the anisotropy by the external electric field [39][40][41].

Two possible spiral states as a function of DM interaction
However, before preceding to the electric-field-induced manipulation of AFMs, we consider stationary states of AFMs as a function of DM interaction energy. Figure 2 shows two spiral structures with different DM interaction energies and these are formed by additional coupling with crystalline anisotropy, which is proportional to ~lz 2 (see Eq. (1)). Under exchange approximation where the exchange energy J is the larger than other energies, or |J| >> Dy and Kz, we can assume that the spiral structure has continuously varying spin texture, , where Δ is the interspacing of the nearest neighbors in Néel space. Therefore, the energy density E1D is described as The a and A are the homogeneous and inhomogeneous exchange constants, respectively, and , where am( | ) u m is a Jacobi amplitude function with the elliptic modulus m and the elliptic integral of the first kind u.
Especially, u is regarded as arc length of the unit ellipse, defined as is related to the phase shift along z axis and C1 modulates u and m, respectively. With exchange interaction and anisotropy fixed, DM interaction modulates the reference from a circle to an ellipse. For example, when DM interaction that prefers to be spiral dominates the effective anisotropy that prefers to be uniform, m approaches to zero. Therefore, l ={cos[ϕ(z)], 0, sin[ϕ(z)]} is described as the trigonometric function where ϕ(z) is linearly proportional to z;  [30].
Each state is characterized by the first condition that is given as ϕ(z = lw/2) = nπ for the S state or ϕ(z = lw/2) = (2n+1)π/2 for the AS state where n is an integer. The second condition becomes Neumann-type boundary condition as Eq. (3a): Notably, as dy/dc is over where the effective magnetic field is obtained from the functional derivative of energy density To set a trial function for l, we introduced the collective coordinates θ(t) for the dynamic phase and k for the pure spiral soliton profile: ϕ(z, t) = k(z-(lw+1)/2)+θ(t), where we arbitrary shift the soliton profile by (lw+1/2) so that θ(t) represents the phase at center or ϕ(z = (lw+1) /2, t) = θ(t).
Inserting Eq. (5) into Eq. (4b) and integrating the sublattice number N from N = 1 to N = lw, the soliton equation of motion is derived as: propagates with steady state velocity v = ws/α as in domain wall motion driven by SOT. Note that in Fig. 3(a), norm barrier E is calculated from the normalized anisotropy difference between two states in Fig. 3(b) and is comparable to Г in the pure spiral regime, as shown in Fig. 3(a). For example, when dy/dc = 3.5, Γ < 0, θ(∞) would be nπ (S state), which is located at potential minimum; thus, θ(∞) = (n+1/2)π corresponds to potential maximum (AS state). Therefore, the anisotropy energy difference between the S and AS states are interpreted as barrier To switch Néel magnetization, our strategy is to modulate potential barriers by controlling ratio dy/J through several steps in which SOT plays a perturbation role. As shown in Fig. 4(b), the stationary soliton state is alternatively changed from S to AS states (Ebarrier < 0 to Ebarrier > 0 in Fig. 4(a)) and then from AS to S states (Ebarrier > 0 to Ebarrier < 0 in Fig. 4(a)) as the DM energy increases. It completes the Néel arrangement switch in the five steps. In dy/J = 0 (step 1), the uniform antiferromagnetic state along the +z axis is interpreted as an S state with ϕ(lw/2) = 0.
Although the DM energy turns on when dy/J = 0.043 (step 2), the soliton state is not changed because ϕ(lw/2) = 0 and Ebarrier < 0. When the DM energy is lowered by dy/J = 0.03 (step 3), the S state is unstable because Ebarrier > 0 [see Fig. 4(a), AS-state] but, interestingly, does not go into the AS state because it is a metastable state located at a potential maximum [see Fig. 4(c)], which implies the necessity of small perturbation such as SOT. Therefore, small SOT with unidirectional polarization is necessary for deterministic switching. For example, with a spin current with -py, the soliton would go to an AS state with ϕ(lw/2) = -1/2π; if spin polarization is of py, AS state would be of ϕ(lw/2) = 1/2π. Next, in the lowered dy/J = 0.015 (step 4), AS state with ϕ(lw/2) = -1/2π is required to go S state with ϕ(lw/2) = -π. Eventually, as DM energy shuts down (step 5), final S state is maintained with θ = -π. All processes are described in Fig. 5(a).
Note that our solitonic approach allows for simplifying the multistep manipulation of AFMs; because the first two steps and the fourth and fifth steps are in the same state of ϕ(lw/2)= 0 and ϕ(lw/2) = -π, so these overlapping steps could be omitted. As shown in Fig. 5(b), only the first, third and fifth steps that form the single pulse shape can switch an AFM. In addition, the dy variation from step 1 to step 2 results in spreading and shrinkage of k, i.e., breathing motion due to inertia. However, this motion does not lead to the phase propagation. In addition, it is desirable to consider the field-like torque taking place during working in the real devices. When the magnetic field is applied along arbitrary directions, we can add the Zeeman interaction energy E1D, Z = γħH·m into the total energy density, where γ is the gyromagnetic ratio and ħ is the reduced Plank constant. And Eq. [43]. If the magnetic field is time-varying, the spiral soliton is driven by field-like torque, ~dhy/dt [46], which is derived after inserting Eq. (5) into Eq. (4b). To suppress field-like torque, the proper strength of SOT should be applied.
As noted in introduction, structural DM interaction strength by asymmetric electrodes could be reduced below dc by engineering its thickness [53] or utilizing symmetric electrodes, compared with electric field-induced DM energy. However, the structural DM interaction, weak enough to form a quasi-uniform configuration, reduces the required electric field strength.
The above statements are also valid in ferromagnetic counterparts because a ferromagnetic spiral structure is formed by competition between anisotropy and DM energy and is excited by SOT; in ferromagnetic nanowire, two conditions are given as ϕ(z = lw/2) = nπ for the S state or ϕ(z = lw/2) = (2n+1)π/2 for the AS state and 1 or y and dc = 4(AKeff) 1/2 /π [30]. Finally, it remains to be seen if there is the electric field effect in different magnetic systems. In magnetic metal system with broken inversion symmetry, the generation mechanisms of DM interaction are two folds: 1) Fert-Levy mechanism [54] and 2) Rashba SOC [39][40][41]55]. In the Fert-Levy mechanism, itinerant electron is mainly exchange-coupled with magnetic ion by RKKY interaction. An additional coupling leads to the DM interaction by scattering of itinerant electron with heavy metal. As aforementioned, the Rashba SOC is related to also itinerant electron in the material with strong SOC. Another electric field induced modulation of anisotropy is reported in the ferromagnetic metal/oxide interface or Ta/ultrathin CoFeB/MgO [56], the perpendicular magnetic anisotropy (PMA) is originated from hybridization of oxygen p-orbital and iron d-orbital. In this case, the electric field induces charge redistribution of electron of magnetic metal, resulting in modulation of PMA [21,56,57]. However, the magnetic insulator is lack of conduction electron and it is hard to expect the charge redistribution by electric field and its related anisotropy modulation.
In conclusion, we investigated spiral dynamics in the presence of DM interaction. In solitonbased spin dynamics, there are two states (symmetric and antisymmetric state) due to competition between anisotropy energy and DM interaction, in which one is stable at a potential minimum, and the other is metastable at a potential maximum, implying that external (or

DATA availability
The data that supports the findings of this study is available from the corresponding author upon request.   The potential barrier that is calculated from norm S, norm < 0 (or norm barrier E > 0), the S state (or AS state) is energetically stable with minimum potential energy and the AS state (or S state) is metastable with maximum potential energy. DM pulse with (a) multistep or (b) single-step profile is applied to induce antiferromagnetic switching, applying weak spin-orbit torque (SOT). Here, electric-field-induced anisotropy is as a function of DM energy. According to potential barrier profiles, the first two steps (dy/J = 0 and 0.043) are stable with a symmetric (S) state. At dy/J = 0.03, the S state is metastable and antisymmetric (AS) becomes a stable state with ϕ(lw/2) = -1/2π (not 1/2π) due to unidirectional SOT with spin polarization -py. In the fourth and fifth steps (dy/J = 0.015 and 0), the AS state is metastable. Thus, the S state has ϕ(lw/2) = -π (not π) due to -py. In our switching scenario, the first two steps and the fourth and fifth steps overlap. Therefore, the Néel order could be switched using a single-step function without the second and fourth processes.